CURRENT ADVANCES IN M E C H A N I C A L DESIGN AND P R O D U C T I O N VII
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CURRENT ADXv~ANCES IN MECHANICAL DESIGN AND P R O D U C T I O N VII Proceedings of the Seventh Cairo University International MDP Conference
Cairo-Egypt February 15-17, 2000
Edited by
Mohamed E HASSAN C~m[-ctcnct, Chairman, Pro]~'.s.sor~.tndHead, Mcchanicttl Dcsign ttn~! Producti{;n Department, Faculty o[ Enginccring, Cairo Univcrsitx; Giza 12316-Egypt and
Cotl lt'lt'~lcc
Said M. MEGAHED Gctlt'~tl[ 5ccFt'ltilV, Editor-in-Chiq[and Prq/cssop;
Mechanical Dcsi,qn ctnd Production Department, Fticttlt\' t!l Erlgint't'ring, Cairo University, Giza 12316-Egypt
Pergamon AMSTERDAM - LAUSANNE - NEW YORK - OXFORD - SHANNON - SINGAPORE - TOKYO
ELSEVIER SCIENCE Ltd The Boulevard, Langford Lane Kidlington, Oxford OX5 IGB, UK
9 2000 Elsevier Science Ltd. All rights reserved. This work is protected under copyright by Elsevier Science, and the following terms and conditions apply to its use:
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First edition 2000
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PREFACE The Mechanical Design and Production Department, Faculty of Engineering, Cairo University has established its series of international conferences on the Current Advances in Mechanical Desilm and production since 1979. This conference is the 7 ih in the series (MDP-7), held in Cairo during the period February 15-17, 2000. The conference brings together engineers and scientists from allover the world with a view of exchanging experience and highlighting the state of the art in the fields of Mechanical Design and Production as well as Technology Transfer. Several distinguished researchers were invited to address the conference with keynote papers to enrich the sessions and to highlight the recent advances on the various fields of mechanical design and production. A total of 160 papers were submitted to MDP-7 conference from more than 21 countries from the 6 continents. All papers were thoroughly refereed by the conference scientific committee. Following the reviewing process, a total of 104 papers in addition to 15 industrial applications and case studies were accepted for presentation. These MDP-7 proceedings contain 10 invited papers together with 54 selected papers. Further papers are published independently by the Mechanical Design and Production Department, Faculty of Engineering, Cairo University. The conference proceedings include basic and applied research papers in Mechanical Design and Production classified into six main categories: System Dynamics, Solid Mechanics, Material Science, Manufacturing Processes, Design and Tribology, and Industrial Engineering and its Applications. We would like to express our gratitude to our colleagues in the Mechanical Design and Production Department, Faculty of Engineering, Cairo University who spend their time and effort in assisting us to have this fruitful work. We also acknowledge the cooperation of the conference scientific committee members, for their immediate response in reviewing all the papers submitted to the conference and for their valuable suggestions. We would like to take this opportunity to express our gratitude to the Academic Institutions and Industrial Organizations for their financial support. This support made it possible to meet part of the conference expenses. Finally, we dedicate our thanks and respects to all authors whose contributions made it possible to maintain the international reputation of the MDP Conferences. We hope that the MDP-7 proceedings present a useful contribution, reflecting the C u r r e n t Advances in Mechanical Design and Production at the onset of the 21 st century.
Said M. Megahed
Mohamed F. Hassan
General Secretary and Editor.in.Chief
Conference Chairman
vi
Current Advances in Mechanical Design and Production, MDP-7
SCIE .NT.IFIC COMMITTEE AND INTERNATIONAL ADvIsoRY BOARD Abbas, A.T. Abdelaal, R. Abdelhakim, M. Abdel-Kader, S. Abdelmaksoud, H. Abdelraouf, H. Abdo, M.A. Abo lsmail, A. Aboelfetooh, M.N. Adli, A. AI-Ashram, A. All, G.M. Allam, M.N.A. Anis, H.I. Arafa, H.A. Ashour, S. Attalla, W. Badran, F. Bahei Eldin, Y.A. Bahgat, A. Barakat, M.A. Bayoumi, A.M.E. Bayoumi, S.E.A. Bazaraa, A.S. Bedewy, M.K. Choi, B.K. Dardiri, M.A. Elarabi, M.E. Elbestawi, M.A. EI-Demerdash, M.F. Eleiche, A.M. El-Hakim, M.A. EI-Hebeary, M.R. EI-Kharboutly, A. EI-Kousy, M.R.
Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt USA Egypt Egypt Egypt S. Korea Egypt Egypt Canada Egypt Egypt Egypt Egypt Egypt Egypt
EI-Mahalawy,N. EI-Raghy,S. EI-Sabbagh,A. Elsayed,E.A. Elsebai,M.G. El-Sheikh,A. EI-SherbinyM.G. Elwani,M.H. EI-Zoghby,A.A. Ezzat,A. Farag,M. Fat-Halla,N. Gommaa,A.H. Hammouda,M.I. Hanafi,A.A. Hassan,G.A. Hassan,M.A. Hassan,M.F. Hassan,M.F. Hassan,S.D. Hassan,Y.K. Hassanien,A. Hegazi,A. Hosni,Y.A. lbrahim,I.A. Jamshidi,M. Kassem,M.E. Kassem,S.A. Khattab,A.A. Khorshid,S.Y. Koura,M. Krempl,E. Mahmoud,F.F. Mansour,A.M.A. Megahed,M.M.
Egypt Egypt Egypt USA Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt USA Egypt USA Egypt Egypt Egypt Egypt Egypt USA Egypt Egypt Egypt
Megahed,S.M. Meguid,S.A. Mokhtar,M.O.A. Mostafa,A.A-F. Moustafa,M.A. Nassar,M. Rabei,G. Radwan,A.A. Ragab,A.R. Ragab,M.S. Renaud,M. Riad, M.S.M. Riad, S.M. Rizk, M. Sabry,Sh. Sadek,E.A. Said, M.E. Salama,A.E. Salama,A.S. Saleh,I. Salem,H. Sallam,M. Shalaby,M.A. Shash,Y.M.S. Sherif,A.O. Soubeah,M.E. Taha,M. Wahdan,A. Wifi, A.S. Wustof,P. Yehia,N.A. Younan,M. Youssef,M.R.
Egypt Canada Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt France Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt S. Arabia Germany Egypt Egypt Egypt
STEERING AND ORGANIZING COMMITTEE Arafa, H.A. Bayoumi, S.E. Elaraby, M.E. Eleiche, A.M. EI-Sherbiny, M.G." Fawzy, I. Hassan, M.F. +
Hassan,Y.K. Kassem, S.A. Khattab, A.A Megahed, M.M. Megahed, S.M." Metwalli, S.M. Mokhtar, M.O.A.
* Faculty Dean + conference Chairman ** Conference General Secretary
Radwan, A.A. Ragab, A.R. Riad, M.S. Said, M.E. Salama, A.S. Shalaby, M.A.
vii
Current Advances in Mechanical Design and Production, MDP-7
INVITED AND KEYNOTE LECTURES Bayoumi, A.M.E. Choi, B.K. EIbestawi, M.A. EIsayed, E.A.
USA S. Korea Canada USA
Hosni, Y.A. Jamshidi, M. Krempl, E.
USA USA USA
Meguid, S.A. Renaud, M. Wustof, P.
USA France Germany
EDITORIAL COMMITTEE
ASSISTANTS TO SERETARY
Atria, M.S. EI-Zoghby, A.A. Gadalla, M.A. Hassan, M.F. § Megahed, M.M. Megahed, S.M." Nassef, A.M.O. Omar, M.A. Zeyada, Y.F.
Gommaa, A.H. Shendi, M.
TREASURERS Bedewy, M.K. Riad, S.M.
EXEXUTIVE COMMITTEE Staff Members of the Mechanical Design and Production Department Faculty of Engineering, Cairo University
Emeri.tus Professors Professors Bayoumi, S.EA.. Elarabi, M.E. Hassan,Y.K. Kandeel, S.E. Kouta, F.H. Mostafa, A.A-F. Riad, M.S.M. Riad, S.M.
Arafa, H.A. Bahgat, B.M. Basily, B.B. Bedewy, M.K. EI-Dalil, S.A. Eleiche, A.M. EI-Hebeary, M.R. EI-Sherbiny M.G.' Fawzy, I.
Hasan, M.F.§ Hassan, G.A. Kassem, S.A. Khattab, A.A. Megahed, M.M Megahed, S.M.'" Metawalli, S.M. Mokhtar, M.O.A. Radwan, A.A.
Associate Professors
Assistant Professors
EI-Habak, A.M. EI-Zoghby, A.A. Fouad, A.E.A. Khorshid, S.Y. Mansour, A.M.A. Mawsouf, N.M. Megahed, H.A. Mohamed, M.A.A. Othman, T.A. Salama, M.S.~ Shalaby, M.A.
AbdeI-Aal, O.M. Abdo, M.Z. Abdrabu, M.M.A. Abou-Hamda, M.M. Adly, M.A.~ Anany, A.A. Azzam, B.Sh.N. Bayoumi, L.S.E. Behnam, M.M. EI-Danaf, E.A. EI-Gamil, M.A.
9 ,
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i
,
Radwan, M.A. Ragab, A.R. Ragab, M.S. Said, M.E. Salama, A.S. Salim, F.B. Shash, Y.M.S. Soliman, F.A. Wifi, A.S. ++
ii
EI-Geddawi, M.E. EI-Hossiny, T.M. EI-Shazly, M.H.Y. Gadalla, M.A. Galal, G.M.A. ~ Hassan, M.E. Kamal, B.A. Nassef, A.M.O. Rashad, R.M. Saleh, Ch.A. Zeyada, Y.F.
" Faculty Dean + Conference Chairmanand Departmen'iHead ~* (2onferencr GeneralSecret~ ~ o n leave
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TABLE OF CONTENTS
PREFACE TABLE OF CONTENTS
ix
Section I: S Y S T E M D Y N A M I C S
I.l"
Autonomous Control of Complex Systems: Robotic Applications Jamshidi, M. (USA)
1.2'
A Simplified Inverse Kinematic Model Calculation Method for All 6R Type Manipulators Renaud, M. (France)
15
1.3
A Systematic Algorithm for Flexible Manipulators Simulation Megahed, S.M. and Hamza, K.T. 0ggyp0
27
1.4
Non-Linear Trajectory Control of Flexible Joint Manipulators Bravo, R.R. and Dokainish, M.A. (Canada)
37
1.5
Theoretical and Experimental Investigation of Integrated Structure/Control Design of High Speed Flexible Robot Arm Fanni, M. and EI-Keran, A.A. (Egypt)
45
1.6
Generalized Path Generation for a Mobile Manipulator Bayle, B., Fourquet, J.-Y. and Renaud, M. (France)
57
1.7
Fuzzy Guidance Control for a Mobile Robot Gharieb, W. and Nagib, G. (Egypt)
67
1.8
Fuzzy Logic Sliding Mode Controller for DC Drive Ibraheem, A.A., Bahgat, A. and Abdel Motelb, M.S. (Egypt)
75
1.9
Driver Modeling using Fuzzy Logic Controls for Human-in-the-Loop Vehicle Simulations Zeyada, Y., EI-Beheiry, E., EI-Arabi, M. and Karnopp, D. (Egypt, USA)
85
1.10
Optimal Active Suspension with Preview for a Quarter-Car Model Incorporating Integral Constraint and Vibration Absorber Abduljabbar, Z.S. and EIMadany, M.M. (S. Arabia)
95
1.11
Dynamic Ride Properties of a Roll-Connected Vehicle Suspension Rakheja, S., Ahmed, A.K.W., Liu, P. and Richard, M.J. (Canada, USA)
105
1.12
Bilinear Control Theory of Smart Damping Systems EI-Beheiry, E. (EgYp0
113
1.13
A Neural Adaptive Approach for Relative Guidance of Aircraft Shahzad, M., Slama, J.G. and Mora-Camino, F. (France, Brazil)
123
1.14
Simulation of Turbulence-Induced Vibration of Loosely Supported Heat Exchanger Tubes Hassan, M., Dokainish, M. and Weaver, D. (Canada)
131
1.15
Effect of Port Plate Silencing Grooves on Performance of Swash Plate Axial Piston Pumps Kassem, S.A. and Bahr, M.K. 0ggyP0
139
Section II: S O L I D M E C H A N I C S II.1"
The Modeling of Inelastic Compressibility and Incompressibility using the Viscoplasticity Theory Based on Overstress (VBO) Krempl, E. and Ho, K. (USA)
151
11.2"
On the Mechanical Integrity of Aeroengine Compressor Disc Assemblies Meguid, S.A. (Canada)
161
11.3
Effect of Void Growth on the Plastic Instability of Uniaxially Loaded Sheets Saleh, Ch.A.R. and Ragab, A.R. (Egypt)
173
11.4
Shakedown Analysis of an Infinite Plate with a Central Hole under Biaxial Tension Attia, M.S., Abdel-Karim, M. and Megahed, M.M. (Egypt)
185
11.5
Experimental and Analytical Investigations of Residual Stresses Induced by 195 Autofrettage EI-Shaer, Y.I., Aref, N.A., AbdeI-Kader, M.S., EI-Maddah, M. M. and Megahed, M.M. (Egypt)
11.6
A Mathematical Model for the Study of the Dynamic-Visco-Elastic Contact 205 Problems Ali-Eldin, S.S., Aly, M. F. and Mahmoud, F.F. (Egypt)
11.7
Elastic-Plastic Analysis of Notch Root Stress-Strain and Deformation Fields under Cyclic Loading Hammouda, M.M.I. and Seleem, M.H. 0ggyp0
215
11.8
Elasto-Plastic Thermal Stresses in Functionally Graded Materials Considering Microstrueture Effects Shabana, Y., Noda, N. and Tohgo, K. (Japan)
223
11.9
Fracture Modeling of a Cruciform Welded Joint During Weld Cooling Behnam, W.M., Alkhoja, J., Recho, N. and Zhang, X.B. (Egypt, France)
233
II.10
Application of Elastic-Plastic Fracture Mechanics Criteria to Specimens Cut From Plastic Pipes EI-Zoghby, A.A. and AI-Bastaki, N.M. (Egypt, Bahrain)
243
II.11
Acoustic Emission Detection of Micro-Cracks Initiation and Growth in Polymeric Materials Abo-EI-Ezz, A.E. 0ggyp0
253
11.12
Evaluating the Stress Concentration due to Elongated Defects in Welded Area Abd EI-Ghany, K.M., EI-Mahallawi, I. And EI-Koussy, M.R. (Egypt)
261
11.13
Mechanics of Fracture in Fibrous Metal Matrix Composites Bahei-EI-Din, Y.A. and EIrafei, A.M. (Egypt)
271
11.14
On Simultaneous Failure of Cross-Ply and Angle-Ply Composite Laminates 281 Khalil, M., Bakhiet, E. and EI-Zoghby, A. (Egypt)
11.15
Modelling of Shape Rolling using Three-Dimensional Finite Element Technique Abo-Eikhier, M. (Egypt)
293
II.16
Shape Optimization of Metal Backing for Cemented Acetabular Cup Hedia, H.S., Abdel-Shafi, A.A.A. and Fouda, N. (Egypt)
303
11.17
Properties of Cementitious Composites Containing Non-Reeyelable Glass as a Fine Aggregate Shehata, I.H., Elsawy, A.H., Varzavand, S. and Fahmy, M.F. (USA)
313
Section III: M A T E R I A L S C I E N C E III.1
A Shape Memory Behavior Newly Revealed in Cu-Be Alloy Masoud, M.I., Naito, K., Era, H. and Kishitake, K. (Japan)
323
111.2
Poisoning of Grain Refinement of Some Aluminium Alloys AbdeI-Hamid, A.A. and Zaid, A.I.O. (Jordan)
331
111.3
Effect of Shot Peening on the Fatigue Strength of 2024-T3 Aluminum Alloy in the Unwelded and Welded Conditions Zaid, A.I.O., Ababneh, M.A. and AI-Haddid, T.N. (Jordan)
339
111.4
Creep Behavior of Solid Solution Alloys: Role of Dynamic-Strain Aging Soliman, M.S. and Almakhdoub, S.A. (S. Arabia)
347
xii 111.5
Influence of Intense Plastic Straining on Room Temperature Mechanical Properties of AI-Cu-Li Base Alloys Salem, H.A. and Goforth, R.E. (Egypt, USA)
357
111.6
Structure and Engineering Properties of Some Ductile Irons Refaey, A., H a f ~ M. and Fatahalla, N. (Egypt)
369
111.7
Conventional Versus Thin Slab Casting: A Numerical Simulation Approach for the Comparison of Microstructural Properties Youssef, Y.M., Megahed, G.M. and Lee, P.D. (Egypt, UK)
381
111.8
Simulation and Control of the Cooling of Hot Rolled Steel Wire Rod Labib, H.F., Megahed, G.M., EI-Mahallwi, I., Dashwood, R.J. and Lee, P.D. (Egypt, UK)
389
111.9
Effect of Alumina Additions on the Mechanical Behavior of PM MMC with Low Strength Matrix Mazen, A.A. and Ahmed, A.Y. (Egypt)
397
III.10
Evaluation of Damping Behavior of Spray Deposited SiC Particulates Reinforced Al Composites Abo EI-Naser, A.A. (Egypt)
407
III.11
Bonding and Properties of Explosively Compacted Copper Powder and Polypropylene Granules Hegazy, A. Abosree. (Egypt)
415
Section IV: M A N U F A C T U R I N G P R O C E S S E S IV.l*
New Trends in CIM: Virtual Manufacturing Systems for Next Generation Manufacturing Choi, B.IC and Kim, B.H. (South Korea)
425
IV.2"
Intelligent Machining Systems: Challenges and Opportunities Teltz, R. and Elbestawi, M.A. (Canada)
437
IV.3
CAM System for Efficient Generation of Part-Programs for Wire-EDM EI-Midany, T.T., EI-Keran, A.A. and Radwan, H.T. (Egypt)
447
IV.4
On the Prediction of Surface Roughness in Turning using Artificial Neural Networks EI-Sonbaty, I. and Megahed, A.A. (Egypt)
455
IV.5
On the Adjustment and Validation of Finite Element Models for Hemispherical Cup Forming Wifi, A.S., Ragab, M.S., Hussein, A.A. and Abdel-Hamid, A. (S. Arabia, Egypt)
467
xiii IV.6
Development of Spot-Weld Bonded Low Carbon Steel Damping Sheets Darwish, S.M.H and Ghanya, A. (S. Arabia, Egyp0
477
Section V: D E S I G N AND T R I B O L O G Y V.I"
Contribution of CAD-CAM and Reverse Engineering Technology to the Biomedical Field Hosni, Y.A. (USA)
491
V.2~
Design for Manufacture and Assembly (DFMA): Concepts, Benefits and Applications Bayoumi, A.M.E. (USA)
501
V.3
Inversion of Frusta as Impact Energy Absorbers Aljawi, A.A.N. and Alghamdi, A.A. (S. Arabia)
511
V.4
Effect of Laser Surface Treatment and Work Hardening on the Fretting Wear Resistance of Zr-2.5Nb Alloy at High Temperature Atria, M.H. (Canada)
521
V.5
In-Vitro Model to Evaluate the Effect of Attachment Designs on Stresses Transferred to Surroundings in Implant-Retained Overdentures EI-Wakad, M.T. (Egypt)
531
V.6
Assessment of Reliability Parameters for Maintenance Based Units using Linearized Weibull Model Mostafa, A.A-F. and Khattab, A.A. (Egypt)
539
Section Vl: I N D U S T R I A L E N G I N E E R I N G VI.I"
Optimal Replacement of Components Subject to Degradation Elsayed, E.A. (USA)
553
VI.2
Feature Recognition Algorithm for Process Selection McCormaek, A.D. and Ibrahim, R.N. (Australia)
563
VI.3
Development of a Genetic Algorithm Based on Fuzzy Logic Sets for Solving Facility Layout Problems Ramadan, M.Z. and Abou EI-Ez, S.R.S. (Egypt)
571
VI.4
Manufacturing Cell Formation Problem: A Graph Partitioning Approach Selim, H.M. (UAE)
579
VI.5
An Investigation of the Group Scheduling Heuristics in a Flow-Line Cell Hozayyin, A.S., Badr, M.A. and Helal, M.E. 0ggyp0
591
xiv VI.6
Scheduling Approach for Mixed Networked Batch Processes Soltan, H.A.M. (Egypt)
603
VI.7
A Fuzzy Reactive Approach for the Crew Rostering Problem E! Moudani, W., Brochado, M.R., Handou, M. and Mora-Camino, F. (France, Brazil, Niger)
611
Vl.8
Effects of Static and Dynamic Mean Variation on the Process Capability Mohammed, H.H. (Egypt)
621
VI.9
Short Term Management of Water Resource Systems Faye, R.M., Sawadogo, S., Gonzalez-Rojo, S. and Mora-Camino, F. (France, Senegal, Mexico)
631
AUTHOR INDEX
641
SUBJECT INDEX
643
LIST OF PARTRICIPANTS
645
KeynotePaper
Section I
SYSTEM DYNAMICS
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Current Advances hi Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, Februa O' 15-17, 2000
AUTONOMOUS CONTROL OF COMPLEX SYSTEMS: ROBOTIC APPLICATIONS
Jamshidi, M. Department of Electrical and Computer Engineering and Autonomous Control Engineering - ACE Center, The University of New Mexico
ABSTRACT One of the biggest challenges of any control paradigm is being able to handle large complex systems under unforeseen situations. A system may be called complex here if its dimension (order) is too high and its model (if available) is nonlinear, interconnected, and information on the system is uncertain such that classical techniques can not easily handle the problem. Soft computing, a collection of fuzzy logic, neuro-computing, genetic algorithms and genetic programming, has proven to be a powerful tool for adding autonomy to many complex systems. For such systems the rule base size of soft computing control architecture will be nearly infinite. Examples of complex systems are power networks, national air traffic control system, an integrated manufacturing plant, etc. In this paper a new rule base reduction approach is suggested to manage large inference engines. Notions of rule hierarchy and sensor data fusion are introduced and combined to achieve desirable goals. New paradigms using soft computing approaches are utilized to design autonomous controllers for a number of robotic applications here. I. INTRODUCTION Since the launching of Sputnik in the former Soviet Union, extensive progress has been achieved in our understanding of how to model, identify, represent, measure, control, and implement digital controllers for complex large-scale systems. However, to design systems having high MIQ| (Machine Intelligence Quotient, registered trademark by Lotfi A. Zadeh), a profound change in the orientation of control theory may be required. Currently, one of the more active areas of soft computing is fuzzy logic, and one of the more popular applications of fuzzy logic is fi~zzy control. Fuzzy controllers are expert control systems that smoothly interpolate between hard-boundary crisp rules. Rules fire simultaneously to continuous degrees or strengths and the multiple resultant actions are combined into an interpolated result. Processing of uncertain information and saving of energy using common sense rules and natural language statements are the basis for fuzzy control. The use of sensor data in practical control systems involves several tasks that are usually done by a human in the * This work was supported, in parts, by NASA Grant number NCCW-0087 This paper represents a set of applications of soft computing approaches to complex systems such as mobile robots, flexible arm, etc. The structure of the paper is as follows: Section 2 gives a brief introduction into autonomy through soft computing. Section 3 introduces two notions ot sensory fusion and rule hierarchy. Section 4 constitutes a few applications of autonomous control for complex systems through soft computing approaches. Conclusions are described in Section 5.
4
Current Advances in Mechanical Design and Production, MDP- 7
decision loop,e.g., an astronaut adjusting the position of a satellite or putting it in the proper orbit, a driver adjusting a vehicle's air-conditioning unit, etc. All such tasks must be performed based on the evaluation of data according to a set of rules in which the human expert has learned from experience or training. Often, if not all the time, these rules are not crisp, i.e., some decisions are based on common sense or personal judgment. Such problems can be addressed by a set of fuzzy variables and rules which, if properly constructed, can make decisions as well as an expert.
I
$ u
No. Rules -390.625 No. Rules = "/5
Fig. 1. The sensory fusion - rule hierarchy structure for fuzzy control systems
2. AUTONOMY THROUGH SOFT COMPUTING Soft Computing is an umbrella terminology used to refer to a collection of intelligent approaches such as neural networks (NN), fuzzy logic (FL), genetic algorithm (GA), genetic programming (GP), neuro computing, etc. Soft computing techniques can allow one to design an a u t o n o m o u s controller through learning (NN), optimization (GA) or reasoning (FL). Neural networks, genetic algorithms and genetic programming are augmented with fuzzy logic-based schemes to enhance artificial intelligence of automated systems. Such hybrid combinations exhibit added reasoning, adaptation, and learning ability. In this article, three dominant hybrid approaches to intelligent control are experimentally applied to address various robotic control issues, which are currently under investigation. The hybrid controllers consist of a hierarchical NN-fuzzy controller applied to a direct-drive motor, a GA-fuzzy hierarchical controller applied to position control of a flexible robot link, and a GP-fuzzy behavior based controller applied to a mobile robot navigation task. Various characteristics of each of these hybrid combinations are discussed and utilized in these control architectures. The NN-fuzzy architecture takes advantage of NN for handling complex data patterns, the GA-fuzzy architecture utilizes the ability of GA to optimize parameters of membership functions for improved system response, and the GP-fuzzy architecture utilizes the symbolic manipulation capability of GP to evolve fuzzy rule-sets. 3. SENSORY FUSION AND RULE HIERARCHY In many real-life problems the number of sensory data is way too many for any reasonable sized rule base. For example for a 4-variable system with 5 linguistic labels per variable, 625 rules are nominally needed. For a 10-variable process, the size of the rule base would be over 9.7 million. In other words, the size of the rule base would quickly approach infinity as the number of variables increase. In an effort to reduce the size of the rule base, many approaches are possible. Two of these approaches are sensory fusion and rule hierarchy which have been detailed in the book by the author [5]. Sensory fusion employs a linear combination of the two variables x and y to form a fused variable z= a. x + b. y, where a and b are arbitrary
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parameters. In such a way the number of sensory inputs to the inference engine would be reduced in half. Another approach is to classify the rule of the rule base in groups in according to the roles they play in the performance of the system, e.g. rules for stability, rules for tracking, rules for optimization, etc. In this paradigm, the two most critical variable, say for stability, could first be input to the first sub-set of rules, then the output of the first sub-group would be joining second most important set of sensory variables to be inputted to the second hierarchy of rules, etc. In practice, it can be assessed that neither method by themselves can reduce the rule set substantial when the number of variables is a large number, say 6 or higher.
|
of FIs
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Fig. 2. GP- Fuzzy Mobile Robot Control Architecture A third possibility is to combine the rule hierarchy and sensory fusion jointly. Figure 1 shows this structure for rule base reduction. Jamshidi [5,6] has applied this approach for the balancing of an inverted pendulum with a glass of wine on top of it.
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Fig. 3. Hierarchical decomposition of mobile robot behavior 4. AUTONOMOUS CONTROL IN ROBOTICS In this section many applications of soft computing towards rendering various degrees of autonomy in control systems will be presented. There are too much details in each of these
case studies that we can cover in this short paper. Relevant references will help the readers in all case. Since 1993, University of New Mexico's CAD Laboratory and later on the ACE Center have been active in designing the autonomous behavior of many configurations of robots.
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F_uzze-GP Control In this application a real-time fuzzy controller was designed to guide a
mobile robot through unstructured environments. The paradigm proposed in [ I 0] is based on a hierarchical fuzzy control architecture with respect to primitive and advance behaviors of the robot. The fuzzy rule bases for various behaviors are optimized using genetic programming. Figure 2 presents the architecture of GP-Fuzzy autonomous controller architecture. It is well known [6] that as the number of sensory variables increase the number of fuzzy rules will reach explosive levels. Therefore, for most complex applications of fuzzy control, such as mobile robot control, a rule reduction approach is a very necessary step of the design process. In an attempt to reduce the rules, here the functional operation of the robot is divided into primitive and advance (composite) behaviors, thereby leading to specialized sets of fuzzy rules to be used on needed basis at different levels. Figure 3 shows one possible hierarchical decomposition of a mobile robot behavior. As shown, goal-directed navigation of a mobile robot can be decomposed as a behavioral function of goal-seek (collision-free navigation) and route-follow (a path-planning type direction). These behaviors can be further decomposed into primitive behaviors such as avoid collision, wall-following, etc. linguistic rule-base. Valid behaviors should conform with the syntactic rules of construction. The above theory has been put into both simulation as well as real-time studies. Figure 4 shows the path of the mobile robot under an evolved steady-state genetic programming (SSGP) for coordination and behavior modulation. As compared with a hand-derived coordination and behavior, this approach resulted in a more direct path to the goal due to higher motivation applied to go-to-xy. The resulting path here is executed about 20% faster than the path taken via hand-derived coordination. It is also noted that the behavior modulation under evolved behavior is more complex. Near uniform bouts of competition and cooperation throughout the task evident in the decision-making, thus leading to similar amounts of behavioral influence for each primary behavior. In this section one of many applications of fuzzy logic and its hybridization with genetic programming is represented. The author's research colleagues at the ACE Center have many theoretical and experimental results which can not possibly be covered here. Interested readers can consult some recent references [5,10]. 12
12
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Fig. 4. GP Performance and Resulting Goal-Seeking Task Trajectory Ne.uro-Fuz~ Control another autonomous control approach stemmed from soft computing is
the hybridization of fuzzy logic and neural networks for real-time learning (system identification) and control. The proposed architecture for autonomous control is based on combining the learning capabilities of NNs and reasoning properties of fuzzy logic paradigms. A neural network learns about the behavior of the plant and uses that knowledge
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to modify the parameters of an adaptive fuzzy logic controller. The adaptability of the fuzzy controller is derived from a rule generation mechanism and modification of the scaling factor or the shape of the membership functions. The rule generation mechanism monitors the system response over a period of time to evaluate new fuzzy rules. Non-redundant rules are appended to the existing rule base during tuning cycles. The membership functions of the input variables are adjusted by a scaling mechanism. A multi-layer perceptron neural network classifies the temporal response of the system into various patterns according to oscillatory behavior, response overshoot, steady state error, etc. This information is used by the decision mechanism which determines the scaling factor of the input membership functions. Another neural network identifies the dynamic system, hence acting as a reference model. This model can be used to determine the stability of the new rules generated before applying to the real system. In order to implement this hybrid controller in real time, it is necessary to have substantial computing power. The TMS320C30 digital signal processor from Texas Instruments, with its powerful instruction set, high-speed number crunching capability, and its innovative architecture is ideally suited for such an application (Akbarzadeh, et al., 2000). There are commercially available boards based on TMS320C30 chips, which can be installed on a personal computer (PC). A board from DSP Research has been utilized for this purpose. The software for the control algorithm is developed in C-language and is compiled and downloaded to the DSP board. Collectively, these computing resources are used to implement the neuro-fuzzy controller architecture in real-time to control a direct drive motor used as a robot actuator. Figure 5 presents the neuro-fuzzy autonomous control system which was applied to a directdrive robot. Figure 6 shows the stabilized response of a typical direct drive motor of a scalar robot. The fuzzy logic controller has completely learned to control the direct drive motor after 300 sampling instances.
Fuzzy-GA Contro.!. Genetic algorithms are robust optimization routines modeled after the mechanics of Darwinian theory of natural evolution [3]. GAs do not require gradient evaluation, hence they are applicable to solving a great range of optimization problems including determination of optimal parameters of a fuzzy logic rule-set. Genetic algorithms have demonstrated the coding ability to represent parameters of fuzzy knowledge domains such as fuzzy rule sets and membership functions [4] in a genetic structure, and hence are applicable to optimization of fuzzy rule-sets. Here, several issues pertaining to such integration of the two paradigms are discussed and illustrated through an application on realtime hierarchical fuzzy-GA control of a single-link flexible robotic arm. To understand the actual mechanism of GAs, one may begin with its three most commonly used operators, namely: reproduction, crossover, and mutation. A member of a given population which has a higher fitness is given a higher chance to reproduce identical replicas of itself in an intermediate population. In this fashion, the optimization routine facilitates reproduction of higher fit individuals and hampers the reproduction of lower fit individuals. After reproduction, crossover randomly mates two individuals from an intermediate population and creates offspring which are made up of a random combination of their parent's genetic code. For each generation, the process of crossover is repeated for all individuals in the population. The population size is often a constant equal to the number of individuals in the initial population. The operations of reproduction and crossover create an environment where every generation benefits from the best genetic codes of the previous generations. However, if the building blocks for the optimal genetic structure is not in the initial
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population, these two genetic operators will be unable to find it. The last genetic operator, mutation, randomly mutates one or more of the values in the individual's genetic code in order to create diversity. The mutation operator allows for exploring n e w structures (directions of search) hence allowing the genetic optimization routine to invent new solution and finally locate the optimal solution even though the individuals in the initial population may not have contained the building blocks for the optimal solution. Delm~ TaT O Tempmm~ AdaptiveFizzy
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Fig. 6. Neuro-Fuzzy C o n t r o l S t a b i l i z i n g R e s p o n s e s o f a Direct Drive M o t o r When applying GA to optimization of fuzzy rule-sets, several questions arise. First is the design of the transformation function (the interpretation function) between the fuzzy knowledge domain (phenotype) and the GA coded domain (genotype). This is perhaps the most crucial stage of GA design and can significantly degrade the algorithm's performance if a poor or redundant set of parameters are chosen for a given optimization problem. Two important general categories of fuzzy expert knowledge consist of domain knowledge and meta-knowledge. Meta knowledge is the knowledge used in evaluating rules such as fuzzification (Scaling or Z, cut), rule evaluation (such as Min/Max) and defuzzification (such as Max Membership, Centroid, or Weighted Average) methods. Relatively little research has been performed to study the effect of optimizing the meta-knowledge [2]. Most of the current
Current Advances in Mechanical Design and Production, MDP-7
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research, as in this work, concentrate on optimizing parameters of the domain knowledge. The domain knowledge consists of the following two categories, 9 Membership Function: General Shape (Triangular, Trapezoidal, Sigmoidal, Gaussian, etc.), Defining points (Center, Max Right, Min Left, etc.) : 9 Rule-Base: Fuzzy Associative Memory, Disjunctive (OR) and Conjunctive (AND) operations among antecedents in the rule base. Even though various methods exist to encode both rule-base and membership functions in one GA representation, such coding can have several potential difficulties. In such situation, in addition to the level of complexity and large number of optimization parameter, the problem of competing conventions may arise and the landscape may unnecessarily become multi-modal. This is an important problem since there are often several (or many) fuzzy rulesets which can represent a given nonlinear function. This means that there are more than one optimal solution to a given optimization problem which raises the issue of multi-modality for fuzzy logic systems, or more specifically competing conventions where different chromosomes in the representation space have the same interpretation in the evaluation space. When designing the interpretation function, therefore, the coding needs to contain fewest possible parameters to avoid the problem of dual representation, and yet the coding needs to have enough complexity to contain all possible optimal or near optimal solutions. Evolutionary approaches such as Niched GA [2] are designed to search in complex multimodal landscapes. As a result, in the present approach, this problem is attended to by limiting the optimization parameter space to membership function parameters only. This was a design decision which was made considering the following two considerations. 9 The problem of multi-modality is introduced when the GA string contains both parameters of membership functions and rules. 9 In control of most physical systems, rules can often be derived either intuitively or through operator experience. The ambiguous and fuzzy portion of a knowledge base is often the membership functions. For simplicity in coding the simulation and the real-time algorithms, only triangular membership functions are coded for optimization here. Figure 7 illustrates a triangular membership function whose three determining parameters (a,b, c)are shown. Assuming a normalized membership function, the three parameters are real numbers between-1 and 1. The coding in GA is performed as follows, the real parameter an is first mapped to an n-bit signed binary string where the highest bit represents the sign. This way, the parameter a can take on 2 n different values. Then the binary number is aggregated with other n-bit binary numbers to construct the phenotype representation. l
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Fig. 7. A Triangular membership functions with its three parameters
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Current Advances in Mechanical Design and Production, MDP- 7
The second issue which arises is how to utilize initial expert knowledge for a better and faster convergence. In other search routines such as hill-climbing, it is clear that starting from a "good" point can significantly improve computation time needed for convergence to an optimal solution. However, the conventional GA applications generate a random initial population without using any a-priori expert knowledge. This, in general, will provide a more diverse population while sacrificing convergence time. This convention can indeed be adequate if there is no a-priori knowledge as to where a "good" solution may exist. However, in fuzzy logic applications, there is usually access to some expert knowledge which, even though it may not be the optimal solution, is often a reasonably good solution. Sehultz and Grefenstette [9] addressed the problem of incorporating a priori knowledge by introducing two types of populations, homogeneous and heterogeneous. The homogeneous population consists of individuals created randomly while their string is augmented with the same priori rules. In this sense, they are all identical and hence homogeneous. He concluded that a trade-off exists between manual knowledge and machine learning. The heterogeneous population consists of members which are not identical. This is also referred to as the "seeding" technique. In [7], the process of seeding the initial population with one or more experts' knowledge is proposed. The few seeded chromosomes have the chance of reproducing through mutation and crossover with other randomly generated chromosomes in the population. This method improves the performance of GA by providing the genetic population with a set of highly fit building blocks, as compared with GAs starting with random initial populations. However, such population still requires a large number of iterations before convergence since the "useful" schemata exist in only one or few seeded members and can only be reproduced as fast as the rate of reproduction. Akbarzadeh [1] proposed the grand-parentingscheme where the initial population is comprised of mutations of the "knowledgeable" grandparent. This scheme takes advantage of expert knowledge while maintaining diversity necessary for an effective search. Through grandparenting, an expert's a priori knowledge can be utilized to improve fitness of GA's initial population, thereby increasing the speed and performance of the search routine. In the present approach, the method of grand-parenting is used to improve the convergence rate of the GA optimization process. The third issue is defining a fitness function. A fitness function is a very important aspect of GA design since it determines the direction of the search. Fitness functions come in as many different forms as the systems which they are optimizing. In general, for a lumped parameter system (such as a flexible robot arm), parameters such as control effort u(t), rise-time tr, overshoot y and steady-state error ess are usually incorporated in a quadratic fitness function. Often, constant multipliers define the relative degree of importance which is given to a certain parameter compared to others. The above concepts were applied to controller optimization of a flexible link robotic system as shown in Figure 8. The flexible link can be represented by a distributed-parameter system with spatial as well as temporal parameters. In other words, the states of a flexible robotic system are functions of both space and time. This complicates the modeling of the system and, consequently, the process of designing the controller. Due to the complexity of a mathematical representation for such systems, fuzzy logic is considered an attractive alternative to their control. One of the issues in development of fuzzy controllers is determining faithful expert knowledge. Expert knowledge, however, is difficult to produce since there is often no human expert to consult and training a human expert may not be a
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feasible alternative due to cost and other practical considerations. Furthermore, human psychological issues may prohibit a faithful reproduction of a rule-base from an expert. In addition, the unstructured operating environments associated with space and waste handling projects require the robot controller to also adapt to changing conditions. In the process of designing fuzzy rule sets, membership functions are often chosen through an ad hoe process of random selection and evaluation. As a viable alternative, good results have been achieved by employing genetic algorithms to tune membership parameters within a fuzzy controller's knowledge base [8]. Genetic algorithms equip the fuzzy controller with some evolutionary means by which it can improve its rule-base when faced with inadequate a-priori expert knowledge or varying circumstances in its operating environment.
i
....
I
Fig. 8. Autonomous Control Architecture Through a Fuzzy-GA Approach The GA-optimized architecture proposed here needs several parameters defined. Spatial variables are fuzzified for use in a rule base at the higher level of hierarchy. Other parameters in the knowledge base are not allowed to vary. The fitness criterion used to evaluate various individuals within a population of potential solutions was based on the error e(t), effort u(t) and 3, as an inverse square function. Consequently, a fitter individual is an individual with a lower overshoot and a lower overall error (shorter rise time) in its time response. Here, results from previous simulations of the architecture are applied experimentally. The method of grand-parenting [ 1] was used to create the initial population. Members of the initial population are created through mutation of the knowledgeable grandparent(s). As a result, a higher fit initial population results in a faster rate of convergence as is exhibited in Figure 9. Figure 9a shows the time response of the GA-optimized controller when compared to previously obtained results through the non-GA fuzzy controller. The rise time is improved by 0.34 seconds (an 11% improvement), and the overshoot is reduced by 0.07 radians (a 54% improvement). The average fitness of each generation is shown in Figure 9. A total of 10 generations were simulated. The mutation rate for creating the initial population was set at 0.1. This value was chosen to increase diversity among members of initial population. GA depends on this diversity to exploit a large number of differing path of solutions in parallel. The mutation rate throughout the rest of the simulation, however, was set to 0.01. Since a high mutation rate delays convergence. The probability of crossover was set to 0.6. Initial experimental results demonstrate that the GA-learned controller is able to control the actual experimental system as in Figure 9.
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Current Advances in Mechanical Design and Production, MDP- 7
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Fig. 9. Computer Simulation and Experimental Results of GA-Fuzzy Autonomous Control System. (a) Simulation, (b) Fitness Function Behavior and (c) Experimental Results. The hardware used to implement the above algorithms is the same as was explained in the previous section on neuro-fuzzy control with a few modifications pertaining to flexible robot control such as a tip end position sensor and several strain gauges distributed evenly across the length of the flexible beam. Control update was performed at 250 Hz. 7. CONCLUSIONS The basic theme of this paper was autonomous control and autonomy through several architectures of soft computing. A number of robotic applications were used to illustrate these architectures. These autonomous controllers are simple to implement in a laboratory environment on either a PC or on a chip-level board. Soon, autonomous control through intelligent paradigms technology will be a matter of economy and not controversy. It features applications in a wide variety of fields such as control, pattern recognition, medicine, finance, marketing, etc. should be given serious considerations as additional tools for the solution of problems which are most suitable for this technology, i.e. problems where a mathematical model is neither available nor feasible. Applications to complex systems require careful considerations of the system's model, its structure, the behavior and means of sensory data, and rules specialization and hierarchy. Finally, new avenues should be opened for new software design and analysis of control systems utilizing the power and efficiency of all tools such as fuzzy logic, neural networks and genetic algorithms. ACKNOWLEDGEMENTS The author wishes to thank his peers and associates; Dr. M. Akbarzadeh, Dr. E. Tunstel, Dr. K. Kumbla, Mr. A. EI-Osery and Ms. Sandi Avrit for their help in preparation of the manuscript. REFERENCES Akbarzadeh, M. R., Fuzzy Control and Evolutionary Optimization of Complex Systems, PhD Dissertation, University of New Mexico, (I 998).
Current Advances in Mechanical Design and Production, MDP- 7
2. 3. 4. 5. 6. 7. 8. 9. 10.
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Akbarzadeh, M., Kumbla, K., Tunstel, E. Jr., and Jamshidi, M. "Soft Computing for Autonomous Robotic Systems," International Journal on Computers in Electrical Engineering", Vol. 27, (2000), (to appear). Goldberg, D. E., "Genetic Algorithms in Search, Optimization and Machine Learning," Addison-Wesley, (1989). Homaifar, A. and McCormick, E., "Simultaneous Design of Membership Functions and Rule Sets for Fuzzy Controllers Using Genetic Algorithms," IEEE Transactions on Fuzzy Systems, Vol. 3, No. 2, pp. 129, (1995). Jamshidi, M., "Large-Scale Systems - Modeling, Control, and Fuzzy Logic", Prentice Hall Series on Environmental and Intelligent Manufacturing Systems (M. Jamshidi, Ed.),Vol. 8., (1996). Jamshidi, M., "Fuzzy Control of Complex Systems," Soft Computing, Vol. 1, No. 1, pp. 42-56, (1997). Lee, M. A. and Takagi, H., "Embedding Apriori Knowledge into an Integrated Fuzzy System Design Method Based on Genetic Algorithms," Proceedings of the 5th IFSA World Congress, (1993). Lee, M. A. and Takagi, H., "Integrating Design Stages of Fuzzy Systems Using Genetic Algorithms," Proceedings of the 1993 IEEE International Conference on Fuzzy Systems, San Francisco, CA, pp. 612-617, (1993). Schultz, C. and Grefenstette, J.J., "Improving Tactical Plans with Genetic Algorithms," Proceedings of the 2nd International Conference on Tools for AI, Herndon, (1990). Tunstel, E. W. and Jamshidi, M., "Intelligent control and evolution of mobile robot behavior," in Applications of Fuzzy Logic - Towards High MIQ(TM) Systems, (Jamshidi, M., Titli, A., Zadeh, L. A., and Boverei, S. - eds.), Prentice Hall Series on Environmental and Intelligent Manufacturing Systems (M. Jamshidi, Ed.), Vol. 9, Chapter 1.
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Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
15
A SIMPLIFIED INVERSE KINEMATIC MODEL CALCULATION METHOD FOR ALL 6R TYPE MANIPULATORS
Renaud, M. LAAS-CNRS, 7, Avenue du Colonel Roche, 31077 Toulouse, Cedex 4, France
ABSTRACT An inverse kinematic model of a manipulator - equipped with an end effector- is a function which allows to calculate a manipulator configuration corresponding to a given end effector location (position and orientation). In this paper, we consider the case of 6R type manipulators having a single open kinematic chain structure and six revolute joints R. The first method, which brought back the determination of this model to the computation of the roots of a polynomial equation whose degree - equal to sixteen - is minimal, was proposed in 1988 by Lee and Liang. Nevertheless, Lee and Liang's technique was extremely complicated and necessitated divisions which prevent it from being used in any case. Here, we present a simplified method for this calculation, which does not involve any division; as a consequence it works in any case. We present also an example of a particular, but non academic, manipulator for which we actually obtain sixteen different real configurations corresponding to a particular given end effector location; this result demonstrates that the degree of the previous polynomial equation cannot be reduced. KEYWORDS Inverse Kinematic Model, Configuration, Location, Generalized Coordinates, Operational Coordinates. I. INTRODUCTION In this paper, we consider manipulators having a single open kinematic chain structure equipped with an end effector (gripper or tool). An Inverse Kinematic Model (IKM)is a function which allows to calculate one configuration of this manipulator corresponding to a given end effector location (position and orientation). The configuration is defined by a set of generalized coordinates and the location by a set of operational ones. In fact, as we will see later in this paper, each IKM associates to one real location (i.e. defined by real operational coordinates) one complex configuration (i.e. defined by complex generalized coordinates). We study the case - well known to be the most difficult among all the six joints manipulators of the 6R type manipulators having six revolute joints R, and for which several IKM exist. Nevertheless each IKM is defined on a particular subspace of the location space; as a consequence the number of complex configurations, corresponding to one given location, is function of this particular location. The number of real configurations - which is, in any case, less than the previous one - is also function of this particular location. This last number-
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Current Advances in Mechanical Design and Production, MDP- 7
which is a very complicated function of the given location - and the number of IKM - which is independent of this location - must not be confused. Here, we are looking for the IKM and, more particularly, for their number. One of the earliest attempts to calculate the IKM has been made by Pieper [1 ] who, in 1968, demonstrated, by naive elimination, that their number was at most 524,288. Several years after, in 1976, Roth [2] conjectured that this number was at the most of thirty two. In 1980, Duffy and Crane [3] proved this conjecture by combining spherical geometry and Sylvester dialytic elimination [4,5]. Afterwards, in 1985, Tsai and Morgan [6] used homotopy continuation and produced several examples of 6R type manipulators for which the number of IKM was equal to sixteen and supposed this result general for all the 6R type manipulators. The demonstration of this result was established for the first time in 1988 by Lee and Liang [7,8] who have, moreover, proposed a computational method of these sixteen IKM. This technique used, extremely complicated, involves divisions and as a consequence fails in some cases. Several solutions were proposed to simplify this technique, but they all share the same drawback, namely the use of divisions. See for example the works developed by Raghavan and Roth, in 1989, [9], Manocha and Canny, in 1992 and 1994, [10,11], Kohli and Osvatic in 1992, [12], or Mavroidis, Ouezdou and Bidaud, in 1994, [13] . . . In this paper, we propose a new method, much more simpler than the previous ones, and which presents no singularities at all; therefore, it applies without restrictions to any kind of 6R type manipulators. Nevertheless our method relies on the original method of these authors and shares the following original ideas: it transforms the 6R open single kinematic chain into an hypothetic 7R closed single kinematic chain, 9
it uses a vector, specified in the sequel, already considered by these authors, it uses the subtle artifice which consists of writing in an affne way cosine and sine (here of the angle q6 ) as a function of the half angle tangent (here of the angle ~-),
9
it uses the dialytic calculation.
Our method consists of bringing back the calculation of all the IKM to the computation of the roots of a sixteen degree polynomial equation whose unknown is a generalized coor- dinate ql
half angle tangent (here of y ). Because of the impossibility to express these roots literally our method will be only numeric, although it allows the literal writing of this equation coeffcients. Nevertheless this literal writing is generally too complicated to be of any use. For a particular 6R type manipulator, the algorithm we have developed allows to compute numerically all the complex configurations corresponding to a particular given end effector location. Among these configurations, we extract the real ones. Finally, we give an example of a very simple, but non academic, particular 6R type manipulator- the RMS of the American shuttle - for which we obtain sixteen real configurations corresponding to a particular given end effector location. This demonstrates that the previous polynomial equation degree cannot be reduced and, as in Lee and Liang's calculation, allows us to prove that our method is minimal.
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2. D E F I N I T I O N OF THE M A N I P U L A T O R T O P O L O G Y
The considered manipulator consists of six rigid links jointed in a single open kinematic chain with revolute joints R. The first link in the chain is connected, via the first joint, to the base and the last one supports the end effector. Links are designated by Ci and joints by L,, i being an index increasing from one to six starting from the base; thus (:6 designates the end effector. The Fig. 1 represents an example of a particular 6R type manipulator: The RMS of the American shuttle.
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Fig. 1: The R~IS 6R type manipulator
3. D E F I N I T I O N OF T H E M A N I P U L A T O R G E O M E T R Y
The orthogonal affine frame R0 = (O0,_Xo, Y-o'z0) is embedded in the base, named link Co and an orthogonal affine frame Ri = (Oi, x i, Yi' zi) is embedded in each link C,i (see. Fig. 2) using the modified Denavit Hartenberg (mDH)iterative procedure [14] [15] [16] (i=l, 2 , . . . 6): 9 0i-1 is the foot of the perpendicular common to the axes of Li-1 and Li, located on the axis of Li-1, while f~i is the foot of this perpendicular located on the axis of Li, 9 xi_ 1 is the unit vector along this perpendicular, oriented from the axis of Li-1 to the one of Li, 9 zi_ ~ is the unit vector (of arbitrary orientation) along the axis of Li-1, 9 -Y~-I is a unit vector such that Ri be direct. To initialize this procedure we consider that L0 is a fictitious joint whose axis is identical to the one of L0 and we fix O0 = O1..XIoreover we assign z__0 = _z1. To end this procedure, we choose 06 = f~6 and we assign z__6 as the unit vector of the axis of L6, with an arbitrary orientation. The location of Ci with respect to its antecedent C~_~ is defined by the four mDH parameters of the Fig. 2: 9 ai-1 algebraic angle between zz_ 1 and z i measured around xi_l, 9 ai-l length of the perpendicular common to the axes of Li-1 and Li (ai-1 3> 0), 9 Oi = q~ algebraic angle between x,_~ and x i measured around z__i, 9 ri = fliOi, algebraic distance measured along z i. Note that a0 = a0 = r~ and r6 = 0, and let Ci = cos 0i = cos qi and & = sin Oi= sin qi. The homogeneous transform matrix T~_I,, [14] [15], from R,-1 to R~, is then defined by:
18
Current Advances in Mechanical Design and Production, MDP- 7 Li.i Z-i~i.i I i xi Oi Oi ai- I
Oi.!
ri
Fig. 2: Modified Denavit Hartenberg parameters Ci
-Si
0
cosc~i_ 1 S/
COSCei-I Ci sin C~i_ 1 Ci
-sincei_l cos O~i- 1
T i - 1,i =
ai-1 - r i - 1 sinai_l
sin ai-1 Si r i - 1 sin a i - 1 " 0 0 0 1 R e m a r k : these mDH parameters must not be confused with the classical Denavit Hartenberg parameters (cDH) used by numerous authors [17] [18] [13]...; modified parameters are much more general than classical ones as, contrary to the classical ones, they can be generalized to manipulators having an open complex kinematic chain structure. Among the previous four mDH parameters three are constants (ai-1, a i - 1 , ri) and one is variable (0i = qi). The geometry of a particular manipulator is then defined by the table of the twenty four mDH parameters among whose eighteen are constants and six variables. For example the table corresponding to the RMS 6R type manipulator is indicated hereafter:
i
1
2
3
4
ai-1
o 0
0
o as
a3
Oi ri
ql 0
q2 0
q3 0
q4 0
.
5
.
.
.
.
6
a4
0
q5 0
q6 0
, ,
4. DEFINITION OF THE MANIPULATOR CONFIGURATION AND OF THE END E F F E C T O R LOCATION The manipulator configuration q is defined by six generalized coordinates: q = (ql q2 . . . q6) t and its end effector location x by six operational coordinates: x = ( x l x~ . . . x6) t. Numerous choices of operational coordinates can be made. Our proper choice is specified in the sequel.
4.1 Transformation of the open 6R type manipulator into an hypothetic closed 7R type one
Let L7 be an hypothetic joint whose axis is defined by point 06 and unit vector z 7 - Y_Y-6" Then a6 = - r / 2 and a6 = 0. Moreover 06 is located in the right place according to the mDH iterative procedure. Furthermore f~7 - 06. Now let us consider the perpendicular
Current Advances in Mechanical Design and Production, MDP-7
19
common to the axes of LT and Lo let O7 and l~0 be, respectively, the feet of this perpendicular on the axes of L7 and L0. and J:7 the unit vector of this perpendicular oriented from the axis of L7 to the one of L0. The location of R6, with respect to R0, (i.e. the location of the en(l effector with respect to the base), is defined by the six independent parameters" 07, /'7. 07. a7. 00, r0,
which correspond to the mDH parameters of the homogeneous transform matrix from /~6 to R7 (for 07 and rT) and from RT to R0 (for aT, aT, 00 and ro). We choose these parameters as operational coordinates. Remark: let us particularize R0. for each end effector location, by choosing x_.0 = x_T. This leads to impose 0o = 0 and the previous location is now only defined by five independent parameters.
5. INVERSE KINEMATIC MODEL CALCULATION 5.1 Loop Equation To each set of the five previous independent parameters correspond several sets of generalized coordinates ql, q2. q3, q4, qs, q6 that verify the loop equation: T61(ql)T12(q2)... T56(q6) -- E where E represents the four order unit matrix. Let us remark that: T6~(q~) = T67TT~(q~) with" c07 -sO7 0 0 C1 - $1 0 a7 COz7S1 c0~7C1 - s o l 7 - r l so~ T 1 r7 and Tzl(ql) = 0 0 -sOTS1 sa~ S1 cat 7"lCaT 0 1 0 0 0 1 the loop equation using the "preferential" index p [18] such that' and as a consequence p 3" T61 (ql )T12(q2)T23(q3) = T65(q6)T54(qs)T43(q4). The first member of the previous equation corresponds to the sub chain A and depends only on q~, q2 and q3. while the second one is related to the sub chain B and is only function of q6. q5 and q4. We are going to eliminate the q3 variable by taking into account only the first two columns of the T 0 matrices. Let To be the 4 x 2 matrix constituted by these first two columns. Then the matrix T23 is constant and we can write successively"
T67 =
0 0 -sO7 -c07 0 0 It is possible to write p - E(~). with n - 6
T6, (q, )T12(q2)T'23 = T65(q6)T54(qs)T43(q4), T6~ (q~)T13(q2) = T65(q6)T53(q5, q4), T63(q,. q2)= T6a(q6, qs, q4). The last matrix equation defines a system of six equations, in five unknowns q~, q2, q6, qs, q4, related by one constraint determined hereafter. Let us write" o~ a b
o
The six equations to be solved are then:
1
.
20
Current Advances in Mechanical Design and Production, MDP- 7
c~(ql,q2)
= =
a(q,, q2)
c~(q6,qs, q4)
/3(q,,q2)
a(q6, qs, q4)
b(ql, q2)
= =
/3(q6, qs, q4)
7(q,,q2)
b(q6, q5, q4)
c(q,, q2)
= =
7(q5, q4)
c(qs, q4),
while the constraint relation is given by: (c~? + ~2 + 72)(q,, q2) = (c~2 + f12 + 72)(q6, qs, q4) = 1.
5.2 Resolution of The Previous Six Equations System Let P-63 be a vector whose origin is Os and extremity 03. In order to solve the previous six equations system we propose to solve an augmented system in which appear eight other equations. To this aim, let us add eight other scalars (u, v, w, l, m, n, s and a) to the six already introduced (c~,/3, 7, a, b and c). The first three represent the vector z__3 x P-63 1 2 components, with respect to Rs, the next three the v e c t o r ~P_63z_3(z3.P63)P63 components, with respect to the same frame, and the last two are: s = z__3.P_s3 = c~a + fib + 7c and 1 2 = ~1 (a2 + b2 + c2)" = ~/2_63
R e m a r k : the ~p_~3z3 ~ 2 -- (Z_3.P_63)P_63vector was already introduced by Lee and Liang [7] [8] and used by Manocha and Canny [10]; nevertheless, as these authors, we were unable to explain its introduction. Then, the a u g m e n t e d system to be solved is given as follows: c~(q,,q2)
a(q,, q2) u(qi, q2) l(ql,q2)
= =
c~(q6,qs, q4)
/3(q,,q2)
=
/3(q6, qs, q4)
b(q6, qs, q4) v(qs, qs, q4)
=
l(q6, qs, q4)
b(q,, q2) v(q,, q2) m(q,,q2)
=
=
a(qs, qs, q4) u(q6, qs, q4)
= =
7(ql,q2)
=
7(qs, q4)
c(q,, q2) = w(ql, q2) = m(q6, qs, q4) n(ql,q2) =
c(qs, q4) w(qa, q4) n(qs, q4)
s(ql, q2) = s(q5, q4) a(q,, q2) = a(q5, q4). R e m a r k : it seems judicious to consider the following six equations in four unknowns sub system: 3'(q1,q2)
=
n(ql,q2)
= n(q5,q4)
7(qs,q4)
c(ql,q2) s(ql,q2)
= c(q5,q4) = s(qs,q4)
w(ql,q2) a(ql,q2)
= w(qs, q4) = a(q5,q4)
Although it is possible to calculate all the IKM starting with this sub system the calculation leads to solve a polynomial equation whose degree is thirty two and not sixteen. That is why we have only worked on the augmented system.
5.2.1 Inspection of the left terms The scalar (~(ql, q2) can be written: ~-
( C2
$2
1 )
c~s ~1
with
c~s Ctl
= [~]
1
and [c~l =
c~
~.
O/lc
C~ls a l l
c~1
.
The [c~] m a t r i x is independent of q2 and ql; it depends on the considered manipulator and on the end effector given location. This matrix can be calculated in a literal way very easily. It is possible to do the same thing with the other thirteen scalar terms and therefore to obtain the other thirteen matrices: [~], ["/], [a], [b], [c], [u], [v], [w], [l], [m],
N,
[ol.
Current Advances in Mechanical Design and Production, MDP-7
5.2.2 Inspection
21
of the right terms
After various a t t e m p t s and tedious calculations it a p p e a r s t h a t it is p a r t i c u l a r l y interesting to introduce the two following vectors: q______. (13 8003 8004 C005 ( ~p6a~3 1 2 . , - (za.p63)P63) + aa r5 saa set4 z 6 x (z 3 x P6a) --r4 (a3 cct3 cot5 -F a5 set3 s o 5 ) sct3 s o 4 P63
+~ (~ a~ ~o~ ~o~ - ~1 (ag - a] + ~,~2 + ,'~ - ~) ~~
CCt5) 80~4 _Z3
--a3 a4 s003 c04 c 0 5 "z3 x P-63 -t- a3 a4 c a 3 8ct3 z 6 x P-63 --t- a3 a4 (r4 + r5 cc~4) s o 3 _z6 x z 3
-?
1,.,2 Z -_ r__ -- 03 8 0 4 (~ff_63_ 3 (Z_3.P_63)P63) + a3 r5 8a4 c0~5 2:6 x (2:3 x /)63 ) a3 r4 col3 8 a 4 P63 2 (aZ3 - a ] - 0 5 + r24 - r 2) s004 z__3 - aa a4 ca,, z_a x P63 + a4 (a3co~3cc~5 + assaasc~5) z6 x P6a
+a3 a,4 (r4 + r5 ca4) ca5 z_6 x z a
Indeed, these two vectors lead to the following f u n d a m e n d a l properties: 9 the first two c o m p o n e n t s of q and L', with respect to R6, can be w r i t t e n as a linear function of some of the fourteen previous quantities, 9 the first c o m p o n e n t s of z_6 x q, q, _.r and _z6 x r_, with respect to Rs, can be written as an affine function of some of the fourteen previous quantities. In order to calculate the first two c o m p o n e n t s of q and r_, with respect to R6, let us use the following table:
m 37(6)
l
--V
a
o'
-Y(6)
m
u
b
/3
z__6x p~a b
z6xza
a
a
v
q
-/3
B
A
r
A' B'
This table lead us, w i t h o u t an3" difficulty, to the c o m p o n e n t s we are looking for: -B
= a3 so3 so4 ca5 1 - a3 r5 saa so4 v - r4 (a3 cc~3 ca5 + a5 sa3 SOs) saa so4 a 1
'
'
2
+ a 3 (a3 a5 c003 s005 -- ~ (a~ -- a~ + a 5 + r~ -- r~) 8 a 3 c a 5 ) 8t~4 o' -- a3 a4 8 a 3 c a 4 c a 5 u --a3 a4 c003 set3 b -- a 3 a4 (r4 + r5 cot4) 8 a 3 j~
.4
- " a 3 8 ( l 3 8 ( 1 4 c o 5 l l l -[- a 3 r5 8 0 3 800'4 u - - r 4
(a3 c~3 c~5 + a5 s~3 s~5) s~a s~4 b
+a3 a4 co'3 s003 a + a3 a4 (r4 + r5 cct4) set3 c~
A' = a3 ~0~ t-~.~ ,.~ ~
co~ ~ , - ~
~ ~0~ .~,~ a - ~ ( ~ - a ] - a g
+~-~)
~4 ~-a~4
c~4
- a 4 (a3 ca3 cc~ + as sa3 sos) b - a3 a4 (r4 + r5 cc~4) cc~5/~ B ' = a 3 8004 Ill +(/3 1"5 8004 C(15 t l - - a 3 r 4 c003 8(I 4 b - ?
((7,2--(/, 2 --a~ + r 4 2 - r 2) 8 ~ 4 ~ - - a 3 a 4 COl4 V
+a4 (a3 c003 c05 + a5 s003 s00s) a + a3 a4 (r4 + r5 cc~4) coe5 oz T h e z 6 x q. q. s and Z 6 X _F first c o m p o n e n t s , with respect to Rs, are d e s i g n a t e d by: _xs.r _ , X = ~ . ( z 6 x q_) 9 l ' = s . .y ' . . and Y' = -xs.(z 6 • r__). Then, after tedious calculations, we obtain"
Current Advances in Mechanical Design and Production, MDP- 7
22
X -- - - a 3 8 a 3 Sa4 S a 5 n ~- a 3 a4 s ~ 3 c a 4 s a 5 w 4- r4 (a3 c a 3 s a 5 -- a5 s a 3 c a s ) s a 3 s a 4 c 1 +a3 (a3 as ca3 cas + 5(a3 2 - a 2 + a 2 + r42 - r 2) sa3 sas) sa4 3' - a ~ (~] c ~ + r~ (r~ c ~ +
~) ~ )
~
Y -- sc~4 (a3 a5 r5 80/3 7 - a3 (a3 c a 3 s ~ 5 - a5 80/3 c a 5 ) s A- r4 82a3 sol5 a -4- a3 r4
x' = ~ ~ ~.~ ~.~ ~ + ~ (~ c~ ~
- ~ ~
as ca3 s~3 ca5
c . ~ ) ~ + a. ~ ( ~ + ~ r
~
Y ' = - a 3 a5 s ~ 4 w - a3 a4 a5 c0~4 3' + a4 so~3 sc~5 t7 -4- a3 a4 a5 c0~3 c0~5 - a4 T4 r5 8~3 c0~4 80~5
+~ (~
- ~ + ~ - ~ - r~)~
~
The interest of the previous components appears when one uses the relation: = c 6 ~ - $6 y__6,
indeed:
X = C6 (x_6, z_s, q) - $6 ( ~ , ~ , q) = - A C6 + B Ss Y = C6x_~.q- S6y_s.q = - B C 6 _ + $6 ~ . r_ = X ' = - C 6 x__s.r
A'
- AS6 C6 + B' $6
Y' = C6 ( 2 , n-s, r) - $6 ( ~ , _z6, r__)= - B ' C6 - A' $6 Let us replace the unknown angle q6 by its half angle tangent ~, according to Lee and Liang's proposal. Then, let be: x6 = tan ~ 9 Taking into account that: x6 - _3a_ - 1-ce 1+C686 ' the previous four equations system can be written as follows: Jx6+K Lx6+I j' x6+K' L' x6 + I'
with: I = K'=Y'
A+X,
J=
+ B ' , L' = Y ' -
A-X,
K = Y+B,
= =
0 0 0 0
L =Y-B,
I' - A ' + X ' ,
J' - A ' - X ' ,
B' .
5.2.3 Inspection of the augmented system equations The quantities I, J, K, L, f , J', K', L' are given by affine functions of the fourteen basic scalar terms. As a consequence it is possible to calculate the matrices [I], [J], [g], [L], [I'], [J'], [K'], [L'] which are given by the same relations as functions of the matrices [c~], [~], [7], [a], [b], [el, [u], [v], [w], [/], [m], [n], [s], [a]. These matrices are independent of q2 and ql; they depend only on the considered manipulator and on the end effector given location. Then it is possible to write, successively:
I~ It
= [1]
S', and I = ( (72 $2 1 ) Is and to do the same thing with: J, 1 I1 K , L, I', J', K ' and L'. If one introduces the q2 half angle tangent" x2 = tan ~ and take into account that C2 = !l+x~ _ ~ and $2 = ~l+x 2 it is possible to write the previous equations as follows:
Current Advances in Mechanical Design and Production, MDP-7
[(J~ [(L, [(J', [(n]
./~).r.~ + 2.1~.r.2 + - Lc).r.~ + 2Lsx2 g:).r.~ + 2Ji.r2 + - L;).v~9 + 2L:x2
(Jr + J~)]x6 + [(K~ - K~)x~ + 2K, x2 + (L, + Lc)]x6 + [(I, - Ic)x~ + 2Isx2 (J; + J~)lx6 + [(K'I - K : ) x 2 + 2Ki.'r2 22 + 2I~x2 + (L,' + L;)]x6 + [ ( I ' 1 - I~)x ' '
23
+ (K, + Kc)] = O, + (11 3t- I~)] = 0,
+ (K', + K:)] = 0, + ( I x' + I'~)] = 0,
In order to be able to use the dialytic calculation let us multiply these equations by x2" [(.S~ - Jc)x 3 + [(L,[(J,' - go) -r3' + [(n] - L:)x~
2J, x.~ + (J, + J~)z2]x6 + [(hq - Kc)x~ + 2K~x~ + (K, + K~)x2I = o, = O, + 2L x ~ + (L, + n~)x2]x6 + [(I, - Ic)x 3 + 2Isx~ + (I, + Ir t
!
t
9g'x~-~ 2 + (J, + g~)x2]x6 + [(K'l - K:)x 3 + 2t(;x~ + (K' 1 + K~)x2] = O, + 2Lix~ + (L', + L;)x2lx6 + [(I; - I:)x 3 + 2Iix~ + (I', + I'~)x~] = O.
Then. we obtain an eight linear equations in eight unknowns system whose second member is zero (the unknowns are: x~.r6, x22x6, x2xe,, xe,, x 3, x~, x2, 1). This system is not a Cramer one and therefore its determinant must be zero. Each determinant element being an affine function of C1 and $1 this leads to a polvnomial~ equation, whose unknown is Xl = tan q~2 if 1-x 2
one uses the relations: C1 = ~ equation is sixteen, and its roots software (which was our choice). values offers no difficulty, and. as
and S1 = ~l+~," The maximal degree of this polynomial can be obtained very easily, for instance using Maple The computation of the others generalized coordinates a consequence, is not presented here.
6. E X A M P L E
We have calculated all the IK~I for the RMS manipulator (see Fig. 1) with the following numerical values: a2 = 1, a3 = 0.1, a4 = 1 and for the given location defined by the following operational coordinates: 07 = 7r. r7 = 0.75, a7 = arcsin 0.8, ar = 0.8, Oo = 0 (obligatory!), ro = 0.45, We have obtained the following sixteen real configurations (in radians)"
So,. 1I a b c d e f g h i j k / m n o p
qi
....
6'.0031 0.0235 0.3411 0.4605 1.4286 1.5433 2.0131 2.0553 -3.138~ .3.1181 .2.800~ .2.6811 .1.713( .1.5982 .1.1284 .1.0862
q3' [ q4 2.3749 -0.9159 2.4950 2.1294 2.2901 0.6670 3.0395-0.1247 2.1570 3.1387 3.1392 1.1040 0.3763 -2.8325 1.0695 0.2953 0.3597 2.1868 -1.0934 2.2177 0.4849 -0.8661 -1.0286 2.6988 0.7667 0.9159 -2.49513 1.0122 -2.2901 -0.667C 0.1021 0.1247 -2.15713 0.0029 -3.1392 .1.104G 2.7653 2.8325 -1.0695 2.8463 -0.3597 .2.1868 .2.0482 -2.2177 .0.4849 -2.2755 1.0286 .2.6988
qs' 1.4795 1.8177 2.3955 0.6458 3.0445 0.0202 -0.583~ -2.430c~ 1.6621 1.3239 0.7461 2.4957 0.0971 3.1214 -2.5581 -0.710(
qs~ -0.0648 2.9808 1.2532 -1.5786 -2.7523 0.5941 -0.7943 2.3144 3.0768 -0.1608 -1.8884 1.5630 0.3893 -2.5475 2.3473 -0.8272
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Current Advances in Mechanical Design and Production, MDP-7
7 CONCLUSION In this paper, we have presented a simplified inverse kinematic model calculation method for 6R type manipulators. Our method simplifies significantly the original one due to Lee and Liang, and contrary to this one, can be used, for all 6R type manipulator without any restriction as it does not involve any division. The inverse kinematic model calculation is brought back to the computation of the roots of a polynomial equation whose degree, equal to sixteen, is minimal and a non academic example shows that every sixteen roots must be real. Nevertheless we think that our method can still be simplified and that the previous sub system of six equations in four unknowns suffices to obtain also a sixteen degree polynomial equation. In spite of numerous attempts we have not succeeded in obtaining this minimal degree polynomial equation using this sub system. ACKNOWLEDGEMENTS We acknowledge V. Cadenat and J.-Y. Fourquet for their help concerning the preparation of this paper. REFERENCES 1. Pieper, D.L., "The Kinematics of Manipulator under Computer Control", PhD. Thesis. Stanford University. Stanford. California. USA. October (1968). 2. Roth, B., "Performance Evaluation of Manipulators from a Kinematic Viewpoint", Cours de Robotique, IRIA, Le Chesnay, France, August (1976). 3. Duffy, J. and Crane, C., "A Displacement Analysis of the General Spatial 7-Link, 7R Mechanism", Mechanism and Machine Theory, Vol. 15, pp. 153-169, (1980). 4. Cayley, A., "On the Theory of Elimination", Cambridge and Dublin Mathematical Journal, 3, pp. 116-120, (1848). 5. Salmon, G., "Lessons Introductory to the Modem Higher Algebra", Hodges and Foster, Dublin, Ireland, (1876). 6. Tsai, L.W. and Morgan, A.P., "Solving the Kinematics of the Most General Six and Five-Degree-of-Freedom Manipulators by Continuation Methods", Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 107, pp. 189-200, June (1985). 7. Lee, H.Y. and Liang, C.G., "A New Vector Theory for the Analysis of Spatial Mechanisms", Mechanism and Machine Theory, Vol. 23, No.3, pp. 209-217, (1988). 8. Lee, H.Y. and Liang, C.G., "Displacement Analysis of the General Spatial 7-Link 7R Mechanisms", Mechanism and Machine Theory, Vol. 23, No.3, pp. 219,226, (1988). 9. Raghavan, M. and Roth, B., "Kinematic Analysis of the 6R Manipulator of General Geometry", The fifht International Symposium of Robotics Research, Tokyo, Japan, (1989). 10. Manocha, D. and Canny, J.F., "Real Time Inverse Kinematics for the General 6R Manipulators", IEEE International Conference on Robotics and Automation, Nice, France, (1992). 11. Manocha, D. and Canny, J.F., "Efficient Inverse Kinematics for General 6R Manipulators", IEEE Transactions on Robotics and Automation, Vol. 10, No.5, pp. 648-657, October (1994).
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12. Kohli, D. and Osvatic M., "Inverse Kinematic of General 6R and 5R,P Serial Manipulators", Journal of Flexible Mechanisms, Dynamics and Analysis, DE-Vol. 47, pp. 619-627, (1992). 13. Mavroidis, C., Ouezdou, F. B. and Bidaud, P., "Inverse Kinematics of a Six-Degree-ofFreedom General and Special Manipulators Using Symbolic Computation", Robotica, Vol. 12, pp. 421-430, (1994). 14. Khalil, W. and Kleinfinger, J.F., "A New Geometric Notation for Open and Closed Loop Robots", IEEE International Conference on Robotics and Automation. San Francisco, California, USA, (1986). 15. Craig, J.J., "Introduction to Robotics. Mechanisms and Control", Addison Wesley, Reading, Massachussets, USA, (1986). 16. Smith, D.R., "Design of Solvable 6R Manipulators", PhD. Thesis, Georgia Institute of Technology, Georgia, USA, May (1990). 17. Paul, R.P.C., "Robot Manipulators. Mathematics, Programming and Control", MIT Press, Cambridge, USA, (1981 ). 18. Gorla, B. and Renaud, M., "Modeles des Robots Manipulateurs. Application a leur Commande", CEPADUES, Toulouse, France, (1984).
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Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
27
A SYSTEMATIC ALGORITHM FOR FLEXIBLE MANIPULATORS SIMULATION
Megahed S.M.'and Hamza K.T.*" * Professor, ** Graduate Student, Mechanical Design and Production Department Faculty of Engineering, Cairo University, Giza 12316- Egypt. E-mail
[email protected] and
[email protected]
Abstract This paper presents a systematic simulation algorithm for flexible manipulators with revolute joints. This algorithm takes into consideration the flexibility of links and joints, structural damping and joints' internal clearance. Flexible multi-body dynamics formulations are employed to obtain the equations of motion of such manipulators by considering their links as a lumped and as a consistent mass system. Simulation of one and two link planar manipulators is provided and comparison between the techniques of lumped mass and consistent mass is performed. In both cases, the system oscillates about the trend of rigid body motion, but better performance is achieved in the case of consistent mass. This algorithm can be readily adopted to simulate the performance of a control system of flexible manipulators.
Keywords Flexible Manipulators, Multi-body Dynamics, Finite Segment, Structural Damping.
1. Introduction Recently, there has been an increasing interest in studying flexible manipulators because of their advantage of high load to weight ratio. Since there is no such thing physically as a perfect joint or a rigid body, control schemes of manipulators that are based on rigid body mechanics often resulted in degraded control performance especially at high speeds. Higher speeds being a constant demand for higher production rates, extensive research work is directed towards developing good performance controllers for flexible manipulators [1-4]. Accordingly, much research work is also directed towards modeling and simulation of flexibility effects on the dynamic behavior of manipulators. The dynamic equations of motion of flexible manipulators are highly nonlinear and are often coupled. Therefore, the problem of simulating the dynamic behavior of a general flexible manipulator is difficult. Moreover, there is no exact analytical solution for such systems except for some limited special cases. Flexible manipulators, being a special class of flexible multi-body systems, have been treated as such [5]. However, there are many techniques and methods for studying flexible multibody systems, and there is a disagreement about accuracy and efficiency of such methods [6,7]. Most popular techniques are the Floating Frame of Reference [6-8], incremental Finite Element [6], Finite Segment [6,9] and Absolute Nodal Coordinate [6,7,10].
28
Current Advances in Mechanical Design and Production, MDP-7
In this paper, a variation of the Finite Segment approach is employed to formulate the dynamic equations. The global position and orientation angle of each node are used as generalized coordinates. The use of finite rotation as nodal coordinates, as demonstrated by Shabana [7] does not lead to an exact representation of flexible body inertia. Shabana proposed the use of slopes as nodal coordinates to define orientation in the Absolute Nodal Coordinate method [7,10]. The formulation utilized in this algorithm, however makes computing stiffness and damping forces easier while providing acceptable accuracy for cases of small deformation per element which is applicable in most manipulators. A complete formulation of the equations of motion of such manipulators is presented along with the algorithm for their implementation. One and two link manipulators are simulated to compare the lumped mass formulation and the consistent mass formulation and to demonstrate the use of the algorithm for simulating controller performance. Both techniques have been incorporated in the algorithm. 2. Equations of Motion
A variation of the Finite Segment approach is used to formulate the dynamic equations of motion. The Finite Segment approach assumes a flexible member to be composed of a number of discrete rigid bodies that are connected by springs. This makes the treatment of a flexible link similar to the treatment of a set of several rigid bodies. Each link in the manipulator is broken into several elements connected at nodes (Fig. 1). Equivalent masses are computed at these nodes, then by applying the principles of rigid body dynamics on those masses, the manipulator equations of motion are: Mi=F
(I)
where M is the manipulator mass matrix, i is the nodal acceleration vector referenced to the global axes and F is the total force vector representing the effect of stiffness, damping and actuator forces, referenced to global axes. The generalized coordinates (Fig. 2) are the global position (xi, Yi) and the angular orientation 0~ of the i~ node in the system. Since the manipulator is a single open chain, it is convenient to assemble the stiffness and damping forces directly rather than first assembling the stiffness and damping matrices and then multiply them by the deflections and velocities respectively. Assembling the stiffness and damping forces directly also facilitates the use of nonlinear expressions for computing the forces. 2.1 Element Stiffness Forces
An element in a general configuration is shown in Fig. 2. The stiffness forces due to this element at nodes i and j are first computed in the direction of element axis ui and vi then transformed to the direction of the global axes. These forces are given by: Fu i
--
Fvi
=
Mzi
-Fuj -F~j Mzj
= = = =
Ka du KbjdV - Kb2d0 -Kb2dv + Kb3 dO -Mzi - FviLo
(2)
Current Advances in Mechanical Design and Production, MDP-7
29
with: EA
Ka =
'
6EI K b 2 = - -L2o
12EI Kbl = ' - L To '
du : c , ( x , - x,)+ s , ( y j - y , ) - L o
&
,
4EI Kb3 = - - ~ ,
dO = Oj -0 i
-x,)+ c,(yj-y,) - - o
d v : -si(x i
where Ka is the axial stiffness constant, Kbl, Kb2, Kb3 are bending stiffness constants, du, dv, dO are the axial, transverse and angular deflection of the element respectively, E is the modulus of elasticity, A is the link cross sectional area and I is its second moment of area. Transformation to the direction of the global axes gives:
LFy,
]I
I
s,
c, JLF,,
_
[ c'
LF,j
s,
c,
jLF~j]
ci=cos(Oi) & s i=sin(oi) 2.2 Element Damping Forces
Several methods are customarily used to express the damping effect of structural members [11,12]. However, in most cases, the exact damping properties of a structural member are uncertain and have to be determined experimentally [ 11 ]. In many cases, it is acceptable to use an equivalent viscous damping, on the other hand, solid (or structural) damping is mainly believed to be proportional to the stress (or deformation), but in phase with the velocity [ 11 ]. An advantage to assembling the damping force directly instead of the damping matrix first is the ease of using multiple formulae that are either linear or nonlinear to express the damping forces. Damping forces of an element are first computed in the direction of the element local axes then transformed to the direction of the global axes. The computation of damping forces is very similar to that of the stiffness forces. Equivalent Viscous Damping
Fui Fvi Mzi
=
=
-Fuj -Fvj Mzj
=
Ca dO
= = =
Cbl d';' - Cb2d0 -Cb2 d~' + Cb3 dO -Mzi - FvjLo
(3.a)
Solid Damping
Fui Fvi
= =
Mzi
=
-Fuj -Fvj Mzj
--- Ca du sign(dti ) = Cbidvsign(d~') - Cb2 dO sign(dO ) = -Cb2dv sign(de,) + Cb3 dO sign(dO ) -Mzi - Fvi Lo
(3.b)
where Ca, Cbl, Cb2, Cb3 are the damping constants. These damping constants are best identified experimentally for the members to be simulated. However, it is sometimes acceptable to estimate approximate values for them such that the simulated response is closely matching similar systems. The time derivatives of the element deflections are given by:
30
Current Advances in Mechanical Design and Production, MDP-7
dE)-'Oj-E)i, du=ci(:~j-:~i)+si(~'j-~'i)
&
d~,=-si(:~j-:~i)+ci(~'j-~'i)
2.3 Joint and Actuator Forces Revolute joints can be modeled as perfect joints (Fig. 3) or elastic joints (Fig. 4). The actuators driving the joint can also be perfect (Figs. 3,4) or elastic (Fig. 5). Perfect actuator joints join the last node of a previous link (Node 1) with the first node of the next link (Node 2). In the case of joints with elastic actuators, an extra node is added in between so as to simulate the internal dynamics of the actuator. Generally, all values of bearing stiffness, damping coefficients, internal lumped mass and bearing clearance should be determined separately for each joint prior to simulation, either experimentally or based on an acceptable model. The following cases are studied: Perfect Joint: The effect of adding a perfect joint is incorporated in the dynamic equations of the system as algebric constraint equations: xt = x2
&
yt = y2
(4)
Elastic Joint: The radial deflection between Nodes 1 and 2 is permitted. Joint stiffness and damping forces are computed as function of this deflection, then transformed to the direction of the global axis and added to the global force vector. A possible formula for computing radial forces is:
Frt = " F,z = {
KsoftSr + CrS, Ksoa 8c + Kstiff (Sr " 8c ) + Cr~r
For 5c-> 8r For 5c < 8r
(5)
where 5c is the joint radial clearance, 5r is the radial deflection between nodes 1 and 2, ~ir is the radial relative velocity, Kson and Kstiffare respectively a small and a large stiffness value used to simulate the effect of internal beating clearance. Perfect Actuator: Actuator torque Ta is added to the global force vector at Node 2 and with the same magnitude but in opposite direction at Node 1 (Figs. 3,4).
Elastic Actuator: To simulate the internal actuator inertia, an extra node with a lumped mass is added between the last node of the previous link and the first node of the next link. The actuator torque is added at nodes 1 and 2 (Fig. 5) in equal and opposite directions. However, the torque is transmitted from Node 2 to Node 3 through a torsional spring and damper whose characteristics can be set to simulate the effect of the actuator transmission system. 2.4 Mass Matrix The system mass matrix is obtained by assembling the element mass matrices of all the elements in the system. Optionally, additional lumped masses can be added at specific nodes to simulate special effects like the presence of a payload, or the mass of an actuator. Equivalent Lumped Masses can be computed at nodes i and j (Fig. 2 ) o f an element by dividing the total element mass (m) and inertia by two (since the element has a uniform cross section). Lumping results in a diagonal element mass matrix given by:
Current Advances in Mechanical Design and Production, MDP-7
I
0
0
0
0
0
0
l_ 2
0
0
0
0
0
0
L', 24
0
0
0
0
0
0
~ 2
0
0
0
0
0
2
M
=m
0
o
0
0
~ 2
0
o
31
0
L',
(6.a)
Lumped mass formulation leads to a fully diagonal system mass matrix so the dynamic equations are uncoupled. Efficient computation of global nodal acceleration vector can be performed. Integration of the acceleration vector gives the system state at the next time step. Lack of dynamic coupling in lumped mass formulation generally results in bad accuracy of simulation results and requires the use of a large number of elements per member in order to achieve acceptable accuracy. Alternatively, Consistent Mass formulation takes into account the dynamic coupling between the nodes of the element. Ignoring the coriollis, centripetal and tangential acceleration terms resulting from element deformation [ 11 ] (usually valid for small deformation per element) can simplify an expression for element consistent mass matrix. The element mass matrix will be [11 ]: m
1
0
0
3
0 o
M=mR T 1
6 0
m
I
6
--"--'
!!L,
35
210
~!L.
e.
210
105
0
0
"~ 70 -13L. 420
13Lo 420 - L~ 140
o o I
3 0 0
0
0
9_~ 70 13L. 420
-13L. 420 - L~ 140
0
0
13 35 -ILL. 210
-ILL. 210 L20 105
R,
Where" R =
c,
s,
0
0
0
0
- s,
c,
0
0
0
0
0
0
l
0
0
0
0
0
0
c,
s,
0
0
0
0
- s~
c~
0
0
0
0
0
0
I
(6.b)
Since the dynamic equations of motion become coupled when the consistent mass formulation is used, a set of linear algebric equations will have to be solved at every time step in order to determine the global nodal acceleration vector. This is however a trade-off of computational speed for better accuracy of solution. 3. Computation algorithm
.
A computation algorithm is developed for simulation of flexible manipulators either as a lumped mass system or a consistent mass system. The flow chart of the computational algorithm is shown in Fig. 6. The required pre-processing includes: input of the problem data; namely the number of links and joints, links' geometry and material properties, joints' stiffness and damping properties, number of elements per link and parameters for the numerical time step integration technique. The next step is to compute the mass, stiffness and damping constants for each element (Sec. 2) and establish element-node connectivity. Next, the initial state (Positions & Velocities) of the system has to be determined. Then, the simulation procedure starts, the current system state is used to compute stiffness and damping forces due to elements and joints and are added to form a global force vector. Then, the actuator forces (torques)are computed according to the control law whose performance is
32
Current Advances in Mechanical Design and Production, MDP-7
Elements
Sta.
Nodes ~
|
yJ/
~
/
Input No. of links, joints, / their properties meshing and time step parameters.
-., '
~.~"
..f-'e,f~a'/ :l ~y". y z/,,'~//,'//,'~oun d
]
I
~._Revolute
Joint
-
I
' Fie. 1. A Flexible Manipulator
. . . .
Compute Mass, Stiffness & Damping Constants of Elements. 'I:
Establish Element Connectivity
vj V
Compute Lumped
du
[
Y~
,
FiR. 2. An Element in a General ConfiRuration
,.
F TM
.........
Compute Initial State Vector. ,,L! v1
I-m~ ~ {
Next L i n k ~
,,
,!
.
Use current state to compute stiffness, damping and actuating forces; Assemble global force vector.
,,_?,0
Previous Link
Fig. 3. Perfect Joint with Perfect Actuator
Next Link---~
T.
N.._ PreviousLink Fig. 4. Elastic Joint with Perfect Actuator
NextLink--~
/.
"rI - ' ' ' Divide - ' ' ' ' ~ - by Compute element [ nodal masses mass matrices, [ to obtain assemble global mass ] global matrix and multiply I accelerations force vector by its [ inverse to obtain global accelerations "l~ ["q .
!
,,
Integrate numerically to get the System State at next time step; Record new State. Time = Time + Step Ye
_ ( E n d ) Fig. 5. Elastic Joint with Elastic Actuator
Fig. 6. Computation Algorithm Flow Chart
Current Advances in Mechanical Design and Production, MDP-7
33
being simulated and added to the global force vector. The global force vector is multiplied by the inverse of the system mass matrix to obtain the global accelerations, which are integrated numerically to obtain the system state at the next time step. The new system state is recorded and the process is repeated with the new state being the new current state until the desired simulation time is reached. It should be noted that in lumped mass formulation, the system mass matrix is constant and diagonal so it is evaluated only once during pre-processing, while in case of consistent mass formulation, it has to be re-evaluated at every time step. A tailored computer program is developed using MATLAB. Fourth order Runge-Kutta [13] is used for time step integration. 4. Example 1: One-Link Manipulator A single link (Fig. 7) of length L = 1.0 m, Mass = 1.0 kg, modulus of elasticity multiplied by second moment of area EI = 100 N.m 2, modulus of elasticity multiplied by cross sectional area EA = 1000 N, is subjected to a constant torque T = 1.0 N.m, (Fig. 8), no damping is considered and the revolute joint is assumed to be perfect. Initial state is 0(0) - 0.0 rad and 6(0) - 0.0 rad/s with no initial strain. Simulation is performed for 4 cases: a rigid body system, 4 and 8 element per link lumped mass, and 4-element per link consistent mass. The expected motion of the link is to be vibrating but its gross motion should be following the motion of the rigid body system. Figs. ( 9 . a - d ) show the recorded system state at various locations of the link for the first 0.5 (s) of motion. It can be seen that the consistent mass formulation is rippling very closely about the rigid body motion, while the simulation using lumped mass formulation has the same trend, but its values tends to diverge. This is further seen in Fig. l0 where the difference between angular orientation of the link tip of the flexible system and the rigid one is plotted against time. This divergence can be understood since lumping is an approximation of mass distribution in the system, it results in a slightly different response, while consistent mass formulation only ignores the inertia change due to deformation, and thus will simulate the gross motion (rigid body mode) very accurately.
5. Example 2: Two-Link Manipulator A two-link manipulator (Fig. 11) is employed to evaluate the performance of a controller based on rigid body dynamics when used with a flexible system. Both the direct and inverse dynamic model of the problem are provided in [14]. Initial state is at rest, with both links aligned on the global X-axis. The first link is to move with constant angular acceleration for a duration of 1.0 (s), constant angular velocity for 1.0 (s), then constant angular deceleration for 1.0 (s), during which, the link will have completed a complete revolution, all the while, orientation of the second link is maintained parallel to the X-axis. Actuator torques for such maneuver of a rigid system are given in Figs. (12.a,b). The actuating torques are employed for flexible systems and their response is simulated. Flexible system data is: lengths Ll - L2 = 1.0 m, Masses MI - M2 - 1.0 kg, modulus of elasticity multiplied by second moment of area EI = 40,100,1000 N.m 2, modulus of elasticity multiplied by cross sectional area EA -- 1000 N, no damping is considered, and the revolute joints are assumed perfect. Simulation using consistent mass formulation is performed with 4-elements per link. Figs. (13.a-d)showthe system state at different locations while Figs. (14.a,b) show the path of the end point of each link. It can be seen that lower link stiffness results in degraded performance of a controller based on rigid body dynamics, which can still give acceptable performance, if the links are stiff enough.
34
Current Advances in Mechanical Design and Production, MDP-7
jo~
@
o.
-
Fig. 7. E x a m p l e 1: O n e - L i n k M a n i p u l a t o r
o.~s o~
o'.= o~+ oi,
o.~
Tim,+ ll )
o s
o.ii
Fig.8. A c t u a t o r T o r q u e
0.4
.....
0.35
,
..
,
.,,
~,
0.3S
,
,
9~
...
.
.
.
,
.
;
:
~ o,
~o. ~
i
o.~1
~
o.1!
0.2
~
o.1
I
ILilid
0.1S
-41-- 4-IEI COlll~lem M u s I
"
"
olol !
0.1
r m o (el
~
- - , -
~._--:
.+ .
.
.
.
.
,
o.s
Time 1-)
s
(b) Link Orientation at Point A
.
0.38
Oils
9
v
.
..
:
.
. .
.
. .
.
v .
.
.
.
.
.
.
+ .
.
03
~oolm
j
"
.+,].,,r//t.
o
, ~.:..
.
o~
(a) Link Orientation at Point G
.~._--_
~
4 - -+ t,,.+ ~ .
o.I
I+o1
1
"-
4.El C m ~ l ~ l
~o~
097
i
0.2
..L__ R ~
:
s o.Is Rillid
--
O.95
.-.-. . .,,..t, . .t.,,. . . .., II
O.94 0.~
0.05
0.1
O. 1S
0.2
\ \ II 0.2S 0.3 T i m * (e)
0 36
0.4
0 45
0 S
(c) X - Position of Point A
-0
O+OS
0.1
0.1S
0.2
0.2S
0.3
0.3S
0.4
0.4S
O.S
(d) Y - Position of Point A
Fig. 9. E x a m p l e I Results [ El--100 N.m 2, E A - 1 0 0 0 N, L = I . 0 m ] O02--
I
.
,
,
,
,
,
,
,
--r--- --
B
0
~ -0.02 11
e
0
0~
o, i
o. t5
o.2
0.2s
Time ( m c )
03
0 3s
o.4
0.4s
0.5
Fig. 10. E x a m p l e 1 Difference in Orientation from Rigid System at point A
Fig. 11. E x a m p l e 2: T w o - L i n k M a n i p u l a t o r
Current Advances in Mechanical Design and Production, MDP-7
i!!.
I-T~
35
J -3
(Sec)
T'~ne (See)
(a) Actuator 1 Torque
(b) Actuator 2 Torque
Fig. 12. Example 2 Actuator Torques
'Fi ,i
///
:L
,
|~
~,~
t~,-,,: Ii l++',',=_.. ! l
/
I,.., 1/ o
[
"~
os
I
i s l~,ne (-)
2
2'S
'
I
.
i
3
3s
(a) Orientation of Linkl at Point A
o
os
t
t
l ' * . e (l)
2
2S
3
3S
(b) Orientation of Link2 at Point B .-!
/y "~
o's
I
is 1 " ~ (i)
~
! 2s
3
o$
0
3s
(c) X-Position of Point B
os
Is
-
-
2
=s
(d) Y-Position of Point B Fig. 13. Example 2 Results
[
4
os oe t.
'
,~ go
os,
i
j ~ .o2i I I
"1t
-oeI
! .1iS-,,
'
.o's
,;
x - POl,llUm - "A" ira)
o's -~T-----~s
(a) Path of Point A
"I .,
o
o$ ! x - PosJoon. " i t (m)
is
(b) Path of Point B
Fig. 14 Example 2: Path of Links' End-Points
3
3s
36
Current Advances in Mechanical Design and Production, MDP-7
6. Conclusion
Simulation of flexible manipulator dynamics is an essential tool in developing and assessing their control in a more accurate manner. This paper presents a systematic algorithm for such a purpose based on the mathematical modeling of flexible manipulators as flexible multi-body systems. The equations of motion of planar single open chain manipulators, with only revolute joints are provided. Minor modifications are required to expand these equations to spatial manipulators. A comparison between lumped mass and consistent mass formulation is performed. Simulation results showed that the lumped mass formulation is only suitable for determining the trend of motion, or qualitative analysis, while the consistent mass formulation provides better accuracy which is more suitable for quantitative analysis. In the second example, consistent mass formulation was employed to simulate the performance of a controller as a demonstration of the basic motive of developing this algorithm. References
1. Hastings, G.G. and Book, W.J., "Experiments in Control of a Flexible Robot Arm", Robots 9, Conf. Proceedings, Vol. 2, Detroit, Michigan, USA, pp. 20.45 - 20.57, 1985. 2. Schmitc, E., "Dynamics and Control of a Planar Manipulator with Elastic Links", Proceedings of 25th conf. on Decision and Control, Athens, Greece, pp. 1135-1139, 1986. 3. Yim, W., Zuang, J. and Singh, S.N., "Experimental Two Axis Vibration Suppression and Control of a Flexible Robot Arm", J. of Robotic Systems, 10 (3), pp. 321-343, 1993. 4. Eisler, G.R., Robinett, R.P., Segalman, D.J. and Feddema, J.D., "Approximate Optimal Trajectories for Flexible-Link Manipulator Slewing using Recursive Quadratic Programming", J. of Dynamic Systems Measurement and Control, Vol. 115, pp. 405-410, 1993. 5. Idler, S.K., "Open Loop Flexiblity Control in Multi-body System Dynamics", Mechanism and Machine Theory, Volume 30, No. 6, pp. 861-869, August 1995. 6. Shabana, A.A., "Flexible Multi-body Dynamics: Review of Past and Recent Developments", Multi-body System Dynamics, Volume 1, pp. 189-222, March 1997. 7. Shabana, A.A., "Dynamics of Multi-body Systems", Cabridge University Press, 1998. 8. Shabana, A.A. and Schwertassek, R., "Equivalence of the Floating Frame of Reference Approach and Finite Element Formulations", Int. J. of Nonlinear Mechanics, Vol. 33, No. 3, pp. 417-432, 1998. 9. Connelly, J.D. and Huston, R.L., "The Dynamics of Flexible Multi-Body Systems: A Finite Segment Approach - I Theoretical Aspects & II Example Problems", J. of Computers and Structures, Vol. 50, No. 2, pp. 255-261, 1994. 10.Shabana, A.A., "An Absolute Nodal Coordinate Formulation for the Large Rotation and Deformation Analysis of Flexible Bodies", Technical Report No. MB96-1-UIC, Department of Mechanical Engineering, University of Illinois at Chicago, March 1996. l l.Sandor, G.N. and Erdman, A.G., "Advanced Mechanism Design: Analysis and Synthesis", Prentice-Hall Inc., New Jersey, USA, 1984. 12.Sun, C.T. and Lu, Y.P., "Vibration Damping of Structural Elements", Prentice Hall PTR, USA, 1995. 13.Woods, R.L. and Lawrence, K.L., "Modeling and Simulation of Dynamic Systems", Prentice Hall Inc., New York, USA, 1997. 14.Megahed, S.M., "Principles of Robot Modeling and Simulation", John Wiley & Sons Ltd., London, UK, 1993.
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
37
NON-LINEAR TRAJECTORY CONTROL OF FLEXIBLE JOINT MANIPULATORS
Bravo, R.R.* and Dokainish, M.A.** *PhD Candidate, ** Professor, Department of Mechanical Engineering McMaster University, Hamilton, Ontario, Canada, LSS-4L7 ABSTRACT It has been known for years that the presence of flexibility in the joints can severely limit the performance of industrial manipulators when performing trajectory tracking tasks. This paper addresses the problem of control of manipulators, taking into account the existence of flexibility in the joints. Dynamic models are developed for the general n-link manipulator and for the particular case of a two-link articulated flexible joint manipulator. The model is used for the simulation and synthesis of the control strategies: Feedback Linearization and Variable Structure (Sliding Mode) control. These techniques are numerically simulated for the tracking of a prescribed trajectory in the wrist space, and the results compared to the case of a manipulator controlled by a PD plus Gravity controller, which does not account explicitly for the presence of flexibility in the joints. The results of the simulations show that the PD plus Gravity controller is not capable of achieving satisfactory tracking, while the Feedback Linearization Controller shows excellent performance for the case of no uncertainties. The Sliding Mode Controller results in excellent performance, and exhibits good robustness in the presence of parametric or modelling uncertainties, showing the advantages of the proposed approach. KEYWORDS Flexible Manipulators, Non-Linear Control, Feedback Linearization, Variable Structure Control. 1. INTRODUCTION Industrial manipulators have been part of our lives for the last 30 years. They are taking charge of activities that are tedious, repetitive or dangerous for human operators. These activities include traditional applications such as material handling, spot welding, etc., which involve only point-topoint control, and more complex tasks such as arc welding, spray painting, mechanical and electronic assembly, etc. which often involve trajectory tracking control and force control. Applications have become more demanding and the manipulators are required to be smaller, lighter, faster and more precise. This evolution results in increased flexibility, in both the structure and the joints of the new generations of manipulators. The induced compliance will degrade the tracking performance of the robots [1 ], since it can cause lightly damped oscillations whose resonant frequencies are low enough to fall inside the bandwidth of the controller. Research has revealed that for most manipulators, the principal source of flexibility resides in the drive system rather than in the links. For this reason, considerable attention is given by the researchers to the problem ofjoint flexibility in manipulators. The appearance, in recent years, of powerful low-cost microprocessors has spurred great advances in the theory and application of nonlinear control.
38
Current Advances In Mechanical Design and Production, MDP-7
This has resulted in rapid development in the areas of feedback linearization, sliding control and adaptive control techniques. Several approaches have been proposed to control flexible joint manipulators: Spong [7] employed a static feedback linearization approach, which was coupled to a variable structure scheme by Sira-Ramirez and Spong [5]. Sira-Ramirez et al [4] used dynamic feedback linearization, while Khorasani [2] and Slotine and Hong [6] made use of a singular perturbation approach that divides the dynamic model into slow and fast subsystems that are controlled separately. 2. MODELLING Figure I shows a schematic diagram of a two link articulated manipulator, as well as the relevant parameters and variables, consistent with the Denavit-Hartenberg conventions. Figure 2 depicts a flexible joint, where the joint compliance is modelled by a linear torsional spring. I Direction of the gravity Vector g ,
~"-----~--~ la, nil i
~..-~~ ! ....... ~.. __~I
-
'P
l-t, m t-:"~ "
Link i Transmission
at
-A, i
) L,~
i
-..~..__ J~__~
Actuator i r,
Fig. 1. Coordinate systems and parameters
Fig. 2. A flexible joint
The symbols shown in Figs. 1 and 2 are defined as follows: qh: qm,:
r,: m~"
I:,: J,,.:
K,: Lc,: ~7,:
angular displacement of gh link (rad) with respect to the x,.! axis. angular displacement of i'h actuator (rad) with respect to the x,.! axis. torque applied by i'h actuator. mass of i'h link moment of inertia of i'h link with respect to its centre of mass moment of inertia of i 'h actuator with respect to its axis of rotation stiffness of i'h joint length of gh link distance from i'h joint to the centre of mass of i'h link Transmission ratio between the i'h link and actuator
The presence of the joint compliance introduces dynamics associated with the link position and the actuator position, thus causing the vector of generalized coordinates of the manipulator to contain both link and actuator positions: q = [qt r, qmr]rwhere qt is the vector representing the angular displacements of the n links and qm is the vector of angular displacement of the n actuators. Defining Tand Uas the kinetic and potential energies of the arm, respectively, then the Lagrangian function of the arm can be defined as"
Current Advances in Mechanical Design and Production, MDP-7
39
L (q,q) = r (q,q) - U (q)
(1)
where q = dq/dtis the vector of link and actuator velocities. The general equations of motion of a robotic arm can be formulated in terms of the Lagrangian function as:
d 0 L(q,q) - 0____ L(q,dl) = F, dt adl e aq ,
i = 1, 2, ..n
(2)
where n is the number of links and/7, is the resultant torque applied to the i 'n joint. The kinetic energy can be expressed as follows
1
r(q, dl) = -~ dl
r
[D(q)] 4
(3)
where the inertia matrix [D(q)] and the coordinates q for links and actuators are given respectively for a two link manipulator by" [D(q~)] [D(q)] =
I._i+ [O(q)]
[0]
2m2atLc2c~
[J.]]
2
2
mlL:!+ /=2+ m2Lc2+ m2a I +
:
to]]
t2)
I:2+ m2L~2§ m2atLc2COs(qt2) '
2
/.2 + m2L~2+ m2aiLceCOS(qt2)
o
1
q={ qt, qt2 qm, qm2 }r
d,,,2
(4)
The potential energy of a flexible joint manipulator contains a gravitational component, as well as a contribution caused by the elasticity of the joints. It is given by"
1
U (q) = -~( qt- [111-Iqm
)T
-!
[g] ( ql- [111 qm ) - g
r ~ mk ck(qt)
(5)
k=l
where [rl] is a n x n diagonal matrix of containing the transmission ratios r b and [K] is the n x n diagonal matrix ofjoint stiffness, c*(q) are the vectors of the positions of the centres of gravity of each link k. For a two link manipulator, they are given by:
cl(q) - { Lcl cos(qi) /
c2(q) = / a ! cos(ql) + L c2 cos(ql + q2)1
[
Lct sin(q!) J
(6)
al sin(ql) + L c2 sin(ql + q2)J
The generalized forces acting on the joints are produced by electric, pneumatic or hydraulic actuators and by friction. Friction is a complex, nonlinear effect that is hard to model accurately. The following model is adopted for the links and for the actuators" v
c
v
c
bt,(dl) = Itt, Clt, + Itt, sgn(dtt) , b,,,,(dl) = It,, dim, + It,,,, sgn(dl,,) i = 1, 2, ..n (7) where Itv and ~rr are the viscous and coulomb friction coefficients. The vector of generalized forces is F = [- b t , [, - bruit] r, or, for the two joint manipulator: F-
{-btl
-bt2
,l-brat
~.2_bm2 }T
(8)
Substituting equations (3) and (5) in (I), and equations (I) and (8) in (2), the following matrix equations are obtained, which represent the dynamic model of the flexible joint manipulator:
40
Current Advances in Mechanical Design and Production, MDP-7
[D(qt)]#~+ c(q~, 4,)+ b~(4)+ h(q)+ [K](qt- [vl]-' qm) = 0 [J,,,]q., + b,.(q.,) - [rl]-' [K](q,- [111-' q,n)
(9)
= 1;
where, for a two link manipulator, the velocity coupling, gravity and input torque vectors are given respectively by the following expressions: 2m2 al Lc2 Clll 012 sin(q/2) - m2 al L c2 ql2 sin(qt2
c(qt,4, ) = m2 al Lr q~ sin(q12) miLe / sin(qtt)+ m2a I sin(qu)+ m2Lc2 sin(qt/+qt2) ! h(q~) = -go'
I
m2 L,:2 sin(q/t +qt2)
0o)
It is important to point out that this model is based on the following two assumptions [7]: the kinetic energy of each rotor is only due to its own rotation, and each actuator is symmetric with respect to its associated axis of rotation. The first assumption removes the dynamic coupling of the actuator and link variables in the inertia matrix while it allows for the modelling of the elasticity in the joints; the second assumption is very easy to justify and it makes the potential energy of the manipulator and the velocity of the rotor centre of mass independent from the actuator position. Defining the state space vector by: 9
T
T
x=tx, ,x, ,x,
T
"
,x{]' = [q,~,4,~,[n]
'q.r,[nl-'#.q~
then the dynamic model of the flexible joint robot in state space form is given by: "fl = X2
"~Z = - [D%)]-I{C(Xl.Xz)+
b,(xz)+ h(Xl)+ [Kl(x t - x3) }
(ll)
.i' 3 = x 4
-~4 = - [3.1-~[rll-~{b..(x4)-
[rll-~[Kl(x,
-x3)-
~}
3. FEEDBACK LINEARIZATION CONTROL It has been demonstrated that the dynamic model of a rigid joint manipulator is linearizable using static state feedback allowing for the familiar computed torque control method. However, for the flexible joint case, this is not straightforward, since the presence of the compliance makes the relation between the actuator input and the link position weak. In order to extend the feedback linearization approach to the flexible joint manipulator, it is necessary to introduce a change in the state space variables of the dynamic model, transforming them from actuator and link states to a link-only state. The state space transformation that accomplishes that is given next: y,-r,(x)=x I yz-T2(x)=~, = x,
Y3=T3(x)=-~z= -[D(x, )]-' {c(x, ,xz) +h (x~)+b,(xz) +[K](x, -%)} y 4 = T4(x)= T3(x)= -
(12)
d([D(x,)]- ' {c(x,,xz) +h (x,)+b ,(xz) +[Kl(x,-x3)})
This mapping from the old to the new state space is globally invertible. The new state space representation of the system can now be written in the feedback linearizable transformed form: .p = [A] y + [B] [M(y0]-I[1: -flY)]
(13)
Current Advances in Mechanical Design and Production, MDP-7
41
with [o1 [fl [o] [o1 [A]-
[0] [0] [/] [0] [o1 [o] [o] [rl
[B]-
[o] [o]
(14)
[o]
[o1 [o1 [o] [o] [fl where [/] and [0] are n x n identity and zero matrices, respectively, and [mo, l)l=[M(Tl(x))l=[N(xt)l:[D(xt) ] [K]-I[rl][Jm] J(Y):J(T(x)):[N(xl)]
~[D(xl)]I{c(xt,x~)+h(xt)+bt(xz)+[K](x I -xs)} + dt z
[-~t2{c(xl,x2)+h(Xl)+bt(x2)}
+[K].~ 2 - [ N ( x , ) ] - '
[rl]-'
[K] (x, - x3)
The control law that cancels the nonlinear terms present in the transformed model is given by: 1: = f(y) + [M(yt) ] v
(16)
Applying the control law to the feedback linearizable model (13), assuming no uncertainties, yields the next system of linear, first order, controllable differential equations: .P : [A] y + [B] v
(17)
The vector v is chosen to stabilize the transformed linear system tracking the desired trajectory: I,' = J ) i d - [ K i ] { Y 4 -
Y4d}-[K2]{Y3- Y3d}-[K3]{Y2- Y2d}-[K4]{Yl- Yld}
(18)
with [K,], [K_~],[K3] and [K4] being gain matrices which are selected in such a way as to place the poles of the closed loop system in the left hand side of the complex plane, andy,a being the desired trajectory of the i'h state variable. Selecting the gain matrices as follows: [KI]
= 4~.[/] ,
[K2] = 6~2[/] ,
[K3] = 4~3[/] ,
[K4] = ~4[/]
(19)
where ~. is a positive number, results in the next uncoupled, exponentially stable error dynamics: e (4) + 4)re (s) + 612e ~2) + 413e (t) + ~,4e = 0
(20)
with e = Yl -Yld, etl)= de/dt, e ~2)= d2e/dt :, etc. 4. SLIDING MODE CONTROL The Feedback Linearization control strategy requires exact knowledge of the kinematic and dynamic parameters of the manipulator to achieve best results. However, most of the times, such information is available only in approximate values, and some parameters may change due to normal operation conditions. For this reason, it is desirable to seek a control system which is able to tolerate some degree of parameter uncertainty and still maintain a good dynamic performance. In a Variable Structure Control system, a discontinuous control law is employed to guide the system to a switching surface. Once the system is on the switching surface, the control law guarantees that the dynamics are insensitive to parameter variations. The feedback linearization control law is employed with a discontinuous outer loop, to construct the new "sliding mode" control law. Starting with the transformed feedback linearizable model (13), one can apply a control law which is evaluated using approximate values for the manipulator parameters:
Current Advances in Mechanical Design and Production, MDP-7
42
:]Cv)
[Mfy,)]
+
v
(21)
where ^ means that the term is calculated with approximate parameters, instead of the actual ones. Implementing the control law leads to the following nonlinear perturbed model" ~= [All, + [B][M(yl)]- I[~(y)_ f(y))+ h~/r0,l)v] = [A]y+ [B][~(y)+ ([Ill(y)] + [/])v]
(22)
where the nonlinear perturbations are defined as ~(V) = [M(Y|)]- | { ](.V) -f(Y) }
[ql(V)] = [M0'|)] -| [/I;/(,Vl)] - [ / ]
(23)
The problem at this point consist of selecting the outer loop control law v to stabilize the system in spite of the bounded modelling errors represented by ~(.V) and [r For this purpose, a Lyapunov based approach is used. A switching vector of the form S : [K,I{ y, - Yd, } + [K21{ Y2 - Ydz } +[/(31{ Y3 - YdJ } + { Y4 - Yd~ }
(24)
is selected. The scalar Lyapunov function is defined as: i =
v(s) : ~
/1
i=l
IS, I
(25)
where St are the scalar components of the switching vector S. The function V(S) >0 for S , 0. The control is chosen such that I;" ~ -p < 0 for a given p > 0 whenever S , 0. This assures stability, according to Lyapunov's second method, and it leads to the following conditions, which are known as the sliding conditions:
S t . S~< 0
i : 1, 2, ..n
(26)
which imply that Si > 0
if
Si < 0
or
Si < 0
if
S i> 0
(27)
To satisfy the condition for the existence of the sliding regime (26), the derivative of S is found:
"~ : [KI]0~2 -Yd2)
+
[K2]0~3 -Yd3)
+
[KI](Y4 -fld4)
(28)
+ (f~4 -Yd4)
= [g,]{v= -Yd~) + [r2]~v~ -YdJ) + [/3]0'4 -Yd,) + (0{v) + ([*(Y)] + [q)v - ~ d ,
The outer loop control v is chosen as v - ~
- [r,]
ly2 - ya2}
- [to21 Lv3 - y ~ }
- [r31 {y4 - yo~} - [ r , = ]
sgn{s}
(29)
where sgn {S} is the vector formed by applying the scalar sign function to the switching vector and [K.,m] is specified below. Substituting (29) in (28) results in the following expression:
S=r
-Yd2} - [KI]{Y3-YaJ}- [KI]{Y4-Yd4}}-{II/{Y} +[l]}[Ksm]$gn(s) (30)
Assuming some bounds in the uncertainty, the matrix of switching gains [Ksm] can be specified: II~(.v)*
[r
[gt]{Y2-Yd2}-
[K2]{Y3-Yd3}-[K3]{Y4-Yd4}}ll<
II[qJ(y)]ll -< ~ < 1
6(y)
(31)
where 5(y) is a scalar function with finite value and ~ is a scalar. From (31) one can obtain: [K,.I
> (1 - ~)-160,) [/]
(32)
If the switching surface gain matrices are designed as follows [Kl] = 3X [/]
[K2] = 3~.2 [ / ] ,
[K3] = Z3 [/]
(33)
where ~ is a positive real number, then when the system is in the sliding regime, i.e. S=0, the dynamics of the system are given by the following reduced order, stable differential equation:
43
Current Advances in Mechanical Design and Production, MDP-7
(34)
e (3) + 3~. e (2) + 33.2 e (t) + ~3 e = 0
which is independent from the manipulator parameters. In practical applications, the presence of the discontinuous switching may result in control chattering. To avoid this the boundary layer concept is used, where instead of trying to keep the system dynamics on S = 0, one tries to keep S close to S = 0, by employing a continuous approximation of the sgn function, the sat function:
: I sgn(S/f3), IS,/13,1 > ] [ s/f~, Is/f~, l <-i
sat(S/~,)
(35)
where 13,is the boundary layer thickness. Using sat instead ofsgn implies that a trade-offis made between the theoretical control law precision and its smoothness, because it leads to tracking within a guaranteed precision rather than "perfect" tracking. 5. SIMULATION AND RESULTS A computer program was written to simulate the closed loop dynamic behaviour of a two-link flexible joint manipulator, according to the state space model (11 ). The reference trajectory to be tracked by the manipulator is a straight line using an eight-order interpolating polynomial in the time domain., with maximum velocity and acceleration of 2 m/s and 2 m/s 2respectively. The wrist point is required to move from (-0.6,0) to (0,0.4). The kinematic and dynamic parameters used in the simulation, given in table 1, correspond to those of the experimental arm FLEXROD (figure 3), built in the Mechanical Engineering Department of McMaster University.
i
Fig. 3. FLEXROD manipulator Table 1. Manipulator parameters. i
i
Parameter
ii
Jlllll
I.,
m, (kg)
(kg-m2) 9
i
i
i
iii
Link 1 i
ii
4.8
Link 2 i
i
4.8 i
ii
illl
i
i
i
iii i
iiiiii
a, ii
i
(m) iiiii ii ii
i
ill ii
ii
0.305
0.025
0.585 i
i
i
i
9.53x10"5
0.22
9.53x10"5 ,
iii
iiii
n, ii
0.145 i
ii
J,, (kg-m2)
(m)
i
0.025
i
ii
i
L,.,
ii iii
-
g, (N/rad) . . . . . .
lO0
lO0
100
100
in
The values of the joint stiffness are much lower than what is commonly found in practice, and they are chosen to exaggerate the effect of the flexibility. Figure 4 shows the results of the simulations of the flexible joint manipulator controlled by a PD plus Gravity controller [8], the feedback linearization controller, and the sliding mode controller.
44
Current Advances in Mechanical Design and Production, MDP-7
0.45
9
0.4
03
"~ = <
i . . . . 9
~"
-
[ ,
! [
.......
. . . . . . . . . . . --4.................. -.................. 4 ........................ ~....................... T.............. ~
. . . .
0.2
r .........................
4
4 ...........
......
...................................
, ....
. . . . . . . . . . . . . . . . . . . . . . . .............. . . . . . .,. .............. . i .i.-" ..... .,,,' ! i ! i .... "'*' 4L .i.,
"
0.15
i .....,7,_ . . . . . . . . . . . . . f__.,.,.._~.z . . . . . 0.1 ~'~"
0.05
----~-------! _,~.,.'~ [
.... !
.....
-0.6
'
-0.5
. . . . . .
-0.4
k
[ .t_
i i ~ , | ! ........ PD+Gravity I - - - - - - - - t - ~ ............. I .......... FeedbackLinearizati~ __]__
| ......... S l i d i n g M o d e s / Reference Trajectory
0 ~..~-.......i ................................4.~ i I ~
-0.05
.................
~ ....
-0.3
i i
~.-.
-0.2
t- .... /
, ......... [
~ ~ ............~..................... ~ 9 i
-0. I
...
i...
0
0. I
X Axis (m)
Fig. 4. Tip trajectory for the different control techniques For the feedback linearization controller, it was assumed that there is parametric uncertainty in the inertial parameters of the second link, where the controller parameters are 70 % of the actual parameters, to mimic the presence of an unknown payload. The tracking performance is noticeably superior to that of the PD plus Gravity controller, with no visible oscillations. The sliding mode control was simulated using the same uncertainty as before. Tracking performance is improved, due to the robustness of the control law once it reaches the sliding regime. 6. CONCLUSIONS The development of the dynamic model of a flexible joint manipulator has been presented, with particular emphasis on the two link articulated case. Simulation results show that conventional controllers, such as PD plus Gravity, may not be able to provide good performance. A feedback linearization approach for the control provided much improved performance, eliminating any oscillations in the tracking task, while a sliding mode controller improved the performance further when uncertainties where present. Control chattering usually associated with the sliding mode control technique was successfully suppressed by using a continuous switching approximation. 7. R E F E R E N C E S 1-
Good, M., Sweet, L. and Strobel, K., "Dynamic Models for Control of Integrated Robot and Drive Systems", J.Dyn.Sys.,Meas. & Cont., 107, pp.53-59, (1985). 2- Khorasani, K., "Adaptive Control of Flexible-Joint Robots", IEEE Trans. Rob.&Aut.,8(2), pp.250267,(1992). 3- Singh, S. N. and Zakharia, Y. N., "Variable Structure Control of a Robotic Arm in the Presence of Uncertainty", J. Rob. Sys., 6(3), pp.111-132, (1989) 4- Sira-Ramirez, H., Ahmad,S. and Zribi,M.,"Dynamical Feedback Control of Robotic Manipulators with Joint Flexibility", IEEE Trans. Sys., Man & Cyb., 22(4), pp.736-747, (1992). 5- Sira-Ramirez, H. and Spong, M. W., "Variable Structure Control of Flexible Joint Manipulators", Intl. J. of Rob. & Aut., 3(2), pp.57-64, (1988). 6- Slotine, J. J. and Hong, S., "Two-time Scale Sliding Control of Manipulators with Flexible Joints", Proe. Am. Cont. Conf., pp.805-810, (1986). 7- Spong, M., "Modelling and Control of Elastic Joint Robots", J. Dyn. Sys., Meas. & Cont., 109, pp.310-319, (1987). 8- Tomei, P, "A Simple PD Controller for Robots with Elastic Joints", IEEE Trans. Aut. Cont.,36(l 0), pp.1208-1213, (1991).
Current Advances in Mechanical Design attd Production Seventh Cairo University hffernational MDP Conference Cairo, February 15-17, 2000
45
THEORETICAL AND EXPERIMENTAL INVESTIGATION OF INTEGRATED STRUCTURE / CONTROL DESIGN OF HIGH SPEED FLEXIBLE ROBOT ARM Fanni, M.* and EI-Keran, A.*
*Assistant Professor, Prod. Eng. and Mech. Design Dept. Faculty of Eng., Mansoura University, Mansoura, Egypt. e-mail:
[email protected],eg and
[email protected]
ABSTRACT An integrated structure/control strategy for design of high speed flexible robot arm is investigated. The application of time optimal control theory, to get the minimum traveling time of a flexible robot arm, results in a multi switch bang-bang control. At first, two modes without damping are used to model the flexible robot arm as often done in the literature. The experimental investigation shows that, the bang-bang control which based on this model, results in a large position error. The air damping is experimentally proved to be the source of this error. The air damping had rarely been reported in the literature of flexible robot arms. A new mathematical model is developed which takes into consideration the effect of air damping. The problems arrived from the complexity of the new developed mathematical model are solved and satisfactory results are obtained. To achieve robustness of the control algorithm, the bang-bang control is followed by PD control. Both the optimal traveling time of the bang bang control and the settling time of PD control depend on the arm structure, namely the inertia, flexibility and damping of the arm. Using finite element analysis and optimization technique, the shape of the robot arm is optimized to minimize the optimal traveling time with constraints on the settling time and end point deflection. So, a minimum traveling time is obtained from both control and structure view points. The test rig, which is built to test the robot arm, is designed taking into consideration low cost, simple hardware and large flexibility. The control program is written mainly in Pascal with some procedures in Assembly Language. The experimental results show clearly that, the traveling time of optimized robot arm is lower than that of non-optimized one. The comparison between theoretical and experimental results shows good agreement. KEYWORDS Flexible Robot Modeling and Control, Finite Element Method, Optimization. I. INTRODUCTION Recently, there are growing demands in the robot industry for light structures in order to increase the speed of the robot motion and thus the productivity. As a slender structure gets lighter, it tends to be flexible, leading to vibration during operation. Large efforts are done to model, control and design such robots.
46
Current Advances In Mechanical Design and Production, MDP-7
Different forms of Euler-Bemoulli model are used in [ 1] to model the flexible robot arm. A beam element of the finite element method is used to model flexible robot arm in [2]. A three dimension solid element is introduced in [3] for structural optimization of such arms. Structural properties are derived in [4] using the lagrange's equations of motion with the assumed modes method. An evaluation of several controller synthesis methods is found in [5]. A non-linear integral-differential equations of Euler-Bemoulli beam with its optimal control are presented in [6]. The question of end effector trajectory control of an elastic macro-micro manipulator is treated in [7] using inverse and predictive controllers. A few works are found in the literature which deal with the structural design of flexible robot arm. In [8,9,10], the construction of the flexible ann is optimized using constant topology / varying size arm cross section. In [3, 11 ], the construction of the arm is optimized using varying topology / varying size arm cross section. Damping is often neglected in the modeling of flexible robot arm [1-3, 5-9]. Even, in the previous works of bang-bang control, damping is neglected for both rigid and flexible modes as in [10, 11] and is considered only for flexible mode in [12]. In [4] joint viscous friction is considered with comments about structural damping of the arm. The only found work [13], that reports about air friction of flexible robot arm did not adopt this damping for structural design of robot arm. In this work, an integrated structure/control design methodology of high speed flexible robot arm is investigated theoretically as well as experimentally to bridge the gap between theoretical and experimental works. A new mathematical model is developed which takes into consideration the effect of air damping. The arm shape, which affects the inertia, flexibility and damping, is optimized to minimize the optimal traveling time. The optimal traveling time is obtained through application of time optimal control theory. Constraints on settling time and end point deflection are also considered. Experiments are carded out to verify theoretical findings. 2. TIME OPTIMAL CONTROL THEORY AND BANG-BANG CONTROL The problem can be formulated as follows: given a single flexible robot arm which moves in a horizontal plane and an actuator with a maximum torque To. How should the actuating torque vary with time to make the arm rotates a given angle 0o, from rest to a final rest state in a minimum time (see Fig. 1). The arm is represented, in the first model by one rigid mode and one flexible mode without damping. The state equations are: i(t) = Ax(t) + bT(t),
(I)
where
01 0 0 A= i
00
00
0
-co ~
Ii '
b=
Lb,
where x(t) is the modal coordinate vector, T(0 is the actuating torque, co is the natural frequency of the flexible mode, b0 is the component of weighted rigid mode shape at a coordinate where the torque is applied, ((l/b0) 2 = J is the moment of inertia of the arm about the rotating axis) and bl is the same as bo but for flexible mode, (bl 2 = q is the flexible mode participation coefficient [9]). Both b0 and bl can be obtained using finite element method.
47
Current Advances in Mechanical Design and Production, MDP-7
Through applying the Pontryagin maximum principle, the optimal solution is found to be a multi-switch bang-bang control, (see [11 ]). The switches are symmetrical about the middle switch so one needs to calculate only two time intervals (ta, tf) from the following two equations in order to calculate the switches which are at (td2-ta), td2 and (td2+ta). ( l / / 2 ) 2 - 2t~ = 0~ J
(2)
To cos(co t / / 2 ) - 2cos(co ta) + 1 = 0
(3)
Figure 2 shows the three switches, where tf is the optimal traveling time. It is seen from equations (2, 3) that, the optimal traveling time, tf, depends on the inertia as well as the natural frequency of the arm. Both quantities depend on the construction of the robot-arm. So, through optimizing the arm shape, one can further decrease the traveling time. So, a minimum traveling time is obtained from control as well as structure view points. Before going into the optimization process, it is desired to carry out experiments for a flexible robot arm to check the validity of the above presented model (two modes without damping) as well as to check the accuracy of the bang-bang control.
Actuating Torque To
Final
0TL~~Initial position at rest
tf ,J r)
Fig. 1. Flexible robot arm
Fig. 2. Bang-bang control of flexible robot arm
3. EXPERIMENTAL TEST RIG The test rig is designed considering low cost, simple hardware and large flexibility. The test rig consists of a flexible robot arm moving horizontally. The robot ann is fixed directly on the axis of a permanent magnetic DC Table 1. Motor specifications. motor. Table 1 shows the specifications of the DC motor which are determined 5 Resistance, R (t2) through measuring of angular velocities and currents for different values of 13 Inductance, L (mH) voltage and applying the governing 0.3475 Back EMF constant, Kb (V.s/rad) equations of the DC motor. Both 0.3475 Torque constant, K, (N.m/A) resistance and inductance are measured Damping coefficient, C (N.m.s/rad) 0.9E-03 directly by inductance and resistance measuring instrument. The motor has no attached velocity feedback tachometer, as usual in servomotors. Instead, a single incremental encoder is used to sense both position and velocity. The encoder is coupled with the end of the motor axis (see Fig. 3). The encoder terminals are connec'~ed directly to a parallel printer port of personal computer (Pentium 233). A Pascal computer program is written to count the output pulses of the encoder and also to determine
48
Current Advances In Mechanical Design and Production, MDP-7
the direction of rotation. This eliminates the need of dedicated PC encoder counter card. A timer software based on PC chip 8254 [14] to give a resolution of 1 ~t sec is written in Assembly Language. A regulated power supply circuit is built to give a voltage of 14 v (which is the linearity upper limit of the used motor). Switch- mode power amplifier [ 15] is also built to drive the motor. Instead of using pulse width modulation integrated circuit (PWM-IC), a Pascal procedure is written to drive the switching amplifier through parallel printer port to give the required voltage.
4. THE FIRST E X P E R I M E N T - SIGNIFICANT EFFECT OF AIR DAMPING A flexible robot arm is manufactured from aluminum (Young's modulus = 70 GPa, density = 2700 kg/m 3) of rectangular cross-section with length of 760 mm, height of 20 mm and thickness of 3 mm. The inertia of arm with its hub is found to be 0.024 kg m 2. The ratio of hub to arm inertia is 0.03. It is to be noticed that, this small ratio allows the robot arm to behave close to hinged-free beam during its vibration. Thus its bending vibration is accompanied with oscillations about its axis. So its bending vibration can be sensed and controlled through sensing and controlling its joint angle directly. The small value of hub inertia is required to facilitate high speed of robot arm. The natural frequency of the arm is calculated through a computer program written according to Euler-Bernoulli dynamic model. This model with its boundary conditions can be found in [l ]. The natural frequency is found to be 16.094 Hz. Same frequency is obtained using finite element method with 3D beam elements of the ANSYS-Program [16]. An accelerometer and FFT analyzer (Yokogawamodel 3655E, shown in Fig. 3) are used to measure the natural frequency. The measured value is 16.5 Hz which is fairly close to the calculated one. Maximum torque of the DC motor is taken as 0.52125 Nm (higher torque can be taken from this motor, itis limited hereto protect the electronics from high current). The angle rotated by the arm is chosen as 180~ By solving equations (2, 3) the time intervals ta ,tf and consequently the switching times can be found. According to Fig. 2, the torque must be constant between each two consecutive switching.
49
Current Advances in Mechanical Design and Production, MDP-7
This requires constant current as it is clear from DC motor governing equations:
(4) (5)
v ( t ) - K b O(t) = L d i ( t ) / d t + Ri(t) T(t) = K,i(t)
where v(t) is the applied voltage, i(t) is the current and 0 (t) is the angular velocity. This can be done using current source. However, it is planned to follow the bang-bang control by a stabilizing position control, that prefers voltage control. So, a controlled voltage is applied which gives the constant required torque. Thus, the control circuit is kept simple. This is done by two methods. In the first method, equations (4, 5)are solved for constant torque using Laplace transforms giving the following volt equation: v(,) = X b b (') + RTo / K, + L ( + T o / X t - Tota / K, )8(0
(6) where 5(t) is the delta function and Totd is the torque before the last switching. In applying equation (6), the angular velocity is measured by means of the encoder and the PC timer and then fed back into the volt equation. The last term in equation (6) is carried out by applying the maximum permissible voltage (+Vo) for a certain interval of time (Vt), such that +Vo*Vt = L(+T0-Totd)/Kl. The practical application of equation (6) shows that, the last term has negligible effect. In the second method, equations (1) are solved for constant torque (+To) and the modal displacements and velocities are obtained as functions of time. Then the angular displacement and angular velocity (vector y) can be found by mode summation:
y=b'x,
01
b'=
b0 0 5 I
The resulted angular velocity is substituted in equation (6) and the new volt equation is written as" v(t) = Kb[x 2 (0) +_Tot / J + (+_qTo / c o - x3 (0)co) sin(cot) + x4 (0) cos(cot)]
(8)
+- 1r / K, + L(+_To - L,~ ) 8 ( 0 / K, where x2(0), x4(0) and x3(0) are the modal velocities and displacement just before the last switch and can be obtained from the solutions of equations (1) by substituting the proper time. Note that, equation (8) requires no feedback and can also be obtained through applying the transfer function between the volt and the I Theoretical ] torque. 200.00
--
"
Two control programs for both methods are written. The results of the experiments are presented in Fig. 4, which shows the variation of robot angle 0 with time during the bangbang control. The two lower curves are for the measured values of the two above mentioned methods. The upper curve is for the theoretical values obtained from the solution of equation (1). This figure shows clearly the large differences, on one hand, between the theoretical and experimental results and, on the other hand, between the results of the two experimental methods. The position error at the end of the bang-bang control between the
Experimental' meth~ 2 /
f
9 Experimental, method 1 j
160.00 -
~ 12o.oo 8 -~ .~ 80.00 o o~
~
40.00 o.oo
!
0.00
i 0.20
0.40 T i m e (see.)
i
! 0.60
Fig. 4. Bang-bang control for model without damDin~
i 0.80
50
Current Advances in Mechanical Design and Production, MDP- 7
theoretical and the experimental results of the first and second methods are 38.2% and 22.8% respectively. The position error between the two experimental results is 20.0%. This indicates that, the used model (two modes without damping) is not sufficient for representing the system dynamics. It is clear that, the rigid mode is responsible for this large position error rather than the flexible mode (maximum displacement of the flexible mode is about 1% of that of the rigid mode). As a concluded result, damping of the rigid mode must be taken into consideration. The measured damping of the motor beating without the arm is very small, see Table 1, but the robot-arm itself may introduce more damping. To examine this idea, the speed of the motor is measured with and without arm at the same applied voltage of 13.3 v. Of course, the speed is measured after being constant (i.e. after the transient response time). The speed without arm is found to be 34.7 rad/s and with arm is only 16.5 rad/s. This decrease in speed (more than 50%) is a direct result of the rigid mode damping. Now, the question is, from where this damping comes? There are two possible answers. The first one is that, the centrifugal force resulted from arm inertia have a large reaction at motor bearing which causes this large damping. The second one is the air damping. To examine both ideas, a weight is attached at the end of the arm in a way, that affects the arm inertia largely (more than 300%) but the air damping slightly. The measured arm speed after that does not significantly change (less than 3%). This result indicates that, the air friction is the source of the noticed damping. To further examine the air damping, the damping torque caused by the air resistance is computed and compared with the measured one. According to [17], the damping force Fd caused by a constant air velocity u is given by: Fa = 0.5pu 2A C o (9) where p is the air density equals 1.23 kg/m 3, A is the exposed area and Co is the drag coefficient which equals 2 for rectangular cross section [17]. Equation (9) is integrated assuming constant drag coefficient to give a damping torque Td for the rotating robot arm: 2
1 pCowot' b
(1o)
where w0 is the height and t is the length of the robot arm. The computed damping torque is found to be 0.559 Nm. The measured damping torque is determined using equations (4, 5) and found to be 0.5257 Nm, which is fairly close to the calculated one. This result assures that the air is the source of the noticed damping. 5. NEW MATHEMATICAL MODEL WITH AIR DAMPING The air damping is modeled by an equivalent viscous damper. The angular velocity of the arm is measured at different voltages and the relation between the ann angular velocity and the damping torque is linearized about an average operating arm velocity. Another way is averaging directly the viscous damping coefficients measured at different operating velocities. Both methods give close results. The second method is considered further because of its simplicity. The equivalent damping coefficient, C, is found to be 0.025 Nm see. It is clear that, the air damping is negligible for the flexible mode, because the deflection of the flexible mode is small [18]. The state equations of the flexible arm is the same as equations (1)except matrix A which becomes:
Current Advances in Mechanical Design and Production, MDP- 7
0 A=
1
0
0
i o 0 i 0
0
0
-co 2
51
(ll)
where (z equals the damping coefficient (C) divided by moment of inertia of the arm (J). The solution of the Pontryagin maximum principle is a multi-switch bang-bang control but not symmetrical about the middle switch as in the previous case without damping. Through applying the final state conditions, which dictate that the angular velocity must be zero and the angular displacement must equal 00, the following equations (in dimensionless form)are derived: I - 2e -~'/~ + 2e -(y'§ - 2e -(~'§247 + 2e -(y,§247247 =0 (12) (13) Y~ -Y2 +y3 -Y4 - z = 0 (14) 1 - 2 cos Y4 + 2 cos(y 3 + Y4)- 2 cos(y 2 + Y3 + Y4)+ cos(yl + Y2 + Y3 + Y4) = 0 2 sin Y4 - 2 sin(y 3 + Y4) + 2 sin(y 2 + Y3 + Y4) where the dimensionless quantities are defined as"
-
sin(y~ + Y2 + Y3 + Y4) = 0
(15)
x = d c o / C , z = O o c o C / T o , y, =ag,, i = 1 ..... 4 (16) where t~,t2,h,t4 are the time intervals between each two consecutive switchings. Note that: yl+y2+y3+y4=y=o~tf. The solution of these nonlinear equations is extremely difficult. This is because there are many adjacent roots 0.80and it is difficult to find the root which gives minimum traveling time, tr. Many -" t3 ~[~,1~~t2 t4 ~J known algorithms, commercial soflwares (MATHCAD for example) ~" o.4oand advanced methods like the z~ homotopy method [19] have been |tT" = applied without success. The solution of ~. ooo * I ' I ' I ' I 0.20 0.40 0.60 80 this problem comes through modifying =~ o )0 .~9 .~ Time (sec.) an algorithm found in [20] in order to locate simultaneous zero crossing of ~ -0.40four functions of four variables. This modification requires grid divisions of four dimension space. The advantage of tf . . . . -0.80 this algorithm is that it gets all the roots Fig. 5 Torque-Time diagram for in certain space, so one can choose the solution the new model which has minimum tf. The disadvantages are the long executing time and the large storage space. The algorithm is further modified to solve the problem of storage space and partially that of the executing time. The solution of these equations for minimum tf is now obtained and is found to have the following characteristics (see Fig. 5): The first and last intervals (tl, t4) are much larger than the inner intervals, which is similar to the previous case without damping. The time of positive torque is larger than that of negative torque (unlike the previous case, where they are equal).
This can be understood by noting that the damping torque resists the acceleration and assists the deceleration. One can use equation (6) as it is but equation (8)must be modified for damping consideration. The new equation is:
Current Advances in Mechanical Design and Production, MDP-7
52
v ( t ) = K b [ + T o I C - ( + T OI C - x 2 ( O ) ) e - ~
x4(O) cos(a,t)] +__RT o / K,
x3 (O)m) sin(rot) +
+ (+qTo I co -
(17)
+ L ( • o - To,j)a(t) / K,
6. THE SECOND EXPERIMENT AND PD CONTROL Figures 6 and 7 show part of the transient response of the flexible robot arm to step volt of 13.6. The theoretical curves are obtained through the transfer function of the flexible robot arm with damping. The theoretical and experimental results show good agreement specially for the displacement response. This indicates the correctness of the data identified from the system as well as the validity of the new developed mathematical model. The effect of non modeled higher flexible modes are clearer in the velocity response. 40.00 --
r
~. m
30.00 -
f
lheoritical
m
_ ~
~d
t~ G)
~-
E
6.0O
L..
,
....
.~
Experimental
1
r
9
o ta
-0
20.00~
~
E |
4.00
.~ o
o Q. tn --9 a
lO.OO -
c: <
0.00
> 1,,. m c <
-10.00 -
,
I-
0.00
'
0.04
I
-i
0.08 Time
I
'
I
0.12
i
0.16
7
2.00
0.0O
-2.00
0.20
0.00
0.04
0.08
(sec.)
0.12
0.16
0.20
Time (sec.)
Fig. 6. Displacement response
Fig. 7. Speed response
The flexible robot arm is now tested for bang-bang control but with the new switching times. The two control methods (equations 6, 17) are applied. The results are shown in Fig. 8. The theoretical curve is obtained through the solution of the new mathematical model with damping. The three curves are close to each other. The two experimental curves are almost coinciding. The position errors between the theoretical and the two experimental methods are
~oooo- [:
} ,=.oo_
*~""
I--
.
]
~oooo
~x,O.n,.,,--,~,q
160.00
120.00
I '=~176
i
.~176176
~
40.00
4o.oo-
o.oo
80.00
-
-
0.00
,
I
0.20
9
I
"
0.40 Time (sec.)
,
I
0.60
Fig. 8. Bang-bang control for model with damping
,
7
0.80
0.00
!
0.00
!
0.40
0.80 Time (sec.)
i
1.20
1.60
Fig. 9. Bang-bang control followed by PD Control for nominal design
Otrrent Advances in Mechanical Design attd Production, MDP-7
53
3.4% and 2.4% respectively. The position error between the two experimental methods is 1%. This indicates the validity of the proposed mathematical model with damping. However There are still position and velocity errors. The bang- bang control is not very robust and accurate. So, a PD control is made to follow the bang-bang control. The experimental results are shown in Fig. 9. It is to be noted that, the time of bang-bang control is 0.777 sec and the measured settling time of the PD control is about 0.633 sec. So, in the aimed optimization process, the settling time must be also taken into account. 7. STRUCTURAL OPTIMIZATION It can be seen from equations (12-16) that the optimal traveling time tf (the sum of the time intervals) depends on the moment of inertia (J), natural frequency (co) and damping coefficient (C). The optimal traveling time decreases with the decrease of J and C and with the increase of co. The three quantities depend on the construction of the arm. Note that the air damping depends on the arm dimensions (se equations 9, 10). So, through optimizing the arm shape using finite element method and optimization technique one can get lower tf. The problem is that one needs to solve equations (12-15) in each iteration of the optimization process, which is impractical. Instead, equations (12-15) are solved first for different values of x and z. After each solution the parameter y, (=~yi), is determined. Then y as a function of x and z are obtained by means of multi-regression technique. The obtained relation, which is used in optimization, is: y = 40.4z223264 + 3.93xz
(18)
Equation (18) agrees very well with the calculated values for z > 1. Another accurate relation could be obtained also for z < 1. The nominal design has z-15.28 and it is not expected that, z will be lower than 1 for the optimal design, so relation (18) is enough for the presented example. In [3, 11] transverse holes are made along the arm length to optimize certain performance index. The used design variables are holes radii, height and thickness of the robot arm. The resulted optimum shape [3] is more efficient than shapes found in other literatures and also easier to be manufactured. The holes here seem to be efficient also in decreasing the air damping because they decrease the exposed area A (see equation 9). The damping coefficient as function of arm dimension is found to be" I
C = D ~r3w(r)dr
(19)
0
where D is constant equals 15.16645677. The above integration is determined for thearm with single row of n transverse holes, the resulted relation is: C = D{l 4w o / 4 - s xp~a~ + 2.3562p, a~}
(20)
i--!
where a~ is the hole radius and pi is the distance between hole center and arm axis. The design variables of the problem is the arm height and the holes radii. Different number of holes (10, 20 and 30) are considered. In each ease, the holes are arranged into five groups. Each group contains equal adjacent holes to reduce the number of the design variables. The optimization is carried out for three different thicknesses of available aluminum sections (3.5, 3.0 and 2.7 ram). The objective function is the minimization of the optimal traveling time tf. The natural frequency of the flexible mode and the moment of inertia are calculated using finite element method of ANSYS-Program [16]. The damping coefficient is calculated using equation (20).
54
Current Advances in Mechanical Design attd Production, MDP- 7
Then tr is calculated by means of equation (18). The static deflection due to a payload of 2.5 N and due to the arm weight is calculated using finite element method and is constrained to be less than 3.5 mm in the optimization problem. Three dimension solid elements are used to model such arm construction [3]. The hub inertia is modeled by concentrated masses at nodes around the axis of rotation. The moment of inertia of the masses around the rotational axis equals to the hub inertia. Using the above elements and technique to model the nominal design to check their accuracy gives a natural frequency of 16.143 Hz which is close to the previous calculated and measured values (see section 4). In considering the settling time of the PD control, the air damping is neglected because the velocity of the arm is small during this stabilizing control (see Fig. 9). In order to keep the settling time of the flexible robot arm small, the first open loop zero and first open loop pole must be kept high [10]. Additional to these requirements, another practical requirement is found in this work. That is: the open loop gain (K) must be kept high also. Because of the limited volt source (14 v) the conversion constant K0 is limited (where V(s) = Ko E(s), V(s) is the actuating voltage and E(s) is the position error). The optimal gain (K0 K), which has large value (based on the method in [9]) could not be reached. The high value of the open loop gain increases the upper limit of the possible obtained optimal gain. The transfer function from the actuating torque to the robot arm angle is given by: O(s___~)= K ( s 2 + Z 2) (21) T(s)
s 2 (s 2 + P2 )
where the open loop gain (K), first zero (Z) and first pool (P) are given by: (22) The above open loop quantities are constrained to be slightly higher than their values of the nominal design in the optimization problem. The optimization method of the ANSYSProgram is used. All associated calculations of optimization problem are carried out within the environment of ANSYS-Program. ANSYS Parametric Design Language (APDL) is used to write the task program. K = (l + qd) / d,
Z = ra~]i"/ (l + qd),
P = co
8. OPTIMIZATION RESULTS AND DISCUSSION The best optimization results are obtained with thirty holes and a thickness of 2.7 mm. The optimum design variables are 19.8 mm for the height and 4.5, 7.9, 7.5, 8 and 8 mm for the holes radii of the five groups. The finite element mesh of the optimal robot arm is shown in Fig. 10. The characteristics of the optimal design and those of the nominal design are shown in Table 2. Table 2. Nominal and optimal design result
Nominal Design Optimal Design
Inertia (kg m 2) 0.024 0.014
Damp.
(N m s/tad)
0.025 0.015
Pole (Hz) 16.143 16.3
Zero ' Gain (Hz) (kg m2)"1 4.8 ......... 458 5.7 565_
Deft. (mm) 3.1 .3.2
Optimal I time (s)_[ 0.777 i1 0.593 J
The optimal traveling time is decreased by 23.7 %. The moment of inertia, the damping coefficient and the mass are decreased while natural frequency (first pole), first zero and gain of the open loop as well as the static deflection are increased.
Current Advances in Mechanical Design and Production, MDP-7
200.00
55
-
f
/
160.00 Q
~ 120.00 E U
.~
80.00
10 t__
<
40.00
0.00
-
f
0.00
Fig. 10. FE-Mesh of optimal arm
.
.
'
.
.
.
.
.
I 0.40
.
.
.
'
.
T i m e (see.)
I 0.80
!
: ---1
1.20
Fig. 11. Bang-bang control followed by PD Control for optimal design
The optimal shape of the robot arm is manufactured and tested. Fig. 11 shows the results of the bang-bang control followed by PD control. The measured settling time of the PD control is 0.501 sec. The settling time is decreased by 20.85%. The total traveling time is decreased by 22.4%. It is to be noted that, the settling time depends also on the end conditions of the bangbang control. Although stress constraint is not considered in the optimization problem as in the literature [8-11], it is calculated here for the optimal shape and found to be safe. This work assures the importance of the integration between theoretical and experimental investigations. 9. CONCLUSION Theoretical and experimental investigation of integrated structure/control design of high speed flexible robot arm is presented. The air damping has been experimentally proved to have large effect on bang-bang control of such arm. A new mathematical model including air damping is developed and its associated problems are solved. The application of this new mathematical model shows a significant decrease of position error of bang-bang control from 38.2% to 3.4%. The comparison between theoretical and experimental results shows good agreement. A relation between arm shape and its air damping is derived. The arm damping in addition to its inertia and flexibility affect the traveling time of the arm. The shape of robot arm is optimized to minimize the optimal traveling time of the bang-bang control with constraints on the settling time of the followed PD control and end point deflection. This is done using finite element method and optimization technique. The test rig is designed with considering low cost, simple hardware and large flexibility. Some hardware control circuits are replaced with suitable software. The experimental investigation shows that the total traveling time of the optimized robot arm is decreased by 22.4% with respect to the nominal design. REFERENCES Wang, F.-Y. and Guan, G., "Influences of Rotatory Inertia, Shear and Loading on Vibrations of Flexible Manipulators", Journal of Sound and Vibration, Vol. 171, No. 4, pp. 433-452, (1994).
56
2.
Current Advances in Mechanical Design and Production, MDP- 7
Kalra, P. and Sharan, A., M., "Accurate Modelling of Flexible Manipulators Using Finite Element Analysis", Mech. Mach. Theory, Vol. 26, No. 3, pp. 299-313, (1991). 3. Fanni, M., "Modeling and Optimization of High Speed Flexible Robot Arm", Fifth Int. Conf. on Production Eng. and Design for Development (PEDD), Cairo, Egypt, (1998). 4. Arteaga, M., A., "On the Properties of a Dynamic Model of Flexible Robot Manipulators", ASMA Journal of Dynamic Systems, Measurement, and Control, Vol. 120, pp 8-14, March, (1998). 5. Chang, P.-M.and Jayasuriya, S., "An Evaluation of Several Controller Synthesis Methodologies Using a Rotating Flexible Beam as a Test Bed", ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 117, pp. 360-373, Sept., (1995). 6. Zhu, W., D. and Mote, C. D., "Dynamic Modeling and Optimal Control of Rotating Euler Bemoulli Beams", ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 119, pp. 802-808, December., (1997). 7. Yim, W. and Singh, S., N., "Nonlinear Inverse and Predictive End Point Trajectory Control of Flexible Macro-Micro Manipulators", ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 119, pp. 412-420, Sept., (1997). 8. Asada, H., Park, J.-H. and Rai, S., "A Control-Configured Flexible Arm: Integrated Structure/Control Design", Proc. IEEE Int. Conf. on Robotics and Automation, Sacramento, California, pp. 2356-2362, April (1991). 9. Park, J.-H. and Asada, H., "Concurrent Design Optimization of Mechanical Structure and Control for High Speed Robots", ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 116, pp. 344-456, (1994). 10. Rai, S. and Asada, H., "Integrated Structure/Control Design of High Speed Flexible Robots Based on Time Optimal Control", ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 117, pp. 503-512, December (1995). 11. Fanni, M., "Design of Non-Traditional High Speed Flexible Robot Arm Based on Time Optimal Control", 6th Int. Conf. on Artificial Intelligence Applications, Cairo, (1998). 12. Pao, L. Y. and Franklin, (3. F.," Time-Optimal Control of Flexible Structures", 29th IEEE Conference on Decision and Control, pp. 2580-2581, Honolulu, Hawaii, (1990). 13. Khrrami, F., Jain, S. and Tzes, A., "Experiments on Rigid Body-Based Controllers with Input Preshaping for a Two-Link Flexible Manipulator", IEEE Transactions on Robotics and Automation, Vol. 10, No. 1, pp 55-64, Feb., (1994). 14. Brey, B., B., "The Intel Microprocessors", Macmillan Publishing Company, (1994). 15. Michael, J., "Industrial Control Electronics - Appl. and Design", Prentice Hall, (1988). 16. ANSYS User's Manual for Revision 5.0, Volume I, Procedures, (1994). 17. Anderson, J. D., "Fundamentals of Aerodynamics" ,Mcgraw-Hill, (1991). 18. Beards, C. F., "Structural Vibration Analysis and Damping", Arnold, (1996). 19. Watson, L. T., Billups, S., C. and Morgan, A., P., "HOMPACK: A Suite of Codes for Globally Convergent Homotopy Algorithms", ACM Transactions of Mathematical Software, Vol. 13, pp 281-310, (1987). 20. Hostetter, G., H., "Analytical, Numerical, and Computational Methods", Prent. Hall, (1991)
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
57
GENERALIZED PATH GENERATION FOR A MOBILE MANIPULATOR
Bayle, B., Fourquet, J.-Y. and Renaud, M. LAAS-CNRS, 7 Ave. du Colonel Roche, 31077 Toulouse, Cedex 4, France
ABSTRACT We present a generalized path generation algorithm for a nonholonomic mobile manipulator built from a n joint manipulator mounted on a nonholonomic mobile platform. The anipulator end-effector path is imposed by the task whereas the natural redundancy of the system is used to avoid singularities. We describe the algorithm which calculates the generalized coordinates for both the manipulator and the platform as a function of the operational coordinates of the end-effector location (position and orientation). It is based on a norm minimization for a direct differential kinematic model combined with a gradient descent method. Simulations have been realized on a planar mobile manipulator, made of a double-pendulum mounted on the platform. KEYWORDS Mobile Manipulators, Nonholonomy, Redundancy, Minimization, Quadratic Criterion. 1. INTRODUCTION Mobile manipulators give rise to an increasing interest in robotics laboratories and in application domains. These mechanical systems are the result of coupling manipulators with mobile platforms. They combine mobility and manipulation capabilities and seem particularly suitable for human-like tasks. As an example, when we write across or erase a blackboard, we locate our body so that our arm is in the most comfortable configuration to write. From the robotics viewpoint, a writing task leads naturally to move the manipulator and the mobile platform at the same time. In this paper, we present a generalized path generation algorithm for a nonholonomic mobile manipulator built from an n joint manipulator mounted on a nonholonomic mobile platform. More precisely, we are looking for the combined eVolution of the platform and of the manipulator so that the location (position and orientation) of the manipulator end effector follows a desired path, defined as a function of a parameter. There is a real need for coordination since the mobile manipulator has to follow its preplanned end effector path, and at the same time, has to conserve good manipulation capability. A typical characteristic of mobile manipulators is their high degree of redundancy created by the addition of the platform degrees of mobility to the manipulator ones. The coordination is
58
Current Advances in Mechanical Design and Production, MDP-7
then based upon the resolution of this redundancy. In the litterature, the generalized path generation algorithms of mobile manipulators are rather limited. It is necessary to distinguish these algorithms in accordance with the problem to be solved; in a first class the end-effector path is imposed whereas in a second class only the final end-effector location is imposed ( in this case the generalized path generation is the consequence of the control of the mobile manipulator). The most important contributions belonging to the first class must be themselves divided into two subclasses. In the first subclass the generalized path generation is obtained by adding tasks in order to solve the redundancy problem. To our knowledge the first work in this area is due to H. Seraji [1,2] which considers an instantaneous kinematic approach. Others works in the same area are due to C.-C. Wang et al. [3], U. M. Nassal [4], F. G. Pin et al. [5] and G. Foulon et al. [6]. Always in this class, Y. Yamamoto [7] considers moreover a dynamic approach. In the second subclass the generalized path generation is obtained by minimizing a quadratic criterion; this is for example the case in the works of C.-C. Wang et al. [3] and ofU. M. Nassal [4]. In the paper [8], J.-Y. Fourquet and M. Renaud present some links between the two different approaches. The works conceming the second class are very limited; a few contributions are due to G. Foulon et al. [9] and C. Perrier et al. [10]. Finally, among a lot of interesting contributions in the field of mobile manipulators, let us mention the works of O. Khatib et al. [11], K. Nagatami et al. [12], T. Sugar et al. [13]. In this paper we propose a generalized path generation based on a pseudo-inversion scheme, that minimizes a quadratic criterion, and we try and modify the configuration evolution - the generalized path - by imposing the free term in the pseudo-inversion scheme from a scalar function determined by singularities. First we introduce the modelling of the mobile manipulator, the problem and the notations. Then, we define the methodology of the generalized path generation. This approach is applied to a planar system built from a double pendulum type manipulator arm. Finally, conclusions and perspectives are presented. 2. MODELLING OF THE MOBILE MANIPULATOR 2.1 Notations
In the general case, the mobile manipulator is built from an n joint manipulator mounted on a nonholonomic mobile platform and the position of the manipulator base, with respect to the platform, is defined by two parameters a and b (see Fig. 1). Let R = (O,s
be the fixed frame in which the path of the manipulator end-effector is
given and R' = (O',s
be the moving frame attached to the platform. Furthermore, let
Ro = (0o,~o, 3'o,io) be the moving frame attached manipulator base (remark that the moving
frames Ro and R' are parallel) and R, =(O,,s a moving frame attached to the manipulator end-effector and rotating around the point On. The point O,,+!, which is the center of the end-effector, is fixed relative to the frame R,.
Current Advances in Mechanical Design and Production, MDP-7
59
Yo
'
0
. . . . . . . . . . .
,
.
.
.
.
.
.
..~
~, --'- x
x
Fig. 1: Mobile manipulator when n = 6
2.2 Differential Kinematics Equations of the Mobile Manipulator 2.2.1 Mobile platform subsystem If we leave the wheels of the platform out of account we can identify the three generalized coordinates- defining its configuration- with the three operational coordinates- defining its location. If we choose the three operational coordinates as: ~, = [Sp~ r 5p3]T = [x y O]T (see Fig. 1), the three generalized coordinates are also defined by: qp = [qr,1 qp2 qps]r = [x y d] r. With the previous abstraction, the motors action on this platform is represented by: du = [dul du2] T = [da dO] r, in which a represents the curvilinear abcissa of the point O', on its path in the plane (O, ~', if). The configuration differential kinematic model [14] of this platform is then: dqp = S(qp)du, with:
[cos !] [:io0!] =
0
From this model the nonholonomic constraint can be written in the form:
Gv(qv)dqv = with" G v ( q p ) = [sin %3 -cosqp3
O] = [sinO
O,
-cosO
(1)
0].
2.2.2 Manipulator subsystem The configuration of the manipulator is defined by the n generalized coordinates: qb = [qbl qb2 ... qb,,]r, and the end-effector location, i.e. the position of the point O,,+1 (defined here by its Cartesian coordinates) and the orientation of the frame 7"r relative to 7"r by the m operational coordinates: ~ = [~t ~b2 ... ~b,,]T. The direct kinematic model of the manipulator [1.5], relative to R0. is: ~b = fb(qb) and the direct differential kinematic mode], with Jb(qb) = ~a/b is [151'
d~b = Yb(qb)d~.
(2)
60
Current Advances in Mechanical Design and Production, MDP-7
As 7~' is obtained from Tq.oby a translation characterized by the two constant parameters a and b the end-effector location, relative to 7~', is defined by the m operational coordinates: I t ~t T " i t ~b -" [~bl ~'b2 "'" bm] , w,th: ~bt = a + ~b,, ~b2 = b + ~b2, ~'b3 -- ~b3, . . . , ~'bm = ~bm, and we can write: d~,b = d~,b = Jb(qb)dqb.
2.2.3 Mobile manipulator The configuration of the mobile manipulator is defined by the u = n + 3 generalized coordinates: q = [qt q2 q3 q4 qs . . . qul T = Ix y 0 qb, qb2 . . . qb,,]r. From Eq. (1) it is then possible to write: G(q)dq = 0,
(3)
with: G(q) = [Gp(cb) 0]. We can characterize the end-effector location, i.e. the position of the point O,+t and the orientation of the frame Tr relative to T~, by the ~t operational coo,ai,,t~.~. ~ = [6 ~ . . . G] r. The direct kinematic model of the mobile manipulator [15], relative to "g, is: t~ = f(q) and the direct differential kinematic model, with J ( q ) = ~ is [15]" d,~ = J(q)dq.
(4)
From Eqs. (3) and (4) we obtain: j(q)]dq=
[:,]
.
But it is more interesting to use the new set of quasi-coordinates [16] - defined by [17]: P=
[agqb,
qb2 ... qb~]T. Then: d q =
M ( q ) d p , with: M ( p ) =
"]S(qp)
-'El where E J
is the n order unit matrix. Taking into account that G(q)M(q) = 0, we obtain, with J(q) = J(q)M(q): J ( q ) d p = d~.
(6)
2.2.4 Application to a planar mobile manipulator Let us consider a horizontal double-pendulum mounted on the platform; its configuration is defined by q = [z y 0 qbt qb2]T and v = 5. In the frame TO, the position of the point O3 is given by the Cartesian coordinates ~ and ~2 and the orientation of the end-effector by the angle ~3; then tt = 3. Furthermore we suppose that a = b = 0 (see Fig. 2). Here, p = [a 0 qb, qb2]T and the matrix J ( q ) becomes: C3 - a t ~ 3 4 - a2~345 -alS34 - a2S345 -{t2S345 1 $3 a1C34 4- a2C345 a1C34 4- a2C34s a2C34s ] , 0 1 1 1
with: $3 = sin q3, C3 = cos q3, $34 = sin (q3 + q4), C34 = cos (q3 + q4), $345 = sin (q3 + q4 + qs), and C345 = cos (q3 + q4 + qs).
Current Advances in Mechanical Design and Production, MDP-7
0~
..... ~ ....................
61
',
~::;, ,.:. . . . .
...."
o-**'*
, q,~,_ a2
.................
, ,,
"',
i? -i"
Fig. 2: A planar system
3. GENERALIZED PATH GENERATION From now on, J( q) which is always a function of the configuration, is denoted as ]. We impose the end-effector location to follow, in the frame R, a desired path, defined as a function of a given parameter s: s 9 " (d(s). At each value of the parameter s it is necessary to calculate the increment 6p(s) corresponding to the imposed increment 6(a(s), that is to invert the relation (f~d(s) = J S p ( s ) , which is an approximation of Eq. (6). The increment (fp(s) gives the increment 5q(s) in accordance with the relation d~q(s) = M(q)(fp(s). This generates the generalized path q(s) corresponding to the imposed operational path
In order to simplify writing we don't use the explicit dependence on the parameter s. Due to the fact that, generally, the matrix j is not square there exist an infinite number of solutions of the previous inversion problem. All the solutions are given by: 5P = J # gt~t + (E - ,]~ J)z,
(7)
where J # and z denote respectively a generalized inverse of 0y and an arbitrary vector with same dimension as 6p. So, as the solutions are parameterized by the vector z and the choice of a generalized inverse, it remains to choose a particular solution based on additional features of the problem" existence of kinematic singularities, obstacles ... For this purpose, we develop a scheme based on criteria minimization and hierarchy of tasks. The main task is still described by Eq. (6). 3.1 A Pseudo-inversion Scheme Let us recall that the particular inverse solution: 5p = J+ 6t~ + (E - J+ J)Sp~,
(8)
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Current Advances in Mechanical Design and Production, MDP- 7
with J+(q) the pseudo-inverse of J(q), is the solution of the following problem of optimization: minimize g(6p) = lll~ p - ~pal[ 2, with 6~ =
Y6p.
P r o o f : The associated Lagrangian l($p,,k) writes: 1
and the first order extremality condition leads to:
k~J
= 0
6p = 6pa + j--r)~.
Since E is the identity matrix, the second order condition for the solution to be a minimum: 0~!
T
-O(~pp) = E
> O,
is verified here. As )~ = ( j j T ) - l ( 6 ~ - J6p~), the solution is given by Eq. (8). The previous class of solutions allows to choose a particular generalized inverse- the pseudo-inverse- and it needs to define an interesting reference vector 6p~. In the next paragraph, this vector is defined by using Eq. (8) successively for two tasks.
3.2 Task Hierarchy
This part is much inspired by Nakamura [18] and uses the same approach to introduce task hierarchy. In the following we will consider tasks defined by: 9 the evolution of some variables (the location for instance) describing the task; 9 the model linking the differential of these variables to the differential of mobile manipulator quasi-coordinates. So, for two tasks rt(s) et r2(s), functions of a normalized parameter s E [0 1]: 8rt = JI6P and ~r2 = J2~P. Here, the end-effector path is imposed and the first task is then d~ = d~rl = j6p. Depending on the environment and on the mobile manipulator, different choices can be made for the second task. In the next paragraph, they are presented in a common framework based on the gradient method. We aim at achieving both these tasks but there is a priority order: the first will be realized while minimizing a criterion relative to the second. Minimizing [[6p- 6p~l[~ leads to: 6p = J+6~ + (E - J+J)~p~. Taking into account the second task, we obtain: 8r2 = J2J+8,~ + J 2 ( E - J + J ) * p . .
Current Advances in Mechanical Design and Production, MDP. 7
63
We choose to solve this equation to get the minimum norm 1 solution for 6p~:
Let L = J 2 ( E - J+J). Then as shown in [191:
(9)
6p = j+6~ + L+(6r~ - j~j+6~). 3.3 Reconfiguration of the System Using the Gradient Method
The problem is to minimize a scalar function 79, depending on the configuration of the system. With the relations between the differential of the configuration dq and the nonexact differential form dp, we can deal with the nonholonomy of the system. Indeed, for a given vector of coordinates c the gradient method consists in choosing dc according to the equation, where tr is a positive scalar:
dc +x
"~c
=0.
(10)
This method is not compatible with the kinematics of our system, i.e. we cannot apply it to our generalized coordinates q, as the platform velocities are constrained by nonholonomy. "Yet we can modifv. Eq. (10)', indeed we can set dp = M T d q = --xMT(O~') T ~ as the dpi are independent. ov 79 being a function of q: d79 = ~ov dq = - x ~ov M M T 5~ 9
()
And so d ~ < 0 if M M T is positive definite. Here M M T is only positive semi-definite but nevertheless, we can show easily, that M M r ( ~r ~ = 0 corresponds to a configuration where (07,T; T;,~,~ is perpendicular to the longitudinal axis of the platform. Of course, in this case it is impossible to follow the gradient for the first two coordinates. Otherwise, the function 79 decreases when its gradient is not zero. If it is, the configuration is stationary and if the Hessian (~yq-rq •_ 0 with an appropriate choice, then it is a minimum. Notice that in the following we will use the increment 6p in our algorithm, as we explained in the previous paragraphs.
3.4 Using Gradient Method
We use the principles shown in 3.2 and 3.3 with the following tasks: 6~ = j 6 p " task 1. and
In this last case 6r2 = _~r
-x
~qi
]
=6p-
task 2.
M)T and J2 = E.
Using the idempotency of E - J + J . we find: 6p = J+ 6( - x( E - J+ J)
M
.
(11)
tFor a system: y = Ax, the minimum norm solution that minimizes [y - Ax]r[y - Ax] is x = A+y.
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64
4. PLANAR MOBILE MANIPULATOR 4.1 Field of Application of the Method
To be aware of the possibilities of reconfiguration we can evaluate 2 Proj = E - J + J . In our example, the computation of Proj can be achieved from the analytical expression of J+, which is not feasible for more complex structures. We find that: 0
Proj =
0
0 -0
0 0
0
0
mod ~').
0 0
The expression of Eq. (11) is: cos 0 - ~ + sin 0 0~'1 5p = J+5~, - ~P--~oj
N
0'1
, when qbt =fl g(
mod ~r).
(12)
Oqt,2
that is to say: 0o7, ] ~fp=J+tf(--~L_~V0+~
:/:~(modlr).
(13)
It is then clear from Eq. (13) that only the third and fourth elements of the gradient vector Td 8~, will be involved in the reconfiguration. Also, only the second and third cornponents of vector 5p will be affected.
4.2 Singularity Avoidance
The system is singular when q3 = qbt = 2( mod 7r). In order to avoid this singularity we can choose the following function P: 1 P = " Iqb, J"
(14)
is an even function of qbi that tends to infinity when qbi increases to ~ or decreases to - ~ . Furthermore P has a unique minimum in zero. Computing its gradient leads to the quasi-coordinates differentials, according to the previous equations. Here are presented some simulations obtained with different choices for ~ (see Fig. 3). The end-effector has to follow the vertical axis, from the origin to the point (0,7). The first simulation corresponds to the solution 5p = J+5~. The path obtained for the platform is a cycloid. The second simulation has been obtained with the function P detailed in this paragraph. Note on obstacle avoidance: from Eq. (13), we can easily catch that we cannot use the linear velocity generation to avoid obstacles. In fact, with the proposed scheme, obstacle avoidance might be possible using 0~ Yet, it is still a perspective of our work. 2p--'r'~oj is a projection on Null(J) as g P r o j = O.
Current Advances in Mechanical Design and Production, MDP-7
65
5.CONCLUSION In this work, we proposed a pseudo-inversion scheme. We explained how to use a secondary task in the generation of the configuration for a mobile manipulator with a nonholonomic platform. Wi.th the help of a planar example we show that this method may have a limited range of application depending on the size of the null-space of the jacobian matrix, that's to say on the redundant degrees of freedom of our system. We apply this scheme to show how to avoid singularities in our exemple. The case of obstacle avoidance is a perspective of our work: this particular problem requires a specific approach as a secondary task doesn't allow any type of action on configurations. REFERENCES 1.
2. 3.
4.
5.
6.
Seraji, H., "An On-line Approach to Coordinated Mobility and Manipulation", 1993 IEEE International Conference on Robotics and Automation, Atlanta, USA, pp. 28-34, May (1993). Seraji, H., "A Unified Approach to Motion Control of Mobile Manipulators", The International Journal of Robotics Research, Vol. 17, No. 2, pp. 107-118, February (1998). Wang, C.-C., and Kumar, V., "Velocity Control of Mobile Manipulators", 1993 IEEE International Conference on Robotics and Automation, Atlanta, USA, pp. 713-718, May (1993). Nassal, U. M., "An Approach to Motion Planning for Mobile Manipulation", 1994, IEEE/RSJ International Conference on Intelligent Robots and Systems, Munich, Germany, pp. 831-838, September (1994). Pin, F. G., Hacker, C. J., Gower, K. B. and Morgansen, K. A., "Including a Non-Holonomic Constraint in the Full Space Parametrization Method for Mobile Manipulators Motion Planning", 1997 International Conference on Robotics and Automation, Albuquerque, USA, pp. 2914-2919, April (1997). Foulon, G., Fourquet, J-Y. and Renaud, M. "Control of a Rover-Mounted Manipulator", in Lecture Notes in Control and Information Sciences 232, Eds. A. Casals, A. T. de Almeida, Experimental Robotics V, Springer, ISBN 3-540-76218-3, pages 140-151, (1998).
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7.
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Yamamoto, Y., "Control and Coordination of Locomotion and Manipulation of a Wheeled Mobile Manipulator", Ph.D. Dissertation, University of Pennsylvania, (1994). 8. Fourquet, J-Y., and Renaud, M., "Coordinated Control of a Non-Holonomic Mobile Manipulator", 1999 ISER International Symposium on Experimental Robotics, Sydney, Australia, pp. 115-125, March (1999). 9. Foulon, G., Fourquet, J-Y. and Renaud, M., "Planning Point to Point Paths for Nonholonomic Mobile Manipulators", 1998 IEEE/RSJ International Conference on Intelligent Robots and Systems, Victoria, Canada, pp. 374-379, October (1998). 10. Perrier, C., Dauchez, P. and Pierrot, F., "A Global Approach for Motion Generation of Non-Holonomic Mobile Manipulators", 1998 IEEE International Conference on Robotics and Automation, Leuven, Belgium, pp. 2971-2976, May (1998). 11. Khatib, O., Yokoi, K., Chang, K., Ruspini, D., Holmberg, R., and Casal, A.,"Coordination and Decentralized Cooperation of Multiple Mobile Manipulators", Journal of Robotic Systems, Vol 13, No. 11, pp. 755-764, (1996). 12. Nagatami, K. and Yuta, S., "Door-Opening Behavior of an Autonomous Mobile Manipulator by Sequence of Action Primitives", Journal of Robotic Systems, Vol. 13, No. 11, pp. 709-721, (1996). 13. Sugar, T. and Kumar, V., "Decentralized Control of Cooperating Mobile Manipulators", 1998 IEEE International Conference on Robotics and Automation, Leuven, Belgium, pp. 2916-2921, May (1998). 14. Campion, G., Bastin, G. and D'Andr'ea-Novel, B.," Structural Properties and Classification of Kinematic and Dynamic Models of Wheeled Mobile Robots", IEEE Transactions on Robotics and Automation, Vol. 12, No. 1, pp.47-62, February (1996). 15. Gorla, B. and Renaud, M.," Modelling of Robot Manipulators. Control Applications", to be published (1999). 16. Neimark, Ju. I. and Fufaev, N.A., "Dynamics of Non-Holonomic Systems", Translations of mathematical monographs, Vol. 33, AMS, (1972). 17. Foulon, G., Fourquet, J.Y. and Renaud, M., "Coordinating Mobility and Manipulation Using Non-Holonomic Mobile Manipulators", Control Engineering Practice, Vol. 7, no 3, pp 391-399 March (1999). 18. Nakamura, Y., "Advanced Robotics, Redundancy and Optimization", Addison-Wesley Publishing, (1991). 19. Maciejewski, A. and Klein, C. A., "Obstacle Avoidance for Kinematically Redundant Manipulators in Dynamically Varying Environments", International Journal of Robotics Research,Vol. 4, no 3, pp 109-117, (1985).
Current Advances in Mechanical Design and Production Seventh Cairo UniversiO' International MDP Conference Cairo, February 15-17, 2000
67
FUZZY GUIDANCE CONTROL FOR A MOBILE ROBOT
Gharieb, W.* and Nagib, G.** * Assistant Professor, Computer and Systems Engineering Dept., Faculty of Engineering- Ain Shams University, 1 EI-Sarayat st., I 1517 Abbassia, Cairo, Egypt E-mail:
[email protected], Fax: (202) 2850617 ** Assistant Professor, Electrical Engineering Dept., Faculty of Engineering, Cairo University - Fayoum Branch, Fayoum, Egypt, Fax: (202) 084- 334031
ABSTRACT This paper presents a simple fuzzy guidance control methodology for a mobile robot. The aim is to navigate its motion in unknown environment. The different navigation problems such as goal seeking and obstacle avoidance are discussed. Every navigator is designed as a separate fuzzy system to compute the recommended steering angle and linear velocity independently. In goal seeking, the distance between the robot position and aim point is used to compute the recommended linear velocity. While the deviation angle to move toward the aim point is used to determine the recommended steering angle. The capturing information from sensors computes these variables to avoid fixed obstacles in the robot working area. Then, the minimum risk criterion detects the final decision to be taken. The simulation results using MATLAB 5.0 affirmed that the proposed control methodology has the potential to navigate in unknown environment and to achieve the goal objectives. KEYWORDS Fuzzy Control, Guidance Systems, Non-linear Systems, Robotics 1. INTRODUCTION Autonomous mobile robots are required to navigate in more complex domains, where the environment is uncertain and a reactive capacity in their navigation system is required. This implies a strong dependency on sensed information about the robot's environment [ 1]. Some researchers have been applied different digital signal processing algorithms to treat the gathered information from the sensors. Others have been used the virtual and associative memory to make the fuzzy controller more robust with respect to temporary loss of sensorial information. While others, have been applied learning techniques to learn the fuzzy behavior from the sensed numeric data [1-4]. Most control techniques depend on a supervisory level to compute the recommended (desired) values for the feedback control loops [5-9]. In real time applications, vision cameras or ultrasonic sensors are used to measure required environmental information. However, these sensors provide an imprecise information due to noise, poor reliability and if perception data is lost. It is known that fuzzy logic control has the ability to
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Current Advances in Mechanical Design and Production, MDP-7
cope with imprecise information in heuristic rule based knowledge without need of a mathematical model [10-12]. Even, if the model is available it will be non-linearandnot easily to be controlled using classical control techniques. Therefore, fuzzy control methodology is able to handle the robot guidance problem with imprecise data provided by ultrasonic sensors and can provide fast navigation. The paper is organized as follows: section 1 presents the motivation to design a fuzzy guidance control. Section 2 demonstrates the concept of fuzzy logic control design. While section 3 is devoted to the development of fuzzy guidance control design. In section 4, A simulation study on a simplified dynamic model for a mobile robot is demonstrated and the obtained results are presented. Some concluding remarks given in section 5 end the paper. 2. FUZZY LOGIC CONTROL Fuzzy logic is developed by Lotfi Zadeh to deal with uncertainty in system representation [10]. This logic has found a variety of applications in various fields ranging from sensors, motors, steam turbines, intelligent controllers, to medical diagnosis and decision making. Mamdani and his research group developed the first attempts to control industrial processes using fuzzy logic [11 ] where ill-defined processes can be satisfactorily controlled using IFTHEN fuzzy rules. Fuzzy logic controller bases its decision on inputs such: (error, variation of error, ..., etc) in the form of linguistic variables derived from membership functions which are used to determine the fuzzy set to which a value belongs and its degree in that set. The variables are then matched with the IF-THEN rules, and the response of each rule is obtained through fuzzy inference. The response of each rule is weighted according to the confidence or degree of membership function of its inputs, and the centroid of responses is computed to generate the appropriate controller output.
Membership
Membership
il
Functions ....Functions R If-ThenFuzzyRules
_
I
[
Crisp
J
inp.ut
t
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Fig. 1. Fuzzy Logic Control.
Generally, fuzzy logic controller consists of four main blocks as in Fig. l:fuzzification, knowledge base, fuzzy inference and defuzzification. Fuzzification block maps crisp inputs into fuzzy sets. Knowledge base includes input/output membership functions and If-Then fuzzy rules. Fuzzy inference computes the corresponding fuzzy decision (action). Defuzzification block maps the output fuzzy action to a crisp output value. This task is achieved by using the output membership functions [ 12].
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3. FUZZY GUIDANCE CONTROL Our goal is to design a simple guidance controller to navigate the robot motion in unknown environment. The robot starts to move in x-y plan from a starting point (0,0) toward the aim point (xa, ya) in avoiding any static obstacle as shown in Fig. 2.
When the robot reaches the aim point, it will be stopped. The robot linear velocity (v) and its direction angle (h) are controlled to achieve the required goal. The recommended steering angle controls the direction angle of robot motion, while the input voltage to a DC motor controls the linear velocity. That means, the guidance controller adds self-autonomy (machine intelligence) behavior for the robot motion. It can plan, reason and perform control objectives without human intervention. The robot has a set of ultrasonic sensors in left, right and front sides to detect the obstacle position in his way toward the aim point. 3.1. Motion dynamics The robot motion is constrained to be in a forward direction, that means, it is not allowed to move back. Ultrasonic sensors are fixed on front, right and left sides to measure the distance of nearest obstacle. Motion dynamics is similar to a moving car and the updating algorithm in [6]. The simplified model is described by: dO
Ov+~,-~-=0~;
dh x3 -~- =0
dv
+'1;2 " ~ = Vr
(1) Where: 0 v
h "~I ,'1;2 1;3
Steering angle, Or Recommended steering angle Linear velocity vr Recommended linear velocity Direction angle (integration of the steering angle) Time constants for steering and velocity servo systems (x~<x2). Time constant for direction angle.
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Current Advances in Mechanical Design and Production, MDP-7
Robot position can be determined by: Vx = v cos(h) ; Vy = v sin(h)
f
(2)
t t Xm(t)= j'v x dt ; Ym(t) = j'Vy dt 0 0 v,, Velocity component in x-axis direction. Vy Velocity component in y-axis direction. Xm,Ym X-Y axes robot position.
Where:
Distance to aim point is computed as d-~(x
a
_xm)2+(ya_Ym)2
(3)
Desired robot direction angle (~) is modified according to robot position as follows:
(4)
~ = tan-l( y-a ) X- xym am
The above equations indicate that the motion dynamics is non-linear and there is a coupling effect between the velocity and recommended direction angle. Where (~-h) is the deviation angle to move toward the aim point. Variables (d) and (~) are time dependent signals to be controlled at each point a decision has to be taken.
3.2. Navigation problem Fuzzy guidance controller is able to resolve the following navigation problems in unknown environment including fixed obstacles: goal seeking, front obstacle avoidance, right obstacle avoidance and left obstacle avoidance as in Fig. 3.
Aim point _[
i
~~
Goal
seeking Direction _angle
Rightsen~_v~
Right obstacle avoidance
21. Front obstacle avoidance
Front sen
Left sensors ! ~ '
Decision making
:>
Mobile robot
Recommended Steer angle & Linear velocity
"~ Left
obstacle avoidance Fig. 3. Fuzzy guidance control.
I-- -PLinear velocity
Current Advances in Mechanical Design and Production, MDP-7
71
Each navigator is a task oriented control algorithm and designed as a separate fuzzy system to compute the recommended steering angle and linear velocity independently. In goal seeking, the distance between the robot position and aim point (d)is used to compute the recommended linear velocity (vr). While the deviation angle (~-h) to move towards the aim point is used to determine the recommended steering angle (0~). In obstacle avoidance, the capturing information from sensors determines the recommended steering angle, while the recommended velocity is reduced to half in order to decrease the robot inertia. Finally, the minimum risk criterion detects the applied control to the robot and the final decision will be taken as illustrated in Fig. 3.
In goal seeking, the crisp input to the fuzzy system is the distance (d) to compute the recommended velocity. Input membership functions are triangular in form (VC, CL, SM, ME and LA). They are very close, close, small, medium and large respectively. Output membership functions are singleton (ZE, VS, SM, ME, LA). They are zero, very small, small, medium and large respectively. Fuzzy rules are very simple as: If d VC Then v~ ZE If d CL Then vr VS If d SM Then v~ SM If d ME Then vr ME If d LA Then Vr LA The recommended steering angle Or is computed as function of the deviation angle (~-h). The input membership functions are triangular in form (NL, NS, ZE, PS and PL). They are negative large, negative small, zero, positive small and positive large respectively. The output membership functions are singleton (LH, LF, ZE, RG and RH). They are turn left hard, turn left, turn zero, turn right and turn right hard respectively. The fuzzy rules are: If (~-h) ZE Then Or ZE If (~-h) PS Then Or LF If (~-h) PL Then Or LH If (~-h) NS Then Or RG If (~-h) NL Then Or RH
In obstacle avoidance, the measured distances RS, LS and FS are fuzzified using membership functions CL, NE and FA. They are Close, Near, and Far respectively. The visual zone for front side is a symmetric 30~ Right obstacle avoidance rules: If RS CL Then If RS NE Then If RS FA Then Left obstacle avoidance rules: 9If LS CL Then If LS NE Then If LS FA Then Front obstacle avoidance rules: If FS CL and If FS CL and If FS NE and If FS NE and If FS FA Then
Or Or Or
LH LF ZE
Or Or Or
RH RG ZE
RS LS RS LS Or
FA FA FA FA ZE
Then Or Then Or Then Or Then Or
RH LH RG LF
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Current Advances In Mechanical Design and Production, MDP- 7
When the robot sensors detect the front obstacle only, either right or left steering can be taken. The decision depends on the situation and which is much better to be taken. In decision making, Each fuzzy system produces its output, then the decision making block
has to select the values of recommended variables. The minimum risk criterion aims to avoid the nearest obstacle by computing the minimum distance as follows:
(5)
J = Min(LS, RS, FS, d) The recommended steering angle is selected as" If J=LS Then Left obstacle avoidance If J=RS Then Right obstacle avoidance If J=FS Then Front obstacle avoidance If J=d Then Goal seeking 4. S I M U L A T I O N W O R K
The robot motion is simulated on a PC using MATLAB 5. The sampling time is chosen as 0.1 see., xl=l see., x2 = 0.5 see. and x3 = 1 see. Input membership functions are given in Fig. 4. _.__
0 V
0.5
FA
1 PL
LA d(m)
(,-h) p~
0
0.1
0.5
1
2
-90
-30 0 30
90
Fig. 4. Input membership functions. While the output membership functions are as shown in Fig. 5. Z
VS
SM
ME
LA
RH
RG
E
LF
LH
9 ...............................................................
0 0.01 0.1
1).3
(1.5
. 3 0 ............. i 5
............
Fig. 5. Output membership functions.
..........
.............
~
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Oarrent Advances in Mechanical Design and Production, MDP-7
First results are obtained to show the goal seeking navigation without obstacles as in Fig 6. The robot starts to move from (0,0) toward the aim point (10,10). 12 1(
y-axis in meters
,
/
,/
x-axis in m e t e r s i
-2
I
0
I
2
,
I
4
,
I
6
,
I
8
10
12
Fig. 6. Goal seeking in working plan (no obstacles).
The obtained result shows that the goal seeking is achieved. Next results show the robot navigation to avoid obstacles at (2,2) and (6,4). We observe that the proposed controller has been succeeded to avoid obstacles and reached to the aim point within a finite time. The sharp turn to avoid the second obstacle is due to the high-speed value at this point. 1; 1(
y-axis in meters t"
/
/
/
0
x-axis in m e t e r s !_
!
-2
l,
0
,
I
2
I
4
I
6
,
8
I
12
10
Fig. 7. Robot navigation to avoid obstacles.
Next figures show the velocity and heading angle curves during obstacles avoidance. O
. ~
.
O1
,
150
.............
Robot velocity
............. Heading angle
10
i" 0
5 seconds |
2O 40 60 Fig. 8. Linear velocity.
,
0
20
,rn-e,n secon,,,, 40
Fig.9. Heading angle.
60
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Current Advances in Mechanical Design and Production, MDP- 7
We observe that the robot velocity is reduced to half two times while it was trying to avoid obstacles. Robot reached the aim point and is stopped after 63 seconds. The robot starts to move with 0 heading angle. It turned left towards the aim point then it turned right to avoid the first obstacle and then turned left to avoid the second one. Finally, it turned right and reached with a steady state 37 degrees heading angle. 5. CONCLUSIONS This paper presents a simple fuzzy guidance control to navigate a mobile robot in uncertain environment. The design objectives are broken down to implement each one independently. Then, the final decision is taken based on which one of all navigation problems has the minimum risk to avoid obstacles. Each navigator is designed using a simple fuzzy system (single-input single-output). The obtained results by simulation affirmed that the proposed control algorithm has the potential to achieve the control objectives such as: goal seeking and obstacle avoidance. In the future work, path planning will be considered as well as the dynamic obstacles avoidance. Also, coupling problem between steering and velocity control loops will be treated to improve the system performance. REFERENCES 1. Arrue, B.C., Cuesta, F., Braunstingl R.and Ollero A., "Fuzzy Behaviors Combination to Control a Non-Holonomic Mobile Robot Using Virtual Perception Memory", Proc.of 6th IEEE Int.Conf. on Fuzzy Systems, vo1.(3/3), PP. 1239-1244, Spain, July, 1-5, (1997). Garcia-Gerezo, A., A-Mandow and Lapez-Baldan, M. J., "Fuzzy Modelling Operator Navigation Behaviors", Proc.of 6th IEEE Int.Conf. on Fuzzy Systems, vol. (3/3), PP. 1339-1345, Barcelona, Spain, July, 1-5, (1997). 3. Yoichiro Maeda, "Beahvior Learning and Group Evolution for Autonomous Multi- Agent Robot", Proe. of 6 th IEEE Int. Conf. on Fuzzy Systems, vol. (3/3), PP. 1355-1360, (1997). 4. Urzelai, J., Uribe, J. P. and Ezkerra, M.," Fuzzy Controller for Wall-Following with a Non-Holonomous Mobile Robot", Proc.of 6th IEEE Int.Conf. on Fuzzy Systems, vol. (3/3), PP. 1361-1368, Barcelona, Spain, July, 1-5, (1997). 5. Steven G. Goodridge, Michael G. Kay and Ren C. Luo, "Multi-Layered Fuzzy Behavior Fusion for Reactive Control of an Autonomous Mobile Robot", Proc. of 6th IEEE Int. Conf. on Fuzzy Systems, vol. (1/3), PP. 579-584, Barcelona, Spain, July, 1-5, (1997). 6. Vaneck, T. W.,"Fuzzy Guidance Controller for an Autonomous Boat", IEEE Control Systems, vol. 17, n~ 2, PP. 43-51, (1997). 7. Bing-Yung Chee, Sherman Y. T. Lang and Peter W. T. Tse, "Fuzzy Mobile Robot Navigation and Sensor Integration", Proc. of 5th IEEE Int. Conf. on Fuzzy Systems, vol. (1/3), PP. 7-11, New Orleans, USA, Sept., 8-11, (1996). 8. Sng Hong Lian, "Fuzzy Logic Control of an Obstacle Avoidance Robot", Proc. of 5th IEEE Int. Conf. on Fuzzy Systems, vol. (1/3), PP .26-30, USA, Sept., 8-11, (1996). 9. Debay, P., Eude, V., Said Hayat and Edel, M., "Fuzzy Control for the Future Automatic Guidance Near the Bus Stations", Proe. of 5~ IEEE Int. Conf. on Fuzzy Systems, vol. (1/3), PP. 660-666, New Orleans, USA, Sept., 8-11, (1996). 10. Zadeh, L. A., "Fuzzy Sets", Inform. Contr, Vol.8, pp. 338-353, (1965). 11. Mamdani, E. H., and Assilian, S., "An Experiment in Linguistic Synthesis with a Fuzzy Logic Controller", Int. J. Man-Machine Studies, Vol. 7, pp.l-13, (1975). 12. Lee, C. C., "Fuzzy Logic in Control Systems: Fuzzy Logic Controller-Part I & II", IEEE Trans., Syst. Man, Cybern., vol. SMC-20, n~ 2, pp. 404-435, (1990). .
Current Advances in Mechanical Design and Production Seventh Cairo University htternational MDP Conference Cairo, February 15-17, 2000
75
FUZZY LOGIC SLIDING MODE CONTROLLER FOR DC DRIVE
Ibraheem, A.A.', Bahgat, A.'" and Abdel Motelb, M.S.* "Electronics Research Institute, National Research Center, Dokki, Cairo, Egypt "Faculty of Engineering, Cairo University, Giza, Egypt,
ABSTRACT Sliding mode control is used now in the speed control of electric drive systems. It provides attractive features such as fast dynamic response, insensitivity to variations in plant parameters and external disturbance. However, chattering is one problem, which limits the use of the sliding mode to control DC motors in industry. This paper presents a new fuzzy sliding mode controller for the speed control of DC motor drives. The fuzzy controller is used to adjust the switching gain constant in the sliding mode controller to guarantee the system stability and practically to eliminate the chattering problem. The performance of the proposed system is compared with that of traditional sliding mode under different operating conditions. KEYWORDS Fuzzy Logic Control, Sliding Mode Control, Robust Control, Speed Control of DC Motor 1. INTRODUCTION The DC motors have been commonly accepted in industrial applications, where fine speed adjustment is needed. Some of applications, such as steel rolling mills and paper mills, are characterized by fast and large changes in operating conditions. Electric traction systems require high starting torque. In other applications such as computer numerical control (CNC) precision speed pattern has to be followed with minimum absolute error. Variable structure system theory has gained attention over the past decade as the method may offer robustness in the control of electric drives and power electronics [1-4]. In the variable structure system control (VSC), the system response is forced onto a predetermined sliding surface and towards a pre-specified point in the state-space, using predetermined switching gains and variables. The switching surface is solely defined by parameters that are independent of the plant model, so the sliding mode dynamics are completely insensitive, or invariant, to bounded plant parameter changes. Under certain conditions, the sliding mode dynamics are also invariant to bounded external disturbances. Included in the basic design approach are the specifications of the switching surface and the design of a control input that meets the reaching conditions, which declare that the control always drives the state onto the switching surface [5-8]. A disadvantage of the sliding mode controllers is that the discontinuous signal produces chattering dynamics, so that the control system can be switched from one value to another in very fast way. In practical systems, it is impossible to achieve the high switching control that
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is necessary to realize most of VSC designs. There are several reasons for this. One reason is the presence of finite time delays for control computation. Another reason is the physical limitations of actuators [9]. Different schemes [4, 10, 12] have been proposed in literature to eliminate chattering. However, these do not solve the problem completely. One of these schemes is to define a boundary layer in the vicinity of the sliding surface and then use a continuous approximation of the switching function in the layer. Although chattering can be eliminated by the use of boundary layer, a loss of robustness can result. The main objective of this paper is to present a new VSC with fuzzy logic, which is used to adjust the switching control gain of the sliding mode. The Fuzzy Variable Structure Controller (FVSC) is simulated using the Matlab. The speed control of a DC motor has been tested under different loading and operating conditions using the developed FVSC. 2. SLIDING MODE CONTROL Behavior based sliding mode control (or variable structure system) control (SMC) is a robust one that depends very little upon the model of the plant to be controlled. The system response in the phase-plane is forced onto a predetermined "sliding surface" and towards a prespeeified point in state-space, using predetermined switching gains and variables as shown in Fig. 1, where x is the state and x is its derivative. In position control, x represents the position error and x" represents speed, whereas in speed control, x presents speed and x presents acceleration. And the controlled system response is insensitive to parameter of the motor and depends only on the slope of the sliding surface s ( x , t ) [4].
S<0
I x.
S >0
X
S =0
S >0
I
S
Fig. 1. Sliding mode state trajectory in phase-plane The design of the sliding mode controller is to obtain the state x for tracking presence of model uncertainties and disturbances. With tracking error e = x(t) - xa(t ) =
(e,e',...,e<"-')) r
xa
in the
(1)
a sliding surface s(x,t) = (d/dt +
2) ~
(2)
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is defined starting from the initial conditions where 2 is a strictly positive constant that can be interpreted as the slope of the sliding surface and n is the order of the controlled system. The tracking problem x = x d can be considered as the state vector e remaining on the sliding surface s(x,t) = o for all t > 0. s(x,t) = 0
(3)
when e(O)= 0
( x = x~ )
The sufficient condition for this behavior is to choose the control value so that [5],[7] 1 d (s2(x,t)) < _qls I "~-~ -
.
'
rI > 0 -
(4)
Considering s 2 (x,t) a Lyapunov function, it follows from (4) that the controlled system is stable. From phase plane, we obtain that the system is controlled in such a way that the state always moves towards the sliding surface. The sign of the control value u must change at the intersection of the state trajectory e(t) and sliding surface. In this way, the trajectory is forced to move always towards the sliding surface, then the existence and convergence condition can thus be re-writing as: Then
s.s<_ -q.lsl
s.sgn(s) < - r /
(5)
3. DYNAMIC MODEL OF THE CONTROLLED DC-MOTOR The schematic block diagram of the separate excited DC motor fed from three-phase full converter-bridge is illustrated in Fig. 2. The three-phase converter-bridge controls the direct current of the armature and the field voltage is kept constant.
4
L
Constant Field Voltage
Three Phase AC Supply
Three Phase Full-Wave Thyristor Converter
Fig. 2. D C m o t o r fed f r o m 3 - p h a s e full w a v e T h y r i s t o r B r i d g e
The electromechanical equations of the drive are given by the following equations:
r = T, + 8co + j d c o dt
(6)
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Current Advances in Mechanical Design and Production, MDP-7
T = K,,~i,,
(7)
d va = Eb + La -'~ia + Raia
(8)
Eb = Ko~co
(9)
where va : io 9 B 9 J : T : T~ : Ko: O: to :
Armature Voltage Armature Current Viscous Friction Constant Moment of Inertia Motor Torque Load torque Motor Armature Constant Flux Motor Speed
Figure 3-a represents the block representation of the motor and the electronic amplifier with the speed controller in form of a sliding mode. The equivalent dynamic block representation is given in Fig. 3-b.
3 ~ACS~
3 I I
"-
ICI~'
Tscke-lpmer~or (a)
(b)
Fig. 3. Block Diagram of a Controlled DC motor 4. SLIDING MODE CONTROLLER DESIGN The control law u (control voltage of the firing circuit) of the sliding mode controller is given by: u = t~ - k sgn(s)
with k>O
(10)
such that k is the switching gain constant and the sliding line s is defined as a function of the dco error signal (e = -x~ = co - co,,/) and the error derivative ( e = x 2 = - - ~ ) .
Current Advances hz Mechanical Design and Production, MDP-7
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(11)
s = 2e+e = (co"~ef - 2 e
- f)
(12)
,JL o
K,,~
Such that f is the compensation term defined as:
(13) f:-
Z - . + 7 o., +
t,oJ
) JK,#
Since the control law (10) is a discontinuous across the sliding surface, it might lead to chattering. Thus the S a t ( s ) function is replaced instead of s i g n ( s ) function for improving the sliding mode controller characteristic [5-7]. (14)
u = fi - k * s a t ( s )
The sliding line slope 2 is taken to be 5 and therefore the sliding line time constant is (0.2 Sec.) and the control switching gain is chosen such that the existence condition in [5-7] given in (5) is always satisfied. The schematic diagram of Simulink program used for simulation of the DC motor, three-phase controlled Thyristor Bridge, and the sliding mode controller is shown in Fig. 4.
I
Torque
v.
~[
Lo~ Tortl~
~
""~1
A.SpeedOol
Nmse
~eT~o,.
[.... i SI I "~i
x
~s)
~...,'mm,d
,..o.o-2, i I
il ~
[
I
~ ~-~~ Sum 1'
Fig. 4. SIMULINK model of Sliding Mode controller 5. DESIGN OF THE FUZZY SLIDING MODE CONTROLLER In the fuzzy sliding mode controller (FSMC) scheme the sliding surface s ( t ) forms the input space of the fuzzy implication of the major switching rule and the switching gain is written in the form of fuzzy rule, given by:
Current Advances in Mechanical Design and Production, MDP-7
80
I F s is Aj and s" is B j T H E N K is C j
where j = 1.... n and n is the number of rules and s(t) is the sliding surface, and the major principal of a FSMC can be represented by the following equation
(15)
u = -KF,,,,(e,e,2).sgn(s )
such that the control gain KF=:z (non-linear, non-continuous and positive function of error e, derivative of error e and 2 ) is driven according to the following rule: "above the switching line a negative control output is generated and a positive one below it (ANBP), where close to the sliding surface the control outputs are smaller than at a larger distance" [13, 14]. The fuzzy input membership functions (on a normalized range of 1) are illustrated in Fig. 5, and the rule base in the fuzzy association matrix (FAM) is shown in Table 1. The output membership functions "N", "Z", and "P" correspond to fuzzy singletons a t - l , 0, and 1, respectively. The composed output is calculated using center of sum approach. NB
I
-1
NM i~ISAZ PS PM
I~B
PB
-0.4-0.2 0 0.2 0.4
1
NM NS
?VV
'1
-0.4 -0.2 0 0.2 0.4
J
Fig. 5. Input membership function of FSMC Table 1. FAM matrix of FSMC
O
rp~
!pB PM PS AZ NS NM NB
NB p N N N N N N
NM P P N N N N N
NS p p p N N N N
S ZE P P ....p Z N N N .,
PS P P P P P N N
PM P P p P P N N
PM P P P P P P N
6. SIMULATION RESULTS STUDY To indicate the effectiveness of the developed FSMC three cases are studied: 1. Transient response in case of motor starting without load 2. Sudden load change 3. System parameter variation
1
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Current Advances in Mechanical Design and Production, MDP-7
In addition a comparison between the FSMC and SMC regarding chattering phenomena is studied. Table 2. Show the parameters of the DC motor for simulation study. Table 2. Parameters of the DC motor
175 Watt rated power 220 V rated armature voltage 1.3 A. Rated armature current 1500 Rated motor speed Armature resistance Armature inductance Voltage constant L Torque constant Moment, , _ of inertia constant Viscous fraction constant
26.6 0.1h 1.47 V/rad/sec. 1.47 N.m./amp 0,0041-kg~.m2 0.00i9 m-/Sec. ~
Figure 6 shows the motor speed in case of starting at no load using each of the two controllers SMC and FSMC. Figure 7 shows a zoom into motor speed at the instant of sudden load change at t =1.5 seconds. The corresponding armature current responses are illustrated in Fig. 8. It is clear from these figures that the speed response is relatively faster to reach a steady state value and has a good disturbance rejection capability for load disturbance in case of FSMC compared to the case of SMC. 1540
12001400 / f
(__
"-"-'-"
;s.c I
.~ IOOO'
I
15.10 1520 1510
Fs.cl
. SMC ]
LoadingIn8tant -: ............r....... l . ~ o ~ . . , ~ "
1480 1470
0
0
--
,!
0'5 Time (See)
Fig. 6. Motor speed transient of the SMC and FSMC
1440 i
"
215. . . . . . ~me ( $ e c )
2
Fig. 7. Zoom in to loading instant of the SMC and FSMC
Figure 9 shows the motor speed and the armature current in case of starting at no load using SMC controller for two different values of motor load inertia. Figure 10 shows the motor speed and armature current in case of starting at no load using FSMC controller for the same study case. It is clear from these two figures that both controllers are robust to system parameter variation, but the peak current drawing in case of SMC is greater than that in case of SMC. Figure 11 shows the motor armature voltage for both controllers SMC and FSMC. It is clear from this figure that the control signal (armature voltage) in case of FSMC is smoother than that of the SMC, because the fuzzy logic controller (FLC) provides better damping and reduces chattering. The oscillations of the armature voltage in case of SMC controller may
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Current Advances in Mechanical Design and Production, MDP-7
cause damage for the motor drive and the control equipment. The amplitude of these oscillations can grow up when increasing the gain in order to reduce the rise time as FSMC. ,-~ 1500 r
2.S
'
~.~-""~
......................
- ? ..............................................
SMC
I ~
FSMC I
~
E
o. . . . 0
0.2
0.4
.
-;
4
o.e
0.8
;-.
ii
9
.
1 "rim, (s,r
.
--
9
1.2
,;. . . .
.
1.4 9 .
-
1.6
1.8
~
.
2 --
.o.o~w~~.. In.,ti(S.C) .
I. ~
Motor Wdh Dol~le Inelli. (SMC, ]
0.$
O0
015
Fig. 8. Armature current transient of the SMC and FSMC
|,oooI./ /
, ,_
o,
~~
'r,m, (s,r
..w.,o__..o
..4
0.2 .
0.4 .
0.6 .
.
0.8
! (S*r
.
-.
o
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o,
"
oe
"
0.,
;
1.2
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,
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,.,
;~
01,,
;
,:,
" 1 ~ (S,r
1:,
6,
11,.
=
I 2
O~ 0
250
. 0.2
.-
,
,i,
2
Fig. 10. Motor speed and armature current of the FSMC for different motorload inertia
0
.
0.4
~
.
0.2
0.4
i'!V ~0
0,
Fig. 9. Motor speed and armature current of the SMC for different motor-load inertia
':I/ 0
o,
.
. 0.6 .
.
.
......
. . . I 1.2 r u m ('S,r
0.8 .
.
.
.
........
0.6
0.8
. 1.4
1
1.2
1.4
1.6
1.8
~,
,
1.6
t.8
l J 2
2
Fig. 11. Motor armature control voltage of the SMC and FSMC
7. CONCLUSIONS This paper presents the design of Fuzzy Sliding Mode Controller for the speed control of DC motor drive. The controller is based on a sliding mode control where its switching control gain can be adjusted by a fuzzy controller. This controller not only improves the system transient response, but also yields better performance in terms of steady state-error, stability, chattering elimination and robustness criteria based on rejecting disturbance. ACKNOWLEDGMENT This research work has been conducted in the frame of FRCU project No. 205. This project is financed through USAID in Egypt and is executed in collaboration with Oakland University and Case Westem and Reserve University in USA.
Current Advances in Mechanical Design and Production, MDP-7
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REFERENCES 1. Emelyanov, S. V., "Theory of Variable Structure Systems", Moscow, Nauka, (1970). 2. Borojrvic, D., Naitoh, H, , and Lee, F. C, " Soft Variable Structure Based Adaptive PI Control for DC Motor Position Control", in Proc., IEEE, Industrial Applications Society Annual Meeting, pp. 283-288, (1986). 3. Utikin, I.V., "Sliding Mode Control Design Principles and Applications to Electric Drives", IEEE, Trans, Industrial Electronics, Vol. 40, NO. 1, Feb., (1993). 4. Habibi, S.R., and Richards, R.J., "Sliding Mode Control of An Electrically Powered Industrial Robot", lEE, Proc.-D, Vol. 139, No. 2, March. (1992). 5. Hung, J.Y., Hung, J.C., "Variable Structure Control: A Survey," IEEE, Trans, Industrial Electronics, Vol. 40, No. 1, Feb (1993). 6. Nandam, P.K. and Sen, P.C., "Accessible-State-Based Sliding Mode Control of a Variable Speed Drive System", IEEE, Trans, Industrial Appl., Vol. 31, NO. 4, Jun/Agu. (1995). 7. Slotine, J.J.F.," Applied Nonlinear Control", Prentice Hall, 2nd edition, (1991). 8. Buja, G.S., Menis, R. and Valla, I., "Variable Structure Control of An SRM Drive", IEEE, Trans, Industrial Electronics, PP. 56-79, Vol. 40, NO. 1, Feb., (1993). 9. Furuta, K., "VSS Type Self-Tuning Control," IEEE, Trans, Industrial Electronics, PP. 3744,Vol. 40, NO. 1, Feb., 1993. 10. Ackermann, J. and Utkin, V., "Sliding Mode Design", from Authors Publications WWW (e-mail:
[email protected]) 11. Spurgeon, S.K. and Patton, R.J., " Robust Variable Structure Control of Model Reference Systems," lEE, Proc., Vol. 137, Pt. D, No. 6, Nov. (1990). 12. Hung, J.Y., Nelms, N. M. and Steven, P.B., "An Output feedback Sliding Mode Speed Regulator for DC Drives", IEEE, Trans., Industrial Appl., Vol. 30, No. 3, May/June (1994). 13. Lo, J.C. and Kuo, Y.H., "Decoupled Fuzzy Sliding-Mode Control", IEEE, Trans., Fuzz., Syst., Vol. 6, No. 3, Aug. (1998). 14. Glower, J.S. and Munighan, J.," Designing Fuzzy Controllers from a Variable Structure Standpoint," IEEE, Trans, Fuzzy Sys., Vol. 5, No. 1, Feb. (1997).
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Current Advances in Mechanical Design and Production Seventh Cairo UniversiO, International MDP Conference Cairo, February 15-17, 2000
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DRIVER MODELING USING FUZZY LOGIC CONTROLS FOR HUMAN-IN-THE-LOOP VEHICLE SIMULATIONS Zeyada, Y.*, EI-Beheiry, E. **, EI-Arabi, M. ~ and Karnopp, D. ~176
*Assistant Professor, ~ Mechanical Design and Production Department, Faculty of Engineering, Cairo University, Giza 12316-Egypt. Lecturer, Department of Production Engineering and Mechanical Design, Faculty of Engineering, Menoufia University, Shebin EI-Kom, Egypt. oo Professor, Mechanical, Aeronautical and Materials Engineering Department, University of California at Davis, CA 95616, USA.
ABSTRACT It is a universal trend that automotive engineers tend to use fast digital computers to develop advanced vehicle systems. They can design, analyze, and test their systems using computer simulation before physically manufacturing them. It is still questionable how these advanced systems would react with the human driver. One way to deal with this problem is to develop a computer model that is capable of controlling the vehicle in a way similar to human driver behavior. Fuzzy logic inference systems are known of their great ability to simulate human reasoning process as well as the possibility of being further trained to mimic specific human control process. This paper presents a new driver model using fuzzy logic controls. The model is designed to control the longitudinal as well as the lateral motions of the vehicle by performing simultaneous steering and braking commands. The model is tested on a vehicle model having an integrated active steering and direct yaw control strategy as developed by this paper's authors in [1]. The results show success of this fuzzy model in simulating driver control actions in curve following and collision avoidance maneuvers. KEYWORDS Driver Modeling, Vehicle Dynamics, Fuzzy Logic, Collision Avoidance, Lane Following. I. INTRODUCTION One of the methods to include the human behavior in vehicle dynamics simulation is to design a driver model capable of performing control task close enough to what an average driver would behave. For such a task, the use of fuzzy inference systems is strongly recommended. This is due its ability to create a decision-making process based on the logical analysis of the system inputs, which suits the way of human analytical thinking procedure, i.e., they can map those functions that have no equivalent mathematical model or whose mathematical models are very complicated, s e e Sugeno [2]. Most recent developments in the application of the neural networks (NN) and fuzzy logic (FL) to vehicle systems have been summarized in a review article by Ghazi Zadeh et al. 1997 [3]. Wuertenburger and Isermann [4] have presented
Current Advances in Mechanical Design and Production, MDP- 7
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supervision of lateral vehicle dynamics by using a model-based scheme. The model is able to monitor the lateral motion of the vehicle utilizing an estimation of the road friction coefficient as determined by a driving state observer. Four wheel steering controllers were studied by Nagai et al [5]. The tire and the suspension nonlinearities were taken into account in the identification of the vehicle motion by integrating a NN parallel to a linear controller to improve overall performance of the controller. A driver model has been presented by Kageyama and Pacejka [6] which utilizes the so-called risk level concept for the course decision making process and a FL controller for the course tracking. A speed controller and steering torque controller are also integrated within the system. A feedforward neurocontroller with one hidden layer was employed to emulate driver's behavior by Nuesser et al. [7]. It has the vehicle speed, yaw angle, road curvature, road width, and lateral deviation of the vehicle from the path as inputs. The model returns the steering angle as an output. A FL driver utilizing yaw velocity and lateral displacement signals to produce the steering angle has been Studied by Xi and Qun [8]. This article is mainly devoted for developing FL model of human driver behavior for humanin-the loop simulations of advanced vehicle systems. The FL model not only performs driver steering inputs, as was the case in the majority of publications, but also provides human driver braking events. The model is trained to react to driving situations in both collision avoidance and curve following. 2. VEHICLE MODEL Two axes systems namely called OXYZ fixed in space axes-system and oxyz moving axissystem are used to describe a comprehensive nonlinear 14-DOF vehicle model as shown in Fig. 1. A condensed model of 4-DOF has been extracted which includes tire nonlinearities. These nonlinearities are essential to the study vehicle dynamics in emergency situations.
Sprung Mass 0
PITCH MODE
ROLL MODE
Side view
Front view
Q~~y Car front
l
!
Suspension Z Tire
X
[
o
I N 3 and N 4
x
I
N I and N 2
o
y
I N2 and N4
Y
I
Nwand Na
Fig. 1. Schematics of vehicle body in pitch and roll modes The 4-DOF nonlinear model includes the longitudinal, lateral, yaw and roll motions. No pitch, heave, or tire deflections are considered. Moreover, the wheel spin rates are determined by implying a targeted slip ratio for each wheel separately. Those slip ratios are inputs to the model and are controlled by an appropriate control system such as the (ABS) anti-lock brake
Current Advances #t ittechanical Design and Production, MDP-7
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system or the (DYC) direct yaw control system. Details of both the comprehensive and the simplified models are in Zeyada [9]. 3. FUZZY LOGIC DRIVER MODEL
3.1. Sensigg System For the frizzy driver model, a sensing system is basically a group of variables that provide simple and enough information about the vehicle position w.r.t, roadway. Consider Fig 2., where there are five rays goes out from the vehicle center in left, right, straight-ahead, straight-left, and straight-right directions. The respective lengths of those rays measured from the vehicle center to the roadway borders are indicative quantities to determine the position of the vehicle. These quantities are referred to as DL, DR, Ds, DsL, and Dsn respectively. These distances can be measured in reality by many techniques such as laser ranging or electromagnetic sensing as implemented in intelligent vehicle highway systems. The angle between Ds and DSL is equal to the angle between Ds and DSR, and is referred to as 19.
DSL
Vehicle
Direction of motion
Fig.2. Sensing road geometry for fuzzy driver 3.2. Decision-Making System The decision-making system is a part of the driver model responsible for determining the amount of steering and braking necessary to keep the vehicle in the right course. It is mainly consisting of two modules, the steering module and the braking module. Both modules will be designed using fuzzy logic systems. The system will utilize the sensing variables described above as inputs and produces the appropriate steering and braking commands.
3.2.1. The steering module Generally, the amount of steering necessary to guide the vehicle in a roadway would be generated by two mechanisms. The first is called the preview mechanism that resulted from the anticipation of incoming curve on the roadway. The second is called the deviation mechanism which resulted from the deviation of the vehicle center from the center the roadway. The steering portion contributed by the first mechanism can be summarized by the following rules:
lf DsL > DSR then left steering is required, If Dsn > DsL then right steering is required. For the deviation mechanism the following rules can be written,
If DL > DR then right steering is required, If DR > DL then left steering is required.
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Current Advances In Mechanical Design and Production, MDP-7
The fuzzy steering module is designed using the above rules. It consists of two main parts, the left steering part and the right steering part. The left steering part is focused on the sensing variables that causes a left steering command. While the fight steering part is concerned with the sensing variables that causes a right steering command. The left steering part is mainly a fuzzy inference system that utilizes two normalized inputs and produces one normalized output that represents the left steering angle required. The normalized inputs are:
[)R
=
DW ~2-DR D w 12
DSR DSR max
,andbsR=
(1)
,
where Dw is the road width and Ds~,~ is the maximum allowable value for DsR. The normalized output is, ~L = 6/./Smax, where ~,~ is the maximum allowable steering angle. The above-normalized variables are only allowed to vary in the range from 0 to 1. Figure 3 shows the membership functions of the inputs and output of the left steering fuzzy system. Table 1 shows the inference rules. Similarly the right steering fuzzy system can be handled. For a vehicle runs on the road center with no incoming curves ahead the left steering system opposes the right steering one, and consequently no net steering output is generated. Now consider the vehicle gets closer to the right shoulder (deviation case), this will cause/)R to increase while /)L is held zero. The result is a left steering command. Also, for an incoming
ll ""
0 '
'
~
'
'
0.1
0.2
0.3
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'
'
'
'
0.1
0.2
0.3
0.4
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~
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,
,
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ol 6 " 0.7 ' '"'
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'
H"
'
'.,
0fi
0.6
0.7
0.8
1
~
!o. I~ 0
...................... 0.9
1
iL
Fig.3. Membership functions for inputs and output of the left steering system
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Current Advances in Mechanical Design and Production, MDP-7
left turn (preview case) DSR will get shorter causing the system to react by producing a left steering command. The above behavior can be justified by inspecting the inputs/output surface for the left steering fuzzy system shown in Fig 4. The figure also shows how the system performs for simultaneous preview and deviation inputs. The above discussion is also applicable for the right steering fuzzy system. Notice the flat zero area (i.e. no steering output is assigned) associated with low values of/gR, it is introduced to simulate the human driver behavior of tolerating a small shift from the center of the lane. Table 1. Fuzzy inference rules for the left steering system
DSR VS
bR
S
M
H
S
vs
vs
S
M
M
vH
H
H
vH
VH
M .
.
.
.
H
s
VH "
vs
s
.
M"
Fig. 4. Input/output surface for the left steering system 3.2.2. The braking module Similar to the fuzzy steering system, the braking system also utilizes sensing parameters in order to generate the braking command. For simplicity, the system uses only one sensing variable in normalized form that's Ds. The system input is, /)s = DS/DsT, where Dsr is the dynamic stopping distance which mainly depends on the vehicle speed. It is the distance covered by the vehicle to come to a complete stop by applying maximum braking. Assuming constant deceleration and by simple manipulation Dsr can be calculated to using the following equation'
90
Current Advances in Mechanical Design and Production, MDP-7 DST
:
(2)
Uo /2aDEC ,
where, u 0 , is the vehicle initial velocity at the time of brake application, and aDE C is the deceleration rate which can be obtained from: aOE C
= F B /mt
,
(3)
where F B is the total braking force and m t is the total vehicle mass. Assuming equal vertical loads on the four wheels and ignoring the dynamic load transfer due to braking, one obtains this simple equation aDE C
:
(4)
4FB(tire )~mr,
where FB(tire ) is the braking force per tire. Using the tire model and assuming 0.15 longitudinal slip the braking force per tire is about 80% of the vertical load applied i.e.: (5)
F B(tire)= O.8(mtg/4).
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1
braking command
Fig. 5. Membership function for the braking command The deceleration rate can now be given by a DEC =0.Sg, where g is the gravitational acceleration. The output of the braking system is the normalized braking command that ranges from 0 where no braking action is taken to 1 where maximum braking is required. The membership functions are shown in Fig. 5 and the inference rules are given in Table 2. Notice that the braking system will respond only for values of Ds less than or equal three times of the expected stopping distance at the instant vehicle speed.
Current Advances #t Mechanical Design and Production, MDP-7
91
Table 2. Inference rules for the braking fuzzy system .
.
VSH
.
.
SH
ME
/-/
M
VLO
LO L
,,,
VH
4. SIMULATION, RESULTS AND DISCUSSIONS The developed driver model here is employed in this section throughout two simulated emergency maneuvers. The first driving situation simulates the case of an attempt to handle a sharp 90 degrees turn with relatively high speed at the turn entrance. Figure 6 shows the road turn, the starting point is 200 meters far from the turn start. At this point a speed of 45 m/s (about 160 km/hr) is assumed. The simulation includes the performance of a vehicle equipped with the fuzzy active-steering differential-braking control system developed in Zeyada et. al. [ 1]. Figure 7 shows the vehicle deceleration due to braking action. Figures 8 and 9 show the breaking and steering commands performed by the fuzzy driver model. Lastly, Fig 10 shows a 3-D plot for the braking command related to the vehicle position within the turn.
40 20 ~" >.
0
Start
-2.0. -40 -60
0
50
100
X (m)
150
200
250
Fig 6. The road turn used in the first simulation --
SO
E
45 4O
w o.
35 30
-~ .u
2S
"
20
>
15 10
0
0
Tim
9(s)
i'o
.......
Fig 7. Vehicle deceleration due to performed by the fuzzy driver
Current Advances In Mechanical Design and Production, MDP-7
92
"o t..-
E E o
0.6
"
0.5
,',
0.4
u
.to
"0
._'2 0 . 3 lu
E '0
(1.2
z 0.1 .,
0
|
5
Tim e (s)
10
15
Fig. 8. The normalized braking command braking
8--
1
0-81
7 6 5
.4~
3 e 1 o -1
o
-
~rm(s)
1'o
Fig. 9. Steering command performed by the fuzzy driver model
Y(nl
-50
0
5O
100
150
200
250
X(nt
Fig. 10. The vehicle path for the 90 degrees turn simulation
The second simulation presented here is performed to simulate the situation where the driver wants to change Lane due to an accident occurrence in his lane ahead. A schematic diagram for this maneuver is shown in Fig. 11. To create a critical situation the initial velocity of the vehicle is assigned to 45 m/s (about 162 km/hr), and the initial distance between the vehicle and the accident blockage is assigned 50 meters. Figure 12 shows the vehicle path for different values of steering sensitivity gain (SSG). This is a premature to adjust the fuzzy driver model to suit various conditions encountered in different simulations. The steering and braking commands performed by the driver model are given in Fig. 13 and Fig. 14. Considering these plots, one can see that the fuzzy driver model managed to steer the vehicle throughout this critical situation with a performance close to what a human driver is expected to do.
93
Current Advances in Mechanical Design and Production, MDP-7
Targeted lane
Original lane
" Accident .4
o
~
~
'~o
..
=
,oo
,
.4
Fig. 11. Changing lane maneuver
o
~
,
9
;o
~
~
,oo
Fig. 12. Vehicle path for different steering sensitivity gain values
However, a problem of this model is the erratic steering and braking behavior observed in the above characteristics. Despite the fact that this behavior could also be existed in a real critical driving situation due to the stressful nature of the maneuver, there are ways to train the fuzzy model to behave closely to an existing maneuvering performance observed by a human driver. Instead of manually tuning the fuzzy rules developed earlier in this chapter, a training procedure can be employed to tune those rules using input-output data pairs obtained from experimental observations. The resulting fuzzy system would give input-output characteristics closer to be experimental observations depends on the accuracy of the training procedure. For simplicity, a tuning process would be established for only the braking module of the fuzzy driver model. Artificial data pairs will be used here due to the lack of experimental measurements. A training scheme is now used to create a fuzzy braking system that performs according to the desired characteristics. The fuzzy model structure used for this purpose is the
.
.
. T
.
.
,
.
.
,
.
.
.
,
.
.
1'.2
i
. . . .
, . . . . . . .
,
o.
0
1
2
3
4
5
6
Fig. 13. Steering command for emergency lane change maneuver (SSG-45)
0
0
.|
1
.
/
2
.
3 ~rre (s)
4
5
6
Fig. 14. Braking command for emergency lane change maneuver (SSG=45)
94
Current Advances in Mechanical Design and Production, MDP- 7 1.4
E0 U
D e s i r e d characteristics
F u z z y model output
0
1.2 1,
0.8 N "i
O.a
0.4
~
ol5
.
1
.
.1.5 .
.
2
.
2.5
b~ Fig. 15. Fuzzy braking module characteristics after training course
Sugeno [2] type where outputs of the fuzzy system are mainly represented by a linear combination from inputs. Using Matlab software, a hybrid training technique is used to train the system. Three training epochs where performed to lower least square error below 1E-5. The resulted characteristics of the tuned braking module are depicted in Fig. 15. 5. CONCLUSIONS On the basis of the above discussions, one can say that fuzzy inference systems can be very powerful tool when it comes to human reasoning. The introduced fuzzy driver model here is capable of guiding the vehicle in both lane following and collision avoidance situations. It provides the vehicle with appropriate steering and braking commands, and further training of this type of models might lead to performance features very close to desired ones. REFERENCES 1. Zeyada, Y., Kamopp, D., EIArabi, M. and EIBeheiry, E., "A Combined active Steering Differential-Braking Yaw Rate Control Strategy For Emergency Maneuvers", SAE Paper No. 980230, (1998). . Sugeno, M., "An introductory Survey of Fuzzy Control", Information Sciences, Vol. 36, pp 59-83, (1985). 3. Ghazi Zadeh, A., Fahim, A. and EIGindy, M., "Neural Networks and Fuzzy Logic Applications to Vehicle Systems", Int. J. of Veh. Des., Vol. 18, No. 2, pp 132-193, (1997) 4. Wuertenburger, M. and Isermann, R., "Model Based Supervision of Lateral Vehicle Dynamics", Proceedings of the American Control Conference, Vol. l, pp 270-280, (1994) 5. Nagai, M., Ueda, E. and Moran, A., " Nonlinear Design Approach for Four-WheelSteering System using Neural Networks" Veh. Sys. Dyn., Vol. 24, No. 4-5, pp 329-342, (1995). 6. Kageyama, I. and Pacejka, H. B., " On a New Driver Model with Fuzzy Control", supplement to Vehicle System Dynamics, Vol. 20, pp 314-324, (1991). 7. 7.Nuesser, S., et al., "Neuro-Control for Lateral Vehicle Guidance", IEEE Trans. On Microprocessors, Vol. 13, No. l, 1993, pp 57-66, (1993). 8. Xi, G. and Qun, Y.," Driver-Vehicle Environment Closed-Loop Simulation of Handling and Stability using Fuzzy Control Theory", Veh. Sys. Dyn., Vol. 23, pp 172-181, (1994). 9. Zeyada, Y., "Integrated Active Systems in Emergency Maneuvers for Automobiles", Ph. D. Thesis, Cairo University, Giza, Egypt, (1999)
Current Advances in Mechanical Design and Production Seventh Cairo UniversiO' hffernational MDP Conference Cairo, February 15-17, 2000
95
OPTIMAL ACTIVE SUSPENSION WITH PREVIEW FOR A QUARTER-CAR MODEL INCORPORATING INTEGRAL CONSTRAINT AND VIBRATION ABSORBER
Abduijabbar, Z.S. and EIMadany, M.M. Mechanical Engineering Department, King Saud University, Riyadh, Saudi Arabia E-mail:
[email protected] and
[email protected]
ABSTRACT An optimal multivariable controller with preview has been designed for suspension control of vehicles. The controller takes the form of a linear quadratic regulator with supplementary states for added integral action. The effect of preview control on the performance of a quartercar model equipped with a passive vibration absorber is examined. The vibration absorber is used to reduce the axle vibration without sacrificing the ride comfort. It is assumed that the road irregularities ahead of the vehicle are measured, and this information is used to generate an enhanced control law in order to provide further improvements in the performance over that without preview. An optimal performance comparison of active systems with preview, optimally designed using full-state feedback with and without a passive vibration absorber is presented and discussed.
KEYWORDS Suspension Design, Vehicle Control, Preview Control. I. INTRODUCTION The design methodology of road vehicle passive suspensions has been well established, and an automotive engineer must make a trade-off between ride comfort and road holding. Active suspension technologies have been investigated and developed in order to improve the ride quality [ 1-3]. Unlike passive systems comprised of springs and dampers, active suspensions use force-generating elements, driven by an external power source. These elements can be made to respond to any set of measurements of the system parameters. Thus, the active suspensions can, in principle, continually supply and modulate the flow of energy, and can be adapted to the instantaneous operating conditions by changing their characteristics accordingly. However, the performance improvements of the active system over the passive ones are limited because of the lack of sufficient information about the incoming road input. This lack of information forces the designer to select the control laws that are acceptable to a large class of inputs, leading to systems that are optimally in an average sense. With look-ahead preview, road elevation information ahead of the vehicle is gathered and utilized in controlling
96
Current Advances In Mechanical Design and Production, MDP-7
suspension actuators. Consequently, the required control force can be synchronized in an efficient way, leading to more relaxation of the trade-off between ride comfort and road holding [4-13]. It is well known that it is not the "optimal" nature of the design alone that makes the linear quadratic regulator (LQR) attractive but the fact that it provides the control engineer with a straightforward technique to obtain a multivariable control law. Although LQR is intended for application as a regulator in its classical form, manipulation of the performance index can extend its use to servo problem [14]. The addition of integral action on the controlled variable of the vehicle suspension deflection can increase the robustness of the suspension design through compensation of vehicle model mismatch and the elimination of steady-state error as a result of constant disturbing forces acting on the system. This paper deals with the synthesis of an optimal finite preview multivariable integral controller for active suspension system based on a three-degree-of-freedom vehicle model. The controller utilizes knowledge about approaching road disturbances to prepare the active suspension system for the incoming input. An integral constraint is included in the performance index to achieve better attitude characteristics. The suspension control problem is described and the optimal control law is presented. The results of the analysis is applied to a quarter car model equipped with a simple vibration absorber to suppress the unsprung mass vibration. 2. VEHICLE MODEL A three-degree-of-freedom quarter-car model is shown in Fig. 1. The sprung and unsprung masses are denoted by mm amd m2, respectively. The active control force is assumed to be in parallel with a passive suspension comprising of stiffness kl and damping cl. The tire stiffness is k2. A vibration absorber of mass m3 is attached to the unspring mass along with a passive suspension having a spring stiffness k3 and a damping c3.
l
o, k
,
....
I,,
im,]
....
I
~
,
Pl t
kz 1
t t I
I
vI
Fig.
I.
,-
L
....
,~
A quarter-car model with passive vibration absorber.
97
Current Advances in Mechanical Design and Production, MDP-7
The vehicle is assumed to travel at a constant speed over a random road surface, which is approximated by an integrated white noise. Hence, the vertical road velocity disturbance, vi, is modelled as a white-noise input and it is specified by E[vi (t)] = 0 and E[vi (tl) vi (t2)] = V8 (tl - t2), where E[.] denotes the expectation operator, 8(.) is the Dirac delta function, and V = 2nx 10 .4 m2/s 3. It is also assumed that the rate of change of road elevation is measured at the distance L in front of the vehicle, i.e., vi (x), xc [t, t + tp] is known where tp = L/v and v is the forward vehicle velocity. The equations of motion can be written in standard state variable form: =Ax+Bu+Ew
(1)
The state variables are the suspension deflection xl, the tire deflection x2, the sprung mass velocity x3, the unsprung mass velocity x4, the integration of the suspension deflection x5 - ~ x! dt, the absorber mass deflection and velocity x6 and x7, respectively. The matrices of the governing equation (1) are given by
m
._.
.
0
0
-1
1
0
0
0
0
0
0
-1
0
0
0
o)!2
0
-2%10) 1
2%10) 1
0
0
0
_pl0)2
0)2
2~10)1Pl
1
0
0
0
0
0
0
0
0
0
1
0
0
-1
0
0
0
2~3o33
0
0)2
_2~30) 3
_2~10)lPl_2~3o)3P3
0-0)2p3-2~30)3p
3
t
B-J0 o 1 - o ,
0 0 0], t
E=[0
1 0
0
0
0
0] , a n d w = v i .
The symbols are defined as follows: u = - x reactive force ml 0) 3 = ~ k 3 / m3 = 23n
input; co I = ffk I / ml = 2n rad/s; co 2 = ffk2 / m2 = 20n rad/s;
rad/s; Pl = ml / m2 = 10; p3 = m3/m2 = 0.1; ~l = el / 2ml0)l = 0.3; ~3=
c3 / 2m30)3= 0.4. The controller design is based upon the minimization of a quadratic performance index including sprung mass acceleration, suspension working stroke, tire deflection, integral action, and control force,
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Current Advances in Mechanical Design and Production, MDP- 7
Ltx Ix]]
=
EI~:] + qtx~ + q2x~ + q3x] + pu2 ]
=
E
TuT
(2)
R u
where o4 + q l
0
-2~!c013
2~1Ol3
0
0 0"
0
q2
0
0
0
-291o~ 3
0
(2%,co!)2-(2~,co,)20
0 0
o
o
Q~
N =[o 2
0 0 o
o
0
0
0
0
q3
0 0
0
0
0
0
0
0 0
0
0
0
0
0
0 0
0 -2~1o I 2~lc01 0
0
0] T
R=l+p ql, q2, q3 and p are weighting factors which reflect the designer's preference. In this study, the corresponding values are ql = 5x10 3, q2 = 7x10 7, q3 = 104, and p = 0. 3. O P T I M A L PREVIEW CONTROL LAW Consider the linear system described by Eq. (1), given that the matrix A is asymptotically stable. Assume that w(t) is an unknown a priori input with zero mean and w(x), x c [t, t + tp] is given deterministically. Consider also the performance criterion of Eq. (2) in which Q is a symmetric and nonnegative definite matrix, R is a positive definite scalar and such that Qn = Q - NR "l N T is a nonnegative definite. Then the problem of determining an input which minimizes the criterion of Eq. (2) is called an optimal preview control problem. It is to be noted that solutions to finite preview control problem for linear active systems have been obtained using spectral techniques and Wiener filter theory [4], dynamic programing [5], and calculus of variation [6]. The control law is given by u='Olx'G2 r where
G! =
(3)
R-l[ N T + BTp] and G2 = R "l B T
Here, P is the symmetric positive definite solution of the Riccati equation P An + ATp - PBR-IBTp + Qn = 0,
(4)
Current Advances in Mechanical Design and Production, MDP-7
with
99
A, = A - BR "l N T.
The vector r is given by
r(t) =
'i e
(5)
PDw(t + a)da
o
where Ac = A - BR "1 (N T + BTp) is the closed loop matrix which is asymptotically stable. The control law of Eq. (3) is composed of a feedback part-R "! (N T + BTp)x which is an often used optimal control strategy for active suspensions without preview, and a preview controller -R "1 B ' r which is a feedforward part, containing the information about the road input obtained from preview sensors. The feedforward part utilizes preview to smooth out the vehicle response. The closed loop system is described by = Acx - BR-tBTr + Ew
(6)
The power spectral density matrix for the system of Eq. (6) is given by S o (co) = VHDD* H * where
H=
(7)
[jcoI-A c ]-1, tp
D = - B R - I B T f e Aca PE eJC~
+ E,
o
and * denotes matrix conjugate transposition. 4. RESULTS AND DISCUSSIONS In this section, selected results of the simulation of the quarter-car model shown in Fig. 1 are presented and discussed. Optimal control theory has been employed to find the optimum control force that minimizes the performance index given by Eq. (2). By changing the weighting factors p, ql, q2, and q3, an infinity of optimal systems can be identified and consequently a wide range of system performance characteristics can be obtained. The wheelhop damping ratio is kept approximately at 0.3 for the different vehicle suspension systems considered. Table 1 shows the rms response of the quarter-car model to random road disturbances, with and without preview. The results are presented for a passive system, and a multivariable proportional plus integral (PI) controller which is based on state variable feedback and integral control of the suspension deflection. The PI controller is considered with and without passive vibration absorber.
1O0
Current Advances in Mechanical Design and Production, MDP- 7
Table 1. RMS Response of the Vehicle to Random Road Inputs
(a) Without Preview . Passive PI . 0.0800 0.0708 0.0096 0.0078 0.0033 0.0032
. ,
rms body acceieration, g rms suspension deflection, rn rms tire deflection, m
_
(b) With Preview PI + PI Absorber 0.0590 0.0560 0.0062 0.0062 0.0021 0.0020
.
.....tp- 0.2s
tp = O.Is ,
,
irms body acceieration, g . rms suspension deflection, m rms tire deflection, m
i
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0.0507
.
9
.
..
0.0050 0.0019 ..
PI + Absorber 0.0570 0.0082 0.0031 .
.
.
.
Pi + Absorber 0.054 0.005 0.002
For the systems without preview, the PI controller shows an 18% reduction in the rms suspension deflection, while the rms values of the other variables are kept almost the same compared to passive suspension performance. However, the PI with vibration absorber provides a significant improvement in the fide comfort which is manifested in the reduction of the rms body acceleration by a 29% reduction in comparison to the passive system. The rms values of the tire deflection quantifies the loss of contact of tire from the road. It may be observed that the rms tire deflection and hence the road contact is the same for the three suspensions considered. Therefore, the PI with vibration absorber provides improvements in body acceleration and suspension deflection without diverse effect on tire deflection. For the active systems with preview, marked improvements in body acceleration suspension deflection and tire deflection compared to active systems without preview are observed. In Fig. 2, the power spectra of the body acceleration, suspension rattle space, tire deflection and control force are shown. The results are presented for systems without preview. In terms of body acceleration, substantial benefits of the addition of the vibration absorber can be observed particularly at frequencies close to the body and wheel-hop frequency. There is a point on the body acceleration characteristic corresponding to the wheel-hop frequency, which is known to be an invariant point, i.e., it cannot be changed regardless of the control law used as in passive and PI controller without absorber. However, the addition of the absorber eliminates such a constraint and an improvement in body isolation is obtained in the whole frequency range of interest. At the wheel resonance frequency, the presence of the absorber reduces tire deflection, but increases it in the low frequency range. The suspension deflections for the PI and PI + absorber are very close. With preview control, tp - 0.2 s, shown in Fig. 3, a substantial reduction in the required rattle space at the low frequency range is obtained, eliminating the drawback of the PI controller without preview. Marked improvements in the sprung mass acceleration are observed compared with passive and active systems without preview. Drastic reduction in the tire deflection is obtained in almost the whole frequency range. In the presence of preview control, the frequency domain performances of the PI and PI + absorber, in general, are very close.
Current Advances in Mechanical Design and Production, MDP- 7
I0 -i
10 4
,
.
,_
l 01
.
N E10 m
qn
k
~" I0.i C) (n o.
o rn
100 I0 i Frequency. Hz
10
10 i
i0 i
10 0 I0 i Frequency. Hz 10 "l N
N
N ~ I O "l
" ~ 10 -?
~ tn o.
'~\
~10 4
\ I0 4
~
t.
10
10o 10 ~ Frequency. Hz
.~
10 2
\
10 o 10 ~ Frequency, Hz
102
Fig. 2. Response power spectra without preview. (--- passive, active (PI), " active (PI + Absorber) 10 -~ N
~
,,,
10 ~
10 "4
I0
/
\
N
E 10 4
4
~
0in 10 4 Q.
(2 a.
Y
I0 "i
10 0 10 t Frequency, Hz
10e 10 t Frequency, Hi:
10=
10 "1 /%
,,,•E
lO-V
,-
,'.', /
10" J
.
10
r~ - I 0 -a n
10 ,,
10o 101 Frequency. Hz
10 2
.
,
10 o 10 ~ Frequency, Hz
Fig. 3. Response power spectra with preview tp - 0.2 s (--- passive, active (PI), active (PI+Absorber)).
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Current Advances In Mechanical Design and Production, MDP- 7
5. CONCLUSIONS Optimal multivariable controllers with and without preview, with and without passive vibration absorber have been designed for the ride control of road vehicles. The performance characteristics of such suspension systems are evaluated and compared with a passive suspension system. In case of no preview, the actively controlled suspension equipped with vibration absorber may be designed to provide excellent ride comfort without diverse effect on suspension travel or tire deflection. The presence of preview control allows significant improvements in the vehicle performance in terms of ride comfort, suspension travel, and road-holding ability. The active systems with and without vibration absorber give similar performance characteristics. ACKNOWLEDGEMENT The authors would like to thank the Research Center, King Saud University for supporting this research. REFERENCES 1. Hac, A., "Suspension Optimization of a 2-DOF Vehicle Model Using a Stochastic Optimal Control Technique", Journal of Sound and Vibration, Vol. 100, No. 3, pp. 343357, (1985). Chalasani, R.M., "Ride Performance Potential of Active Suspension Systems- Part I: Simplified Analysis Based on a Quarter-Car Model", ASME Monograph, AMD - Vol. 80, DSC-Vol.2, pp. 205-221, (1986). 3. EIMadany, M.M., and Abduljabbar, Z., "On the Ride and Attitude Control of Road Vehicles", Computers and Structures, pp. 245-253, (1992). 4. Bender, E.K., "Optimum Linear Preview Control with Application to Vehicle Suspension", Trans. ASME Journal of Basic Engineering, Ser. D., Vol. 90, No. 2, pp. 213221, (1968). 5. Tomizuka, M., "Optimal Linear Preview Control with Application to Vehicle Suspension Revisited", ASME Trans. Journal of Dynamic Systems, Measurement, and Control, Vol. 98, No. 3, pp. 309-315, (1975). 6. Hac, A., "Optimal Linear Preview Control of Active Vehicle Suspension", Vehicle System Dynamics, Vol. 21, pp. 167-195, (1992). 7. Huisman, R.G.M., Veldpaus, F.E., Voets, H.J.M., and Kok, J.J., "An Optimal Continuous Time Control Strategy for Active Suspensions with Preview", Vehicle System Dynamics, Vol. 22, pp. 43-55, (1993). 8. Pilbeam, C., and Sharp, R.S., "On the Preview Control of Limited Bandwidth Vehicle Suspensions", Proc. I. Mech. E., Vol. 207, pp. 185-193, (1993). 9. Abdel-Hady, M.B.A., "Active Suspension with Preview Control", Vehicle System Dynamics, Vol. 23, pp. 1-13, (1994). 10. EI-Demerdash, S.M., and Crolla, D.A., "Hydro-pneumatic Slow-active Suspension with Preview Control", Vehicle System Dynamaeis, Vol. 25, pp. 369-386, (1996). 11. Senthil, S., and Narayanan, S., "Optimal Preview Control of a Two-dof Vehicle Model Using Stochastic Optimal Control Theory", Vehicle System Dynamics, Vol. 25, pp. 413430, (1996). .
Current Advances in Mechanical Design and Production, MDP-7
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12. Kok, J.J., van Heck, J.G.A.M., Huisman, R.G.M., Muijderman, J.H.E.A., Veldpaus, F.E., "Active and Semi-active Control of Suspension Systems for Commercial Vehicles, Based on Preview", Proceedings of the American Control Conference, Vol. 5, pp. 2992-2996, (1997). 13. Li, Qin, Yoshimura, T., and Hino, J., "Active Suspension with Preview of Large-Sized Buses Using Fuzzy Reasoning", Int. Journal of Vehicle Design, Vol. 19, No. 2, pp. 187198, (1998). 14. ElMadany, M.M., and Abduljabbar, Z.S., "Linear Quadratic Gaussian Control of a Quarter-Car Suspension", Accepted for Publication in Vehicle System Dynamics, (1999).
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Current Advances hz Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
105
DYNAMIC RIDE PROPERTIES OF A ROLL-CONNECTED VEHICLE SUSPENSION
Rakheja, S. ~ Ahmed, A.K.W. *, Liu, p.'t and Richard, M.J. t *CONCAVE Research Center, Concordia University, Montr6al, Qu6bec, Canada H3G 1M8 tPresently with Goodyear Tires, Akron, Ohio, U.S.A. tDepartment of Mechanical Engineering, Universit6 Laval, Qu6bec, Canada G 1K 7P4 ABSTRACT Static and dynamic properties of a vehicle suspension comprising hydraulic struts interconnected in the roll plane are investigated. The feedback effects due to fluid flow through the connecting pipes on the suspension stiffness and damping properties are derived and discussed. Fundamental properties of the interconnected suspension are compared with those of an unconnected suspension with and without an anti-roll bar, in terms of load-carrying capacity, suspension rate, roll stiffness and damping characteristics. The anti-roll performance of the interconnected suspension is analyzed for centrifugal acceleration excitations encountered during directional maneuvers. The ride quality performance is evaluated for excitations occurring at the tire-road interface. It is concluded that the interconnected suspension with inherent enhanced roll stiffness and damping characteristics can significantly improve the heave ride performance and limit the body roll motion. KEYWORDS Interconnected suspension, feedback damping, anti-roll suspension, vehicle ride, roll stability. 1. INTRODUCTION Vehicle suspension designers are challenged to achieve a compromise between the conflicting dynamic ride, handling and control performance requirements [1]. While enhancement of rollover threshold and directional control performance requires relatively stiff suspension, softer suspension is desirable to achieve good ride performance. Passive vehicle suspensions are thus invariably designed with soft springs coupled with auxiliary roll stiffness mechanisms to attain a compromise between ride, handling and control performance and rattle space requirements of the vehicle. Auxiliary roll stiffeners, achieved through antiroll devices, however may affect the vehicle ride in an adverse manner due to coupled vertical and roll dynamics of the vehicle. Inter-connected hydro-pneumatic suspensions with progressive rate stiffness characteristics and almost constant natural frequency offer considerable potential for ride amplitude control, self-leveling, anti-roll and anti-pitch control [1,2]. Such system may therefore, be tuned to offer an improved handling and control properties, while good ride quality of the suspension is retained. Although roll and ride properties of various passive, semi-active and active suspension systems have been extensively reported, such properties of interconnected suspension have been reported in only few published studies. Moulton and Best [3] proposed and analyzed a
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Current Advances in Mechanical Design and Production, MDP-7
concept of an interlinked suspension system with rubber springs. Meller [4] proposed an interlinked hydro-pneumatic suspension comprising a self-energizing load-dependent intemal pump, energized by the relative movement between the unsprung and sprung masses, which facilitated the ride height control. Felez and Vera [5] investigated the ride and roll characteristics of passive and active interlinked hydro-pneumatic suspensions. The study demonstrated an improved anti-roll performance of the inter-connected suspension and instability problems associated with the active suspension. Tanahashi et al. [6] investigated an interlinked hydro-pneumatic suspension with three stage damping force modulation based on the vehicle response and road conditions. Rosam and Darling [7] presented the performance potentials of interconnected suspension coupled with an active roll control concept. In this paper, a nonlinear analytical model of a vehicle suspension, inter-connected in the roll plane, is analyzed to examine its properties in the vertical and roll modes. The interconnected suspension system is modeled incorporating the fluid compressibility, nonlinear damping characteristics, multiple chambers of pressurized fluid and gas, and the multidegrees-of-freedom (DOF) vehicle model. The feedback effects of inter-connecting fluid pipes on the vehicle roll stability and ride performance are established from the simulation results. The dynamic properties of the interlinked suspension with a beam axle are compared with those of an independent strut suspension to demonstrate its performance potentials. 2. ANALYTICAL MODEL OF AN INTERCONNECWED SUSPENSION The ride and roll performance of a vehicle suspension can be conveniently analyzed using the four-DOF roll plane model of the vehicle. Figs. 1(a) and 1(b) illustrate the roll plane models of a vehicle with interconnected and unconnected hydraulic suspension, respectively, in conjunction with a beam axle. The tires are modeled as a parallel combination of a linear spring and a viscous damper, assuming point contact in the roll plane, while the unconnected suspension employs an anti-roll bar. X~t(t ) and x~,(t) represent the excitations due to road roughness at the left-and fight-wheels, respectively. T,(t) is the roll moment excitation encountered by the sprung mass during a directional maneuver. The differential equations of motion of the four-DOF vehicle models can be expressed in the following matrix form:
§ tr( ,
try}
(1)
where [m] is the 4x4 mass matrix and {re } is the excitation force vector due to the tire-terrain interactions and roll moment. The force vector {f(z,~,t)} comprises forces due to suspension and tires. Vector {z}represents the coordinates for relative motions. 2.1. Ride and Roll Properties of Interconnected Suspension The interlinked suspension is realized by connecting the upper and lower chambers of the struts as shown in Fig. 1. The suspension is analytically modeled assuming turbulent fluid flows through the damping valves, laminar flows through the interconnecting pipes, polytropic process of confined gas, incompressible fluid, and negligible strut friction [9]. The fluid couplings between the two suspension struts yield a feedback effect that influences the static as well as dynamic characteristics of the suspension forces. The performance characteristics of vehicle suspension are strongly related to its static and dynamic properties, such as load-carrying capacity, bounce suspension rate, effective roll stiffness, and bounce and roll damping. The interconnected suspension model is thus analyzed to derive its static and dynamic properties [9]:
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Current Advances in Mechanical Design and Production, MDP-7
L _ F -
L~
~s(t) _ -V
L,
L
j q
...........]l ~
,.j
............ ~1:2.
L, .....
,F .
X.j.(t)
Lr
.J
T
x'W"-
59:
Fr'
F,' ..... I Ktl
Ctl
0.(t)
Xi~(t) L
Ktr
.... ~_
Ktl
tr
Ctl
Xi,(t)
Xil(t)
J
0.(t)
Kt,
tX,,(t)
L ....
Lw, -I I- " Lwl Lwr -I (a) (b) Fig. I. Roll plane models of vehicle with (a) interconnected and (b) unconnected suspension. F
l-~.t
T
Load-Carrying Capacity: Assuming identical struts, the load-carrying capacity, defined as the load supported by the beam axle suspension at the design ride height, can be expressed as: W = 2N,(po -po)A
where N,
(2)
is the number of suspension struts used on each side, P0 is the nominal gas
pressure at the design ride height and po is the atmospheric pressure. Area A represents the piston rod area for an interconnected system, and the piston head area for an unconnected system. The load-carrying capacity of an interconnected suspension is thus considerably smaller than that of an unconnected suspension with identical struts and charge pressure. The nominal pressure or strut size of the interconnected suspension therefore must be appropritely selected to achieve load-carrying capacity identical to that of an unconnected suspension.
Suspension Rate: Assuming polytropic process of the confined gas, the spring rate of the axle suspension corresponding to the static ride height of the vehicle can be derived as [9]: K,,~ = 2N, n p~ A 2 (3) Vo where n is the polytropic exponent and v0 is the nominal gas volume of a single strut. Since A represents piston rod and head areas for interconnected and unconnected systems, respectively, the suspension rate of an interconnected suspension is considerably lower than that of an unconnected suspension.
Roll Stiffness: Static suspension roll stiffness can be derived from the restoring moment ( M ) developed by the suspension which is subjected to a static roll angle (~o). During roll motions, the left and right struts develop unequal forces leading to certain heave motion of the sprung mass ( x ). The roll stiffness can thus be expressed as: dM OM ,3M dx K~, . . . . + ~ ~ (4)
d~o
O~o
& rico
where dx/dtp can be computed from the corresponding static equilibrium equation. Solution of Equation (4) yields the roll stiffness corresponding to its static ride position: K~0 = K .0 e 2 (2a - l )2
(5)
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Current Advances in Mechanical Design and Production, MDP-7
where g is half the suspension track, and a is piston head to rod area ratio, which is greater than 1. The roll stiffness of the unconnected suspension at design ride height is derived as: 0 2 K~0 = K=g , without anti-roll bar;
0
0
K@ = K ~ g
2
+ K s , with anti-roll bar
where K s is the auxiliary roll stiffness of the anti-roll bar. Equation (5) reveals that the static roll stiffness of the interconnected suspension can be effectively enhanced by selecting larger piston area ratios. Damping Characteristics: The hydro-pneumatic suspension interconnected in the roll plane yields additional damping due to fluid flow through the interconnecting pipes, and due to the coupling effects. The damping force developed in a strut is strongly influenced by the movement of the interconnected strut. The damping properties of the interconnected suspension can be explored for different vibration modes. In the bounce mode, the damping force can be expressed as:
f~, = , lel + (6) where c I and c 2 are damping parameters associated with turbulent orifice flows and laminar pipe flows given by cl = 0.5#43,/(Caa) 2 and c 2 = 128g/~ 2,/OrD4). A , , A, and a are crosssection areas of piston rod, interconnecting pipes and damping orifice, respectively, p and/l are fluid density and viscosity, and L is the pipe length. Ca is the discharge coefficient. In the roll mode, the damping force developed by a strut can be expressed as: f~, = c, (2a - 1)3 ilil + c2i
(7)
Equation (7) reveals that the damping force developed by a strut in the roll mode is always larger than that in the bounce mode due to the coupling effects of the left and fight struts. For an unconnected strut, the damping force is represented by the first term of Equation (6), where ct is the damping parameter associated with turbulent flows through the orifice. 3. COMPARISON OF SUSPENSION PROPERTIES
In view of ride and handling performance, the stiffness and damping properties of four different configurations of vehicle suspension are evaluated and compared to demonstrate the coupling effects of an interconnected suspension. The selected suspension configurations include: (i) conventional unconnected suspension (UC); (ii) unconnected suspension with anti-roll bar (UCR); (iii) interconnected suspension I (IC-1); and (iv) interconnected suspension H (IC-2). The parameters of different suspensions are selected to achieve identical load-carrying capacity of 14110 kg at design ride height. The suspension rate and roll stiffness of IC-I configuration are selected as 960 kN/m and 470.4 kNm/rad (ot=1.125). The IC-2 suspension, however, is configured to yield low suspension rate (610 kN/m) and high roll stiffness (735 kNm/rad) by selecting a=1.284. The static properties of the four suspension systems are evaluated and compared as functions of the relative vertical and roll displacements. Fig. 2 illustrates the vertical and roll stiffness properties of the four suspension configurations as a function of the relative vertical and roll displacement across the struts, respectively. All the suspensions exhibit progressively hardening vertical rate due to the force-deflection characteristic of gas springs. Unconnected suspension with and without anti-roll bar, and IC-I suspension yield identical vertical rates over the defection range of _+0.03 m, whereas the IC-2 suspension yields lower vertical rate.
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Current Advances in Mechanical Design and Production, MDP-7
The roll stiffness rates of the interconnected suspensions are considerably larger than those of the unconnected suspension, due to the coupling effects of the interconnected struts. The roll stiffness of unconnected suspension is considerably enhanced by the anti-roll bar. The roll stiffness of all the suspension configurations exhibits progressively softening property. 1.,1~0 t .2QO ~
! ..
1.000
..........'--::- :;:i:i-:iii.::i.-:?i;iii
i ~
!~ l
l
0
,~,
,l
.
I
JL
.__
0
...... '
DEFLECTION (fit
.....
I
8PRUNG MASS ~
.....
i
I, , 0.~
!
"
).08
ANGLE (rid)
(a) Vertical suspension rate (b) Roll stiffness Fig.2.Vertical and roll stiffness properties of different suspension configurations. Beside stiffness, the suspension damping plays an important role in the ride and handling characteristics of a vehicle. The relative vertical and roll damping properties of the suspension configurations are thus evaluated assuming constant area damping orifices. Figs. 3 illustrates the damping forces of a single strut in bounce and roll modes. Although the suspension systems are configured with identical orifice areas, the resulting damping properties of various suspension systems are considerably different as illustrated in the Figures. Since anti-roll bars do not contribute to damping properties of a suspension, the damping properties of the two unconnected suspensions are identical. Fig. 3 demonstrates that the bounce mode damping force of IC-1 suspension strut is similar to that of the unconnected suspension, while the damping force of lC-2 strut is considerably lower. The damping force of an unconnected suspension strut in the roll mode is identical to that in the bounce mode. The roll mode damping forces due to the interconnected struts in all the configurations, however, increase considerably due to feedback effect. The interconnected configurations thus offer significantly larger damping force in the roll mode. 60
,|
l
I
......
80
..."Y"
~
,,..Y ...... ...""""
,J ~.,..o
, , ~ . I "r
10 O0
0,0$
~1
0,1S
I l m t l r VEI.~:~I"r (mtl)
0.|
r
0.(:6
O.T 0..16 8 r o U T Via.OOZe Ovr
0.1
O.S
(b) Roll mode (a) Bounce mode Fig.3. Damping force developed by a single strut in bounce and roll modes. 4. PERFORMANCE OF THE INTERCONNECTED SUSPENSION The equations of motion of the four-DOF vehicle models are solved to determine the performance characteristics of the four different suspension configurations. The simulation parameters used in the study correspond to a modern highway bus, and are given by:
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Current Advances in Mechanical Design and Production, MDP-7
ms=1411Okg,
ls=19300kgn~, I,,=381~_:_~ kgrn2, ee=e,.=O.7m
m..=3616kg,
k,t = k,r =37548 kN/m, C,e =c,, =9500 Ns/m, h.z =0.62 m, e,, =ew,. =l.03m
The roll and ride properties of the interconnected suspension system are evaluated and compared with those of the vehicle models employing unconnected suspensions with and without anti-roll bar. The roll performance is analyzed in terms of the sprung mass roll angle response of the vehicle subject to a lateral acceleration excitation encountered during a directional maneuver. The dynamic ride performance is established in terms of both heave and roll shock isolation characteristics of the suspensions subject to road excitations. 4.1. Roll Response Performance The roll response of the vehicle employing different suspension systems is investigated under two different lateral acceleration excitations encountered by the sprung mass: (0 rounded step acceleration encountered during a steady turn; and (ii) transient lateral acceleration encountered during a typical lane-change maneuver [9]. The roll angle response of the sprung mass subjected to rounded step and transient lateral acceleration excitations are shown in Fig. 4. The results under steady turning acceleration excitation show that the steady state roll angle response of the sprung mass with unconnected suspension (0.065 radians) is quite large due to its low roll stiffness. The roll angle response approaches a steady value of 0.038 radians, when interconnected as well as anti-roll bar suspensions are employed. The peak roll angle response of the vehicle employing interconnected suspensions, however, are lowest due to their high roll mode damping. The results further reveal that the roll oscillation frequencies are also different reflecting the difference in roll stiffness as presented in Fig. 2. For the interconnected suspension systems, faster decay rates of the sprung mass roll response is also observed due to larger damping rates in the roll mode.
I-----
I oc u,,c.,,c_ I
o.1
l~
I~
O.O4
o
1
|
s 111,E(a)
4
8
e
a. Rounded step lateral acceleration.
gt)
0
!
l
II
4
$
t
?
9~ o0 b. Transient lateral acceleration.
Fig.4. Roll angle response of sprung mass under lateral acceleration excitations. The roll response characteristics of the sprung mass under transient lateral acceleration excitation corresponding to a typical lane change maneuver reveal that the suspension system interconnected in the roll plane can effectively limit the body roll motion. The anti-roll bar suspension also demonstrates its superior anti-roll capability. A comparison of the peak response amplitudes of interconnected suspensions further reveals the effect of roll mode damping in reducing the body roll. The effect, however, is not as significant as that observed for step response.
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Current Advances in Mechanical Design and Production, MDP-7
4.2. Ride Response Performance Dynamic ride characteristics of the vehicle employing unconnected, unconnected with antiroll bar and interconnected suspensions are analyzed in terms of transient vertical and roll response of the sprung mass, when subjected to half-sine displacement excitations at the tireterrain interface. The vertical shock attenuation characteristics are evaluated for an in-phase excitation, while the roll response is evaluated for an out-of-phase excitation. Fig. 5 presents the vertical displacement and acceleration response of the sprung mass employing different suspensions, subjected to an in-phase half-sine bump input. As the results show, the peak displacement response of the IC- 1 suspension is similar to that of the unconnected suspension and considerably larger than that of the IC-2 suspension. The large peak response of unconnected and IC-I suspension is attributed to the high bounce mode damping of these suspensions. All the suspension configurations, except the IC-2, exhibit sprung mass bounce resonance near 1.2 Hz, while the corresponding frequency of the IC-2 suspension is near 1 Hz. These results further show that the IC-2 suspension produces nearly 50% of the acceleration response of the unconnected and IC-1 suspensions, under half-sine bump input.
o.oo4
| IC/I IC[~..IC- L . . _
O.OOG I 0.00~ o.00t 0
1
la
--
Io i
/~
I
1 ~
w ~'~o
~
~
~,aE(,1 s
....
o
.......
............. ". . . . . ~ -
(~ 4
8
e
0.~
o.~
o.~
'n~uE1,) o.~
0,~
o.~2
o.~4
a. Displacement b. Acceleration Fig.5. Vertical displacement and acceleration response of the sprung mass of the vehicle under in-phase bump excitation. The roll angle and roll acceleration response characteristics of the sprung mass of the vehicle employing different suspensions, subjected to an out-of-phase bump, are compared in Fig. 6. The results clearly reveal that the peak roll displacement and acceleration response of the vehicle with IC-l suspension is considerably large due to its high suspension damping in the roll mode. The roll displacement response of the vehicle, however, decays relatively rapidly. The results further demonstrate that the anti-roll bar has no significant influence on the roll ride response since the body roll angle is considerably small under such an excitation. Although the peak roll response of the IC-2 suspension is slightly larger than that of the antiroll bar suspension, its roll response decays at faster rate. It should be noticed that the IC-I suspension, which yields superior anti-roll performance, results in excessive roll acceleration of the sprung mass. It is thus concluded that similar to suspension stiffness, high suspension damping is desirable for good vehicle handling, while lower damping is preferred for enhancement of vehicle shock attenuation performance. 5. CONCLUSIONS A vehicle system roll plane model is developed and investigated to examine the fundamental properties of a roll connected hydro-pneumatic suspension. The properties are evaluated in terms of load-carrying capacity, suspension rate, roll stiffness and damping characteristics.
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Current Advances in Mechanical Design and Production, MDP-7
The roll and ride response characteristics of a highway bus employing the interconnected suspension is evaluated for lateral acceleration excitations encountered during directional maneuvers and for transient excitations occurring at the tire-road interface. It is shown that the interconnected suspension has inherent enhanced roll stiffness, which can be effectively controlled by the roll stiffness amplification factor determined by the strut configuration. The suspension rate of the interconnected suspension exhibits a progressively hardening property, while the roll stiffness exhibits a progressively softening characteristic. The interconnected suspension also yields higher damping rate in the roll mode than in the bounce mode. The simulation results reveal that the interconnected suspension with enhanced roll stiffness and damping can significantly restrict the body roll motion for improved vehicle handling performance. The larger roll stiffness and damping, however, deteriorate the roll ride performance of the vehicle. The property of relatively large roll damping of the interconnected suspension is a positive factor for restricting body roll, but a negative factor for good vehicle ride quality. The interconnected IC-1 suspension with enhanced roll stiffness and reduced suspension rate can provide a better compromise between ride quality and handling performance of a vehicle. o.o~
I
I
o.ool
"~
:-,
I
o
~ fo.oo~)
[0: 0
1
2
$ TME (a)
4
S
11
0
J
0.C~ 0.G4 0,08 0.~ 'flUE Iq
0.1
Gll
0.1'4
b. Acceleration a. Displacement Fig.6. Roll displacement and acceleration response of the sprung mass of the vehicle under out-of-phase bump excitation. REFERENCES 1. Bank, T. A., "Some ABC's of Air Spring Suspensions for Commercial Road Vehicles," SAE paper No.800482.,(1980). 2. Chu, Y. and Li, Z., "The Dynamic Response of Vehicles with Hydro-gas Suspension to Roadway Undulation," ACTA Armamentari, No.2, pp. 30-42, (1984). 3. Moulton, A.E. and Best, A., "Rubber Springs and Inter-connected Suspension Systems," Engineering Design Show Conference, Paper No. 15a, (1970). 4. Meller, T., "Self-Energising, Hydropneumatie Levelling Systems," SAE No. 780052. 5. Felez, J. and Vera, C., 1987, "Bond Graph Assisted Models for Hydro-Pneumatic Suspensions In Crane Vehicles," Veh. Sys. Dyn., Vol.16, pp. 313-332, (1978). 6. Tanahashi, H., Shindo, K., Nogami, T. and Oonuma, T., "Toyota Electronic Modulated Air Suspension for the 1986 SOARER," SAE Paper No. 870541, (1987). 7. Rosam, N., and Darling, J., "Development and Simulation of a Novel Roll Control System for Interconnected Hydragas Suspension", Veh. Sys. Dyn.,V.27, pp. 1-18,(1997). 8. Mayne, R. W., "The Effects of Fluid and Mechanical Compliance on the Performance of Hydraulic Shock Absorbers," ASME J. of Eng. for Industry, pp. 101-106, (1974). 9. Liu, P.J., An Analytical Study of Ride and Handling performance of an Interconnected Vehicle Suspension", M.A.Sc. Thesis, Concordia University, Canada, (1994).
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
113
BILINEAR CONTROL THEORY OF SMART DAMPING SYSTEMS
EI-Beheiry, E.M. Lecturer, Department of Production Engineering and Mechanical Design, Faculty of Engineering, Menoufia University, Shebin EIKom, Egypt
ABSTRACT Methods for optimal and sub-optimal bilinear control of smart dampers are derived in this article. The smart dampers are considered in the form of (CVD) continuously variable semiactive dampers which are capable of adapting their damping forces to reasonably match fullor limited-state control (inputs) forces generated by broad-band actuators. During the operation of the smart dampers, it is assumed that some other locations in the vibratory system are controlled by simultaneous broad-band force generators. The System is then a piecewise (bilinear) vibrator in which the control inputs appear additively and multiplicatively. The chosen performance index is constrained by (i) the inability to measure all the system state variables that are necessary for the operation of the broadband control input and (ii) the necessity to consider the switching states of the smart damping elements. A bilinear control theory is first developed and then applied to an automobile model of 4-DOF having front suspension of smart damper and rear suspension of broadband ideal actuator. Performance comparisons with (FLSC) full-state control and (LMSC)limited-state control designs show the effectiveness of the (FLBC) full-state bilinear control and the (LSBC) limited-state bilinear control designs derived in this work. KEYWORDS Bilinear Control, Output Feedback, Vibration Control, Vehicle Suspensions. 1. INTRODUCTION Active suspensions are close to be in mass production in the coming few years, s e e Hac et al. [3,4]. Several studies like those made by Kamopp and Margolis [7], Kimbrough [8] and Youn and Hac (9) have shown that adaptable suspensions of variable structure in the sense of damping and stiffness elements might lead to a remarkable vibration control of vehicles on roads of varying surface conditions. These systems are the ones we call "smart" in this work because they are always capable of adapting themselves to the designer's desires of performance. However, when we come to the optimization of adaptable semi-active dampers the task is not as easy as the calculation of optimal control laws of active suspensions. Semiactive dampers are rapidly switched in nature and therefore they have strongly nonlinear dynamic behavior. Many attempts have been made for the optimization of semi-active dampers, but the contribution of researchers in this area is not as much as their contribution in the optimization of active suspensions. EIBeheiry [1], Hac and Youn [2], Hedrick et al. [5] and Hrovat, et al.[6] have all considered the optimization of semi-active dampers with/without preview control algorithms.
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Current Advances in Mechanical Design and Production, MDP-7
This paper addresses the problem of developing optimal and sub-optimal bilinear control methods for the design of adaptable semi-active dampers, which simultaneously operates along with ideal broadband actuators mechanized at different locations in one system. The problem is complicated by the inability to measure all the system state variables that are necessary for optimizing the force generated by the actuator. This situation clearly appears in vehicles where tire deflections represent the most difficult parameters to measure or to estimate. 2. A P P L I C A T I O N M O D E L AND F O R M U L A T I O N The model of an in-plane 4-DOF half-car model is schematically shown in Fig. 1. mb is the sprung mass and db is the sprung mass pitch moment of inertia. If Yb and Ob represent the bounce and pitch displacements of the sprung mass, respectively, then Yl = Yb + llOb and Y2 = Yb-120b will be the sprung mass displacements at the front and rear suspension connections, respectively. Masses m I and m 2 represent front and rear axles; their absolute displacements are Y3 and Y4, respectively, k 1, k 2 and Cl, c2 denote stiffness and damping factors of passive front and rear suspension, respectively. Linear springs k 3 and k 4 represent front and rear tire stiffness. Active control forces at the front and rear suspensions are denoted by u I and u 2 , respectively, l I is the distance from the vehicle C.G. to the front suspension and 12 is the distance from the vehicle C.G. to the rear suspension. Yol and Yo2 are the road elevations imparted at the front and rear tire. Values of both basic vehicle and optimized passive suspension parameters are shown in Table 1. The model equations of motion are:
mbf~b + kl (Yl - Y3 ) + Cl (J'i - Y3 ) + k2 (Y2 - Y4 ) + c2 (J'2 - Y4 ) = Ul + u2 JbOb + klll (Yl - Y3 ) + ClII ()I - . P 3 ) - k212 (Y2 - Y 4 ) - c 2 1 2 ( ) 2 - J'4 ) = llUl -12u2 (I) ml.P3 + k3 (Y3 - Yol ) + kl (Y3 - Yl ) + Cl ()'3 - .Vl ) = -Ul m2.P4 + k4 (Y4 - Yo2 ) + k2 (Y4 - Y2 ) + c2 (3;'4 - JP2 ) = -u2
Table 1. Basic and optimized vehicl e parameters Basic vehicleparmmters
Yv
Y2
1 2 ~ l
mb = 5747 kg
!
db = 7689 kg.m 2 n~ =m2 =59.5 kg k3 =
~
=
190000N/m
I I = !.38 m
~ ~U Au2omor
Actuator
kl, Cl
Ul
,.
12 = 1.36 m
0pt~
paraS, ers
r kI = 10202N/m k2 = 10114N/m
r4 ~ ~ ]
Yo2
K~ ~ .
T Yol
cn = 1 5 0 0 N M m
c2 = 1500N.s/m
Fig. 1. A 4-DOF in-plane half car model Introducing the following state variable, control input and disturbance input vectors
Current Advances in Mechanical Design and Production, MDP-7 t
x(t)=[x,,
x2.........
t
x81,
u(t)=[u,,
115 t
u 2] , w ( t ) = [ w , ,
(2,3a,b)
w 21
x! =(Yl -Y3),
x2 =(Y2 - Y 4 ) ,
x3 =(Y3 - Y o l ) ,
x4 =(Y4 - Y o 2 ) ,
X5 = Yl,
X6 = P2,
X7 = ))3,
X8 = ))4,
(4)
equations (1), (2) and (3) can be reformed in a linear state space form as follows
(5)
x(t) = Ax(t) + BlUl (t) + B2u2(t)+ Dw(t), where A,
B 1 , B 2 and D are the constant state space matrices, w(t) is a vector of road white
noise inputs where w ! = Yol and w 2 = Yo2 .In this study, the limited-state control forces at the front or the rear suspensions are assumed to be of local-state measurements, i.e., the feedback signals are confined to only local signals at each actuator's location. One can say that the limited-state control forces are subjected to control structure constraints such that : u!=KlMIX,
(6,7)
u 2 =K2M2x
where K 1 and K 2 are constant feedback gain matrices, M 1 and M 2 are local measurement matrices of dimensions 3 • 8. In the case when a (FCVD) front continuously variable semiactive damper is used at the front suspension, the control force u I (t) is given by
(8)
U l(t ) = v I (t)(x5 - x 7 ),
where v 1(t) is a variable damping coefficient that can be varied within a given range of values (9)
v imin < v I (t) _
where V I = [ 0
0
0
0
1 0-1
(10)
0]
The substitution of equation (10) into equation (5) leads to a bilinear state-space form (11)
x(t) = Ax(t) + BlV 1(t)VlX(t) + B2u 2 (t) + Dw(t). 3. F U L L - S T A T E C O N T R O L (FLSC)
In order to account for all the important response measures, the following performance index is chosen for this study
2
2
2
]
E[.]
i=1 i=1 i=1 stands for expected values or variances, a b is the sprung mass bounce
acceleration,
ap is the sprung mass pitch acceleration, asi is the sprung mass bounce
where,
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Current Advances in Mechanical Design and Production, MDP- 7
accelerations obtained at both connection points with the suspension, Asi are front and rear suspension deflections, and A w i are front and rear tire deflections, a I ,~ ,or 3, and Pl ,P2, and T l, T 2, are weighting factors which reflect designer's preference. The performance index of equation (12), after simple algebraic manipulation with the system equations of motion (1), can be rewritten in a more convenient, new form for our problem as follows
J=
~_S~ RSlI Qc S2lI Ui x t t/ tli,m/2~! I[x' U[ U~ j-QI Q~ R2 u2
(13)
The matrices Q I, Q c, R l, R2, S I and S 2 are all constant weighting matrices come as a result of the manipulation of equation (12) and the model equations of motion. The optimization problem of the full-state controller of the system described by equation (5) is to find the optimum feedback control laws ufl and u f2 which minimize the generalized performance index in equation (13). If one introduces the following matrix notations ~RI = R! _ Qc R~IQ~:, =S~ - Q c R ~ I s [ , =Bi - QcR~IB[,
R2 = R 2 - Q~:R~iQc s~ = s [ - Q~:R~Isi
(14)
fi~ =B~z - Q~:R~IBi
A n = A - BIR, i-Is[ - a 2 ~ l g ~ _ ,
Qn =QI - SIRllSi - S2R21S2,
R n = B l ~ ' l ~ i + B2R~IB~, and if either (An,BI) or (BI,B2)isastabilizablepairand (An,Qln/2) is detectable pair, the optimum control laws are then given by : (15a,b) Ufl =KflX, uf2 =Kf2x where K fl and K f2 are optimum constant feedback gains of Ufl and Uf2,respectively, (16a,b) and Pn is a non-negative definite solution of the following algebraic Riccati equation PnAn + AnPn - PnRnPn + Qn = 0 ,
(17)
Because of space limitations the proof of these new results obtained here are omitted. 4. FULL-STATE BILINEAR CONTROL (FSBC) Now, we turn to the bilinear control system of equation (11). This system has independent additive and multiplicative control inputs. The additive control input, u2(t ), at the rear suspension is scalar and magnitude-unconstrained, but the multiplicative control input, v I (t),
Current Advances in Mechanical Design and Production, MDP-7
117
at the front suspension is scalar and magnitude-constrained. The damping coefficients v ! (t) of a FCVD is to be permanently varied to ensure that the front semi-active damping force acts as close as possible to the fully active forces Ufl(t ) given by equation (15a). Strictly speaking, in every damping state the fully active control force u f2 (t) of the rear suspension is to be adapted all the time according to the updated value of the damping coefficient v 1(t). If one considers the matrix notations in equation (14) in addition to the following ones, Am(vl(t))= A *
_B2R~l(g~ +QcRi-lsi)+(Bl' _B2R~lQc)v * ' l(t).V1,
Qm(v~(t)) =Q!-S2R21(S2 + Q~Ri'ISl) + SlV]'(t)Vl + V~v~'(t)Sl + V~v~(t)~,lV~ (t)Vl Rm =B2R~I(B~ + QcRi'IBI), and if either ( A m ( V r ) , B I ) o r (B! , B2)isastabilizablepairand
(Am(v*I),Q lm/2(v *!)) is
detectable pair, the damping coefficient of the front suspension and the rear fully-active force are then given by : Vlmin if Ufl(t)(x5(t ) - x7(t))
Vlmax(X5(t)-x7(t)) 2 u fl(t) / (x5(t) - x7 (t)) elsewhere uf2 (v ; (t)) = - R ~1[~ , B2Pm (v ~'(t))]x(t)
(18)
(19)
where Pm is a non-negative definite solution of the following matrix Riccati equation Pm (v*I(t))Am(v!(t))+A * 'm (v*(t))P m (v*l(t))-Pm(Vl * (t))R mPm (v*I(t))+Qm (v*l(t))=0 l
(20) and the closed loop matrix, Acl(V~(t))=Am(Vl(t))-RmPm(vl(t)) is to be asymptotically stable due to the non-negative definiteness of the Riccati solution in equation (20) and the positiveness of the the front semi-active damping coefficient. The proof of these new results can be obtained by following the same procedure in Appendix A. 5. LIMITED-STATE CONTROL (LMSC)
Theorem 1 Consider the optimization of the system described by equation (5) and the performance index of equation (13) with control inputs u! (t) and u 2 (t) subjected to control structure constraints defined by equations (6) and (7), respectively. If one introduces the notations of equation (14), then the suboptimal feedback control laws will be Ull = KIIMIX,
Ul2 =K/2M2x
(21a,b)
where K 11and K 12 are optimum feedback gains of the front and rear control inputs : KII = - R / l [ S l X U i +~lrXMi ](MiXMi) -1 ,
(22a)
118
Current Advances in Mechanical Design and Production, MDP- 7
(22b)
K/2 =-R21[S2XM~_ +B2FXM~_](M2XM[) -! , which requires a consecutive solution of the following two Lyapunov matrix equations: AX + XA' + DWD' = 0 , A'F +FA +M~K~S~ + S I K I M I + M ~ K ~ S ~ +S2K2M 2
(23)
+NiKiRiKlN
(24)
l +N~K~R2K2M
2 +Q} - 0 ,
for the calaculation of the minimized performance index
[
J* = tr F(DWD'
)] =tr [(M~K~S~ + S I K I M I +lv,2r~2o 2 +S2K2M 2 ""
~
+ M i K i R i K I M i + M ~ K ~ R 2 K 2 M 2 + Qi)X],
(25)
where X and F are the state and costate covariance matrices, and the closed loop matrix A = A + B IK i Ml + B2K 2M2 must be stable in any iteration during the minimization process.The proof of these results is a minor modification of the proof of (Case I) Theorem 2 in Appendix A. 6. LIMITED-STATE BILINEAR CONTROL (LSBC) Theorem 2 Consider the minimization of the bilinear control system described by equation (11) with respect to the performance criterion (13) in which u l(t ) is replaced by Vl(t)VlX(t ) . This performance is subjected to two constraints: (i) the rear active control force u 2 (t) is confined to local-measurements as in equation (7) and, (ii) the range of values of the damping factor v i (t) is embraced by upper and lower bounds as in equation (9). If one considers the notations of equation (14), and upon the satisfaction of the stabilizability and detectability conditions, the sub-optimal front damping factor and rear feedback gains will be
Vlmin if U l l ( t ) ( x 5 ( t ) - x7 (t)) < Vlmin(X5(t)- x7(t)) 2 v~ (t)=)Vlmax if Ull(t)(x5(t ) - x 7 ( l ) ) > V l m a x ( X s ( t ) - x7 (t)) 2 lUll (t) / (x5(t) - x7 (t)) elsewhere
(26)
K2(v;(t)) =-R21[S~X'M~ + B2F'X'M~ + QcR?t(SiX'V[ + B I r ' x ' v l +P,I v
(t)(V!XV~))x(VIXV[ )-1 (VIXM [)](M2 XM )-i
(27)
which requires a consecutive solution of the following two Lyapunov matrix equations ~' *I (t)) + DWD' = 0 I (t))X + XA'(v
~A(V *
(28)
A,'(v l*(t))F + F.A(v 1.(t))+ M~K~ (v I*(t))S~ + $2K2 (v*i (t))M2 + VIV I ' * (t)Si t * +SlVl9 (t)VI + M~K~(Vl* (t))R2K2(vl* (t))M2 + VlVl (t)RlVl* (t)Vl +Qi =0
for the calaculation of the minimized performance index
(29)
Current Advances in Mechanical Design and Production, MDP-7
J* :
tr [ F(DWD' )j= tr [ (M~K~(Vl 9(t))S~
119
+S2K2(vl 9(t))M2 + ViVl9(t)S ~ +SlVl"(,)v
+M ~ K [ (v I (t))R 2 K 2 (v i (t))M 2 + V~v I (t)R i v I (t)Vl + Q[ )X],
(30)
where, .X.(v*1(t)) = A +BlVl* (t)V 1 +B2K2(v*I(t))M2, is the closed loop matrix which the controller, if there is a feasible solution, must stabilize it. It is obvious from equation (26) that the front semi-active damping force will exactly follow the sub-optimal (limited-state) control force u tl of equation (21 a) when the later calls for energy dissipation. Otherwise the front damping is set to the maximum or the minimum bound depending on which one can provide front damping force closer to the desired control force. The proof of theorem 2 is given in Appendix A. 7. RESULTS AND DISCUSSIONS Figure 2 shows a comparison of behavior of various suspension types in response to a bumpy road. It is obvious that in Fig. 2 the (LMSC) behaves the same as the (FLSC) in terms of all the performance measures. But both of them provide better vibration control capabilities in terms of the sprung mass accelerations. The thing that is really noticeable in Fig. 2 is the remarkable reduction in the front tire displacement for both the (FSBC)and the (LSBC) designs. Although this reduction is on the expense of little deterioration in the pitch acceleration, one can say that reducing the front tire displacement will lead to reducing the frequent bottoming phenomenon on rough or bumpy roads. On the other hand, the (LSBC) provides better bounce acceleration than the passive system and even better than the (FLSC) systems for some peaks as seen in Fig. 2. The well-known facts about the behavior of the (FLSC) and (LMSC) suspensions can also be deduced from Table 2. They always provide distinguished riding qualities in terms of the sprung mass accelerations, but this time without usual severe deterioration in tire-to-road holding and suspension travel capabilities. Figures 3 and 4 depict the behavior of the (CVD) of the (LSBC) in response to a hole followed by a bump and a stochastic road input, respectively. These two figures show that continuously variable dampers can only produce their damping forces in only the dissipation mode. 8. CONCLUSIONS Bilinear control methods of vehicle suspensions with/without tire deflection measurements are developed and applied to an in-plane 4-DOF car model. These methods suggest the use of hybrid suspensions consisting of simultaneous semi-active dampers and broadband actuators in one vehicle system at different locations. The simulation predictions made here show that the suggested suspension designs could be comparable to full and limited-state controllers which only employ broadband actuators. These designs are an attempt to gather both features of semi- and fully-active suspensions in a hybrid suspension design. They might be a forward step towards the resolution of the inevitable conflict between cost and quality of performance in the design of suspension systems. Furthermore, the bilinear control methods developed in this work are not confined for application to vehicle suspensions but they can be extended for application to so many engineering problems.
Current Advances tn Mechanical Design and Production, MDP-7
120
!0
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-0.08 | I , I , I , I , I , I L I , 1 . I , I . ~ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 I.I 0.08
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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 I.I 1.2 Time (see)
,
t
I
1.2
Fig. 2. Vehicle response to a hole followed by a bump. Table 2 Comparison of various suspension designs in response to stochastic road input. Optimal passive system (OPSV) is taken a s 100 percent. System J~b w xt x2 x3 x4 OPSV FLSC LMSC FSBC LSBC
20001.
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o . -I00 0.0
0.5
i.0
1.5
2.0 2.5 3.0 T i m e (sec)
3.5
4.0
4.5
Fig. 4. Response of the LSBC to a stochastic input
$.0
Current Advances in Mechanical Design and Production, MDP-7
121
REFERENCES 1. EIBeheiry, E.M., "A Method For Preview Vibration Control of Systems Having Forcing Inputs and Rapidly-Switched Dampers", Journal of Sound and Vibration, Vol. 214(2), pp 269-283, (1998). 2. Hac, A. and Youn, I., "Optimal Design of Active and Semi-Active Suspensions Including Time Delays and Preview," Trans. of ASME, Journal of Vibration and Acoustics Vol. 115, pp 498-508, (1993). 3. Hac, A., Youn, I. and Chen, H. H., "Control of Suspensions for Vehicles with Flexible Bodies- Part I: Active Suspensions", Trans. of ASME, Journal of Dynamic System Measurement and Control, Vol. I 18, 508-517, (1996). 4. Hac, A., Youn, I. and Chen, H. H., "Control of Suspensions for Vehicles with Flexible Bodies- Part II: Semi-Active Suspensions," Trans. of ASME, Journal of Dynamic System Measurement and Control, Vol. I 18, pp 508-517, (1996). 5. Hedrick, J. K., Rajamani, R. and Yi, K., "Observer Design for Electronic Suspension Applications," Vehicle System Dynamics, Vol. 23, pp 413-440, (1994). 6. Hrovat, D., Margolis, D. L. and Hubbard, M., "An Approach Toward the Optimal SemiActive Suspension," Trans. of ASME, Journal of Dynamic Systems Measurements and Control, Vol. 110, pp 288-296, (1988). 7. Karnopp, D. C. and Margolis, D., "Adaptive Suspension Concepts for Road Vehicles", Vehicle System Dynamics, Vol. 13(3), pp 145-160, (1984). 8. Kimbrough, S., "Bilinear Modeling and Control of Variable Component Suspension", in Proceeding of the American Society of Mechanical Engineers, Winter Annual Meeting AMD 80, 235-255, (1986). 9. Youn, I. and Hac, A., "Semi-active Suspensions with Adaptive Capabilities," Journal of Sound and Vibration, Vol. 180(3), pp 475-492, (1995). APPENDIX:
PROOF
OF THEOREM
2
It is worth noting that the constraints of equation (9) can be rewritten in the form of two inequality constraints as follows - v 1 (t) + v Imin -< 0,
v i ( t ) - v Imax <- 0 .
(Al)
The Hamiltonian approach, for the problem of theorem 2 is given by 11
H
= 2[tr(QiX) + 2tr(ViXSlv I ) + tr(VIXVfv~R iv l ) + 2tr(M2 XS2K2) +tr(M 2 XIVI~K ~R 2 K 2 ) + 2tr(VlXM 2K ~QeVl) + t r ( F ' ( ~ . X + X~' + DWD')) + 2tr(l I (v Imin - v l)) + 2tr((ql (Vl "v lmax))],
(A2)
where 11 and q l are scalar Lagrangian multipliers corresponding to the inequality constraints in equation (AI), and F is the co-state covariance matrix. Substituting for the closed loop matrix A (as defined in the text), we get the necessary conditions for optimality 9
(OH/c~1)=O.5[2SIXVf + 2RIviVIXV f +2QeK2M2XV f + 2Bir'XVf + 2(I1-ql)]=0 (A3) (r
= 0.5[2S~XM ~ + 2R2K2M2XM~ + 2QcVlVlXM [ + 2B~F'XM~]= 0, (A4)
Current Advances in Mechanical Design and Production, MDP-7
122
(OH~OF)--X
=
A,X + XA'
+
DWD',
(A5)
(aH/aX) =-1~ = Qi +M~K~S~ +S2K2M 2 + ViviS', + S l v l V 1 + M ~ K ~ R 2 K 2 M 2 +Vfv~RlVlV i + A'F + F A = 0 .
(A6)
Equations (A3) and (A4) yield: v I = - ( R I - Qc R 2IQ c)-I[(S i - QcR~Is~ )XVl +(B[ -
Qc R~I B~ )F'XVf - (/I - ql)],
(A7)
K 2 = - ( R 2 - Qc R I'lQc)-I[(s2 - Qc RI'IS[)XM ~ + (B~ - Qc R IIB~)F'XM +Qc Ri -I (lk - qk )(V! XVf) -I (VIXM ~ )](M 2 XM~)-i.
(A8)
Case I. Non of the constraints (AI) is active. In this case it is convenient to write
l ! =q! = 0 .
(A9)
Substituting equation (A9) into equations (A7) and (AS) with the matrix notations of equation (15) considered, it follows that the optimum constant feedback gains for the limited-state regulator is given by equation (22), where
v, -
r'
xv; +
rxv ,](v, xvf)-',
KI2 = -ff,~l[g2XM [ + B2FXM~](M2XM[) -1 ,
(A 0) (A11)
Since the system is time-invariant, as t-+oo the matrices X and F tend to be zeros, and therefore equations (A5) and (A6) become AX + XA' + DWD' = 0, A'F + FA + M ~ K ~ S , ~ + S 2 K 2 M 2 + Vfv~S~ +SlVlV I + M ~ K ~ R 2 K 2 M 2 +VfV~RlVlV 1 +Q~ =0.
(A12) (A13)
C a s e II. One of the constraints (A l) is active. For example, I 1 # 0, or q l * 0. Strictly speaking, either Hll = 0 or Hql = 0, and hence either v I9 =Vlmin or v *I =Vlmax.
(AI4)
From equation (A3), one obtains
II - q l
= +(R2 - QcRllQc)V~VlXV/+ (S~ - QcRi-Isi)xv~ + (B9 - Q{:Ri'lBi)F'XVf
(A15) Substituting equations (A7), (AS), (Al4) and (AI5) into equations (A5) and (A6), and after simple but lengthy algebraic manipulation, it is easy to show that the optimum values of the constant feedback gains of the input control force and the optimum concurrent damping value are as in equations (26) and (27). The proof of sufficiency of this problem is easy but lengthy.
Current Advances ha Mechanical Design and Production Seventh Cairo UniversiO' International MDP Conference Cairo, February 15-17, 2000
123
A NEURAL ADAPTIVE APPROACH FOR RELATIVE GUIDANCE OF AIRCRAFT Shahzad, M.*, Slama, J.G. ~ and Mora-Camino, F. "§
* LAAS du CNRS, Toulouse, France. # Programa de Engenharia Mecfinica, COPPE/UFRJ, Rio de Janeiro, Brazil. 0- Air Transportation Department, ENAC, Toulouse, France.
ABSTRACT In this study a neural adaptive approach is examined for aircraft trajectory generation. The relative guidance of an aircraft, which is aimed to join in minimum time the track of a leader aircraft, is particularly considered. Neural networks are applied to generate on-line optimized aircraft trajectories associated to the current relative position. In a first place, the minimum time optimization problem is considered. Then the synthesis of a neural approximator of optimal trajectories is discussed. Trained neural networks are used in an adaptive manner to generate intent trajectories during operation. Finally simulation results involving two wide body aircraft are presented where the pursuit trajectory is generated with and without knowledge of the leading aircraft's flight plan. KEYWORDS Relative Guidance of Aircraft, Trajectory Generation, Neural Networks. 1. INTRODUC'FION In order to absorb the increasing air traffic flows, the innovative concept of Free Flight ("Safe and efficient flight operations in which the operators have the freedom to select their path and speed in real time") has been under study over the last years [l ]. The implementation of Free Flight has been made possible by the emergence of new navigation technologies such as GPS, ADS-B, TCAS and of new onboard computation capabilities [2]. Hence the absolute position of an aircraft as well as its short term intents can be communicated to other aircraft in the neighborhood through data links, while precise relative positions can be computed on line. Then aircraft should be able to realize relative maneuvers such as minimum separation crossings, mergings and meterings along common airways. In this communication, the case of the pursuit maneuver is considered more particularly. Time and cost optimization of aircraft trajectories has been of great interest for many decades and various numerical solution techniques have been developed [3-8]. However these techniques are not in general compatible with an on-line operation which is here a necessity since the leading aircraft may modify at any time its guidance parameters (speed, heading and flight level) in accordance with new atmospheric conditions (wind and temperature) or following instructions issued by the traffic control system. On the other hand, simple proportional navigation
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Current Advances in Mechanical Design and Production, MDP- 7
techniques, developed for missile homing, provide direct solutions to a similar problem [9-10]. However air transportation regulations (load factor limitations and standard maneuvers) as well as economical, structural and comfort considerations, prevent their utilization in the case considered in this study [11 ]. The control strategy proposed in this communication is adaptive and makes use of a neural network structure to get an on line approximation of the optimal trajectory associated to the current relative situation [ 12]. In section 2, the minimum time optimization problem considered is displayed and analyzed. It is shown that the adoption of a simplified flight mechanics model for its formulation allows the analysis and subsequent characterization of the optimal trajectory in terms of turns and straight line segments. In section 3, it is shown that the pursuit maneuver is at most composed of the following ordered phases: avoidance turn, distancing, redirection, convergence and tracking which are characterized by a few parameters. Then the synthesis of the neural approximator of optimal trajectories is discussed in section 4. A time consuming but simple iterative process of generation and evaluation of the optimal parameter values is developed to build up such optimal trajectories for any possible initial relative situation. This process is run over a selected set of initial situations to cover, through neural generalization, a region corresponding to 10 minutes of flight. The resulting input-output data file is used to train a set of feed forward one-hidden-layer neural networks through a classical back propagation algorithm. Then, starting from the current relative situation, the neural network structure provides the parameters of the corresponding optimal trajectory which can be submitted to the pilot on its navigation display and the current directives for navigation which are passed to the auto pilot (auto flight mode) or to the flight director (manual mode). In section 5, simulation results of the proposed approach to different situations (initial relative positions of the two aircraft, wind conditions supported by both aircraft and flight plan followed by the leader) are displayed. The results obtained show that this neural adaptive approach provides promising results and seems able to overcome the rigor of airworthiness requirements. 2. PROBLEM FORMULATION AND OPTIMAL SOLUTION In the case considered, a pursuer aircraft (P) is attempting to follow a leader aircraft (L) with the following simplifying assumptions" 9 the pursuit is bi-dimensional i.e. in the horizontal plane 9 the aircraft have constant velocities lip and VL 9 the follower aircraft exactly knows the position and the present route of the leader aircraft. The geometry of the planar pursuit is shown in Fig. 1. The equations of motion, in relative polar co-ordinates, are: d = VLcos(8 - 9'L)- Ve cos(0 - 9r = {-VLsin(8- 9,~ )+ g~, sin(O-gre)}/d
~ =r~
(1) (2)
(3)
where d is the relative distance between the two aircraft, 9the line of sight direction, 9rp and 9'L respectively the pursuer and the leader headings. 0, 9rp, and r are measured with respect to a common earth reference.
Current Advances in Mechanical Design and Production, MDP-7
125
L
~
.'"L
x
Y ~ ' / p
x
Fig. 1. The planar pursuit An optimization problem is formulated to generate a minimum pursuit time trajectory for the Pursuer: t!
minimize Idt with (1), (2), (3) and limit conditions: 0
t = 0: d ( 0 ) - d o, 0 ( 0 ) - 0 o , gte (0)= ~e0
(4)
(5) where l/ is the unknown final time. Other constraints are also imposed : ~m,. -< ~P -< ~m~, and d > dm~. r dm~"- d _<0 to take into account maximum turn rates and minimum distance separation. The Hamiltonian for this problem can be written as 9 u = l + ,t~d + ~o0 + ~ r + u(d~,. - d )
(6) and (7)
(8)
where 2 are the adjoint variables and # is an influence variable :
~,_~0 :/f(do,. -d)=0 /.t = O "if(dm, . - d ) < 0
(9)
the conditions of optimality are given by Euler-Lagrange equations : 2~ {- Vz sin(O -~,,. )+ v~ sin(O -~,~ )}+/.t ~ =2v .
~o {_V~co~(O-9tL )+ Vp cos(O2o = ---d--
ao {v os(O
~,~)}--2d{--VLsin(O-- ~,,.)+ V.sin(O- ~,~)}
(10)
)
with the final conditions of adjoint variables : 2 o (t/)= v o , 2~, (t,)= v,,, 2u (t,)= 0 as there is no fixed constraint on final d. vo and v~, are the constants which must be specified in order to satisfy final conditions of 0 and g. The above system of equations should be solved numerically in order to get desired minimum time trajectories, anyhow, by interpreting conditions of optimality, H can be written as" g H = H(O,d, v e .... )+2~,r e with re =~-~-tan0e, the minimization of H leads to the conditions 9 if 2~ > 0 then r = ~.,., if 2~, < 0 then r = ~.... The case where 2~, = 0 is more complex to analyse, however this singular situation can be clarified through the manipulation of the optimality conditions [13], leading to the conclusion that when 2~, = 0 then ~bp = 0 and Vp remains constant. Hence, a time-optimal planar trajectory consists of a sequence of maximum bank angle turns and straight line segments.
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Current Advances in Mechanical Design and Production, MDP- 7
3. OPTIMAL T R A J E C T O R Y GENERATION In this study, a simple case with two aircraft is considered for trajectory generation, as shown in Fig. 2, with the following considerations 9 9 The cartesian co-ordinate system is oriented with the Leader at origin, and flying along positive x-axis direction (heading angle of 900). 9 The Pursuer aircraft may have any position and heading angle in the horizontal (x, y) plan. 9 The aircraft are flying at the same constant speed (V= VL=Vp).
-
!
YP
Ve
! !
xe
L~X p,-
VL
Fig. 2. Relative horizontal position of two aircrafts 3.1. Trajectory Elements It has been shown in the previous section that all optimal trajectories of the Pursucr consist of sequences of arcs and straight line segments. The set of optimal maneuvers can be divided into subsets depending upon the initial relative positions and heading angles of the two aircraft" 9 The Pursuer is within the protected zone around the Leader and its first providence is to get clear from the Leader. 9 A direct maneuver is feasible : the Pursuer can arrive behind the Leader at a distance greater than or equal to minimum separation distance in minimum time. 9 No direct maneuver is feasible : it covers situations where the Pursuer must delay its trajectory so that minimum separation is effective. So, the optimal trajectory is composed of different elements organised in sequence. Dynamic Programming can be used here to globally optimise the parameters of each element. The algorithm for an optimal trajectory generation is given as" Step 1. Compute the relative distance 'd' between the two aircraft. Step 2. Determine if d < din,,. If no, go to step 3. Otherwise, determine the aircraft trajectory with left as well as right turn. The selected turn direction is one which results in larger value of the minimum in between distance during maneuver. Step 3. Determine if a direct maneuver is feasible. If yes, go to step 4. Otherwise, determine the aircraft trajectory with left as well as fight turn till the miss-distance becomes equal to the minimum separation distance behind the Leader. Afterwards, the aircraft flies straight ahead with the present heading till the point a direct maneuver becomes feasible. Step 4. Determine the aircraft trajectory with left as well as fight turn till the point from where convergence to the Leader's route may be possible. Step 5. Determine the straight segment of the aircraft trajectory for convergence towards the Leader's route. Step 6. Determine the aircraft trajectory for joining the Leader's route. Step 7. Out of generated trajectories (two solutions are possible for step 2 and step 3), that one is selected with which the Pursuer joins the Leader's route in minimum time.
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Current Advances in Mechanical Design and Production, MDP-7
Hence, following the above algorithm, the Pursuer trajectory can be built up from a set of basic parameterized maneuvers : 9 Avoidance Turn : tA, time at the end of Avoidance Turn; grA, final value of heading angle attained at the end of Avoidance Turn. 9 Distancing : tD, time for Distancing. 9 Redirection : tn, time for Redirection; VR, final value of heading angle attained at the end of Redirection. 9 Convergence :tc, time for Convergence. 9 Joining" tj, time for Joining. Some examples of generated optimal trajectory parameters are presented in Table 1 and the corresponding trajectories are shown in Fig. 3.
Table 1. Some examples of the optimal trajectory parameters
1 2 3 4 5
20.0 '20.0 30.0 -7.0 7.0
20.0 20.0 30.0 7.0 7.0
225" 135 90 135 225
2 0 !10 22.4 36.5
222 -245 166.6 276.5
52 0 21 0 0
125 31.9 148 83.5 186.34
46.4 180 36.1 48.8 13.6
0 17.6 0 0 0
Fig. 3. Examples of optimal rejoinder trajectories
31 63.8 38.1 29.1 54.2
211.5 133.3 317.3 135 277
17.493 33.987 24.545 14.197 47.274
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Current Advances in Mechanical Design and Production, MDP- 7
4. ON LINE NEURAL TRAJECTORY GENERATION The Leader may modify at any time its guidance parameters (speed, heading and flight level) in accordance with new atmospheric conditions (wind and temperature) or following instructions issued by the traffic control service. Thus on line trajectory generation is a necessity here. During this on line process it becomes impractical to calculate the optimal trajectory at each instant of time, in fact the need is to have some solution which could be immediately displayed to the pilot or taken into account by the autopilot. A neural network fulfils this requirement as it can be trained to memorize input-output relationships for a system and afterwards can be used as an interpolator to generate outputs corresponding to current inputs. Since the early 1990s, neural networks have been used in a variety of applications and there has been a growing interest in using neural networks to solve optimal guidance problems [ 14]. 4.1. Training Data The optimal Pursuer trajectories are generated for many different initial conditions. The overall input/output structure of the program used for this purpose is shown in Fig. 4. Inputs:
9 Initial position and heading angle of the Pursuer (xp, yp, r Outputs: 9 Avoidance tum: tA, ~a, da (da=0 for left turn and da =1 for right turn) 9 Distancing'to 9 Redirection:ts, ~s, dR 9 Approach : tc 9 Joining 9tj Input situations considered for training are xp(km) : - 100, -50, -30, - 10, -4, -2, 0, 2, 4, 10, 30, 50, 100 ye(km) 1O 0 , 5 0 , 3 0 , 1 0 , 4 , 2 , 0 : 9(o)0 o , 2o, 4 o, 6 ~......... 360 o. Since the infinite set of optimal trajectories is symmetric about the x-axis, the training has been performed only for positive y co-ordinate values of the initial position of the Pursuer. Only time required for each phase of maneuver and turn directions have been used for training as corresponding heading angles can be calculated analytically. The general structure of the neural networks used in this application is classical [15] and is composed of three layers. The transfer functions associated with the neurons are selected as hyperbolic tangent functions. The Levenberg-Marquardt's training method has been used here, it combines elements of steepest descent and Gauss-Newton methods. The general structure of the designed neural networks is given in Fig. 6. Each trajectory parameter is computed by a separate neural network to get a good accuracy in terms of the sum-squared error between the network outputs and the desired outputs. Table 2 summarizes the hidden layer composition of each neural
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Current Advances in Mechanical Design and Production, MDP-7
network designed for trajectory generation. Training is made efficient by scaling the training data, so that it falls in the range [- 1 1]. The input-output data pairs have been used in a random manner in order to improve efficiency and speed of the training process [ 15]. Table 2. The construction of hidden layers NeuralNetworks
~qumberof neuronsin ....... the hidden!a~er Time for AvoidanceTurntA(sec) 40 Directionof AvoidanceTurnd~ 40 Time for Distancin[tD(sec) 35 Time for Redirectionta (sec) ........ 40 Directionof Redirection.dR 30 -- " Time for Convergencetr (sec) 20 Timefor.Joinin~tl(sec) 5
input
. Hidden
Laver
Output
L _ . J ~
Layer
_.a~
..a'
Fig.6. The structure of neural networks
5. SIMULATION RESULTS AND CONCLUSIONS A simulation study has been performed considering two Airbus A300 aircraft. It has been supposed that the Leader aircraft makes a right tum to take a new constant heading. The guidance system of the Pursuer makes use of a neural trajectory generator to define, every second, new references (either a turn rate or a constant heading) for the autopilot. The guidance laws implemented in the autopilot are classical superposed PID loops (a fast piloting loop and a slower guidance loop) similar to those encountered in aircraft of this class [ 16]. 8o
Y (kw)
gllll~
dlt~lkts)= .avsonl
~V (km)
1~1.4t.e?4 (.)s
o.]Js,
t]~aa.I
~111~ (~,): d| tlYmlk till
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r
r i iill
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6
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o
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.
-44)
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ii
o
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~1 r vw,Q z~mr
(.)~
C*ts),
(ft)=
9,stg
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os* r ~,,,~o zMr
(*ts)~
(ft)=
13},gO t~4.tt
46e.91 10998
o (.): dDt(,v'.): dlse (n):
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(ft)=
9
O.OOe lSeY.: s:S.O0 X:4,99 4141. ti
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Simulation results are shown in Fig. 7a and 7b. Initial position of the Leader is taken as coordinate (0, 0) with a heading of 90 ~ The Pursuer is initially at 40 km behind and 40 km North of the Leader, with an absolute heading of 90 ~ In the case of Fig. 7a, it is assumed that the Pursuer has no knowledge of the Leader's flight plan. With the activation of the relative guidance mode, the Pursuer makes an initial right turn (Redirection) and starts Convergence. When the Leader modifies its heading to 135 ~, the Pursuer changes progressively its own heading (Redirection) and then after convergence turns right (Joining) to follow the Leader. This results in a very swaying trajectory, which is much undesirable in ATC standards. While
iii~il ~II~I~I~I~ i~iii
~i ~
~i
i i i i i i ii~i ~ ii//ii~ i~i i~ ~i~ ~:
i~!ii~ ~i ~!~
130
Current Advances in Mechanical Design and Production, MDP- 7
in the case of Fig. 7b, the Pursuer knows exactly the intent of the Leader so, the pursuit trajectory can be generated taking into account the final route of the Leader. In the second case, the Pursuer avoids excessive maneuvering while the pursuit time is smaller than the previous case. Extensive simulation experiments show that the proposed approach provides an efficient device to cope with relative guidance of civil aircraft. REFERENCES 1. RTCA, Inc., Report of the RTCA Board of Director's Select Committee on Free Flight, Washington, (1994). 2. Krozel, J., and Peters, M., "Conflict Detection and Resolution for Free Flight", Air Traffic Control Quarterly Journal, Special Issue on Free Flight, 1997. 3. Erzberger, H., and Lee, H.Q., "Optimum Horizontal Guidance Techniques for Aircraft", Journal of Aircraft, Vol. 8(2), pp 95-101, (1971). 4. Slattery, R. and Zhao, Y., "Trajectory Synthesis for Air Traffic Automation", Journal of Guidance, Control, and Dynamics, Vol. 20(2), pp 232-238, (1997). 5. Shapiro, I., and Ben-Asher, J.Z., "Near-Optimal Horizontal Trajectories for Autonomous Air Vehicles", Journal of Guidance, Control, and Dynamics, Vol. 20(4), pp 735-741, (1997). 6. Grimm, W., and Hans, M., "Time-Optimal Turn to a Heading: An Improved Analytical Solution", Journal of Guidance, Control, and Dynamics, Vol. 21 (6), pp 940-947, (1998). 7. Clements, J.C., "Minimum-Time Turn Trajectories to Fly-to-Points", Optimal Control Applications and Methods, Vol. 11, pp 39-50, (1990). 8. Betts, J.T., "Survey of Numerical Methods for Trajectory Optimization", Journal of Guidance, Control, and Dynamics, Vol. 21 (2), pp 193-207, (1998). 9. Guelman, M. and Shinar, J., "Optimal Guidance Law in the Plane", Journal of Guidance, Control, and Dynamics, Vol. 7, No. 4, pp 471-476, (1984). 10. Yuan, Pin-Jar., "Optimal Guidance of Proportional Navigation", IEEE Transactions on Aerospace and Electronic Systems, AES-33, Vol. 33(3), pp 1007-1011, (1997). 11. Hale, F.J., 'Introduction to Aircraft Performance, Selection and Design", Wiley, New York, 1984. 12. Shahzad, M., Slama, J.G., and Mora-Camino, F., "A New Approach for the Automation of Relative Guidance of Aircraft", 13th International Conference on Systems Engineering, Las Vegas, USA, Aug. 1999. 13. Bryson, A.E., and Ho, Y.C., "Applied Optimal Control", Hemisphere Publishing Corporation, Washington, D.C., (1969). 14. Youmans, E.A., and Lutze, F.H., "Neural Network Control of Space Vehicle Intercept and Rendezvous Maneuvers", Journal of Guidance, Control, and Dynamics, Vol. 21, No. 1, pp 116-121, (1998). 15. Demuth, H., and Beale, M., "MATLAB Neural Network Toolbox", Math-Works Inc., Natick. MA, (1994). 16. Mora-Camino, F., "Syst6mes de Conduite Automatique et de Gestion du Vol", Ecole Nationale d'Aviation Civile, (1995).
Current Advances in Mechanical Design and Production Seventh Cairo Universi O' International MDP Conference Cairo, February 15-17, 2000
131
SIMULATION OF TURBULENCE-INDUCED VIBRATION OF LOOSELY SUPPORTED HEAT EXCHANGER TUBES
Hassan, M.*, Dokainish, M.** and Weaver, D.** *PhD Candidate, ** Professor, Department of Mechanical Engineering McMaster University, Hamilton, Ontario, Canada, L8S-4L7 ABSTRACT Flow-induced vibrations of exchanger tubes result in tubes impacting and rubbing against their supports which leads to tube failure due to fretting wear damage. Fretting wear is correlated to the tube response obtained by analytical techniques. In this paper, a finite element model is presented and utilized to analyse the nonlinear dynamic behaviour of heat exchanger tubes subjected to crossflow turbulence. The tube/support interaction model that accounts for the effect of various tube parameters, such as clearances, friction at the supports, support stiffness, and damping has been incorporated in INDAP (Incremental Nonlinear Dynamic Analysis Program). An extensive parametric study was conducted. Simulations of the tubes with lattice-bar supports were carried out. The effect of the permanent tube-support preload arising from the crossflow drag and the tube-to-support clearances on the tube response is investigated. The results obtained show that the tube exhibits two different responses in the drag and lift directions. Tube response, impact force, and contact ratio are effectively presented in a dimensionless form. KEYWORDS Nonlinear Dynamics, Flow-induced Vibrations, Finite Element Analysis, Impact. INTRODUCTION Tube and shell heat exchangers used in nuclear and chemical power plants contain bundles of tubes exposed to parallel and/or cross-flow. Each of the tubes is supported at a number of locations to stiffen the structure and hence reduce the vibration amplitude due to fluid excitation. Supports are typically assumed to provide perfect pinned boundary conditions forcing vibration nodes at the support locations. This type of support is called support-active (SA). With this assumption, simple beam analysis can be used to predict the tube natural frequencies as well as tube responses to different excitations. However, in practice, clearances have to be allowed at the support locations to provide the manufacturing tolerances required for tube/support assembly, to allow for thermal expansion, and to facilitate construction. These clearances are a source of nonlinearity in the tube's boundary conditions. Clearances can allow tubes to vibrate freely in the support space without contacting the support. This type of support is called support-inactive (SI). Tubes with support-inactive can be susceptible to fluidelastic instability forces due to their low natural frequencies. Moreover, clearance allows impact and rubbing between tubes and their supports to take place at the support locations which can lead to tube fretting wear. In extreme cases, tube failures occur due to fatigue or thinning and splitting at mid-span as a result of tube-to-tube clashing. However, most plant-shutdowns are attributed to failures due to fretting wear at the tube supports. Problems typically arise in U-bend regions where the tube's natural frequencies tend to be low due to the ineffectiveness of supports, or in areas producing localized high velocities such
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Current Advances in Mechanical Design and Production, MDP- 7
as entrance and exit nozzles. U-tube supports need special design consideration due to the tubes' curvature. Various types of supports are being used in the U-bend region. Weaver and Schneider [ l ] reported a variety of support geometries suitable for the U-tube supports. Anti-vibration bars (AVBs) have shown effectiveness in supporting the tubes in the U-bend region by minimizing the risk of failure due to fluidelastic instability. There is virtually no reference in the literature to the dynamics of tubes with lattice-bar support configuration (triangular array). This type of tube-support arrangement is considered highly nonlinear and its dynamics are not well understood. Excitation due to fluid flow can be categorized as (a) turbulent buffeting, (b) Strouhal periodicity, (c) fluidelastic instability, and (d) acoustic resonance. The fluidelastic instability mechanism often leads to failures on a short term basis. However, turbulence buffeting can cause low amplitude tube motion resulting in a long-term fretting wear or fatigue. Taylor et al. [2] attributed the tube response found in the U-bend region of operating steam generators to the turbulence-induced forces and not to the fluidelastic instability forces. In addition, experimental investigations carded out by Iwase et al. [3] showed that when the tubes are supported according to the proper design specifications, they are only subjected to the random excitation mechanism. In addition, progressive long term wear and chemical cleaning processes cause enlargement of the tube-to-support clearances which in turn alters the support effectiveness. To increase the design reliability and to assess the service life of heat exchanger components, a study of the tube support-interaction behaviour and associated tube dynamics has to be established. Analytical tools are required to calculate the tube response and tube support interaction of actual geometries. To date, the determination of tube response and impact forces has usually been restricted to specific flow and tube conditions that resemble some practical cases. There is a need for a generalized and a systematic analysis which gives a better understanding of the nature of nonlinear tube dynamics. TUBES/SUPPORT INTERACTION Analysis of vibrations of tubes with loose supports is a nonlinear boundary condition problem. This nonlinearity in the boundary conditions results in a system with an unknown set of natural frequencies which is the main parameter used in assessing the design adequacy. The impact-sliding behaviour of heat exchanger tubes at their supports is very complex due to its dependence on the tube/support geometry as well as the fluid excitation modelling. In this paper, INDAP was used to simulate the nonlinear dynamics of tubes in crossflow. INDAP is an in-house general purpose finite element program capable of solving a large variety of nonlinear dynamics problems. The code structure and organization are described in [4]. The tube/support interaction model was implemented in INDAP to simulate the nonlinear dynamics of multi-span heat exchanger tubes. INDAP simulates tube response to external multi-sinusoidal or random excitations. The beam equation of motion is discretized in space into finite elements and in time by Newmark's method. The resulting set of equations of motion can be written as:
Equation 1 represents equations of the unknown vectors {//}, { ~ }, and { u } representing all possible displacements and their temporal derivatives. [M], [C], and [K] are the global mass, damping, and stiffness matrices respectively and { Fe(t) } is the external fluid excitation due to crossflow. The system matrices are, in general, nonlinear due to contact. The nonlinear contributions due to the contact are added to the system equations through a displacement and velocity dependent external force (Pseudo-force). The contact forces are then calculated based
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133
on the support geometry and tube response. The Pseudo-force vector for the dynamic equation is given by" {F.vt}
(2)
= - [ C x t ] {ti} - [ K NL ] {U}
Hence the equation of motion becomes: (3)
[ M ] {//} + [ C t ] {ti} + [ K t ] {u} = {F(t)} + {FNL(t)}
where subscript L refers to linear parameters and subscript NL refers to nonlinear parameters. Tube/support impact is modelled by introducing a spring and a damper to the system. When contact takes place, the tube wall and the support both deform and the impact force (F~) results. This impact force is given by the product of the equivalent tube/support stiffness and the normal tube/support overlap. Impact forces are not known since they are dependant on the displacement and the velocity of the tube. Therefore, an iterative procedure is used to calculate the time history of displacement and impact forces. Modelling of impact forces is very dependent on the contact configuration dictated by the support geometry. The mathematical modelling of the fiat-bar support used in this analysis was described in detail by Yetisir and Weaver [5]. SYSTEM DESCRIPTION Figure I shows a finite element model that consists of a straight tube fixed at one end and loosely-supported at the other end. Three tube configurations were used in this analysis. In the first configuration, the overall tube length is 2 m with an outside diameter of 0.015 m and a wall thickness of 0.0015 m. The tube mass per unit length and Young's modulus are 0.7397 kg/m and 230 GPa respectively. . . . . - -
x/L = 1.0
Fig. 1. Finite element model of the cantilever tube The tube is loosely supported by a lattice-bar support which consists of a group of bars arranged to form a diamond shaped support clearance. Lattice-bar supports can be categorized according to support offset as shown in Fig 2 where d is the tube diameter, Cr is the radial clearance of a centred tube, and e is the offset of the supports. Tubes are discretized into twenty elements. Each element is a twelve DOF beam element. d
a) No offset
b) With offset
Fig. 2 Lattice bar supports
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Current Advances In Mechanical Design and Production, MDP- 7
TURBULENT EXCITATION FORCING FUNCTION Tubes were subjected to random excitation forces in two orthogonal directions arising from turbulence in the flow field. Turbulence forces are usually characterized by their power spectral density (PSD). The bounding spectra given by OengOren and Ziada [6] gives the PSD of the fluctuating forces S r ~ ) . The PSD presenting the turbulence must be translated into a force versus time record which has the same statistical properties of the actual forcing function [7]. This is accomplished by presenting the fluctuating forces per unit length by Fourier series. The power spectral density of the fluctuating forces in the drag and lift directions are assumed to be the same. Fluctuating forces in the drag direction are composed of a steady state component representing the steady drag and a random component representing the fluctuation due to turbulence. The fluctuating forces were imposed as fully correlated in space. However, excitation forces for the two orthogonal directions (lift and drag) were fully uncorrelated. RESULTS Twelve radial clearance values ranging from 0.001 mm to 1 mm were examined. Figure 3-a shows the contact ratio (Ct) of the tube with lattice-bar support for various flow velocities. The contact ratio is defined as the ratio of the contact duration to the total time. For all flow velocities, increasing the clearance (Cr) decreases the contact ratio. The steady drag component keeps the tube in continuous contact with the support for small clearances. Further increase in the clearance results in a sharp decrease in the contact ratio then it levels off asymptotic to zero. For any clearance size, increasing the flow velocity increases the contact ratio. Data can be effectively displayed in dimensionless form, as shown in Fig. 3-b. This is accomplished by normalizing the clearance by the RMS support inactive resultant response (drs~). RMS supportinactive response is the linear response of the tube without support at the end. RMS impact force (F~mp)depends on both the level of the impact peaks and the contact duration. Figure 4-a depicts the RMS impact force values versus clearance for various flow velocities. RMS impact forces decrease linearly as support clearance increases, then levels off asymptotic to zero. For any clearance size, increasing the flow velocity increases the RMS impact force. The dimensionless RMS impact force is presented as F=FimJ(Ftur L). F, Fturand L are the dimensionless RMS impact force, RMS distributed turbulence force per unit length, and the tube length respectively. Using these dimensionless parameters, the RMS impact forces collapse on one single curve for the same support configuration, as shown in Fig. 4-b. I
Ilell Ilell
I
Ib ell ~
II
9 Confi 8. I - 4 m/s 9 Config I - 6 m/s Config l - $ m/s
...06
04
0.~ 2
~
Confilt Coati| --.i-- Coatis
00
.....
0
OI
0.2
03
04
05
06
07
011
Cr [mm]
a) Contact ratio versus clearance
09
0
0 I
02
03
0.4
05
06
0.7
0.8
0.9
Cr/drsi
b) Contact ratio versus the dimensionless clearance
Fig. 3. Effect of clearance on the contact ratio
Current Advances hr Mechanical Design and Production, MDP-7
135
O4 035 03 025
~t
l
d'mO
. co,,f,s I.~m/s 9 Config 9 Config
I - 6 m/s I - 8 m/s
At e
f
J015 01 005 0
o 0
01
02
03
04
05
06
07
Og
07
0q
Cr [rnrnl
a) R M S Impact forces versus clearance
og
09
Cr/drst
(b) Dimensionless Impact forces versus the dimensionless clearance
Fig. 4. Effect of the clearance on the RMS impact forces Figure 5-a shows the RMS tube response (dy)in the lift direction as a function of the support clearance for various flow velocities. Initially, increasing the support clearance causes the RMS tube response to decrease up to a certain support clearance value. Then the RMS tube response increases gradually. At large support clearances, the RMS response is expected to approach the RMS response of the tube with support-inactive. The point of minimum RMS response, however, shifts to a larger clearance as the flow velocity increases. For any clearance, increasing the flow velocity increases the response level. The RMS lift response and support clearance are normalized by the RMS lift and the resultant support-inactive response respectively. Data for various flow velocities are represented by a single curve using these dimensionless parameters (Fig. 5-b). Figure 6-a shows the RMS mid-span tube response (dz) in the drag direction as a function of the support clearance for various flow velocities. The RMS tube response increases linearly as the clearance increases. This behaviour is maintained up to a certain clearance after which the response approaches the RMS tube response with support-inactive. As both axes are replaced by dimensionless parameters, a single curve can be obtained, as shown in Fig. 6-b. Figure 7 shows a sample of the PSD of the tube response with zero-offset lattice supports. The spectra presented are for different clearances and all with the same flow velocity (4 m/s). For a small clearance (Fig. 7-a) the PSD of the tube response contains peaks at frequencies which correspond to the natural frequencies of the fixed-hinged configuration. For larger clearances (Fig. 7-c). the PSD of the tube response contains peaks at frequencies which correspond to the natural frequencies of the fixed-free configuration. Figure 7-b depicts the PSD of the tube response for an intermediate clearance where the mode switch takes place. Moreover, the point at which dimensionless lift and drag response change their trend corresponds to the dimensionless clearance at which mode switch takes place. The tube response spectra for the case of the support with a 5% offset of the tube length (e/L=0.05) are shown in Fig. 8. For small clearances (Fig. 8-a), the PSD contains peaks at frequencies corresponding to a combination of two linear modes. These modes correspond to the pinned configuration at the support locations. For larger clearances (Fig. 8-c), the PSD of the tube response contains peaks at frequencies corresponding to the natural frequencies of the fixed-free configuration. Figure 8-b depicts the PSD of the tube response for an intermediate clearance where the mode switch takes place.
Current Advances in Mechanical Design and Production, MDP-7
136
4O
~
C o n f i S . I . 4 m/s !
35 ~ ' 4 - 30
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* 9
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.
.
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b) Dimensionless RMS tube response versus the dimensionless clearance
a) RMS Tube response versus clearance
Fig. 5. Effect of the clearance on the RMS tube response in the lift direction ,oo - ' - - c - , , 700
+
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tram]
: . . . . 0.7 0.11
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b) Dimensionless tube response versus dimensionless clearance
b) RMS tube response versus the clearance
Fig. 6. Effect of the clearance on the RMS tube response in the drag direction
o
enxluen~
1o
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do
g
to me oo e~iuency
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~m
/m
40
Io
to
",oo
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a) Cr = 1.0 mm
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o
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,0
m
m
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u.
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Fig. 8. Response spectra of the tube with lattice-bar support (5% offset)
-
.
9
Current Advances in Mechanical Design and Production, MDP-7
137
To investigate the applicability of scaling the dimensionless results reported herein, additional configurations with different tube lengths and diameters were analysed. These configurations are designated as configuration 2 and 3 respectively. The geometric properties are listed in Table I. Configuration
Tube Length mm
Outer diameter mm
308
6.35 15.88
Lineal density kg/m
Modulus of elasticity GPa
3.86
0.16912
105
14.25
0.32
106
Inner
diameter mm ......
2
..........
........
617
..
Table. l Geometric and material properties of test cases Figure 9-a-d show contact ratio, dimensionless RMS impact forces, lift response, and drag response for all configurations. The general behaviour of the response is unchanged and an excellent agreement with the first configuration is obtained. 04
Config Config Config
~~i~o - _
9 Config *
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-
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m/s m/s m/s m/s
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b) Impact Forces
09
08 07
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-
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0.6
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f) 4
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d) Drag response
Fig. 9. Dimensionless tube response
0 7
0.8
l) 9
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Current Advances In Mechanical Design and Production, MDP- 7
CONCLUSION Simulations for a loosely supported heat exchanger tube subjected to crossflow turbulence excitation were presented. The effect of the clearance which exists between the tube and the lattice-bar supports was investigated. The study indicates that increasing the support clearance results in decreasing the impact forces and the contact ratio. On the other hand, lift response of the tube slightly decreases for a range of small clearances then increases producing a point of minimum response at a certain clearance. Drag response increases linearly as clearance increases. The tube response, impact force, and contact ratio were effectively represented in a dimensionless form. It was demonstrated that the results can be scaled to predict the nonlinear response of tubes with different geometrical and material properties and subjected to different flow conditions. The proposed dimensionless parameters are of practical interest since the tube nonlinear-response can be correlated to random excitation due to turbulence. The proposed dimensionless parameters can be used as a guideline for design purposes and for assessing the in-service heat exchangers. Ultimately the method would be extended to account for both turbulence and fluidelastic excitations. REFERENCES 1- Weaver, D. S., and Schneider, W.,"The Effect of Flat Bar Supports on the Crossflow Induced Response of Heat Exchanger U-Tubes," ASME Journal of Pressure Vessel Technology, Vol. 105, pp 775-78 l, (1983). 2- Taylor, C., Boucher, K., and Yetisir, M.,"Vibration and Impact Forces Due to Two-Phase Cross-Flow in U-bend Region of Nuclear Steam Generators," Flow Induced Vibrations, Bearman, P. (ed), Balkema, A., Rotterdam, pp. 401-412, (1995). 3- Iwase, T., Sunami, T., Matsutani, K., Nakamura, T., Mureithi, N., Tsuge, A., Watanabe, Y., Tomomatsu, K., and Takaba, O.,"Flow-Induced Vibration of a Tube Array in the Inlet Region of A High Performance Steam Generator (Part l : Turbulence Induced Vibration)", ASME Winter Annual Meeting, Dallas, Texas, Vol. l, pp 265-272, (1997). 4- Dokainish, M. A.,"Incremental Nonlinear Dynamic Analysis Program: Verification Manual," INDAP Manual, McMaster University, Hamilton, Ontario, Canada, (1987). 5- Yetisir, M., and Weaver, D. S.,"The Dynamics of Heat Exchanger U-Bend Tubes With FlatBar Supports," ASME Journal of Pressure Vessel Technology, Vol. 108, pp 406-412, (1986). 6- Oengoeren, A., and Ziada, S.,"Unsteady Fluid Forces Acting on a Square Tube Bundle in Air Cross-Flow", Symposium on Flow-Induced Vibration and Noise, Vol. l, pp. 55-74, (1992). 7- Shinozuka, M.,"Digital Simulation of Random Process in Engineering Mechanics with the Aid of FTT Technique," Stochastic Problems in Mechanics, University of Waterloo Press, pp. 227-286, (1974).
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
139
EFFECT OF PORT PLATE SILENCING GROOVES ON PERFORMANCE OF SWASH PLATE AXIAL PISTON PUMPS Kassem, S.A.* and Bahr, M.K.** *Professor, **Graduate student, Mechanical Design and Production Department Faculty of Engineering, Cairo University, Giza 12316- Egypt.
ABSTRACT In this paper a mathematical model is developed for the kinematics, piston chamber pressure and output flow rate of swash plate axial piston pumps with conical cylinder blocks. Software is developed to study the effect of the port silencing grooves on these variables. The study led to a proposed geometry of the suction and delivery silencing grooves that causes gradual rise and drop of the piston chamber pressure, with cavitation avoided, for a pump with certain dimensions. The recommended groove dimensions render less fluctuation in the output flow rate for a nine-piston pump. The developed model can be used to conduct a similar study for any swash plate axial piston pump. KEYWORDS Swash Plate, Axial Piston Pump, Chamber Pressure, Silencing Grooves, Instantaneous Flow Rate. 1. INTRODUCTION Various aspects of the static and dynamic characteristics of constant and variable geometric volume swash plate axial piston pumps were investigated during the last three decades. The variation of cylinder pressure during one complete revolution of the cylinder block was studied and shown to be much influenced by the geometry of the silencing grooves existing at the ends of the suction and delivery ports [ 1,2,5,6]. Effects of sloping and non-sloping, square or semi-circular silencing grooves on the variation of cylinder pressure were investigated theoretically [5] and shown to yield poor characteristics, especially when the initial depth is large or the slope is small. Triangular cross-section grooves were found to create low reverse flow and less overshoots and undershoots of cylinder pressure [ 1-6]. When the pump suction pressure was atmospheric, cavitation in the chamber was recorded, in some cases for very for small intervals. The investigations dealt mainly with pumps with cylindrical cylinder blocks, where the piston line of stroke is parallel to the pump axis of rotation [1-6]. Pumps with conical cylinder blocks are now widely used in both industrial and mobile applications. In these pumps the piston line of stroke is inclined to the pump axis of rotation in order to reduce the piston inertia force that tends to detach the piston from the slipper pads or the swash plate. Such a technique allows driving the pumps at higher speeds, which increases the pump specific power.
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Current Advances in Mechanical Design and Production, MDP- 7
Edge and Darling [ 1] investigated the cylinder pressure transients in pumps with sliding valve plate using a developed model for axial piston pumps of cylindrical cylinder blocks which incorporates bi-directional port plates with sloping semi circular cross section silencing grooves. The pump suction pressure during their analysis was boosted while the delivery pressure was constant. The evaluated cylinder pressure overshoot and undershoot was considerable. The drop of cylinder pressure during the start of suction stroke is evidently not suitable for pump operation, if the pump suction pressure is nearly atmospheric; the case widely met in industrial applications. Modification of silencing grooves to suit this case was not studied. Marring [2] investigated the variation of the pump driving torque assuming a shape for the silencing grooves that causes linear variation of the cylinder pressure between the valve plate delivery and suction ports at a certain value of the delivery pressure. At the other values of the delivery pressure, the assumed shape of the grooves was shown to yield noticeable overshoots and undershoots in the cylinder pressure. Kaliafetis and Costopoulos [3] studied both theoretically and experimentally the static and dynamic characteristics of a standard variable geometric volume swash plate pump with constant pressure regulator. Modeling and designing a variable geometric volume axial piston pump was carried out in [4], and the effect of some design parameters on the pump performance was studied theoretically. Effect of the suction pressure, rotational speed, shape and sealing condition of the valve plate on cavitation in an axial piston pump was studied experimentally by Atsushi [5]. Calculation of the moment acting on the swash plate is presented in [6]. In the present work, a comprehensive theoretical study is carried out to investigate the effect of the triangular silencing grooves dimensions on the piston chamber pressure and pump flow rate fluctuation for a nine-piston swash plate pump with conical cylinder block, in order to reach a recommended port plate configuration. 2. PUMP DESCRIPTION Figure 1 shows schematically the pumping mechanism of the investigated swash plate axial piston pump, which has an odd number of pistons nested in a circular array within the block at equal intervals. The cylinder block is rotated by means of the pump driving shaft, and is held tightly against the valve plate under the effect of the supply pressure and the force of the cylinder block compression spring. A ball-and-socket joint connects the end of each piston to a slipper pad, which is kept always in contact with the swash plate. During cylinder block rotation, each piston periodically passes over the discharge and intake ports on the valve plate. Since the slipper pads are held against the inclined plane of the swash plate, each piston undergoes an oscillatory motion in and out of the cylinder block. As any piston passes over the intake port, it withdraws from the cylinder block and fluid is drawn into the piston chamber. As the piston passes over the discharge port, it advances into the cylinder block and pushes the fluid out of the piston chamber. The pivoting point, around which the swash plate swings, is the center of the pitch circle of the pistons spherical heads when the swash plate angle of inclination is zero. 3. PUMP MODELING 3.1 Pump Kinematics The displacement Sk of the kt~ piston as a function of its angle of rotation Ok, measured from the top dead center, can be deduced as follows. Take the swash plate pivoting point as the origin of the initial frame of reference of the pumping mechanism, with Z0 coinciding with the axis of the driving shaft and Y0 the axis around which the swash plate is swinging, as
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shown in Fig. 2. The figure shows also the other needed five frames of reference XIYIZ~ to XsYsZs. Five steps of rotations and translations are then carried out to get the coordinates
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Current Advances In Mechanical Design and Production, MDP-7
of the k th piston spherical head center at the angular position Ok relative to the initial frame of reference. These coordinates must satisfy the equation of the swinging plane, which is inclined by an arbitrary angle ot to the vertical plane. The first transformation matrix Tol represents the transformation from the initial frame of reference to the first one, which governs the translation along the Z0 axis for a distance Lt. The matrix Tt2 represents the rotation around Zt by an angle Ok. The matrices T23, T34and T45 represent the translation along the X2 axis a distance R2 ( R2 = 0.5 D2 ), the rotation around Y3 through an angle [3, and the translation along Z4 for a distance-L3, respectively, the values of the transformation matrices are given in Appendix 1. Now,
[cos0~ co~
Tos=To,xTI2xT2,xT34xT4,=/si~cosl3
-si~
cosO
IS,ore
0
0
-cos0~ sin{3 -sin0k si~ co~ 0
cos0 sinl3+ cos0k'] L3ksin0kco + sin0k|
and relativeto the initial frameof reference: Xsk = L3k COS0k sinl~ + R2 cos0~
(l)
Zsk =-- L3k cosl3 + L,
(2)
These coordinates must satisfy the equation of the swash plate inclined surface (3)
x 0 tan a + z o = 0
Substituting the values of XSkand Zsk from equations 1 and 2 into equation 3 results in, L3k = ( R z cos Oktan ot + L, )/(cos [3- cos Ok sin [3tan ct)
(4)
The displacement sk of the kth piston given by
(5)
s k = L3k - L I / cos [3 where : Ok = tot + 2n (k-l)/z
and
[3 = tan "~0.5(D,- D2)/L ~
A program was developed to calculate the displacement sk of any piston as function of Ok using the aforementioned relations. The program shows simultaneously, in an animated way, the swash plate and the piston movements during one revolution of the drive shaft, and plots the piston displacement, velocity and acceleration against 0 k for any constructional parameters of the pump. 3.2 Piston Chamber
Pressure
The variation of each piston chamber pressure during one complete revolution of the pump shaft is derived assuming that the pump rotational speed is constant, the pump suction and delivery pressures are constants, the inertia effect of the oil column inside the piston chamber is negligible, and that the total instantaneous leakage flow rate out of the piston chamber QLk is proportional to the piston chamber instantaneous pressure Pk. Applying the continuity equation to the control volume of the piston chamber shown in Fig. 3, we get:
Current Advances in Mechanical Design and Production, MDP-7
where
and
143
dsk Pk Vck dpk Osk + Ap "-~'- = Odk + ~ + ~ " R L B dt
(6)
Qdk =Cd" Adk~][2[(Pk-Pd)]/P]" sign( Pk "Pd )
(7)
Qsk = Cd" Ask~/[2i(Ps Pk)t/ P]" sign (Ps "Pk )
(8) (9)
Vck = Ap (0.5 k - s k) + Vo
To evaluate the values of Adk and Ask at any Ok, the dimensions of the port plate suction and delivery ports should be known. Knowing the variation of Adk and Ask with Ok, and for given constructional and operational parameters, the value of each piston chamber pressure pk and delivery flow rate Qdk at each Ok can be calculated using equations 6 to 9. Now, the total pump delivery flow rate Q at any Ok is given by z
Q = ~-'~Qdk
(lO)
k---!
4. PUMP SIMULATION A software package based on Matlab was developed to calculate first the displacement, velocity, and acceleration of the piston at any angular location Ok using equations 1 to 5. The program plots the piston displacement, velocity, and acceleration versus Ok. This was carried out for a 9-piston pump with the following dimensions in mm; D!=71.75, D2=60.20, D3=54.70, dp=17.00, L1=76.60, L2=66.10, Lc=57.30, Lp=59.10 and 0p=10 ~ (semiangle subtended by the cylinder port ) when running at 1450 rpm. Simulation results are shown in Fig. 4, which shows that the motion is nearly harmonic. The piston motion is simple harmonic only when 13= 0. The developed software also calculates the piston chamber pressure and the pump delivery flow rate. For this purpose, the shape of the suctionand delivery ports should be known. Figure 5 shows the shape of a port plate, which is of practical importance. The two silencing grooves at the ends of each port are triangular, with two equal sides that meet at right angles. The angles at which the silencing grooves start and end, as well as the other angles of interest on the port plate, are also shown in this figure. A separate program first calculates and plots the values of the ports areas Adk and Ask for one complete revolution of the cylinder block. The calculated areas are saved in data files to be read when needed, to reduce the required memory size and computational time when pump simulation is in run. The program allows changing any angle on the port plate using a user interface. Figure 6 shows this interface and the variation of the porting areas for the shown values of the angles. With the piston motion determined, and Adk and Askcalculated, the pump variables concerned with can be calculated by solving equations 6 through 10 numerically. This has been carried out for the pump with the aforementioned parameters. The obtained results are presented in Figs. 7 to 11. It is to be noted that the simulation process is terminated when the pressure Pk is of any value less than zero, since cavitation would then develop within the piston chamber, and the mathematical model would be invalid.
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Current Advances in Mechanical Design and Production, MDP- 7
Fig. 6. User interface for calculating port areas. Figure 7 shows the variation of chamber pressure with Ok for three values of the groove angle to. It shows that the chamber pressure increases with pump rotation and overshoots the delivery pressure, then it decreases to the value of the delivery pressure. This can be explained as follows. At the early stages of the delivery stroke the piston chamber is isolated from both the suction and delivery ports. During this stage, the piston advances in its chamber and compresses the fluid causing its pressure to rise at a rate depending upon the piston velocity, piston area, chamber volume, oil bulk modulus and leakage flow rate. After a certain angle of rotation the piston chamber communicates with the delivery port through the silencing groove. Fluid flows from the delivery port into the piston chamber if the delivery pressure is higher than the chamber pressure, and in the opposite direction if the delivery pressure is less than the chamber pressure. In either case the flow rate depends upon the groove dimensions and the differential pressure. The chamber pressure during this stage can be controlled by the proper choice of the dimensions of the grooves and the angle through which this precompression period; namely 0 = 01d - 0p, occurs. When the chamber is fully connected to the delivery port, the chamber and delivery pressures are equal. Figure 7 shows that with narrow and shallow grooves, i.e when to is small, the overshoot in the chamber pressure is high. When the piston approaches the end of the delivery stroke, the chamber pressure is seen to rise again above the delivery pressure, since the silencing groove area is now decreasing with cylinder block rotation till it attains zero, while the piston is still forwarding in the chamber compressing the oil inside it. Maximum chamber pressure occurs when 0 = 180~ During the initial stages of the suction stroke, the chamber pressure is seen to decrease to values even less than zero, indicating the occurrence of cavitation. Cavitation conditions are worse when to is small, since the notch effective area through which suction occurs increases in this case at a low rate. This would not allow filling the chamber with fluid at the appropriate rate which copes with the piston velocity. Cavitation occurrence clearly depends on the silencing groove angle to and the angles at which the silencing grooves start and terminate. A wide and deepo notch is seen to be better regarding the variation in the chamber pressure. Increasing to from 5 to 15~ is seen to improve the variation in the cylinder pressure, which would be reflected positively on pump performance and noise. It also improves the fluctuations in the output flow rate. For the values of to higher than 15" ( results are not presented here ) no further improvement was recorded. On the other hand, even for to - 15" cavitation is seen to occur at the start of the suction stroke. To eliminate this, the effect of the other constructional parameters on the chamber pressure are investigated. Figure 8 depicts the effect of the angles 0~d and 0~s on the piston chamber pressure, when the notch length is constant and to = 15". Increasing 010 and 0ms while keeping the valve plate symmetrical is seen to be of detrimental effect on the chamber pressure and the output flow rate. Several computational runs were
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145
carried out to find the valve plate geometrical parameters which yield performance. Some of these results are presented in Figs. 9 to 11. Figure 9 cavitation can be avoided when 0nd = 0is = 10 ~and 02d = 0:s = 20 ~ , for a delivery 10 MPa. Keeping 01d = 0is = 10 ~ and increasing the notch length; namely 02d and to cause cavitation at the beginning of the suction stroke as seen in cases 8 and 9.
acceptable shows that pressure of 02s, is seen
When the pump supply pressure was increased to 20 MPa and 30 MPa, the assumed geometry of the grooves in case 7 yielded poor performance. Better perform.ance at the various delivery pressures was recorded for the following angles: q~ = 15* 0nd = 15,02d = 25", Ois = 10* and 02s = 19", and the silencing grooves at the ends of the ports are eliminated, with Oad = O 4 d - 165* and 03s = 04s = 169 , as shown in Fig. 10. In this case, however, the fluctuation in the output flow rate is seen to be high when the delivery pressure is 30 MPa. If this is not acceptable, the dimensions given in Fig. I 1 can be adopted since the resulting flow rate fluctuations are less. Case I tp:5 ~
Case 1
= 9,,o
I
Case 2 q ~ : 100
,, 9
,
'
Case 3 q ~ : 15 ~
01d : eli
=
15
Cne 3
ii-,
',
Max Overshoot
:-"
~
.,:;,
~so
j,. ,,-
9171%
i
'<-'.'<-".2
Max Overshoot
_
I
90 1 %
-g Max O v e r i h o o t
920%
I
.,,,,_,,/_.L,.,~:. ,i
'L
d
".
0~d= 02, 35 ~ =
03d = 03 s
--
145 ~
O4d = 0 4 s
-
165
ml
:i
~ ":
% F l u c t u a t i o n = 49 % i _.
% Fluctuation = 30% . . . . . .
I
'
% F l u c t u a t i o n = 15% ,n oo ~s ~oo m Plllell .sulpd~rd l s l p l t C m N i t , Ill Ikllp'ml}
....
Fig. 7. Effect of notch angle on piston chamber pressure and pump flow rate. Case 4 Old = Os, = 15 ~ 02d = 02s = 25 ~ 03d-- O3s -- 155 ~ 04d = O4s = 165 ~ Case 5 Old =
0Is
""
Case
6
i
i. Max Overllhoot ]
-- 2 0 ~ ..t
02d = O2s = 3 0 ~
C,.4
i
&$
9
.
a~
9
21%
~)i
i,
i
Max Overshoot
180
PIt4oeiqulor dbpl~aneet, i( ~IV~ ) _
'
I t
99 0 %
u
i
d0 /
J3
,
re
Max Overshoot 41~
= ISS
(10
!
200% 1;0'
"
PJetonelqJ~derdJ~cemelt, 0( delp'ee) .
.
.
.
.
))*~
.
O3d-" 03s -- 150 ~ O4d-" 04s Case
--"
160
25 ~
=
09,
:
02d =
02s
"- 3 5~
0,d
~
6
1 ~1
. %n.~u.,on-43,~
03d = 03s ---- 145 ~ O4d-- 04s
-"
155 ~
for tp = 15 ~
F i g . 8. Effect of transition periods on piston chamber pressure and pump flow rate.
Current Advances In Mechanical Design and Production, MDP-7
146
Case 7
02d-- 02s -- 20*
i "0
i:
Case 8 02 d "- 02S "- 25 ~
I
c,, ,!
Max Overshoot
90.16%
i n,
i:
I
c_.
Max Overshoot
ii
92%
c_.
Max Overshoot
92%
Case 9 02d = 02s -" 3 0 ~ for
q~= 15 ~
01d--'- 0Is -" 100
li
03d = 03, = 169*
04d----" 04s = 169
~
1
"0!
% Fluctuation ~ F ~ ? = l18 ~ e%/ ' ~
~.
% Fluctuation = 10%
% F l u c t u . l o n - 10%
,!'
Fig. 9. Effect of notch length on piston chamber pressure and pump flow rate. Case I0
Case11
~rselO
Pd = lO0 bar C a s e 11 Pd = 2 0 0 bar
Max Overshoot
i
91.7
Max Overshoot
%
i: i "*
90.11%
Case 12
I
CalVe 12
Ma. o v . . , o o , . 0 0,%
!
Pd -- 3 0 0 bar For
I
I 9
q~= 15 ~
~s
m
,35
,CO ~
~o
,,~
9
~s
w
,~s
m
, m
~0
3,s
m
9 r~.~.g~,~..~.,~k~,-
Otd = i 5 ~
O~s = 10 ~ 02d --'- 25*
02s = 19 ~ 03d = 04d = 165*
%~
I
9 14%
% Fluctuatlon = 20%
03s = 04s = 169 ~
Fig. 10. Effect of load pressure on piston chamber pressure and pump flow rate. Case Pd Case Pd = Case Pd---
ZI
13 100 bar 14 2 0 0 bar !5 3 0 0 bar
itr.i 9 ;-.~~.;~',~::,
For
Cgls C a s e 14 Max Overshoot Max Or 9
3,.
"I
90.7%
"1 i
Max Overshoot u 0.14%
"
q~= 15 ~ Old = 2 0 ~
Ozs = 10 ~
J:
02d = 3 0 ~ 02s = 19 ~ ~
%F I ~
03, = 04s = 169*
g.-~.,~,,,~-.~.
O3d
-- 04d "- 165
9 17%
1
. . . . . . . .
Jr-C::, "
Fig. 11. Variation of piston chamber pressure and pump flow rate for the proposed groove geometry.
Current Advances in Mechanical Design and Production, MDP. 7
147
5. CONCLUSION In this paper a mathematical model describing the motion of the pistons of a swash plate axial piston pump with conical cylinder block was developed. The mathematical model describing the piston chamber pressure variations and the pump output flow rate was developed as well. Simulation of the performance of a pump with nine pistons was conducted using a developed software. The shape of the suitable suction and delivery ports was determined theoretically in view of the simulation results. This shape of ports showed to cause gradual rise and drop in the piston chamber pressure, with cavitation inside chambers avoided, for the whole range of delivery pressures. Pump noise can thus be significantly reduced. The determined ports configurations, on the other hand, showed to render relatively small fluctuations in the output flow rate. The developed models and software can be used at the design stage of swash plate axial piston pumps with any dimensions. AKNOWLEDGMENT The authors would like to thank Prof. Dr. G. Rabie and Eng. K. Hamza for their help. REFERENCES 1. Edge, K.A. and Darling, J., "Cylinder Pressure Transients in Oil Hydraulic Pumps with Sliding Plate Valves", Proc. Instn. Mech. Engrs., Vol. 200, No. B 1, (1986). 2. Marring, N.D., " The Torque on the Shaft of an Axial Piston Swash Plate Type Hydrostatic Pump", Journal of Dynamic Systems, Measurement, and Control, Vol. 120/57, (1998). 3. Kaliafetis, P. and Costopoulos, T., " Modeling and Simulation of an Axial Piston Variable Displacement Pump with Pressure Control ", Mechanical Design and Control System Section & Machine Design Laboratory, Mechanical Engineering Department, National Technical University of Athens, Patission 42, 106 82 Athens, Greece, (1994). 4. Marring, N.D. and Johnson, R.E., "Modeling and Designing a Variable-Displacement Open-Loop Pump ", Journal of Dynamic Systems, Measurement, and Control, Vol. 118/267, (1996). 5. Atsushi, Y., " Cavitation in an Axial Piston Pump ", Bulletin of the JSME, Vol. 26 No. 211, (1983). 6. Kiyoshi, I. and Masakasu, N., " Study of the Operating Moment of a Swash Plate Type Axial Piston Pump ", The Journal of Fluid Control, Vol. 22, Issue 1, (1994). NOMENCLATURE
Symbol
Description
Ad(As) Ap ap B Cd D!
Delivery ( suction ) porting area. Piston cross-section area. Piston acceleration, Effective bulk modulus Coefficient of discharge. Diameter of cylinders pitch circle at the cylinder block base. Diameter of cylinders pitch circle at the cylinder block top. Diameter of pitch circle of valve plate
D2 D3
Value
Unit
1.3xl 09 0.611 0.07175
m2 m2 m/s 2 Pa m
0.0602
m
0.0547
m
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148
Piston diameter Piston number in the arrangement of the piston group. Length. Length. Variable length. Cylinder length. Prime-mover drive speed Pump delivery pressure. Piston chamber pressure. Pump suction pressure. Pump total output flow rate. Delivery flow rate from one cylinder. Leakage flow rate out of one cylinder. Suction flow rate into one cylinder. Resistance to leakage out of cylinder. Piston displacement. time Additional volume. Cylinder volume. Piston velocity. Number of pistons. Swash plate angle of inclination. Cylinder block conical angle. Silencing groove notch angle. Angular position of the k"* piston. Oil density.
dp k L! L2 L3 Lc n
Pd Pk
Ps
Q
Qd
QL Q~ RL Sk
t
Vo Vc Vp z ct
Ok P
0.017
In
0.0766 0.0661
m m m m
0.0573 1450 100x 105
rpm Pa Pa
Pa
0.05xl 05
I x I 013
m3/s m3/s m3/s m3/s Pa/(m3/s) m s
m3 m3
1x 10.6
m/s 9
850
kg/m 3
APPENDIX 1
ilOOoI FlcOSOo sinOo01
0 1 0 0 Tot = 0 0 1 L, 000 0
100
T23_ 0 1 0
R2
0
00 1 0 000 0
T, 2 = si~0
[cosl3
T34=[
O
[Sio[3
cos0 0 0
0-sinl3
0
1
0
0
O cos[3 0 0
1
0 0 1 0 0 1
100 0 10 0 T45= 0 1 -L 3 O0 0
Section 11
SOLID MECHANICS
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Current Advances in Mechanical Design and Production Seventh Cairo UniversiO' International MDP Conference Cairo, Februar)' 15-17, 2000
151
THE MODELING OF INELASTIC COMPRESSIBILITY AND INCOMPRESSIBILITY USING THE VISCOPLASTICITY THEORY BASED ON OVERSTRESS (VBO) Krempl, E. and Ho, K. 1
Mechanics of Materials Laboratory Rensselaer Polytechnic Institute Troy, NY 12180-3590 [email protected] _
ABSTRACT An isotropic, small strain theory of viscoplasticity based on overstress (VBO) with inelastic compressibility is formulated that can easily converted to inelastic incompressibility. The theory consists of a flow law the inelastic part of which depends on the overstress, in addition two tensor-valued, stress-like state variables and a scalar, stress like state variable are introduced together with their growth laws. A simplified version needs five constants to represents kinematic hardening, cyclic neutral behavior and nonlinear rate sensitivity, creep and relaxation in the inelastic incompressible case. This version is applied to the modeling of uniaxial monotonic and cyclic loading and ratcheting of AISI 1026 steel with good results. KEYWORDS Viscoplasticity, Plasticity, Inelastic Compressibility, Ratcheting, Numerical Experiments. 1. INTRODUCTION Before the advent of economical computing power inelastic stress analyses of components was not frequently performed. Detailed inelastic analysis was too time consuming and too expensive. Engineering practice was then and still is to a large extent to perform an elastic analysis and to apply large safety (or ignorance) factors. This procedure did not result in the economic use of materials. The electronic revolution has brought powerful workstations to the design office and inelastic analysis is now possible within the economic constraints. Up-todate material models, the constitutive equations are needed. They should reflect the demand for safety and reliability, computational needs, the vastly improved mechanical testing and recording procedures, the progress in materials manufacture and the advancements in materials modeling. A constitutive equation that reflects this reality does not yet exist. Indeed it has been stated that the description of the inelastic material behavior is now the weakest link in inelastic analysis. Present material models for inelastic analysis are rate-independent plasticity for low homologous temperatures, say below 0.4. (Homologous temperature is defined as the ratio of the test temperature t ~ the melting temperature measured in degree Kelvin.) At temperatures wNow at the Yeungnam University, Korea.
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Current Advances in Mechanical Design and Production, MDP- 7
exceeding this ratio creep theory is used as a model. It has a different characteristic than plasticity. In some instances plasticity and creep theory are combined and used in inelastic analysis. When the predictions of these theories were compared with newly performed experiments systematic deviations were observed, especially for plasticity-creep interaction, see [1]. This and other observations led to the development of so-called "unified" state variable theories. These theories are labeled "unified" since no separate repositories for creep and plasticity are introduced. Rate dependence of inelastic deformation is fundamental and rate independence is modeled by making rate dependence very small. The modeling of plasticity-creep interactions causes no problems due to the unified nature of these theories, see [2] for a recent review. The senior author and his students have developed the viscoplasticity theory based on overstress (VBO). It has been applied to metals and alloys at low and high homologous temperatures and has recently been extended to the modeling of solid polymers. The purpose of this paper is to introduce the isotropic, small strain theory with and without inelastic incompressibility. A simplified version of VBO which needs only five constants for the characterization of inelastic behavior is then introduced and applied to uniaxial, monotonic and cyclic loading as well as ratcheting of AISI 1026 steel at room temperature with good results. The overstress dependence of the inelastic strain rate can model many observed, "paradoxical" deformation behaviors, see [3,4] among others. The nonlinear overstress dependence is the basic reason why nonlinear rate dependence, creep and relaxation can be reproduced. 2. THE VBO MODEL The model is presented for isotropy, compressible and incompressible inelastic behavior, low and high homologous temperature for isothermal and variable temperature conditions. BOX 1 contains the basic equations valid for any constant inelastic Poisson's ratio. The model does not contain any strain measure and is written in terms of rates with the current configuration as reference configuration. The finite deformation version differs from the small strain version only by the objective stress and strain rates, see [5]. These objective rates contain spins derived from the velocity gradient or its anti-symmetric part. In some motions the spins are zero. This is the case for the important state of uniaxial deformation. For this special state of deformation the finite deformation version and the small strain version are identical. It is therefore important to remember that all stresses and strains are to be interpreted as true quantities. The true (Cauchy) stress and the true strain rate, the rate of deformation, the symmetric part of the velocity gradient, appear in the flow law, see [6]. By using the current configuration as the reference configuration several important tests can be performed numerically, see [7]. The differences between a constant load and a constant stress creep test can thus be simulated. In situations where spins are nonzero the strains have to be restricted to be small and material derivatives are used. The VBO theory consists of the flow law that is written in rate form. The total rate of deformation, the true strain rate, is the sum of the elastic and inelastic true strain rates. The elastic strain and the inelastic strain rates are given in terms of stress rates and a function of the overstress, the difference between the stress and the equilibrium stress, respectively. A dependence of the inelastic rate of deformation on the drag stress is also possible but not
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BOX 1. Isotropy; inelastic compressibility; variable temperature; static recovery; positive, neutral and negative rate sensitivity. FLOW LAW f [x] = function ofx
l+r/
"
d l+v'
dt
Ek[r/o]
E ~,t
E
(o'- g ) -
T/
(tr(o'-g))l
Ek[r/D]
+aT1 ~h
~,lSeOStty function
~,,,
INVARIANTS F2-----((l + 7/)2tr ((dr- g)(dr- g)) + (,1' - 2T/)(tr (dr - g))2 )/(1 + 2,/2 )
o- /" .
.
F[~DI
~
~
= ((1 + r/) 2tr(gg)+ (7/2- 2r/)(trg) 2)/(1 + 2~72) GROWTH LAW FOR THE EQUILIBRIUM STRESS
~r r
-
_()
...... h.,p~ late
,OT .
t'UIICl|OII
.
cr-g k[F/D]
r g- f k[F/D] A+
~anable temperature
nr~
d ~ a m l c recoven'
GROWTH LAW FOR THE KINEMATIC STRESS .it"
= E, Ek[1-'/D]
F
kmcmatic stress rate
SIMPLEST GROWTH LAW FOR THE ISOTROPIC STRESS = A,. (A I - A)~, or rate dependent form isotropic stress rate
static recoveD'
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Current Advances in Mechanical Design and Production, MDP-7
implemented here. When after a deformation all rates have died out the stress is equal to the path-dependent equilibrium stress, which can be nonzero. VBO is therefore a solid. The evolution law of the equilibrium stress contains two other state variables, the tensor valued kinematic stress, and the scalar valued isotropic stress. They have the same function as the quantities of the same names in rate-independent plasticity. The kinematic stress also sets the tangent modulus at the maximum strain of interest. The general equations are displayed in BOX 1. The growth laws for the state variables are written in stress formulation rather than the usual strain rate formulation. Strain rate quantities can be easily introduced when the definitions of the elastic and inelastic strains are taken from the flow law and substituted in the growth laws for the state variables. We prefer to work with the stress formulation, see [8]. The set of equations represent a modified standard linear viscoelastic solid where the linear response for an infinitely slow loading has been changed to the nonlinear, hysteretic growth of the equilibrium stress. The viscosity constant has been replaced by the viscosity function k [G/D], which is a decreasing function of the overstress invariant/drag stress ratio. (The flow function F[ ] can be used instead, see BOX 2) The inelastic rate term is similar to the elastic part except for the absence of rates, tensorial dependence on the overstress (s - g ) and the nonlinear viscosity function.
BOX 2. Simplified VBO. Isothermal, inelastic incompressibility, positive rate sensitivity; low homologous temperature. THE SET OF DEVIATORIC EQUATIONS
=-f-
r r'
A+
+ O-
/ ' = E,r[
F" :1 = 0; no isotropic hardening; A = A,
Hydrostatic hypo-elastic relation
l-2v
tr~ = - - - - - - t r & E
with B=2.5x10 "s l/s, universal due to dimensional considerations; D, m, !h, E, ~:, E, and A
to be determined, see Table.
A power function is selected for F[ ]. Other functions can be used provided they fulfill the following conditions: F[0] = 0; and increasing. Examples: sinh[ ] and (sinh[ ])m, m>0.
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155
The constant inelastic Poisson's ratio h allows for inelastic compressibility. This capability is needed for modeling solid polymers and other materials that are compressible in the inelastic range, see [6,9]. When h is set to 0.5, inelastic incompressibility is reproduced, see BOX 2. Although not necessary, the inelastic strain rate can then be written using the usual stress deviators, see BOX 2. The growth laws for the equilibrium and the kinematic stresses can be split into a deviatoric and a hydrostatic part, see BOXES 1 and 2. When h = 0.5, the inelastic Material Constan~
hydrostatic relations between the state variables are not needed. The inelastic strain rate is written in terms of deviators. The Af 170 MPa overstress invariant G d is normalized so that it is equal to the B 2.5• overstress magnitude in a uniaxial setting. Normalizing for D 101 MPa E 181 517 i P a torsion would alter the divisor (1 + 2h2), as is the case in regular E, 1380MPa plasticity. The use of the elastic modulus E in formulating the ,m, 15 y 0.45 inelastic strain rate is for convenience and symmetry. The value ...... of the product Ek[] matters in modeling. Details are explained in [10] and in [9]. The modeling of a particular material the constants and functions of the model must be determined. Experimental results are needed here. This will be explained using the simplified version of VBO. It is listed in BOX 2. i for Simulation s
2.1 Simplified Version Deleting all embellishments such as variable temperature, static recovery terms, negative rate sensitivity and isotropic hardening yields the simplified version listed in BOX 2. The equations in BOX 2 can be derived from those in BOX l by setting h = 0.5 and introducing the deviators. The widely used power law is used to represent the function F. Due to isotropy, the models discussed need two elastic constants, the flow or the viscosity function and the shape function in addition to the growth law for A and the tangent modulus at the maximum strain of interest. The flow or the viscosity function is responsible for reproducing nonlinear rate sensitivity and nonlinear creep and relaxation. The shape function controls the transition from the initial quasi-elastic region to fully developed flow. A constant can replace this function, see [l l] and [12] and BOX 2. An accurate modeling of the transition from the quasi elastic region to the region of fully established inelastic flow might now not be possible. The simplified model is made to reproduce kinematic hardening and the hysteresis loop closes after one cycle, see Figs. l and 2. The number of constants that have to be determined for modeling the inelastic behavior reduces to five, see Table. In this case the VBO model represents nonlinear kinematic hardening with a final nonzero slope and no isotropic hardening. The model is cyclically neutral and the rate sensitivity together with creep and relaxation is nonlinear. The constants were determined from test data of AISI 1026 carbon steel tested in a heat treated and pre-cycled condition by [13]. The effect of rates was not explored in this paper. It is known that carbon steels show rate sensitivity at room temperature, see [ 14]. Consequently the viscous part of VBO was estimated. Fig. 3 shows the stress strain diagram reproduced with VBO at two strain rates differing by a factor 100. The spacing of the stress-strain curves is a measure of rate sensitivity. The equilibrium stress is also plotted and it is seen that its evolution is rate independent after a transition at the knee of the stress-strain curve. For large strains the linear kinematic behavior is clearly demonstrated. The rate-independent kinematic stress is responsible for modeling monotonic work hardening and it also sets the tangent modulus at the maximum strain of interest.
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Current Advances in Mechanical Design and Production, MDP-7
(im! "l
45 " - ' ~ ' ~ Simulation
o "~ -10 >( <: -20
-40~ , m m m m ~ , s i , ' - -1.0 -0.8
-0.6
-0.4
-0.2
0.2
0.4
0.8
0.8
1.0
A x i a l Strain
Fig. 1. Hysteresis loops of AISI 1026 steel from [13] showing almost cyclic neutral behavior and the simulation of the first cycle bv the simolified VBO (dashed curve~. "
300
(/) '~'
'
9
i
",
v
i
-
-v
' ;. . . .
v
-
,
I ) - I . E - 5 l/s
v
0
9
o~
o.o,
a /
0
-IQ0
-200
-.o-oo,
- o r'e
~o.;o,
-o.~
- o . '~
;
o ;=
Axial Strain
' ooo,
''o . ~
Fig. 2. Hysteresis loops at the first cycle for two different amplitudes. The loops close after one cycle. The evolution of the equilibrium stress is also plotted. ~ - 1 . E - 3 lls - - ~ .11E-5
-0
ooos
o.oi
I/s
o.oTs
9
00"2
.
............
002s
0.03
o
0.035
0.04
o.oLs
Axial Strain Fig. 3 Stress strain diagrams for two different strain rates. The kinematic and the equilibrium stress are not affected by the rate change. The "kinematic type" hardening behavior modeled by VBO is evident.
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157
Next the model is exercised for cyclic, completely reversed strain controlled loading at two amplitudes namely at 1.0 and 0.25 % strain, see Fig. 2. The hysteresis loops close after one cycle due to the absence of isotropic hardening which was not "turned on." It is seen that the equilibrium stress undergoes a variation that is very similar to that of the stress. Where the two curves intersect the inelastic strain rate is zero and the slope of the stress-strain diagram is equal to the modulus of elasticity E. The hysteresis loops plotted in Fig. 5 of[13]are also shown in Fig. 1 demonstrating a good fit
. . . .
r
9
-
!
I
s "" " sp C/1~/I T~ fl
t = 1.E-5 1/s - O = 7.6 MPa/s t oo,
o ~"
"'~176
-
i oois
_
x
-
002
Axial Strain
! o.o~s
003
Fig. 4. Following the experimental method used in [13] the max stress was determined by loading and unloading in strain control up to point A. Load control with a, = 220 MPa and a.. = 45 MPa causes ratcheting.
--'-r'----r
~
t
"
i
,
'
,
'
~a=220 MPa
o
C/)
.~_
O2.~
Om=45 MP~
,, , , "
/
00~
- - - i - -
",
~m--63 MPa.
t"
/
~
/
p
f
~
-
*
-
t
,"["
x
xt
/
,jr ~ ' ~ 1
s /
/
/
~
.
"
~,=2eMP~_
i
f
o
""
.
-
"
-
,..
-
-
-
-
-
"
-
"
I
"+o Experiments --. Simulation %
__...--_sL
_ _
-L . . . . I0
a
IS
I
20
,
-
-
i
25
!
,30
Number of c y c l e s
-
_L_.__-__J
35
. . . .
40
I . . _ _
,15
50
Fig. 5. The computed axial strain accumulation 'for three mean stresses. The data taken from [13] are well simulated.
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Current Advances in Mechanical Design and Production, MDP-7
Ratcheting with mean stress is computed next. To determine an appropriate stress level for ratcheting in a laboratory test, the first loading and unloading is under strain control and is stopped at zero stress, see the dashed curve and point A in Fig. 4. In laboratory ratcheting tests this information is used to set the mean stress and amplitude. The same procedure has been followed in numerical tests as seen in Fig. 4. The initial strain control and the subsequent stress-controlled cycles are shown together with the continuing ratcheting. (The computation uses true stress control whereas the experiment conducts the test under load control; because of this difference in control discrepancies between the laboratory and the numerical experiments may come up at large strains.) Although the strain advance per cycle reduces, there is no indication that ratcheting will cease entirely. A comparison of the simulated ratcheting strain with the experimentally found data taken
from Fig. 8a of [ 15] is given in Fig. 5. The correlation is very good. 3. CONCLUSIONS The VBO model for compressible and incompressible inelastic deformation is introduced. The appropriate model for metallic materials is obtained by setting the inelastic Poisson's ratio to 0.5. The simplified version of VBO for metals and alloys that needs five conw is shown to reproduce kinematic hardening, cyclic neutral behavior and nonlinear rate sensitivity, creep and relaxation. Uniaxial ratcheting simulations compare well with experiments on AISI 1026 steel at room temperature. 4. ACKNOWLEDGEMENT This research was supported in part by the US Department of Energy, Office of Basic Energy Sciences, Grant DE-FG02-96ER14603. Ozgen U. Colak performed the numerical simulations. REFERENCES 1. Pugh, C.E., et al., "Currently Recommended Constitutive Equations for Inelastic Design Analysis of FFTF Components", Oak Ridge National Laboratory: Oak Ridge, TN, (1972). 2. Krausz, A. and Krausz, K., "Unified of Constitutive Laws of Plastic Deformation", San Diego: Academic Press, (1996). 3. Krempl, E., "The Overstress Dependence of the Inelastic Rate of Deformation Inferred from Transient Tests", Materials Science Research International. l(1): pp. 3-10, (1995). 4. Krempl, E., "Models of Viscoplasticity. Some Comments on Equilibrium (back) Stress and Drag Stress", Acta Mechanica., Vol. 69, pp. 25-42, (1987). 5. Gomaa, S., "Computational Procedures for Finite Deformation Rate-Independent Plasticity and Visco-Plasticity Based on Overstress", in Dept. Mechanical Eng., Aeronautical Eng. and Mechanics, Rensselaer Polytechnic Institute: Troy, NY. pp. 156, (1999). 6. Ho, K., Application of the viscoplasticity theory based on overtsress to the modeling of dynamic strain aging of metals and to solid polymers,specifically Nylon 66, in Mechanical Engineering, Aeronautical Engineering and Mechanics. 1998, Rensselaer Polytechnic Institute, Troy, N.Y. 7. Krempl, E., "Creep-Plasticity Interaction, in Modeling of Creep and Damage Processes in Materials and Structures", J.J.Skrzypek and H. Altenbach, Editor, Springer Verlag: Udine, Italy, (1999).
Current Advances in Mechanical Design and Production, MDP-7
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8. Krempl, E., "A Small Strain Viscoplasticity Theory Based on Overstress, in Unified Constitutive Laws of Plastic Deformation", A. Krausz and K. Krausz, Editors, Academic Press, San Diego. pp. 281-318, (1996). 9. Krempl, E. and Ho, K., "An Overstress Model for Solid Polymer Deformation Behavior Applied to Nylon 66", ASTM STP 1357, STP 1357, pp. 118-137, (1999). 10. Krempl, E., "From The Standard Linear Solid To The Viscoplasticity Theory Based On Overstress", Computational Engineering Science, Hawaii, USA, Springer, (1995). I I. Maciucescu, L., S. T.-L., and Krempl, E., "Modeling the Deformation Behavior of a PnPb Solder Alloy Using the Simplified Viscoplasticity Theory Based on Overstress (VBO)", Journal of Electronic Packaging, Vol. 121, pp. 92-98, (1998). 12. Tachibana, Y. and Krempl, E., "Modeling Of High Homologous Temperature Deformation Behavior Using The Viscoplasticity Theory Based On Overstress (VBO): Part III-A Simplified Model", Journal of Engineering Materials and Technology, Vol. 120, pp. 193-196, (1998). 13. Hassan, T. and Kyriakides, S., "Ratcheting in Cyclic Plasticity", Part I" Uniaxial behavior. International Journal of Plasticity, Vol. 8, pp. 91-116, (1992). 14. Sakui, S., Nakamura, T. and Ohmori, M., "The Effects of Grain Size and Deformation Rate on the Tensile Properties of Mild Steel at Low Temperature", Journal of the Iron and Steel Institute Japan, Vol. 49, pp. 996-1003, (1963). 15. Bari, S. and Hassan, T., "Anatomy of Coupled Constitutive Models for Ratcheting Simulation", Manuscript submitted for publication, pp. 38, (1999).
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NOMENCLATURE
r
-[rl A Ac ,4, ,4 /
D E,E,
k R
o',o"
s,k
(Y
Coefficient of thermal expansion Rate sensitivity coefficient equal zero for positive rate sensitivity Overstress invariant Effective inelastic rate of (t,,,E,,,) deformation, ~/= t~ + ~ i
1/temperature No dimension
Box l Box l
Stress l/time
Box l Box l
Shape function, I ___~- > E, / E Elastic Poisson's ratio, constant Inelastic Poisson's ratio, 0_
No dimension
Box 1
No dimension No dimension
Box l Box 1
Stress 1/time
Box l Box 1
Stress
Box 1
Stress
Box 1
Stress
Box 1
l/time
Box 1
Time
Box l
l/time
Box l
l/time
Box l
Stress, Stress/time Stress, Stress/time Stress Stress
Box l
Stress Degree Kelvin
Box l Box l
Effective equilibrium stress Temperature
Box 2 Box l Box l Box 2
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
161
ON THE MECHANICAL INTEGRITY OF AEROENGINE COMPRESSOR DISC ASSEMBLIES
Meguid, S.A. Wallace G. Chalmers Endowed Chair Engineering Mechanics and Design Laboratory University of Toronto- Canada Email" [email protected]
ABSTRACT In this study, we examine the effect of the critical geometric features and interface conditions in the dovetail region of aeroengine disc assemblies upon the contact stress distribution at the blade/disc interface. Three aspects of the work are accordingly examined. The first is concerned with the finite element modelling and analysis of the contact stress field. The second is concerned with the assessment of the mechanical integrity and the damage tolerance capability and the "least damage" potential of the examined blade/disc configurations. The third is devoted to the difficulties associated with the use of user-defined contact stiffness parameters. To overcome these difficulties, we propose a new method, which is mathematically consistent and accurate, known as variational inequalities (VI). The outcome of the work reveals: (i) the significant effect of the different geometric features and interface conditions on the resulting contact stresses, (ii) the ability to predict the crack propagation with reasonable accuracy, and (iii) the advantages of the newly developed VI modelling technique. I. INTRODUCTION The safety of an aircraft has always been the main concern of engine certification authorities. Economic pressure resulting from the reduced availability of strategic materials, the high cost of engine components and the continual demand, by all gas turbine users, for longer component life and higher thrust-to-weight ratio engines continue to provide a stimulating challenge for aeroengine designers. Figurel shows the complex and intricate geometry of a typical aeroengine compressor disc. During the past decade, significant changes have been made in the design criteria employed and material property requirements for critical structural components in aeroengines. The demands for an improved engine performance have led to an increase in component operating temperature and stress. At the same time, the development of new high strength materials, the improvement of analysis methods and enhanced non-destructive inspection techniques have encouraged aeroengine designers to allow for higher stress levels in order to meet the weight saving requirements, and to fully exploit the potential of higher strength materials. The need
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Current Advances tn Mechanical Design and Production, MDP-7
to achieve high performance, low structural weight and minimum manufacturing cost have to be reconciled with the requirements for a long operational life and inherent safety [1-4].
Fig. 1. Typical aeroengine compressor disc. Aeroengine designers have, for many years, adopted the safe life design criterion (SLC). In this case, critical aeroengine components are designed and used for a specified safe life at the end of which they are retired from service. The cyclic life limits are established by a statistical analysis of all available specimens and component life data to provide an acceptably low probability of service failure. This has resulted in the specified safe life being defined as the time span after which one in 1000 discs will have a fatigue induced crack of approximately 0.75mm in length. The fact that the crack could propagate safely to a size somewhat larger than 0.75mm is not usually taken into account, since the additional life is regarded as a hidden safety factor. The SLC is the only procedure being presently covered by the British Civil Airworthiness Requirements (BCAR) regulation and the American Federal Aviation Authorities (FAA). These requirements include: (i) engine performance specification, (ii) thermo-mechanical stress analysis, (iii) determination of loading cycles, (iv) establishment of the design curves from experiments, and (v) full scale component tests to demonstrate the LCF life. The safe life design criterion has resulted in conservative underestimation of useful lives. It has been estimated that, with the use of SLC, about 90% of discs are removed from service having consumed less then 50% of their safe life capability [3]. On the other hand, a few discs in the remaining 10% may fail prematurely because the life criteria used are inflexible and unable to cope with conditions that fall outside the strict boundaries within which they have been formulated. These considerations have led to the development of an alternative procedure called retirement for cause (RFC) and damage tolerant designs (DTD), wherein controlled fracture permits a significant safe flight time after the onset of initial cracking or other types of damage. Whilst the RFC requires an analysis of existing engine components, the DTD addresses the design of new components. The implementation of the above criterion is dependent on the designer's ability to ensure that catastrophic large crack growth will not occur between inspections and that the final crack size does not endanger the integrity of the component. In this case, every attempt should be made by the designer to limit the effect of any unforeseen damage. This concept is known as the fail safe-least damage condition. This can be achieved by either ensuring structural redundancy for load sharing or load shedding should catastrophic failure occur. Clearly, the
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163
structural redundancy option is commonly associated with a weight penalty and may not be suitable for aeroengine design. In this study, attention is devoted to examining the structural integrity of compressor disc assemblies. Three aspects of the work are examined. The first is concerned with the 2D and 3D finite element analysis of the stress field at the blade/disc interface. The second is devoted to the initiation and subsequent propagation of cracks in regions exhibiting maximum stress concentration. The propagating cracks were then tracked using an appropriate crack tracking criterion. The third is concerned with the development of variational inequalities based formulations for the accurate analysis of frictional contact problems. 2. FE MODELLING OF BLADE-DISC ASSEMBLY The stress analysis and mechanical integrity assessment of aeroengine discs have received the attention of several investigators. For example, the work of Kenny et al. [5] and Nurse and Patterson [6] was concerned with stage two fatigue crack growth paths in fir-tree fixtures. Parks and Sanford [7,8] conducted two and three dimensional photoelastic analyses of the firtree region of a turbine disc. Ruiz et al. [9] used high sensitivity Moir6 interferometry to determine relative normal and tangential displacements at the dovetail boundary. Meguid and his co-workers have performed extensive numerical and experimental investigations of the stresses in the dovetail and fir-tree regions of compressor and turbine blades and some of their efforts can be seen in Ref. [ l 0-12]. 2.1. Finite Element Modelling
The main emphasis of this study is the three dimensional finite element analysis of the dovetail region in aeroengine compressor disc assemblies. However, in order to provide a reference for comparison with the three dimensional analysis, and ensure the completeness of the present article, additional results concerning the two dimensional model have been included. Accordingly, two and three dimensional nonlinear finite element analyses were conducted in order to assess the effect of the critical geometric features on the stress field of the dovetail region of an aeroengine compressor disc assembly. Figure2 shows an example of the disc and blade geometries considered in the current study.
o'
tT
+2.o+
17.9"9 I 905 F
13.! I
Di~ geongtry
Di.memie, os in m m
p
13.o~
4 -
1 q
Blade geongtry
Fig. 2. Disc and dovetail geometry examined.
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Current Advances tn Mechanical Design and Production, MDP-7
The two dimensional analysis was conducted for the mid-section of the disc, thus pertaining to plane stress conditions. In view of disc symmetry, only one sector of the disc was modelled. Because of the curved boundary of the model, eight-noded quadrilateral and six-noded triangular plane stress elements were selected. The two dimensional model is incapable of predicting the stress variations across the thickness of a disc. Furthermore, it is incapable of assessing the effect of the skew angle on the stress field. Accordingly, the accurate prediction of the stress state in the disc requires three dimensional modelling of the compressor disc assembly. For the case of a disc with a straight dovetail slot, the same sector of the disc as in the two dimensional analysis was modelled. Due to lack of symmetry, the finite element analysis of a disc with a skew dovetail slot was conducted using a sub-modelling technique. Both twenty-noded hexahedral and ten-noded tetrahedral elements were used in the three dimensional analysis. In order to estimate the optimum number of elements in each model, convergence tests were carried out. Both the accuracy of the solution and computing time were taken into consideration. Figure3 shows typical three dimensional meshes used to test convergence.
/
Fig. 3. Convergence tests for the three dimensional geometry. The contacting surfaces in the dovetail region were modelled using contact elements. These elements allow for the modelling of gap and friction conditions at the interface. Two methods of satisfying nonlinear contact conditions are available: a penalty method and a combined penalty-Lagrange multiplier method. Both methods involve the use of user defined parameters: normal and tangential stiffnesses, Kn and Ks. These stiffnesses play a major role in dictating the convergence and accuracy of the solution (see earlier work by Meguid et al. [ 10-
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165
12]). Large stiffness values of K, are required to avoid interpenetration, while large values of Ks are needed to accurately characterize the stick and slip states of the contacting bodies. However, the use of excessively high values of K, and Ks could result in ill-conditioned global stiffness matrices, leading to numerical errors and divergence. On the other hand, the use of smaller values of K, and Ks may result in convergence to the wrong solution, allowing for interpenetration and incorrect estimates of the stick and slip regions. In this study, the penalty method was assumed and stiffness values were chosen such that the criteria of minimum interpenetration and convergence were satisfied. The minimum interpenetration was selected to be 5% of the smallest element size. Furthermore, the solution was accepted if the change in the maximum contact stress between two successive iterations did not exceed 5%. The stiffness values, determined from those tests, were found to be 10 ~ N/m and 109 N/m for K, and Ks, respectively. The material properties selected for the modelling of the blade and the disc were that of titanium alloy Ti-6AI-4V, as shown in Table 1. The majority of the work examined in this study was conducted using/~=0.25, which is an average value of the coefficient of friction during the fretting fatigue life of the component [13]. However, to investigate the effect of different interface friction conditions at the blade/disc contact region, the coefficient of friction was varied from 0.0 to 1.5. Table 1. Mechanical properties of Ti-6AI-4V
Young's modulus E (Gpa) 114
Poisson's ratio v 0.33
Density p (kg/m3) 4429
Since the compressor disc assembly does not experience high thermal gradients, all the models examined were subjected to centrifugal loading only. No attempt has been made to accurately model the blade except insofar as providing the necessary centrifugal loading and the associated effect at the interface. 2.2 Two Dimensional Finite Element Results
In order to evaluate the effect of the flank length of the blade and flank outer and inner radii of the disc upon the resulting stress field, several geometries were modelled (not shown). The effect of friction at the blade/disc interface for sliding and sticking contact upon the resulting stress field was investigated. Sticking contact between the blade and the disc occurred at the lower contact point for/z.>.l.O, while the two bodies were found to be entirely in sliding contact for smaller values. The results obtained from the two-dimensional analysis reveal the following: i. an increase in the outer radius of the disc increases the stresses at the upper contact point, whereas the maximum value changes only by a small amount, ii. a change in the inner radius of the disc does not affect the stress distribution, iii. a change in the flank length of the blade relocates the point of maximurn stress concentration on the disc boundary, but leaves the values of the peak stresses virtually unchanged, iv. in the case of sliding contact, an increase in the coefficient of friction decreases the disc boundary peak stresses, and
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V.
in the case of sticking contact, the peak stresses increase due to separation at the upper part of the interface.
2.3 Three Dimensional Finite Element Results 2.3.1 Straight dovetail slot To create the three dimensional model for a straight dovetail slot, use was made of the two dimensional model. The three dimensional geometry was created by extruding the two dimensional sector in the direction normal to its plane. In view of symmetry of geometry and loading, only one half of the disc (tl 0 - mm) was modelled. The von Mises stress distribution along the blade/disc interface at two different thickness locations is shown in Fig.4 together with the stress distribution obtained from the two dimensional analysis. The stress distribution for the three dimensional model reveals that the stress level in the middle of the disc is much higher than that at the disc surface. As can be seen more clearly from Fig.5, the magnitude of the peak stress at the lower contact line increases by as much as 40% from the disc surface to the disc central plane. This trend of the stress field was verified using three dimensional photoelastic stress freezing technique [12]. The two dimensional analysis underestimates the stress level along the contact region. The maximum value of the von Mises stress obtained from the two dimensional investigation at the lower contact point is 7% smaller in magnitude than the maximum value predicted by the three dimensional analysis (Fig.5). The three dimensional model with the straight dovetail slot was also analyzed for different values of the coefficient of friction. The results reveal that the blade and the disc experience sliding contact for all values of the coefficient of friction investigated (0 to 1.5). As can be seen in Fig.6, an increase in the coefficient of friction decreases the disc boundary peak stresses. This effect of the coefficient of friction on the stress field at the blade/disc interface was identified earlier by the two dimensional model.
30
~':25
. . . . . . . . . . .
!
~
q. m
d;sc t h k : k n e n = d;sc surface
c
....
20
. ........
28
1
centre
..P 2e a
20-anob~;s
"~ o24
tt
9
t..
i
t.
i I
~ -~ lO "6
E o
Z
"-""
5 .
.......-
"
0
I
2
"'
""
3
"
""
E
....
~18
-"
4
"o ._N20 "6 2O-~olys;s
7
5
6
7
8
g
10 11 12
O;stonce Along Interfoce (ram)
Fig. 4. Von Mises stress along the interface at different thickness locations
16
-5
_
,
-4
.
.
.
-3
.
.
.
-2
i
-1
-
9
0
.,,
a,
I
-
.
2
.
.
.
3
.
.
4
5
D;stonce Through Thickness (turn)
Fig. 5. Von Mises stress across the thickness at the lower contact line
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167
% 35
%
m 25 10 o N
o20 E 0
z
15 10
0
1
2
3
4
5
Oistonce Through Thickness (ram)
Fig. 6. Effect of coefficient of friction on the von Mises stress across thickness at the lower contact line. 2.3.2. Skew Dovetail Slot For the case where a skew angle of 20 ~ was assumed, use was made of sub-modelling. The entire disc assembly was at first discretized using a coarse mesh, where the blades and the disc were joined at the interface and no allowances were made for contact effects, as depicted in Fig. 7(a). A submodel with a refined mesh was then developed with the appropriate boundary conditions for the complete analysis, as depicted in Fig. 7(b). Due to anti-symmetry, only the stresses at one side of the blade/disc contact region were examined. The distribution of yon Mises stresses along the blade/disc interface for the front and back surfaces is shown in Fig.8.
The figure indicates a large difference between the corresponding surfaces in the stress values at the lower contact region. Figure9 demonstrates the increase of the von Mises stress at the lower contact line from the front surface to the back surface of the disc, where the peak value occurs. It is clear from Fig.8 that the two dimensional analysis underestimates the maximum equivalent stress along the interface by as much as 40%. This could have serious implications concerning the safety margins of the disc assembly.
(a)
(b)
Fig. 7. Discretized geometry of (a) the full model, and (b) the submodel.
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168
:
.;3o
38
/
: front surface
34
30 in
: : -
E :761
: 3 O - ~ a ~ ; , (2o" .Sk~) : 3O-~a~,~s (O'Skm,)
-
/
20-Anal~fs
/ ,
22 I
F~~0
}, 14
Z
5
10 0
1
2
3
4
5
6
7
8
9
I0
1I
-0.5
I2
I~JaaceAlenqInte#oce (men) Fig. 8. Variation ofvon Mises stress along the interface of a skew dovetail slot
-0.2
(1.0
0.2
0.,5
Nom~llzed D'~once Through Zhk:ke~'s (ram)
Fig. 9. Variation ofvon Mises stress across at the lower contact line of a skew dovetail slot
3.1NCREMENTAL CRACK TRACKING The importance of crack tracking in aeroengine compressor discs is shown schematically in Fig. 10. It is clear that both the initiation site of the crack and the crack trajectories determine the resulting failure. Experience shows that cracks are most likely to initiate at the lower contact point in the dovetail region due to the presence of a high stress concentration and the fretting action between the disc and the blade. These cracks are predominantly throughthickness cracks. In view of this, only two-dimensional analysis of crack tracking will be conducted. 3.1 Criteria For Mixed-Mode Crack Growth The three most important criteria currently adopted in predicting both the direction of crack growth and the critical applied load under monotonic loading conditions for mixed-mode problems are: (i) the maximum tangential stress (MTS) criterion, (ii)the maximum strain energy release rate (G) criterion, and (iii) the minimum strain energy density (SED) criterion. These criteria are also used to predict the direction of fatigue crack growth. In view of its simplicity and reliability for the current application, we focus our attention to the use of the maximum tangential stress criterion. In this criterion, Erdogan and Sih [ 14] postulated that the crack will propagate along a plane normal to the maximum tangential stress o0 in the vicinity of the crack tip, where
,
O0 = (2~,r)l/2 cos
ol K ! cos 2 - - - K2. 2
sin0
(1)
and K[ and K. are mode I and mode II stress intensity factors, respectively. 3.2 Results and Discussion
Cracks were initiated at point c in the dovetail region of Fig. 10. These cracks were modelled by creating two appropriate surfaces with unrestrained nodes. To model the singularity at the crack tip, special crack tip elements were used. Initially, an incremental crack of length 0.5ram was allowed to grow in that region with its direction of propagation being dictated by the
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different tracking criteria used. For the case of the maximum-minimum principal stress criterion the newly developed maximum-minimum principal stress trajectories were evaluated using the finite element method (Fig. l 1 (a)). A further increment of crack growth was then given in the new direction and the whole process was repeated several times to track the propagation of these cracks. m
.....
, sos
of two blades
rim tear
~'
DISC; Fig. 10. Importance of crack tracking in a compressor disc.
Fig. 1 l(a) clearly indicates that for the considered dovetail geometry, propagation will take place across the reduced section of the dovetail region and not through the disc. One can therefore conclude that, provided other aspects of the design are maintained properly, the geometrical features of the dovetail section examined will ensure "contained" or limited damage should failure occurs. This has been verified experimentally, as shown in Fig. 11 (b). 4. FRICTIONAL CONTACT USING VARIATIONAL INEQUALITIES During the analysis of the compressor disc assembly, it became apparent that existing finite element formulations and commercial codes cannot accurately predict the field variables in cases involving contact and friction. In the following section, we provide some of our most recent work pertaining to these difficulties and the techniques adopted in overcoming them. Existing finite element codes rely on the use of the traditional variational approach to treat contact problems. In this context, contact elements are developed and are assembled within
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the original finite element code. This approach employs the concept of virtual work or that of minimum potential energy to extend the original finite element method into solving a set of equations by formulating a contact element. However, experience with these elements indicates that they involve mathematical inconsistencies when describing frictional forces of the Coulomb's type, and that they require user defined parameters, which influence the solution. These parameters, which are provided on a trial-and-error basis, affect the convergence and accuracy of the results. 4.1 Variational Inequalities Formulation To overcome these difficulties, contact problems are formulated using a more rigorous and consistent approach, known as the variational inequalities (VI) approach [15-17]. This approach offers many advantages over the traditional contact element formulation, including an improved and enhanced computational economy, as a result of the elimination of the use of special contact elements; existing mathematical inconsistencies are removed as a result of the rigor in the formulation of the problem; and the elimination of user defined parameters which affect the accuracy of the results and its rate of convergence. The contact problem is formulated in terms of (i) equilibrium, (ii)a friction law and (iii) contact and boundary conditions. The general variational inequality for frictional contact problems is provided by the following: a ( u , v - u) + J ( u , v ) - J(u,u) >_f(v - u) with,
and
VvaH,
(2)
a(u, v) = ~oij(u)eij(v)dxdydz,
(3)
J(u, v) = ~c - ~toN(u)~vrlds ,
(4)
f(v) = ~ ~ vidxdydz + -,[ ti v ~ds
(5)
Adopting the VI approach, the mathematical formulations describing frictional contact in elastic and elasto-plastic solids have been developed and the necessary solution techniques have been implemented. In addition, the new approach has been applied successfully to a variety of engineering problems involving contact in gears, aeroengine disc assemblies, robotic dextrous grippers and simulation of metal forming processes. In the example that follows, we demonstrate the advantages offered by the VI approach over that of traditional contact elements. 4.2 Verification Example Let us consider the problem of frictionless contact between an elastic cylinder and a rigid foundation, as depicted in Fig. 12(a). The problem is solved using the newly developed VI approach as well as using a commercial finite element package employing a traditional contact element. This element employs a penalty method with implicit contact constraints iterations and generally requires the user to supply two penalty parameters representing the normal and tangential contact stiffnesses, K. and K,, respectively. In this case, however, the problem was solved using different values of K. only. The results, which are summarised in Fig. 12(b), show the normal contact stress distribution in the contact zone obtained by VI in comparison with the theoretical Hertz solution and contact element predictions. The figure shows that the accuracy of results obtained using contact elements are governed by the choice of the penalty
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171
parameter Kn. The figure also reveals that only when Kn is of the order of l 0 s N/m or higher does the results of contact element approach those obtained by VI.
Fig. 12. Contact between an elastic cylinder and a rigid foundation: (a) discretized geometry of cylinder, and (b) variation of normalised contact stress versus contact displacement.
5. CONCLUSIONS This study reveals the following: 1. the maximum stress occurs at and just below the lower contact point along the bottom tooth of a compressor disc, 2. the stresses vary through the disc thickness and cannot be predicted by a two dimensional analysis, the straight slot geometry experiences maximum stresses at the outer surfaces, 4. the coefficient of friction influences the peak stress value, 5. in the skew model, relatively large stress variations exist at the contact region as well as through the thickness, 6. the maximum-minimum principal stress crack tracking criterion can reliably predict the general direction of the propagating cracks into the dovetail region, 7. for the dovetail geometry considered, crack propagation will take place across the finger region and not through the disc, thus containing possible fretting damage, and 8. the newly developed VI approach is highly effective in modelling contact. ,
ACKNOWLEDGEMENT The author wishes to thank Dr. N. EI-Abbasi for his assistance with the preparation of this article.
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REFERENCES
1. Larsen, J.M. and Nicholas, T., "Cumulative-Damage Modeling of Fatigue Crack Growth in Turbine Engine Materials", Engineering Fracture Mechanics, 22, pp. 713-730, (1985). 2. Singh, G.D. and Rawtani, S., "Fir-tree Fastening of Turbomachinery Blades", Int. J. Mech. Sci., 24, pp. 377-384, (1982). 3. Liu, A. F. and Ekvall, J.C. "Materials Toughness and Residual Strength of Damage Tolerant AircraR Structures", ASTM, STP 486, pp. 98-121, (1970). 4. Meguid, S.A., "Engineering Fracture Mechanics", Elsevier Applied Science, London, (1988). 5. Kenny, B., Patterson, E., Said, M., and Aradhya, K., "Contact Stress Distributions in a Turbine Disc Dovetail Type Joint- A Comparison of Photoelastic and Finite Element Results", Strain, 27, pp. 21-24, (1991 ). 6. Nurse, A.D., and Patterson, E., "Experimental Determination of Stress Intensity Factors for Cracks in Turbine Discs", Fatigue Fract. Engng Mater. and Struct., 16, pp. 315-325, (1993). 7. Parks, V.J., and Sanford, R.J., "Experimental Stress Analysis of the TF-30 Turbine Engine Third-stage Fan-blade/Disc Dovetail Region", NRL Report, NRL 8149, (1977). 8. Parks, V.J., and Sanford, R.J., "Three-dimensional Photoelastic Stress Analysis of the Dovetail Region of the TF-30 Turbine Engine Third-stage Fan, NRL Report, NRL 8276, (1978). 9. Ruiz, C., Post, D. and Czamek, R. "Moir6 Intefferometric Study of Dovetail Joints", ASME J. Appl. Mech., 52, pp. 109-114, (1985). 10. Papanikos, P., and Meguid, S.A., "Theoretical and Experimental Studies of FrettingInitiated Fatigue Failure of Aeroengine Compressor Discs", Fatigue Fract Eng. Mat. Struct 17, pp. 539-550, (1994). 11. Papanikos, P., "On the Structural Integrity of Dovetail Joints in Aeroengine Discs", M.A.Sc. Thesis, University of Toronto, (1992). 12. Papanikos, P., Meguid, S.A., and Stjepanovic, S., "Three Dimensional Nonlinear Finite Element Analysis of Dovetail Joints in Aeroengine Discs", Finite Elem. Anal. Des., 29, pp. 173-186, (1998). 13. Hamdy, M.M., and Waterhouse, R.B., "The Fretting Wear of Ti-6AI-4V and Aged Inconel 718 at Elevated Temperatures". Wear, 71, pp. 237-248, (1981). 14. Erdogan, F., and Sih, G.C., "On the Crack Extension in Plates under Plane Loading and Transverse Shear", Journal of Basic Engineering, 85, pp. 519-527, (1963). 15. Refaat, M.H., and Meguid, S.A., "On the Elastic Solution of Frictional Contact Problems using Variational Inequalities", Int. J. Mech. Sci., 36, pp. 329-342, (1994). 16. Refaat, M.H., and Meguid, S.A., "A Novel Finite Element Approach to Frictional Contact Problems", Int. J. of Num. Meth. in Eng., 39, pp. 3889-3902, (1996). 17. EI-Abbasi, N., and Meguid, S.A., "Large Deformation Analysis of Contact in Degenerate Shell Elements", Int. J. Numer. Meth. Engrg., 43, pp. 1127-1141, (1998).
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-I 7, 2000
173
EFFECT OF VOID GROWTH ON THE PLASTIC INSTABILITY OF UNIAXIALLY LOADED SHEETS Saleh, Ch.A.R. * and Ragab, A.R.** Assistant Professor, ** Professor, Department of Mechanical Design and Production, Faculty of Engineering, Cairo University, Giza, 12316-Egypt
ABSTRACT Plastic instability for uniaxially loaded bars for materials undergoing void growth is investigated. These voids may already exist initially in the material or initiate earlier with deformation at sites of second-phase hard particles. A simplified version of GursonTvergaard's yield criterion, amenable to analytical derivation is used in the analyses. The predicted limit strains compared with that obtained experimentally show the same trend. Limit strains at localized necking is also determined for sheet with a defect in the form of a voided zone inclined to the loading axis. It is found that necking will not develop in the defected zone unless the initial void volume fraction combined with the appropriate angle of inclination produces limit strain less than that predicted by Hill. Other wise necking will occur according to Hill's condition. KEYWORDS Limit Strains, Necking, Plastic Instability, Void Growth. I. INTRODUCTION Limit strains are often determined analytically by considering plastic instability where uniform plastic deformation gives way to one in which deformation becomes localized. Necking in a tensile bar is the most familiar example, where the analysis results in Consid6re's condition of uniform tensile strain c u equal to the strain-hardening exponent in the law
o=Kc"
(I)
i.e.c u = n . This limit strain defining the onset of diffuse necking is identified by the maximum tensile load condition which has been applied by Swirl [ 1] to include other biaxial straining conditions. A further development by Hill [2] showed that localized necking in a sheet subjected to uniaxial tension is identified by assuming maximum load condition together with plane strain condition at a specific plane. Hill's analysis defines a limit strain ors a = 2n at a plane inclined 54 ~ 44' to the tensile axis using the simple power law; eq. (1). In this work the effect of voids and void growth on the plastic instability of voided metals are displayed using a yield criterion for voided material. This surpasses the shortcomings that conventional plasticity theories and their associated flow rules do not rigorously fit the
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Current Advances in Mechanical Design and Production, MDP-7
analysis of plastic deformation of voided (porous) solids, for which the condition of volume constancy as well as the independence of yielding on the hydrostatic stress component are not valid. 2. CONSTITUTIVE MODEL FOR VOIDED SOLIDS Voids may be found in metals as in sintered powder compacts or exist at grain boundaries as a material defect. They also may be nucleated around inclusions or at second hard phase particles. With excess plastic deformation, voids link-up to form macroscopic defects. Gurson [3] in 1977 suggested a yield condition for voided solids, expressed as
]j
(2)
where UM is the effective stress of the matrix material, ~ and Om are the apparent effective and mean stresses respectively. The void volume fraction Cv defined as the ratio of the volume of voids to the total volume of porous solids, could be determined by microscopic examination for specimens cut from the deformed sheet metal or from measurements of the relative density change with strain. The above mentioned methods used to evaluate the initial void volume fraction, showed that the void volume fraction ranges from 1x 10"4to 5x 10-3 for conventional metallic alloys [ 4 and 5]. However it could reach 0.2 for sintered powder compacts [6 ]. For superplastie alloys the current volume fraction of a value 0.3 is not uncommon at tensile elongation in excess of 500% [7]. Gurson's model is modified by Tvergaard [8] to the following form
(3) The parameters q l and q2 were introduced to bring predictions of the model into closer agreement with full numerical analysis for materials with periodically distributed voids. It is found that assigning a value in the range of 1.5 to 2.5 to ql and taking q2 equal to unity improves agreement between experimental results for several different powder materials with numerical simulation of plastic flow of porous ductile solids [9]. A simplified version of Gurson-Tvergaard's amenable to analytical derivations has been suggested [10] by expanding the hyperbolic term and neglecting the terms of (3q2Om/2~M) of order higher than 2. Since the loading is uniaxial or biaxial, it is expected to obtain results identical to that when using the original modified Gurson's model [11] i.e. eq. (3).The effective matrix stress is thus related to apparent stress according to the simplified GursonTvergaard's model by =
+9qlq t 0'
1 O-qlCv)
2Cv(l' m
(4)
For uniaxial loading this relation becomes
/ 1+ q~q~ev 4 cM-
(l-qlCv)
~l
(5)
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7
175
The matrix strain can also be related to the apparent strain ct by substituting equation (A-9) into eq.(A-3) (see Appendix A) for Xxy= 0 to give that
(
(1-q,Cv) 1+ q~q~Cv E:M= ( l - C v )
c,
(6)
4
The current void volume fraction corresponding to a certain level of cl is obtained by integration of eq.(A-3) ( for uniaxial loading and after substitution of dL by dCv from eq. (A-
8)) as e,
lln
fray/' [ ,q (C'v-i i ) ~Cv. -11
=3 [[,C )
qlq2
(7)
L
In this work the matrix material is assumed to obey the power law --
nM
6 M =KM(E)M
--
YM
(e)M
(8)
where the subscript M refers to the matrix material, i.e. KM,n M and 7M are the strength coefficient, strain-hardening exponent and the strain-rate sensitivity of the matrix material respectively. These parameters are assumed to be initially known. 3. TENSILE L I M I T STRAIN OF UNIAXIALLY LOADED BARS
In the following the plastic instability of an axially-loaded bar having a heterogeneous distribution of voids shown in Fig. 1 is analyzed. The initial void fraction of voids for zones (i) and (u) are C vo' and C vo,. respectively where zone (i) has a higher concentration of voids than the average, zone (u) i.e. (Cvo, > Cvo, ). The zone of
.Zone (i), Cvoi bo O o
P ,qt--
O o o
,--)~ p
.o. ~ . . ~ne (u); Cvou=0
higher void concentration is assumed to be extended in a plane normal to the applied Fig. 1 - Schematicdrawing for a specimen with an initial load. As the matrix material supports the void inhomogeneityunder uniaxial loading. load P, the equilibrium requires that P = OM A M = OM AM, (9) where AM is the cross-sectional area of the matrix material. If the original apparent crosssectional area is Ao, and AoM is the original cross-sectional area of the matrix material, then eq. (9) may be written as OM=e-CU=AoM. = O'M,e -eMiAoM' Since the following relation holds approximately between areas AoM ~ A o ( 1 - C v o ) hence,
(10)
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If the matrix material - deforming at constant volume - behaves according to eq. (8), then substitution of eq. (8) into eq. (11) and the integration give ~....//~ - ~ . / / ~ dc M, r E ; M, e
=
F ( l - C v o , ) ] ~'*~' .~/ ,.//, [ j e~'ed~ M l
(12)
The calculation proceeds by determining the current void volume fraction C, corresponding to an apparent longitudinal strain c i from equation (7). Then the matrix strain CMbeing expressed by equation (6) allows the determination of the value of CM. and hence c u by integration from equation (12). The limit strain c= is defined at the condition ds=/dc i --->0 i.e. when deformation becomes localized in the heavily voided zone (i) Fig. 2 shows limit tensile strains in uniaxially-loaded bars with different values of Cvo where zone (u)is considered to be free of voids. 3.1 Influence of Strain-Hardening Exponent The effect of both the initial void volume fraction and the strain-hardening exponent on tensile limit strain are shown in Fig. 3, where zone (u) is taken to be free of voids. The experimental points -describing the strain at fracture- are shown in the figure are for 1% Cr, 0.25% Mo cast steel [12]. It can be shown that the limit strain decreases with increasing the initial void volume fraction. Also -as expected- the limit strain increases with increasing the strain-hardening exponent (riM). 0.5 . 0.3 -
n = 0.3,]k= 0.01
Cv~
1
9Experimenlal;Caststeel [12 ]
~
0.4
q,-'q; 1
y=O.Ol,q,= q= I
=i
'
,~ .=_ I
0.3
0.2
e~ t.., ,,..,
.,., I
o.,[
0.1
0.0 0.0
,,
0.1
0.2
0.3
Strain "r Fig2- Limit strains in an tmiaxialy loaded rod with initial void inhomogeneity as predicted by Gurson Tvergaard's model.
0.0 0.00
I
I 0.02
,
I 0.04
0.06
Initial VoidVolumeFraction- C
0.08
0.10
VO
Fig. 3-Comparison between experimental fmchu~ strain and the calculatedlimit strain at necking for different values of initial void volume fraction.
3.2 Influence of Strain Rate-Sensitivity In order to examine the strain rate-sensitivity, the material is considered to be perfectly plastic i.e. nM = 0. In this ease the integration of eq. (12) gives
Current Advances in Mechanical Design and Production, MDP- 7
eu=-'/ln(l-Cvo
'
-
177
(13)
+
Again for each value of •i ( the apparent strain in zone ( i ) ) , the current void volume fraction Cv and hence the corresponding matrix strain is defined using eqs. (7)and (6) respectively. Thene u is
35 . . . . . 9 Zircaloy-4
--
C,
//
* T P bi --6S nA IE- u4 lVe c l i r
n:0
2s/
9
/ ~/_ ~-i ~ <,.(~ ,% / _ c,fl/
[,31..d [,,l
~ ~ 2.0 - determined fr~ eq" (13)" ' i is ~.~ 1.5 progressively increased and the limit .~ strain is the value of ~;u when
dlZu/d• i --> 0. Figure 4 shows the increase in limit strain rate with higher strain rate sensitivities for a given initial void volume fraction Cvo. The experimental points shown in Fig. 4 are for different superplastic materials [13] and [14]. They are indicated on the plots, to warrant that the theoretical model follows the same experimental trend.
/
9 Ti-5 A I - 2 5 S n
30
_ ~/ '-' ~@ / / ~ / / ~ / / ~
,:,
/
./
y, ~,,,/ ~, 9 c,~
9
i.o
*
0.5-=
-__='
o.o o.o
o.i
~:~==
0.2
0.3
0.4
0.5
0.6
0.7
Strain-Rate Sensitivity-~M Fig. 4 - Comparison between experimental fracture strain and calculated limit strain at necking for superplastie materials.
4. T E N S I L E L I M I T STRAIN OF UNIAXIALLY LOADED SHEET M E T A L 4.1. Effect of the Orientation of Void Defect
In the following analysis, the limit strain at localized necking is determined for a loaded sheet with a zone of voids inclined to the loading axes as shown in Fig. 5. The zone of higher void concentration is oriented at an angle ~ to the minimum principal strain direction. For simplicity, it is assumed from the beginning that zone (u) is free from voids while the voids are concentrated in zone (i) with an initial void fraction C vo. The sheet may also possess an initial geometric non-uniformity i.e. to, < t o . The stresses corresponding to X-Y coordinate system are obtained according to the stress transformation law
Y
v~ s
C
/I/ "/' tou
/~toi o2
Fig. 5 - Schematicdrawing for a stretched sheet metal having an inclined void inhomogeneity.
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o x. =
yu
o , + o 2 + (~o ,c- oo2 )s 2 2
2,,
Zxv. = ( o , - o 2 ) sin2~ 2
(14)
The sheet is then subjected to the following stress ratios with respect to X-Y coordinates. Or.l=
o Vu _ (1 + a ) - (1 - a)cos 2~ , Oxu (1 + a)+ (1 - cz)cos 2r
OL
- -
2
=-
z XVu (l - or)sin 2~ = OXu (1~.+. a)+(l - cx)cos 2r
(15)
where ot = o2 / or. The equilibrium condition gives Oxu Au= OXMiAMi
(16)
txvu Au = ~XYMi AMi
and
Defining132 as 132 =tXVMi/~
'
(17) henceeqs. (16) and (17) give a 2 =132.
The sheet is assumed to have normal anisotropy, according to Hill's yield function which gives the matrix effective stress as 2 ~2 =O.~:M +O2VM _~.OxMOV M +2 0 +RR ) I:2XYM+2 ( RR- l ) [O2M--OZM(OXM +O VM)] (18)
where R = (1 + i)/i and i is the average strain ratio for normal anisotropy. The developed flow rule equations are given in the Appendix A for normal anisotropy, however for simplicity the analysis will be confined here to isotropic condition only i.e. i = l i.e. R = 2.Then OXMi and Ox. are related to ~M, and ~u respectively using eq. (18), for plane stress condition ( i.e. Oz = 0 )as m
OMi =[1+13~--13, +3a~] ~
(19)
OXM i
and
~" = [1+ a~ - a , + 3a;]~
(20)
OX u
where 131 = OYM i / O X M i . Substituting eqs. (19) and (20) into eq. (16) yields
~176
-_. 2
+ oc, - a , + 3a 22
=
WMi A-i o. A.
(2 I)
where A M, is the cross-sectional area of the matrix material in zone (i) which is related to total cross-sectional area as AMi =Ai (1-Cvi)=bti (1-Cvi)=bt0i.(l-Cvi)e ~i
(22)
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where C v, is the current void volume fraction, b is the width of the cross-section, ti is the current thickness and ez, is the apparent thickness strain for zone (i). Also for the unvoided zone A u = bt Oue~ Zu
(23)
The flow curve of the matrix material is assumed to obey the power law represented by eq. (8). Substituting eqs. (22), (23) and (8) into eq. (21) yields
[<,. L~M~j
Ll+~ +o<,+3orI
]~
L-Co+~,,
exp (ezi-~:zu)
(24)
The sheet deformes such that the following continuity condition is satisfied de v, = de v.
(25)
where de v, is the apparent strain in the Y- direction. The limit strain at localized necking will be determined assuming that the plastic deformation of the voided material is described by Gurson Tvergaasd's-model. The relation between devi and d~M, is described by eqn. (A-12) in the Appendix A for Gurson-Tvergaard's model, then eq. (25) can be written as 0.5 _( 1 +131)qlq~Cvi d~, = [l~v_Cvi/ 13i-2+.1 4 1 d~Mi (i- qiCvi ) r 2
ml
1 +{x~ -ct I + 3ct~ 1+13~-fl, +3or 2 +
qiq~Cvi (1+~, )2
(26)
4
The limit strains at localized neck el. and 62u are obtained at instability, i.e. when d~u --~0. d~ To arrive at the solution, the stress ratio 13i is firstly assumed for an incremental change in ~u,. Hence ~u is then calculated by solving eq. (26). oez,and the current void volume fraction Cv, are also calculated using eqs. (A- 13) and (A-15). The value of131 is then checked and corrected for that increment of ~r by using eq. (24). The above procedure is followed for each increment until
d~u d~i --~ 0. The strains 6"x~,6~ and 6xv,, are then
determined according to eqs. (A-11), (A-12) and
(A-14) for Cv. - 0. The limit strains
61~ and 62. at localized necking are then calculated by strain transformation as
Zlu 2u =
I~Xu + I~yu + 2 --
I/
l~Xu
-6Vu
2
/
2 u + 13Xy
(27)
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The solution of uniaxially loaded sheet (i.e. ot = 0 ) with void defect oriented an angle ~ to the second principal stress direction is displayed in Fig. 6, which shows the influence of the initial void volume fraction on the limit strain for different orientations of this void-like defect.
n~ 0.22, ~ -1.5,
q= q~ I
0.8
For an anisotropic sheet having no defect, Hill [2] indicated a limit strain ofr = n (1+~) as a limit strain due
,~
to localized neck occurring at an angle
.~
= n/2 -
arc tan 4 ( r + l ) / ~ ;
a
o
1.0
i
\
0.6
x.-'o.
~
I
Hill'slimit:e~=(l+Dn
condition which appears in Fig. 6 as a curve degenerated to a point located "~25" on the ordinate. Hence it is expected 0.4 that necking will not develop in the defected zone unless the initial void volume fraction combined with the appropriate angle of inclination 0.2 produces limit strain less than n 0.00 0.01 0.02 0.03 0.04 0.05 0.06 Initial Void Volume Fraction - Cvo (1+~). Table 1 gives the strain at necking for different materials (obtained from forming limit diagrams Fig. 6 - The influence of the inclination of void inhomogeneity on at uniaxial loading conditions) and the the calculated tensile limit strain at localized necking for various initial void volume fraction using Gurson-Tvergaard's
calculated limit strains,
model.
Appropriate values of void volume fractions and inclination angles are assumed to estimate the limit strains which are seen to be close to that obtained experimentally.
Table 1. Comparison between experimental and calculated uniaxial limit strains. Material 0.2:~. 0.2:Z 0.25 0.2~ 0.2.4
0.01 0.01 0 0.01 0.01
Experimental limit strain 0.788 0.536 0.42 0.51 0.445
Copper [17] 0.36 Austenitie 0.41 Stainless steel
0 0.01
0.47 0.405
A-K steel [15] LPG steel [16] 2036-T4al [15] AI [17] AI-Mn [16]
[18]
"t
.
.
.
.
Calculated limit strain 0.783 0.53 0.47 0.508 0.487 0.486 0.427
Remarks ~ = 350, Cvo= 0.005 ~ = 25 ~ C,o = 0.0045 ~ = 35 ~ Cvo = 0.0003 t~= 35 ~ Cvo =01003 ~ = 35 ~ C~o = 0.004 Cvo = 0.0016 (exp.) ~ = 25 ~ Cvo = 0.01 ~ = 25 i;, Cvo = 0.01
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4.2 Correlation of C vo with Thickness Homogeneity
I .OE-!
l'he correlation between the initial void volume fraction Cvo and thickness inhomogeneity fo to maintain the same predictions of limit strain at uniaxial loading is shown in Fig. 7. In order to attain the same limit strain, it is noticed that a smaller value of C vo-within the range which may exist in conventional metallic alloys (=- 10"4)- gives the same effect when using an exaggerated value for geometrical inhomogeneity (almost one order of magnitude higher).
_-
n~ 0"2,F= I, YM--0
//%~~
1.0E-2 ~"~ ._m .~ -
!
.0E-3 :-
~ l / , 1.0E-4 1.0E-4
//
, ,,l
I.OE-3
/
,
~-Gurson-Tvergaardq=2, S' model;q2=l+
, , ,
l-fO
~Gurso
n'
:mod71'
I.OE-2
,,
I.OE-I
Fig.7 - Correlation between the initial void fraction Cvo and the initial thickness irdaomogeneityfo giving same limit strain for uniaxial stretching. 4.CONCLUSION Application of Gurson-Tvergaard's model for voided models to predict limit strains for uniaxially loaded bars shows its capability to display the effect of voids and void growth on plastic behavior. It is found that the limit strain is highly affected by the presence of small initial void volume fraction.; a finding which is supported experimentally. Limit strains for sheets with a defect in the form of a voided zone inclined to the loading axis is determined. It is found that necking will not develop in the defected zone unless the initial void volume fraction combined with the appropriate angle of inclination produces limit strain less than twice the strain-hardening exponent. Otherwise necking will occur according to Hill's condition i.e., at a plane inclined with an angle equal to (arc tan x / ( r + 1)/r ); to the tensile axis. REFERENCES 1. 2. 3.
4.
Swift, H.W.," Plastic Instability under Plane Stress", J. Mech. phys. of Solids, 1, pp. 1-18, (1952) Hill, R., "On Discontinuous Plastic States with Special to Localized Necking in Thin Sheets", J. Mech. Phys. of Solids, 1, pp. 19-30, (1952) Gurson, A.L., "Continuum Theory of Ductile Rupture by Void Nucleation and Growth: Part I- Yield Criteria and Flow Rules for Porous Ductile Materials", J. Mat. Tech., Transaction of ASME, 99, p.2, (1977). Parmar, A. and Mellor, P.B., "Growth of Voids in Biaxial Stress Fields", Int. J. Mech. Sci., 22, pp. 133-150, (1980).
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Current Advances in Mechanical Design and Production, MDP-7
5. Melander, A., "A New Model of the Forming Limit Diagram Applied to Experiments on Four Copper-Brass Alloys", Mat. Sci. Eng., 58, pp.63-88, (1983). 6. Becker, R, Needleman, A., Richmond, O and Tvergaard, V, "Void Growth and Failure in Notched Bars", J. Mech. Phys. Solids, 36, pp. 317-351, (1988). 7. Patterson, I.W.D. and Ridley, N., "Effect of Phase Proportions on Deformation and Cavitation of Superplastic od13Brass", J. Mat. Sci., 16, pp. 457-464, (1981). 8. Tvergaard, V., "Influence of Voids on Shear Band Instability under Plane Strain Conditions", Int. J. Fracture, 17, pp. 389-407, (1981). 9. Tvergaard,V., "Effect of Yield Surface Curvature and Void Nucleation on Plastic Flow Localization", J. Mech. Phys. Solids, 35, pp.43-60, (1987). 10. Abdou, M.Z., "Deformation, Cavitation and Fracture of Superplastic Alloys with an Application to od13 Brasses", Ph. D. Thesis, Cairo university, Faculty of engineering, (1990). 11. Ragab, A.R. and Saleh, Ch., " Evaluation of Some Constitutive Models for Voided Solides", Accepted for Publication in Int. J. Plasticity (1999). 12. Kalpakjian, S,. "Manufacturing Process for Engineering Materials", Addison Wesley, p. 131, (1984). 13. Lee, D. and Backofen, W. A.," Superplasticity in Some Titanium and Zirconium Alloys", Trans. Met. Soc., AIME, 239, p. 1034, (1967). 14. Avery, D.H. and Backofen, W. A., Tran. Q. ASM, 58, pp.551-562, (1965). 15. Hecker, S. S. "Simple Technique for Determining Forming Limit Curves", Sheet Metal Industries, pp. 671-676, (1975) 16. Date, P. P. and Padmanabhan, K. A., "On the Prediction of the Forming-Limit Diagram of Sheet Metals", Int. J. Mech. Sei., 34, pp. 363-374, (1992). 17. Kohara, S., Katsuta, M., Aoki, K. and Suzuki, K.," Forming-Limit Curves for Aluminum, Copper and their Alloys", Advanced Technology of Plasticity, 1, pp. 765-768, (1984). 18. Marques, M.J.M. and Baptista, R. M. S. O., "Formability Characterization of Austenitic Stainless Steel", Elsevier Science Publishers B. V., pp.233-242, (1992). NOMENCLATURE AM
Cvo, Cv P V VM, Vv dW b
fo i, u n r
to, t
EM
Cross-sectional area of the matrix material. Initial and current void volume fraction respectively. Load. Total volume of the porous solid. Volume of the matrix material and voids respectively. Increment of work done by the matrix material per unit volume. Width. Thickness imperfection. Subscripts indicating zones of higher and average concentration of voids respectively. Strain-hardening exponent. Average strain ratio for normal anisotropy. Initial and current sheet thickness respectively. Stress ratio in uniform and voided zone respectively. Angle of inclination of voided zone. Strain and effective strain respectively. Effective matrix strain.
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Volumetric strain. Strain-rate sensitivity exponent. Proportionality factor in the flow rule. Density of the porous solid and matrix material respectively.
~y
T dk P, P,, ~ ' "if,
183
Stress, effective stress and mean stress respectively Effective stress within the matrix material. Shear stress Coordinates axes along inclined voided zone.
~ m
13M
T
X,Y,Z
APPENDIX A MODEL, FLOW RULE AND
MODIFIED G U R S O N - T V E R G A A R D ' S VOID GROWTH CHARACTERISTICS The simplified form of this criterion is
{
=
9
1 ]2 ~2+'4 q'q~Cvcrm ~ [l_qlC v For planar isotropic material ~ has the following form
)] [4-oz(ox+o
+o _2R ffxoY +2 0+R) R -
(A-l)
A-2)
where R = (1 + ~)/~ and x xz = x YZ = 0 The flow rule is derived by partially differentiating eq. (A-1) with respect to stress ( and then cr z is substituted by zero for plane stress condition) as d~ x =d~. 2o x - ~ - ~
+ ................... 2
d~u =d~. 2o v --~-a x +
2
+~
(A-3)
+av
(A-4)
)
(A-5)
d~z= dL[-2 (RR 1)+ qlq~C v ]( d~xv = d~'[ 4 (1 +R)R "lrXy]
(A-6)
Hence the volumetric strain de v =d~ [ 3q',q,]Cv (o x + o v )] = dCv
2
(A-7)
0-Cv)
Eq. (A-7) shows that dL could be related to the change in void volume fraction as 2 dC v d~ = -(A-8) 3 (c x + c v ) q l q ~ C v ( l - C v ) The plastic work done per unit volume of porous solid must be identical to the work done by the matrix material having volume of ( 1- Cv ), hence:
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dW-Eo'i~: i = d~(2~'~X1- q,C v
-
(A-9)
2 (1-qlCv) 2 ~M
Oy -
-
1 0-Cv)d~.
Then d~. = If O~I
)2=(1 CV)~Md~M
a 2 = -~xY
and
-
{~X
O'X
[
Eqs. (A-8) and (A-9) with eq. (A-I) yields d~ M 4 (1- q ,_.Cv__) l+o~t2_ 2 (I+R) q,q2C v (l+t~,)2 = ~ q~q~Cv(l +ctiXl-Cv) 2 ~o~ I + 2 R ot2 +- 4
dC v
(A-10) Eqn. (A-10) can be rearranged to give the void growth with respect to the effective matrix strain. Eq. (A-9) with the flow rule give dt:___xx= (l Cv) deM
(1-qtCv)
qiq2Cv(l+
R
[
l+ot~ - "2~ l
+
4
R
-
~1
(l_Cv)
-
(l-Cv)
I
(1-qICv) l+ct~
d~x v = 0-Cv) dE M
(l-q,Cv)
4
1 R
qlq2Cv(l+ 4
_+
l+ot 2 ~-ot I +2
dr: z d~M
(A-ll)
(1 R) q 2 + or22 + iq2Cv (1 +ott) 2
-
de v
.... o,,]
1 ~
-
R
(R-l) ~ + R
o~ !
)]
q,q v
R)ct22 + . . . .
4
(A-12) (l+ct I
q,q2Cv~ ( ,, l+ot~) 4
) 2I -- ~tx I +2(I+R) -01.22+ qlq~Cv ( 1 + o~ R
[2(I§R
4
10.5
j 2
(A-13)
(A-14)
[
2or (1 + R) o~22+ q,q~Cv (1 +or,) 2 l+ot2- R ~+2 R 4
The void growth rate with respect to the strain in Z-direction is obtained by substituting eq. ( A - 8 ) into eq. (A-5) to give dC----Y-v= 3q'q~Cv(1-Cv) (A-15)
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
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SHAKEDOWN ANALYSIS OF AN INFINITE PLATE WITH A CENTRAL HOLE UNDER BIAXIAL TENSION
Attia, M.S.', AbdeI-Karim, M.'" and Megahed, M.M."" Graduate Student, "* Professor, Mechanical Design and Production Department Faculty of Engineering, Cairo University, Giza 12316- Egypt "'Associate Professor, Industrial Engineering Department Faculty of Engineering, Cairo University, Fayoum- Egypt E-mail' mme_gahed@wne)l.worldnet.com.ez, .msaateya@alphaI-eng,cairo,eun.eg
ABSTRACT In this paper, the shakedown domain for an infinite square plate with a central hole under cyclic biaxial tension is evaluated by a number of numerical and analytical techniques. These techniques are: full elasto-plastic finite element solution coupled with line search technique and an iterative elastic technique known as the elastic compensation method which relies on successive reduction of elastic modulus in regions where elastic stresses exceed yield strength. In addition, an analytical technique proposed recently in the literature, which relies on a gradient optimization technique to determine the shakedown load factor, is employed. The shakedown domains obtained through different solution techniques are compared. The comparison showed contradiction between literature solutions and the solutions presented herein. The reasons behind this discrepancy were scrutinized and certain recommendations are made to ensure validity of shakedown solutions proposed in literature. KEYWORDS Shakedown, Biaxial Tension, Infinite Plate, Finite Element Method (FEM), Iterative Elastic Techniques, Cyclic Loading I. INTRODUCTION When structures are subjected to cyclic loading, there is a certain limiting load, which causes plastic deformation in the first cycle, but the behavior remains elastic in the subsequent cycles [ 1]. This load is known as the shakedown load. When the shakedown load is exceeded, the structure may undergo ratchetting in which plastic strain accumulates in subsequent cycles leading eventually to exhaustion of ductility. In addition, beyond the shakedown limit, the structure may undergo cyclic-plasticity in which the plastic strain alternates between negative and positive values leading to accumulation of low-cycle fatigue damage. Due to the stress concentration present at the hole of a central plate, Fig. 1(a), both elastic and plastic analyses have been reported. Several authors have produced stress function solutions of this problem in the linear elastic domain [2]. Figure 1(b) shows the elastic hoop stress, 60, distribution around the hole edge for uniaxial horizontal traction. It should be noticed that for PI>P2, the maximum stress state always occurs at point "A". Also, the sign of t r0 changes
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from negative to positive until the ratio of P2/P! is equal to 1/3 at which the minimum value of o0 is zero. For higher values of P2/PI, the stress distribution is always positive. For the limiting value of P2/PI= l, the hoop stress distribution is circular and o0= 2Pi = 2P2. The problem, however, becomes more complicated as the plate undergoes plastic deformation as shown by Chakrabarty [3]. The determination of limit and shakedown domains for an infinite plate with a central hole under cyclic biaxial loading conditions has been investigated in the literature. In 1972, Belytschko [4] has solved this problem using FEM and a non-linear optimization technique. The FEM was used to evaluate residual stress fields which, when added to elastic stresses, the state of maximum stress is contained in the elastic range. The optimization was directed towards finding a magnification factor by which the elastic domain is enlarged such that the addition of the residual stresses to the magnified elastic stresses keeps the structure in the elastic range. Limit and shakedown domains as obtained by Belytschko [4] are shown in Fig. 1(c). In 1994, Stein [5] developed an analytical method and used the Kuhn-Tucker optimization theorem [6] to obtain the shakedown domain (Fig. 1(c)). In Stein's work, residual stresses represented a shift in the yield surface and the optimization algorithm searched for a magnification factor by which the elastic domain is magnified such that the enlarged domain is still included in the shifted yield surface. In this paper, the shakedown domain of the infinite square plate with a central hole is obtained using more than one technique. First, the classical elasto-plastic (EP) finite element analysis is used. Next, an iterative elastic technique known as the elastic compensation method (ECM) is applied [7-10]. Third, Stein's solution [5] for this problem is used. Finally, the obtained domains are compared to verify their correctness and accuracy. 2. PROBLEM LOADING AND GEOMETRY Consider the problem of an infinite square plate with a central hole subject to a system of horizontal and vertical uniform tensile stresses, P l and P2 respectively as shown in Fig. 1(a). P~ and P2 were assumed to vary cyclically - in phase - between zero and certain maximum magnitude, P, as shown in Fig.l(d). The plate material is assumed to be steel having the following properties: Yield strength; Y= 210 MPa, Young's modulus; E= 210 GPa, Plastic modulus; El,= 0.005E = 1050 MPa and the hardening behavior is assumed to be kinematic. 3. ELASTO-PLASTIC SOLUTION (EP) In this section, the problem will be solved using the multi-purpose nonlinear FE package COSMOS/M. Due to double symmetry, only one quarter of the plate was considered in the FE mesh as shown in Fig. 2. In this mesh, average element size in the vicinity of the hole is chosen to be very small (0.013 D) in comparison with an average element size of 0.757 D away from the hole; D being the hole diameter. The problem was modeled using four-node isoparametric plane stress elements with a total of 1600 elements and 1706 nodes and then solved for various ratios of P2/Pl ranging from 0 to 1. The solution was performed for 5 successive loading-unloading cycles for each value of Pz/P,. In EP analysis, a number of iterations are performed for each ratio of P2/Pm until a certain maximum traction value, at which neither ratcheting nor reversed plasticity occurs, is obtained [11 ]. This value defines the shakedown load at this traction ratio. The resulting shakedown domain is shown in Fig. 3.
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4. ELASTIC COMPENSATION METHOD (ECM) The elastic compensation method is an iterative elastic method, which is capable of solving elasto-plastie problems using simple elastic techniques [7-9]. This method depends on performing several elastic iterations while reducing the elastic modulus in regions where the effective stress exceeds the yield strength. Also, Young's modulus in the elastic regions is also modified but the procedure is concerned with the elasto-plastie regions. This process is repeated until convergence is achieved. The method is implemented in conjunction with FEM and has been employed in evaluating shakedown and limit load domains and proved to give fairly accurate results [7-10, 12]. In limit and shakedown analyses, a certain initial load, Pa, is randomly adopted since the procedure is completely elastic and is held constant throughout the iterations. In each iteration, Young's modulus is modified for each element according to the following formula:
~o
Ei+ I = E i ~
(l)
where, Ei+l = Young's modulus in the i+l iteration, Ei = Young's modulus in the i'th iteration, ao = nominal stress, usually taken as 1/2 or 2/3 of yield strength, Y, and ~i - maximum unaveraged nodal effective stress for the element in the i~ iteration. For this purpose, a computer program in QBASIC language was developed for use in conjunction with the FE package to apply the ECM procedures. It has been noticed that COSMOS/M package (V. 1.75A) allows for direct modification of Young's modulus for a maximum of 90 elements, i.e. much less than the current problem size. To overcome this problem, a simple but effective technique was used. This technique, simply, depends on creating a fictitious relation between Young's modulus and temperature and associating all the elements in the model with this relation. Hence, if a fictitious temperature is applied to the elements in the FE model, this will indirectly perform the required modification in Young's modulus. The QBASIC program includes a module for creating this fictitious temperature. Figure 4 shows a flowchart representing the ECM implementation with the FE technique. The FE mesh used in the ECM is shown in Fig. 5. In this mesh, the element size around the hole is chosen to be very small (0.018 D) in comparison to an element size of D away from the hole. Although the element size is greater than it in the EP solution, it should be mentioned that when a finer mesh was used, no remarkable accuracy was achieved. A total of 900 four-node plane stress isoparametric elements and 966 nodes are used in the ECM analysis. An initial value of the horizontal traction; Py, is assumed and the vertical traction; P2D, is taken as a ratio ranging from 0 to 1 from Pi~ ECM estimates of the shakedown domain are shown in Fig. (6) in comparison with the elasto-plastic solutions. Figure 6 reveals that the shakedown load for uniaxial traction is approximately equal to 0.67Y for elastoplastic solution while it is about 0.52Y for iterative elastic solution giving a difference of 0.15Y. This difference can be attributed to the steep decrease in the elastic stresses in the iterative elastic solution from maximum stress (3P) at point "A" in Fig.l(a) to zero at 0 = 30 ~ This decrease affects the solution accuracy in the FEM since it causes the denominator in equation (l) to approach zero. This feature of ECM solution has been indicated in [6]. This difference, however, decreases gradually as P2/P! increases from 0 to 1. When P2/PI = 1, this error becomes 0.015Y which implies much greater correlation between the ECM and the elasto-plastic solution.
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5. STEIN'S SOLUTION [5] In 1994, Stein [5] has formulated the shakedown load evaluation problem for linear kinematic hardening materials as an optimization problem of the form: 13- , max
(2) Vj r K
r [poe(j)+y] < y2
(3)
where 13 is the shakedown load factor which is to be found, tOthe yon Mises yield function, oe(j) the elastic stress vector at load vertex j, K the set of load vertices. The effective residual stress, y, is defined in [ 13] and Y is the initial yield stress of the material. It is interesting to note that this problem has the following geometrical interpretation: Find the maximum enlargement of the elastic stress domain and the corresponding shift in the yield surface on condition that the enlarged elastic domain is still included in the shifted yield surface. This maximum enlargement factor, 13, is evaluated as 13= min 13j
Vj r K
(4)
The mathematical formulation of the optimization problem is as follows [5], u
~
o,a
,
0,
j_>0,
0L oy, -
~___
gig j - 0 ,
,
0,
0L oy~ = 0,
,gi <0,
j---1,2,3,4
,
0L 0y, - 0, m
(5)
(6)
where L is the Lagrangian function defined as: 4
L(13,~., y) = -13 + ~ ~.jgj
(7)
j-i
~. are the Lagrangian multipliers and gj are given by : g j - 6[13o ~ (J) + y ] - O o .
2
The resulting eight non-linear equations are depicted in [5]. Furthermore, he proposed the following assumption to simplify the problem: Si3-Si2+Si4
i- 1, 2, 3
(8)
In equation (8), Si3 represents the ith principal stress at load vertex number 3. The same notation applies at load vertices 2 and 4. This assumption is valid only where all principal stresses are in the same direction. The small number of load vertices and the plane stress behavior allowed for the analytical solution of the problem. The problem was solved using the program MACSYMA and the resulting shakedown domain is shown in Fig. 1(c) Comparing both elasto-plastic and iterative elastic solutions, Fig. 6, with the original Stein solution as obtained in [5], a large discrepancy is noticed at Pl=P2 where the difference in the shakedown load is about 0.5Y. The reasons behind this discrepancy are thoroughly
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investigated and scrutinized. Stein [5] considered the load domain to be rectangular consisting of 4 vertices, Fig. 7, at which the principal stress vectors were evaluated at the maximum stress point (Point "A" in Fig. 1(a)). These principal stress states were then used to evaluate the shakedown domain. In ref.[5], the principal stress vector was constructed as = {0"0,0"r,0"z}T" The theory behind Stein's solution depends on the definition of principal stresses such that: a~ > o 2 > 0 3. This could be in contradiction with the way in which principal stress vector; ~ = {0.o,Or,0",}T is constructed. Also, this will violate the postulation embedded in equation (8) since it is not guaranteed that there will be coincidence between the two arrangements as can be seen in table 1. The error in the original Stein solution was corrected as listed in Table I. Then, the problem was re-solved using the symbolic math toolbox in MATLAB 5.1 program. Fig. 8 compares both corrected and original Stein solutions. As seen from the figure, the shakedown domain size is exactly twice that of the elastic domain for any combination of Pt and P2. It is noticed that the two solutions are identical for the case of uniaxial traction. This is because the principal stress vector in this case depends only on one traction component (either P t or P2) which is not the case in biaxial traction where the principal stress vector depends on the two traction components. Hence, the discrepancy between the two solutions approaches its maximum as P2/P! approaches unity.
Table 1. Original and corrected Stein solutions in terms of the stress state at point "A" for Pt>P2
Load vertex l
2
4 3 Corrected Original Corrected -P2 0 3PI -P2
O' 1
Original 0
Corrected 0
Original 3P!
Corrected 3P 91
Original 3PI -P2
G2
0
0
0
0
0
0
0
0
t~ 3
0
0
0
0
0
0
0
"P2
_
,
0.1,0.2,0"3 are the principal stresses
6. DISCUSSION The ECM results agree with both EP and corrected Stein solutions particularly for higher ratios of P2/PI. Compared with the full EP solution, the ECM technique has the advantages of being simple, modest in its CPU and computer storage requirement. As can be seen from Fig. 8, remarkable differences are noted between the above family of solutions and both the original Stein solution and Belytschko solution. It was noticed that original Stein solution is correct only for the case of uniaxial traction. In Belytschko solution, a mesh containing only 26 elements with only 4 elements at the hole were used. Clearly, this is a very rough mesh in comparison to the meshes adopted here. Figure 9 illustrates the shakedown domain as obtained by all previously mentioned solution techniques. As seen from the figure, there is very good agreement between the elasto-plastic
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Current Advances in Mechanical Design and Production, MDP-7
and the corrected Stein solutions. The strong agreement between these two solutions indicates that the correction made to the original Stein solution is reasonable. To verify the validity of the obtained solutions, two loading points were selected. The first point lies at Pn=P2=0.75Y, i.e. outside the domain obtained by Stein and the second point lies at P n=P2=l.05Y, i.e. outside the domain obtained by both the EP and ECM solutions; see Fig. (8). The problem was solved at these points for 10 consecutive cycles using EP method and the resulting stress strain curves are shown in Figs. 10(a), 10(b) respectively. In Fig. 10 (a), the structure will move along the path 2-3 after the first cycle, which means that the behavior is completely elastic. This contradicts with the original Stein's solution [5] and agrees with the EP, ECM, and corrected Stein solutions developed in this paper. Figure 10(b) shows that the structure will exhibit cyclic plasticity once the loading point exceeds the shakedown domains obtained by EP, ECM and corrected Stein solutions obtained here. 7. CONCLUSIONS Analyses of the shakedown domain of an infinite plate with a central hole by a number of numerical and analytical techniques have shown the following: 9 Care has to be taken when constructing the principal stress vector in the analytical method proposed by Stein (1994) to avoid misleading results. 9 Both the elasto-plastic and corrected Stein solution yielded almost identical results 9 There is a fair agreement between shakedown domains obtained by the elasto-plastic and elastic compensation method. 9 The elastic compensation method has shown to be simpler due to its modest computing requirements. Artificial means for modifying Young's modulus should, however, be sought and incorporated into the ECM. 8. REFERENCES 1. Megahed, M. M., "Cyclic Plastic Behavior of Structural Components" International Journal of Mechanical Engineering Education, Vol. 10, No. 4, pp.235-257, (1982). 2. Timoshenko, S. P. and Goodier, J., "Theory of Elasticity", McGraw-Hill, New York, (1951). 3. Chakrabarty, J., "Theory of Plasticity", McGraw-Hill, (1987). 4. Belytschko, T., " Plane Stress Shakedown by Finite Elements", International Journal of. Mechanical Sciences, Vol. 14, pp. 619-625, (1972). 5. Stein, E. and Huang, Y., "An Analytical Method for Shakedown Problems with Linear Kinematic Materials", International Journal of Solids Structures, Vol. 31, No. 18, pp. 2433-2444, (1994). 6. Moustafa, A. A., "Mechanical System Design", Cairo University Press, (1996). 7. Mackenzie, D. and Boyle, J.T., "A Simple Method for Estimating Shakedown Load for Complex Structures ", Trans. ASME, Journal of Pressure Vessels Technology, Vol. 265, pp. 89-94, (1993). 8. Mackenzie, D. and Boyle, J. T., "An Iterative Elastic Procedure for Estimating Lower Bound Limit Loads" ASME, Journal of Pressure Vessels Technology, Vol. 230, pp. 129134, (1992). 9. Mackenzie, D. and Boyle, J. T., "A Method of Estimating Limit Loads by Iterative Elastic Analysis", International Journal of Pressure Vessels Technology, Vol. 53, No. 1, pp. 7795, part I, pp.77-95, part II, pp.97-119, part III, pp.121-142, (1993).
Current Advances in Mechanical Design and Production, MDP- 7
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10. Marriott, D. L., " Evaluation of Deformation or Load Control of Stress Under Inelastic Conditions Using Finite Element Stress Analysis", International Conference of Pressure Vessels and Piping, ASME, Pittsburgh (1988). 11. COSMOS/M V.I.75A manuals, Structural Research and Analysis Corp., SRAC, (1996). 12. Mohamed, A. I., Megahed, M.M., Bayoumi, L. S., Younan, M. Y. A., "Applications of Iterative Elastic Techniques for Elastic-Plastic Analysis of Pressure Vessels", ASME, Journal of Pressure Vessel Technology, Vol. 121, No. 1, pp. 1-6, (1999). 13. Zarka, J. et al., " A Practical Method to Determine Elastic or Plastic Shakedown of Structures"- Simplified methods in pressure vessel analysis, ASME/CSME Pressure Vessels and Piping Conference, Montreal, pp. 47-60 (1978).
Current Advances in Mechanical Design and Production, MDP-7
192
L
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1% Fig. l(b). Elastic stress distribution around the hole under uniaxial horizontal traction
Fig. l(a). Schematic representation of the plate problem -
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Current Advances in Mechanical Design and Production, MDP- 7
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194
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Current Advances in Mechanical Design and Production, MDP-7
|
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Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
195
EXPERIMENTAL AND ANALYTICAL INVESTIGATIONS OF RESIDUAL STRESSES INDUCED BY AUTOFRETTAGE
El-Shaer, Y. I.*, Aref, N. A.**, AbdeI-Kader, M. S.*, EI-Maddah, M. M.* and Megahed, M. M.*** * Assist. Lecturer, Assist. Prof. and Prof., Military Technical College, Egypt ** Department of Weapons and Ammunition, Egyptian Armed Forces *** Prof., Dept. of Mechanical Design and Production, Cairo University, Egypt ABSTRACT In this paper, experimental results of residual stresses induced by autofrettage of thick-walled tubes to different overstrain values ranging from 50 to 100 % are presented. Electro-chemical boring (ECB) is used together with Sach's technique are employed in order to avoid the complications associated with traditional mechanical boring methods.. The paper also presents the predictions of three mathematical models, Hill's, Chen's and Megahed-Abbas', developed to estimate the autofrettage stresses. Comparisons between theoretical predictions and test results show that the three models are generally capable of simulating different behavioral facets of residual stresses with varying degrees of accuracy. However, MegahedAbbas model provided residual stress distributions that are very close to test results. Since the original version of the latter model deals with specific nonlinear hardening responses during reverse loading, which differ from those experimentally obtained, the need arise to enhance the predictive capabilities of Megahed-Abbas model. Nonlinear hardening during reverse loading and the change of the Bauschinger effect factor with plastic strain were properly taken into account using numerical techniques. Sensitivity of the model to small excursions in the value of reverse loading hardening exponent and the change of the Bauschinger effect factor was elaborated on and the results show strong dependence of model accuracy on some parameters, particularly at the tube bore. KEYWORDS Autofrettage, Residual Stresses, Thick-Walled Tubes, Electro-Chemical Boring, Reverse Yielding, Bauschinger Effect. 1. INTRODUCTION Residual stress distribution in autofrettaged thick-walled tubes is a very important factor in design, fracture analysis, and fatigue life estimation. Therefore, measurement as well as modeling of residual stress distribution have met a widespread interest of several investigators such as Hill [1 ], Parker et al [2], Chen [3], Gamer [4], Megahed and Abbas [5], Perl and Arone [6], and Cheng and Finnie [7]. Aref et al [8] introduced the ECB as a new technique that assists in measuring the residual stress distribution in autofrettaged thick-walled tubes and showed that the model of Megahed
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and Abbas [5] could be used to predict the residual stress distribution with fairly high accuracy. El-shaer et al [9,10] indicated that the accuracy of Megahed-Abbas model could be further improved by using numerical integration and equation solvers when nonlinear hardening, during reverse loading, and the change of the Bauschinger effect factor with plastic strain were considered. In the present paper more insight into Megahed-Abbas model is attempted by studying the effect of changing the elastic range following load reversal and the strain hardening exponent during reverse loading on the predicted residual hoop stress distribution for different radius ratios and overstrains. Results are then contrasted to experimental measurements and relevant conclusions drawn. 2. EXPERIMENTAL WORK 2.1.
Material Characterization
The material chosen for the present study is a high strength medium alloy steel with the chemical composition listed in Table 1. Table I. Chemical composition in weight percent as given by the manufacturer C
Si
Mn
Ni
Cr
V
S
P
Fe
0.27 -.0.40
0.10 -4).35
0.20 -4}.70
3.0 -3.6
0.7 -1.2
0.1 -4).3
<.012
<.012
aem.
Loading, unloading and reverse loading characteristics were also measured experimentally. The material was found to experience linear hardening during loading, that is: o=Y+Kep
(1)
where Y is the monotonic yield strength, K is the hardening coefficient, and Cp is the plastic strain. On the contrary, reverse loading was characterized by nonlinear hardening described by: Ao=2Y*+21"mM(A~p)s,
(2)
where m, M are material constants determined from reverse loading curves, and Y* is half the elastic range following load reversal which varies with ~ according to:
(3)
Y*/Y=C+(I-C) exp(-Dcp),
where C, D are material constants determined from reverse loading curves. Table 2 lists the aforementioned material constants of the material Table 2. Mechanical properties and material constants of as-received material
HRC
v
r (~/lpa)
E (GPa)
K (MPa)
37
0.28
1000
196
2250
M m C .(MPa) . . . . . . . . 4730
0.43
0.61
D 440
Current Advances in Mechanical Design and Production, MDP- 7
2.2.
197
Autofrettage of Specimens
Autofrettage was carded out by means of an oversized mandrel of diameter 21.53 mm, which was made from high speed steel (DIN 17350) with hardness (HRC 65) in order to maintain its original dimensions during autofrettage. This mandrel was forced using a hydraulic press into thick-walled tubes with different internal diameters to produce different overstrains, O, as shown in Table 3. Table 3. Dimensions and autofrettage parameters of test tubes
Spc. No.
a~ (mm)
.
1
10.6563
2 .
3 4 @ Innerradius 2.3.
.
.
.
.
.
.
.
15.1030
19.7390 .
.
10.6414
.
49
15.8863 .
.
19.7670
10.6165 & Outerradius
~,~ (%)
*(mm)
. . . . . . . . .
19.7523
10.6446 .
c
b~ (mm)
.
58 .
.
16.1909
19.7522 *Pl~iicffontradius
61
17.6196 77 #Percentovers~am=(r
Electro-chemical Boring (ECB)
A special design of an electro-chemical boring machine was developed, which avoided the side effects accompanied with traditional mechanical boring applied in Sach's method. This machine enables controlled removal of the bore metal by anodic dissolution of an electrolytic cell, in which the tube is the anode and the tool is the cathode. The electrolyte (sodium chloride) is pumped through the gap between the tool and the tube, while direct current is passed through the cell at low voltage. The process parameters are so adjusted to cut a suitable layer in the inner surface of the tube without inducing any mechanical or thermal effects. The ECB cell is shown in Fig. 1. For more details refer to Refs. [8, 11 ].
Tool
F l o w
Workpiece
Flow
Drive
-----
Main Frame
Fig. 1. Electro-chemical boring cell
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2.4.
Current Advances in Mechanical Design and Production, MDP. 7
Measurement of Residual Stresses
In Sach's technique, the residual hoop stress is calculated from [12]:
cro = l _ v 2
A + f s~ 2f
(4)
J'
where fb=nb2, f=nr2, df=-f(aiter cutting)- f (before cutting) =2nrdr, S=eo+Vez, E is Young's modulus, v is Poisson's ratio, e.o and ~ are the measured hoop and axial strains, respectively. 3. MATHEMATICAL MODELING Three mathematical models developed to calculate the autofrettage stresses are considered herein; these are Hill's [1], Chen's [3], and Megahed-Abbas' [5]. These models are based on different simplifying assumptions, with Hill's model ranking the simplest and MegahedAbbas the most sophisticated. Whereas Hill's model considers elastic-perfectly plastic isotropic material response, Chen's and Megahed-Abbas models encompass more realistic material response characteristics, such as strain hardening and induced anisotropy. In order to successfully handle changes of the Bauschinger effect factor with plastic strain and nonlinear hardening during reverse loading, a computer program in C-t+ language was constructed to calculate the residual stresses, using the trapezoidal rule of numerical integration and the halting-root method for solution of equations. The closed form solution of Megahed and Abbas derived for particular values of the material constant m (cf. Eqn. (2)) was extended to encompass any arbitrary value of m, and to account for changes of the Bauschinger effect factor with plastic strain (cf. Eqn. (3)). The procedure for calculation of residual stresses is shown in Fig. 2. 4. RESULTS AND DISCUSSION Residual stress distributions using the three models were compared with the experimental residual stress distribution obtained from the electro-chemical boring technique for tubes with a radius ratio of 2 and different degrees of autofrettage. The comparison is shown in Fig. 3 for 49% overstrain. The three models are seen to yield almost identical results except at and near the bore, where discernible differences do exist. However, Megahed-Abbas model was found to generally provide the closest results to the experiment. 4.1. Effect of Y'/Y Ratio
The exponential decay of Y'/Y value was taken into consideration in the modified model rather than taking it as a step function in the original Megahed-Abbas model (cf. Fig. 4). The effect of Y*/Y representation was studied using the original and modified versions of Megahed-Abbas model at different radius ratios and overstrains. The comparison is shown in Fig. 5 for b/a=2, m=0.5 and different overstrains. It can be seen that both versions yield almost identical results, except at and near the bore, where significant differences exist, especially for low overstrains. For instance, at | and r/a--l, the modified model predicts ~0s /Y = -0.46 which is 6.5% higher (in absolute value) than its counterpart predicted by the original model. This effect was found less pronounced for greater radius ratios.
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I
.... Input c tube dimensions . a[~b,and plastictnorf
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.
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.
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Fig. 2. Procedure of calculation of residual stresses in Megahed-Abbas model. 4.2. Effect of Strain Hardening Exponent m For a constant Y'fY value of 0.61, different m values (0.45, 0.5 and 0.55) were considered and the effect on the unloading plastic hoop strain (A~ p) and the normalized residual hoop stress (o0R/5") was studied for different radius ratios and overstrain combinations, a sample is shown in Fig. 6. It can be seen that Ac0P decreases with r until it vanishes at the reverse plasticity front and increases with m. For instance, for b/a=2 and O=50%, it is seen that at the bore AcoP corresponding to m=0.5 is equal to 0.16, which is 25% smaller than for m=0.55 and 25% greater than for m=0.45. These results show the strong dependence of A~0p on m. Hence, the use of an exact value of m would contribute significantly to accuracy of computation using Megahed-Abbas model. In Fig. 7, a change in o0R/y of about 11% for the same combinations of b/a, m and 9 is shown at the bore.
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Current Advances in Mechanical Design and Production, MDP-7
4.3. Combined Effect of Y'/Y Ratio and Strain Hardening Exponent m
The combined effect of Y'/Y ratio and m was studied for b/a=2 only, as this effect was found to be less pronounced for greater radius ratios. As shown in Fig. 8, at the bore and for q>=50%, o0R /Y = -0.46 for m- 0.5. For m=0.45, o0R /Y is-0.52, which is 13% greater, whereas for m=0.55, o0R/Y is -0.42, which is 9~ smaller. 4.4. Comparison between Experimental and Predicted Results
Results using the original and modified models are compared with the experimental residual stress distributions obtained using the electro-chemical boring technique, as shown in Fig. 9. A fairly good accuracy is generally observed between the experimental results and the modified model, which indicates the validity and accuracy of the modified model. 5. CONCLUSIONS Based on the above results and discussions, the following conclusions can be drawn: a) There is a strong dependence of A~0P on m; use of exact value of m would contribute significantly to the accuracy of computation. b) The change of Y*/Y with plastic strain has a significant effect on the predicted residual hoop stress, especially for low overstrain values. This effect was found less pronounced for greater radius ratios. c) A fairly good agreement is obtained when comparing the results of the modified model with the experimental results obtained using the ECB technique. REFERENCES 1. Hill, R., "The Mathematical Theory of Plasticity", Clarendon Press, Oxford, (1950). 2. Parker, A., Underwood, J., Throop, J., and Andrasic, C., "Stress Intensity and Fatigue Crack Growth in a Pressurized, Autofrettaged Thick Cylinder", ASTM STP 791, American Society for Testing and Materials, Philadelphia, pp. I-216 - 1-237, (1983). 3. Chen, P., "The Bauschinger and Hardening Effect on Residual Stresses in an Autofrettaged Thick-Walled Cylinder", Journal of Pressure Vessel Technology, Vol. 108, pp. 108-112, (1986). 4. Gamer, U., "The Expansion of Elastic-plastic Spherical Shell with Non-linear Hardening", Int. J. Mech. Sc., Vol. 30, pp. 415-426, (1988). 5. Megahed, M. and Abbas, A., "Influence of Reverse Yielding on Residual Stresses Induced by Autofrettage", Int. J. Mech. Science, Vol. 33, pp. 139-150, (1991). 6. Perl, M. and Arone, R., "An Axisynunetric Stress Release Method for Measuring the Autofrettage Level in Thick-Walled Cylinders-Part I: Basic Concept and Numerical Simulation", Journal of Pressure Vessel Technology, Vol. 116, pp. 384-388, (1994). 7. Cheng, W., and Finnie, L., "Measurement of Residual Hoop Stress in Cylinders Using the Compliance Method", Journal of Engineering Materials and Technology, Vol. 108, pp. 87-92, (1986). 8. Aref, N., Senbel, H., Abdel-Kader, M., EI-Maddah, M., and Megahed, M., "Measurement of Residual Stresses in Autofrettaged Thick-Walled Tubes Using Electrochemical Boring Technique", 70, AMME Conference, Military Technical College, Cairo-Egypt, pp.145155, (1996). 9. EI-Shaer, Y. I., Mostafa, M. M., and Abdel-Kader, M. S., "On Modeling Autofrettage Residual Stresses", 8th Int. ASAT Conf., Military Technical College, Cairo, Egypt, (1999).
Current Advances in Mechanical Design and Production, MDP-?
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10. EI-Shaer, Y. I., "Effect of Residual Stresses on Fracture Toughness and Fatigue Crack Growth in Thick-Walled Tubes", M. Sc. Thesis, Military Technical College, Cairo, Egypt, (1998). 11. Aref, N., " Residual Stress Assessment of Elastic-Plastic Thick-Walled Tubes Using Electro-Chemical Machining", Ph. D. Thesis, Military Technical College, Cairo, Egypt, (1998). 12. Kovac, M., Miyano, Y. and Woo, T. C., "Residual-Stress Measurement in SS 304 Seamless Tube", Experimental Mechanics, pp. 209-213, (1989). 0.3 0.2 ~
/
0.1 o
a. -0.1 O O J= -0.2 m
//
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-----
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.0
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Fig. 3. Experimental and predicted residual hoop stress, b/a=2, 0--49%
o
Experiment Original model Modified model
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=
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o
i
0
)
~
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2
P Fig. 4. Variation of Y*/Y ratio with monotonic plastic strain
Current Advances in Mechanical DesiEn and Production, MDP-7
202
0.6
r r
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Fig. 5. Effect of Y*/Y representation on the normalized residual hoop stress
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~
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Current Advances in Mechanical Design and Production, MDP-7
203
0.3 0.2
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.o ffl
0 1.5
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,"
,'
Radius ratio, rla Fig. 8. Effect of m and Y*/Y changes on the normalized residual hoop stress ( b/a=2, (1)=50%)
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Current Advances in Mechanical Design and Production, MDP-7
0.3 ~ - " 1
0.2 0.1 e
0
Q ,
~ -0.1
"i
= -0.2
,.
j
_
_L
.[,,
t
/ Z,
.
.
.
.
.
i~5,s~l
i
9
,,j
/
/
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to-0.5
"
r~---~,[:"
Z
Odgi"hal M e O ~ A b b a s P l U m
41.6 1.0
1.2
1.4
1.6
Radius ratio, r/a
1.8
2.0
-0.6
1.0
1.2
1.4 1.6 Radius ~to, ~a
(a) 0.3
~
1
- --
I
0.1
,
9 I
/ d~
-0.1
~1-o.2
~
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/
9Exempt ~
:~ -0.5 -0.6
J~
./...
0
-0.3
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Co)
0.2
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~
:f"
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! 1.0
1.2
1.4 1.6 Radius ratio, rfa
(c)
1.8
2.0
1.0
1.2
/ Modified Megahed-Abbas Procedure -Original Megahed-AbUas Procedure 1.4 1.6 Radius ratio, rfa
(d)
Fig.9. Experimental and predicted residual hoop stress, b/a=2.
1.8
2.0
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
205
A MATHEMATICAL MODEL FOR THE STUDY OF THE
DYNAMIC-VISCO-ELASTIC CONTACT PROBLEMS Ali-Eldin, S.S.*, Aly, M.F. ** and Mahmoud, F.F. *
*Department of Mechanical Design and Production Zagazig University, Zagazig ,Egypt **National Center for Nuclear Safley and Radiological Control Atomic Energy Authority, Egypt ABSTRACT The main objective of the present research work is to develop a computational model to simulate the visco-elastic dynamic contact problems. The proposed mathematical model exploits the incremental convex programming in the framework of a finite element model. The main concern of the model is to study basically the effect of the internal material viscosity on the dynamic characteristics of the system such as rate of decaying and response amplitude. Rayleigh damping model is adopted for simulating the internal material viscosity. For model verification and testing two examples of different natures are solved. KEYORDS Dynamic Contact, Integration
Viscoelasticity, Convex Programming, Active Constraints, Direct
I. NTRODUCTIOIN Mechanical contact is of particular interest in numerous engineering applications ranging from metal forming, nuclear reactors, machine design to soil-structure and structure-structure interaction under dynamic excitation. The complexity of a contact problem is in three aspects. The first one is the nonlinear boundary condition of the contact region. The second aspect is how to describe truly the friction phenomenon on the contact surface. The last aspect is the non-linearities of material behaviors and geometric deformation. The importance of the contact phenomena has brought about extensive research work over the years. At present, there are three categories of study methods: analytical, iterative, and mathematical programming method. The analytical method can give a closed form solution and is convenient in practice. Nevertheless, the fitting range for the analytical method is much smaller and only a few problems with regular configurations and special loads can be solved. In the iterative method, the fundamental principle is to take the contact boundary conditions into the finite element equations and then adopt a trial and error method to solve the problem. The third is the mathematical programming method, whose fundamental principle is to turn the contact problems into a special example of mathematical programming. In general, the contact problem can be treated as the minimization of the systematic potential energy under some contact constraints.
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In the aspect of dealing with contact constraints, the existing methods can be mainly classified into two major branches. One is the Lagrange multiplier method, suggested by Carpenter, et al [1] and Taylor, et al. [10], which can be explained as a hybrid variational principles. Another is the penalty function method, which can be thought of as a kind of artificial element, Hunek, [3]. Mahmoud et al. [6] have developed an incremental convex programming model for solving mathematical programming models consisting of a quadratic objective function and a set of linear inequalities, with non zero free coefficients and the design variables could have positive or negative signs. This developed model has been applied on non-conformal contact problems, Mahmoud et al. [5]. Recently, Mahmoud et al. [7], extended the incremental convex programming model to deal with the inequalities of zero free coefficients, which corresponds to the conformal contact , and applied on multiphase frictional contact problems. The present work extends the model to deal with elasto-dynamic contact problem where both inertia and material damping are taken into account. 2. DAMPING MODEL Damping exists in most mechanical systems or structures. Damping predominantly controls the dynamic response and attenuation of vibration. Therefore, using an assumption about damping in the analysis is a justified compromise. Even for experimental purposes, the assumption of damping has to be incorporated so that the obtained data can be processed. There are several damping assumptions available, which are based on single degree of freedom system and extended to multi degree of freedom systems. For single degree of freedom system, damping theories are established to include; for example, viscous damping, coulomb damping and structural damping. A dashpot is frequently idealized as a viscous damping element, in which the damping force is assumed to be proportional to the relative velocity between the two ends of it. The relative motion of surfaces sliding against each other causes coulomb damping. Structural damping caused by the relative motion of surfaces sliding against each other's, it is introduced to model energy dissipation of continuous systems. For a continuous system, energy dissipation within materials is supposed to be caused by thermoelasticity and grain boundary. In dealing with a multi degree of freedom system, all the above damping assumptions are applicable. Amongst these, viscous damping and structural (hysteretic) damping models are most commonly used. Due to the problems of obtaining a system's damping information, there is no practical way of forming the physical damping matrix by using the finite element method The damping properties of a system can vary according to the structure design or the material used. In the analysis of the dynamic response of any system the damping information is essential for satisfactory results. The common Rayleigh damping [8] given by
* , k=0
is a reasonable approximation for small levels of damping, where p is an integer. Using the Rayleigh will consequently result in the existence of real modes that can decouples the system's equations of motion. The simplest case is that consisting of only two terms such as:
[cl-.o [zl +..[-1. where,
ao and a l are
arbitrary constants.
(2)
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The common Rayleigh damping described by Eqn.(1) is proposed by Caughey [8], for which the two-term model (2) is only a special case when p=2, where aoand at are the material and structural damping coefficients, respectively [2]. They are typically determined from two given damping ratios corresponding to two unequal natural frequencies of vibration. If natural frequency, oi and a~ corresponding modal damping ratio, ~i are selected, a0 and at should satisfy the following relation :ao
at
~'s = 2oh + m2 a~, , where i -- 1,2
(3)
3. STATEMENT OF THE DYNAMIC CONTACT PROBLEM Consider two elastic bodies, as shown in Fig. 1,being subjected to dynamic load. Each of the two bodies occupies a bounded domain f2 in RN,N = 2,3 with open interior if2. Assuming that the material of the two bodies is isotropic and homogeneous and the contact between adjacent surfaces is frictionless. Figurel depicts the contact system and illustrates the different types of boundaries.
r~o _ , t ~ , l ,
FD Fig.l. Contact of two bodies
3.1. Strong Form of The Initial- Boundary Value Problem The problem is considered as an initial boundary value problem The deformation is governed by the following equation of motion of the system, boundary conditions, and initial conditions, which are known as the strong form of the problem (i) Equilibrium equations:.. Gij,j + fbi = P u i + ~ u i where, the constitutive equation is given by Gij = Eijkl.Uk,I ii) Initial and Boundary conditions:a) Initial conditions
(5)
b) Displacement boundary conditions
u(x,O) = 0 9 J u(x,O) = 0
(4)
u(x)=u(x,t) ~' X E I"D Vx~IU~2
c) Loading boundary conditions:crijnj= qi 'V' xE FF
d) Contact boundary conditions :un-g<0 V x~rc qn-<0
m
where, ui,ui ,ui, u(x,t) and qi prescribed
;are the displacement ,\l velocity, acceleration,the
displacement and prescribed boundary traction, respectively. Furthermore, fbi,
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208
qij, P and nj are respectively, the body force, the Cauchy stress component, density, and the component of a unit normal vector n on the boundary. 3.2 The Variational Formulation of The Problem The prementioned classical formulation is equivalent to a dual formulation of minimizing the following functional of the total energy: (6) subjected to" Un - g
Vx 9
EFC
9
(7) 9
where, a ( u , u ) and d ( u , u ) are the kinetic energy operator and the energy dissipated by damping, respectively. Furthermore, S(u,u) and w(P,u) are functions representing the strain energy and external work done. The equivalent Lagrangian formulation of the above model is given as follows :Min L(u, u, A) = E + J A.(u n - g)+ ds
(8)
Fc
where, L is the Lagrangian function ,( un- g )+ represents the level of interpenetration and 2 is the Lagrange multiplier factor. 3.3 T h e D i s c r i t i z e d M o d e l By the finite element discretization scheme, the functional E defines by Eqn. (6) would take the following form"
1 E(u'u)=-2 {u}T[M]{u} -{u}r[C]{u} +-'! 2 {u } T[K] {u} - {u } T {p}
(9)
and contact constraint may be reformulated as follows :T [GI {u}-{g}_< 0
(10)
Therefore, the equivalent Lagrangian formulation of the contact problem would take the following form :Min
1 " T 9 L(u, u , X ) = ~ - l u } [Ml{u}+{x}T({G}T{u}-g
1
T
9 1 [C]{u}+ ]-{u }T
)-{u}T{P}
[Kllu} (11)
where; [M],[K] and [C) are the overall mass, stiffness and damping matrices, respectively, {u} is the displacement vector , {X} is the contact force vector which stands as the Lagrange multipliers vector, {P} is the total external force vector, {G}, is the initial gap vector of
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contact pairs. Since the problem defined by Eqn. [ 11 ] is a convex One, then the Kuhn-Tucker (K-T) necessary conditions are also sufficient, and any local minimum is also a global one. The global minimum point, defined by v, v and Z should satisfy the following (K-T) necessary and sufficient conditions "-
[M
]{ v } +
{G }T { v }
[C ] { v } = {g}-
+ [ K ]{ v } + {G }T {~--} = { p } {S}
(12) (13)
where, {S } is a non negative value which represents the slack variable. Also we have to notice that the Lagrange multiplier {X} should be a non- positive vector. m
4.
SOLUTION P R O C E D U R E
The two bodies are discretized into a finite element grid. The mesh is refined at the critical area throughout the contact regions. Displacement and traction boundaries should be described. A strategy of how to determine the active and inactive set strategy will be exploited as given by Mahmoud et al [6]. Apply the incremental convex programming procedure to determine the incremental displacement vector{u}, velocity vector { u } and the incremental Lagrange multiple's vectors { L } which represent the contact pressures at time instance T(k+l) a) Contact detection 9 - The displacement responses should satisfy the contact inequality constraints 9 u k9+ l - _Ujn k+l
,k+l -< c"-'ij
on inactive
Fc, i,j =1,2,... N C
where uikn+1 and ujkn+1
are the normal components of displacements of the node pair i and j, G .k+l
is the gap of the candidate pairs i and j, and NC is the number of candidate contact pairs.
" Compute violation function V k+l = umk+l- ujnk+i - G,/ on inactive Fe. - The violation function plays a unique role such that: if V K§ < 0 the pair of nodes i, j are separated: if V K+l - 0 the pair of nodes are in contact and ; if V K§ > 0 , the pair of nodes i ,j are interpenetrated.
- Among the set of node pairs associated with positive values, the pair with the maximum V K§ value is selected to be the next contact. This pair is assigned by ij such that Vij -'- MAX (Vu)
on inactive Fc
- Using a suitable interpolation function relating the displacements and velocities at time T(k) and T ( k + l ) , the instant of time TC at which the gap is closed can be calculated.
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Current Advances in Mechanical Design and Production, MDP- 7
b) Separation detection :
- The separation is detected when the contact pressure between two pairs of nodes reaches a zero value at the contact point. The identification of a candidate active constraint to be changes as an inactive one depends on the determination of the violation vector: sk
_ k ~k-I
=~A I
- For all candidate active constraints having the following condition: _ k
k-I
Sgn(~.Ai ) ;~ Sgn(~Ai
)
where each element of the violation vector is given as" _ k
k-I
- Among the set of constraints having a tendency to release, the one with maximum value ,'T', is selected as the next active constraint turns into inactive one such that
- Compute the adaptive scale factor time ,TS, required to establish this new inactive pair of nodes, where TS, is the release time. - Among the set of node pairs which are associated with tensile force at instant T k+t, the pair with minimum separation time selected to be the succeeding release points such that :TS = MIN ( TSil ),
for active F c
c) Time marching :
- The instant of time TI for the new state should be selected such that TI - MIN (TC,TS) - Calculate the displacement, velocity and acceleration at TI using an interpolation function. - The stiffness matrix and the mass matrix including damping must be modified to accommodate the new state. - The problem is reinitialized with time marching ahead until the domain of interest is covered or the velocity vector reaches zero. 5. NUMERICAL SIMULATION Numerical simulations of two dynamic contact impact problems are presented in this section. The first is the impacting of two bars and the second is a beam impacted by an impactor. 5 . 1 . Impact of T w o Bars
This problem, although simple to solve analytically, is a source of considerable insight as regards impact and separation phenomena. Many researchers have considered the solution of this problem,J1,4,9,10]. The given values are dimensionless and are topically as given by Carpenter, et al. [ I].
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a) Impact of two identical bars: Two identical bars, one initially stationary and the other moving with constant velocity v = 1.0, impacts each other at time t=0; see Fig.2. Simulation is preformed by two-nodes, one dimensional linear elastic elements. A uniform mesh of 20 elements is considered for each bar. The time step used is set to 0.0001 msec. Damped as well as undamped solutions are considered . The damping parameters are assumed to be (a0=0.00 I) and (an=0.065). Figure 3.a displays the history of displacement, velocity and contact force for tips of the two contacting bars. The results are compared with Carpenter, et el., [1 ] and a good agreement is found. A comparison between damped and undamped solution is shown. Oscillations in the velocity are noticed in the undamped solution and vanished for damping case. V1
[
I
I,,
I
E= l . 0 L=10.0 A=I.0 p=l.0 Fig. 2. Impact of two identical bars 40
-.-, r.
-
_
I ......... oom,,;S . . . . . Ref. (
~
=-- : =--
~
...... -1 ~
/
Undamped
...... ; ~a~--~0
10
20
30
.........
40
0
80
TUUl~lul
0
10
20
Time 1 .s
.~ o.,~:" : :K -o.I
Bor 1..I
rj~v
.....
0.0
10
-
20
'
Time
~
4O
: ,
.
.
I
30
40
8O
.
_o, iii iii
IVV,, !
30
.
Time
0.7
--0.6
-O.S : lUWUwiVUVUlUUU,hlUUlOInUUnUlWlltlUrIIVVUnlVUUUUI, IIll 0
~
~, ~o Q.
TnaVitvU~l~UVVUUUUilUV~vvwnUUliinivuuT,i~uuuu~il
0
Ref. (1) Undamped
5O
0
0.0 j
=
10
I
..
20
Time
nln I I I I I l | n i l t / i l l 30 40
~ . ~
--
I
nI
8o
--"
eD
u0o -o.2
o
~
..... ---
:
'~c --0.4
m,r P ~
~
Undarnlmd
o
-0.:
. . . . . n,~. ( 1 ) . =;:== U n d a m L ~ m ,
~
-0.4 "-~~-_,.,_..,_~- ~ _
_
8 --O.e
lllllldljl|lllllllll|jlljllllllllll,llWlU, 0
10
20
Time
30
l/iWllllll 40
SO
--O.t
iiIlllllllllllllillillllliillllllJ'liillillllllilllla
0
10
20
Time
30
40
BO
(a) (b) Fig.3. Impact of two; (a) identical bars and (b) dissimilar bars; history of bar tip displacement , velocity and contact force for damped and undamped conditions and comparison with reference (1).
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b) Impact of two dissimilar bars Two dissimilar bars coming into contact at time t=0. The properties of the two bars are the same as the previous example except that E1=0.49. Damped and undamped solution is preformed with the same damping parameters as similar bars. The displacements, velocities and contact force at the contacting tip are shown in Fig. 3.b. Good agreement with the result of Carpenter, et el., [ 1] is exhibited. 5.2 A beam Impacted by an Impactor A beam impacted by an impactor is simulated in this study. Figure.4 shows the beam and the impactor dimensions and material properties. Material properties of the beam and the impactor are the same. The beam and impactor are discretized using three nodes triangular element. The numerical integration is preformed using an explicit (central deference) scheme. The time step used is 0.2 psec. Both damped and undamped simulations are preformed. For damped simulations the damping parameters are chosen to be: a o = 0.0009, al = 0.0003. The impactor initial velocity is 10 rn/sec. Fig.5.a and Fig.5.b, shows the history of the beam displacement, velocity at the contact point, for the two cases of damped and undamped. Highfrequency oscillations of the velocity of the beam are seen in the undamped condition, where no oscillations are found in the damped condition. Slight material damping eliminates these oscillations as shown from the figure. Attenuating beam transverse displacement and velocity with time is shown also for the damped condition.
Impactor E = 10.0 G Pa v = 0.25
p= 1000 k g / m 3 V
Omm
Beam
t_.m . .q
600 mat
"i i.... IT,--
~ y
60 mum
Fig.4. Dimensions of beam and impactor
Impactor displacement and velocity at the contact point is also shown in Fig.5.c and Fig.5.d, for undamped and damped conditions, respectively. In addition, for undamped conditions it is noticed that, no dissipation in the velocity and transverse displacement has occurred. The solution including material damping are more realistic because no real material have zero internal damping. It can be noticed that damping reduces, or eliminates the level of oscillations of the velocity.
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213
..........................
20
20
A
12
4. --4-
_o ._m r
1
0
~
--12
-20
-20
1 2
.............
i ll +iii ili8+'2
+
8 +
o
--4 -12
ll~nnlij,jliUVnUnnVilUUUU,HVVlUWUV,iV~l~VVvv~ v
0
40 80 120 160 Time (sec X 10-=)
200
+I,+-V-+
0
"ri~lul~ql i i i i i l
,tO
-
.....
--8
l ~
u ul ui'i'l I u = u i I i i i1 ii~in i i i i i i i i u v i i -+ " - 1 2
80
120
Time (sec X lO -=)
160
200
(b) Beam damped
(a) Beam Undamped
160
,'~ 160' E E 120-
i"
"120
-80
cQ) 80 E o 40 o
.e_ Q
-
40
-0
0 -40
i/vii
v I r l l i ul I l l i i
v IV J l i i v v vl
li"1ililfJlliiillilUl
-40
iiiil=lli
12-
E
+-
0-4-8-
.................
...... ~ 0 -4 -8
-12-
0
10 20 30 40 50 Time (sec X 10-3)
(c) Impactor undamped
60
I I II i II i t ['l I l I I I I+i-I+i] I i I I l f i l l
0
10 20 30 40 50 Time (sec X 10-s)
i
60
--12
(d) Impactor damped
Fig.5. History of beam and impactor contact nodes displacements and velocities for damped and undamped conditions.
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Current Advances in Mechanical Design and Production, MDP- 7
6. CONCLUSION For more accurate simulation of the behavior of contacting/impacting system internal and external damping should be accounted for. The present paper extends an incremental convex programming model to deal with elasto-dynamic contact problems. Material damping is taken into consideration and the linear conventional Rayleigh model is adopted. The proposed mathematical model exploits the incremental convex programming in the framework of a finite element model. The model is applied on two problems of different nature. The first is a typical problem of an impact of two bars. The results of the proposed model are compared with that published in the literature. Good agreement are noticed for both undamped and damped contact problems. The second is a two dimensional simulation of a beam impacted by an impactor. More realistic results are obtained by taking the internal damping into consideration. REFERENCES 1.
.
3. 4. 5.
6.
7.
8. 9. 10.
Carpenter, N. J., Taylor, R. L. and Katona, M. G., "Lagrange Constraints for Transient Finite Element Surface Contact", International Journal for Numerical Methods in Engineering, Vol. 32, pp. 103-128, (1991 ). Jiang, L. and Rogers, R. J., " Proportional Material Damping in Finite Element Impact Analysis", Computers and Structures, Vol. 3 l, No. 2, pp.235-247, (1989). Hunek, I., " On a Penalty Formulation for Contact-Impact Problems" Computers and Structures Vol. 48, No. 2, pp. 193-203,(1993). Mahmoud, F. F., Hassan, M. M. and Salamon, N. J.," Dynamic Contact of Deformable Bodies " , Computers and Structures Vol. 36, No. 1, pp. 169-181, (1990). Mahmoud, F. F., AI-Saffar, A. K. and EI-Hadi, A. M., "Solution of Non-conformal Unbounded Contact Problems by the Incremental Convex Programming Method ", The Arabian Journal for Science and Engineering, Vol. 16, No. 2B, pp. 325-332, (1991). Mahmoud, F. F., AI-Saffar, A. K. and Hassan, K. A. ,"An adaptive incremental Approach for the Solution of Convex Programming Models ", Mathematics and Computers in Simulation", Vol. 35, pp.501-508,(1993). Mahmoud, F. F., Ali-Eldin, S. S. and Emam, S. A.," An Incremental Mathematical Programming Model for Solving Multi-Phase Frictional Contact Problems ",Computers and Structures, Vol. 68, pp. 567-581,(1998). Man Liu and Gorman, D. G., " Formation of Rayleigh Damping and its extensions" Computers and Structures Vol. 57 No. 2 pp. 277-285, (1995). Ning Hu ," A solution Method for Dynamic Contact Problems", Computers and Structures Vol. 63 No. 6, pp. 1053-1063, (1997). Taylor,R. L. and Papadopoulos, P.," On Finite Element Method for Dynamic Contact/Impact Problems" International Journal for Numerical Methods in Engineering, Vol. 36,
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
215
ELASTIC- PLASTIC ANALYSIS OF NOTCH ROOT STRESS-STRAIN AND DEFORMATION FIELDS UNDER CYCLIC LOADING H a m m o u d a , M.M.I. * and Seleem, M.H. **
* Professor, Mechanical Engineering Department, AI-Azhar University, Cairo, Egypt ** Assistant Prof., Engineering Materials Department, Zagazig University, Zagazig, Egypt
ABSTRACT Two dimensional elastic-plastic finite element model was used to simulate the stress-strain and the deformation at the root of a notch under both monotonic and cyclic loading. The analysis was performed for the plane stress state and constant amplitude loading of zero stress ratio. Twenty one single edge notches having different depth and radius were studied at different maximum stresses. Relevant kinematic parameters corresponding to loading and unloading phases of a cycle were computed and correlated. Elastic-plastic analyses of the stress concentration factors at the notch root were invoked. The monotonic and cyclic notch root plastically deformed zones are presented KEYWORDS Finite Element Method, Cyclic Loading, Notch Plastic Deformation, Stress Concentration Factors I. INTRODUCTION Geometrical irregularities are of central importance in the life assessment of structural components, since they act as local stress and strain raisers. The majority of fatigue failures occur at such discontinuities [1]. A survey of serious accidents involving fatigue fracture clearly underlined the importance of stress concentrators as crack initiation sites [2] In designs against static loading conditions, an elastic stress concentration factor, Kt, is used to estimate the local stress at the notch root. For high cycle fatigue, it has been suggested that there is a relationship between Kt and a fatigue strength reduction factor, Kf, obtained via a material constant called the material notch sensitivity factor, q. Estimating Kt, Kf and q appeared to work reasonably well for infinite life estimation, but for finite life this approach does not provide a good correlation with experimental results [3]. The widely used strain-life approach [4] is based on the knowledge of the notch root strain history. Neuber's rule [5] is a widely accepted analytical technique for the nonlinear analysis of notched bodies. Although this technique received considerable attention [6], there is some doubt about its applicability. It has been reported [7] that for low cycle fatigue-life prediction,
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Current Advances In Mechanical Design and Production, MDP- 7
this technique is not always accurate. For moderate stresses, Neuber's rule tends to overpredict notch strains [8-10]. Methods of estimating stress and strain at the notch root have been developed to overcome some of the limitations of the Neuber's approach. Glinka [I 1] assumed that the local elasticplastic energy density in the localized plastic region is proportional to the nominal elastic energy density. However, relatively good fatigue-life predictions were obtained from Neuber and Glinka approaches for the maximum strain [12]. It has been found that the elastic-plastic strain concentration factor, the strain distribution ahead of notches and the plastic zone size in their front are important quantities for estimating the fatigue-life of notched specimens [13]. The objective of this work is to investigate numerically the monotonic and cyclic stress-strain and deformation accommodated at the root of notches having different geometries. 2. NUMERICAL ANALYSIS An elastic-plastic two-dimensional finite element, FE, model as detailed in [ 14] was utilized to simulate the stress-strain and deformation fields at the root of a notch of different geometries. The von-Mises yield criterion and the Prandtl-Reuss flow rule were adopted. Baushinger's effects were considered through the kinematic hardening model modified by Table 1. Numerical program in the present work Notch geometry Depth, D . Radiuslp " Kt 0.11 mm 0.11 mm 3 0.11 0.0275 5 i0.16 0.16 3 0.3 0.3 i3 . 0.3 0.075 5 0.3 0.01875 9 0.6 0.6 3 0.6 0.3 3.83 0.6 0.15 5 !0.6 0.0375 9 I1 1 3 1 0.5 3.83 1 0.25 5 1 0.0625 9 1.5 1.5 3 2 2 3 2 1 3.83 2 0.5 5 2 0.125 9 4 i4 3 L, 8 !8 3 U
i
l
'
,,i
~,
,,
9
,
.
Omx/% . I 0.69, 0.74, 0.8, 0.86, 0.91 0.51, 0.6, 0.67, 0.8, 0.89 0.67, 0.7, 0.73, 0.76, 0.8 0.77, 0.8, 0.83, 0.89, 0.91 0.4, 0.46, 0.49, 0.54, 0.86 0.29, 0.34, 0.4, 0.46, 0.54, 0.86 0.69, 0.74, 0.77, 0.8, 0.86 0.69, 0.74, 0.8, 0.86 0.37, 0.38, 0.4 0.23, 0.29, 0.34, 0.4 0.69, 0.74, 0.77, 0.83, 0.89 i 0.57, 0.66 0.37, 0.43, 0.51 0.23, 0.29, 0.34, 0.89 0.67, 0.69, 0.71 0.69, 0.7, 0.71 0.46, 0.57, 0.66 0.37, 0.43, 0.51 0.23, 0.29, 0.34 0.67, 0.69, 0.71 0.67, 0.69, 0.7 ........ 9
,
,
i
i
,
9
.
.
.
.
.
Dt
Details of the U-notch
Current Advances in Mechanical Design and Production, MDP- 7
217
Ziegler [15]. In the plastic regime, the stress-plastic strain behaviour of the material was assumed to obey a simple power law. The two types of semicircular and U-shaped notches listed in table 1 were analysed to cover a reasonable range of notch size and acuity. The notches were subjected to different applied maximum nominal stresses, 6max. The analysis was carried out for the plane stress state on a plate having a symmetrically single edge notch which had the dimensions of 200 mm height and 100 width. The plate was subjected to either monotonic or cyclic constant amplitude axial loading with a stress ratio R=0. A mesh generation technique was developed to model one half of the plate with constant strain triangular elements. A small element size enough to accurately capture both the monotonic and cyclic plastic deformation existing around the notch root could be realized. The minimum element size at the notch root was generally 0.040. An example of the FE idealisation used in the present work is shown in Fig.1. The plates had the following mechanical properties : yield stress, try = 350 MPa, modulus of elasticity, E--206 GPa, monotonic and cyclic hardening exponent, n = 0.2 and Poisson's ratio, v = 0.3. In the course of a stress cycle, the stress- strain and deformation parameters at the root of the notch were incrementally traced. The size of the notch root plastically deformed zone, A, was calculated as the diameter of a circle having the same area as the plastically deformed elements. The extent of A at the maximum stress was Am. The extent of the cyclic notch root plastically deformed zone, Ar was given by the plastically deformed elements commonly generated at maximum re-loading and minimum un-loading. 3. RESULTS AND DISCUSSION 3.1. Notch Root Stress-Strain Field
The effect of notch acuity as described by the ratio D/p on Kt is shown in Fig.2. The solid line in Fig. 2 represents the well-known relation between Kt and D/p for elliptical notches, i.e. Kt = 1 + 2 (D/p) ~ The value of Kt was estimated in the elastic regime as the ratio of the normal stress in the element just close to the notch root and the corresponding nominal applied stress. A general increase is observed in Kt with the increase in D/p. Figure 3 presents the stress concentration factor, K,,, and the strain concentration factor ,K~, beyond the commencement of notch root plastic deformation. Here, K~ and K~ are defined respectively as [16] trn/tr and en/e, where On and gn are respectively the notch root stress and strain, and tr and e are the applied nominal stress and strain. Due to the development of plastic deformation, with increasing applied stress, Ko decreases with a decreasing rate until it reaches approximately a constant value while Ke shows an opposite trend. The results of K~ for the different notches are plotted against the notch-stress parameter, p = Kt c/try, in the form shown in Fig. 4. Curve fitting of the data in this figure lead to the following relation to estimate K~ in term of Kt and tr/try 1 - (K,,/Kt) 0'7
--
0.83 ( 1 -
!,1.-0.7 )
(1)
The limiting conditions for this equation are as follows. When p approaches unity, i.e. tr/O'y = l/Kt, K~ approaches Kt. When ~t equals Kt, i.e. a/ay = 1, Ko approaches a constant value dependent only on Kt as given by K~~ = 0.83+0.17 Kt07.
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Current Advances in Mechanical Design and Production, MDP-7
An attempt to fit Kt , Ko and Kc with the well-known Neuber's approximation rule failed to gather the present data of all the notches analysed. The effect of Ko and Kt on K~ is presented in the form shown in Fig. 5. The data in Fig.5 could be best-fitted by the following relation K,;/Kt - 1 = 0.29 [ (Kt/Ko) L8 - 1 ]
(2)
3.2. Notch Root Plastically Deformed Zone The progress of the monotonic plastic zone shape with the applied stress for semi-circular notch of 1 mm depth is represented in Fig. 6. The figure illustrates that plasticity took place first at the notch root and spread more rapidly along the notch surface in the form of a long narrow strip and this is confirmed with the results of Bowie et al [17].
The development of the monotonic plastic zone, Am, at the notch root with ~t is presented in Fig. 7 in the form of 4Am/p against (~t2-1) for the present notches. The continuous dashed line represents Neuber's analytical solution given by 4Am/p = (~t2-1). This figure illustrates that up to a certain stress level the present finite element results can be fairly represented by the Neuber's analytical approximation. Obviously, Neuber's solution underestimates Am for all the present notches, since balancing of the applied load is not ensured. The scatter in the estimation of Am is slightly small up to a certain stress beyond which the Neuber's solution fails completely in the estimation of Am, i.e. rapid increase in the values of Amwith an increasing rate is observed. The stress at which the results start to deviate from the Neuber's solution depends on Kt. In Fig. 7, a decrease in this stress is indicated with the increase in Kt. Thus, the accuracy of the Neuber's solution in estimating Amis a function of Kt and O/Oy. The results presented in Fig. 7 are plotted in the form shown in Fig. 8. The extent of AmCan be predicted in terms of stress level and Kt by making use of the following best-fitted relation g 4Apm = 0.76(/z 2 . - 1 0.75 + 0.21 __
(3)
When p approaches Kt, general yielding occurs. When ~t approaches unity, Ambecomes zero. The extent of the notch root cyclic plastic zone, Ac, with ~t is shown in Fig.9. The figure illustrates an increase in Ac with an increase in the applied stress range and notch acuity. The data in the figure can be best-fitted by the following form 4A~ : 1.627(/12 _ 1)o.6 P
(4)
The term ~t is defined for cyclic loading as KtAo/AOcy,(AOcr~2 cry ), where Ao is the applied stress range and Acrcyis the cyclic yield stress range. 4. CONCLUSIONS Finite element analysis provided reasonable prediction of notch root elastic stress concentration factor.
Current Advances in Mechanical Design and Production, MDP- 7
2i9
2.
Up to a certain stress level dependent on the notch acuity, the monotonic plastic zone is represented by the Neuber's analytical approximation. Neuber's solution underestimates Am at higher stress levels. 3. Simple relations were suggested to estimate the notch root stress and strain concentration factors and the extent of both monotonic and cyclic plastic zones. REFERENCES
1. Fuchs, H. O. and Stephens, R. I., "Metal Fatigue in Engineering", John Wiley, New York, (1980). 2. Campbell, G. S. and Lahey, R. A., "Survey of Serious Aircraft Accidents Involving Fatigue Fracture", Int. J. Fatigue, Vol. 6, pp 25-30, (1984). 3. Sherrat, F., "Fatigue Life Estimation: A Review of Traditional Methods", J. Soc. Environ. Engegrs, pp 23-30, (1982). 4. Dowling, N.E., Brose, W.R. and Wilson, W.K., "Notched Member Fatigue Predictions by the Local Strain Approach. Fatigue under complex loading", SAE, Advances Engng,, Vol. 6, pp 55-84, (1977). 5. Neuber, H., "Theory of Stress Concentration for Shear Strained Prismatical Bodies with Arbitrary Nonlinear Stress-Strain Law", ASME Journal of Applied Mechanics, Vol. 28, Dec., pp 544-550, (1961 ). 6. Seeger, T. and Heuler, P., "Generalized Application of Neuber's Rule", J. Test. Eval., 8(4), pp 199-204, (1980). 7. Krempl, E., "The Influence of State of Stress on Low Cycle Fatigue of Structural Materials", ASTM STP 549, (1974). 8. Sharp, W.N., Jr., Yang, C.H., and Tregoning, R.L., "An Evaluation of the Neuber and Glinka Relations for Monotonic Loading", ASME journal of Applied Mechanics, Vol.59, June, pp 550-560, (1992) 9. Harkegard, G., and Stubstad, S., "Simplified Analysis of Elastic Plastic Strain Concentration in Notched Components under Cyclic Loading", Fatigue design, ESIS 16, Solin, J. et al., eds., Mechanical engineering publications, London, pp 171-186, (1992) 10. Harkegard, G., and S~rb~, S., "Applicability of Neuber's Rule to the Analysis of Stress and Strain Concentration under Creep Conditions", ASME, Journal of Engineering Materials and Technology, Vol. 120, July, pp 224-229, (1998). 11. Glinka, G., "Energy Density Approach to Calculation of Inelastic Strain-Stress near Notches and Cracks", Engng Fract. Mech., 22(3), pp 485-508, (1985). 12. Shatil, G., Ellison, E. G. and Smith, D.J., "Elastic-Plastic Behavior and Uniaxial Low Cycle Fatigue Life of Notched Specimens", Fatigue Fract. Engng Mater. Struct. Vol. 18, N0.2, pp 235-245, (1995). 13. Shingai, K., "Elastic-Plastic Strain Concentraion a and Plastic Zones of Notched Specimens under Tensile Load", Proceedings of the International Conference on Engineering Against Fatigue, Sheffield, UK, 17-21 March, pp 423-430, (1997). 14. Hammouda, M. M. I. and Sallam, H.E.M., "An Elastic-Plastic Finite Element Simulation of Crack Tip Deformation in Fatigue", Presented at ICF8, Kiev., 1993, Advance in Fracture Resistance in Materials, Int. Congress on Fracture, Vol. 2, pp 3-10. (1996). 15. Ziegler, H. A., "Modification of Prager's Hardening Rule", Q. Appl. Math., (17), (1959). 16. Klesnil, M. and Lukas,P., "Fatigue of Metallic Materials", Elsevier Science Publ., Amsterdam, (1992). 17. Bowie, O. L and Freese, C. E., "Elasto-Plastic Plane Strain Analysis for a Circular Hole in a Uniaxial Tensile Field", Trans. ASME, J. Appl. Mech. pp 712-72 l, (1971).
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Current Advances In Mechanical Design and Production, MDP-7
K
\/\/ /\/\ \/\/ /\/\ \/\/ /\/\
Bou,ndary conditions : 1. Along J K P = Z vertical nodal forces 0 = Nodal horizontal disp. 2. Along ML. 0 = Nodal vertical disp. 3. Otherwise
\/\/ /\/\ \/\/ ./\/\ \/\/
0 = External nodal forces
-._j , , ~ ~ ~ M
L Fig.1
I
I
"I
I
l
I II
I
I
Finite element idealisation
I"
I
I
1 I~ I I
l
I
"v
I ''~
w
'1
~
D/p 30
10
,, 0
1 2
, o
4 16
I
w
a o
a
K,
o~ o 0
2o
I
.
O
Kt
Kt= t +2(D/p) ~
Present FE i
i
1
i
I ii!
I
1o Notch acuity,
l
i
i
D/p
I i
i1
100
Fig. 2 Elastic stress concentration factor against notch acuity
K,** ***
jr
,,
0 .... o.o
0 0 0 0 K.
' 0.2
'
',' o.~
' 0.8
'~
0.8
:~
1.o
(7/(Yy Fig. 3 D e v e l o p m e n t of I~ a n d K, with the applied stress
Current Advances in Mechanical Design and Production, MDP-7
0 . 8
" 6
, '
=
f
-
l
221
......
0.6
4
I 0.4 I ',--I
2 0.2
0.0 d~u.0.0
! 0.2
t
.
0.4
l-(z/~)
I
i
0.6
0
0.8
2
~
4
6
8
notch-stress parameter p=l.00
,,
, ,-
~
_
~=1.03
! Notch root ~t = 1.16
~
g = 1.28
b
..
_
,
!
12
Fig. 5 Relation b e t w e e n strain and stress concentration factors
l
""
--
(Kt/~)~'a- 1
Fig. 4 Development of I~ with the
-~-~.~
,
10
Aj "
. . _
1
cA
Fig. 6 Development of monotonic plastic zone shape in a plate having a semicircular notch of I mm depth for various notch-stress parameter
Current Advances In Mechanical Design and Production, MDP-7
222
20
,.,
!
I
30
,
l
4A=/p= z_ 1
I
o
i
I
'
Present FE
D
#
0
20
@ 0
o
I0 0 s
s
S ~ s
.s I
10
i
D/p=
D/p= l O0
Q-
60
~.
i
,
,
0
,
5
10 ,
,
s
t
I
,
0
5
2 I
10
15
200
,
8 0
150 40 @
0 0
g
100
u
20 5O
D/p= 4 I
10
I
D/p= 16
I
O
20
30
w
30
0
60
90
2
Fig. 7 D e v e l o p m e n t
o f /Xm w i t h
160
the
notch-stress
0
!
|
!
iiii
parameter
4
!
|
! |~111|
!
!
! ill!
120 ck
80
0.1
40 o 0
50
100
Eq. (3) Present FE
150
200
( ~ - I ) ( 0 . 7 5 + 0 . 2 , ( ~ - Z) / ( K ~ - ~ ) ) Fig. 8 D e v e l o p m e n t of l~, w i t h t h e notch -stress parameters
0.01 ........ J 0.01 0.1
Eq. (4) o P r e s e n t FE . . . . . . . . ' ' ' ' ..... 1 I0 2
/~-1 Fig. 9 D e v e l o p m e n t of ~ with the notch -stress parameter
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
223
ELASTO-PLASTIC THERMAL STRESSES IN FUNCTIONALLY GRADED MATERIALS CONSIDERING MICROSTRUCTURE EFFECTS
Shabana, Y.M.*, Noda, N.** and Tohgo, K.** * Ph. D student, Graduate School of Science and Engineering, Shizuoka University, Japan ** Professor, Mechanical Engineering Department, Shizuoka University, Japan
ABSTRACT Functionally graded materials (FGMs) have been developed as ultrahigh temperature resistant materials for aircraft, space vehicles and other engineering applications. Most of FGMs are particle reinforced FGMs and their compositions depend on position. In the heat-resistant FGMs which contain particles (ceramics) in the matrix (metal), the matrix will be subjected to plastic deformation, particles will be debonded, and finally cracks will be generated. In the past many studies of FGMs, a macroscopic combination law has been adopted to express the material properties. However in order to obtain more correct thermal stresses in the FGMs, it is necessary to analyze thermal stresses by use of the microscopic combination law. In this paper, we use the thermal stress constitutive equation of a particle-reinforced composite taking temperature changes and damage process into consideration. The volume fraction of the ceramic and the metal are evaluated using the power law distribution. Elastoplasticity is considered in the formulation and a finite element model of the formulation is developed. The effect of the particle volume fraction and the initial temperature condition on the stress variations are discussed in the manufacturing process. KEY WORDS Thermal Stress, Functionally Graded Material, Constitutive Equation, Elastoplasticity, Incremental Theory, Temperature Dependent Properties. I. INTRODUCTION FGMs [ 1] are microscopically inhomogeneous composite materials in which the mechanical properties vary smoothly and continuously from one surface to the other. This is achieved by gradually varying the volume fraction of the constituent materials. FGMs are made from a mixture of ceramic and metal or combination of different metals. The ceramic constituent of the material provides the high temperature resistance due to its low thermal conductivity while the metal constituent prevents fracture caused by stresses. In many studies of FGMs, the material properties due to the macroscopic combination-law have been adopted [2-5]. However in order to obtain more correct thermal stresses in the FGMs, it is necessary to analyze thermal stresses by use of the microscopic combination-law. The microscopic combination-law without the thermal effects have been discussed [6-9]. Tohgo and Chou [8] developed incremental theory of particulate-reinforced composite including debonding damage. Asakawa et. al. [10] derived a thermal stress constitutive
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Current Advances in Mechanical Design and Production, MDP-7
equation for particle-reinforced composite taking temperature changes and damage process into consideration. Noda et. al. [11] studied the elastic thermal stresses in an FGM plate subjected to heating and the number of damage particles and the possibility of crack generation taking the manufacturing process into account are also studied. Reddy and Chin [ 12] studied the thermoelastic response of functionally graded cylinders and plate taking the thermomechanieal coupling into consideration. They used the power law distribution to vary the volume fraction of ceramic and metal. There are three micro mechanical approaches that have been widely used for predicting the effective properties of FGMs: the rule of mixture, mean field micro mechanics (MFM) and self-consistent micro mechanics (SCM) [13]. The temperature-dependent material property is one of the most important factors in the accurate evaluation of thermal stresses [14]. In this work, the elastic-plastic thermal stresses induced in an FGM rectangular plate during the manufacturing process are presented using the stress constitutive equation of particle reinforced composite taking temperature change and damage process into consideration. The particle volume fraction is assumed to vary according to a power function along the thickness direction only. The temperature-dependent properties of the constituent materials are considered. The effect of the particle distribution parameter on the macroscopic stress, stress in the matrix and stress in the particle in the FGM are discussed and shown in figures. Also the effect of the initial temperature on the maximum tensile matrix stress is discussed. 2. GOVERNING EQUATIONS 2.1. Heat Conduction Governing Equations Two dimensional transient heat conduction, when the material properties are dependent on both position and temperature, is given by
0 cgT C(T,y)p(T,y)~Tt =-~{2(T,y)-~-}
0 OT +~{x(r.y)~-}
(1)
where T, t, C, p , 2 are the temperature, time, specific heat, density and the heat conductivity of the material respectively. 2.2 Thermal Stress constitutive Equations In this study, the effects of different parameters on the thermal stress are discussed. The considered parameters are the initial temperature, the maximum particle volume fraction and the exponent of the composition power law. The thermal stress constitutive equation, considering the damage process and temperature change, proposed by Asakawa et al. [ 10] can be written as:
ds~ --
3k~
[ {(k, - ko)v + ko} {(1 - v)(1 - fp - f~ ) + f~ + dfp } + (1 --v)(fp - dfp)
1_
(2) l
Xko]d~ +
--
_
{(kl-ko)v+ko}dfpcr~ +{60
3ko(1-v)Ah dz~ =
I
2/,o(1-v')A.
--
k,(fp-dfp) (60-a,)}3dT Ah
[{f/j,-po)V" +/~o} {(1- ~u')(l- fp - f , ) + f, +af,,}
+ O - v ' ) ( f p - d f .)/zoldcr~ +
1
2/Zo(1
-
v"
)A d
{(/~. -/zo)v + /~o}df, a,~'
(3)
Current Advances in Mechanical Design and Production, MDP-7
where
225
m
Ah = ( l - f p -
f~){k o + ( k , - k o ) v I + ( f v - d f p ) k .
Aa = (l-f.-
f~ll/.t o + ( . u . - , u o ) v ' } + ( f p -dft,)12 ,
de**, d6~,
(4)
do'** and do'ij are the hydrostatic component of incremental macroscopic strain,
deviatoric component of incremental macroscopic strain, the hydrostatic component of incremental macroscopic stress, deviatoric component of incremental macroscopic stress, do'~, and do',~' are the hydrostatic and deviatoric components of microscopic stress of panicle, dT is the incremental temperature change, a 0 and a~ are the coefficients of thermal expansion of matrix and panicles, fp, f, and dfr are the particle volume fraction, the void volume fraction and the damage panicle volume fraction in the incremental process, v and v' are Eshelby's tensor for spherical inclusion in the elastic field. l+v 0 v=~ (for the hydrostatic component) 3 ( l - v 0) -
. 2 ( 4 - 5 v 0) v = (for the deviatoric component) 15(1- v 0) In plastic field the Eshelby's tensor can be written as: _. l+v'~ v = ~ (for the hydrostatic component)
(5)
30-V'o)
. 2(4- 5V'o) v =
150-V'o)
(for the deviatoric component)
(6)
ko,/~o, k~ and/6 are given by
k, = ~ , E,/ 1 , = ~ E, (i=0, 1) (7) 3(1 - 2v, ) 2(1 - v, ) where E 0 , E~, v 0 and v~ are Young's modulus and Poisson's ratio of the matrix and the panicle respectively. The incremental stress of the matrix and panicles, do" ~ = (do-~,,do'~') and do "p = (do-~ ,do-:'), are given by
dag = /Co + (k, -/Co)V- (do'** + a~df,,) + 3kok , (1 - v)(fp - df,,)(a, - ao) 3 d T An
(9)
defT,'=/"o + (,u,-/~o)V" (aa~ + a:,'df,,) Aa k !
do;-16 (do-lj + o-;'df p )
(8)
An
3kok, (1 - f , - L ) ( a ,
- ao)(1 - v)
3dT
(10)
An (11)
For the particle statisticaldistribution in their size, the statisticalproperty of the debonding or fracture of" a panicle is described as a function of the m a x i m u m tensile stress of the particle and its size. From the weakest link theory, the cumulative probability P,(o-P,x) of the particles debonded or fractured can be written as
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Current Advances in Mechanical Design and Production, MDP-7
P, (r p ) = 1 - exp{-(
a;o
)" }
(12)
Where n and So are the shape parameter and the scale parameter respectively. 2.3. Finite Element Formulation. The finite element formulation of the heat conduction equation can be expressed as:
(13) where [O], [~] matrices and {f} vector are referred to the whole structure. The finite element formulation of thermo-elasto-plastic deformation is {Atr} = [ D]({A6} - {A6} r - {A6}~, (in elastic region)
(14)
{Act} = [D]~p({A6} - {Ac}r - {A6}dp
(15)
(in plastic region)
where [D], [D]~p are the elastic and elastic-plastic stiffness matrices of each node, {A6} is the incremental macroscopic strain, {Ac}r is the increment of thermal strain produced by incremental change of temperature and {A6}dn is the incremental strain produced by damage particles. [D],{A6}r,{Ae}~ , can be evaluated using equations (2) and (3) as a function of the material properties, void volume fraction, particle volume fraction and damage particle volume fraction. The effective load at each node is expressed as {AF} = ~[B] r [D]({A6}r + {A6}~)dV (in elastic region) (16) {AF} = ~[B]r[D]~p({A6}r + {A6},o,)dV
(in plastic region)
where [B] is the strain matrix of the element. The governing equation is expressed by [K]{Ad} = {AF} where [K] is the stiffness matrix and {Ad} is the incremental displacement vector.
(17)
(18)
3. NUMERICAL ANALYSIS Using finite element method, the thermal stresses in an FGM plate with thickness s and width b shown in Fig. 1 are studied. All the plate surfaces are exposed to natural convection and we assume that the convective heat transfer coefficient a is l Ow/m2K in the numerical calculations. The volume fraction of the particle changes along the thickness direction continuously from 0% on the lower surface to its maximum value on the upper surface. The initial particle volume fraction at each node fp (Y) is given by
f p(Y) = f por= (19) where Y(= y / s) denotes the non-dimensional position normalized by the thickness, fpo and m are the initial particle volume fraction on the upper surface and the distribution parameter of the composition. Thermal stresses are evaluated using different values of the distribution parameter of the composition. The particle volume fraction along the thickness of a FGM plate with the upper surface containing 30% particles are shown in Fig. 2 for different values of m. The FGM plate is assumed to deform under the plane strain conditions. The element used in the finite element analysis is 8-node-isoparametric element [15]. The number of elements along X(= x / s) and Y axes are 80 and 70 elements respectively. The element dimension is
Current Advances in Mechanical Design and Production, MDP-7
227
chosen to vary along each axis. The element dimension along X-direction is chosen to be smaller near the boundary surfaces. The element dimension along Y-axis is chosen according to the value of distribution parameter of the composition (m). For example, when m = 10 the number of elements from Y = 0 to Y = 0.8 is 30 and from Y = 0.8 to Y = 1 is 40 elements while for the case of m = 0.1 the number of elements from Y = 0 to Y = 0.2 is 40 elements and from Y = 0.2 to Y = 1 is 30 elements. The materials of the matrix and the particles are assumed to be aluminum (AI) and partially stabilized zirconia (PSZ), respectively. The temperature dependent material properties of AI and PSZ are shown in Fig. 3 [11 ]. The temperature dependent material properties are given at each node of the element and they are interpolated by quadratic shape function within the element. In order to discuss the thermal stresses in the FGM due to fabrication process, we assume that the plate is cooled from the fabrication temperature to the room temperature by natural convection. The mechanical boundary conditions of all surfaces are free. 4. N U M E R I C A L R E S U L T S AND DISCUSSION Figure 4 shows the macroscopic stress o'~, the matrix stress cr7 and the hydrostatic stress of particle crff for m = 1 after achieving the steady state, when the aspect ratio, initial particle volume fraction on the upper surface and initial temperature are s / b = 0.1, fpo =0.3 and T, = 800K respectively. It can be seen that the macroscopic stress takes its maximum value on the upper comers of the plate. The stress in the matrix and hydrostatic stress in the particle vary linearly along the coordinate Y and independent on coordinate X . The maximum value of the matrix stress occurs on the upper surface of the plate while the maximum hydrostatic particle compression stress occurs on the lower surface of the plate. The effect of the parameter m on the maximum values of the macroscopic stresso'~m ~ , the hydrostatic particle stress tr hpm~, and the matrix equivalent stress Cr,qm~ ,"
can be shown in
Figs. 5-7. It can be seen from Fig. 5 that the macroscopic stress changes from tension to compression as the composition parameter increases and has value near zero for m = 1. The maximum value of the particle hydrostatic stress takes its minimum compression at m = 0.1 and increases with increasing the parameter m as shown in Fig. 6. The maximum value of the matrix equivalent stress equals near zero at m = 1 and increases as the parameter m changes from unity. The maximum value of the equivalent stress occurs at m = 10 as shown in Fig. 7. Figure 8 shows the maximum tensile matrix stress o'xm~ versus the parameter m. The highest value occurs for m = 0.4 and sharply decreases till m = 1. Beyond m = 3 the stress decreases with small rate. Figure 9 shows the maximum tensile matrix stress Crxma~ versus the initial particle volume fraction on the upper surface for different values of the parameter m when the initial temperature T, = 800K. The figure shows that the severe value of the parameter m is 0.4 from the point of view of the maximum tensile matrix stress. The relation between the maximum tensile matrix stress and the particle volume fraction is about linear.
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Current Advances in Mechanical Design and Production, MDP-7
Figure 10 shows the effect of the initial temperature on the maximum tensile matrix stress when the initial particle volume fraction on the upper surface f ~ =0.3. Also the maximum tensile matrix stress occurs for m = 0.4. The relation between the maximum tensile matrix stress and the initial temperature is about linear. 5. CONCLUSIONS The effect of the composition parameter, initial temperature and the particle volume fraction on the upper surface of FGM plate on the thermal stresses are presented considering the temperature dependent properties of the constituent materials. Concluding remarks are made as follows: The maximum tensile macroscopic stress occurs at the smaller value of the distribution parameter of the composition while the maximum compression occurs at the highest value. .
The maximum compressive particle hydrostatic stress occurs at the composition parameter m = 3 and beyond this value the maximum particle hydrostatic stress can be considered as a constant. The minimum value of the maximum matrix equivalent stress occurs for the parameter m = 1. The relation between the maximum value of the tensile matrix stress and the particle volume fraction on the upper surface of the plate is about linear. The maximum tensile matrix stress occurs for the parameter m = 0.4. The effect of the initial temperature on the maximum tensile matrix stress is same as the particle volume fraction on the upper surface of the plate.
REFERENCES 1. 2.
3.
4.
5.
6. 7.
Yamanouchi, M. et al., Proc. of the First Int. Sympo. on Functionally Gradient Materials, Sendai, (1990). Noda, N., and Tsuji, T.,"Steady Thermal Stresses in a Plate of Functionally Gradient Material", Proc. First Int. Symposium of Functionally Gradient Materials, Sendai, pp. 339344, (1990). Obata, Y. and Noda, N., "Transient Thermal Stress in a Plate of Functionally Gradient Material", Vol. 34 Ceramic Transactions, Functionally Graded Materials, Edited by J.B.Holt, M. Koizumi, T. Hirai and Z.A. Munir, The American Ceramic Society, pp. 403410, (1993). Fuchiyama, T., Noda, N., Tsuji, T. and Obata, Y., "Analysis of Thermal Stress and Stress Intensity Factor of Functionally Gradient Materials", Vol. 34 Ceramic Transactions, Functionally Graded Materials, Edited by J.B.Holt, M. Koizumi, T. Hirai and Z.A. Munir, The American Ceramic Society, pp. 425-430, (1993). Obata, Y. and Noda, N., "Steady Thermal Stresses in a Hollow Circular Cylinder and a Hollow Sphere of a Functionally Gradient Material", Journal of Thermal Stresses, Vol. 17, No. 3, pp. 471-487, (1994). Eshelby, J. D., "The Determination of the Elastic Field of an Ellipsoidal Inclusion and Related Problems", Proc. of the Royal Society, London, Vol. A241, pp.376-396, (1957). Tandon, J. P. and Weng, G. J., "Stress Distribution in and around Spheroidal Inclusions and Voids at Finite Concentration", ASME, Journal of Applied Mechanics, Vol. 53, pp. 511-518, (1986).
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8. Tohgo, K. and Chou, T. W., "Incremental Theory of Particulate-Reinforced Composites Taking Damage Process into Consideration", JSME, Series A, Vol. 39, No. 3, pp. 389397, (1996). 9. Mura, T., "Micromechanics of Defects in Solids", Martinus Nijhoff, The Hague, (1982). 10. Asakawa, A., Noda, N., Tohgo, K. and Tsuji, T., "Constitutional Equations of Thermal Stresses of Particle-Reinforced Composite", JSME, Series A, Vol. 60, No. 575, pp. 16321637, (1994). 11. Noda, N., Nakai, S and Tsuji, T., "Thermal Stresses in Functionally Graded Material of Particle-Reinforced Composite", JSME, series A, Vol. 41, No 2, pp. 178-184, (1998). 12. Reddy, J. N. and Chin, C. D., "Thermomechanical Analysis of Functionally Graded Cylinders and Plates", Journal of Thermal Stresses, Vol. 21, No. 6, pp. 593-626, (1998). 13. Wang, J., Deng, J. and Zhang, Q., "Thermoelastic-Plastic Analysis of Ceramic-Metal Graded Materials", Proc. of ICCM-10, Whistler, B.C., Canada, August, pp. 695-702, (1995). 14. Naobumi, S. and Yoshihiro, S., "Thermally Induced Stress Waves in Functionally Graded Materials with Temperature-Dependent Material Properties", Journal of Thermal Stresses, Vol. 20, No. 3-4, pp. 281-294, (1997). 15. Hinton, E. and Owen, D. R. J., "Finite Element Programming", Academic Press, (1977).
Current Advances in Mechanical Design and Production, MDP-7
230
Yak ~k
Macroscopi c stress . . a .s for . . m=. 1.. . "7-~g0-___.) "~ r--.4.6 p/,
....4
....
~3 ~2
S '.
. . . .
_ \
!
oO
~i
b
0.1
Fig. 1. Configuration of the FGM plate.
:!-2 0
0.2
0.4 0.6 pos~t~nV
0.3
I
0.25
.ql
0.1
.....
0.8
1
m
Matrix, stress 0"~ f o r m a l .
0.2 0.15
,
g ' -'-.~ x'~
~0.05
m 100
~
t. . . . . . . .
__
0 0
0.2
0.4
0.6
0.8
Pos~k~(V)
1
0
Fig. 2. Volume fraction of the particles along the plate thickness for different values of the composition parameter m. on Ov v (MPa) (MPm)
200
E (GPal
a x C xlO'*(l/KI (~V/mK) (J/kI [ )
I
200 0.
0
0.2
-50O
0.4 0.6 PositionY
" 0.8
1
0.8
1
f
I1.
~-ooo
o
~" -700 0
g
~s
G.
-8OO 0.
0
0
0.31. m
am
qm
m
Temperature
TtB
m
0.4
a ~. C xl0*(I/K) (W/InK) (JUts K)
(GPm) 200
I
mo ~-
mo ~,,. 0.35
~
] c.-- ' ' ~ ~ ,
S "
"e~ ~
~"
)
SO0
)
.400
140 9 i 0.3
~00 300
! 400
500 600 Tgnq3,,~
700
800
0.4 0.6 ~ilionY
Fig. 4. Stresses for m = 1 . .
(m) Temperum~dependenceot'^l pmcemes. v
0.2
l
Co)Tempemea~clepende.:eo f PSZ pmpen.es.
Fig. 3. Temperature dependence o f material properties [ I I ]. ( E " Young's modulus, o" 9 Yield stress, or," Tensile stress, v- Poisson's ratio. C" Specific heat, ,t" Thermal conductivity, or- Coefficient of linear thermal expansion)
Current Advances in Mechanical Design and Production, MDP-7
-~
Maximum tensile matrix stress 0" ~,.,.,..
Maximum macroscopic stress (7"r....
2OO
'!
i
" .
0
~ o ~
-100
E ,~
-200
t~
=g" 230 / i ~, ,-
220
~
210
=~
-300 0
2
4
6
m
8
10
I
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o
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4
6
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I
[
"---
4
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0.1
~
'
~ ' b
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.
.
I
0.15
.
. . . . . . . .
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0.3
fp
10
n!
"
,.
,
-..-o--m.0.1'
~'-; '~176 50 ~.
;
10
Maximum tensile matrix stress 0"" Xmaa
Fig. 9. Relation between the maximum tensile matrix stress and the particle volume fraction on the plate upper surface.
Maximum matrix equivalent stress Creq t
8
6
Fig. 8. Relation between the maximum matrix stress and the distribution parameter m.
'~
8
.
m
Fig. 6. Relation between the maximum particle hydrostatic stress and the distribution parameter m.
Maximum tensile matrix stress cr~],,.
~,
300
i m
E ~
g~
. . . .
2
9_.i: ,5o - -._ m-,O
-900
.
,
~ m - |
o. ~ -800
"~
.
9
I.
250
v
25O
.
~
0
Maximum particle hydrostatic stress O"p _m
]
. . . .
Fig. 5. Relation between the maximum macroscopic stress and the distribution parameter m.
-6OO
v
250-
tO0
b v
231
250
150 = ~ 200
x
~
_=
m-1
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150
~
,oo
0 0
2
4
m
6
8
10
Fig. 7. Relation between the maximum matrix equi~.alem ,;tress and the distribution parameter m.
800
850
T(K)
Fig. 10. Relation between the maximum tensile matrix stress and the initial temperature.
,oo
This Page Intentionally Left Blank
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
233
FRACTURE MODELING OF A CRUCIFORM WELDED JOINT DURING WELD COOLING
Behnam, W.M.*, Alkhoja, J.**, Recho, N.*** and Zhang, X.B.** * Assistant Professor, Mechanical Design and Production Department Faculty of Engineering, Cairo University, Giza 12316 - Egypt. ** Conference Teacher, *** Professor, Laboratoire d'Etudes et de Recherches en Mecanique des Structures, Universit6 Blaise Pascal, Clermont II, 03100 Montlu~;on- France. e-mail : behnam@,alpha l-eng.cairo.eun.eg [email protected]!ermont, fr alkhoj [email protected], fr zhang~moniut,univ-bpc!ermont.fr
ABSTRACT The aim of this work is to analyse numerically the fracture behavior of a cruciform welded joint during weld cooling. This welded joint presents the particularity of the existence of two types of cracks: a defect at the weld toe and a lack of penetration at the weld root. The cooling process begins from a weld temperature at 800 ~ for steel E36, till ambiant temperature taking into account the different types of heat transfer. The welding process, and the metallurgical phenomenon of the material during cooling are outside the scope of this study. This study evaluates the fracture risk on the basis of the energy release rate (G) associated with existing cracks. After verification of the validity of this fracture parameter (G)in a thermoelastoplastic medium, a parametric study of the welded joint, after the welding process, is carried out as a function of the joint geometrical parameters and the displacements boundary conditions in a transient thermomechanical medium, taking into account the variation of the mechanical characteristics as a function of the temperature. This work allows to establish nomographic charts which can guide the welder of this type of joints. KEY WORDS Welded Joints, Fracture Mechanics, Thermomechanical Analysis, Finite Elements Method, Heat Transfer. 1. INTRODUCTION It is difficult to evaluate the fracture risk which occurs during a localized heat transfer. This heat transfer generates an important stress gradient amplified by the weld geometry and the boundary conditions. Although it is difficult to establish a fracture criterion for a material situated in a transient thermomechanical medium, the energy release rate during the crack growth constitutes a solid
Current Advances in Mechanical Design and Production, MDP-7
234
parameter to estimate the fracture risk in this medium. This fracture parameter has been validated on a plate with thermal gradient load for a bilinear elastic-plastic law behavior material [ l]. After this validation, this parameter will be applied on the case of a cruciform welded joint in a transient thermomechanical medium. 2. ENERGY RELEASE RATE CALCULATION METHOD The energy release rate is expressed by a contour integral J, Fig. 1 [2]. This integral is defined by the following equation" au G = J = ~w d y - o i j n j ~ - d S
(1)
F
where w is the strain energy density and u is the displacement.
yI
\
/
~b
Li
F
Fig. 1. Contour integral Equation (1) presents the classical case of the J integral. In the case of a gradient thermal load [3], the mechanical properties change from a point to another due to the presence of a temperature gradient, which results in a nonhomogenous mechanical behavior of the structure. In this case the J integral will be 9
J = J'(w dy - crijnj ~ dS) + j'J'o'iict 1"
dxdy + J'J'o"iiT'~'dxdy &:t
A
"
Fig. 2. Geometry of the cruciform joint with the retained defects (ap and 2ar)
A
dT Ox d x d y - H( ]" tYI"dceq) A
dxdy
(2)
0
where the first integral corresponds to the classical case, the second and third integrals correspond to the case of thermal loading with a variation of thermal expansion coefficient (r as a function of the temperature (T). The two last integrals correspond to the case of the variation of mechanical properties (such as Young's modulus E, elastic limit Oy) as function of the temperature.
Current Advances in Mechanical Design and Production, MDP-7
235
Numerically, to calculate these integrals, it is necessary to change these integrals to volume integrals using the method G_Theta (G_Theta = J) proposed by the finite elements code "Castem 2000" [4, 5]. 3. APPLICATION TO THE C R U C I F O R M JOINT The cracks, in a welded joint, can exist at the weld toe where the stresses are maximum, or at the weld root. The cruciform welded joint presents the particularity of the existence of these two types of defects, Fig. 2. During cooling of steel E36 [6], whose carbon content is about 0.156% [7], the first recristallisation, passing from iron 5 to iron ),, exists at 1400 ~ C. The second recristallisation, passing from iron 3, to iron ct, exists at 900 ~ In this study, only the metallurgical behavior of the joint from 800 or 700 ~ will be taken into account, which means that the influence of the recristallisation phenomenon will not be considered. 3.1. Geometrical
Parameters
The geometry of the considered joint is presented in Fig. 2, on which we have" - b and L : the thickness and the length of the discountinuous plate respectively, - b~ and Lt : the thickness and the length of the continuous plate respectively, -e :the weld thickness, - 2ar : the lack of penetration, considered in this analysis as an initial crack length at the weld root, - ap : the defect at the weld toe, considered here as an initial crack length, and
- 0 : weld toe angle (taken as 45 ~ The geometry of this joint is defined by the following non-dimensional ratios' b~/b, e/b, 2ar/b and ap/b (%) and (2ar) are considered in this joint as initial defects at the weld toe and the weld root respectively. The initial defect dimension is equal to the crack length causing the crack propagation [ 8]. In this study, the following non-dimensional ratios are considered
9
bl/b = 1
arfo= 0.01 and 0.02
e/b -- 0.35, 0.57 and 0.71
3.2. Thermal
Parameters
2ar/b - 0.2, 0.4 and 0.6
The heat transfer during weld cooling is affected by" a) Conduction
The conduction exists between the weld and the joint. The temperature field at time (t)is calculated by the Fourier heat equation. This equation, in two dimensions, has the following form"
236
02T
Current Advances in Mechanical Design and Production, MDP-7
02T
pc Off'
Ox2+ 0 7 -
(3)
~ Ot
The thermal characteristics of steel E36, at room temperature, are given as follows [9] 9 - thermal conductivity ( 2 ) = 43 W/m ~ - density (p) = 7.8 x 103 kg/m 3
- specific heat (c) = 470 J/kg ~
b) Convection The convection exists between the external contour of the joint and the environment. During cooling, the convection exchange coefficient (1~) changes according to the external joint contour temperature variation. To calculate this exchange coefficient, we have assimilated the joint external surface by the four contours shown in Fig. 3. The studied weld is situated on the contour 1. The thermal properties of the environment (air) are given as follows [9] : - thermal conductivity ( A ) = 0.03 W/m ~ - Prandtl number (Pr) = 0.7
- kinematic viscosity (v) = 3x10 "5 m2/s
c) Radiation In this study, we consider only the radiation between the external joint surface and the environment. The form factor (F) is equal to 1 and the emissivity (c) is equal to 0.94 for a rough surface [9].
i-..... "'~
contour 2
contour 3
contour I
contour 4
Fig. 3. The four contours considered during the thermal modeling
Fig. 4. The mesh of the crucilorm joint
4. MODELING The welded cruciform joint is meshed on Fig. 4. This mesh represents one from 18 meshes corresponding to different non-dimensionnal ratios (ap~, 2ar/b and e/b) for a plate thickness ratio (bl/b = 1). For each value of weld thickness (e/b), three meshes corresponding to the three values of lack of penetration ( 2 a ~ ) are constructed. These meshes are realized for each defect
Current Advances in MechanicalDesign and Production, MDP-7
237
at the weld toe (ao/b). All these meshes are realized with triangular elements of 6 nodes. The total number of nodes varies between 3225 and 3747 nodes for different configurations. 4.1. Thermal Modeling The calculation is done by the use of the finite elements code "Castem 2000" developed by CEA (Commissariat a l'Energie Atomique - France). To calulate the convection and radiation exchange coefficients (he and hr respectively), we consider the four contours on Fig. 3. The calculation of each coefficient for each contour is presented in the following paragraph :
a) Calculation of convection exchange coefficient The Rayleigh number is defined by the following equation"
RaLc =
gfldT L~ I,,2 x Pr
(4)
where dT is the difference between the average temperature Tm of the external surface and the environment medium (Too =20~ Lc is the length on which the heat transfer by convection is effected and g is the gravitational acceleration (9.81 m/s2). The coefficient 13is calculated by the following equation"
[3 -
l %
where
T~+T~ 2
Tr = ~
(K)
(5)
The convection exchange coefficient he is calculated from the relation between the Nusselt number and the Rayleigh number as follows :
NULc -
hcLc _ m(RaLc)n 2
(6)
where the m and n values change according to the heat direction [10] for each contour as shown in the following table 9 M 0.154 0.27 0.59
1/3 1/4 1/4
contour (Fig. 3) 1 and2 horizontal side of 3 and 4 vertical side of 3 and 4 .
b) Calculation of radiation exchange coefficient The radiation exchange coefficient is a function of the average temperature (Tin)for each contour, it is calculated as follows 9 h, = 6oF(Tm2 + T~ )(T m + T| ) where ~, F values are defined previously and o is defined by the Stefan Boltzman constant.
(7)
238
Current Advances in Mechanical Design and Production, MDP-7
The total heat exchange coefficient for each contour will be equal to the sum of the two exchange coefficients, as followsh = h~ + h~
(8)
For all the meshes, we have chosen the value of (X/b), on Fig. 2, equal to 3.5, in order to be far enough from the stress local effect at the weld toe. c) T h e r m a l calculation steps
In order to establish the temperature field at the beginning of the cooling process, we mention the following steps: a) at t=O, we impose the weld temperature at 800 ~ (situated on contour 1, fig. 3) and the remained joint at 20 ~ b) from this temperature distribution, we can calculate the total exchange coefficient for each contour, considering T| = 20 ~ c) we take a time interval (dt), with the same weld temperature of 800 ~ and from the last total heat exchange coefficient calculated and the material thermal conductivity, we can obtain the new temperature distribution. d) we repeat steps b and c till t=5 minutes (corresonding to the welding time). e) from this time, we remove the imposed temperature at the weld to begin the cooling process. We repeat steps b and c with a time interval (dt) till t=180 minutes (corresponding to the cooling time). At this time all the joint is nearly at room temperature. Fig. 5 shows the temperature and the energy release rate evolution at the weld toe during cooling time. This figure is established for the joint with the following geometry" ap~ = 0.01
2aJb = 0.2
e/b = 0.35
and
'Rmpemtu~(oo 800
G( N/ mm) ~
700
,,~
600
o
5OO
b = bl = 20 mm
o
20
o- -~176
18
a ....
16 14
[ --,*-'l~mpemtum
a
-o-G
12
400
10
3011
8 6
200
4
lO0
2
0 0
60
120
180
Omrmt tkne (nmmes)
Fig. 5. Temperature and energy release rate evolution at the weld toe during cooling time. 4.2. Mechanical modeling The material of this joint is considered as steel E36 with the following mechanical characteristics at room temperature 9
Current Advances in Mechanical Design and Production, MDP- 7
239
- Young's modulus (E)= 200 GPa - Poisson's ratio (v) =0.3 - Thermal expansion coefficient (tx)= 10 -5 ~ - Elastic limit (6y)= 302 MPa - Tangent modulus (Et) = 0.01 E = 2 x 103 MPa (for a bilinear elastic-plastic behavior law as shown in Fig. 6). At a time (t), knowing the temperature distribution by mesh element, we introduce the material mechanical behavior law taking into account the evolution of its mechanical characteristics as function of the temperature [ll to 15].
o
Oy
6
From Fig. 5, we have no creep phenomenon since the temperature decreases quickly from 800 to 400 ~ at nearly 5 minutes.
Fig. 6. Bilinear elastic-plastic behavior law
Modeling assumptions The following assumptions are considered : - Same material behavior law for the weld and the base metal (steel E36). - Plane strain conditions - The existence of the two cracks on one weld. The cracks on the other welds are not considered. - No stresses or strains at the beginning of cooling. - No mechanical loading on the joint. The thermomechanical calculation steps are shown in Fig. 7 9 At any cooling time t = tn
Time ( ~ Temperature field: To (x, y)
Initial conditions : ao = 0 Uo = 0
Time (t=t~ Temperature field : T, (x.y)
[ dT (x, y ) = ( T . - To) (thermal load) ie~ to,
- Bilinear elastic-plasticmaterial 1~_ behaviorlaw ~(mechanicalproperties as function of the temperature) 'NNN~ '
-Displacementconditions. boundary .
[ DisplacementsU,
G_Theta (Go) = J
Fig. 7. Thermomechanical calculation steps
]
240
Current Advances in Mechanical Design and Production, MDP-7
5. RESULTS The results concerning the effect of geometrical parameters on the fracture risk are based on the unfavorable bridle case. We found that the bridle of the four sides of the joint generates the maximum value of maximal principal stresses at the weld toe. During cooling, we calculate the energy release rate for the crack points at the weld toe (p) and the weld root (r), as shown in Fig. 7. The value of the energy release rate increases at the beginning of cooling either on the weld toe or the weld root, followed by light increase till reaching a stable value dependent on the cooling time. This is due to the fact that at the beginning of cooling, there is a fast decrease of the temperature (Fig. 5), so the value of the thermal load (dT=Tn-T0) will be very important.
I
I Fig. 8. Crack points on the studied weld.
The geometrical ratios retained in this work are (Fig. 2) : - ap/b : related to the influence of the defect length at the weld toe (0.01 and 0.02) - 2a~/b : related to the influence of the lack of penetration at the weld root (0.2, 0.4 and 0.6) - e/b : related to the influence of the weld thickness (0.35, 0.57 and 0.71) 18 numerical tests have been made for constant plate thickness (bl = b =20 mm). The results are presented on the nomographic charts (Fig. 9 and 10) for ~b=0.01. On these figures, we represent iso-values of the energy release rate G in N/mm at the end of cooling process, where we have observed a stabilization of the energy release rate value. This stable value at the end of cooling has been retained to express the fracture capability for each case of geometrical configuration. During non-homogenous and transient thermal loading, it is impossible to define a fracture criterion to cover the whole studied geometries. However, the calculated energy release rate at the end of cooling time represents the fracture capability. By changing the defect length (ar at the weld toe from 1 to 2% of the plate thickness (b), the value of the energy release rate increase by 27% at the weld toe and by 4% at the weld root. We observe, on Fig. 9, that for a lack of penetration (2a~) less than 37% of the plate thickness (b), the fracture risk at the weld toe is weakly influenced by the weld thickness variation (e). On the other hand, for a lack of penetration (2a~) more than 40% of the plate thickness (b), the fracture risk at the weld toe decreases strongly with the incease of the weld thickness (e). For a lack of penetration (2a~) between 37 and 40% the plate thickness (b), the variation of the weld thickness (e) has no influence on the fracture risk at the weld toe, Fig. 8.
Current Advances in Mechanical Design and Production, MDP- 7
241
On Fig. 10, for a joint with a geometrical configuration situated in zone A, the reduction of the lack of penetration (2at) or the increase of the weld thickness (e) has the same effect on the reduction of the fracture risk at the weld root. On the other hand, outside this zone, for a lack of penetration (2a~) lower than 40% of the plate thickness (b), reducing the lack of penetration (2a,) will be more efficient than increasing the weld thickness (e). However, the reduction of the fracture risk at the weld root is least.
a~
,,~
---
,
,,,, . . . . . . . . . .
a6J
,,~.~
in
a~
ill
a.Jl
14J
9
a65
"-'
.
.
.
.
.
~
"
,~.r "
<
2.;
......
:#+,,+.,,+.,,,
II i
aJd:o azs
,*
II II
II II
OJ~
~ g ? O
II II II
--
"'~ U
[ + A
r
|
a)b aJs 04o a'~s ~50 o.'~J 06o
Lack of penetration (2ar)/plate thickness (b) Fig. 9. Iso-values of the energy relase rate at the weld toe for ap/b-O.O1
03~
a:~o
,
o.zs
o.Jo
,
o.Js
.
o.~o
i
i
o.~s
I
_
s
o.so
s
,J_
~ss
~, .
0.60
Lack o f penetration (2ar)/plate thickness (b)
Fig. 10. Iso-values of the energy release rate at the weld root for ap/b=O.Ol
When the non-dimensional ratios (ap/b, 2ar/b and e/b) are known, Fig. 8 and 9 will constitute a reference point to predict the fracture risk of a cruciform welded joint after welding process. 6. CONCLUSIONS From this work, we can conclude that the energy release rate (G) is a good parameter allowing the evaluation of the fracture risk for the case of non-homogenous and transient thermal load. Also, this work allows to establish a manufacturing strategy in order to prevent or to reduce the fracture risk of a cruciform welded joint. This strategy is function of" - mechanical parameters related to displacement boundary conditions (bridle). - geometrical parameters expressed by the non-dimensional ratios (2at/b, ap/b, e/b and bdb). As future extension for this work, it will be preferred to enlarge this study for different types of welded joints, also to compare numerical results with experimental results. Later, this work allows to deal with problems related to thermal fatigue. REFERENCES 1. Wassef, M., "Mod61isation /l la rupture d'un assemblage soud6 lors de refroidissement du cordon de soudure", Th~se de Doctorat, Universit6 Blaise Pascal, Clermont-Ferrand II, Laboratoire LERMES, (1998). 2. Rice, J.R., "Mathematical analysis in the mechanics of fracture", fracture academic press, New York, vol.2, pp. 192-308, (1968).
242 3.
4. 5.
6. 7. 8.
9. 10. 11.
12.
13. 14. 15.
Current Advances in Mechanical Design and Production, MDP-7
Cheissoux, J.L., "Calcul de J en bidimensionnel par int6gral de contour", M6canique de la rupture, m6thodes num6riques pour l'ing6nieur, Institut pour la Promotion des Sciences de rIng6nieur (IPSI), Paris, 51 p, (1986). Lorenzi, H.G., "On the energy release rate and the J integral for 3D crack configurations", Int. J. of fracture, pp. 183-193, (1982). Brochard, J. and Suo, X.Z., "Le taux de restitution de l'6nergie G en m6canique de la rupture non-lin6aire, formulation de la m6thode Go et description de la programmation dans Castem 2000", Commissariat a l'Energie Atomique (CEA), Rapport DMT/94-640, 109 p, (1994). Godfroid, H., "M6tallurgie pour m6caniciens", Soci6t6 des publications m6caniques, 378 p., (1959). Varisellaz, R., "Soudage-El6ments de conception et de r6alisation", Editions Dunod, Paris, 230 p., (1987). Recho, N, "Distribution de la taille du d6faut initial dans les soudures d'angles, Assemblage en croix et assemblage de tubes", Revue Construction m6tallique, n~ 8 p., (1983). Sacadura, J.F., "Initiation aux transferts thermiques", Editions Technique et Documentation, Paris, 446 p., (1980). Petit, J.P., "Transfert thermique par convection", Techniques de l'ing6nieur, Al 541, pp.15-16., (1990). Kruppa, J., Law, M.and Twilt, L.T., "Eurocode n~ - calcul des structures en acier", Projet pr6par6 pour la Commission des Communaut6s Europ6ennes, Document du Centre Technique Industriel de la Construction M6tallique (CTICM), (1990). DTU (Document Technique Unifi6), "M6thode de pr6vision par le calcul du comportement au feu des structures en acier", Revue Construction m6tallique, n~ (1982). Goursand, J.P., "Donnt~es physiques sur quelques aciers d'utilisation courante", Office Technique de rUtilisation de l'Acier (OTUA), (1975). Muzeau, J.P. and Lemaire, M., "Mod/fle num~rique de comportement d'ossatures en acier sous forte 616vation de teml~rature", Revue Construction MtStallique, n ~ 3, (1980). Kaneko, H., "Etude par la m~thode des ~lt~ments finis du comportement m6canique d'616ments plaques en acier soumis h l'incendie", Revue Construction mt~tallique, n ~ 1, 13 p, (1990).
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-I 7, 2000
243
APPLICATION OF ELASTIC-PLASTIC FRACTURE MECHANICS CRITERIA TO SPECIMENS CUT FROM PLASTIC PIPES
EL-Zoghby, A.A." and AI-Bastaki, N.M.'* * Mechanical Design and Production Department, Faculty of Engineering, Cairo University, Giza 12316-Egypt. ** Department of Chemical Engineering, College of Engineering, University of Bahrain, P.O. Box 32038, Isa Town, Bahrain. ABSTRACT Elastic-plastic fracture mechanics testing of specimens cut from finally produced plastic pipes may help in determining the fracture toughness of the pipe material under the same geometrical and working conditions of the pipes. To determine the fracture toughness of PVC, C-shaped three-point bend specimens were cut from commercial PVC pipes, pre-cracked and tested according to the elastic-plastic test criteria applied to standardized metallic specimens. Multi-specimen J-integral and CTOD testing techniques were used. The same specimens were used to construct J-R and ~-R curves. Validity criteria were applied and checked according to the ASTM and BS test procedures. Tests showed a good agreement of the obtained results with those previously published by other investigators. This indicates that the applicability of elastic-plastic fracture criteria to specimens cut from finally produced plastic pipes is possible. KEYWORDS Elastic-plastic Fracture Mechanics, Plastic Pipe Specimens, Polyvinylchloride, J-Integral, Fracture Toughness, J-Test, Crack Opening Displacement Test. I. INTRODUCTION Plastic pipes have been extensively and increasingly used in constructional as well as industrial applications. They are frequently used in water and gas distribution networks and chemical plants. The need for these pipes has increased rapidly in the last few years. This necessitates the determination of the mechanical properties, specially fracture toughness, which are used in designing these pipe networks against catastrophic failure. However, the materials used for these pipes such as polyvinylchloride (PVC) and polyethylene (PE), show elastic-plastic behavior when tested to determine their fracture toughness under normal testing conditions of loading rates and temperatures [1,2]. In the testing of these materials, large plastic deformations are formed at the crack tip. This makes the linear-elastic fracture mechanics (LEFM) testing procedures invalid. The brittle behavior that is needed for LEFM testing procedures is not obtained under the normal testing conditions in this case. To get this brittle behavior, high-testing rates should be used [3]. This may affect the fracture mechanics characteristics of these materials. Thus the need for applying elastic-plastic testing techniques in the testing of tough polymers becomes of increasing importance.
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The J-integral testing procedures, based on the original work of Rice [4], proved to be successful and have been applied to a wide variety of tough polymers [ 1,2,5-7]. Many of these investigations were done to determine the fracture toughness properties under elastic-plastic testing conditions using molded and manufactured standard specimens from the pipe materials [5-7]. Some other attempts were made to use specimens cut directly from finally produced plastic pipes [ 1,2]. These specimens are cut as curved segments from the wall and loaded in three-point bending with a pre-crack on the inner surface of the pipe. Some of these specimens are not standard and some others are standardized. The use of specimens cut from plastic pipes was studied in previous investigations [3,8] trying to apply the LEFM testing techniques to plastic pipe products. These specimens were either cut in the form of C-shaped segments loaded by three-point bending with a pre-crack on the concave side (C-shaped SENB) [3] or as pipe specimens with an axial pre-crack on the outside surface and loaded by internal pressure [8]. There are some advantages of cutting specimens directly from the pipe. In one hand, they help in determining the fracture toughness of the actual pipe material after production. On the other hand, pre-cracks are machined in the axial direction of the pipe in order to determine the fracture toughness in the hoop direction, which is the weakest direction due to the extrusion of the pipe during the manufacturing process. Recently, many investigations [5-7] have adopted the ASTM E813 - 89 J-integral standard testing procedure used for metallic specimens [9], to test plastic materials having elastic-plastic behavior. Crack tip opening displacement (CTOD) testing techniques have been also used and standardized for metallic materials with elastic-plastic characteristics. The BS has published BS 5762:1979 [ 10] for CTOD testing that was the first test standard to measure the fracture toughness in the elastic-plastic behavior. The ASTM has also published E 1290-89, an American version of the CTOD standards. These standard test methods can be applied to ductile and brittle materials, as well as to materials in the ductile-brittle transition. In the present investigation a multi-specimen testing procedure as proposed by Haung [5] and Chung and Williams [6] based on the ASTM J-integral testing technique is applied to curved three-point bend specimens cut from the pipe wall. The crack tip opening displacement test techniques of the BS [ 10] and the ASTM [ 11] are also applied to these specimens. 2. EXPERIMENTAL WORK
2.1. J-integral Test Tests were carried out using C-shaped three point bend curved specimens cut from an extruded commercial eight inch PVC pipes purchased from a local manufacturer. The outside diameter, D, of the pipe is 220 mm and the average wall thickness, W, is 14 ram. The pipe was first machined in the form of tings with the required specimen width, B, using a lathe at a low cutting speed. Cooling fluid was used to avoid excessive heating of the specimens during machining. Specimens were then cut to the required shape shown in Fig. 1 and then precracked in two steps. They were first V-notched on the concave side using a 60 ~ counter-sunk cutter on a milling machine. Notched depths were further sharpened by pressing a razor blade to the final pre-crack depth, a. This razor blade pre-cracking method was recommended by many investigators [3,12] to avoid the plastic deformations formed at thecracktipduring fatigue, which is the usual pre-cracking technique used by the ASTM E 813-89 standard method for metallic materials. Pre-crack depth to specimen thickness ratio, ao/W, was
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245
controlled to be in the range of 0.45-0.55 according to the ASTM standards. The net ligament after pre-cracking is about 7 ram. P
-~
,\\, ,
Dimensions in mms.
"\\
/
s
i Fig. l-a. C-shaped SENB test specimen.
Fig. l-b. Clip gage mounting.
Tests were carried out using the ASTM E 813-89 multi-specimen J-integral technique as adopted by Hating [5] and Chung and Williams [6]. According to the ASTM standard the specimens were loaded to different sub-critical crack depths. As proposed by Haung and Chung and Williams, the final crack depth was marked by freezing the test specimens in liquid nitrogen and then breaking these specimens by applying an impact load using a hammer. The crack size and crack growth were measured at nine equally spaced points along the specimen width (see Fig. 2) as suggested by ASTM E 813-89 standard method using a travelling microscope. Measurement stations 1
2
3
4
5
6
7
8
9
-
Machined notch ~ i
~...~1/
~
/
/
/
/
/
/
/
i
--d
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-t
Post-test fracture
-,~.ll-'~1
Fig. 2. Fractured test piece surface.
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For three-point bend specimens (SENB), the corresponding J-integral, J, was calculated as 2U J=~ B ( W - a)
(1)
where U is the energy consumed in stable crack growth, W is the pipe thickness and a is the final crack depth. The same equation is herein used for the C-shaped curved specimens cut from the PVC pipe. This energy was obtained by measuring the area under the load-load line displacement curve using a polar planimeter and then converted to energy units. The energy U was used in the above equation after a correction made to account for the specimen local deformation at the loading and support points [5,6]. It also accounts for the friction between the specimen and these supports during loading. This correction was estimated as 10% of the total energy, UT, measured under the curve. The J-R curve was then constructed by plotting J values versus crack extension, Aa, following the procedure recommended by the ASTM E 813-89 standard method. Data points were fitted according to an exponential approximation by a power law defined by the equation J = m (Aa)"
(2)
where m and n are fitting parameters. A blunting line representing the blunting behavior of the material at the crack tip was also constructed. This line is defined by the equation J = 2 ov Aa
(3)
where ov is the yield strength determined by a separate tension test. PVC pipe was axially split, heated above its glass transition temperature in an oven, flattened between two parallel plates and gradually cooled in 24 hours [3]. Standard test specimens were cut in the hoop direction of the pipe. The test was carded out at a range of crosshead speeds suitable to an estimated local displacement rate at the crack tip. This rate was calculated by dividing the crack tip opening displacement, 8, by the corresponding test time. A crack initiation value, JQ, was estimated at the intersection of the power law curve and a 0.2 mm offset blunting line defined as J = 2 ov ( A a - 0.2)
(4)
The validity of specimen dimensions as well as the validity of J-test results were checked. The test is considered valid and JQ = JIc if the following conditions are satisfied B, ( W - a 0 ) dJ da
<
O'y,
J 25 : Q
>
and
(5)
O'y
at
AaQ
where a0 is the pre-crack depth and AaQ is the crack growth at the initiation point.
(6)
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Moreover, the J-controlled crack growth condition must be fulfilled, which is achieved by limiting the crack extension to less than 6% of the un-cracked ligament. Based on this condition, the ASTM E 813-89 assumes a 1.5 mm offset blunting line to be used for data exclusion. Combining these two conditions, it can be seen that the pipe specimen should have a minimum wall thickness, W, of 50 mm and a width, B, of 25 mm. This is not satisfied in our case and may impose problems in producing specimens of such a size for polymers. Chung and Williams [6] realized this problem and considered the 6% criterion too stringent and proved that a geometry independent R-curve can exist at crack growths of up to about 10% of the un-cracked ligament. It has been also proposed that this limit can be relaxed to 15% for metals and a size independent R-curve can still be obtained [15]. Therefore, it is herein suggested that a maximum crack extension of 12.5% of the ligament is considered acceptable for polymers.
2.2. Crack Tip Opening Displacement Test The crack mouth opening displacement, vg, (namely COD) is usually measured by attaching a clip gauge to knife edges glued to the mouth of the specimen pre-crack, in order to hold the clip gage, as shown in Fig. 1. A record of P versus vg is obtained and analyzed as shown in Fig. 3-a. Experimental CTOD, 5, estimates are made by separating vg into elastic and plastic components, ve and Vp respectively. The elastic part of vg is not used but the elastic contribution to 5 is calculated according to the LEFM for plane strain conditions. The plastic component, ~ip,of CTOD is obtained by assuming that the test specimen rotates about a plastic hinge as shown in Fig. 3-b. This component is related to the plastic displacement at the crack mouth, Vp, through a similar triangles construction. v. i,,
i
P
v,
Fig. 3-a. Load vs clip gage displacement.
Fig. 3-b. Hinge model for SENB specimen.
Having obtained the required value of the clip gage displacement, it is necessary to convert this to the relevant CTOD, 8, using the following relationship for the C-shaped SENB specimens having 0.45 a0/W 0.55. To calculate 8 2
8 = Kt (1 - 0 2) + rp ( W - a 0 ) v p 2Or E rp (W - a0) + a0 + z
(7)
248
where
Current Advances in Mechanical Design and Production, MDP-7
YP KI = _i\---'--~
(8)
~WB
and where KI is the stress intensity factor, v is the Poisson's ratio, E is the modulus of elasticity, r o is the plastic rotational factor, z is the thickness of the knife edge pieces (see Fig. 3-b) and Y is the specimen geometry factor which can be obtained from Ref. [3,8]. The rotational factor rp is a constant between 0 and 1 that defines the relative position of the apparent hinge point. According to the BS 5762: 1979, rp = 0.40 while the ASTM E 1290-89 specifies rp = 0.44 for the SENB specimen. The BS 5762:1979 and the ASTM E 1290-89 standards use graphical procedures forthe determination of the initiation value, 8i. They use a crack growth resistance (8-R)curve obtained by plotting 8 against the crack extension, Aa. In the ASTM technique, two vertical exclusion lines are drawn at 0.15 mm and 1.5 mm of crack extension. In analyzing the test results, the BS standard assumes a linear regression to determine 8i. This value is determined at the intersection of the linear relation with the vertical axis, i.e., at Aa = 0. The ASTM fits the data points to an offset power law expression as 8 = C I (C2 + Aa) c3
(9)
where Cm, C2 and C3 are fitting parameters. The initiation value, ~5i,is then determined as the CTOD corresponding to Aa = 0.2 ram. The only specimen size requirement of the BS and ASTM CTOD standards is a recommendation to test full section thickness. The CTOD standard test methods [10,11 ] were also used in this investigation. As in the case of J-R curve, the exclusion lines are drawn at 0.15 mm and 12.5% of the net ligament aRer pre-cracking when applying the ASTM standard in this investigation. A Jic value which is considered equal to G~c in the LEFM regime can be calculated from the crack initiation CTOD value, 8, obtained from the ~i-R curves as [ 16] Jic = Glc -- ovai
(10)
3. RESULTS AND DISCUSSIONS Test specimens were prepared and tested according to the conditions previously mentioned. They were conditioned at 22~ and 60% relative humidity for 48 hours before testing. Tests were carded out on a displacement-controlled machine at a crosshead speed of 5 mm/min. The load versus load-point displacement were recorded using a personal computer interfaced to the testing machine. The load versus crack mouth opening displacement were also recorded using an x-y plotter connected to the clip gage and the load cell of the testing machine. The results of the tension test are shown in Fig. 4 at displacement rates ranging from 0.01 mm/min to 1000 mm/min. At a rate of 0.1 mm/min, which is the estimated average displacement rate at the crack tip, the yield strength is obtained as 40 MPa.
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J-test results were plotted and shown in Fig. 5 as a J-R curve. A set of eleven data points was used in fitting Eq. (2) as shown in the figure. The ASTM standard requires a minimum of four points to fit the curve. The initiation value, Jo, is obtained as 4.32 kJ/m 2. All validity conditions are satisfied, hence JQ is considered to be Jlc. This value is a good estimate specially if it is compared to the LEFM Glc value obtained in previous investigations as 3.5 kJ/m 2 [3, 8]. Crack growth resistance curves were obtained by plotting the CTOD against crack growth, Aa, for the same test specimens used in plotting the J-R curve. The CTOD test results were also plotted, as previously described, according to the BS and ASTM standards and shown as 8-R curves in Figs. 6 and 7 respectively. Fig. 6 indicates that the crack initiation value, 8i, is obtained from the BS 8-R curve as 90 ~tm. A similar value is obtained from the ASTM 8-R curve shown in Fig. 7 as 105 lam. Based on Eq. (10) hc is 3.6 kJ/m 2 and 4.3 kJ/m 2 for the BS and the ASTM techniques respectively. All JIc test results are also shown in Table 1. It was found that the ASTM E 813-89 for the Jtest and the ASTM E 1290-89 for the CTOD test yielded the same results of hc. The Jic value obtained by the BS testing procedure is more conservative than that obtained using the ASTM testing technique. The obtained hc values may be considered as a material property which does not depend on the specimen dimensions.
Table 1. Jtc test results of the tested PVC
-Test method
J-integral ASTM E813-89 4.32 .
JIc
.
.
.
.
.
.
.
CTOD ASTM E 1290-89 4.30
CTOD BS:5762:1979 3.6
LEFM Ref. [3, 8] 3.5
KJ/m 2
4. CONCLUSIONS From the previously obtained results it can be concluded that the multi-specimen standard elastic-plastic testing techniques, used for the straight SENB specimens, can also be applied to the C-shaped SENB specimens cut from plastic pipes. The C-shaped specimens have the advantage that they are cut directly from an actual pipe and pre-cracked in the axial (extrusion) direction which is the weakest direction in the pipe. A J-R curve can be drawn using these specimens and a good estimate of hc is obtained. An 8-R curve can also be obtained using the test procedures of the BS and ASTM standards. The ASTM E 1290-89 can be used as a good graphical method to determine 8i in the elastic-plastic behavior of plastics. Same specimens can be used to obtain J-R as well as 8-R curves. Although all the requirements of the ASTM standard J-integral and CTOD test procedures are satisfied in this work, except the 1.5 mm exclusion line, it is recommended to use larger pipe sizes of larger thickness to satisfy all validity conditions of these tests.
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REFERENCES 1. Abou-Hamda, M.M.," Evaluation of Elastic-plastic Fracture Criteria as Applied to High Density Polyethylene Pipe Materials ", M.Sc. Thesis, Cairo University, Giza, Egypt, (1989). 2. EI-Badawy, A. A., " Elastic-plastic Fracture Mechanics for Two Dimensional Stable Crack Growth and Instability as Applied to PVC Pipe Materials ", M.Sc. Thesis, American University in Cairo, Cairo, Egypt, (1995). 3. Darwish, A. Y., "Fracture Analysis of PVC Pipe Materials ", Ph.D. Thesis, MIT, (1981). 4. Rice, J.R., " A Path Independent Integral and the Approximate Analysis of Stain Concentration by Notches and Cracks ", Trans. of ASME, vol. 35, pp. 379-386, (1968). 5. Haung, D. D., " The Application of the Multi-specimen J-integral Technique to Toughened Polymers ", Elastic-plastic Fracture Test Methods: The User's Experience, ASTM STP 1114, J. A. Joyce, Ed., ASTM, Philadelphia, vol. 2, pp. 290-305, (1991). 6. Chung, W. N., and Williams, J. G., "Determination of J~c for polymers Using the Single Specimen Method ", Elastic-plastic Fracture Test Methods: The User's Experience, ASTM STP 1114, J. A. Joyce, Ed., ASTM Philadelphia, vol. 2, pp. 320-339, (1991). 7. Bernell, C. R., and Frontini, P. M.," Fracture Toughness Determination of ABS Polymers Using the J-method ", Polymer Testing, 11, pp. 271-288, (1992). 8. EL-Zoghby, A. A., "Fracture Toughness Evaluation of Polyvinylchloride Pipes ", Ph.D. Thesis, Cairo University, Giza, Egypt, (1982). 9. E 813-89, " Standard Test Method for Jmc,A Measure of Fracture Toughness ", ASTM Annual Book of Standards, Philadelphia, Part 10, (1989). 10. BS 5762: 1979, " Methods for Crack Opening Displacement (COD) Testing ", British Standards Institution, London, (1979). 11. E 1290-89, " Standard Test Method for Crack Tip Opening Displacement Testing", ASTM Annual Book of Standards, Philadelphia, Part 10, (1989). 12. Fltieler, P., Mandell, J. E., and McGarry, F. J.," Applicability of Plane Strain Fracture Toughness Test Techniques to Plastic Pipe Materials ", Report 79-1, Cairo University/ Massachusetts Institute of Technology, Technological Planning Program, (1978). 13. Bakker, A., " Investigation into the Validity of J-based Methods for the Prediction of Ductile Tearing and Fracture ", Ph.D. Thesis, Delft University, (1986). 14. Shih, C. F., Andrews, A. H., and Delorenzi, H. G.," Crack Initiation and Growth Under Fully Plastic Conditions: A Methodology for Plastic Fracture ", Report NP-701-SR, Electric Power Research Institute (EPRI), Palo Alto, CA, pp. 6.1-6.63, (1978). 15. Gordon, J. R., and Jones, R. L.," The Effect of Specimen Size on the J-R Curve Behavior of a Titanium Alloy ", Fatigue and Fracture of Engineering Materials and Structures, vol. 12, No. 4, pp. 295-308, (1989). 16. Ewalds, H.L., and Wanhill, R.J.H.," Fracture Mechanics ", Edward Arnold (Publ.) and Delflse Uitgevers Maatschappij, pp. 156-158, (1984).
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80 70 rt
60
v
U
.c:" E:
50
(D
L.
u) "10
40
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>. 30
0 01
0 10
1 00
10 00
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10
8 -
Blunting line J=2cyAa
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9 ~
9
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/
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.
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10
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E v
Current Advances in Mechanical Design and Production, MDP-7
200
::L
c,O
c-. (D
160
E
(D O Q. (D r r
.,...,
= 9 0 + 48 A
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120
80
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40
r L_
r
0
00
I
I
I
I
I
I
I
I
I
01
02
03
04
05
0.6
07
0.8
0.9
10
Crack extension, A a (ram)
Fig. 6. Determination of c5i from CTOD test data (BS 5762: 1979)
E v
200
::L
uo .,..; c.-
160
(D
E
(D O
C1 u~ ..=. "(3
r" e~D (D.. O (3. .bE O
120
12.5% Exclusion line
0 1 5 mm Exclusion line I
I
I
I I
8 i
I I
= 133 ~/0.3"+ A a
80
I I I
0.2 mm Offset line
40
I I I
L_
O
0 0.0
. . . .
0.2
I
I
I
04
0.6
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I
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Fig. 7. Determination of 8 i from ~-R curve (ASTM E 1290-89)
1.0
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
253
ACOUSTIC EMISSION DETECTION OF MICRO-CRACKS INITIATION AND GROWTH IN POLYMERIC MATERIALS Abo-EI-Ezz, A.E. Associate Professor, Force and Material Metrology Department National Institute for Standards, P.O.B. 136 Giza, Egypt.
ABSTRACT In this study, an Acoustic Emission (AE) monitoring system was developed and employed to detect the AE signals originating at the near vicinity of a single edge crack in a PMMA sheet. Different loading modes were applied in one loading spectrum on each test specimen. Throughout the entire loading spectrum the AE signals were recorded simultaneously with the load. The relationship between the crack advancement and AE signals was discussed. Examination of the fracture surface of test specimens revealed surface marks due to microcrack advancement. The number and the size of marks were matched with that of AE signals, which support the validity of AE technique. KEYWORDS Crack Detection ,Cracked Polymer, Crack Propagation, Crack Initiation, Cyclic Loading, Acoustic Emission, Stress Intensity Factor. I. INTRODUCTION The Acoustic Emission (AE) signals are stress waves. They are generated at critical sections of the stressed materials as a result of micro-cracks advancement or deformation process. The theme is very important as a non-destructive testing technique for observing critical parts in the components of structures or machinery. Utilizing the AE signals produced by the stressed components during normal working enables us to monitor the condition of the important element continuously. When the material is loaded or strained, it may produce a certain acoustic characteristic pattern. If the component begins to fail, its acoustic pattern will change [1]. The detection of such change may serve as a warning for the replacement of the component before it causes sudden failure of the entire structure. The AE technique has been widely applied to hard materials such as metallic alloys [1], concrete and ceramics [2, 3]. However the applications of the technique to polymeric materials or polymer composite materials are very few [4]. In polymers the amount of released energy due to the advancement of micro-cracks is relatively small and hence the AE signals are weak. It should be noted that the amount of released energy depends on the elastic modulus of the material, which is very low in polymers as compared with those of metals (several ten times smaller than those of metals). The differences in the amount of the released energy are thought to be in the same order of the difference in Young's modulus [5]. In
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Current Advances in Mechanical Design and Production, MDP-7
addition to that, the attenuation of these waves in polymers in the frequency range of AE signals, from 10 kHz to 500 kHz, is very large compared to that in metallic materials. These two factors cause some difficulties for detecting AE signals in polymeric materials. There are several non-destructive testing techniques in use to detect material deformations. The strain gauge technique is the most common one, since it could be used with different types of engineering materials. However, the strain gauge should be placed very close to suspected defects, which are unknown in many cases. The small angle X-ray scattering method is another detecting technique that could be used with a thin film placed into vacuum. [6]. While the scattering of visible light (laser beam)could be used only with transparent materials [6]. There are two other new techniques, magnetic and electric field techniques, which are in use for metallic materials [7]. This work has been carried out aiming at implementing the AE technique to detect the initiation and growth of cracks in a polymeric material using the AE signals originating at the near vicinity of a single edge crack. 2. EXPERIMENTAL Single edge notched (SEN) specimens were prepared from cast sheet of Poly-methylmethacrylate (PMMA). A sharp edge notch was generated on one edge of the specimen by chisel impact followed by fatigue pre-cracking (vibration crack) [8]. This procedure resulted in well-finished single crack with the desired length. The specimen geometry was 200 mm in length, 40 mm in width and 4.8 mm in thickness. An Instron-type universal-testing machine was used for specimen loading with a constant cross head speed of 0.02 mm/s. This very slow speed was selected to minimize the mechanical noise from the machine. To investigate the condition at which the AE signals may be studied in stressed PMMA, each specimen was loaded under three different loading modes in one loading spectrum. The modes are monotonic tensile loading, cyclic loading with continuous increase of mean load and finally cyclic tensile loading with a constant mean load. The tensile loading (initial setting load mode) was first applied to a certain stress intensity factor [9] below Kii (Kli is the stress intensity factor for slow crack initiation) to avoid crack propagation. The second, cyclic triangle waveform with continuos increase of mean load was applied (by increasing upper and lower loading stresses every cycle) until the value of Kli is reached. In the third loading mode, the cyclic loading with the constant mean load was continued until final fracture occurred. The AE detection was made via three identical AE sensors of resonance type (190 kHz resonance frequency for each). The sensors were fixed, on the surface of the specimen, at three different positions from the crack tip as shown in Fig. 1. This arrangement was developed as an attempt to distinguish AE signals (true signals) from untrue signals (false) which may be produced from various types of noises, mechanical or electrical noise. Each signal detected by the three AE sensors 1, 2 and 3 were amplified (100 dB in each channel) in three sets of amplifiers; pre and main amplifiers. The output of the AE system was stored in two digital memories A and B, of two channels each, and then displayed on two oscilloscopes A and B, of two channels each. The output of the two sensors I and 2 were stored in the digital memory A and then displayed in oscilloscope A. While the output of the two sensors 1 and 3 were stored in digital memory B and then displayed in oscilloscope B. A schematic diagram for the AE measuring system is shown in Fig. 1. Using such a system would help taking the right decision about the true and false signals.
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Current Advances in Mechanical Design and Production, MDP-7
Machine Fixed Head --~
~
Force Transducer
+
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,+
L~___ ~re-am'l
2
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~
,,m
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]TwoChannels
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]
I
[X.y' ;iotter] [Signai Analyzer& Memory for Machine Movable Head 0.02 mm/sec.
Fig. 1 A schematic diagram for the measuring system.
Output of sensors 1,2 from oscilloscope A
Output of sensors 1,3 from oscilloscope B
...,.
<
tl||Jil A
~2
..=,,
<
e2
o~
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Time
Signal due to mechanical noise, false single < o" 2
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'
:
Zero
Amplitude Time
Signal due to crack advancement, true AE signal
Fig. 2 An example when the detected signal was due to mechanical noise and another one for the case when the signal was due to AE waves.
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The decision was made based on the time correlation between the output signals from the three sensors 1, 2 and 3. By comparing the time difference between the signals of the three sensors, the true signals should reach the sensor, which is nearer to the tip of the crack, channel 1, before reaching the other two sensors, channels 2 and 3. On the other hand, false signals (from mechanical noise) arrive first to the sensors, which are nearer to the specimen clamps, channel 2 or channel 3. However, if the signal was due to an electric noise, a false signal, it may arrive to the three transducers nearly at the same time. Figure 2 show an example for the case when the detected signal was due to mechanical noise and another example for the case when the detected signal was due to a true AE wave. Accordingly, only AE signals were recorded. These procedures were completed quickly within few seconds and the digital memory was again set for the next events. The recorded data were fed later (after failure of test specimen) to a signal analyzer for AE wave analysis. The load variation and the occurrence of AE events were simultaneously recorded on X-T recorder to investigate the correlation between them (Fig. 1). Macroscopic crack advancement during the loading was also monitored, simultaneously, via a TV-camera system as illustrated in Fig. 1. This would help in the investigation of the relation between crack advancement and the generation of AE signals. After final failure of tested specimen the fracture surfaces of the specimen were examined carefully using an electron microscope aiming to find such a relation. 3. RESULTS AND ANALYSIS Values of K~i and Kic (the critical stress intensity factor) of this grade of PMMA were experimentally obtained as 1.16• MN/m 3r2 and 1.57+0.06 MN/m 3r2, respectively. These values are in agreement with previous results [10]. Depending on these values, the loading limits of the three loading modes were defined. Then, the procedures of specimen loading and AE detection were performed several times as explained in the previous section. Figure 3 shows a good example for the obtained results. The whole applied load spectrum and the positions at which the AE signals were detected are shown. Firstly, initial setting tensile mode up to 0.7 Kli (at load of 0.9 kN) with no evidence for detecting AE signals is plotted. Secondly, the cyclic triangle loading form with continuous increase of mean load until the value of stress intensity reached Kli at the upper limit of the load cycle (mean load and load amplitude were 1.2 and 0.3 kN, respectively) is denoted as zone 1 in the figure. Thirdly, the cyclic loading continued keeping the maximum and minimum stress levels constant until the final fracture occurred, is denoted as zone 2. In zone 1, only few AE signals were detected while in zone 2, the signals were detected each cycle (Fig. 3). In both zones, the signals were detected while the load level was approaching its upper level. Finally, at fracture, very strong AE signals were detected. The AE signals detected at different loading stages were quite different. It should be noted that, the advancement of crack tip as observed by the TV cameramicroscope and the detected AE signals, via AE system, were always synchronized. Figure 4 shows an example for three different AE signals detected in zone 1, zone2 and fracture. The signals were quite different in their characteristics. Signals detected in zone 1 are very weak in intensity, small amplitude, and short in duration than those detected in zone 2. While signals detected at fracture are the strongest in intensity (large amplitude)and longest in duration more than 400 ~s. Analysis of the frequency magnitudes and the intensity of the AE signals also revealed distinguishable distribution as illustrated in Fig. 5.
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AE Sianals :
. . . . . . . . . . . . .
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.
.
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.
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200
400
600
800
1000
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.
.
.
.
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200
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Fig. 4 Typical AE signals detected m zone 1,zone 2 and at fracture The frequency range of AE signals detected at zone 1 and at zone 2 are wider (20 - 200 kHz) than those detected at fracture (l l 0 - 170 kHz). Further detailed analysis should be carried out using wide band AE transducers rather than that resonance type ones, which were used in this study for the analysis of AE waves. To define the origins of the detected AE signals, fracture surfaces of tested specimens were examined in details after final fracture. Figures 6 and 7 show an optical photograph and a schematic drawing of the photograph for the fracture surface of tested specimen whose AE results are obtained from load spectrum shown in Fig. 3. Careful examination of fracture surface (Fig. 6) revealed different crack propagation regions as schematically drawn in Fig. 7.
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9
9
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~
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Frequency, [kHz] Fig. 5 Frequency range of the AE waves shown m Fig. 4
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The initial crack, the fatigue crack region, the slow crack propagation region and the onset of the fast fracture region were recognized. The fatigue crack region (A)is the region of our interest where the detected AE signals originated. The length of this region is about 400 ~tm. Its length is comparable to the total length of crack advancement during the cyclic loading, until the onset of the slow crack propagation, as observed via the TV- monitor system. Closer examination of the fatigue crack region A, Fig. 8, revealed several important points. First, many striations could be seen in this region, the direction of which is perpendicular to the crack propagation direction. Second, the total number of the striations (Fig. 8) are nearly the same as the number of cyclic loading applied to the specimen. Third, there are two kinds of striations seem to exist. Light striations with narrow separations of about 2-5 ~tm and more heavy marked striations with separation distances of about 10-50 ~m. The former (light)is found only in the region that matches zone l (Fig. 3). It should be noted that the heavy marked striations is progressively increasing. The light striations and the heavy marked ones sometimes interfere with slight directional deviation (small angle of intersection) in zone 1. Zone 2 is started when the light striations are disappeared and the heavy marked ones could only be seen (Fig. 8).
Fig. 8 Optical photograph offi'acture surface shows enlarged image of the fatigue crack propagation region (region A). 4. CONCLUSION From the results described above, the following mechanisms may be suggested for the single edge crack growth and the emission and the detection of acoustic waves during the cyclic loading of cracked PMMA. At low stress level (lower than KIi), crazes first developed step by step upon application of cyclic loading producing light striation on the fracture surface. When the developed crazes reach a critical length (>l 0 ~tm) the crack proceeds through the crazed region producing the heavy marked striations. At this level of loading the AE signals are not always detectable
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because the tips of craze and crack advance rather slowly in a plasticized material without a sudden release of stored energy. On the other hand, at high stress levels (near K1i)the advancement of craze and crack takes place almost at same time releasing high energy accompanied by strong AE waves. The AE signals are always detectable at this condition. It may be concluded that the AE method is applicable for the detection of an advancing of micro cracks of an order of a few tens of microns in brittle polymers. ACKNOWLEDGMENTS: The author is grateful to Prof. K. Takahashi, Director of Research Institute For Applied Mechanics, Kyushu University, Japan and to Japanese government for giving him the opportunity to do part of the experimental study at Kyushu University. REFERENCES: 1. Williams, R.V., "Acoustic Emission", Adam Hilger LTD, Bristol U. K. (1980). 2. Ohtsu, M., Shigeishi, M. and lwase, H, "AE Observation in the PuU-out Process of Shallow Hook Anchors", Proc. of JSCE, No 408/V-11 (Concrete Eng. and Pavements) Aug., pp. 177- 186, (1989). 3. Shigeishi, M. and Ohtsu, M. "Observation of Mixed-mode Fracture Mechanism by SIGMA-2 D", J. of Acoustic Emission, Vol. 11, No. 4, pp. S 57- S 63, (1993). 4. Nagoh, S., Choi, N. and Takahashi, K., "Acoustic Emission in SMC Specimens with Various Notch Trip Radii", RPPJ, Vol. 32, pp. 311- 314, (1989). 5. Shirouzu, S., "Studies on Acoustic Emission in Polymers", Master Thesis, Kyushu Uni. Japan, March (1984). 6. Zhurkov, S.N. and Kuksenko, V.S., " The Micro-mechanics of Polymer Fracture", Inter. J. ofFrac. Vol. 11, 4, pp. 629- 939, Aug. (1975). 7. Winkler, S. and Winkler, W.D., "Field Emission in Charpy Tests", Roell-Amsler Symposium, May (1998). 8. Abo-EI-Ezz, A.E., Abd-el-hakeem, H.M. and Takahashi, K., "A Study of Fatigue crack Propagation in Glassy Polymer by the Optical Method of Caustics", JSME Int. J., Series A, Vol. 37, No. 4, pp. 466-471, (1994). 9. Tada, H., Paris, P.S., and Irwin, G.R. "The Stress Analysis of Cracks", Handbook, Del Res. Co., Hellertown, Pennsylvania, p. 2.10, (1973). 10. Begueiin, P, "Experimental Approach of Mechanical Behavior of Polymers Under High Rate of Loading", Report No. 1572, Ecole Polytechnique Federale, Lausanne, pp. 1-26, (1996).
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
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EVALUATING THE STRESS CONCENTRATION DUE TO ELONGATED DEFECTS IN WELDED AREA
Abd EI-Ghany, K.M.', EI-Mahallawi, I.'" and EI-Koussy, M.R. "** Cairo University Center for Advanced Software Development Associate Professor, Department of Mining, Petroleum and Metallurgy Professor, Department of Mining, Petroleum and Metallurgy Faculty of Engineering, Cairo University, Giza, 12316 - Egypt
ABSTRACT In welded pipes, a welding defect can cause a disaster. Among the commonly known welding defects, incomplete penetration and fusion, porosity and slag inclusions are the most common to affect welding strength. The acceptance and rejection criteria of discontinuities are the roles of the standard codes such as API, ASME, ASTM, AWS, etc. The standard specifications are restricted and force the inspection engineers to reject or re-weld pipes that may be "Fit for their Purpose". In many cases, these pipes may contain welding defects and discontinuities that produce stress concentration values that may not affect the service life of the material. This paper describes the employment of three-dimensional linear static finite element analysis to find out the values of stress concentration that develop due to the presence of elongated volumetric discontinuities in the welded region. Also, it seeks for the relations among the length and the diameter of the discontinuity, the effective weld and wall sizes and the triaxial stress region induced around each discontinuity. The finite element software package I-DEAS [ 1] was used for this work. About 200 simulation cases were modeled and solved by the finite element software. The modeled cases were identical in the geometry, applied stresses, boundary conditions and meshing size and type, but they varied in weld thickness and discontinuity length and diameter. The applied stresses were biaxial to simulate the welded piping applications and uniaxial to simulate the general purpose applications. The results indicate that, I) The average stress concentration factor when the discontinuity length is parallel to the maximum applied stress is about two, and about three in the normal direction, (yon Mises stress), 2) The acceptance and rejection criteria of the codes AWS D 1.1 and ASME section V (for boilers and pressure vessels) are conservative to some extent, 3) The calculated values were found closer to the criteria o f A P I - 1 1 0 4 and 4). The effect of the triaxial stresses induced around the curved surfaces of the discontinuities is remarkable and should be considered while judging the discontinuities effect. KEYWORDS Welding, Discontinuities, Finite Element Analysis, Concentration, Acceptance and Rejection Codes
Standard Specifications, Stress
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1. WELDING DISCONTINUITIES The steel pipes are likely to suffer from welding imperfections and discontinuities that affect the pipes and the whole structure strength and resistance to the applied stresses and environmental conditions. For example, a welding discontinuity only a few centimeters long can reduce the strength of the pipe by a factor of 5 and lead to disastrous failure [2]. The welded pipe inherits its durability and long life from the following factors: 1. The material properties of the base metal, weld metal and the heat affected zone. 2. The pipe design, dimensions and wall thickness, joint design and weld throat size. 3. The proper selection of welding procedure specification (WPS), Procedure Qualification Record (PQR) and qualified welders. 4. The type, size, and position of welding imperfections and discontinuities in the welded region and in the base metal. Welding discontinuities are classified in the documentation of ISO 6520 and the parameters characterizing flaws are defined in the documentation of IIW/IIS-340-69 [3] into three general types: - Linear (one dimensional) discontinuities, such as slag lines. - Planer (two dimensional) discontinuities, such as cracks, lack of fusion, lack of penetration and other like flaws. - Volumetric (three-dimensional) discontinuities, such as porosity, inclusions and similar flaws. - Imperfect shape, such as reinforcement, angular distortions, misalignment and similar flaws and geometric discontinuities. The determination of the presence, dimensions and position of each type of these discontinuities is the matter of using non-destructive examination techniques such as radiographic, ultrasonic, eddy current, magnetic particle testing, etc. While the cumulative damage due to their presence in the welded region should be carefully evaluated, referring to the published analytical and experimental studies for fracture mechanics [4,5]. On the other hand, the presence of volumetric discontinuities such as porosity and inclusions produces stress concentration around the discontinuities. The apparent material damage occurs when the elevated stress level due to stress concentration exceeds the yield strength of the material. In the ideal cases and under only static monotonic stress, a plastically deformed layer of material covers the discontinuity with no further damage occurring. However, this plastically deformed layer is not desirable under cyclic stress because it can host finite cavities that grow and form dangerous planer defects and cracks (Fig 1.) 0"00"
~
(a) Cyclic Load
0. 13" 0. Plastic deformation region. 0
(b) Static loading
Fig. 1. The behavior of the discontinuity under cyclic and static stresses.
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2. FITNESS FOR PURPOSE CRITERIA The techniques of non-destructive examination provide the engineer with the necessary tools to judge if the welding region contains discontinuities or not. The acceptance and rejection criteria for a discontinuity are the role of the standard codes and specifications such as API, ASME, ASTM, AWS, etc. The implemented acceptance criteria are based on empirical data and experimental case studies, as a result, they may be conservative in some applications in order to gain global acceptance and reliability for their proposed scope of engineering applications. The criteria of Fitness for Purpose of a product means that the product containing discontinuities has sufficient strength and that it does not fail prematurely during service. It is assumed, of course, that the product in not being misused and that it is eventually maintained or renovated when the stipulated lifetime has passed. Any pipe should be designed, manufactured, inspected, and evaluated in such away that it is fit for its intended purpose. The introduction of the Fitness for Purpose criteria is linked to the progress in material science, welding processes and non-destructive examination techniques, in addition to the advanced computer and stress analysis systems. This will allow for the acceptance of larger discontinuities and save the money and time consumed for repairing or rejecting the weldments. 3. WELDING RELATED STANDARD SPECIFICATIONS The standard codes define the quality assurance and quality control of any manufactured product. For example, the standard acceptance codes relevant to welding process define the whole process from selecting the welding consumables (e.g. BS 639, DIN 1913, ANSI/AWS A5.1-81), welding procedure approval (e.g. BS 4870, ASME Section IX), welder approval (e.g. BS 4871, BS 4872, ASME Section IX, DIN 8560) and finally the quality of a completed fabrication (e.g. BS 5500, ASME Section VIII, AWS Structural Welding Code) [6]. Table 1, summarizes the specified acceptance criteria for the elongated and rounded discontinuities (elongated discontinuity is one in which its length exceeds three times its diameter), where t = wall thickness, O.D = pipe outer diameter, L = Discontinuity length, W = Discontinuity width and D = discontinuity diameter.
Table 1. The maximum allowable discontinuities dimensions
Standard Code
API 1104
ASME
AWS
Allowable elongated dimension For O.D > 60.3 mm L <= 50 mm or W<= 1.6 mm For O.D < 60.3 mm L <= 3 t mm or W<= 1.6 mm L <= 3.2 mm (t--10 mm) L <= 1/3 t (t =10-57 mm) L <= 19 mm (t > 57 mm) L <= 2/3 t .....
Allowable rounded dimension D<= 0.25 t D <= 3.17 mm
D <=0.2t
L < - 3.2 mm L < = 1/3t
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4. W O R K PROCEDURE The objective of this work is to use the method of three-dimensional linear finite element analysis to find out the stress distribution in the pipe wall that contains a welding volumetric discontinuity. In the finite element software, the parent part, which was simulating a piece of the wall pipe that was welded, was assumed to be fiat and rectangular (Fig. 2). Its dimensions were 100 mm in length and 50 mm in width, while the thickness was changed among the different runs. Biaxial stresses were applied to the part in the form of two distributed forces. The applied forces were normal to the upper and the left sides of the rectangular part. This simulates the real case of pipe welding, which is subjected to hoop and longitudinal stresses.
(a)
~z
I~
(b)
~z
i~
Fig. 2. The position of the analyzed part in the welded pipe; (a) Longitudinal position and (b) Cireumferencial position. The hoop and longitudinal stresses are defined as (O'h = PDi/2t ) and ( O ' L - - PDi/4t ), where P - pipe intemal pressure, t = pipe wall thickness, Di = internal diameter. The amplitude of the first distributed stress (hoop) was assumed to be 200 MPa, while the other one (longitudinal) was 100 MPa. The values of the applied stresses were based on the use of high strength steel with yield strength about 400 MPa. So, the used maximum applied stress was about one half of the yield strength. About 200 runs, arranged in six different groups, were made. Each run was aimed to find the stress distribution due to certain discontinuity length and diameter, parent part (pipe wall) thickness and specific applied stress conditions. The following list describes the groups:
1- Group one: in this group, the maximum stress (200 MPa) was applied parallel to the 2345-
6-
major axis (length) of the embedded cylindrical discontinuity, while the longitudinal stress (100 MPa) was applied normal to the major axis of the discontinuities. Group two: the previous configurations were reversed. Group three: similar to group one but the discontinuities were located at the surface. Group four: similar to group two but the discontinuities were located at the surface. Groupfive: additional runs to find the effect of larger diameters and the presence of more than one discontinuity. Group six: only one stress (200 MPa) was applied to some selected discontinuity cases in order to find the stress concentration under only uniaxial stress.
The objective of the first two groups is to study the effect of discontinuity orientation on the induced stress concentration. The objective of the third and fourth groups is to study the
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change in the stress values when the discontinuities are located on the surface of the parent part. The runs of the fifth group are designed to find out the stress concentration values due to the presence of two neighbor discontinuities and large discontinuity diameters. Group six was added in order to compare the results of the uniaxial applied stress with the results of the biaxial applied stress. The relations between the stress concentration factor and the ratios of the pipe wall thickness to the discontinuity diameter and length (D/t and L/t)were also studied under the uniaxial applied stress. Some standard codes such as AWS are general purpose and are not designed for piping applications (biaxial stresses), therefore, the results of group six could be used when judging similar codes. The discontinuity had the shape of elongated cylinder with hemispherical caps (Fig. 3). The parent part, applied stresses, boundary conditions and discontinuity shape and position were fixed during the different runs, while the variables were the length and diameter of elongated discontinuity and the parent part thickness. In each group of runs, the discontinuity diameter was varied between 0.4, 0.8 and 1.6 mm. The discontinuity length was varied between 2, 4, 8, 12, 20, 30, 40 and 50 mm and the pipe wall thickness was varied between 3.2, 6.4, 9.5, 12.5 mm (these values were selected to be comparable with the standard codes and field applications). Higher diameter values were applied in the runs of groups five and six. Also, uniaxial stress was applied for some cases in group six.
4.1 The Analysis, Software and Meshing. An expert system was developed to facilitate the modeling and analysis of volumetric welding defects using the finite element software I-DEAS master series 5. Moreover, to assure consistency of the model and accuracy of the results, some verification runs where made on the software NASTRAN using the same geometry, boundary conditions and mesh size. Results were almost identical. The element type selected for the whole part was the three-dimensional quadratic tetrahedron (second order with ten nodes) solid element. To achieve reliable results, and due to curved geometry of the cylindrical discontinuity, very fine elements were used around the curved discontinuity surfaces. The element sizes around the discontinuity were varied between 0.05 mm (for the discontinuity diameter =0.4 mm) and 0.15 mm (for the discontinuity diameter = 1.6 mm). While it was about 10 mm at the parent part free surfaces. Fig. 3 illustrates the part and mesh refinement. Distributed Loads
c~ y
1,1 Cylindrical discontinuity
Fig. 3. Mesh distribution around the discontinuity.
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Because of the symmetric geometry of both the parent part and the cylindrical discontinuity, only one eighth of the problem was modeled. The total number of elements for the whole part was averaging between 15000 and 40000 elements. 5. RESULTS AND DISCUSSION Values of the different stress components resulting from analysis were collected, verified and presented in Figures 4 to 11. The recorded values were the maximum (non-averaged) values in the stress plot regardless of their position in the part. It is important to mention that the variation in mesh sizes around the different discontinuity dimensions could introduce accuracy factor up to +5% of the presented results. The presented values of stress concentration factor were calculated considering the von Mises stress component. The nominal applied yon Mises stress to the same part without discontinuities was 173.2 MPa (under biaxial stress) and 200 MPa (under uniaxial stress). In the first group of runs, where the major axis of the discontinuity was located parallel to the maximum applied stress, the average stress concentration factor was found to vary from 1.5 to 2.0. While in the second group of runs, where the major axis of the discontinuity was located normal to the maximum applied stress, the average stress concentration factor was found to vary from 2.5 to 3.0. In the third and fourth groups, having the discontinuities on the surface increased the stress concentration factor by about 10%. The stress region surrounding each defect was shown to have an average radius of about one half of the length of the discontinuity in all positions. For steel pipes subjected to static stress and having wall thickness from 3.2 mm to 12.5 mm, the presence of an elongated cylindrical discontinuity located parallel to the maximum applied stress (i.e. cireumferencially) and has the maximum diameter of 1.6 mm and length of 50 mm produces stress concentration factor of about 2. (These dimensions of the discontinuities are the highest allowable according to API-I 104). This means that if the applied stress is less than one half of the yield strength, the material will not yield plastically due to such discontinuities. Even, if limited yielding occurs nothing will affect the pipe performance. Stress concentration is not significant in the case of static stress of a ductile material because material will yield in the region of high stress and, with the accompanying redistribution of stress, equilibrium may be established and no harm is done. However, if the stress is cyclic the material may failure due to fatigue [7]. On the other hand, the discontinuities located normal to the maximum applied stress (i.e. longitudinally) produce stress concentration factor of about 3.0. Consequently, the working stress should be less than one third of the yield strength to prevent failure. In this case the acceptance criteria should also be stricter. When the discontinuities are located near the inside or outside surfaces of the pipe wall, the stress concentration increases by about 10% compared to the embedded discontinuities. This is explained by the absence of the third stress component near the free surfaces (see Table 4 and 5 and compare the values of Crz with that in Table 2 and 3). The values of stress concentration factors produced by the finite element analysis are close to the values obtained by Durelli [8] and Dally [9]. Using the experimental method of3D
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photo-elasticity, they analyzed the stress concentration factor due to the presence of cylindrical inclusion with hemispherical end subjected to triaxial stresses. Figure 4. and Fig. 5. present the values of stress concentration factors as a function of the discontinuity length, for wall thickness values of 6.4 mm, 9.5 mm and 12.5 mm, for the discontinuities parallel and normal to the maximum applied stress. Fig. 6. and Fig. 7. present the values of stress concentration factors vs. the discontinuity diameter for the same wall thickness values. While, Fig. 8. compares between the values resulted from the four groups of results for the specific cases of wall thickness = 12.5 mm and diameter = 0.8 mm. Figure 9. shows the increase in the stress concentration factor as a function of the ratio of the discontinuity diameter to the wall thickness (D/t) under biaxial stress. While Fig. 10 shows the higher effect of D/t under only uniaxial stress. Fig. 10 is also close to the curves obtained by Durelli [8] and Dally [9] by using the experimental 3D photo-elasticity method. When two discontinuities are located parallel and near by each other (the distance between their centerlines are less than 2D), the stress concentration values are not affected if the plane containing their major center lines is parallel to the maximum applied stress. This can explained from the plot of the stress contour showing that the maximum stresses are induced at the quadrant points normal to the plane of the maximum applied stress. The calculated results support the criteria of fitness for purpose, presented in the beginning of this paper. The acceptance of discontinuities inside a welded pipe should be a function of the applied stresses not only an extraction from the specifications. For low pressure pipeline applications much longer discontinuities can be tolerated compared with the standard criteria. The calculated results (using biaxial stresses for piping applications and uniaxial stress for general purpose applications) showed that the acceptance and rejection criteria of the codes AWS D 1.1 and ASME section V (for boilers and pressure vessels) are conservative to some extents. For example, the AWS specifications reject any discontinuity if its length is greater than two thirds of the pipe wall thickness, while ASME code rejects any discontinuities that have length greater than one third of the wall thickness. The calculated values were found to be closer to the criteria o f A P I - 1 1 0 4 , which permits maximum diameter up to 1.6 mm and length up to 50 mm in circumferencial welds. Even when the discontinuity length is less than 0.67 of the wall thickness (as required by the AWS standards specifications), the orientation and position of the discontinuities should be considered before applying the design factor of safety. The stress region around each defect was shown to have an average radius of about one half of the length of the discontinuity. This means that if the distance between two neighbor discontinuities (from all spatial directions) is less than the sum of their maximum radii, the stress concentration due their presence in the material will be greater than the calculated results in this work. The shorter the distance, the greater the stress values. The type of the applied stress has remarkable effect on the induced stress concentration. For example, discontinuities under biaxial loads in piping applications produce different stress concentration than the same discontinuities located in plates under uniaxial stress (Fig. 11).
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6. CONCLUSION The following conclusion can be drawn: 1. The calculated average stress concentration factor when the discontinuity length is parallel to the maximum applied stress, is about 2. (von Mises stress). On the other hand, the calculated average stress concentration factor when the discontinuity length is normal to the maximum applied stress, is about 3. 3. The ratio of the discontinuity diameter to the wall thickness (D/t) had shown greater effect under uniaxial stress than under biaxial stress. 4. The acceptance and rejection criteria of the codes AWS D I.1 and ASME section V (for boilers and pressure vessels) seem to be conservative to some extent. 5. The calculated values were found closer to the criteria of API 1104. The large discontinuities accepted by API 1104 (L--50 mm, D=I.6 mm) are reasonable since they produce stress concentration factor in the circumferencial weld direction of about 2. 6. The effect of the triaxial stresses induced around the discontinuities should be taken into account while judging their effect. 7. The calculated results support the criteria of fitness for purpose that requires that the acceptance and rejection of the elongated discontinuities should be evaluated as a factor of the applied stress and the intended use of the pipe .
REFERENCES 1. 2. 3.
4. 5. 6. 7. 8.
9.
IDEAS Master Series 5 Reference manual- SDRC publications- (1997). Owen, D. R. J. and Fawkes, A. J, "Engineering Fracture Mechanics: Numerical methods and Applications", Pineridge Press Ltd., Swansea, U.K., (1983). Gallagher, M. E. and Bosward, I. G., "Quality assurance applied to non destructive testing", "Non Destructive Testing", Vol. 1, A conference organized by the British Institute of Non Destructive Testing, (1987). Sih, G. C., "Mechanics of Fracture - Volume 7: Experimental evaluation of stress concentration and intensity factors", Martinus Nijhoff Publishers, (1981). Broek, David, "Elementary Engineering Fracture Mechanics", fourth revised edition, Martinus Nijhoff Publishers, (1986). Chuse, Robert and Eber, Stephen M., "Pressure Vessels: The ASME code simplified", 6th edition, Mc Graw Hill book company, (1984). Shigley, Joseph Edward, " Mechanical Engineering Design", First Metric Edition, Mechanical Engineering Series, Mc Graw Hill, (1986). Durelli, A. J., "Stress Concentration" contribution work for "Mechanics of FractureVolume 7: Experimental evaluation of stress concentration and intensity factors", Martinus Nijhoff Publishers, (1981). Dally, Janes W., "Experimental Stress Analysis", 3rd ed., Mc Graw-Hill, (1991).
Current Advances in Mechanical Design and Production, MDP-7
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912.6 mm
2.U
I 0.8
Discontinuity Diameter
0.4
1.|
0.8
Discontinuity Diameter
Fig. 6. Stress concentration factors vs. the Fig. 7. Stress concentration factors vs. the discontinuity diameter for different wall discontinuity diameters for different wall thickness. Results from the first group of runs. thickness. Results from the second group of runs.
3,S
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ue
(J
.
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j
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12
Discontinuity
50
Fig. 8. Stress concentration factor when the discontinuity is deep and on surface (t-12.5 mm and D--0.8 ram.) under the two stress cases.
2
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Dis3~ontinuity Dian~ter4 s
ss
Stress concentration factor vs. larger Fig. 9. discontinuity diameters (t-6.4 and L--50 mm) under biaxial stress.
270
~:
Current Advances in Mechanical Design and Production, MDP-7
14.00
i
12.00 Q
u 0 r
t0.00
j
6.~
7.00
12.29
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Fig. 10. Stress concentration factors to different D/t ratios under uniaxial stress.
Bi4xkd (pendhd)
BI4xliall ( ~ q
Fig. I 1. Comparing the stress concentration factors for different stress types.
Table. 2. The values of stress components resulted from the first group of runs where the discontinuities were located parallel to the maximum applied stress. The shown results are only for wall thickness = 6.4 mm. 'Defect Diam. 0.4
Defect . . . . .
Length. (~vmSC .
4 8 12 20 30 40 50
0.8
,
(~y
[
(~'z
"Cmax
1.81 1.72 .~ 1.70 ' 1.73 1.74, , 1.71 1.65 l
314.22 298.69' 293.89 ' 300.12'' 301.95 , , 296.26 286.6 'i i
361.38 342.79' 343.49 i 345.83 350.1 , 344.33 338.6 'i
',
. . . .
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361.15 341.06 ' 286.33 '-104.75 348.45 ' 337.48 ' 291.88 ' -104.1 ' 346.27 335.98 286.12-103.75 348.23 ' 339.82 i 288.23 ' - 1 0 7 . 4 8 ' 339.66 333.43 283.09 -94.8 296.4 1294.71 '. 291.3.2 , -88.5 . 337.69 328.06 288.72 -90.13
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314.43' ' 310.4 i 309.19 ' 299.65 ' 298.19 1267.03 '. 294.42
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345.29 283.83 -90.75 ! 175.14 336.$ ' 2 6 6 . 5 4 ' -97.05 " 1 6 1 . 5 6 333.5 ', 2 8 6 . 0 6 ', -96.63 ,' 164.52337.64 290.9 -10{i.02 165.85 338.6 , 291.45 ,, -96.4 170.87 331.8 289.5 -92.15 166.9 3 2 3 . 3 7 ' i 2 8 3 . 9 9 ' -l 1 0 1 . 4 6 l 161.35li
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i
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353.4 344.5:266.37 -107.75 344.131339.12. 291.92 -104.76 356.3 ,! 348.52 295.2 -102.42 355.64 344.29 309.47 , -106.86 348.8 344.64 302.92 - 1 0 5 . 9 1 355.24 345.69 313.52 -108.33 348.7" ' 340.45"1 304.6'7 "104.2 I ....
, .......................
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|
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.
.
.
.
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
271
MECHANICS OF FRACTURE IN FIBROUS METAL MATRIX COMPOSITES
Bahei-EI-Din, Y.A. and Elrafei, A.M. Structural Engineering Department Cairo University, Giza, Egypt
ABSTRACT An experimental and analytical study of crack-tip plastic deformation fields and fracture of three unidirectionally reinforced metal-matrix composites, B/As FP-Ae203/As and SiC/Ti is described. Behavior of the first two materials with a 6061 aluminum matrix was evaluated at room temperature, while that of the titanium matrix composite was evaluated at 650 ~ The experiments revealed the existence of discrete plastic zones at the notch tip, which cause significant stress redistribution in the notched part before failure. Computed with a modified finite element scheme which incorporates the experimental observations, the stress field suggests that failure of the fracture specimens is controlled by a stress ratio between the maximum axial normal stress in the unnotched ligament and the ultimate strength of the composite. KEYWORDS Fibrous Composites, Fracture, High Temperature, Experiments, Finite Elements. I. INTRODUCTION It is well established that standard or modified fracture mechanics approaches could not in general be used to predict the fracture strength of notched metal-matrix fibrous composite materials [1]. The difficulty lies in the development of inelastic strains which cause substantial stress redistribution and alter the magnitude of the stress concentration factor. The effect of the inelastic response on the fracture behavior of a notched part depends on the yield strength and the strain-hardening rate of the matrix. For example, long plastic zones are found in notched boron-aluminum specimens in the as-fabricated condition [2,3], but not in the overaged T6 condition [4]. While the fracture behavior of this composite system in the latter condition can be estimated from the elastic field, it must be founded on a continuum plasticity theory of the composite material for the former condition. Moreover, the effect of loading rate is significant at high temperature and must be considered in realistic estimates of the fracture load. Attempts to predict the fracture strength of notched polymer and metal matrix composites have centered mainly on the shear lag model with various modifications [5-7]. The present work utilizes the finite element method with feed back from experiments to study the fracture behavior of metal matrix composites. Three unidirectionally reinforced composites are considered, B/Ag, FP-Ag203/Ag and SiC/Ti. Behavior of the first two materials with a 6061 aluminum-matrix was evaluated at room temperature, while that of the
272
Current Advances In Mechanical Design and Production, MDP-7
titanium-matrix composite was evaluated at 650 ~ The experiments on as-fabricated and annealed B/At specimens were conducted by Babel-El-Din et al. [3,4] and revealed the existence of discrete plastic zones at the notch tip, which blunt the tip and cause significant stress redistribution in the notched part before failure. This led to a fracture criterion [8] which is examined here for the FP/At and SiC/Ti specimens. The stress and strain fields in the tested specimens were computed with a modified finite element scheme which incorporates the experimental observations, and the elastic-plastic and/or rate-dependent behavior of the matrix. The analysis predicted long plastic zones of length similar to that observed experimentally in the as-fabricated and annealed B/At composite specimens. The plastic zones computed in the FP/At composite were much shorter, but sufficiently long to blunt the notch tip and modify the local stresses. Slow loading and the low matrix yield strength found at high temperature caused substantial inelastic strains at the notch tip in the SiC/Ti composite as well. In all three composite systems, the computed stress field suggests that failure of the fracture specimens is controlled by a stress ratio between the maximum axial normal stress in the unnotched ligament and the ultimate strength of the composite. The experimental program is described in Section 2. Analytical evaluation of the local stresses in the notched specimens using the finite element method is given in Section 3 together with results for the J-integral and G~ energy release rates and validation of the fracture criterion proposed in [8]. 2. EXPERIMENTAL PROGRAM 2.1 Materials and Testing Procedure The experimental results presented here include three different materials, one is a B/At composite which has been tested earlier by the first author and his coworkers [3,4] and is used here for comparison, and the other two are FP/At and SiC/Ti. All materials were fabricated by diffusion bonding of unidrectionally reinforced plies using hot isostatic pressing at a certain temperature and pressure. The matrix material for both the B/At and FP/At composites is 6061 aluminum, and for the SiC/Ti composite is a Timeta121S titanium. The fiber diameter for the boron and the Sigma SiC filaments is 142 and 100 tim, respectively. The number of plies is six for the B/At composite and four for the SiC/Ti composite, which amounts to a laminate thickness of 1.067 mm and 0.635 mm, respectively. On the other hand, the alumina (FP) fiber has a diameter of only 20 tim. The specimen thickness is 2.5 ram. The fiber volume fraction cf estimated by the manufactures of these materials and verified from micrographs is 0.5, 0.35, and 0.3, for B/At, FP/At and SiC/Ti, respectively. All composite panels were obtained in the as-fabricated condition. Specimens were cut from the composite panels with a diamond saw. The specimen length was the same as the side length of the composite panel; no saw cuts were made perpendicular to the fiber. A center notch was made in each specimen, in the direction perpendicular to the fibers, using the electrostatic discharge machining technique. These notches were typically 0.2-0.4 mm wide. Tables 1-3 show specimen and notch dimensions, where W is the specimen width and 2c is the notch length. Several unnotched specimens were also used for strength measurements. The ends of all specimens were bonded to thick annealed aluminum tabs. The free length between the grips was 203 mm for the B/At, 87 mm for the FP/Al and 127 mm for the SiC/Ti.
Current Advances in Mechanical Design and Production, MDP-7
273
T a b l e 1. C e n t e r - n o t c h e d , B/A~ c o m p o s i t e s p e c i m e n s tested at r o o m t e m p e r a t u r e [3,4]. i=
i
Specimen
W (mm)
AI
25.3
A2 A3 A4 A5 A6 A7
25.3 25.5 25.7 24.4 25.2 25.6
A8 A9 AI0
BI B2 B3 B4
B5 B6 B7 O1 02
'
,
,,
,
t~ (MPa/s)
0.204 I'
37.0
25.3 25.5 25.5
0.708 0.703 0.706
37.0 37.0 37.0
322 329 295
50.8' 50.9 50.6 50.5
0.302 0.301 0.503 0.504
18.5 18.5 18.5 18.5
586 618 455 456
II
18.5 18.5 18.5 37.0 37.0
0.197 0.197 0.194 0.401 0.401 0.402 0.601 0.602 0.602
,
Glig(MPa)
883 . . . . . . . . . .
863 831 642 630 396 454
0.702 0.700 0.701 0.112 0.I 15
,,
Oult(MPa)
(Yult(MPa)
37.0 37.0 37.0 37.0 37.0 37.0
25.4 25.3 25.4 25.3 25.3 25.2 25.4 25.4 25.3
L
i
0.200 0.199 0.395 0.416 0.606 0.592
50.9 50.8 50.9 25.3 25.4
03 04 05 06 07 08 09 OI0 Ol I
,,
2c/W
859
425
602
37.0 37.0 37.0 37.0 37.0 37.0 37.0 37.0 37.0 I
I
1071
862 917
987 1007 963 1041 1033
295 919
769 753 788 568 557 555 368 378 364
1059
840 884 915 919
456
560
370 I
986 i
I
ill ii
1037
958 938 978 948 930 928 922 950 915
770
1075 1070
1103 1108 1003
315
294 302 288 924 914
'
1079 1037 1061 1079 1005 1113
636
'"
~lig (MPa)
1109
958
935
929
T a b l e 2. C e n t e r - n o t c h e d , F P / A / c o m p o s i t e s p e c i m e n s tested at r o o m t e m p e r a t u r e . ii
i
iil
_
Specimen
W (ram)
2c/W
Pl P2 P3 P4 P5 P6 P7 P8
22.') 22.7 22.7 22.7 22.7 22.7 22.7 22.7
0.1 0.1 0.3 0.3 0.5 0.5 0.7 0.7
t~ (MPa/s)
i
i
i
i
i
i
i
i
i
i
i
ii
i
j
i
i
O'uit (MPa)
~ult (MPa)
O'lig(MPa)
~lig (MPa)
316 237 167 211 139 140 88 78
277
351 263 239 301 278 280 293 260
307
.... 33.0 33.0 33.0 33.0 33.0 33.0 33.0 33.0
189 140 83
270 279 277
T a b l e 3. C e n t e r - n o t c h e d , S i C / T i c o m p o s i t e s p e c i m e n s t e s t e d at 650 ~ 9
Specimen ........S i
2c/W
t~ (MPa/s)
18.9
0.096
0.50
560 . . . . . . . . . .
6i9
0.50 0.50 0.50 0.50 0.50 0.50 50.0
366 373 373 260 252 275 402
518 528 528 527 511 556 573
$2 $3
15.2 18.9
0.114 0.102
$4 $5 $6 $7 $8 $9
19. I 19.1 19.1 18.4 18.5 18.8
0.293 0.293 0.294 0.507 0.507 0.505 0.298
q~
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Oult (MPa)
0.50 0.50
i,iii
i
537 523
i
i
~ult (MPa)
(Ylig(MPa)
W (ram)
540
606 582
371 262 ,,,,
402 .... ,,
,
~lig (MPa) 602 525 531 ,,j
573
274
Current Advances in Mechanical Design and Production, MDP- 7
Before testing, each specimen was installed, and symmetrically aligned with flat grips which were then attached to the testing machine. Loading was applied by a 50 kN MTS testing machine, in simple tension, at the rate of about I kN/min for the B/At and FP/At specimens, and 36 kN/min and 0.36 kN/min for the SiC/Ti specimens. All specimens were loaded up to failure, and the ultimate load was recorded. The B/Ag and FP/At specimens were tested at room temperature, while the SiC/Ti specimens were tested at 650 ~ Heating of the SiC/Ti specimens was achieved by a steel susceptor tube that enclosed the specimen and was heated by direct induction using a 5-kW solid-state heating unit. The susceptor tube was 25.4 mm in diameter and 100 mm long. This allowed cooling-water pipes to be installed adjacent to the grips. Temperature measurements and control were performed with thermocouples. A uniform temperature was maintained in the 50.8 mm center zone, while the grips were kept at room temperature.
2.2 Experimental Results Notched specimens of all three materials failed by non self-similar fracture of the unnotched ligaments. Fracture of the B/At and FP/At was accompanied by long splits at the notch tips in the fiber direction. This was preceded by formation of discrete, plastic shear zones at the notch tips which were detected for the B/At specimens by Bahei-EI-Din et al. [3,4] using a bar code technique. Splitting in the fiber direction was not observed in the notched SiC/Ti specimens, but formation of plastic zones at the notch tips is expected to take place prior to fracture as predicted analytically in Section 3. Figure 1 shows a photograph of specimen $7, Table 3, a~er failure. Tables 1-3 list the strength magnitudes found for the fracture specimens. The overall stress at failure ault and the net ligament stress at failure at~g are given for each specimen. Also given in Tables 1-3 is the average ultimate stress Uu~t and average ligament stress UJis at failure for specimens with a constant nominal 2c/W. The results in Table 1 indicate that, in principle, the measured fracture strength is unaffected by the specimen width or by annealing. The loading rate appears to have a small effect on the fracture strength even for the SiC/Ti specimens tested at 650 ~ For a given notch length to specimen width ratio, the measured fracture strength is substantially different for the three tested materials since failure is eventually controlled by the fiber strength. Since deformation of the tested specimens from all three materials had similar features, their fracture strength is best compared on a relative strength scale in which the unnotched strength is the normalizing factor. The measured ultimate strength for the three composite materials listed in Tables 1-3 are normalized by their respective unnotched tensile strength and plotted versus the 2c/W ratio in Fig. 2 for the B/Ag and FP/Ag specimens, and in Fig. 3 for the SiC/Ti specimens. At room temperature, the unnotched tensile strength was measured experimentally at 1659 MPa for the B/Ag composite and 451 MPa for the FP/Ag composite. The ultimate tensile strength for the SiC/Ti composite material tested at 650 ~ is 804 MPa. Apart from experimental scatter,
Current Advances in Mechanical Design and Production, MDP-7 1.0: O.9
~
0.8
'-' ~I
o.7
IZ c
s~
0.5
c ~,~
0.4
~
0.3
m
0.2
N
o.1
7~
o.o
m
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B/AI-F, c~ = 0.5, W = 25.4 mm B/AI-F. cf = 0,5, W = 50,6 mm
A
~,~
275
B/AI-O, Cf = 0.5, W = 25.4 mm
9
FPIAI, c4 = 0.35, W = 23.0 mm
~
- - - - - r = 11 - 2c I W) ~ ~4P~.._ ------- r = 0.57 - 0.56 (2cNV) + 0.43 e ('le (2cNV))
~,~,~,
(correlation coefficient = 0.982)
0.0
0.1
0.2
0.3
Notch
0.4
0.5
0.6
Length~SpecimenWktth,
0.7
0.8
2c/W
Fig. 2. Measured relative strength reduction caused by a center notch in B/Ae and FP/A/specimens. 1.0;
.r
"
o~ C
0.9
-' or)
0.8
~
~
o.6 o.5
.O" ~
0.4
Or)
SiC/Ti, ct = 0.3, W = 19 mm, 650~ ,,,.
or
=~ ~
.~
~"-.....
0.3 0.2
~
o.1
z
o.o o.I
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~.~.
0.5 MPa/s 50 MPaJs
- - - - - r=(1 - 2 c l W ) --..--- r = 0.65 - 0.65 (2c/W) + 0.34 e('14 (2c~) (correlation coefficient = 0.996) ,
I
0.1
,
h
0.2
,,
L
0.3
,
~
0.4
,
I
0.5
,
I
0.6
,
I
0.7
,,
,
,
0.8
Notch Length/Specimen Width, 2c/W
Fig. 3. Measured relative strength reduction caused by a center notch in SiC/Ti specimens.
which is largest for FP/At', the data points in each of Figs. 2 and 3 fall into a narrow range. Moreover, the fracture strength measured for specimen Q1, Table 3, at a high loading rate also falls in the range of the S-specimens tested at a slow loading rate. This suggests that predictions of the relative strength can be derived from a single model of the fracture process, which is a function only of the 2c/W ratio and the material properties. As suggested by Fig. 2, and on a relative strength scale, the fiber material does not play a role in the fracture criterion other than restricting plastic shear deformations at the notch tip to a narrow zone that is parallel to the fiber direction. The dashed line shown in Figs. 2 and 3 indicate the variation of the net ligament strength ratio with 2c/W. The reduction from the net ligament strength is due to stress concentrations caused by the notches. Specifically, plastic zones emanating from the notch tips in the fiber direction induce additional local stresses and cause the reductions in overall strength of the notched specimens from the net ligament strength values [4,8].
276
Current Advances in Mechanical Design and Production, MDP-7
3. ANALYSIS
3.1 Finite Element Model The local stresses in the tension specimens containing a center notch were evaluated with a modified finite element analysis which recognizes the effect of the discrete plastic zones present at the notch tips on the local stresses [4,8]. The finite element domain is shown in Fig. 4. It represents one quadrant of the notched specimen. Displacement boundary conditions derived from symmetry of the specimen geometry and the applied load with respect to the planes x l = 0 and x2 = 0 were applied together with a uniform displacement condition at the boundary x2 = L/2, where L is the specimen length.
l
1~
uniform
displacement
"'
,)
F,
L/2
g ..J a,
t~,:
-
.~ .f
Xl
The finite element solution was simplified by Fig. 4. Finite element domain and limiting plastic or viscoplastic deformation in boundary conditions of centerthe domain to one row of elements which were notched specimens. designated to represent the plastic shear zones. All other elements were assumed to remain elastic. This was suggested by the existence of two plastic deformation modes in metal matrix composites, one is matrix-dominated and the other is fiber-dominated [9]. Stresses outside the plastic zone promote the fiber-dominated mode in which the inelastic strains are negligible compared to the elastic strains. This was confirmed by a nonlinear finite element solution of the whole domain for one B/Ag specimen and one SiC/Ti specimen, which provided local stresses that are identical with the stresses computed for the modified domain of Fig. 4. Finite element solution of the B / A t and FP/At specimens was obtained by the ABAQUS program [10]. In the elastic range, the composite was regarded as a homogeneous, transversely isotropic medium. Elastic-plastic response of the plastic zone elements was specified with the help of Hill's anisotropic yield criterion. A rail shear test provided the stress-strain response of the composite materials. The initial yield stress in shear was measured at 40 MPa and 34 MPa for the B/At composite in the F and O condition, respectively, and 49 MPa for the FP/At composite. Finite element analysis of the SiC/Ti specimens on the other hand was performed with the VISCOPAC program [ 11 ] which derives viscoplastic response of the plastic zone elements from a rate-dependent, micromechanical model of fibrous composites. Properties of the Timetal 21S matrix measured by Dvorak et al. [12] at 650 ~ provide initial yield stress in shear of 26 MPa. Refinement of the stress field computed at the notch tip followed two slightly different procedures. In the SiC/Ti specimens, a refined mesh was selected in the vicinity of the notch tip such that the element dimension in direction of the xl-axis is in the order of the width Wry of a representative volume element (RVE) of the composite. The latter was evaluated from a hexagonal composite cylinder containing a fiber of diameter df as Wry = df ~/(2~t/3~/(3)cf). For df = 0.1 mm and cf = 0.3, we find Wr~ = 0.2 ram. Actual meshing of the domain in Fig. 4 provided elements of width 0.25 mm at the notch tip. The finite element mesh was also refined in the fiber axial direction particularly in the vicinity of the notch tip.
Current Advances in Mechanical Design and Production, MDP-7
277
Calculation of the local stresses in the B/Ae and the FP/Ag specimens on the other hand followed the procedure described by Bahei-EI-Din et al. [4,8] which combines the numerical solution provided by the finite element method, and an analytical solution of the notch tip stress field. The latter is obtained from solution of an elastic half-space under boundary shear tractions equal to the shear flow stress generated in the plastic zones. The half space represents the unnotched ligament ahead of the notch tip, and the shear tractions are applied at the boundary xl = c, starting from the notch tip and extend in the x2-direction for a predetermined distance. Details of this method are provided in references [4,8]. In this case, however, the finite element domain shown in Fig. 4 is modified by evacuating the plastic zone elements within the predetermined distance at which the shear tractions are applied in the analytical solution. The local stresses are then determined by superposition of the numerical and analytical solutions. 3.2 Finite Element Results The length of the plastic zones R was determined by identifying the first element in the plastic zone region, Fig. 4, for which the shear strain is smaller than the elastic limit. The latter was evaluated from the shear rail test for B/Ag and FP/A~, and estimated from the Mori-Tanaka [13] micromechanical model for SiC/Ti. Bahei-EI-Din et al. [3,8] detected long plastic zones in the B/Ae specimens using the bar code technique, and computed the plastic zone length R in the order of 3-7 times half the notch length c. Similar plastic zones were computed for the SiC/Ti specimens with the R/c ratio of 3-5. The plastic zones computed for the FP/Ae specimens however were much smaller with R/c in the order of 0.4 to 2. In any case, the computed energy release rate for the tested specimens indicates that the plastic zones were sufficiently long to blunt the notch and modify the elastic stress field.
It was shown by Bahei-EI-Din et al. [4,8] that the elastic energy release rate GI computed for the B/Ae fracture specimens at the measured ultimate load is different from the J-integral, and neither quantity can be considered a material property since it varies with the 2c/W ratio. The difference between G! and the J-integral is attributed to the plastic zones which modify the elastic field in the notched specimens. Our calculations for Gi and the J-integral for the SiC/Ti specimens are shown in Fig. 5. Here too, the plastic zones result in a J-integral that is quite
20 A E ~ Z v
~ ID
~ llJ
-~
18
SiC/Ti, cf = 0.3
650~
0
J-integral ,
alh
16 14 12 m
10
a""
8
B
4 uJ
9
r'l 0
C
o.s.P~/. 50 UPa/s
2 0
,
0.0
|
0.1
J
I
0.2
,
I
I
0.3
0.4
,
|
O.S
9
9
0.6
2c/W
Fig. 5. J-integral and elastic energy release rate G! computed at fracture stress for SiC/Ti specimens.
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Current Advances in Mechanical Design and Production, MDP-7
different from the elastic energy release rate. The J-integral was not computed for the FP/Ar specimens, but the Gt values, in the order of 1.4 to 3.7 N/mm, were dependent on the 2c/W ratio. Pending verification with more experiments, it can be concluded that the critical energy release rate of the notched unidirectional metal matrix composites tested here is geometry dependent, and is substantially different from the elastic energy release rate. Bahei-EI-Din et al. [8] examined the local stresses computed at the fracture load for the center-notched B/At' specimens listed in Table 1 and suggested that the onset of fracture is controlled by a critical stress ratio of the normal stress in the unnotched ligament at the notch tip and the ultimate strength of the composite material. When the local normal stress was computed at the notch tip as an average over the width Wryof a representative volume of the composite material which contains a single fiber, the critical stress ratio was found to be 1.0, and independent of the speeimen's width. Moreover, the same critical stress ratio was computed for both the as-fabricated and annealed specimens, and for B/At' specimens with a center hole and edge notches [4]. This fracture criterion is expected to be valid only for unidirectional composites with a low yield stress in shear such that discrete plastic shear zones can be formed and blunt the notch tip. For example, Bahei-EI-Din et al. [4] found that the proposed fracture criterion was not applicable for 6061-T6 At'/B specimens with shear yield stress of 74 MPa and flow stress of 188 MPa. The fracture criterion was examined for the FP/At' and SiC/Ti specimens of Tables 2 and 3. Figure 6 shows the local normal stress computed in the SiC/Ti specimens at the fracture loads and normalized by the unnotehed strength of 804 MPa. Two groups of points are shown for these specimens, one computed with the local stresses averaged over a representative volume of width Wry = 2.5 dr, and the other for Wry= 8.0 dr, where dr = 100 ~tm. This result suggests that although the magnitude of the critical stress ratio varies with the selected RVE width, it is a constant, within an experimental scatter, for 2e/W > 0.3. The lower values found at 2c/W = 0.1 suggest that the elastic field is dominant as also indicated by the close values of the J-integral and GI, Fig. 5, for this crack length to specimen width ratio. Also shown in Fig. 6 is the ligament stress, normalized by the unnotched ultimate strength. The difference between the total local stress and the ligament stress is the stress caused by the plastic zone shear stress [4,8]. Figure 7 shows the local normal stress computed at the fracture load for the FP/At' specimens normalized by the unnotched strength of 451 MPa. Here, the stress ratio is plotted against the width of the representative volume over which the local stress is averaged, normalized by the fiber diameter. For a selected Wr, value, the stress ratios corresponding to the fracture load fall within a narrow band that is consistent with the experimental scatter, and suggest that fracture in the FP/At' specimens is also controlled by a critical stress ratio. The implication is that the fracture strength of notched, unidirectional composites which develop significant plastic deformations at the notch tip can be predicted analytically by comparing the local stresses averaged over a selected representative volume to a critical value that is considered a material property. The latter is evaluated analytically for a notched specimen of the material at the overall fracture stress determined experimentally.
Current Advances in Mechanical Design and Production, MDP-7
SiC/l'i, cf = 0.3
2,50
~
total stress, Wn/df = 2.5
650Oc
2.25
0
279
.00
~
1.75
o9 -~
1.25
total stress, Wrvldf = 8.0
1.50
~
-
-
,
1.00
0
z --
0.75
o .J
0.50
~'-"-'-'~-0
-
ligament stress
[ m [ 2 O v ' ' O s M O I ~5/0$M ' ] O "~ J
0.25 0.00 0.0
1.,i
,
l
0.1
0.2
.
|
,
,,
0.3
I
i
I
0.4
,
,
* 0.6
0.5
2c I W
Fig. 6. Local normal stress ratio computed at fracture stress for SiC/Ti specimens. 2.2
FP/AI, c,f = 0.35
2.0 0
m n,
1.8 1.6
u) m
1.2
m
1.0
o
0.8
1.4
Z "~ o _j
0.6 0.4 0.2 0.0
r
oA~
0
J
|
I0
20
.
i
_
30
,
40
,
9
~n
o~
i
50
,
,
60
,
r
,
70
,
|
80
,
I
90
,
*
100
,
|
110
RVE Width / Fiber Diameter
Fig. 7. Local normal stress ratio computed at fracture stress for FP/AI specimens.
4. C O N C L U S I O N S
The results confirm that the onset of failure in center-notched, unidirectionally reinforced B/Ag and FP/Ae tested at room temperature, and SiC/Ti tested at a high temperature is controlled by a critical stress ratio of the normal stress in the urmotched ligament and the ultimate strength of the composite material. They also confirm that the fracture process is not self-similar; long plastic shear zones form at the notch tip before fracture but no such zones appear along the fracture surface. Formation of the plastic zones blunts the notch tip and modifies the local stresses which can be computed analytically by a modified finite element scheme. This is likely to take place in fibrous composites with a low yield strength, which favor matrix-dominated plasticity such as those reinforced by fibers of large longitudinal shear stiffness such as the boron, silicon-carbide and alumina (FP) fibers.
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Current Advances in Mechanical Design and Production, MDP-7
ACKNOWLEDGEMENTS The experiments and ABAQUS computations were conducted by YAB while visiting at Rensselaer Polytechnic Institute. Financial support from the Office of Naval Research and the Air Force Office of Scientific Research is acknowledged. REFERENCES 1. Reedy, E.D., "On the Specimen Dependence of Unidirectional Boron/Aluminum Fracture Toughness", Journal of Composite Materials Supplement, Vol. 14, pp. 118-131 (1980). 2. Post D., Czamek R., Joh, D. and Guo, Y., "Deformation of a Metal Matrix Tensile Coupon with a Central Slot: An Experimental Study", Journal of Composites Technology and Research, Vol. 9, pp. 3-9 (1977). 3. Dvorak, G.J., Bahei-EI-Din, Y.A. and Bank, L.C., "Fracture of Fibrous Metal Matrix Composites - I. Experimental Results", Engineering Fracture Mechanics, Vol. 34, pp. 87-104 (1989). 4. Bahei-EI-Din, Y.A. and Nigam, H., "Fracture of Fibrous Metal Matrix CompositesIII. Effect of Imperfection Geometry and Heat Treatment", Engineering Fracture Mechanics, Vol. 37, pp. 1207-1232 (1990). 5. Naim, J.A., "Fracture Mechanics of Unidirectional Composites", Journal of Reinforced Plastics and Composites, Vol. 9, pp. 91-101 (1990). 6. Dharani, L.R., Venkatakrishnaiah, S. and Dupuy, R.A., "Effect of Specimen Width on the Fracture of Unidirectional Metal Matrix Composites", Composites Science and Technology, Vol. 45, pp. 117-123 (1992). 7. Tsai, W.T. and Dharani, L.R., "Non-Self-Similar Fiber Fracture in Unidirectional Composites", Engineering Fracture Mechanics, Vol. 44, pp. 43-49 (1993). 8. Bahei-EI-Din, Y.A., Dvorak, G.J. and Wu, J.F., "Fracture of Fibrous Metal Matrix Composites - II. Modeling and Numerical Analysis", Engineering Fracture Mechanics, Vol. 34, pp. 105-123 (1989). 9. Dvorak, G.J. and Bahei-EI-Din, Y.A., "A Bimodal Plasticity Theory of Fibrous Composite Materials", Acta Mechanica, Vol. 69, pp. 219-241 (1987). 10. ABAQUS Finite Element Program, HKS, Inc., Rhode Island. 11. Bahei-EI-Din, Y.A., "Finite Element Analysis of Viscoplastic Composite Materials and Structures", Mechanics of Composite Materials and Structures, Vol. 3, pp. 1-28 (1996). 12. Dvorak, G.J., Nigam, H. and Bahei-EI-Din, Y.A., "Isothermal Fatigue of Sigma/Timetal 21S Laminates: I. Experimental Results", Mechanics of Composite Materials and Structures, Vol. 4, pp. 113-130 (1997). 13. Moil, T. and Tanaka, K., "Average Stress in Matrix and Average Elastic Energy of Materials with MisfiRing Inclusions", Acta. Metall., Vol. 21, pp. 571-574 (1973).
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
281
ON SIMULTANEOUS FAILURE OF CROSS-PLY AND ANGLE-PLY COMPOSITE LAMINATES Khalil, M. *, Bakhiet, E. * and EI-Zoghby, A. **
* Assistant Professor, Aircraft Engineering Department Institute of Aviation Engineering and Technology, Embaba Aerodrome, Giza, Egypt. ** Associate Professor, Mechanical Design and Production Department Faculty of Engineering, Cairo University, Giza 12316-Egypt.
ABSTRACT This paper presents a particular class of bidirectional laminated composites where all layers fail simultaneously. The loading conditions, laminate configurations and the corresponding failure envelopes for simultaneous failure are obtained for cross-ply and angle-ply laminates under in-plane loads. The stiffness and strength characteristics of the laminates are discussed based upon the concept of lamination parameters. The optimal laminate configurations to maximize the in-plane strength are also obtained using the Tsai-Wu criterion as a first ply failure strength criterion. The failure surfaces of the studied laminates in stress space and in strain space are drawn and the optimal configurations are then determined. It is shown that the existance of laminate configurations for simultaneous failure depends mainly on the loading conditions. KEYWORDS Cross-ply Laminates, Angle-ply Laminates, Simultaneous Failure, In-plane loads, Lamination Parameters, Failure Strength. 1. INTRODUCTION Composite materials have become an increasingly attractive alternative to metals for structural applications where high strength-to-weight and stiffness-to-weight ratios are required. These materials are strong, durable and damage tolerant. They meet design and certification requirements and offer significant weight advantages. This work mainly concentrates on the strength and stiffness of symmetric bidirectional composites, namely cross-ply and angle-ply laminates, which are orthotropic and hence easy to handle. When used in the fight way, they may offer weight advantages over quasi-isotropic lay-up [ 1]. The failure strength characteristics of these laminates depend highly on the fiber direction of the individual plies. Thus, it is important to tailor laminate configurations for enhancing the strength of composite structures. For the analysis of the required stiffness and strength of laminates, the use of the lamination parameters is convenient. Some work on the optimum design of laminates using lamination
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parameters have been made, such as the evaluation method of the in-plane stiffness [2], the stability optimization of laminated composite plates [3], the strength optimization of laminates [4] and the optimum design of laminated cylindrical pressure vessels under both stiffness and strength constraints [5]. The present paper first shows the failure surfaces of cross-ply and angle-ply laminates using the Tsai-Wu failure criterion. The optimal laminate configurations which give the highest failure strength of the studied laminates are determined. Based on the lamination parameters [4-6], the loading conditions and the laminate configurations for simultaneous failure are then derived under in-plane loads. The corresponding simultaneous failure envelopes for Carbon / Epoxy composite laminates are drawn and the results are finally analyzed. 2. STRENGTH OF BIDIRECTIONAL LAMINATES 2.1 Failure Criteria The quadratic failure criterion of Tsai-Wu is based on firing an ellipse to the experimental failure strengths of a unidirectional lamina. The form of the equation accounts for interaction between the stresses causing failure. As in most laminate failure criteria, each ply in the laminate must be interrogated separately in order to determine if failure has occured. In this paper, first-ply failure is adopted in contrast to a progrssive failure model [7]. The axis system convention adopted here is shown in Fig. 1. The x, y and s subscripts denote properties in the ply axis system, and 1, 2 and 6 denote properties in the laminate axis system.
Fig. 1. Laminate and ply axis systems The quadratic failure criterion in stress-space takes the form: F~.cri + F#oio j
=
1
( i , j = 1, 2, 6)
(1)
where the strength coefficients F# and F i are given in Ref. [8]. On the other hand, stating the failure criteria in terms of strain is convenient. In strain space the failure envelopes stays fixed even if the ply ratios of the laminate are changed. The strain limits of a ply are independent of the laminate stiffness [7]. The failure criterion in strain-space can be rewritten as:
Current Advances in Mechanical Design and Production, M D P - 7
Gis i § Gijsis j - 1
( i,j = 1, 2, 6 )
,
283
(2)
where the G's are material constants determined by the reduced stiffness Qij [6]. Introducing the strength ratio, R, for a unidirectional composite: R =
cr allowable = 8 allowable
applied
(3)
F,applied
The failure criterion can be solved for R
[co , j]R
(4)
Because of the quadratic form, there will be two solutions R + and R- corresponding to strain vectors piercing the failure envelope in opposite directions. The in-plane strength of a unidirectional laminate will have multiple strength ratios; one set (R + and R- ) for each ply orientation. The ply with the lowest strength ratio fails first. Two factors control the ply failures in a laminate: the in-plane compliance and the specific ply orientation. There is an envelope for each ply orientation; the first ply failure (FPF) envelope is the innermost boundary of the superposed ply failure surfaces. The plies with higher stress ratios will fail later, when the externally applied stress is increased. This successive ply failure progresses until the last ply failure (LPF) occurs [6]. Figure 2 shows the failure surfaces in strain space for T300/5208 graphite/epoxy unidirectional composites [4]. (2' 10- 3 10
-30
10-3
o
Fig. 2. Failure envelopes in the strain space for T300/5208 composites 2.2 Failure Surfaces of Cross-Ply and Angle-Ply Laminates The failure envelopes in normal stress resultant space - or the principal stress plane- for various cross-ply configurations are shown in Fig. 3. Basically quadrants I (tension-tension) and III (compression-compression) are similar, and there is similarity between II (compression-tension) and IV (tension-compression).
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Current Advances in Mechanical Design and Production, MDP-7
For clarity purpose, the [03 / 902 ] laminate is taken as an example, its envelopes are taken in bold lines, the upper ellipse presenting the 90 ~ plies, the lower one the 0 ~ plies. The shaded area is the FPF envelope [1].
NU/h 3OOO0]
[NLARG(MINTOF [03/902]ENVF..LOP($.OUAORANT I
r Nw/h
1 O0 0
Fig. 3. [0 m /90 n ] Cross-ply failure envelopes in stress space, m/n ratio in 20%-steps
If simultaneous failure of all plies is an optimum condition for a laminate design, it is possible to achieve it only in the quadrants I and Ill where the two failure envelopes intersect. Of all possible stress ratio vectors in both quadrants, those to the intersecting points are the longest, which means an optimum condition for laminate design. There is no coincidence of failure envelopes in quadrants II and IV and consequently no simultaneous failure. But as before the optimum design is characterized by the longest stress ratio vectors. For a given stress ratio, the cross-ply arrangement whose inner failure envelope is the most distant from the origin of the coordinates is the optimum. There is no FPF for angle-ply laminates as all plies must fail simultaneously with the absence of shear. The failure envelopes are narrow, making angle-ply configurations rather susceptible to failure for small changes in stress ratio or ply angle. An optimum ply angle is found in quadrants I and III. However, angle-ply laminates behave poorly in quadrants II and IV as compared to cross-ply laminates [1]. 3. SIMULTANEOUS FAILURE LAMINATES 3.1 Strain Condition for Simultaneous Failure
We consider the laminated composites under uniform in-plane strains (r axis system convention shown in Fig. 1, the strain transformation in double angle function for the layer with the fiber direction 0 can be expressed as [6]:
Current Advances in Mechanical Design and Production, MDP-7
t 6 x t II cos20 % = 1 -cos20 cs 0 -2sin20
2sin20 l i p ] -2sin20/J q 2cos20.][rJ
285
(5)
where p=(61 + e 2 ) / 2 (6)
q = ( 6 ! - ~r2)/2 r = 6 6/2
It is shown that the strain components are independent of the fiber direction 0 when q = 0 (61 = 62 ) and r = 0 (e 6 = 0). Then, the strain and stress components become: ~'! =~162=6"x =~
,
O'x =(Qxx +Qyy)o~x
,
66 = 0
O"r =(Qxy +Q.yy)oex
(7)
o"s = O0 where Qij is the reduced stiffness [6]. It can be seen from Eq. (7) that all layers fail simultaneously for the laminate satisfying the relations of q = r = 0. Such laminate can be considered as possessing the configuration for simultaneous failure. 3.2 Condition of Lamination Parameters for Simultaneous Failure
We consider the laminated composites under in-plane loads (N t ,N2,N6) as shown in Fig. 1. Transforming the original coordinates (1-2) into the principal coordinates (x-y)where the shear component N 6 vanishes, the principal loads become [4]:
NI =(NI +N2)/2 + [(N,-N2) 2/4+N~I '2 Nil =(N, +N2)/2 - [(N,-N2) 2/4+N261 '2
(8)
where the angle, tx, between the principal and the original coordinates is defined by: tan2a = 2N6 / ( N I - N 2)
(9)
Assuming the laminate to be symmetric with respect to the mid-plane, the bending-extension coupling vanishes, and the in-plane stress-strain relations of the laminate in the principal coordinates will be given by:
t = At2 A22 A26 c2
AI6 A26 A66 ~'6
(1o)
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Current Advances in Mechanical Design and Production, MDP-7
The in-plane stiffness in Eq. (10) can be rewritten as follows:
All AI2 ,'122 ,=h d6~
"1 o
~:1 0
1
-
0
,A2~
~1 0
~X2 -~2 ~:2
-~2
0 O" Ui ! 0 U2 0
0
0
1
(11)
U4
0 ~212
~:4
0 0
0 ~3/2
-~4
0 O
U5
where h is the thickness of the laminate and U i are the stiffness invariants [4, 5]. For the symmetric laminate, the lamination parameters ~:i are given by: 1 h/2
n
j" = -h -hi2c~176
1 h/2
n
: j'
1 h/2
r =~
n
1 h/2
I cos2 2O(z)dz = Z h , cos 2 20, i=l
r =~
-h/2
Ic~ 2O(z)dz = .__,hi cos2 20 i
-hi2
i=!
where h i and ~i are the thickness and orientation of the ith layer, and the total laminate tl
thickness h is given by h = ~ h i . i=l For the balanced laminates where + 0 and - 0 layers have the same thickness, we can obtain the relations of ~3 = ~4 = 0 , and hence, r = 0 (c 6 = 0). Then Eq. (10) becomes:
{N,}:h[U,+U,+U, No
UI + U 4 - U2 ~t
U4 - U i +U2
~!- 2U3 ~2 Jl.qJ
(13)
The lamination parameters, ~1 and ~2 govern the in-plane stiffness characteristics of the symmetric balanced laminate. It can be shown that: - I _ < ~ , _
,
~2 <~J2 _
(14)
It is also shown from Eq. (13) that the lamination parameters satisfying the relation of q = 0 (e I = 6 2) are given as follows: ~! - UI + U4 N# - N #
U2
(15)
Nt + N n
and ~2 is an arbitrary constant; ~2 _<~2 <-1. Since - 1 _<~1 -<1 from Eq. (14), the loading condition for which the simultaneous failure laminate configurations exist becomes:
Current Advances in Mechanical Design and Production, MDP-7
U2
<_ N t - Ntt <_
UI +U 4
N t + N#
U2 U ! +U 4
287 (16)
For the loading condition to satisfy Eq. (16), the lamination parameters for simultaneous failure are given by Eq. (14). Then, the relations between the failure strength and the failure strain P become:
{N I}= I= {v)rt2U (+ IU N{pN 4(')/+ tN }h)nt
an
N#/(N t+Ntt
(17)
where x + G y ) 2 +4(Gxx +2Gxy + G ~ )
-(G x +Gy)•
P=
2(Gxa + 2Gxy + Gyy )
(18)
3.3 Laminate Configurations for Simultaneous Failure Here, the laminate configurations corresponding to the lamination parameters for simultaneous failure are obtained. In the derivation of Eq. (13), the laminate is assumed to be symmetric and balanced in the principal coordinates x-y. Hence, 2j3 =~:4 =0 and the lamination parameters (~l, ~2) for the n-layered laminate are obtained from Eq. (12)as follows [4, 5]: cos20i
(19)
n
where the total thickness h = ~ h i = 1. i=!
Figure 4 shows the relation between the lamination parameters (~l, ~2) and the n-layered laminate on the ~:! - ~:2 plane. A point A, on the parabola ~Jl - ~22 corresponds to the layer with the fiber angle + 0 i . For example, the point B 1 corresponds to 0 ~ layer, the point C 1 to 90 ~ layer and the point 0 to • 45 ~ layer. A point P with the lamination parameters (~:1, ~2 ) is expressed as a linear combination of the vectors a i = (cos 2Oi, cos 2 20 i ). If the vectors a i and the lamination parameters (~l, ~2 ) are given, the thickness of the layer, h i , is determined by Eq. (19). For the case of symmetric and balanced laminate (+ 0 A / + Oe)s with two different angles, we have four degrees of freedom (OA, OB, hA, hB). A point ,4 = (cos20 a , cos 2 20A) can be selected arbitrarily on the parabola ~:l - ~=2 then the other parameters can be determined as:
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Current Advances in Mechanical Design and Production, MDP-7
=
with
(20)
hA ~B_~A
'
hB =hA-I
~A = 2 COS2.0A
,
~B - ~2 --~l~A
(21)
4- A ~2
.-i, B,
I[ s
s
p
S
-P.. / ....
SJ S
~ C2
-I
0
~x
I
Fig. 4. Characteristics of the lamination parameters (~l, ~2 ) The laminate configurations are then transformed into the original coordinates: (22) 4. SIMULTANEOUS FAILURE ENVELOPES As it has quadrants envelopes failure of
been indicated in section (2.2), simultaneous failure of all plies exists only in I (tension-tension) and III (compression-compression) where the two failure intersect. Following the procedure stated above, we have studied the simultaneous balanced and synunetric cross-ply laminates (0/90) s and angle-ply laminates
(+_0A /+_OB) s made of carbon/epoxy. For cross-ply laminates, Figs. 5 and 6 show the simultaneous failure envelopes related to the percentage of plies and the final ply orientation respectively. For angle-ply laminates, the effect of the orientation angle _+0 A ( 0 B is fixed to 30 ~ on the percentage of layers and final ply orietations is shown in Figs. 7 to 9. From these figures we can conclude that: *-
The optimal configurations corresponding to simultaneous failure of all plies are obtained only for the loading conditions defining this type of failure. *- For positive values of N 2 , the percentage of 0 ~ layer is higher than that of 90 ~ layer (except the case where N I = N 2 - 1 and the two layers have the same thickness). These results are completely inversed for negative values, and are correct also for the failure strengths crT and cr11.
Current Advances in Mechanical Design and Production, MDP-7 0-
289
The simultaneous failure configurations of 0 ~ layer (thickness, orientation, and failure strength) are related to that of 90 ~ layer. For example, for N 6 = 0.2 we have:
ho~
= h9oo(quad.tll)
h90~162
= h O~ quad./tt )
h9oo(quad.t)
= 1-
hoo(quad./)
13lf O~quad.l) -- t2gf O~quad.ll/ ) ,otf90~
=
t~f 90~
90 ~ + afO~
= ogf 90~
+ 2 Glf O~
and O'I( quad.! ) -" O'll(quad./)
0-
O'H( quad.lll )
= O'l(quad./l/)
For small orientation angle _+ OA 9 - the percentage of layers corresponding to simultaneous failure is high and the final orientation angle is small. - the area of the failure envelope is greater than that corresponding to high orientation angles, which means that the probability of simultaneous failer, for a given loading condition, is greater for small orientation angles.
[ i .......Z,Ve ope,ho. +l
% o f Plies 100,+--,.,_ ...............+....................+~.................+..................+........G , ,
8o
~+"
'~:"-" ~.I/"
•
t.,+,,.,+
Envelope ;_.+- :: h+, 9o01 ,
......................................
//:~.
60:70
5o:
I.....N,+: , i : o l
40, 30 :t-.-
0 "I'--72 ,ill il lO
-1.0
-0,8
+
+ 7 ........................ -0.6 -0.4 --0.2 0,0
0,2
0,4
0.6
0.8
1.0 N2
Fig. 5. Simultaneous failure envelopes related to the percentage of 0 ~ layer and 90 ~ layer of cross-ply laminate of carbon/epoxy
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Current Advances in Mechanical Design and Production, MDP-7
Fig. 7. Simultaneous failure envelopes related to the percentage of 0~ layer of angle-ply laminate (-30/30)s of carbon/epoxy
Current Advances in Mechanical Design and Production, MDP-7
Fig. 9. Effect of the angle ( 0 ) on the final orientation envelopes of angle-ply laminate (+_ 8 A / +_30) s of carbon/epoxy
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Current Advances in Mechanical Design and Production, MDP-7
5. CONCLUSIONS The laminate configurations for simultaneous failure have been examined for cross-ply and angle-ply composite laminates under in-plane loads. The concept of lamination parameters is used for the analysis of the stiffness and strength characteristics of the laminates. The following conclusions are pertinent: 1.
2. 3. 4.
5.
6.
7.
The existence of laminate configurations for simultaneous failure depends on the loading conditions. This type of failure exists only for loads of quadrant I (tension-tension) and quadrant III (compression-compression) of the failure envelopes. There is no coincidence of failure envelopes in quadrants II and IV and consequently no simultaneous failure. The failure strength characteristics of the studied laminates depend highly on the directions of the individual plies. The optimum design of composite laminates is characterized by the longest stress ratio vectors. For a given stress ratio, the cross-ply arrangement whose inner failure envelope is the most distant from the origin of the coordinate is the optimum. 5. For simultaneous failure of cross-ply laminates, the positive values of N 2 causes the percentage of plies, and the failure strength of 0 ~ layer to be higher than that of 90 ~ The opposite is also correct for negative values of N 2 . There is no FPF for angle-ply laminates as all plies must fail simultaneously with the absence of shear. Their failure envelopes are narrow, making their configurations rather susceptible to failure for small changes in stress ratio or ply angle. The probability of simultaneous failure of angle-ply laminates, for a given loading condition, is greater for small orientation angles.
REFERENCES 1. Wurzel, D. P., "On the Design and Optimization of Bidirectional Composites", Journal of Reinforced Plastics and Composites, Vol. 2, pp. 178-196, (1983). 2. Miki, M., "Material Design of Composite Laminates with Required In-Plane Elastic Properties", Proc. ICCM IV, pp. 1725-1731, (1982). 3. Fukunaga, H. and Hirano, Y., " Stability Optimization of Laminated Composite Plates under In-Plane Loads", Proc. ICCM IV, pp. 565-572, (1982). 4. Fukunaga, H. and Chou, T. W.," On Laminate Configurations for Simultaneous Failure", Journal of Composite Materials, Vol. 22, pp. 271-286, (1988). 5. Fukunaga, H. and Chou, T. W., '~ Simplified Design Techniques for Laminated Cylindrical Pressure Vessels under Stiffness and Strength Constraints", Journal of Composite Materials, Vol. 22, pp. 1156-1169, (1988). 6. Tsai, S.W., and Hahn, H.T., " Introduction to Composite Materials", Teclmomic Publishing Company, Inc., (1980). 7. Flanagan, G. and Palazotto, A. N., " Composite Laminate Optimization for Microcomputers", Computers and Structures, Vol. 22, No. 6, pp. 995-1009, (1986). 8. Kim, R. Y. and Crasto, A. S., "Failure of Carbon Fiber-Reinforced Epoxy Composites under Combined Loading", Proceedings of the Ninth International Conference on Composite Materials, Vol. 5 pp. 15-22, (1993).
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
293
MODELLING OF SHAPE ROLLING USING THREE-DIMENSIONAL FINITE ELEMENT TECHNIQUE
Abo-Elkhier, M. Associate Professor, Dept. of Prod. Eng. and Mech. Design, Faculty of Engineering, Menoufia University, Shebin El-kom, Egypt, 32511
ABSTRACT This paper presents details of a quasi three-dimensional finite element formulation for the analysis of shape rolling. A computer program FESHRL is developed based on the incremental updated .Lagrangian formulation. A mathematical representation for the roll surface profile is developed The material behavior at elevated temperature is assumed to be elastic-viscoplastic To verify the capability of the developed program, a square-to-oval isothermal bar rolling process is simulated. The predicted cross-sectional shape at the exit plane is compared with other finite element analysis reported in the literature. The detailed results on metal flow characteristics, the distributions of strain, strain-rate and pressure, and 888finally contact area are presented and discussed. KEYWORDS Finite Element Modelling, Shape Rolling, Metal Flow Characteristics, Square-to-oval Rolling Pass. 1. INTRODUCTION Requirements for the quality and the properties of the rolled products have been increased continuously, It has, therefore, become increasingly important to understand metal flow characteristics and development of plastic deformation in shape rolling. Metal flow has a significant effect on the material behavior and the microstructure development in the deformation zone. Several experimental investigations of shape rolling have been published in the literature. Due to the difficulty of developing theoretical analysis for three-dimensional severe plastic deformation as well as the continuous changes in boundary conditions, most efforts made so far on this subject are either empirical or experimental. As large numbers of process variables involved and the difficulty in investigating metal flow in shape rolling, the use of numerical techniques as an effective tool becomes extremely attractive. The finite element technique has been extensively used for the analysis of flat or strip rolling processes.[ 1-7] In case of shape rolling, some interesting investigations have been made. Most of these studies apply what is called generalized plane strain condition, in which a uniform strain over the cross-section is assumed in the longitudinal direction [8-10]. This assumption results in constant component of strain and strain-rate in the rolling direction and in zero
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component of shear strain related to that direction. Detailed three-dimensional finite element simulations of shape rolling are presented by Park and Oh [ 11], and Hartley et al.[ 12] but its practical applications is limited due to large computation time and complexity of data preparation. In this investigation, a three-dimensional finite element computer program FESHRL is developed for the analysis of shape rolling based on the incremental updated Lagrangian formulation. The formulation takes into account the geometrical ( finite strain) and the material nonlinearities. A mathematical representation for the roll surface profile is developed. The program is structured in such a way that different processes can be analyzed by changing the geometric conditions describing the tooling and specifying the appropriate material behavior. Material behavior at elevated temperature is assumed to be elasto-viscoplastic. To verify the predictive capability of the developed program, a square-to-oval isothermal bar rolling process is analyzed. The detailed results which is not currently available from experiments on metal flow characteristics, the distribution of the strain, the strain-rate and the normal pressure and finally the contact area are presented and discussed. 2. PROBLEM FORMULATION The developed finite element computer program is based on the updated Lagrangian formulation, in which the fiJture state of deformation is referred to the present one (last calculated one). The solution is carded out incrementally. In each increment all the field variables such as; stress and strain tensors are transformed to the current state.. It is worth mentioning that, analysis of metal forming processes which operate under steady-state may be facilitated by employing the Eulerian fixed mesh technique as pointed out by the author [2]. In shape rolling, however, there is severe three-dimensional plastic deformation as well as continuous changes in element boundaries, which may violate the nature of the Eulerian fixed mesh technique. Utilizing fully nonlinear kinematic relations within a linear increment yields the final incremental equilibrium equations. Only the final form will be outlined here and details of the derivations and finite element incrementation procedures can be found in earlier publications [13,14]. The final form of the incremental equilibrium equations may be expressed in the form: ap + "'ap
"~ap
where
K(')
~P=
Sv~D~uO~xs
~x t
dV
(2)
is the usual small displacement, or the incremental stiffness matrix x'(2)=
j
So
0~,p 0~,a
dV
(3)
is the initial stress or the tangent stiffness matrix, ap,~
P K(4) "P = ~S~ o%
jdA
(4)
is the load correction matrix, AU~ is the increment of the nodal displacement vector, AR~ is the increment of the load vector. ~r are the usual finite clement shape functions D~jklis the
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fourth order Hookean material tensor, S Ois the second Poila-Kirchhoff stress tensor, V is the current volume and finally A in the loaded area. Considering two consecutive states of deformation S 1 and $2, the stress and the strain tensors referred to the present state S 1 ( last calculated one ) can be transformed to the current state $2 using the following formulae at each integration point : 2Sq 2 = I~-~.-_l ~-i-7-
02
02
's*' o'x,
2 ~,j =
t~k~ 0t
x,
(5)
where the left superscripts and subscripts indicate the configuration to which the quality is measured and referred to respectively. Also, the fourth order constitutive material behavior tensor oDou can be transformed according to the following relation: ]0~ 2D~*,
02x' 02x'
D,~
aox. oox, o
c32x* 0~-x'
a~
a~
(6)
3. FINITE ELEMENT DISCRETIZATION Owing to symmetry, only one-quarter of the bar is modeled. For the finite element simulation, the quarter is divided into 15x25 eight-nodes isoparametric brick elements along the rolling direction and the cross-section of the bar as shown in Fig. (1). The Gauss-Legndre integration numerical technique of order 2x2x2 points is used for the calculation of the stiffness matrices. For friction modelling at the roll/bar interface, the friction layer technique is employed [ 15]. In this technique, an imaginary layer of elements is added on the upper surface of the bar which model the appropriate lubrication conditions. At the start of rolling, the leading edge of this layer enters the roll-gap and moves with the roll. The bar will be then drawn into the roll-gap. It was necessary to push the bar before it can be pulled into the roll-gap by the friction layer. The pushing velocity is taken equals to the horizontal component of the tangential velocity of the roll. The stiffness of any friction elements outside the contact region is set to zero and has no effect on the analysis. 4. BOUNDARY CONDITIONS The boundary conditions at the roll/bar interface are mixed, the velocity is given zero in the direction normal to the contact surface. The changes in the boundary conditions of nodes which are included in the folding and vise-versa if node is separated from the roll surface are considered carefully. When updating the bar geometry after each time increment, the surface nodes are checked to find out whether they are touching the roll surface or not. A subroutine is designed to determine which of the bar surface node gets in contact with the roll in each time increment. Separation of a node from the roll is decided by checking its horizontal movement inside the roll-gap, if it equals to the contact length, the node set to be free. Along the left hand side surface nodes of the model, the horizontal displacements as well as the shear stresses vanish. Also, along the bottom surface nodes the vertical displacements as well as the shear stresses vanish.The roll profile is represented by a surface of revolution with two radii. This surface can be defined mathematically by a function z(x,y) which represents the coordinate of any point on the roll surface, as follows, see Fig.(2) 9 z(x, y) = R - 4 i R
h(y)] 2 +(L~ - X)2
(7)
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where z(x,y) is the coordinate of the upper roll surface, R is the largest roll radius, h(y) equals half the roll separation at the exit plane and is defined from, the roll groove shape, Lc is the projected contact length ~'~~ b
Ex p
l
~
,%
Fig.l Finite element model used in the analysis
Fig. 2 Schematic illustration of the shape rolling process.
5. RESULTS AND DISCUSSIONS As stated above, the program is structured in such a way that different processes can be analyzed by changing the geometric conditions describing the tooling and specifying the appropriate material behavior. In what follows, only one example problem is simulated; square-to-oval rolling process. The primary reason for this selection is the availability of the numerical results presented by Park and Oh using SHPROL [11 ]. The same finite element model on the cross-section (25 elements) employed in SHPROL is used, whereas, in the rolling direction only 15 layers are utilized instead of 40 layers. The data of the square-to-oval rolling pass are shown in Fig.(3). The cross-section of the bar is 50.8x50.8 mm and the roll profile is as shown in the figure. The bar length is taken to be 75 ram, which is about 2 times the projected contact length. The descriptive geometry is employed to determine the projected contact
R E -9
101.6 m m
81.28 m m d50:8
i
L
mm,,.-~
X;t
t
'
I Undelormed ROLL
__
,,
Fig. 3 Geometry of the square-to-oval rolling pass.
,j
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length, as 36.4 mm. The rolling process is carried out isothermally at 1100 ~ with speed of 6 rpm ( average roll speed of 56.65 mm/sec ) and time increment of 0.01 see. The bar material is AISI 1045 with flow stress at 1100 ~ is expressed as 6=62.7(e ~.)0.16 MPa. The friction factor at the roll/bar interface m (the ratio of shear yield stress of the lubricant to that of the bar) is taken as 0.7. Figure (4) compares the predicted cross-sectional shapes at the exit plane using the developed program FESHRL and SHPROL [11]. It is seen that the presented result shows hour-glass-bulge shape at the right hand side (the maximum width at the top of the section ), whereas, SHPROL depicts double-bulge shape (The maximum width lies between the top of the cross-section and the center ). The hour-glass-bulge shape does generally occur [ 16,17 ] in square-to-oval passes. Typical calculated cross-sectional shapes A, B, C and D at different positions in the roll-gap; x=9.1 ( one-quarter of the roll-gap), 18.2, 27.3 and 36.4 mm respectively are given in Fig. (5). It is shown that the reduction in the cross-section area is slow at the entry plane and most of the change takes place between sections A and B. After section B, the deformation of the cross-section is almost homogenous. Fig.ure (6) shows the deformed mesh of the upper half a) at 36 time increments or in the middle of the roll-gap and b) at 110 time increments. It is seen that the deformation and distortion at the comers are larger than at the centre. Fig. (7) depicts the metal flow on vertical transverse plane perpendicular to the rolling direction in the middle of the roll-gap. It is shown that the spread is concentrated at the element of the free right hand side while those inside the bar elongated longitudinally in the rolling direction. The normalized tangential velocity (v/v0 where v0 is the initial velocity of the bar ) of three selected points as a function of the roll-gap coordinate is plotted in Fig. (8). At the entry plane, the surface of the bar moves faster than the center while at the exit plane, the cross-section of the bar travels with the same velocity. Fig.ure (9) illustrates the top view of the deformed mesh. The hatched area represents the contact area between the roll and the bar. The contact area is determined as 1576 m_rn2. On the other hand, the contact area is calculated using the description of the pass geometry, by intersecting bar section with roll geometry as 1525 mm 2. The deformed width at the exit plane is used in the calculation. The two values are close to each other. The effective strain-rate contours (s "l ) are given in Fig. (10 a and b). At the entry plane, large local strain-rates occur at the comers of the bar and the strain-rate gradient towards the center is large. The predicted normal pressure and shear stress distribution ( normalized by the average flow stress, f)aver --55 MPa ) at steady-state is presented in Fig. (11). A long the sides of the bar, the pressure is higher at the entry plane due to biting action then decreases due to side spread. Along the width direction, the pressure value increases from the center to the side as much as three times. The distribution along the sides and the center line shows one peak near the entrance. Figure (12) depicts the calculated cross-sectional shapes A, B, C, and D at the exit plane for squares of side h = 50.8, 55, 60, and 65 mm respectively. It is shown that, as the square side increases the width spread at the top increases. Increasing the square side causes numerical difficulties especially in case D where the percentage reduction in area equalS 30% as shown in the Figure. On the other hand, this kind of pass is used as breaking pass ( rough rolling) in which the percentage reduction in area should not exceed 30% as recommended in [ 18]..
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3O
3(1.0 -
25.0
25
I I I
.,. une,oeO i!! / s.o-~ -
-
: /
deformed
II
10.0
0.0
'
0.0
I
5.0
v
I
10.0
'
I
IS.0
'
I
i
20.0
I
e
o o e
oe,~
o
o o Q
~ e o
o o
o
e ~ m
15
15.0 -
o o
20
20.0
E E
e o o
10
/
" *
25.0
9
I
0
30.0
y (mm) a) Present program FESHRL
-
.
10 15 20 25 30
5
b) SHPROLl111
Fig.4 Calculated cross-sectional shapes of the square at exit plane after oval shape. 311.1t -
g E
20.1 15.""
D
N
I o.o
5.O
1131 - - r - " T - " r - - - l ~ - ' l " I-- "--1" ' ' r ' I I1.11 5.11 II1.11 15.11 ZII.II .~5.11 .111.11
y tmm)
Fig.5 Calculated cross-sectional shapes at different positions in the roll-gap
a) at 36 increments
b) at 110 increments
Fig. 6 Deformed mesh of the upper half of the bar
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30.0 >~
25.0 -
~, 1.25
20.0 -
~
~
~
~
E E
1.3
1
2
1.2 1.15
]5.o-~
1o.o-
~N
"g"
_
1.1
~, 1.05
_
5.0-
1
o
0.0 f 0.0
i
'
5.0
'i""
'
t
"
!
'
!
'
!
0.95
- - ;
. . . . . .
,
0
10.0 15.0 20.0 25.0 30.0 y (ram)
- ,
5
10
,
,
,
I
i
15
20
25
30
35
|
40
Roll-gap Coordinate (mm)
Fig.8 Normalized tangential velocity in the rolling direction.
Fig. 7 Metal flow Pattern on vertical transverse plane in the middle of the rollsad
Fig. 9 Top view of b a r at s t e a d y - s t a t e . 30.0 25.0 - -20.0
25.0 t
-
"~" ~
15.o
15.0
N
N
10.0
I0.0
5.0
5.0().l) 0.0
'
i
5.0
' i
~ i "
10.0 y
i
~
I'
~-i
15.0 20.0 25.0 30.0 ( m m )
a) at the entry plane
0.0
I
0.0
,
J
5.0
'
w
"-
10.0
I
'
I
"-
r-~-
l
15.0 20.0 25.0 30.0
y (m m) b) in the middle
Fig.10 The effective strain-rate contours (s "t)
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5 4
3
9
2 1 0
-1 -2
-2
X (mm)
Y (mm)
a)long the sides
b)long the width
P/or J
I
c) three-dimensional normal pressure distribution Fig.11 Normalized normal pressure and shear stress distributions at steady-state
25
ROLL
20
A
E15 E ~10
B
N
0
5
10
15 20 25 y (mm)
30
35
40
45
Fig (12) Calculated cross-sectional Shapes at exit plane for different squares.
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6. CONCLUSIONS A three-dimensional finite element program FESHRL is developed based on the incremental updated Lagrangian formulation for the analysis of shape rolling. The program is structured in such a way that different shape rolling processes can be analyzed by specifying the appropriate boundary conditions of tooling and the material behavior. The program provides great details on geometrical parameters and metal flow characteristics To verify the predictive capability of the code, a square-to-oval rolling pass has been simulated under isothermal condition. Comparison of the predicated cross-section profile at the exit plane with that of SHPROL[ 11] shows some discrepancies. The present result shows hour-glass profile which does occur in this pass [ 16,17]. Good agreement is found between the obtained value of the contact area and that calculated using the descriptive geometry. The normal pressure distribution along the sides and the center shows one peak near the entry plane. REFERENCES 1. Abo-Elkhier, M., "Thermo-Mechanical Modeling of Strip Hot Rolling Using Finite Element Technique", Alexandria Eng. J., Faculty of Eng., Vol. 37, No.5, pp. A 271-279, (1998). 2. Abo-Elkhier, M., "Elasto-Plastic Finite Element Modelling of Strip Rolling Using Eulerian Fixed Mesh Technique", Int. J. of Finite Element in Anal. and Design, Vol. 27, pp. 324-335, (1997). 3. Lin, Z.C. and Shen, C.C., "Three-Dimensional Finite Element Method for a Nonisothermal Aluminum Flat Strip Rolling", J. of Materials Eng. and Performance, Vol. 5, pp. 452-461, (1996). 4. Wifi, A.S., Hamrouch, H.and Abdel-Hamid, A.,"A Three-Dimensional Finite Element Study of the Flat Strip Rolling", Proc. of the sixth Cairo University Int Conf. on Mechanical Design and Production (MDP-6), pp. 125-135, (1996). 5. Malinowski, Z., Pietrzyk, M. and Lenard, J.G.," Analysis of the Flat Rolling Process; OneDimensional and Finite-Element Models' J. Mater. Proc. Tech., Vol. 37, pp. 373-387. (1993). 6. Park, B.H., and Hwang, S.M., "Analysis of Front End Bending in Plate Rolling by the Finite Element Method", J. of Manufacturing Sci. and Eng., Vol. 19, pp. 314-323, (1997). 7. Hwang, S. M. and Park, B.H.," Analysis of The Strip Rolling by a Penalty RigidViscoplastic Finite Element Method", Int. J. Mech. Sci., Vol. 34, pp. 971-984, (1992). 8. Mori, K., Osakada, K. and Kobayashi, M., "Finite Element Simulation and Expert System for Shape Rolling Rolling", 1st Int. Conf. on Modeling of Metal rolling Processes, Imperial College, U.K., pp. 692-706, (1993). 9. Giowaki, M., Kuziak, R., Mailowski, Z. and Pietrzyk. M., "Modeling of Heat Transfer, Plastic Flow and Microstructeral Evolution During Shape Rolling", J. Mater Procc. and Tech. Vol. 53, pp. 159-166, (1995). 10. Kim, N., Lee, S.M., Shin, W. and Shivpui, R.," Simulation of Square-To-Oval Single Pass Rolling Using Computationally Effective Finite and Slab Element Method", J. Eng. for lndust., Vol. 114, pp. 329-335, (1992). 11. Park, J.J. and Oh, S.I.,"Application of Three-Dimensional Finite Element Analysis to Shape Rolling Processes", J. of Eng. for Indust, Vol. 112, pp. 36-46, (1990). 12. Hartley, P., Wen, S.W., Pillinger, I, Sturgess, C.E.N., and Petty, D.," Finite Element Modelling of Section Rolling", Ironmaking and Steelmaking, Vol. 20, pp. 261-263, (1993).
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13. Abo-Elkhier, M., "Finite Element Predictions of Cyclic Plasticity in an Axisymmetrically Notched Bar", Eng. Research Bull. J. of the Faculty of Eng., Menoufia University, Vol. XIV, pp. 127-140, (1991). 14. Gadala, M.S., and Abo-Elkhier, M., "A Comparison of Finite Element Formulations in Nonlinear Structural Mechanics", Res. Mechanica, Vol. 18, pp. 157-177., (1986). 15. Rowe, G.W., Sturgess, C.E.N., Hartley, P. and Pillinger, l.,"Finite Element Plasticity and Metal Forming Analysis ", Cambridge University Press, (1991). 16. Kennedy, K.F., " An Approximate Three-Dimensional Metal Flow Analysis for Shape Rolling", J. of Eng. for lndust., Vol. 110, pp. 223-231, (1988). 17. Aoyagi, K., " High Reduction Ratio of Billets, Bars, and Rods", Nippon Steel Technical Report. No. 16, Dec., (1980). 18. Wusatawski, Z.," Fundamentals of Rolling", Pergamon Press., Oxford, (1969).
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
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SHAPE OPTIMIZATION OF METAL BACKING FOR CEMENTED ACETABULAR CUP Hedia, H.S.*, Abdel-Shafi, A.A.A. ** and Fouda, N.***
*Lecturer, ** Associate Prof, *** Demonstrator Production and Machine Design Engineering Department Mansoura University, Mansoura, Egypt
ABSTRACT Stresses are generated in implant materials and bone, and at their interfaces. These stresses may affect the structural properties of the implant/bone system, or bring it to failure at some time in the postoperative period. Due to these stresses, acetabular cup loosening becomes an important problem for long term survival of total hip arthroplasty. It was found that metal backing would tend to reduce stresses in the underlying acrylic cement and bone. Yet, recent studies of load transfer around acetabular cups have shown that metal backing generates higher stress peaks in cement at the cup edges, while generates lower stress peaks in bone at the central part of acetabulum (dome), thus the bone at the dome becomes more stress shielded. In this study a numerical shape optimization procedure in combination with an axisymmetric finite element model was used in order to optimize the shape of a stainless steel metal backing shell. The design was to minimize fatigue notch factor in cement along cement/bone and cement/metal backing interfaces in order to prevent failure of cement mantel and loosening of acetabular components, at the same time increasing fatigue notch factor in bone at the center of acetabulum to prevent stress shielding. The results of this study indicate that cemented acetabular cup designs can be improved by using metal backing shells of nonuniform thickness, thick at the dome and thin at edges. Fatigue notch factor in cement was reduced by 1.1% at cement/metal backing interface and increased by 3.1% in the central bone of acetabulum. Von Mises stresses in the cement edge were reduced by 17.8% and 19.3% along cement/bone and cement/metal backing interfaces respectively. Thus the optimal design will reduce the possibility of fatigue fracture of cement and decrease the stress shielding effect and the likely incidence of bone resorption, whereby extend the expected life of the prostheses. KEYWORDS Optimization, Fatigue Notch Factor, von Mises Stress, Stress Shielding, Metal Backing Shell. I. INTRODUCTION The load transferred through the hip joint is one of the major forces occurring in the human body. After the replacement of this joint in total hip replacement (THR) arthoplasty, the load transferred through the implant to the acetabular bone. The most important long term problem
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of cemented total hip arthoplasty (THA), limiting its average functional life span, is mechanical aseptic loosening. Hence, this is also the most important issue in implant safety. In the cement fixation concept, the acrylic cement and the cement/implant and cement/bone interfaces are relatively weak links [1-2]. There is no doubt that finite element method (FEM)has established itself as an important research tool in orthopedic biomechanics. The stress patterns in a THA structure depend on four principal classes of characteristics: the external load, the material properties, the interface properties, and the geometry. Using the FEM the stress patterns can be evaluated and the relationships between the stresses and the above characteristics were studied by Huiskes and Chao [31. Numerical optimization in an finite element (FE) model was used to find a distribution of the local stiffness in the backing of an acetabular cup by Dalstra, et al [4], which would result in an optimal load transfer from cup to bone. They used a FE model representing a pelvic bone inserted with a cemented acetabular cup combined with a numerical optimization routine. The objective was to minimize loading of cement mantel for a given external load. The local stiffness of the elements representing the backing o f th e implant were the design variables allowed to range between the stiffness of polyethylene (700 MPa) and TiAI 16V4 (110 GPa). The converged solutions represent backings with only a small area of high stiffness material along the antero-superior edge, at the other locations the stiffness should be as low as possible. The present study differs from the above in that the fatigue notch factor is used in the objective function and constraints of optimization. The fatigue notch factor is considered more meaningful in determining the fatigue life of components consisting of relatively complex geometry since the calculation includes not only the peak stresses but also some measure of the volume over which they act [5]. The present analysis was restricted to a cemented acetabular component with metal backing shell. A numerical shape optimization procedure in combination with an axisymmetric FE model was used in order to optimize the shape of a stainless steel metal backing (MB) shell. The design minimizes the fatigue notch factor in cement along cement/bone and c e m e n ~ B interfaces in order to prevent failure of cement mantel and loosening of acetabular components, at the same time increasing fatigue notch factor in bone at the central part of acetabulum to prevent stress shielding and subsequent bone resorption. 2. FINITE ELEMENT IDEALIZATION The real left hip bone of an old man was selected for modeling. A mid- frontal plane section of the model is shown in Fig. 1. The section is taken in a plane inclined superiorly about 15~ posteriorly, shows reasonable inferior and superior symmetry of the pelvic bone. The section is passing through the ilium, acetabulum, and pubis at one cut. Other planes would reveal considerably less symmetry for bone regions other than that immediately adjacent to the acetabulum [6]. The femoral component head diameter is taken as 28 mm. The layer thickness of Ultra High Molecular Weight Polyethylene (UHMWPE) is taken as half of the femoral head radius equals to 7 mm ; this relationship between the femoral head radius and UHMWPE thickness was
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only for metal-backed medium thickness UHMWPE cups [7]. A metal backing shell of 3 mm uniform thickness, and a cement layer thickness of 2.5 mm are assumed for initial design. The metal backing acetabuler component model of Chamley prosthesis cemented into the previous acetabular cavity was used. The model was divided into the cortical bone and subeortical bone, trabecular bone (which was divided into three regions with varying moduli of elasticity), cement layer, metal backing shell, and the acetabular cup as shown in Fig. 2. An axisymmetric finite element model was considered in this study, as shown in Fig. 3. All materials are assumed to be isotropic, linearly elastic and homogenous. Values for the mechanical properties of bone, cement, MB shell, and UHMWPE were taken from literature. The values assumed are shown in Table I [6]. The model was fixed superiorly at the ilium [6,7]. A fixed load of 2500 N which corresponds to 3 to 4 times body weight was chosen [8]. In clinical practice the anatomical position of the acetabular cup and the direction of force can limit the contact angle to 60 ~ in the superior aspect of the cup. The value of the contact pressure was calculated by specifying a contact half angle of 60 ~ corresponding to the anatomical position of the acetabular cup, and this contact pressure is only a function of the load and the femoral head radius, calculated by the following expression [8] 12W P - 5fir 2
(1)
where W is the chosen load (2500N), and R is the radius of femoral head (14 mm). 3. METHOD OF ANALYSIS The above axisymmetric finite element model was used in combination with a numerical shape optimization procedure in order to optimize the inner contour of the uniform metal backing shell. It was concluded from the literature that there was two types of metal backed acetabular cups; the first type was metal- backed medium thickness UHMWPE with d = 0.5R, and the second type was metal-backed thin thickness UHMWPE cups with d =0.25R, where d and R describe the thickness of metal backing shell and the radius of femoral head respectively [8]. Thus, it was estimated that the thickness of UHMWPE can't be less than 3.5 ram. Therefore, a maximal shell thickness of 6.5 mm and a minimal thickness of zero were taken as boundary constraints in the numerical shape optimization (NSO)procedure. Two design variables were used, the thickness of the shell at the edge and at the dome of the cup denoted by XI and X2 respectively. The design objective was to minimize the fatigue notch factor in cement along cement/bone interface in order to delay or prevent cup loosening and migration in total hip replacement patients, while constraining fatigue notch factor in cement along cemen~MB interface at or below its initial level, and maintaining fatigue notch factor in bone at the central part of acetabulum (dome) at or above its initial level to prevent stress shielding. The finite element method and optimization procedure was carried out using the ANSYS general purpose software [9]. The theoretical method used to calculate the fatigue notch factor is the" statistical hypothesis" which is valid for general optimization problems, because during optimization arbitrary shapes appear [10] using the following equation:
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Current Advances in Mechanical Design and Production, MDP-7 --I
!
t.[, l(4x,y))' a 1
(2)
crn A!i where t is the thickness of the specimen, crn is the nominal stress, A~ is the surface area of the smooth specimen, and s is the length along the surface of the notched specimen.
A linear distribution of von Mises stress between adjacent finite element nodes on each surface was assumed. The integral along a line of nodes is carried out analytically using the following equation [11,121.
-'/'
/=E
i-!
+(r,§
)
(3)
2
where o" is the von Mises equivalent stress, X, Y are the coordinates of the nodes, and k is the statistical parameter (not a material constant) which can vary from I to 24. In the present work the minimum value of k was taken (k=1) since the variable nature of the cemented interfaces suggests that the population of voids and defects in the cement is likely to be of random size and distribution. Thus a low value of k which reduces the stress along the whole of the interfaces will be more effective in reducing the probability of fatigue failure than a high value of k which only decreases the peak stresses [5]. To calculate fatigue notch factor (Kf) along each interface a specific program was written using the ANSYS Parametric Design Language (APDL) [9]. To minimize Kf according to the statistical hypothesis, the above integral should be minimized. The resulting Kf can be calculated as follows" I
t k K I = ----'Z-_ x I 0", Ai x
(4)
Thus the objective function for this problem is to minimize the fatigue notch factor Kfin cement along cement/bone interface. The constraints for this problem are" To maintain K/in cement along cement/MB interface at or below its initial value O< Kf <_Ky~where K/~ is the initial value. To maintain K/in the bone at the dome of acetabulum at or above its initial value Ky~ < K/< any arbitrary higher value than KIt where Ky~is the initial value. To maintain the design variables XI and X2 within the allowable range , to ensure the thickness of UHMWPE will not be less than 0.25 radius of the femoral head i.e. 0 < X1 < 63ram and 0 <_X 2 <_6.Smm 4. RESULTS Numerical shape optimization procedure is carried out using ANSYS program, 6 iterations with minimal CPU time were required to reach the final shape of MB shell. The optimal shape
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of MB shell was found to be thin at the edge with minimum thickness of 1.24 mm, and thick at the dome with maximum thickness of 5.43 mm. The optimum shape is represented in Fig 4. A comparison between the results for both the initial design and the optimum design can be explained below: 1.
2.
3.
4.
5.
6.
With reference to Fig. 5, von Mises equivalent stress in bone at the dome of acetabulum along cement/bone interface was increased at all nodes by 3.5%. Fatigue notch factor in bone at the dome region was increased by 3.1% due to the increase of load transfer to the central part of acetabulum. With reference to Fig. 6, the maximum von Mises equivalent stress in cement along cement/bone interface at the edge point was decreased from 1.9to 1.57 MPa(17.8%) compared to the initial design. Fatigue notch factor along this interface was approximately unchanged. With reference to Fig. 7, the maximum von Mises equivalent stress in cement along cement/MB interface at the edge point decreased from 2.44 to 1.96 MPa (19.3%) compared to the initial design and fatigue notch factor along this interface that decreased by l . l % . With reference to Fig. 8, the von Mises equivalent stresses in MB along c e m e n ~ B interface for the optimal shape are higher than those for the initial design near the edge of acetabulum as a result of reduction of the cross-section in this region. However, the stresses near the dome were decreased. Fatigue notch factor in MB at this interface was increased by 11.5%. With reference to Fig. 9, along MB/UHMWPE the von Mises equivalent stresses in MB at the edge increased by 47.5% for the same reason as above. On the other hand increasing the cross-section at the dome would decrease von Mises stress at this region. Fatigue notch factor at this interface decreased by 1.2%. With reference to Fig. 10, along MB/UHMWPE the maximum von Mises equivalent stress in UHMWPE was reduced at the dome by 14.6%, and fatigue notch factor at this interface was decreased by 1.4%.
5. DISCUSSION Compared to a full polyethylene cup, a conventional metal-backed cup has two negative effects on the acetabular load transfer: the stresses in cement along cup's edge are higher and in the central part of acetabulum the bone becomes more stress shielded, these effects were eliminated by the new design of MB shell. The results of this study indicate that using MB shell with non-uniform thickness, thick at the dome and thin at edges will reduce fatigue notch factor along cement/MB interface and in the same time the stresses in cement at cup edges will be decreased. On the other hand the stresses and fatigue notch factor in bone at the central part of acetabulum are increased, which tends to reduce stress shielding and subsequent bone resorption. The CPU time is very much reduced due to the use of fatigue notch factor as the objective function and constraints instead of using loads and stiffness as used before by Huiskes, et al [4]. In the present work only 2 constraints, 2 design variables and 6 iterations have been used to obtain the optimum shape of MB shell. The above optimization was rerun with the objective function to maximize Kf in the bone at the central part of acetabulum rather than to minimize Kf along cement/bone interface. The constrain number 2 in the new optimization
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was therefore to maintain Kf in cement along cement/bone interface within the limits 0< Kf < K~ where Ky~is the initial value. 5 iterations were required to reach the optimal MB shape. The optimal shape was the same as that found from the first optimization, as well as the resulting stress distribution and fatigue notch factor were nearly identical compared with the first optimization. 6. CONCLUSIONS The results of this study indicate that cemented acetabular cup designs can be improved by using metal backing shells of non-uniform thickness, thick at the dome and thin at the edges. The optimal design will decrease the fatigue notch factor and the stresses in cement at the cup edge , as well as increase the fatigue notch factor and stresses in bone at the central part of acetabulum (dome). Thus the optimal design will reduce the possibility of fatigue fracture of cement and decrease the stress shielding effect and the likely incidence of bone resorption, whereby extend the expected life of the prostheses. REFERENCES 1. Huiskes, R., "New Approaches to Cemented Hip Prosthetic Design", In: Technical Principles, Design and Safety of Joint Implants, edited by Gottfried H. Buchhorn and Hans-Georg Willert, Hogrefe and Huber publishers, Seattle, Toronto, pp. 227-236, (1994). Beckenbaugh, R.D., and Ilstrup, D.M., "Total Hip Arthroplasty", J. Bone Joint Surg., Vol. 60A, pp. 306-313, (1978). 3. Huiskes, R.and Chao, E.Y.S., "A Survey of Finite Element Analysis in Orthopedic Biomechanics: The First Decade", J. Biomechanics, Vol. 16,No.6, pp. 385-409, (1983). 4. Dalstra, M., Huiskes, R. and Kuiper, J.H., "Numerical Optimization for the Backing of a Cemented Acetabular Cup", Second World Congress of Biomechanics, July 10-15, Vol. II, Amsterdam, the Netherland, (1994). 5. Hedia, H.S., Barton, D.C., Fisher, J. and lbrahim, A., "Shape Optimization of Charnley Prosthesis Based on the Fatigue Notch Factor", Bio-medical Materials and Eng. 6, pp. 199-217, (1996). 6. Douglas, R., Crowninsheld, R., Brand, R., and Johnston, R., "An Axisymmetric Model of Acetabular Components in Total Hip Arthroplasty", J. Biomechs. 15, pp. 305-315, (1982). 7. Ries, M.D., and Harbough, M., "Acetabular Strain Produced by Oversized Press Fit Cups", Clinical Orthopaedics, Number 334, pp. 276-281, (1997). 8. Jin, Z., Dowson, D. and Fisher, J.,"A Parametric Analysis of Contact Stress in Ultra High Molecular Weight Polyethylene Acetabular Cups", Med. Eng. Phy., Vol. 16, pp. 398-405, (1994). 9. ANSYS User's Manual, Version 5.0A, Vol. 1, (1992). 10. Hedia, H.S., Barton, D.C., Fisher, J., and EI-Midany, T.T., "Effect of FE Idealization, Load Conditions and Interface Assumption on the Stress Distribution and Fatigue Notch Factor in the Human Femur with an Endoprosthesis", Bio-Medical Material and Eng. 6, pp. 135-152, (1996). 11. Hedia, H.S., Barton, D.C., Fisher, J. and Elmidany, T.T., "A Method for Shape Optimization of a Hip Prosthesis to Maximize the Fatigue Life of the Cement", Med. Eng., Phys., Vol. 18, No. 8, pp. 647-654, (1996). 12. Fanni, M., Schnack, E. and Grunwald, J., "Lifetime Maximization Through Shape Optimization of Dynamically loaded Machine Parts", International Journal for Engineering Analysis and Design, Vol. 1, pp. 25-41, (1994). .
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Table 1. Properties of materials represented in the FE model ~,,..
'Material Cortical bone
..... Young's modulus Mpa
. . . . . . . . . . .
,'
Poisson's ratio
(1)
17000
0.3
Subchondral bone (I)
17000
0.3
Trabecular bone a. Intramedullary (2) b. Medium (3) c .Denset (4)
1000 1500 3000
0.3 0.3 0.3
Acrylic cement
(5)
2300
0.32
Stainless Steel
(6)
200000
0.3
UHMWPE
(7)
1000
0.3
'
,. . . . . . . .
,
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Current Advances in Mechanical Design and Production, MDP-7 'I
"
Cortic~ bone
I-
"
~erwnedu_Jl~ 3
s
Stak'dess
slee4
2
UHIdWPE
Fig. 2. The model of the initial design
Fig. 3. The finite element mesh for the model
~elmtum.,
t
~
htf~
Fig. 4. The model of the optimal design.
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.......................................
311
2
2.9
~.
,~
2.7
~.
1.6
2.5 2.3
-=..
1.2
2.1
._~
0.8
1.9
8
=:
I. 5
.... 0
2 (Dome)
4 6 distance(mm)
8 (Edge)
0
10
. . . . . 0
4
- o - Optimal design,kf=-55"c]!
8 12 16 20 24 28 32 36 40 (Done) distance(rrrrl (F:dge)
Fig. 5. Comparison of the stress in bone at Fig. 6. Comparison of the stress in cement the center of acetabulum along along cement/bone interface for initial and cement/bone interface for initial and optimal design. optimal design.
2.5
~.
40
.............................................................................................................
[
r
2
---e.--Initial
design,kf=524*c
]
--0-~--0 ptim a! des!g.n,kf= 585"C_~
30
I
1.5 ~ 9-
= 0.5 :~
.
0
.
.
4
.
~
~lnital design,kf=53.4*c ---o-- Optimal desin,kf=52.8*c .
.
~
-
,,~
' == lo
, :-~_
8 12 16 20 24 28 32 36 (Dome) distance (mm) (Edge)
,,
o
!
40 design,kf=445"c
I
t
~
4
8
12
30
i
I
I
!
t
16
20
24
28
32
distance
(mm)
36
(Edge)
................................................
~. 6
IJ
"
Fig. 8. Comparison of the stress in MB along cement/MB interface for initial and optimal design.
7
---e.--- I n i t i a l
,, 0
(Dome)
Fig. 7. Comparison of the stress in cement along cement /MB interface for initial and optimal design,
g
20
i
! r
Initial design, kf=79*c
t
~ 5 ~ 4 ~ 3
2o
>~ lo , 0
4 (Dome~
8
12 d~stance
16 19 (turn)
23 27 (Edge)
31
Fig. 9. Comparison of the stress in MB along MB/UHMWPE interface forinitial and optimal design,
0
,
0
4
,
~
,
,
,
!
~
--'~-?~"
8 12 16 19 23 27 31 (Dome) distance(n~ (Edge)
Fig. 10. Comparison of the stress in UHMWPE alongMB/UHMWPE interface for initial and optimal design.
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Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
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PROPERTIES OF CEMENTITIOUS COMPOSITES CONTAINING NON-RECYCLABLE GLASS AS A FINE AGGREGATE Shehata, I.H. *, ElSawy, A.H. **, Varzavand, S. § and Fahmy, M.F. ++
* Mech. and Plant Eng., John Deere Engine Works, Waterloo, IA 50704-5100, USA. ** Professor and Chairperson, Department of Industrial Technology, Tennessee Technological University, Cookeville, TN 38505-0001 USA + Associate Professor and ++Professor and Department Head, Department of Industrial Technology, University of Northern Iowa, Cedar Falls, IA 50614-0178, USA E-mail: [email protected], [email protected], [email protected], [email protected]
ABSTRACT This study addresses the use of non-recyclable granulated glass waste as partial aggregate substitutes to the fine aggregate (i.e. sand) in concrete. Portland cement was mixed with the aggregates to produce cementitious concrete composites with favorable results, satisfying both solid waste disposal and natural resources challenges. The compressive and splitting tensile strengths as well as moduli of rupture and elasticity of the concrete composites were determined. The results of the mechanical properties were statistically analyzed and graphically presented in relation to the control specimens. Scanning electron microscopy (SEM) was used to study the relationship between the mechanical properties and the interfacial morphology in relation to the fracture behavior of the concrete composites. The results of this study revealed that the average values of compressive and splitting tensile strengths of the concrete composites containing glass were comparable to those of the control ones. Moreover, it was also found that the moduli of rupture and elasticity of the specimens containing 20% glass are considerably higher than those of the control ones. KEY WORDS Waste Reduction, Glass, Concrete, Fine Aggregates, Recycling, Mechanical Properties. 1. INTRODUCTION The solid waste materials disposal became one of the major environmental, economical and social problems [ 1]. Recycling is the most promising solution to dispose materials in the waste stream [2]. The construction industry is the most attractive market to use recycled solid waste materials successfully. Glass is one of the four highest materials in the waste stream in the U.S. There are three standard types of glass products in this waste stream: clear, green, and brown [3]. The majority of these types of products are recyclable. However, other glass products are not easy to recycle such as mirrors drinking glasses, Pyrex, window glasses, and light bulbs. This is due to the difficulty in, and high cost of, identifying the compositions of these products. So far, no widespread recycling programs have been established for these types of glass [ 1].
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In general, the potential uses of most recyclables, especially in the construction industry, are almost endless. Many virgin and waste materials are widely used in concrete composites as fiber reinforcements (e.g. steel, glass, plastics, sisal and jute) [4-8]. However, the idea of using municipal waste materials as aggregates in structural concrete did not appeal to Mehta [9]. He believed that the presence of crushed glass in aggregates tends to produce unworkable concrete mixtures and, due to the high alkali content, affects the long-term durability and strength. On the other hand, some researchers investigated different types of waste materials (e.g. vegetable and plastic materials) in concrete composites as aggregates [10,11 ]. Results of such research efforts showed the possibility of using these materials in concrete composites effectively. Therefore, the ultimate goal of this paper is to report the research findings on using some of the non-recyclable granulated glass waste materials as aggregate in concrete composites. 2. EXPERIMENTAL PROCEDURES The glass waste material used in this study was a combination of both clear glass window and fluorescent bulbs with a small amount of contaminants. This combination was used as aggregate substitutes for a portion of sand in the cementitious concrete composites. In addition to the control concrete composite, four different volume fractions of the glass waste were used to produce four concrete composites. According to ACI 211 standards, Table 1 shows the materials used to prepare the tested concrete composites and their conditions.
Table 1. Materials used and their conditions Material Water Cement Coarse aggregates
Sand aggregates
Glass aggresates
Condition Tap water at room temperature 23~ Type IA Portland cement (specific gravity = 3.15). smooth and round with average maximum size of 9.5 mm diameter. The oven dry rodded unit weight was 16.65 KN/m 3 and the specific gravity was 2.64. Absorption and moisture content were 0.31% and 0.51% (ASTM C 127-88). Fineness modulus and specific gravity were 2.7 and 2.7 in accordance with ASTM C 136-84a and ASTM C 128-88 standards. Absorption and moisture content were 1.01% and 4.85% respectively. specific gravit), was 2.13 based on ASTM C329-75 standard.
The recommended slump considered in this study was between 25.4 and 76.2 mm. In addition, the average specified 28-day (f'r of the glass cementitious concrete composites was 20.7 MPa. Since no previous data was available to establish a standard deviation for this f'c, some allowances were added before calculating the amount of each constituent in the concrete mix. Therefore, the average strength required f ~ = 28.9 MPa was considered in the mix design. The procedure followed in making and curing all the tested concrete specimens was in accordance with ASTM C 192-90a standards. In this investigation, three mechanical testing methods were conducted. Compression test, ASTM C39-86 standard, was used to measure (f'c) of the control and glass-containing concrete composites. Molded and capped concrete cylinders, 76 mm diameter X 152 mm height specimens were used. The splitting tensile strengths (T)of the control and glass-containing concrete composites were measured by conducting splitting tensile test (ASTM C496-90 standard). The standard cylindrical specimens used in this test were similar to those used in the compression test. The test method for both the modulus of rupture (R) and elasticity (E) of the concrete composites was conducted using simple beam with threepoint bending in accordance with ASTM C293-79 standard. The beam dimensions were 50 mm
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width X 50 mm depth X 152 mm span length with one 25.4 mm overhanged distance from both ends. After conducting each mechanical testing, all the fractured specimens were preserved to generate photographs to study the general fracture modes of these composites. Representatives of fractured compressive samples were taken for further SEM study. 3. EXPERIMENTAL RESULTS
3.1. Mechanical Properties Results showed that the average f'r for the control concrete composite is 36.5 MPa. This is higher than both the average f'r and for (20.7 and 28.9 MPa respectively) considered in this study. Results also showed that the average fr values of the concrete composites containing 5, 10, 15, and 20% glass aggregate substitute were 29.6, 28.1, 27.9, and 34.7 MPa respectively. These values were higher than fr of the control composite by about 43, 36, 35, and 68% respectively. Furthermore, these composites were lighter in weight than the tested control composite by 0.4, 0.8, 1.2, and 1.60%. However, the f'r values of these glass concrete composites were 19, 23, 24, and 5% lower than that of the tested control composite. Results also showed that the average T value for the control specimen was 3.9 MPa. This value, which was about 11% of the obtained fc of the same composite, was within the expected range [12], 8% to 12%. The percentages of reduction in the T values of the concrete composites containing 5, 10, 15, and 20% glass aggregates, compared to that of the control one were 12, 15,23, and 14% respectively. These T values ranged from 10 to 12% of their f'c values. Results also showed that the average R value for the control specimen was 6.1 MPa (about 12 times the square root of the obtained f'r for the same composite). This value was higher than the maximum expected value of the range by 2 times the square root of the obtained ft. On the other hand, the average E value for this composite was 464 MPa. It is also to be mentioned that at 5, 10, 15, and 20% glass aggregate substitutes, the R values for these composites were between 13 and 14 times the square root of their fr values. Meanwhile, these R values ranged from 7% lower than (in the case of 10% glass substitute) and 3% higher than (in the cases of 5 and 20% glass substitutes) that value of the control composite. The values of E for the glass concrete composites were 494, 431, 566, and 624 MPa respectively. These values ranged from 7% lower than (in the case of 10% glass substitute) to 35% higher than (in the case of 20% glass aggregate substitute) that of the control one. It is to be noted that the E values of the 5, 15, and 20% glass concrete composites were higher than that of the control one. This indicates that these composites are stiffer (especially the 20% glass concrete composite) than the control one. The one-way analysis of variance with Tukey HSD by Howell [13] statistical method was applied to the four sets of data (f'c, T, R, and E) individually. The significant differences among these values of the control and glass concrete composites were determined. Results revealed that at a 95% level of confidence, there were significant differences between the f'c value of the control composite and those for the concrete composites containing 5, 10, and 15% glass aggregates. Moreover, no significant difference was identified between the 20% glass concrete composite and both the control and 5% glass concrete composites. This indicates that both the control and 20% glass concrete composites have almost the same f'r value. In addition, the T value of the control specimen was statistically different from that for 15% glass concrete composite. However, no significant differences were identified between the T value of either the control or 15% glass composite and those values of 5, 10, and 20% glass composites. No statistical significant differences identified among all the average values of R for the tested concrete composites. Results showed that the E value of the 20% glass composite was higher and better than that for the 10% glass composite. No significant differences were identified
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Current Advances in Mechanical Design and Production, MDP-7
between the E value for either the 10 or 20% glass concrete composite and those values for the 5%, and 15% glass concrete composites as well as control composite specimen. Figures 1 through 4 show the graphical representation of the experimental results of the f'c, T, R, E values for the glass concrete composites versus the control ones. The best curve fittings obtained using the polynomial curves of the second degree. Figure I shows that the average f'c A
~.
.7
~_,~_ _c_~.~_ ~
4~ I r
.......
~
! ut | . . . . . . . . . . . . . . . . . . . . . . . ~ :i0,1'
2m.7
'eree~t~e
of qrrq~ta
ma~ltuta
Figure 1: The compressive strength Versus the percentage of aggregate substitutes in the glass-containing concrete composites.
Figure 2: The splitting tensile strength versus the percentage of aggregate substitutes in the glass-containing concrete composites.
I, OO
'~~.14
L44i"'"'-"-6
--
iJ
-11)
tO
U
]Peggg~te4pl opt aqgglmlglL~~lt]hett~qlJ
Figure 3: The modulus of rupture versus the percentage of aggregate substitutes in the new cementitious concrete composites.
Figure 4: The flexural modulus of elasticity versus the percentage of Aggregate substitutes in the new concrete composites.
of the control specimen was declining by adding more glass waste material up to about 10% glass aggregate substitute. However, less declining rate was found in the range between 10 and 20% glass aggregate substitute. No significant differences were identified between the control and 20% glass concrete composites. Figure 2 shows that the T value of this concrete composite declined only in the range between 5 and 15% where more strength was gained by increasing the glass aggregates up to 20%. It can be also seen that the T values at 5, 10 and 20% aggregate substitutes were the closest values to that of control composite. In figure 3, the R value decreased only in the range between 5 and 13% glass aggregate substitute and then gained more flexural strength when these aggregates were increased up to 20%. In addition, the range of the R values of the glass specimens indicates that the behavior of these composites was almost
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identical to the control one. In fact, two concrete composites (containing 5 and 20% glass aggregate) had higher R values than that of the control one. Finally, figure 4 shows that all the glass concrete composites are at least as stiff as (or in some cases stiffer than) the control one. 3.2. SEM Examinations of Fractured Specimens A SEM micrograph of the morphology of the sand aggregates showed that the shapes and surface textures of these aggregates were a combination of rounded and smooth particles; equi-dimensional crushed rocks; and rough and angular particles. This combination of fine and coarse sands worked together to produce a concrete composites with satisfactory workability and strength requirements. The used standard sand had a fineness modulus (FM) of 2.7. On the other hand, a SEM micrograph of the morphology of the used crushed glass aggregates showed predominant angular shapes with sharp edges. The surface textures of these particles were a mix of smooth and rough surfaces. The roughness appeared on some of these surfaces (as serration marks) was partially attributed to the crushing process. These glass particles were hard and durable but mixed with a small amount of contaminants. The FM value of these aggregates was 2. l, which means that they were finer than the sand ones. It is to be mentioned that the majority of the used sand and glass sizes were between sieve #100 (150 mm) and sieve #8 (2.36 mm). Figure 5 shows the SEM micrograph of a fractured control sample. It can be seen that the coarse aggregates (GR) and voids were dispersed in a matrix of hydrated cement paste (hcp). These GR and hcp phases of the concrete structure were not homogeneously distributed with respect to each other. The micrograph also shows that the cracking systems (C) occurred in two areas: the hcp phase and the interfacial region (transition zone or tz) between the GR and the hcp. Other micrographs of the same composite showed that the C extended in the tz phase and underneath the GR which were pulled out (upon debonding) from the composite. Some dehydration products in the form of calcium hydroxide (CH) crystals also existed and were scattered in the empty grooves (resulted from pulling out the GR), voids, and the surface of hcp. The type of fracture behavior observed in case of compression testing for the specimen was of cone and split shape. This conforms to the fracture types sketched in the ASTM C39-86 standard. Main cracks propagated along the loading axis separating the tested specimen into a few chunk pieces were also observed. Multi microcracks were also initiated and propagated in the hcp and tz phases causing failure in this composite. The general fracture behavior of the control sample under center-point loads suggested that almost all the tested flexural specimens experienced In-Plane shear fracture on the compression surface, which was in contact with the applied load. However, the side surfaces were fractured with a shear angle of about 20~ to the axial load. Main cracks due to brittle failure along the axis of the diametral load were created and caused the control specimen to split equally when it was tested under splitting tensile loads. Some microcracks were also seen branching from the main cracks due to the possibility of the existence of dense areas of voids near these cracks. These microcracks might have produced other main crack systems across the specimen diameter. In the case of the tested glass samples, the overall FM was decreased with the increase of the glass percentage. From the concrete mix design standpoint, this led to these composites to have less water and gravel and surplus of sand and glass. The continuous reduction of the water content, and consequently water/cement ratio, directly affected the porosity in the tz and hcp phases and their strengths in these composites. In general, the shortage of coarse aggregates and the surplus of fine ones increase the void contents especially if these fine aggregates are angular in shape [ 12]. Due to the fact that the used glass particles were finer than the sand ones and had predominant angular shapes with sharp edges, this might have led the glass samples to have more voids in the tz and hcp phases that could have affected the strengths of these composites.
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Current Advances in Mechanical Design and Production, MDP-7
SEM micrographs for the microstructure of the four tested glass concrete composites showed interracial bonding between the glass aggregates and cement paste. In addition, the amounts of porosity, CH crystals and crack growth and branching were directly related to the contained glass amount in these samples. In the cases of 5 and 10% glass samples, the glass aggregates were far from each other due to their low amounts while these aggregates were closer and filled some voids in the cases of 15 and 20% glass samples. This may explain the reduction in the t~c in the former samples and the sudden increase in the latter ones. Figure 6a shows an initiation of microcracks in the tz phase (specifically from the tips of the aggregates which bring them in contact with the cement paste) and branched in the hcp phase. It can be also seen that these glass aggregates arrested cracks and microcracks coming from surrounding directions. This is because the strength of these aggregates was higher than both phases. Figure 6b enhances this observation and shows the pull-out phenomenon. The figure shows that one glass particle (gray dark area) was pulled out from its place where a propagated crack was approaching from the fight bottom part in the figure towards this aggregate. Aiter pulling this glass aggregate out, the propagated crack was arrested by another glass aggregate (shiny gray area with serration marks across). It was also observed from the generated micrographs for all the fractured samples that
Figure 5: SEM micrograph of a fractured control concrete composite. there were some similarities between the microstructure of the control and 20% glass concrete composites (e.g. cracking systems, interfacial bonding, and voids content). This may rationalize why the average values of f'c of these two composites were significantly indifferent. The types of fracture of the 15 and 20% glass concrete composites tested under uniaxial compression load were of shear and a combination of shear and cone modes respectively. On the other hand, the 5 and 10% glass concrete composites were failed by a shear and a mix of shear and columnar modes respectively. These types of fracture conform to the fracture types sketched in the ASTM C30-86 standard. It is also noticed that the failure modes of these specimens were different from that of the control sample. However, the differences in the fracture behaviors of both the control and 20% glass concrete cylinders were minimal. It is to be mentioned that the glass concrete cylinders were shattered into small pieces upon failure which can be attributed to the role played by the glass waste material in these samples. In other words, the brittleness of the glass material might have assisted these composites to be at least as stiff as, or even stiffer than, the control sample. This interpretation, in general, coincides with the trend of the E values obtained from the statistical analysis of the collected data.
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Figure 6: SEM micrograph of a fractured concrete composite containing glass aggregate substitute. The fracture behavior of all the tested glass concrete composites under splitting tensile loads was almost identical. The initial cracks propagated transversely to the direction of the splitting tensile stress and branched causing a shear fracture mode to exist. Available areas carrying these loads assisted the glass specimens to arrest cracks before failure. Yet, the failure of these specimens needed just a few bridging cracks (due to the decrease of the frequency of crack arrests) than those found in the compressive specimens. This observation was repeatedly seen in all the tested glass and control specimens under splitting tensile stress. This noticing matches the results obtained from the statistical analysis of the collected data. It was obvious that the fracture behaviors of 5 and 20% glass specimens were slightly closer to that of the control one. Similar to the behavior of the control composite, almost all the tested flexural glass specimens experienced In-Plane shear fracture on the upper surfaces while the side surfaces were fractured at a shear angle that ranged from about 0 ~ to 20~ respect to the axial load. This range of shear angles resembles that of the control composite. This observation may support the conclusion drawn earlier through the statistical analysis of the collected R data for all the tested concrete composites. However, it seems that drawing conclusions or comparing the R and E values of different concrete composites based on the visual analysis of the photographs showing their fracture behaviors should be supported by other methodologies. For example, generating and analyzing SEM micrographs for representative samples for these concrete composites may help in relating their general fracture behaviors to their mechanical properties. 4. CONCLUSIONS Based on this study it can be concluded that granulated glass waste material can be used as partial aggregate substitutes to the fine aggregates in Portland cement concrete mixture. The cementitious concrete composites developed containing non-recyclable glass waste materials have acceptable workability, consistency, and plasticity suitable for different job conditions. They are durable, meet the strength requirements, and have a uniform appearance as well as they are more economical than the normal concrete composite. The success of this study suggests the need for further research to examine the use of other waste materials (e.g. plastics, fiberglass,
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need for further research to examine the use of other waste materials (e.g. plastics, fiberglass, etc.) as partial aggregates to sand in the Portland cement concrete mixture. ACKNOWLEDGEMENT Financial support for this investigation was provided by the Recycling and Reuse Technology Transfer Center (RRTTC) at the University of Northern Iowa. REFERENCES 1. Carless, J., "Taking Out the Trash: a No-nonsense Guide to Recycling", Washington, D.C.: Island Press, (1992). 2. Bell, J., "Plastics: Waste Not, Want Not", New Scientist, Vol. 128, pp 44-47, (1990). 3. Duston, T. E., "Recycling Solid Waste: The First Choice for Private and Public Sector Management", Westport, CT.: Quorum Books, (1993). 4. Magdamo, R., "An analysis of the Abaca NattLral Fiber in Reinforcing Concrete Composites as a Construction Material in Developing Countries", Doctoral dissertation, University of Northern Iowa, DA8919070, (1988). 5. Fahmy, M., Egger S., and Varzavand S., "Concrete Reinforcement Using Recycled Polyethylene Filaments", 91st Annual Meeting of the American Ceramic Society. From Abstracts, (1989). 6. Vaverka, J. "An Analysis of Reinforced Concrete Composites Using Recycled Polyethylene Terephthalate Thermoplastic." Doctoral Dissertation, University of Northern Iowa, LD2585.2.V39, (1991). 7. Rebeiz, K., Fowler D., and Paul D., "Time and Temperature Dependent Properties of Polymer Concrete Made with Resin Using Recycled PET", ANTEC Proceeding, pp 2146-2149, (1991). 8. Shah, S., "Concrete Composites, Fiber Reinforced", S. M. Lee (Ed.), Handbook of Composite Reinforcements, New York, NY: VCH Publishers, Inc., pp155-170, (1993). 9. Mehta, P. K. "Concrete Structure, Properties, and Materials." Englewood Cliffs, N J: Prentice-Hall, Inc., (1986). 10. Cook, D. J., "Concrete and Cement Composites Reinforced with Natural Fibers", D. J. Hannant (Ed.), Proceedings of the Symposium of Fibrous Concrete. New York: The Construction Press, pp 99-114, (1980) 11. Rebeiz, K., "Recycling Plastics in the Construction Industry", Waste Age, Vol. 23, pp 35-37, (1992). 12. Kosmatka, S. H., and Panarese W. C., "Design and Control of Concrete Mixtures", 13th. Edition, Skokie, IL: Portland Cement Association, (1988). 13. Howell, D.C., "Statistical Methods for Psychology", 3rd edition, Boston: PWS-KENT Publishing Company, (1992).
Section llI
MATERIAL SCIENCE
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A SHAPE MEMORY BEHAVIOR NEWLY REVEALED IN Cu-Be ALLOY Masoud, M.I. *, Naito, K. *, Era, H. ** and Kishitake, K. ***
* Graduate Student, **Associate professor and *** Professor Department of Materials Science and Engineering, Faculty of Engineering, Kyushu Institute of Technology, Tobata, Kitakyushu, 804-0015, Japan.
ABSTRACT An anomalous phenomenon has been newly found in a copper-l.90mass% beryllium alloy containing 0.28mass% cobalt through an investigation of the aging behavior of the alloy under external stress. A plate of the Cu-Be alloy was elastically bent by an external load and aged at 633K under the stressed condition. The plate warped to the direction of loading over the initial elastic bending during aging. The warped plate recovered the original shape on reheating for solutionizing at 1073K in a few minutes. Thus the Cu-Be alloy exhibited an interesting behavior similar to the shape memory effect. It was found, however, that the bending behavior of the Cu-Be alloy was not caused by martensitic transformation seen in shape memory alloys but closely related to the age-hardening process of the alloy. KEYWORDS: Copper-Beryllium Alloy, Shape Memory Effect, Elastic Stress, Aging-induced Deformation, Solutionizing-induced Restoration. I. INTRODUCTION It is known that Cu-Be alloys exhibit an excellent hardening through aging after solution treatment. The aged Cu-Be alloys have the highest hardness, tensile strength and proof stress among the copper alloys and possess an excellent electric and heat conductivity. Thus Cu-Be alloys are widely used as electric parts due to the high elastic limit and electric conductivity [ 1]. The authors have found a newly revealed phenomenon similar to the shape memory effect in a sense during an investigation of aging behavior of Cu-Be alloys. The shape memory effect was firstly found in a Au-Cd alloy about fifty years ago [2] and subsequently in In-Tl [3] and Cu-AI-Ni alloys [4]. After that, Buehler et al [5] widely demonstrated the shape memory effect revealed in a Ti-Ni alloy and the phenomenon has been accepted all over the world [6-8]. The shape memory effect is characterized by a reversible shape change accompanied by a large deformation due to the martensitic transformation [9]. The newly revealed phenomenon of Cu-Be alloy is similar to the shape memory effect in respect to the reversible shape change with a large deformation. However, the phenomenon is caused by a small amount of initial strain and subsequent aging under the externally stressed condition. Therefore, the mechanism
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is not the same as those of the shape memory effect. The present paper reports the phenomenon newly revealed in a Cu-Be alloy. 2. EXPERIMENTAL PROCEDURE Cold-rolled plates of Cu-Be alloy (Japanese Industrial Standard; JIS #C 1720) which contains beryllium of 1.8 to 2.0 mass% and a small amount of cobalt and other trmnp elements were used in this investigation. Rectangular test pieces with a size of 80 mm in length, 10 mm in width and 2.5 mm in thickness were cut out from the plates. The cut-out test pieces were heattreated for solutionizing at 1073 K for 2 hours in air. Before aging treatment, the test piece was bent by loading at the end of the test piece using a screw fabricated in a hand-made cantilever jig, as shown in Fig. 1. The strains induced by the load were measured using strain gauges attached at the points A, B and C. Figure 2 shows strains at the points of A, B and C as a function of bending deflection. Strains at the points of B and C increase linearly with increasing the bending deflection and are rather small compared to the strain at the point A. On the other hand, strain at the neck of the cantilever (point A) increases with increasing the bending deflection showing a larger tangent, then deviates from the linearity at Hb = 4mm, i.e., s = 2x 10.3 which corresponds to the proof stress. Therefore, when the strain at the neck of the cantilever, which is denoted c^, is less than 2x 10"3, the test piece is under the elastic condition. 3. RESULTS AND DISCUSSION
3.1. Aging-Induced Deformation of Cantilever The solutionized test piece was aged at 523 K, 633 K or 723 K, giving a constant bending deflection at the end of the cantilever using the screw see Fig. 1 and the bending deflection of the test piece was observed. Hardness was also measured by means of Vickers hardness test (load: 5kgf) using load-free specimens with a size of 10xl 0x5 mm. Figure 3(a) and Co) show a test piece set to the cantilever jig and subsequently bent by the screw, respectively. The bending deflection at the end of the cantilever is Hb = 4.5 mm and it corresponds to a strain at the neck of the cantilever of cA = 2.5x10 "3 which slightly exceeds the elastic limit. A small amount of plastic deformation may be introduced at the neck of the cantilever. After aging at 633 K for 3 hours, the test piece warped further to leave the screw as shown in Fig. 3(c). The warping deflection, Hw, in this case is 9 mm and the total deflection of the natural bending and warping results in 13.5 mm. The test piece warped during aging and held the shape when cooled down to the room temperature. It is natural that metals and alloys deformed elastically recover the original shape after releasing the stress. Even if metals and alloys were subjected to some stress relaxation by a structural change, they would not warp over the initial elastic deformation, but hold the shape prior to aging [ 10]. From this point of view, the warping behavior of the Cu-Be alloy is anomalous and interesting phenomenon. The shape change after aging looks like plastic deformation induced by aging. Therefore, the phenomenon is termed "Aging-Induced Deformation"(AID). As it is considered that the phenomenon is closely related to the precipitation through aging, the test piece was re-heated to remove the aging effect and to return to precipitate-free structure. Figure 3(d) shows the test piece after the solution treatment where the screw used for the initial bending was removed in advance. The test piece almost recovers the original shape, see
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Fig. 3(a) although it possesses a slight residual deflection. The slight deflection may be due to a little plastic deformation on the initial bending. Thus the Cu-Be alloy has a property of the reversible shape change and the phenomenon of AID is similar to the shape memory effect in this point. The warping and recovering phenomena include an initial elastic strain, aging and solution treatment, indicating that the phenomenon of AID is quite different from that of the shape memory effect. Figure 4(a) and (b) show the effect of aging on the hardness and warping deflection, respectively. The hardness was measured using specimens under the stress-free condition because hardness of the externally stressed specimens was hardly found to be different from that of the stress-free specimens. All the test pieces exhibit a low hardness of 100 DPN after solutionizing. On aging at a low temperature of 523 K, hardness gradually increases with increasing aging time and reaches 170 DPN in 5 hours, indicating that the test piece heattreated at 523 K up to 5 hours is in an early stage of aging. On the other hand, at a high temperature of 723 K, the hardness rapidly increases in a short time of a half-hour and reaches a maximum of 210 DPN by 1 hour followed by gradual decreasing. The decrease in hardness corresponds to a later stage of aging. Hardening is most pronounced on aging at 633 K among the temperatures where an enhanced hardness of 340 to 380 DPN is obtained after aging for 1 hour or longer. As shown in Fig. 4b, the curve of warping deflection is similar to that of hardness change during aging at each temperature. The warping begins at a half-hour and becomes rapidly large by 1 hour when aged at 633 K, exhibiting the maximum warping deflection (Hw = 9mm). It is noted that the most pronounced AID is accompanied by the highest hardness. Thus the warping behavior is closely related to the hardness change, indicating that the AID is caused by aging. Figure 5 shows the effect of initial bending strain, CA, prior to aging on the warping deflection, Hw, after aging at 633 K for 3 hours. The warping deflection induced by aging increases with increasing the initial strain and reaches the maximum at the strain of about 2x10 "3which nearly equals the elastic limit of the alloy as solution-treated. The warping deflection then decreases a little or holds a constant around 8 mm. It is seen from the result, the AID takes place also in the case of plastic strain. This is because the plastic strain always accompanies elastic strain, as expected from the stress-strain curve. Thus the warping deflection depends mainly on the amount of the elastic strain and the warping occurs even in a plastically deformed specimen. This suggests that the warping easily occurs when the specimen is subjected to initial stress or strain and the direction of warping depends on the direction of external stress.
3.2. Reversible Shape Change Caused by Aging and Solutionizing As mentioned before, the AID occurs at an aging temperature around 630K and the alloy shows an extremely high hardness after aging. The shape change by the warping phenomenon proceeds rather gradually at a wide range of temperatures due to utilizing aging and solutionizing. This means that alternating the temperature and/or time of aging besides the initial strain can control the warping deflection. Moreover, it is possible to obtain a desired shape to some extent only by controlling the amount and direction of initial strain. Therefore a desired deformation is expected to obtain by controlling the initial strain. The features of deformation can be demonstrated as follows using test pieces having various shapes.
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1. A plate of 0.35 mm in thickness was bent to deform plastically into a semi-circular ring and heat-treated for solutionizing, eliminating the plastically deformed structure at the same time, as shown in Fig. 6(a). The solutionized semi-circular ring was put inside an alumina tube with a diameter of 40 mm in order to give an external stress in the radius direction from the outside to the center, see Fig. 6(b). The ring shrank in the radius direction to close after aging, as shown in Fig. 6(c). After solutionizing, the closed ring almost recovers the original shape, resulting in an opened semi-circular ring, see Fig. 6(d). 2. An almost closed ring (Fig. 7(a)) was heat-treated for solutionizing and then an alumina rod with a length of 42 mm was put to open the ring, as shown in Fig. 7(b). The ring stretched out to be flat in the middle of the ring aRer aging, see Fig. 7(c). The opened ring recovered the original shape, resulting in an almost closed ring after solutionizing, as shown in Fig. 7(d). 3. A wire with a diameter of 0.6 mm was wound into a coil by deforming plastically. The coil was subjected to a solution treatment as same as the above cases, see Fig. 8(a). The coil was put between two plates, resulting in a slight shrinkage of the coil with a small amount of deformation, as shown in Fig. 8(b). The coil drastically shrinks after aging, and recovers the original shape after solutionizing, as shown in Fig. 8(c) and (d) respectively. Thus the warping phenomenon revealed in a copper-beryllium alloy would be very useful in future if the phenomenon is properly applied to some mechanisms and products. 4. CONCLUSIONS An anomalous and interesting phenomenon has been found during an investigation on effect of external stress on aging behavior in Cu-l.9mass%Be alloy and termed as "Aging-Induced Deformation". The characteristics of AID and the feature of shape change are summarized as follows. When a cantilever of Cu-Be alloy is aged under external stress, it markedly warps to the direction of the initial loading. The warping deflection caused by aging is closely related to the behavior of hardening process of the alloy and the maximum warping deflection is obtained at the maximum hardness. The warping deflection depends on the amount of the initial strain before aging and the maximum warping deflection is obtained at an initial strain near the elastic limit of the alloy or more. The deformed alloy by the warping through the aging recovers the original shape on re-heating at the solutionizing temperature in a few minutes, indicating that the alloy possesses a property of reversible shape change. ACKNOWLEDGMENT The authors' thanks are due to N.Shinohe, student of Kyushu Institute of Technology, for his aid of this experiment. The authors also acknowledge the Nippon Gaishi Co. for supplying the alloy used in this investigation. REFERENCES 1. 2. 3. 4.
Rosenthal, Y., J., Materials Science, 27, pp. 2193-2198, (1992). Chang, UC. and Read, T.A., J. Metals, Trans., AIME, 191, pp. 47-52,(1951). Basinski, Z.S. and Christian, J.W., Acta Metall., 2, pp. 759-761, (1954). Chen, C.W., J. Metals, pp. 1202-1203, Oct., (1957).
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5. Buehler, W.J., Girfrich, J.V. and Wiley, R.C., J. Appl. Phys., 34, pp. 1475-1477, (1963). 6. Otsuka, K., KINZOKU, (Aug., 1993), pp. 16-24. 7. Meyers, M.A. and Chawla, K.K.,(Ed),"Mechanical Metallurgy", Prentice-Hall, pp. 467514, (1984). 8. Ishida, A., Ogawa, K., Sato, M. and Miyazaki, S., J. Metall and Mat Trans.,28A, pp. 1985-1991,(1997). 9. Miyazaki, S., Otsuka, K. and Suzuki, Y. J. Scripta Metailurica, l 5pp. 287-292, (1981). 10. Miki, M., Tenma, A., Ishikawa, S. and Ogina, Y., J. Mat Trans., JIM, 39,No.4,pp. 455462, (1998).
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Fig.2 Relationship between bending deflection, H b, and strains at the points of A,B, and C shown in Fig.l.
Fig.3 Shape change of cantilever. (a): before bending, (b): bent by screw, (c): aged at 633K for 3 hours, (d): solutionized at 1073K for 5 minutes
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Bending Strain, e A/10 .3 Fig.5 Effect of initial bending strain, e A, on the warping deflection, HW, in test piece aged at 633K for 3 hours.
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Fig.6 Closing/opening system. (a): deformed into semi-circular ring followed by solutionizing at 1073 K for 2 hours, (b): put inside alumina tube, (c): aged at 633 K for 3 hours, (d): solutionized at 1073 K for 5 minutes.
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Fig.7 Opening/closing system. (a): deformed into almost closed ring followed by solutionizing at 1073 K for 2 hours, (b): stretched out using alumina rod, (c): aged at 633 K for 3 hours, (d): solutionized at 1073 K for 5 minutes.
Fig.$ Shrinking/elongating system. (a): wound into coil followed by solutionizing at 1073 K for 2 hours, (b): put between top and bottom plates, (c): aged at 633 K for 3 hours, (d): solutionized at 1073 K for 5 minutes.
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POISONING OF GRAIN REFINEMENT OF SOME ALUMINIUM ALLOYS AbdeI-Hamid, A.A." and Zaid, A.I.O. §
* Mechanical and Industrial Engineering Dept., Faculty of Engineering, Applied Sciences University, Amman 1193 l, Jordan. + Industrial Engineering Dept., Faculty of Engineering and Technology, University of Jordan, Amman, Jordan.
ABSTRACT In this paper, the poisoning effect of Zr, Cr and Ta, present in the AI melt, on the grain refining efficiency of the commercial AI-Ti and AI-Ti-B master alloys is reviewed. The poisoning mechanism is established and three different approaches are proposed to overcome the grain refinement poisoning problem in the Al alloys containing Zr. KEY WORDS Solidification, Grain Refinement, AI-Alloys, Master Alloys, Poisoning, Zr, Cr, Ta. I. INTRODUCTION A fine and equiaxed grains structure is obtained in AI and its alloys by adding a small amount of AI-Ti or AI-Ti-B master alloys into the melt before casting. Many commercial grain refining alloys with different compositions are available but the most efficient one is the AI-5% Ti-1% B alloy [ 1]. In order to obtain satisfactory grain refinement the Ti-level in the A1 melt must be > 0.15% (the peritectic limit in the AI-Ti phase diagram) when using Ti alone, whereas it can be lowered to a value of 0.005% in the presence of B. The grain refinement mechanism is still a controversial matter although many attempts have been made to explain it since 1949 [e.g. Ref. 2-12]. It was reported that the grain refinement of Al is due to the nucleation ofct - AI by the aluminide particles [2], by boride/carbide particles [3-5] or by TiB2 particles coated with TiAI3 "duplex particles" [6-9]. Abdel Hamid [10] has discussed the mechanism and showed that a high local Ti-concentration exists in the vicinity of TiB2 particles making them high efficient nuclei for AI grains and this has also been suggested later by others [ 11,12]. The grain refining practice showed that the efficiency of grain refinement depends upon: The composition (Ti-concentration and Ti/B ratio) of the grain refining master alloy and its microstructure (size and morphology of TiAI3 and TiB2 particles). The rate of addition of the master alloy (level of Ti added). 3. The melt temperature and holding time (contact time) before casting. 4. The purity of AI or the melt composition, i.e., level of the impurities such as Fe and Si or the presence of alloying elements which can enhance grain refinement or "poison" it. .
It was found that the grain refining efficiency deteriorates in the following casses: a) Superheating the AI melt (high pouting temperatures) from the normal temperatures. b) Prolonged holding (contact) times after adding the grain refiner (fading).
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c) Passing certain gasses (N2, Ar, air or H2) through the melt after adding the grain refiner. d) Presence of certain elements, namely, Zr, Cr, and to less extent Mn, Ta, Li, and high levels of Si, in the Al melt. Zr and Cr are known as poisoning elements since they adversely affect the grain refining efficiency of the master alloys, and may cause early fading and incomplete conversion from a columnar to equiaxed grain structure. The grain coarsening from a to c are well understood and can be avoided. They can be attributed to the removal of active nuclei from the bulk of the AI melt by dissolution, gravity segregation/decantation or adherence of the nuclei with the floating gas bubbles. Many detailed information about this subject was reported in the literature {see e.g. Refs. 4,5 }. On the other hand, the exact reasons for the grain coarsening caused by the presence of the poisoning elements such as Zr and Cr, are not well known and are still a controversial matter. Although many hypotheses have been proposed to explain the poisoning mechanism, no work was devoted to overcome this problem or to find a solution for it. Therefore, this paper is devoted to provide a deep understanding of the poisoning effect ofZr, Cr, and Ta through reviewing the previous work and reproducing some of our previous results dealing with the presence of other elements including Zr and Ta in the AI melt [13-16]. Based on this understanding and some preliminary results obtained from our new experimental work, some suggestions to overcome the problem are proposed. 2. POISONING OF GRAIN REFINEMENT OF AI-ALLOYS CONTAINING Cr Cr (0.25-0.30%) is present in some Al-aUoys, e.g., alloy 5052, alloy 6061 and alloy 7075. Cr is added as a corrector for Fe and to produce a golden color in anodizing and for other purposes in AI-Zn-Mg alloys [17]. Addition of Cr alone in the AI melt has no or very limited grain refining effect [ 17]. It was assumed that the effect of Cr on the grain refinement of Al-alloys grain refined by Ti is similar to that of Mn because Cr also forms a complex carbide [3,18]. The poisoning of grain refinement caused by the presence of Cr in AI melts grain refined by the AI-STi-IB master alloys was reported by many investigators [1,13,19-22]. It hasbeen observed that the Cr-content and purity of the AI melts play an important role in deciding the grain refining efficiency of the AI-Ti-B master alloy [20-22]. In general, the poisoning effect of Cr was explained in the same manner as that of Zr (1,13,14,19). This effect was attributed to the coating of TiB2 by a layer of CrB2 which is a poor nucleant for AI [1,4]. It was also explained in terms of the atomic size mismatch of Cr with the solvent AI according to the nucleation theory of Jones [ 19]. Abdel-Hamid [ 13,14] proposed a new poisoning mechanism to explain the poisoning effect of Cr and Zr. He believes that some of the Ti atoms in TiAl3 and TiBz crystals, which are active nucleants for AI, are replaced by atoms of the poisoning elements such as Cr, Zr, etc. producing less effective ternary compound nucleant particles. Recently, Arjuno et al [21-23] have explained the poisoning effect of Cr with experimental evidence which corroborates Abdel-Hamid's mechanism. 3. POISONING OF GRAIN REFINEMENT OF AI-ALLOYS CONTAINING Zr Zr is a minor addition (0.1-0.2%) to some AI alloys (AI-Mg, AI-Zn-Mg, and AI-Zn-Mg-Cu alloys) to reduce the stress corrosion susceptibility (17). Zr has some grain refining effect when added alone in Al-alloys but its effect is much less than that of Ti and is not enhanced by B addition (2,4,23-25). It was reported (17) that the grain refining effect of Ti and Zr are not additive and the effect of the two elements together is less than the effect of each element separately. Very little work was published on the effect of Zr on the grain refining efficiency of
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the binary AI-Ti master alloys [13,14]. However, its effect on the grain refining efficiency of the ternary AI-5Ti-1B master alloys was reported in various papers in CPAI, HPAI and AIalloys [1,14,20,23-29]. It was reported [1 ] that the grain refinement of the AI-5Zn-I.5Mg alloy containing 0.2%Zr by AI-5Ti-1B master alloy was difficult at an addition rate of 0.005%Ti but it was not difficult at an addition rate of 0.020%Ti. The grain refining behavior of the 7050 AIalloy containing 0.1%Zr using AI-5Ti-IB alloy was studied by others [26,27]. It was also reported that the poisoning effect of Zr is influenced by the presence of other elements such as Li which increases its effect [28]. Spittle and Sadli [29] have reported that Zr when present with other elements like Fe and Si results in a poisoning effect whereas the presence of Zr alone in high purity AI (99.99%) is beneficial towards grain refinement by the AI-5Ti-1B alloy (at a rate of addition of 0.01% Ti). Arjuna et al [23] have studied the grain refinement of Al0.2Zr and AI-2%Zr alloys using an AI-5Ti-IB alloy at an addition rate of0.01%Ti. They reported that the poisoning effect was milder at higher concentrations of Zr and in the presence of Si and Fe impurities in AI. It was also reported that the poisoning effect of Zr is slow and depends on the contact time [23,31 ]. The poisoning effect of Zr was attributed to the formation of a coating layer around the TiB2 particles which prevents these particles from nucleating ot-Al [1,32]. The coating layer was assumed to be Zr [32] or ZrB2 [1 ]. The latter is a poor nucleant for a-Al (1,24). However, no evidence of ZrB2 formation was shown by Jones and Pearson [ 1] or by any other investigator so far. It was also reported that the poisoning effect of Zr is due to the mismatch between the Zr solute with the solvent AI [19]. Abdel-Hamid's mechanism, mentioned in the case or Cr, is valid also for explaining the poisoning effect of Zr as will be shown later. This new mechanism was supported and confirmed in the case of Zr by Johnsson [28] and Arjuna et al [23]. For better understanding of the effect of Zr on the grain refining efficiency of the commercial master alloys and the poisoning mechanism, it was needed to reproduce some results from the comprehensive experimental work on the effect of the presence in the commercially Al-melts of different contents of Zr, Ta, Mo or V, (from 0.0 to 0.3%) on the grain refining efficiency of AI-4%Ti and AI-5%Ti-I%B commercial alloys [14]. The crystals formed from AI-Ti melts containing one or two of the above elements were also separated and analyzed for the chemical composition, crystal structure, size and growth morphology [ 15-16]. Figure l shows the effect of Zr-content (0.0-0.3 wt%) on the average grain size of commercially pure Al grain refined by the AI-4.18%Ti alloy (at an addition rate of 0.15 wt% Ti) or by AI-5.1Ti-l.12B master alloy (added at a rate of 0.045%Ti). The addition of grain refiner was carded out into the AI-Zr melt in a graphite crucible at - 1023K while stirring the melt for 2 min and then the alloy was allowed to cool and solidify outside the furnace. The average grain size (taken on 15 fields) was determined by the intercept method. It can be seen from this figure that: 9 The presence of Zr in the Al melt causes grain coarsening of the solidified ingots. This means that Zr has a poisoning effect on the efficiency of the two master alloys allover the investigated range of Zr-content even at high levels of grain refiner addition and short contact times used in this study. This dose not agree with the results of Arjuna et al. [23] obtained on AI-0.2Zr alloy at 2-min contact time. They reported that the addition of Al5Ti-1B to CPAI-0.2Zr alloy has resulted in an initial improvement in grain refinement up to 5min followed by fading. 9 The effect of Zr on the efficiency of the AI-5Ti- l B alloy is much greater than its effect on the efficiency of Al--4Ti alloy. The grain size increases steeply and almost linearly with increasing the Zr-content in the case of ternary alloy whereas in the binary alloy case, the
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increase in grain size with increasing Zr-content in sharp up to 0.05% Zr after which the increase is gradual and small. These results will be discussed later with the mechanism. 9 The grain size exceeds the grain refinement acceptance level (220~m as the average grain size [ 1]) at a Zr-content in the Al melt of-_- 0.1% or 0.15% Zr when the AI melt was grain refined (under the conditions reported above) by the AI-4Ti alloy or By AI-5Ti-IB alloy respectively. 4. GRAIN COARSENING CAUSED BY Ta Ta is a good grain refiner in AI alloys and it is approximately as efficient as Ti [17]. To the best of our knowledge except the work of Abdel-Hamid [ 14], there is no published work reporting the effect of Ta in the melt of Al or Al-alloys on the grain refining efficiency of the binary or ternary master alloys. Abdel-Hamid studied the effect of Ta-content up to 0.3% on the efficiency of the AI-4Ti and AI-5 Ti-l B alloys under the same conditions and addition rates shown in the case of Zr reported above. His results are reproduced in Fig.2 which shows that:
/
DO T IO IN
/
o ~TSB~ool+xo/,.,,,..~.....
~300[
="
.....
300
9 AT/. ADDITION O ATSBAOOITION
Q: w ~ zsc z < n, o
AC,CEPTANC lr LgYEL
0W 2
0
1
5 00
0
0 01
~
~
i
,
vs.,.T,
00
~
02 03 Zr - CONTENT , wt "I.
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Fig.l. Effect of Zr addition on the grain size produced by adding AT4 and ATSB master alloys into A! melt (0.15%Ti and 0.045%Ti+0.01%B respectively)
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0.1 To
02 03 - CONTENT , wt %
Of.
Fig.2. Effect of Ta addition on the grain size produced by adding AT4 and AT5B master alloys into A! melt (0.15%Ti and 0.045% Ti+0.01%B respectively)
9 The effect of Ta-content in the AI melt on the average grain size and consequently on the grain refining efficiency of the two master alloys is similar. 9 In the two cases the average grain size increases by increasing the Ta-content up to a maximum at = 0.12% (the peritectic limit in the AI-Ta phase diagram [17] then it starts to decrease again by further increase in the Ta-content. In case of the AI-5Ti-IB alloy the initial grain size (without addition of Ta) is restored at a Ta-content of 0.2%. Beyond this level the presence of Ta becomes beneficial for the grain refining efficiency of the AI-5Ti1B. The initial grain size in the case of the AI-4Ti alloy cannot be restored at 0.30%Ta and it can be expected that a Ta-content of 0.35% can achieve this objective. This means that only the Ta present in solution (below the solubility limit) has adverse effect on the efficiency of the two grain refiners. The decrease of the average grain size observed beyond the Ta-content of 0.12% can be attributed to the presence in the Al melt of the TaAl3 crystals which are known to be as efficient as TiAI3 crystals in nucleating t~-AI (17,33). Therefore, the contribution of these TaAI3 crystals in grain refinement can compensate the deteriorating effect of Ta in solution in the molten Al.
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9 According to the acceptance level of grain refinement, the presence of Ta up to 0.3% in the AI melt has so little grain coarsening effect that the grain size remains in the acceptable range. Therefore, we cannot consider Ta as a poisoning element under the conditions prevailing in this work. 5. EXPERIMENTAL WORK An extensive experimental program is being carried out to find a solution for the grain refinement poisoning effect of Zr. Some of the new experiments were designed to study the effect of the presence with Zr of other elements, namely, Mo, V, Ta, and B on its poisoning effect in the case of using AI-4Ti or AI-5Ti-IB master alloys as grain refiners. Only some preliminary results are available now and the result will be published in due course. The following materials and facilities were used in this work: 9 Commercially pure aluminum (99.8%) containing 0.09% Fe, 0.05% Si, 0.008% V, 0.005% Cu, 0.005% Zn, 0.005 Na, 0.004% Ti, 0.004% Mg and 0.001% Mn. 9 Pure Zr, Ta, V and Mo metals (>99.8% purity) for preparation of laboratory AI-1%Zr and AI- 1%Ta, AI- 1%V and AI- 1.3%Mo master alloys. These master alloys were prepared at 1373K by adding and stirring these metals into the molten AI bath covered with cryolite. 9 Commercial AI-4%Ti, (AT4), and AI-5% Ti-1%B, (AT5B, both wire and ingot types) grain refining master alloys and commercial AI 3%B master alloy. These alloys contain Fe (0.015%) and Si (0.08%) as main impurities. 9 Graphite crucibles and stirring rods for melting and stirring the desired alloys. 9 Automatic control electrical furnace (vertical type) with an accuracy of_+ 3K. The experiment of grain refining was done as follows: Melting the predetermined quantity of the commercially pure AI in a graphite crucible. - Adding the calculated amount(s) of the desired binary AI-Me master alloys (Me=Zr, Ta, Mo, V or B) into the Al-melt with stirring and the bath temperature was raised to about 1093K and then lowered to 1023K. - Adding the commercial AI-Ti or AI-TiB master alloy at 1023K with stirring for 2 minutes. - Allowing the alloy to cool and solidify in the crucible outside the furnace. - Cutting the ingot at 1 cm from the bottom and preparing the surface for the metallographic examination by conventional methods. Determining the average grain size on 15 fields by the intercept method. -
-
6. DISCUSSION OF RESULTS AND POISONING MECHANISM As mentioned before, Jones and Pearson [1] attributed the poisoning effect of Zr on the efficiency of the AI-5Ti-IB alloy to the coating of TiB2 with ZrB2 layer according to the reaction: Zr (in AI melt) + TiB2 ~ Zr B2 + Ti (in AI melt) They also concluded from their thermodynamic calculations that the above reaction is only possible when the mole fraction of Zr-content in the melt exceeds about 4 times that of excess Ti remaining in solution in the melt (excess Ti--total Ti content- Ti in TiB2). In our experiments the AI-5Ti-IB alloy was added at a rate of 0.045% Ti, therefore, this condition does not apply except at a Zr content of about 0.2wt% or more, but the poisoning effect of Zr occurs at 0.05wt% and rises steeply with increasing Zr-content from 0.0 to 0.3wt%. The coating of TiB2 with a monolayer of Zr or ZrB2 requires very little amount of Zr to isolate and prevent the good nucleant TiB2 particles from being active nuclei for ~ AI (only ZrB2 required
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is 0.2% TiB2 content [ 1]). This fact does not agree with the continuous and steep effect of Zr content on the grain coarsening as shown in Fig.l. Finally, the mechanism based on the formation of ZrB2 or Zr coating does not agree with the slow action of Zr and its dependence on the contact time before casting [ 1, 23, 31 ]. On the other hand the experimental results are well explained by the mechanism proposed by AbdeI-Hamid [ 13, 14] which not only explains the poisoning effect of Zr but also explains the effect of other elements such as Cr and Ta. According to this mechanism, the poisoning element (Zr, Cr, Ta) in the melt diffuses into the active nucleants TiAI3 and TiB2 present in the grain refiner producing ternary compounds which are less effective nucleants for cz-Al (probably due to unfavorable lattice prameters [16]). Here, the free energies of formation of the ternary phases are lower than that of their corresponding binary aluminides or borides. The formation of the ternary compounds as well as their chemical formulae can be represented by the following reactions, into which Zr, Cr or Ta is replaced by Me: ,, Me (in AI melt) + TiAI3 --~ (Mex, Til-x) AI3 + Ti (in AI melt) x Me (in AI melt) + TiB2 -0 (Mex, Til.x) B2 + Ti (in AI melt) According to Abdel-Hamid's mechanism, increasing the poisoning element concentration in the melt and/or increasing the contact time before casting will increase the concentration of this element in the ternary aluminide or boride particles. This will increase the deviation of their lattice parameters and grain refining properties from that of the initial good nucleants TiAI3 and TiB2 particles (towards that of MeAl3 and MeB2 which are poor or less active nucleants for AI). Therefore, it can be expected that the grain refining efficiency of the master alloy will deteriorate continuously with increasing the contact time and the poisoning effect will be greater at higher concentration of the poisoning element, and at higher superheats in the AI melt. These facts are in good agreement with the experimental results and grain refining practice. Finally, the difference in the effect of Zr on the grain refining efficiency of AI- 4Ti and AI 5Ti- 1B master alloys can be attributed to the large difference in the grain refining power of TiB2 and ZrB2 particles, compared with the difference in the grain refining properties of TiAl3 and ZrAl3 which is lower. This agrees with the fact that addition of B to AI-Ti and AI -Zr alloys increases the difference in grain refining efficiency of the two alloys [24]. To confirm his mechanism, Abdel-Hamid [15,16] has studied the formation and mode of crystallisation (crystal structure and growth morphology) of some complex (ternary and quaternary) Ti aluminides containing one or two of the refractory elements, namely, Zr, Mo, Ta, and V. These complex crystals are formed during slow cooling from diluted Al- Ti melts containing one or two of the above elements (the total solutes concentration was less than 0.7 at%). From this study the following facts were obtained: a) The crystal formed from the ternary AI-Ti-Me melts and those formed from the quaternary melts (AI-Ti-Me-Me\) have chemical analyses according to the formula (Til.x ,Mex)Al3 and the formula (Til.x.y. Mex -Mey) Al3 respectively. b) These ternary and quaternary phases have the same crystal structure of the binary Al3 Ti phase with some changes in the lattice parameters (according to the X- ray diffraction analysis carried out on the powder of the crystals separated by dissolving of the Al matrix). c) The presence of Zr, Ta or V separately or in a combination of two elements affects the growth morphology of the TiAl3 crystals to some extent but Mo has a greater effect. This fact has also a direct relation to the grain refining action as the morphology of TiAl 3 crystals in the grain refiner affects its grain refining efficiency [34].
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All the above facts support the mechanism proposed by Abdel-Hamid to explain the grain refinement poisoning caused by Zr, Cr and Ta. Direct metallographic evidences showed that the particles exising at the cystallisation centres in the AI-6Mg-0.15 Ti-0.1Zr alloy (grain rfined by AI-Ti alloy) were (Ti,Zr) AI3 [35] and in the AI - Z r - Mg- Cu containing 0.1% Zr grain refined by A I - 5Ti- 1B were rich in Ti and Zr [32]. Recently, this mechanism was supported by further experimental evidences [21, 22, 28]. For example Arjuna et al [21, 22] have explained the poisoning effect of Cr with experimental evidence which corroborates this mechanism. Also, Johnsson [28] in the case of Zr, argued that the poisoning effect of Zr is associated with the phase ZrAI3 in which some Ti may dissolve to form the complex (Til.x ,Zrx) AI3 resulting in the loss of grain refining effect of Ti. Finally, in 1997 Arjuna et al [23] have identified complex Ti aluminides containing Zr at the grain centers, Zr- aluminides containing Ti and Fe and Fe - aluminides containing Ti and Zr (at the grain boundaries) in the microstructure of A1-2% Zr alloy grain refined by the AI - 5Ti-I B master alloy at an addition rate of 0.01%. 7. PROPOSALS FOR OVERCOMING THE POISONING EFFECT OF Zr Three different approaches are proposed to overcome the poisoning effect of Zr: 9 The first one is based on the experimental results obtained from combining Zr with one or two elements which can stop its poisoning effect. Our preliminary experiments showed that Mo and V are not suitable candidates for this task whereas Ta seems to be a promising element. This approach must be extended to include other possible elements. On the other hand it was reported that the presence of Li [28] or Fe and Si [29] increase the poisoning effect of Zr. Therefore, this fact must be taken into consideration for AI alloys containing these elements. 9 The second approach is based on the treatment of the AI melt containing Zr with AI-B master alloy before the addition of grain refiner master alloy. It is expected that addition of B causes the precipitation of ZrB2 and hence very little Zr will take place into TiAI3 and TiAI3 and TiB2 and therefore their nucleation power and grain refining properties will not be greatly affected by the remaining Zr in the melt. However, this treatment may have a drawback on the AI- alloy properties affected by the Zr present in a-Al. This drawback must be studied in parallel. 9 The third approach is simple and can be applied in industry without difficulties. It is based on optimisation of the addition rate of the grain refiner and the contact time before casting. 8. CONCLUSIONS The presence in the Al melt of certain elements such as Zr and Cr have adverse effect on the grain refinement of Al and its alloys by the Al- Ti and AI-Ti-B commercial master alloys. The poisoning effect of Zr increases by increasing its content in the Al melt in the case of the two types of master alloys. The effect of Zr on the efficiency of the ternary alloy is higher than its effect on the binary alloy. The presence of Ta up to 0.1% causes grain coarsening but beyond this level the grain size starts to decrease until the initial grain size is restored. However, the grain coarsening in the case of Ta is not danger since at all Ta- contents the grain size remains in the acceptable range. The grain refining poisoning of these elements are well understood and explained by Abdel-Hamid's mechanism. According to this mechanism the poisoning elements diffuse into the active nucleants particles, TiAl3 and TiB2 present in the master alloys and form ternary compounds such as (Til.x,Mex) Al3 and (Til.x, Mex) B2 where Me = Zr, Cr, Ta. These complex particles are less active nucleants than the initial ones. Therefore, the grain refining deteriorates with time and with increasing the concentration of the poisoning element
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in the AI melt which led to more diffusion. Some proposals to overcome the problem were presented and an experimental work is now in progress to evaluate these proposals. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35.
Jones, G .P. and Pearson, J., Metall. Trans., Vol. 7B, pp. 223-234, (1976). Crossley, F. A. and Mondolfo, L. F, J. of Metals, Vol. 191, PP. 1143-1148, (1951). Cibula, A., J. Inst. Metals, Vol. 76, pp. 321-360, (1949/50). Cibula, A., J. Inst. Metals, Vol. 80, pp. 1-15, (1951/52). Moriceau, J., Rev. Aluminium, Vol. 413, pp. 977-987, (1972). Backerud, L., Jernkontorest Ann, Vol. 155, pp. 422-424, (1971). Maxwell, I. and Hellawell, A., Metall Trans., Vol. 2A, pp. 1733-1738, (1972). Cornish, A. J., Met. Sci., Vol. 9, pp. 477-484, (1980). Guzowski, M. M., Sigoorth, G. K and Sentner, D. A., Metall. Trans., Vol. 18A, pp. 603619, (1987). Abdel- Hamid, A. A., Proc. 2nd Arab Aluminium Conf., (Arabal 85), Cairo, 23-26 Oct., (1985) paper no 111-4. Johnsson, M., Light Metals, 1993, pp. 769-777, (1993). Backerud, L., Gastafson, P. and Johnsson, M., Aluminium, Vol. 67,pp. 910-915, (1991). Abdel-Hamid, A. A., Z. Metallkde, Vol. 80, pp. 566-569, (1989). Abdel-Hamid, A. A., Z. Metallkde, Vol. 80, pp. 643-647, (1989). Abdel-Hamid, A. A., Z. Metallkde, Vol. 81, pp. 601-605, (1990). Abdel-Hamid, A. A., Z. Metallkde, Vol. 82, pp. 383-386, (1991). Mondolfo, L. F., "Aluminium Alloys: Structure and Properties". London, Boston, Butterworths., (I 976). Glasson, E.L., and Emley, E. F., The Solidification of Metals, ISI, pp. 110-, (1968). Jones, G.P., "Solidification Process 1987", London, Inst. of Metals, pp. 496-499, (1988). Birch, M. E.J. and Cowell, A.J.J., "Solidification Process 1987", London, The Institute of Metals, pp. 149-152, (I 988). Arjuna Rao, A., Murty, B.S. and Chakraborty, M." MetaU. Mater Trans. A, Vol. 27A, pp. 791- 800, (1996). Arjuna Rao, A., Murty, B.S. and Chakraborty, M." Cast Met., Vol. 9, pp. 125-132, (1996). Arjuna rao, A., Murty, B. S and Chakraborty, M; Material Sci. &Teeh.,. Vol. 13, pp. 769777, (1997). Delamore, G. W. and Smith, R.W., Metall. Trans., Vol.2, pp. 1733-1738, (1971). Naess, S.F., Z. Metallkde, Vol. 69, pp. 33-34, (1978). Birch, M. E.J. and Fisher, P., "Aluminium Technology", London, The Institute of Metals, pp. 149-152, (1986). Ahmady, S.M., McCartney, D.G. and Thistlethwahi, Light Metals 1990, pp. 837-843, (1990). Johnsson, M.: Z. Metallkde, Vol. 85, pp. 786-789, (1994). Spittle, J. A. and Sadli, S., Cast Met., Vol. 8, pp. 247-253, (1995). Birch, M.E.J., "Aluminium -Lithium Alloys III", London, The Institute of Metals, Vol. III, pp. 152-158, (1986). Kawecki, Billiton Information Sheet No. 114, Oct, (1976). Koch, H. and Reif, W., Proc., 1st. Int. Conf. Production Eng. and Design for Development, Cairo, Ain shams Univ., Vol., 2, pp. 345-360, (1984). Grebe, W. and Grimm, H.P., Aluminium, Vol. 43, pp. 673-679, (1964). Arnberg, L., Backerud, L. and Klang, H., Metals Technology, Vol. 1, pp. 1-7, (1982). Lashko, N.F., Matveeva, G.N., Morozova, G.L. and Zhorova, L.P., IZV Akad Nauk SSSR Met. Vol. 1, pp. 121 - 123, (1978).
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, Februat 3, 15-17, 2000
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EFFECT OF SHOT PEENING ON THE FATIGUE STRENGTH OF 2024-T3 ALUMINUM ALLOY IN THE UNWELDED AND WELDED CONDITIONS Zaid, A.I.O.*, Ababneh, M.A.** and AI-Haddid, T.N. ***
* Professor, Industrial Engineering Department, Faculty of Engineering & Technology, University of Jordan, Amman, Jordan, e-mail: [email protected] ** Eng., Industrial Engineering Department, University of Jordan, Amman- Jordan *** Dr. Maintenance Department, Royal Jordanian Airlines, Amman - Jordan
ABSTRACT In this paper, investigation of the use of aluminium alloy 2024-T3 in the welded condition is carried out. The effect of shot peening on the improvement of fatigue strength and fatigue life of this alloy in the welded condition is studied. Two types of welded joints are studied, the butt single-U and the double Vee. The effect of the shot peening intensity on the fatigue strength and fatigue life of these two welded joints is also investigated. It was found that for both joints, the fatigue strength and life were improved, and that the improvement increases as the shot peening intensity increases. Furthermore, it was found that the butt single-U welded joint has higher fatigue strength and life than the double Vee welded joint. Metallurgical examinations were carried out and revealed that the depth of the induced compressive stress layer increases with the increase of shot peening intensity. The microhardness survey showed that the heat affected zone, HAZ, is of higher hardness than the base metal, which in turn is higher than that of fused metal zone. KEYWORDS Aluminum Alloy 2024-T3, Shot Peening, Fatigue Strength, Welded Joints, HAZ. I. INTRODUCTION Aluminum and aluminum alloys are widely used materials in construction and design of machine parts due to their high strength - weight ratio and their good resistance to wear and corrosion. Aluminum alloy 2024-T3 is a widely used material in the aircraft industry. It is used in the as received condition (un-welded), and most of the published work on this alloy is at this condition. The possibility of using this alloy in the welded condition widens its use and applications in the aircraft and automobile industries. However, the restrictions of its use in the welded condition are due to its liability to stress corrosion cracking, and its poor fatigue strength. Welding in general provokes a non uniform localized temperature field, which involves a speed heating of the base metal to the melting temperature followed by kinetic cooling.
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Depending on the welding parameters, metallurgical, mechanical and geometrical modifications are created in the welding joints. They involve a brittleness, residual stresses and stress concentration, which decrease the fatigue performance of the welded parts [1]. Wohlfahrt, et al [21 reported that shot peening can be used to enhance the performace of welded parts. Enhancement in fatigue strength stems from several effects, especially from the generation of compressive residual stresses and the increase of the surface hardening. The fatigue strength of weldments can be improved by a number of methods especially by utilizing and using of a preferable distribution of residual stresses. Some treatments such as local compression, spot heating, shot peening and some times prior overloading improve the fatigue strength by creating residual compressive stresses at locations where fatigue cracks are likely to initiate as reported by Masubuchi, [31. Haagensen, [4], studied some methods for fatigue strength improvement and repair of welded offshore structures, and suggested several methods for in-service repair and improvement of welded details are reviewed. Nguyen and Wahhab, [5] developed a theoretical analysis for the effect of residual stresses on the fatigue strength of the butt welded joints by using linear elastic fracture mechanics. Chow and Wei, [6] presented a fatigue damage model for crack propagation in characterizing the fatigue crack growth behavior of cracked structures made of 2024-T2 aluminum plates. Although shot peening is probably the most well known method available for improving fatigue strength and has been widely used in aircraft and automobile industries for the treatment of mechanical parts such as coil and leaf springs, axles and crank shafts, its application to welded components has been fairly limited. However, experimental work has indicated that significant improvement in fatigue strength can be obtained. Measured in terms of fatigue strength at 2x106 cycles the magnitude of these improvements for steel ranged from 15 to 100% as reported by Gurney, [7]. The most extensive test program that has been carried out on the effect of peening on welded joints is that by Nacher, [8]. His work induced tests on both butt welds and fillet welds. The greatest increase in strength obtained was 87% for a transverse butt weld in mild steel. The actual strengths at 2x106 cycles under pulsating tension being 147 and 247 N/rnm 2 in the as welded and peened conditions respectively. However, this improvement was exceptional since in other tests on butt welds, the increase in fatigue strength ranged between 15 and 25%. Konishi et al, [9] obtained an improvement of 100% for high strength structural steel, when treated by normal shot peening and similar improvement has also been achieved for a longitudinal butt weld in AI-5Mg alloy. The only tests which seem to have been reported on transverse butt welds in aluminum alloys, are those of Hedstrom, [10] who obtained an increase in strength of 50%, both at R=0 and R= -1 on specimens fabricated from AI-Zn-Mg alloy. Harison, [11] in his work on transverse fillet-welded specimens, tested few specimens which were peened and subsequently stress relieved. The fatigue strength at, 2xl 06 cycles, of about 127 N/mm 2, was obtained which represents an increase in strength of about 20% as compared with an increase of 85% due to peening without subsequent stress relief.
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A very comparable result, for the same type ofjoint, has also been reported by Booth, I12], in which similar work on transverse butt welds gave an increase of only 6% after stress relief, compared with 46% due to peening alone, where as tests on T-butt welds showed no improvement in strength at all after peening followed by stress relief. Dorr and Wagner [13] reported data on the residual life of fatigue pre-damaged aluminium alloy 2024 in the stretched and naturally aged condition after shot peening. Depending on the shot peening intensity, fatigue induced initial crack with a depth of up to 1500~tm did not reduce residual life as compared to the crack free electropolished reference. The main objective of this work is to investigate the effect of shot peening on the fatigue strength of 2024-T3 aluminum alloy in the welded condition, which if proved effective will widen its use and enhance its cost effectiveness. 2. MATERIAL, EQUIPMENT AND PROCEDURE 650
2.1. Material
600
The material used throughout this work is aluminum alloy 2024-T3 of the-chemical analysis shown in T a b l e 1. It is a copper-magnesium alloy of high mechanical and fatigue strengths. In addition, it can be easily worked and machined to a good surface finish which makes it applicable and widely used in the aircraft industry. It is also a heat treatable alloy and used in the T3 temper condition, which is a solution heat treated, cold worked and aged to a stable condition to improve its tensile strength. Its mechanical behavior as representative stress, ~ versus representative strain, z is shown in Fig. l. Cast steel shots, grade $230 having a mean diameter of 0.6mm conforming to MIL-S-851 were used.
o
It; . O3 ~ ~ ~ ~~ ~ 2
tu
550
500 ~5o ,.oo 350 300 2so 200 15o 1 O0
i :
so % _
|
l
j
i
i
9
=
002 004 006 oo8 olo o12 o14 REPRESENTATIVES'rRAIN ~
Fig.1. Mechanical behaviour of A12024-T3. Table 1. Chemical composition of 2024-T3 aluminum alloy by weight.
I E m nt" '
!
Weight(~A) !-90.9/93.2
Cu
I t si
3.8/4.9 '0.5
Fe [
1
Mn . . . . . i M,'g !.2/1.8
0. 5 ! 0.3!0.9
I i I Ti' Zn i "Cr '. 0.25 .......0,i 0.15 I
2.2. Equipment The mechnaical behaviour of aluminum alloy 2024-T3 as representative stress,~ versus representative strain, ~ was determined using the hydroservo universal testing machine MTS model, of 250 kN capacity at a cross head speed 3x10 "3 ram/s, giving a strain rate of 5x I 0"5 s"i . The shot peening was carded out using a pangborn ES-1580. In this machine a compressed air jet is fed into the nozzle to create a low pressure, high velocity air flow in the suction line leading to the gun. The shots are fed to the gun by gravity from an overhead hopper, where they merge with high velocity air stream. The shot velocity is varied by varying the air
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Current Advances in Mechanical Design and Production, MDP-7
pressure fed to the nozzle. The workpiece is mounted on a rotating table with special fixture and the table speed is controlled by a gear and motor system. Details of the machine and the Almen gauge system are given by Zaid, et al [14], and schematic drawing is shown in Fig.2. The shot peening parameters which include shot velocity, pressure, peening time, stand off distance and table rotational speed were altered until the required intensity was achieved. The fatigue tests were carried out on a standard 9choler rotating fatigue apparatus, which uses a cantilever rotated at 2850 rpm electric motor through a 2:1 step drive. The microhardness survey on the welded specimens was carried out using a microhardness tester MUT-1 at a load of 50gf. 2.3. Procedure
2.3.1. Welding joints Aluminum alloy 2024-T3 welded joints were obtained without post welding thermal treatment and by using Gas Tungsten Arc (TIG) welding. Two types of welded joints were used, the first one was butt double Vee joint, and the second one was butt single-U joint with root faces. The test pieces were constrained to avoid any deformation caused by welding. The welding joints were checked up against any voids and cracks by X-ray method. The welding conditions at which the specimens were prepared are those used by the maintenance department of the Royal Jordanian Airlines and given in Table 2.
Table 2. Welding conditions of joints Item
Sizec~mm)LXWxT a(0)
Value .
125x12.5x12.5 . .
60
h(mm) .
,
(mm)d p(0)
3
.
3
D(mm)
40.
6.25
r(mm)- power 5
AC
Shieldgas Filler wire i Argon
AMS-4'190[,,
Fatigue specimens were machined from a 10ram diameter of the welded joints using a Computer Numerical Control (CNC) lathe to the dimensions shown in Fig.3, taking into consideration that in all specimens of both types of joints the welded region falls in the middle of the test specimen. o< LJ~ ,~,~.~
] ~]
u , f
~ui l~Jl J ~--~ "41 ~:::-'~" /
,
~i~i ~i~
p
c s~!
"
.... ...... / ~ - - .,.,., s,,, -
-
:"
,i L . . ~ I
I
~.o ~-.~,,, /~----~ -1 0~tSO.'~OII /'L~ " "X\ .,.~ s,,~, [ ((~)/
L '
i I
,",
\~._ ~
J/
II ]
Fig.3(a). Welding notations.
I I
,
'"
[ - -'i
1
Note: All dimensions are in inches
Fig. 2. Schematic drawing of shot peening system.
Fig.3(b). Fatigue test specimen.
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Current Advances in Mechanical Design and Production, MDP-7
Four sets of specimens were prepared for each joint. Each set consists of 10 to 12 specimens. Three sets from each joint were shot peened at three different intensities 8A, 10A and 12A respectively, and the fourth set was left un-peened but polished longitudinally in a direction parallel to the rolling direction. Finally, rotating bending fatigue tests were carried out on each set of specimens for each type of joint in the unpeened and peened conditions at the different intensities, and different stress levels. The life of each specimen, at a given stress level, was determined in terms of the number of cycles to failure. When the specimen did not fail within 10 7 cycles it was then considered as not failed. The S-N curves were then obtained and compared. Each test was carried out at least twice to ensure the repeatability of the results. Microhardness measurements were taken along the three zones of the welded joints in the transverse direction to the welding direction i.e. along the rolling direction, starting from the weld metal zone to the base metal through the heat affected zone, HAZ.Microstructural examination has been carried out on the fused metal, HAZ and base metal at the three different peening intensities and finally, electron scanning microscopy was carded out to highlight the depth of the compressive surface layer in the weldment caused by shot peening. 3. RESULTS AND DISCUSSION
3.1. Effect of Shot Peening on Fatigue Life of Welded Joints The S-N curves for the butt single-U welded joints are presented in Fig. 4, from which it can be seen that shot peening improved the fatigue strength at all shot peening intensities and that the improvement in fatigue strength increases as the shot peening intensity increases, for example the fatigue strength at 2xl07 cycles increased from 150 N/mm 2, for unpeened joint to 173 N/mm 2 for the peened joint at 8A intensity with an increase of 15% whereas an overall increase of 35% was achieved at shot peening intensity of 12A. Similar results were observed for the double Vee welded joints, Fig.5. This is attributed to the compressive residual stresses at the surface caused by shot peening. These induced residual comprssive stresses, as established, will reduce and hinder the propagation of the surface cracks eventually leading to fracture. The induced compressive stresses and the strain hardening resulting from shot peening will partially or completely compensate for the existing tensile stress within the specimen. 35O BUTT DOU%LE V E E
35(
o - j ~ E E~E--G'-~---o[ -_--~ A
30~
2
:~ 250
<
- o
9
-
YO|NT
~ ] = 12A 5O
A
200
'.00
150
5O
o
100 53
i
0
10~
I0 s
136
107
CYCLES TO FATLURE , N j
Fig. 4. S-N curves, unpeened and peened conditions for butt single U welded joint.
I0"
105
106
107
108
C~'CLES TO FAILURE , Nf
Fig. 5. S-N curves, unpeened and peened conditions for double Vee welded joint.
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Current Advances in Mechanical Design and Production, MDP-7
The increase in the fatigue strength with the increase of shot peening intensity is attributed to the fact that the affected depth of the induced compressive residual stresses increases with the increase of shot peening intensity. This can be clearly observed in the photoscans of Fig.6 and Fig.7, where the depth of the plastically compressed region below the surface is of larger value at the higher intensity, because higher intensity results from higher shot velocity, hence higher compressive residual stress.
Fig. 6. Photoscan showing depth of compressive layer at intensity =8A.
Fig.7. Photoscan showing depth of copmressive layer at intensity =I2A.
These results are in agreement with previous experimental and theoretical results of other researchers Hamidi, et al [1], and Dorr and Wagner, [13]. Furthermore, it can be seen from these figures that an increase of about 47% is achieved for the double Vee joint at 12A peening intensity which is higher than that of the butt single-U joint at the same shot peening intensity. This may be attributed to the fact that the heat affected zone, HAZ, is wider and of larger size in the double Vee joint, than in the butt singleU joint.
3.2. Metallurgical and Microhardness Examination Examination of Fig.8 reveals that the highest hardness was observed at the heat affected zone, HAZ, although the cooling rate at this zone is slower than the fused metal zone which should result in lower hardness level. This increase in hardness in the HAZ is attributed to the hard structure precipitates of the CuAI2 within the grains in the HAZ as demonstrated by the photomicrograph of Fig.9(a), in addition to that aging might have taken place during the welding process. In the fused metal zone, however, the CuAI2 has preciptated at the grain boundaries, Fig.9(b), resulting in lower hardness in this zone than in the HAZ. Regarding the base alloy zone, Fig.9(c), it possesses higher hardness than the fused metal zone. This is due to the fact that the alloy has been solution heat treated, cold worked and naturally aged to a substantially stable condition, Fig.9(c), which improves its strength and hardness. This explains the experimental observations that fracture, in most fatigue tests, has occured in the heat affected zone, HAZ. This agrees with the findings of Hamidi, et al [1]. Furthermore, the response of the three different zones namely: the fused metal zone, heat affected zone, HAZ and the base metal zone shown in Fig.8 may be different depending on the local microstructure which in turn will affect the fatigue crack initiation and growth.
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120
> -I--
100
9
u") b..l Z O
80-
<
60r
I
0
I
u
~E
9
-
9
9 9
9
9
FUSED METAL ZONE
9
DIRECTION
/'1
I
9
D I R E C T I O N OF W E L D I N G
WELDING JOINT
1
9
9
200 0
9
BASE METAL ZONE
HAZ
9
40-
9
I
2
....
I
3
, ,, ,, ,, t
4
.
.I
5
........
I,,
6
DIRECTION
t
7
OF R O L L I N G
OF M E A S U R E M E N T
_
_t
8
.
DIRECTION OF MEASUREMENT
Fig.8. Vickers micro-hardness measurements in the different welding zones.
CONCLUSIONS Fatigue life and strength of aluminum alloy 2024-T3, in the as received and welded conditions have been improved by shot peening, and the improvement increases with increase of the shot peening intensity. The butt single-U joint is found to possess higher values of fatigue life and strength than the doube Vee joint at the different shot peening intensities. Microhardness surveys at the different welding zones revealed that the heat affected zone, HAZ, possesses higher hardness than the base metal, which in turn possesses higher hardness than the fused metal zone which explains why fracture in most fatigue tests occurs at the HAZ.
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Current Advances in Mechanical Design and Production, MDP-7
REFERENCES 1. Hamidi, T. Fathallah, R. Barralier, L. Sidhom, H. and Costex, L., "Behavior of Shot Peened Welded Steel Under Cyclic Loading", The Sixth Conference on Shot Peening, ICSP6, San-Francisco, pp 130-140, (1996). Wohlfahrt, H. Nitschke, T. and Zinn, W. , "Optimization of the Fatigue Behavior of Welded Joints by Means of Shot Peening a Comparison of Results on Steel and Aluminum Joints", The Sixth Conference on Shot Peening", ICSP6, San-Francisco, pp 243-250, (1996). 3. Masubuchi, K., "Analysis of Welded Structures", 1st Edition. Pergamon Press Ltd, U.S.A, (1980). 4. Haagensen, J., "Methods for Fatigue Strength Improvement and Repair of Welded Offshore Structures", Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering, OMAE Vol.3, Housten, TX, U.S.A. pp 419-427, (1994). 5. Nguyen, T. N. and Wahab, M. A., "Theoretical Analysis of the Effect of Residual Stresses in Welded Structures on the Improvement of Fatigue Strength", Proceedings of the 5th Australian Aeronautical Conference, 2(93), Crows Nest, Aust. pp 463-469, (1993). 6. Chow, Chi L. and Wei, Y., "Fatigue Damage Model For Crack Propagation", ASTM Special Technical Publication, 1292: pp 86-99, (1994). 7. Gumey, T. R., "Fatigue of Welded Structures", 2nd Edition. Cambridge University Press, Britain, pp 317-322, (1979). 8. Nacher, A., "Influence of Local Heating and Surface Peening on Fatigue Behaviour of Welded Joints", British Welding Journal, Vol.7, pp 513-516, (1950). 9. Konishi, K. Takagi, O. and Shimada, K., "The fatigue Strength of Welded Joints of High Tensile Structural Steel", Proc. 1st Jap. Conf. on Testing Materials JSTM, pp 10-14, (1958). 10. Hedstr6m, A., "Cumulative Damage in Specimens of AI-Zn-Mg tested by a New Method for Reducing the Scatter", Conf. on Fatigue of Welded Structures, Brighton, (1970). Cited in Fatigue of Welded Structures, pp 317-322, (1979). 11. Harison, J.D., "Further Fatigue Tests on Fillet Welded Specimens subjected to Prior Overloading", British Welding Journal, Vol.12, No.5, pp 255-258, (1965). 12. Booth, G. S., "The Effect of mean stress on the Fatigue Lives of Ground or Peened Fillet Welded Steel Joints", Weld. Inst. Pep. 34/1977/E. Cited in Fatigue of Welded Structures, pp 317-322, (1979). 13. Dorr, T. and Wagner, L., "Influence of Stress Gradient on Fatigue Behavior of Shot Peened Timetal 1100", The Sixth Conference on Shot Peening, ICSP6, San-Francisco, pp 174-183, (1996). 14. Zaid, A.I.O, AI-Syouf, I.A.A., AI-Haddid, T.N.M, "Experimental Investigation of ShotPeening Parameters", Current Advances in Mechanical Design and Production. Proe. of 6th Cairo University International MDP Conf., pp 405-412, (I 996). .
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
347
CREEP BEHAVIOR OF SOLID SOLUTION ALLOYS: ROLE OF DYNAMIC-STRAIN AGING Soliman, M.S.* and Almakhdoub, S.A. **
*Professor, **Student Mechanical Engineering Department, King Saud University, Riyadh, Saudi Arabia E-mail: [email protected]
ABSTRACT Creep experiments at constant stress were conducted at 673 K using an AI-5.6 at .% Mg alloy. The experimental resutls of the alloy when plotted as steady-state creep rate vs. the applied stress on a double logarithmic scale, show the presence of two transitions in the stress dependence of creep rate at constant temperature. The stress exponent n changes from a value of 4.1 at low stresses (region I) to a value of--- 3 at intermediate stresses (region II) and then increases again to a value higher than 3 at high stresses (region III). The present analyses suggest that the model of dynamic strain aging proposed to describe the creep transitions in AI-Mg alloys, is not in full agreement with the present and previous experimental results, including creep rates at high stresses and activation energies, for AI-5.6 at. %Mg and AI-10 at. %Zn solid solution alloys. KEYWORDS Creep, AI-5.6 at .%Mg, AI-10 at .%Zn, Dynamic Strain Aging. 1. INTRODUCTION The steady .state creep rate, k, of pure metals and solid solution alloys has usually been described as a power function of the applied stress tr, by the Dorn-type equation: ~ = A ' Gb (o. / G )"exp (_ Qc / RT ) kT
(1)
where G is the shear modulus, b is the Burgers vector, k is the B01tzmann constant, T is the absolute temperature, R is the gas constant, Qc is the activation energy for creep, n is the stress exponent and A' is a constant. Based on the value of n and other creep characteristics, the creep behavior of solid solution alloys has been divided into two classes. First, the "pure metal-class" solid solution alloys which essentially have similar characteristic as a pure metal with n value equals to --5. Second, the "alloy-class" solid-solution alloys, which have creep characteristics different from that of pure metals, with n value equals to-~3. However, for alloy-class alloys (n~3), it was shown experimentally by several investigators [ 1-6] that when
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Current Advances in Mechanical Design and Production, MDP-7
the applied stress increases above or decreases below a critical value, the n value tends to increase above 3, reaching values close to --5. Therefore, transition from n---5 (at low stresses) to n~-3 (at intermediate stresses) and to n~-5 at high stresses in the power law regime was observed in some AI-Mg alloys [ 1-3] and in an AI-Zn alloy [4]. This change in the value of n was attributed to a corresponding change in the rate-deforamtion mechanism [4-6]. However, it has been suggested recently that this behavior in AI-Mg alloys (containing Mg content <_ 3%), could be explained using a single equation based on a model of dynamic-strain aging [79]. It has been noticed that this model can not completely describe the creep behavior of these alloys [ 10]. In this investigation, the present experimental results of AI-5.6 at .% Mg alloy obtained at 673 K, along with previous data of the alloy [11], which showed 5-3-5 transitions will be examined using this model. In addition, previous experimental results of an AI-Zn alloy [4] which was carried out over various temperatures and showed similar transitions, will be examined in detail. The attributes and drawbacks of this model will be discussed. 2. EXPERIMENTAL PROCEDURE The material used in the present experiments is an AI-5.6 at .% Mg alloy. It was prepared by melting metals of 99.99% purity and supplied in the form of hot-rolled strips, 3 mm thick. Tensile specimens of 3x 12.5 mm cross-section and 25 mm gauge length were machined from the strip such that the tensile axis of the specimen was parallel to the rolling direction. Specimens were hand polished and prior to testing, all specimens were annealed in situ for 2 hrs at 773 K to remove the effect of machining and to produce a stable uniform grain size of 1 mm as measured by the line intercept method. Atter annealing, specimens were cooled to the testing temperature of 673 K. Creep experiments were carded out at constant stress using Mayes electronic creep testing machine. Details of testing are given elsewhere [12]. 3. EXPERIMENTAL RESULTS 3.1 Creep Curves Examples of creep curves at a low stress o = 1.56 MPa and an intermediate stress c - 15 MPa are shown in figure 1. While extensive normal primary-creep stage is observed at the low stress, inverted primary-creep stage is noticed at the intermediate stress. These stages are followed by steady-state creep.
3.2 Stress Dependence of Steady-State Creep Rate The stress dependence of the steady-state creep rate, i , at 673 K, is shown in Fig. 2, by plotting ~ vs. c using double logarithmic scale. The present data points overlap with previous data at intermediate stresses [11] and are extended to lower stress region which was not covered previously. The data points at low stresses fall on a straight line and a linear regression analysis gave a stress exponent, n of 4.1 in this region. At intermediate stresses, the value for n was - 3.1. At higher stresses n value tends to exceed the value of 3.
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349
4. DYNAMIC-STRAIN AGING MODEL 4.1 Stress Dependence of Creep Rate The model of dynamic strain aging as suggested by Hong [7-9], was used to describe the stress dependence of the steady state creep rate in AI-Mg solid solution alloys [ 1,2]; the stress exponent for creep in those alloys exhibits a minimum value of about 3 at intermediate stresses. The creep behavior of AI-Mg alloys over the entire range of stresses can be described by a single rate equation of the form k = K'(cro - cr o ~'exp (- Q * / RT )
(2)
where K' is a constant (for constant temperature), oa is the applied tensile stress, OD is the frictional stress, n* (--5) is the stress exponent at low and high stresses, and Q* is the activation energy in the absence of dynamic strain aging (activation energy for self-diffusion in pure aluminum). Dynamic strain aging occurs during the creep of AI-Mg alloys, and this leads to strengthening effect which is reflected in the presence of frictional stress OD that opposes the movement of dislocations. The frictional stress oo has the form of a statistical distribution which is given by ~
(3)
=cr~ e x p { - ( T - T)Z/B}
where o ~} is the maximum value of OD, B represents the width of distribution about T and T is the temperature at which a ~) occurs. The value of o ~) is independent of temperature and strain rate. The frictional stress (~D due to dynamic strain aging also depends on strain rate, ~; in Eq. 3, T is predicted to be a function of strain rate as given by the following equation
(4)
&=Aex~-Q/RT)
where Q is the activation energy for diffusion of solute atoms in the alloy and A is a constant. Based on Eqns. (3) and (4), OD is negligibly small at very low and high strain rates but exhibits a maximum value (cr ~) at some intermediate strain rate. This reflects on the value of stress exponent (in Eq. 2) which is close to 5 at both low and high stresses and it decreases to a value close to 3 at intermediate stresses. In order to compare the prediction of the model with experimental results at various temperatures, K' value is calculated as a function of temperature by comparing Eq. 2 with the Dorn's equation (Eq. 1) using the effect stress (6a6D) in lieu of applied stress, i.e. =
A 'Gb
exp-
cr~
G . . . . . TG .'-!
).. o
T/ o" -~
(5)
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Current Advances in Mechanical Design and Production, MDP-7
Comparing Eqns. (2) and (5), we find that At
TGn*-I where A' is a constant independent of temperature. 4.2 Apparent Activation Energy In order to examine the validity of dynamic strain aging as a possible mechanism for controlling creep in solid solution alloys, comparison between the experimental and calculated apparent activation energy is carded out. The apparent activation energy, Q~,p is defined as ~ 81n~ /
t
J=
Taking the natural logarithm ofEq. 2 and differentiating w.r.t (l/T), an expression for Qapp is obtained as:
Q app = Q * +
2n* RT 2 (T - T ) c O (6)
5. COMPARISON BETWEEN THE PREDICTION OF DYNAMIC-STRAIN AGING MODEL AND EXPERIMENTAL RESULTS 5.1 AI-5.6 at .% Mg In order to examine the correspondence between Eq. (2) and the creep behavior for AI-5.6 at. % Mg, it is required to estimate values for K', o ~, B and A in Eqns. (2) - (4). Since the present experimental results at 673 K and the previous data [11 ] were not extended to high strain rates where o ~ and B can be calculated, the values for o ~ and B are taken from the analysis presented by Hong [6] for AI-Mg alloys. Hong [6] has deduced that o ~ for AI=Mg alloys is a function of Mg concentration, CMg, i.e. o~(MPa) = 984.6 CMg
(7)
Therefore, for the presnt alloy, CMg --- 0.056, and o ~ is calculated to be 55 MPa. The value for B is taken as 4x 103 K 2 as deduced by Hong [6] for AI-Mg alloys. Present calculations give the following values for these parameters: K' = 6.8x 101S/(TG3"l); A = 5.4x l 0 g s"1. In addition, the values for n*, Q* and Q where taken as 4.1 (present investigation), 143.4 kJ/mol [13] and 130 kJ/mol [ 14], respectively. In Fig. 3, the prediction of Eq. 2 is plotted as a full curve and the experimental data of AI-5.6 at. % Mg were included for the purpose of comparison. The figure shows the presence of
Current Advances in Mechanical Design and Production, MDP-7
351
excellent agreement between the prediction and the experimental data for the alloy at low and intermediate stresses at 673 K; however this agreement is not surprising since the experimental data of the alloy at these stresses were used to determine the values of the adjustable parameters such as K' and A. By contrast, there is poor agreement between the prediction of Eq. 2 and the experimental results at high stresses at 673 and 623 K. In addition, this poor agreement exists at intermediate and at high stresses at 573 K. Based on Eq. 6, the apparent activation energy in AI-5.6 at.% Mg is calculated at 623 K and plotted against the applied stress in Fig. 4. The values for Qapp has a maximum value of 140 kcal mol l at cr = 50 MPa ( ~ = 7.4 x 10-4 s -t ). The experimental values of Qapp [11 ] is also included in the figure; it has a value of 33.5 kcal mol "t and is independent of stress. Present calculations and Eq. 6, suggest that Qapp based on dynamic strain aging model is a function of stress and temperature. This means a plot of ln~vs.(l / T) will be a curve rather than a straight
ln~vs.O/T)for
line. The plots of AI-5.6 at. % Mg [ll] overa wide range ofstresses gave straight lines. The discrepancy between Qapp as calculated (Eq. 6) and experimental resutls is clear. 5.2 AI-10 at. % Zn
The creep experimental results of Al-10 at. % Zn tested at various temperatures and over a wide range of stresses are available [4]. These data when presented as ?~ (shear strain rate) vs. (shear stress) using double logarithmic scale showed two transitions: the first transition from n = 4.5 at low stresses to n = 3 at intermediate stresses and the second transition from n - 3 at intermediate stresses to n~5 at high stresses [4]. This change in stress exponent is similar to that observed in AI-Mg alloys containing 0.5 to 3 at.% Mg [1-3]. The shear data for Al-10 at. % Zn was converted into tensile stress and tensile strain rate using Tresca criterion: ~=2z
and
o~= 27;/3
The expreimental results of Al- l 0 at. % Zn at 573 K extends over a wide range of strain rates and are used to calculate the model parameters: K', t~~, B and A. The values of n*, Q* and Q are taken as 4.5, 120 kJ/mol and 112 kJ/mol, respectively [4]. The values of Q* and Q represent the activation energy of diffusion of Al and Zn in the alloy. Present calculations, using the creep data of the alloy at 573 K, showed that O'D reaches its maximum value or~ D (15.55 MPa) at ~a = 36 MPa and 6 - 5.33 x 10 -3 s"1 . Fig. 5 shows the dependence of frictional stress, crD on the applied stress era and strain rate ~ at T = 573 K. The values of A and B were calculated as 8.6x 107 s"! and 6.75x 103 K 2. Taking the shear modulus of the alloy as (G = 2.926x104 - 17 T (MPa)) [3], the value of K' in Eq. 2 was estimated as 3.84x 102~ In Fig. 6, the prediction of Eq. 2 is composed with the experimental results of the alloy at various temperatures. The figure shows the presence of good agreement between the model prediction and the experimental data at 573 K and at low and intermediate stresses at other temperatures. This agreement because the data of alloy at 573 K is used to find the adjustable parameters K', A and B. At other temperatures, the values for CrD is negligible at low stresses
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Current Advances in Mechanical Design and Production, MDP- 7
14
10
12 10 o-,e, c~
8
L-
6
.~
03
4 2 I
0
_
I
I
I
, , I
I .
0
I OL,.
100 200 300 400 6(X) 6~3 700 800
0
5
10 15 Time, min
Time, hr
20
25
Fig. 1. Examples of creep curves f o r AI-5.6 at. % Mg at 673 K and stresses. (a) a - 1.56 MPa; (b) a = 15 MPa. 1E-1 =_ _
~E-2
S'm.nXX~( i / , )
,oo
_ _
.
I0
.
0
.
T=673
.
.
.
.
/,//
K
~ 0.01
1E-3 -_~ -
r
rY .c_
1E-4 _=-
,
IE-4
I
L
1s IE-6
1E-5
(/)
-
p
Ig-?
IE-8
1E-6
IE-D 0.1
I
I0
1
10
I000
100
STRESS (MPa)
1E-7
Fig. 3. Comparison between the prediction of the model of dynamic strain aging ( - ) and the experimental data of AI5.6 at. % Mg at various temperatures.
100
Stress (MPa)
Fig. 2. Stress dependence of steady-state creep rate at T = 673 K. a (present investigation); o (ref. 8). Qspp (keal/mol) . . . . . .
160 140
120 I00
80
Fig. 4. Comparison between the calculated values (--) and the experimental data of the apparent activation energy for A1-5.6 at. % Mg.
.oi 0
0
..-~
.
, 20
_._
! 40
........ 60
~ O0
APPLIED STRESS (MPa)
....
i ! O0
120
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Current Advances in Mechanical Design and Production, MDP- 7
and small at intermediate stresses. However, the agreement at high stresses is unsatisfactory; for example at T = 633 K, the deviation in strain rate between the experiment and the prediction is a factor of about 3 at an applied stress of 27 MPa. The apparent activation energy for AI-10 at .% Zn is calculated using Eq.(6) at T = 573 K and compared with the experimental results [ 15] at various applied stresses. As seen in Fig. 7, the calculated Qapp is in good agreement with experimental value in region I (aO ~ 0). However, in region II, Qapp increases sharply and reaches a maximum value (---70 kcal/mol) at o~l 7 MPa and decreases to a value of 20 kcal/mol at 40 MPa. This prediction is not in harmony with the experimental results as presented in Fig. 7. The values of Qapp based on Eq. 6 are calculated as a function of temperature at three different stresses and plotted in Fig. 8. The stresses are 2 MPa (region I), 7.5 MPa (region II), and 35 MPa (region III). From the figure, Qapp at aa = 2 MPa has almost constant value very close to Q* because 60 is negligible at region I. At Ca = 7.5 MPa (region II) Qapp has a high value (55 kcal/mol) at low temperatures and decreases with increasing temperature. At 6~ = 35 MPa, Qapp decreases with temperature and reaches value less than Q* as T decreases below T. This change in Qapp with temperature is inconsistent with experimental results; activation energy for creep in each region is constant and independent of temperature [4,15]. m
A main characteristic of the prediction of Eq. 2 as shown in Figs. 3 and 6 is that at very high stresses the curve representing the dependence of strain rate on stress, when plotted on a logarithmic scale, concides with the extrapolation of the line representing low-stress region. This feature which reflects the characteristics assumed for 6D, could not be easily checked in the present experiments on AI-5.6 at. % Mg because it occurs at very high strain rates which can not be measured accurately with the present equipments. The inflection in stress dependence of creep rate at high stresses was not confirmed by the experimental data on A12% Mg at 573 K [10]. CONCLUSIONS Solid solution alloys such as AI-5.6 at % Mg and AI-10 at .% Zn, exhibit transitions in stress exponent from n~5 at low stresses to n~3 at intermediate stresses and to n-5 at high stresses. .
The change in stress exponent has been usually attributed to change in rate-controlling mechanism" dislocation climb at low and high stresses and viscous glide at intermediate stresses. The prediction of dynamic-strain aging model as represented by Eqs. 2-6, is not in harmony with the experimental results because of two observations: (i) there is poor agreement between the creep rates of the alloys and the prediction of the model at high stresses, (ii) the model predicts higher values for the activation energy at intermediate stresses. In addition, the predicted values for Qapp changes with temperature, which is inconsistent with experimental results.
354
Current Advances in Mechanical Design and Production, MDP-7 jodoo tO00 i I
100
0"o~
I
10 J 0.l 0.01 Ig-3
IE-4 Is IE-6 IE-7 9 lJlllllll I E - 8 __ _ 0.001 0.01
l
9
l
0.1
I
I ttllllt
10
l
I trill
100
1000
C~'o
Fig. 5. Frictional stress dependence on applied stress (o.) and strain rate (l~) for AI-10 at.%Znat T=573K.
. . . . . . .
I00 I0
qepp
(keel/tool)
60
t eo
O.l 0.01 IE-3 ls IE-5 I1E-6
T=633
K
1'--673
K
T:OO0
K
j
m l l Ill
leO
I
IE-7 IE-8 l
IE-O
0.1
l mlmlll
l
!
l
I t~llml
J
10
STRESS
*_1
l l llll
100
(MPa)
O
Oapp
-' tO
IS
' 20
' 25
APPUs STni:SS (Upe)
' 30
* 35
- ' 40
(kcallmol)
.
.
.
.
.
" ........
~
6o
~
4o
"v
2O 10 0 500
_
9 650
.....
9600
45
Fig. 7. Comparison between the calculated values ( - ) and the experimental data of the apparent activation energy for AI-10 at. % Zn.
Fig. 6. Comparison between the prediction of the model of dynamic strain aging (--) and the experimental m e l t s of AI-10 at. % Zn at various temperatures. 60
' 5
O
1000
a 650
,,
T
I 700
.,.
i 750
8O0
850
(K)
Fig. 8. Dependence of apparent activation energy on temperature based on the Prediction of dynamic strain aging model at o = 2 MPa (o), o = 7.5 MPa ( ~ ) and o = 35 MPa (o).
Current Advances in Mechanical Design and Production, MDP-7
355
REFERENCES 1.
2.
3.
4. 5. 6. 7.
8. 9.
10. 11. 12.
13. 14. 15.
Oikawa, H., Honda, K. and Ito, S., "Experimental Study on the Stress Range of Class I Behavior in the Creep of AI-Mg Alloys", Materials Science and Eng., Vol. 64, pp. 237245 (1984). Oikawa, H., Sato, H. and Maruyama, K., "Influence of Temperature on the Transition of Deformation Characteristics of AI-I Mg Alloy in the Power Law Creep Regime", Materials Science and Eng., Vol. 75, pp. 21-28 (1985). Sato, H. and Oikawa, H., "Further Experimental Study of Deformation Characteristics of AI-Mg Alloys in the Power-Law Creep Regime", Scripta Metallurgica, Vol. 22, pp. 87-92 (1988). Soliman, M.S. and Mohamed, F.A., "Creep Transitions in an AI-Zn Alloy", Metallurgical Transaction A, Vol. 15A, pp. 1893-1904 (1984). Mohamed, F.A., and Langdon, T.G., The Transition from Dislocation Climb to Viscous Glide in Creep of Solid Solution Alloys", Acta Metallurgica, Vol. 22, 779-788 (1974). Yavari, P. and Langdon, T.G. An Examination of the Breakdown in Creep by Viscous Glide in Solid Solution Alloys", Acta Metallurgica, Vol. 30, pp. 2182-2196, (1982). Hong, S.I., "Influence of Dynamic Strain Aging on the Stress Exponent and the Dislocation Substructure for Creep of AI-Mg Alloys", Materials Science and Eng., Vol. 82, pp. 175-185 (1986). Hong, S.I., "On the Creep Activation Energies of Alloys", Materials Science and Eng., Vol. 86, pp. 211-218 (1987). H o n g , S.I., "Influence of Dynamic Strain Aging on the Transition of Creep Characteristics of a Solid Solution Alloy at Various Temperatures", Materials Science and Eng. A, Vol. A 110, pp. 125-130 (1989). Chaudhury, P.K. and Mohamed, F.A., "Creep Characteristics of an AI-2 wt.% Cu Alloy in the Solid Solution Range", Materials Science and Eng. A, Vol. 101, pp. 13-23 (1988). Kucharova, K., Saxl, I. and Cadek, J., "Effective Stress in Steady State Creep in an A15.5 at .% Mg Solid Solution", Acta Metallurgica, Vol. 22, pp. 465-472 (1974). Soliman, M.S., "Effect of Cu Concentration on the High-Temperature Creep Behavior of AI-Cu Solid Solution Alloys", Materials Science and Eng. A, Vol. 201, pp. 111-117 (1995). Lundy, T.S. and Murdock, J.F., "Diffusion of AI and Mn in Aluminum", J. Applied Physics, Vol. 33, pp. 1671-1673 (1962). Rothman, S.J. et al., "Tracer Diffusion of Magnesium in Aluminum Single Crystals", Phys. Status Solidi B, Vol. 63, pp. K29-K33 (1974). Kucharova, K. and Cadek, J., "High-Temperature Creep in AI-9.5 at. % Zn Solid Solution", Phys. Status Solidi A, Vol. 6, pp. 33-44 (1971).
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Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
357
INFLUENCE OF INTENSE PLASTIC STRAINING ON ROOM TEMPERATURE MECHANICAL PROPERTIES OF AI-Cu-Li BASE ALLOYS
Salem, H.A." and Goforth, R.E.'" "Visiting Assistant Professor, Mech. Eng. Dept., Faculty of Eng. American University in Cairo-Egypt. *'Associate Professor, Mech. Eng. Dept., Faculty of Engineering, Texas A&M University, College Station, Taxes, USA ABSTRACT Equal Channel Angular Extrusion (ECAE) is a relatively new metal working process which is capable of producing ultrafine sub-micron grain (SMG) structure by means of intense plastic straining without a change in shape or dimensions of the worked material. In the current research work, the influence of ECAE processing on the room temperature mechanical properties of AI-Cu-Li-Mg-Ag-Zr alloys were investigated in the T4 and T6 temper conditions. A conventionally processed alloy via rolling with similar composition to those ECAE processed was also investigated for comparison purposes. Microstructural analysis were assessed by means of optical microscopy (OM) and transmission electron microscopy (TEM). An ultrafine SMG structure of 0.2 to 0.4 ~tm was produced for the ECAE processed alloys from initial grain size of >100 ~tm, while a minimum of 1.2 ~tm was revealed for the rolled alloy. A significant improvement in the mechanical properties at room temperatures were accomplished by ECAE processing in comparison with conventional processing. KEYWORDS Weldalite Al-alloys, ECAE Processing, Conventional Processing T4, T6 Tempering, Room Temperature Mechanical Properties, Microstructure Evolution I. INTRODUCTION Aluminum alloys went through series of developments for aircraft and aerospace applications since 1970s. The demand for ultrahigh strength without prior cold working (stretching), very strong and rapid natural aging capability, good fracture toughness at room and cryogenic temperatures, good weldability, and stress corrosion cracking resistance led to the development of AI-Cu-Li base alloys by late 1980s. These alloys are called Weldalite alloys which were first designed by Martin Meriata, and Reynolds Metal Company (RMC). Weldalite alloys were designed for applications such as cryogenic tankage on space launch systems where fuels are contained at -196~ (77K). Weldalite alloys are characterized by their high Cu:Li ratio with Cu content > 2.5 wt%, and Li-content < 1.6 wt%, in addition to other alloying elements such as magnesium, silver, and zirconium [1-3]. Weldalite alloys contain controlled amounts of different alloying elements in order to achieve the required ultrahigh strength and good fracture toughness for cryogenic aerospace
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applications through the precipitation of different second phase particles. Examples of which are ~5/(AI3Li), [3] 0/(AI2Cu), TI(AI2CuLi), T2(AI6CuLi), and TB(AIzsCusLi2) phases, S/(AI2CuMg), and [3/(Al3Zr) which all precipitate at different morphologies and along different planes and directions [3-5]. Commercially extruded plates and rolled sheets of AlCu-Li base alloys are typically subjected to a cold stretch operation prior to natural (T3 temper) and artificial (T8 temper) aging to introduce dislocations that act as preferable sites for the nucleation of the strengthening 0/, TI, and SI phases. Since ingot Al-base alloys are mostly formed superplastically at relatively high forming temperatures (~0.8Tm) and relatively low forming strain rates (10"3 to 10.4 s"~) lots of efforts were done aiming towards the increase and/or decrease of the SPF strain rates and temperature, respectively and hence achieving cost effectiveness. According to Langdon et al., 16] grain refinement to the sub-micron level offers potential advantages in increasing the overall ductility while increasing strain rate and/or decreasing the SPF temperature. Subjecting cast metals to severe plastic deformation through homogeneous simple shear can produce SMG structure. Equal channel angular extrusion (ECAE) was found to be effective in producing SMG of 0.2-0.3 ttm [71. Accordingly, characterization of the room temperature tensile properties of the ECAE processed Weldalite alloys versus sheets processed via rolling conducted in comparison with previously reported room temperature properties of similar alloys in the T4 and T6 temper conditions investigated by the different researchers. 2. MATERIALS AND EXPERIMENTAL PROCEDURE Weldalite 2095 and 2195 alloy ingots were direct chill (DC) cast at RMC, Richmond VA. The 2095 and 2195 alloys were received in the as-cast condition in the form of plates, and a third alloy was also investigated which was 2095 Weldalite received in the form of superplastic dynamically recrystallized sheets. These sheets were conventionally processed by rolling by RMC (thermomechanical treatment details are classified), and was selected to compare ECAE with conventional processing techniques. Measured compositions o f th e investigated alloys were determined using wet chemistry analysis as given in Table 1. The as-cast alloys were homogenized, solution heat treated (SHT) at 510~ (783K) for 70 min followed by water quenching (WQ) then peak aged at 160~ (433K) for 20 h. Billets with initial dimensions of 2.5x2.5x12.7 cm 2 from the peak aged plates were plastically deformed using ECAE. In ECAE processing, uniform deformation is imposed throughout massive billets without a change in the cross-sectional area. Billets are subjected to shear deformation by extruding them through 2-channels of equal cross-section area intersecting at an angle of 7t/2, as shown in Fig. 1 [7,81. Table 1. Chemical Composition of As-Cast Weldalite Plates and SP Rolled sheets ALLOY
Thickness (in)
209"5as-east plate
0.89
LI 0.92
ca
Mg
Zr
4.18
0.39
0113
0.32
0.02
0.07
"0.44
0.026
01079
219"5as-cast plate
0.77
0.89
3.89
0.28
0.11
2095 SP sheet
0.063
1.08
4.52
0.31
0.15
Ag
0.34 .
.
.
.
Ti
Zn
0 . 0 2 8 0.065
.
Some minor additions: Mn, Fe, and Si less than 0.06 wt% each
The extrusion operation can be repeated several times, since there is no dimensional changes encountered by the material after each pass. Moreover, different SMG structures and textures can be created by changing the plane and direction of shear at each pass. Routes A is when the
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orientation of the billet is unchanged at each pass, and route C is when the billet is rotated 180~ about its extrusion axis after each pass. The ECAE processed billets via routes A and C were extruded 4 passes at 350~ (623K) followed by 4 passes at 200~ (473K) with a total effective strain of 9.36 at 0.5 crn/s cross head speed. Post forming heat treatments were performed to investigate the room temperature mechanical properties of the ECAE versus conventional processing of the Weldalite alloys in the T4 and T6 temper conditions. Post forming heat treatment was conducted by SHT of the as-extruded and as-rolled alloys at 510~ (783K) for lh followed by WQ then natural aging at room temperature for >1000 h, (T4 temper). Peak strengthening (T6 temper) was achieved by monitoring the artificial aging response of the solutionized billets and sheets at 180~ (453K) for different time intervals up to 30 h. For the characterization of ECAE processing for grain refinement in comparison with conventional methods, different analysis such as hardness measurements, room temperature tensile properties in the T4 and T6 temper conditions and microstructural evolution were conducted. Rockwell Hardness (RB) was performed under loads of 100 Kg during 15 sec impressions with an indenting steel ball with 1/16 in diameter. For the extruded samples, hardness measurements were done on a plane sectioned parallel to the flow plane in the middle of the billet. Yield strength (or), ultimate tensile strength (OUTS),and % elongationsto-fracture (%EF) were determined for the ECAE and SP sheets respectively, in the T4 and T6 temper conditions. An MTS tensile testing machine with 20 kips loading capacity was employed. Specimens sliced from the deformed billets parallel to the extrusion flow plane, and from the rolled SP sheets parallel to the rolling direction were machined for the tensile testing. The tensile properties of each condition were based on average results obtained from a minimum of 2-specimens. Microstructural analysis was conducted through the employment of OM and TEM to observe the effect of ECAE processing on the size shape of the produced microstructure under bright field. In addition, particle morphology within the grains and at grain boundaries was also analyzed. TEM analysis was conducted using a JEOI JEM-2010 TEM at high voltages of 200 kv. In order to determine the sub-cells, subgrain, and grain sizes for the as-cast and processed alloys linear intercept method was employed using OM and TEM images, respectively. A double-jet electropolishing technique was employed for the final thinning of the disc. The electrolyte used for the final thinning of the samples was a 25% nitric acid and 75% methanol mixture. Final thinning was performed at an operating voltage of 12 volts, which corresponds to a value of current between 0.275 and 0.3 A and at an operating temperature of-15~ (158K)[9]. An Energy Dispersive Spectroscopy (EDS) attached to the JEOl JEM-2010 TEM was utilized for the composition analysis of the different second phase particles for the ascast, as-extruded, and as-rolled structures. In addition, diffraction patterns were obtained for the different samples to derive useful information about the different second phase particles precipitating within the structure in the SHT, T4, and T6 temper conditions, due to the fineness of the phases. For the identification of the different second phase particles the EDSIM program was utilized through matching the spot patterns generated for the different particles by the TEM with those simulated by the program. 3. EXPERIMENTAL RESULTS 3.1 Hardness Measurements The ECAE processed alloys exhibited an extremely rapid natural aging response when aged at room temperature over time intervals up to 1000 h. Figure 2, shows RB-variation with aging time at room temperature for the ECAE processed billets processed via 4C, 350~ (623K) + 4C, 200~ (473K) versus the rolled alloy. Most of the hardness increase for the ECAE
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processed alloys occurred over the first 24 h of aging time. On the other hand, rolled sheets displayed more gradual increase in hardness-values up to > 1000 h of aging time. Figure 3, shows the variation of RB-values with time for 2195 alloy ECAE processed via routes A and C where route C displayed higher hardness values compared to route A at all aging intervals. Similar behavior was exhibited during artificial aging (T6 temper) at 180~ over time intervals from 25 min up to 30 h. RB variations with aging time at 180~ for the ECAE processed alloys versus the rolled one are shown in Fig. 4. It was revealed that after 15 to 30 min of artificial aging, a slight decrease in hardness occurred. A rapid increase in hardness of the ECAE processed alloys took place up to 4 h of artificial aging followed by gradual and slight increase up to 1Oh of aging time. Artificial aging produced maximum hardness-values at the top of the RB-scale of 95.6 and 92.5 for 2095 and 2195, respectively which occurred at technological practical times of 12 h. Conversely, the SP rolled 2095 sheets exhibited gradual increase in RB-values up to 24 h with a highest displayed value of 93.8 occurring at 28 h. Similar to natural aging behavior, route C exhibited higher artificial aging response than route A.
3.2 Room Temperature Tensile Properties Table 2 lists the ours, and %EF experimental values for 2095 and 2195 ECAE processed alloys via routes A and C versus the 2095 rolled sheets in the T4 and T6 temper conditions. In addition, other results of previously published data by Pickens et a/.,[31 on AI-(4.5-6.3)Cu1.3Li-0.4Ag-0.4Mg-0.14Zr. Weldalite alloys conventionally processed via rolling or extrusion are listed in the table for the sake of comparison [3]. Note that, the Weldalite alloys were naturally aged (T4) for 1000 to 1200 h at room temperature, and artificially aged (T6 & T8 with prior stretching) at 180~ for 12 h. Superior tensile properties were displayed by the ECAE processed alloys regardless of Cu and Li-contents. Moreover, route A processing exhibited a noticeably higher %EF than route C with a slight decrease in UTS. Table 2. Room Temperature Tensile Properties of Weldalite alloys Alloy composition& processing Cony.i~xtrusion 049(6..3 Cu-.2Li)'" Cony. Extrusion2094 (4.6 Cu-l.2Li)'" Rolled2094 Shee.ts (4.89 Cu-1.3Li)'" Rolled SP sh_eets2095 (4.3Cu-l.lLi)" ECAE.2195(3.89 Cu-0.89L!),RouteA" ECAE 2195 (3.89 Cufl.89L!), RouteC" ECAE 2095(4.18 Cu-l.0Li), Route A~' ECAE 2095 (4.18 Cu-l.0Li), RouteC"
"Investigated alloys,
T6 UTS(MPa) %EF 690 4.3 614 10.5 665 5.2 18 715 6.9 15.6 707 12.2 25.3 715 !1 24.0 741 10.5 764 9.1 1 2'1.2
T4 UTs(MPa) %EF 593 16 ,
544 532 531 543
. 591
T8 UTS(Mpa) %EF
,,
712 ,
712
5.1 ,,
6-9
,
""Previously published [3]
,.
UTS units in MPa
3.3 Microstruetural Analysis In order to derive valuable explanation for the superior mechanical properties exhibited by the ECAE processed alloys in comparison with those conventionally processed, it was necessary to study the microstructural evolution and particle morphology of the deformed and post heat treated alloys. This section will be included in the discussion
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4. DISCUSSION 4.1 As-Extruded and As-Rolled Mierostructures
Optical microscopy investigation revealed that the as-east average grain sizes of 2095 and 2195 alloys were 123 and 78 ~tm,respectively. TEM investigation of the ECAE processed Billets deformed at an effective total strain of 9.23 (4 passes, 350~ (623K)+ 4C, 200~ (473K)) revealed the formation of SMG structure. Figure. 5 shows TEM photomicrographs of the as-extruded 2095 billets ECAE processed via routes A and C, and the as-rolled SP sheets. Examination of the ECAE processed alloys via route A revealed the formation of fragmented structure made out of elongated sub-cells about l x0.3 ~tm2 (marked A&B), and equiaxed subgrain structure about 0.25 ~tm (marked C) as shown in Fig. 5a & b. Formation of well defined subgrains and high dislocation activity within others, is an indication of energetic structure (marked B, Fig. 5b) was evident. Examination of the billets ECAE processed via route C revealed the formation of similar structure to route A except for the disappearance of the elongated sub-cells which were replaced by a fine equiaxed subgrains and dislocation cells about 0.18 ttm in average size, (Fig. 5c). A very limited fraction of a dynamically recrystallized subgrain structure about 0.2 ~tm in average size with extinction contours at their boundaries was also observed (marked by arrows, Fig. 5d). The 2095 as-rolled SP sheets examination revealed the presence of equiaxed grain structure with well defined, sharp, and relaxed boundaries about 1.5-2 ~tm in average size and with little dislocation activity within the grains (Fig. 5e). Coarse particles with different compositions were identified in the ECAE and rolled alloys using EDS, such as 8(AILi), 0/(AI2Cu), TI(AI2CuLi), Tz(AI6CuLi), Ta(AllsCusLi2), S/(AI2CuMg), and 13/(AI3Zr)phases. These phases were present within grains and mostly at grain boundaries and triple points with different sizes, volume fraction and distribution. 4.2 Solution Heat Treated Microstructures
TEM analysis of 2095 and 2195 billets processed via ECAE similar condition revealed the formation of a roughly equiaxed-SMG structure with an average size of 0.98 and 1.1 lain respectively. Figure 6 shows a bright field TEM photomicrograph of 2095 billets processed via route C + SHT + WQ. Some of the grains revealed relaxed well defined boundaries, while others showed extinction contours at some of their boundaries which was indicative of an energetic structure. In addition, the TEM photomicrographs revealed the presence of the 13/(AI3Zr) dark spherical precipitate within the grains and at grain boundaries with an evidence of grain boundary pinning (marked by small arrows, Fig. 6a) [10, 11]. Higher Zr-content in the 2095 alloy compared with that of 2195 ECAE processed alloys (Table 1) resulted in the precipitation of higher volume fraction of 13/-phase and hence finer grain structure. Examination of a SAD pattern corresponding to the [112]matrixzone axis, revealed no evidence of the presence of any phases in the matrix with an exception of 8/-phase superlattice reflections and that of 13/-phase, marked by arrows in Fig. 6b. Both phases have LI2 structure with cube-on-cube orientation relationship with the matrix with (111)S/orl3///(111)A~ [3, 12]. The absence of the coarse second phase particles formed in the structure during the extrusion process, indicates their dissolution during the SHT process which was intended. Examination of 2095 SP sheets processed via rolling in the SHT condition, revealed the development of a well defined and relaxed grain structure with an average size of 4 ~tm. Fig. 7a, shows a bright field TEM image of 2095 SP rolled sheets SHT. Similar precipitations to that of the ECAE processed alloys in the SAD pattern corresponding to the [110]m~ix zone axis was evident, (Fig. 7b).
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Current Advances In Mechanical Design and Production, MDP-7
4.3 Natural Aging The rapid natural aging response of ECAE processed alloys in the T4 temper condition (Fig. 2) indicates that most of the second phase particles which are responsible for strengthening has precipitated within the first few hours of aging at room temperature which agrees with the observations made by Pickens et al.,[101. It is suggested that, the equiaxed grain structure developed by route C processing could be providing the structure with higher resistance to deformation than route A which develops elongated grain structure in the direction of flow (parallel to the to the tensile axis of deformation) as shown in Fig 5a. This explains the lower natural aging hardness and strength values displayed by route A versus C, and the significant increase in ductility (Table 2). Therefore, route A processing provides the material with excellent combination of strength and fracture toughness which are the major mechanical properties required for cryogenic applications. Work done by Pickens et al.,[3] on Weldalite 049 (6.3%Cu) extruded plates displayed almost equivalent strength in the T4 temper and significantly lower ductility than that obtained for 2095 (4.18% Cu) billets ECAE processed via route C (Table 2). Considering the difference in Cu-content between the conventionally extruded Weldalite 049 and the ECAE processed alloys, a significant improvement in the room temperature tensile properties was accomplished by ECAE processing. The observed faster natural aging response of the 2095 ECAE processed billets (Fig. 2). could be explained by the higher Li, and Cu-contents in the 2095 compared to 2195 as-cast plates (Table 1). Figure 8 shows low and high magnification TEM photomicrographs of 2195 ECAE processed billets in the T4 temper condition. 8/ and 13/ precipitates were evident by the presence of the superlattice reflection spots in the SAD pattern corresponding to [ 100]m~x zone axis (marked by large arrows, Fig. 8b). In addition, 0/ (AI2Cu) phase, which presence was evident in the structure of the ECAE and rolled alloys in the T4 temper condition, was manifested by the vertical and horizontal streaks for the two variants lying parallel to the beam direction (marked by small arrows, Fig. 8b). Similar observations were made by Kumar et al.,[5] who reported the precipitation of 8/ (AI3Li), and 0/ (AI2Cu) phases in the T3 and T4 temper AI-Cu-Li-Ag-Mg-Zr alloy. Although, the 2095 alloy processed via ECAE had lower Li and Cu-contents (0.92Li-4.18Cu) than that of 2095 SP sheets processed via rolling (l.08Li-4.52Cu), the ECAE processed billets exhibited higher and faster natural aging response than the rolled alloy (Fig. 2). This could only be related to the microstructure developed in each processing condition. Since the 8/(AI3Li) precipitation is oRen associated with dislocations,J131 it could be assumed that higher dislocation density and activity retained in the structure of the ECAE processed billets even after SHT, compared to that of the rolled sheets resulted in more rapid aging response and hence higher exhibited mechanical properties. This was evident by comparing the SHT structures of the rolled 2095 alloy (Fig. 7) with that of 2195 ECAE processed one (Fig. 8a) in the T4 temper condition. 4.4 Artificial Aging Artificially aged samples processed via ECAE displayed hardness values at the top of the RBscale, (Fig. 4). The observed drop in hardness within the first 30 rain is attributed to the reversion taking place by the dissolution of the 8/ (AI3Li) phase which precipitated during the quenching process and during sample preparation. Similar observation was reported by Kumar et al.,[5] and Gayle et al., [14]. The T6 temper billets and sheets were characterized by a higher YS, UTS, and lower ductility than the T4 terrier condition (Table 2), which was consistent with the investigations made by Kumar et al. This is due to the different second phase particles that precipitate in the structure of the T6 and which contribute into the high
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strengthening effect [3, 141. Work done by Pickens, et al.,131 on Weldalite 049 2094 (4.65%Cu) and 049 (6.3%Cu) conventionally extruded plates revealed lower strength/ductility combination in the T6 and T8 temper than that obtained for 2095 (4.18% Cu) billets processed via route C in the T6 temper condition (Table 2). This is indicative of the outstanding room temperature mechanical properties obtained by ECAE processing compared to that of the conventional methods (rolling and extrusion), regardless of the difference in Cu and Li-contents. Billets processed via routes A and C revealed extreme grain size stability against grain growth after artificial aging. A maximum average grain size of 1.2 and 1.07 ~tm were obtained for 2195 and 2095 ECAE processed billets via route C in the T6 temper conditions, respectively. Figure 9 shows TEM photomicrographs of 2195 billets ECAE processed via route C in the T6 temper condition. On the other hand, an average grain size of 4.2 ~tm was obtained for the rolled 2095 SP sheets (Fig. 10). Examination of the TEM photomicrographs for the ECAE processed billets in the T6 temper condition revealed the development of a relaxed grain structure about 1 ~tm in average size (Fig 9a). A well defined wide boundaries with preferred precipitation of Ti-phase (marked by arrows), while other sharp boundaries were precipitation free were observed (Fig. 9b). Note that, no PFZs were observed close to grain boundaries. Investigation of the second phase particles responsible for strengthening of the T6 temper alloys revealed the presence of similar phases to those reported by Pickens et al.,[3, 141, Huang et al.,[15], and Langdon et al.,[16l. Figure 9b shows a SAD pattern corresponding to the [112]matrixzone axis which revealed the following (1) Presence of one variant of Tl-phase normal to zone axis and a second 19.5~ from the zone axis which were represented by the streaks in between the spots (marked by arrows) and elongated spots (marked a), respectively, and the other 2-variants of Tl-phase make 61.9 ~ with the zone axis which made them steeply inclined and which were also represented by the elongated spots, (2) Presence of faint spots from the superlattice reflections of 5/-phase (marked b), (3) Presence of faint streaks on the t420] mauixdirections for [112]matrixzone axis (marked c), was an evidence of the presence of S-phase. Examination of other SAD patterns also revealed the presence of Ti, 8/, I3/, S/ phases, and the absence 0/phase, which dissolution in favor of Ti-phase is suggested 13-51. Similar second phase particle morphology was observed for 2095 rolled alloy except for the absence of evidence for S/phase, which in turn resulted in the formation of PFZs as marked by the arrows in Fig. 10a and b. Previous studies revealed that an optimum mechanical properties can be achieved only when the material is stretched prior to aging. According to Sanders, S/ [3] and TI-phases precipitate both at high and low angle grain boundaries while being the dominant phase. Consequently, the absence of S/ in the structure of the T6 temper billets or sheets whether processed via ECAE or rolling should be expected since no stretching was involved prior to aging. Evidence for the precipitation of S/ in the structure of the T6 billets processed via ECAE, is consistent with the behavior of a T8 temper condition; although stretching was not involved. This can only be explained by the high dislocation activity retained in the structure even after one hour SHT at 510~ and 12 h exposure to 180~ during artificial aging, knowing that S/ precipitation is usually associated with dislocations 13, 14]. Accordingly, the combined precipitation of the S/ and the Ti-phases in addition to the fine grain structure resulted in an ultra-high strength and ductility of the processed alloys at room temperature.
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Current Advances in Mechanical Design and Production, MDP-7
CONCLUSIONS From the current research analysis, ECAE processing has proven its capability of developing a SMG structure <0.6 ~tm in AI-Cu-Li base alloys from initial grain sizes >100 lam. Route C processing produces equiaxed SMG structure slightly finer than that produced by route A. ECAE also Proved its capability of producing extremely energetic structure after post forming HT which resulted in a rapid aging response through the precipitation of 8/, 131,and 01 phases in the T4 temper condition and 8I, 131, and Tin,and S/ phases in the T6 temper condition. In addition, the elimination of PFZs near grain boundaries in the T6 temper condition by S/phase precipitation without prior stretching was evident. Finally, ECAE processing has significantly enhanced the ambient temperature mechanical properties of Weldalite alloys in comparison with conventional processing techniques and hence is most suitable for cryogenic structural parts in aerospace, where ultra-high strength and fracture toughness are demanded. REFERENCES 1. Moore, W. M., Zelin and Chaudhury K., "Superplasticity and Superplastic Forming", eds., Ghosh, A. and Bieler, R., The Minerals, Metal and Materials Society, Denmark, pp. 267275, (1995). 2. Cho, A., Greene, M., Fielding, P. and Skillingberg, H., "Report Reynolds Metal Co.", Richmond, VA, pp. 3-8, (1995). 3. Sanders, T. H., "In Aluminum-Lithium Alloys II", eds., Sanders, T. and Starke, E. A., The Metallurgical Society of AIME, Monterey, CA, pp. 1- 15, (1984). 4. Lavernia, E., Srivalsav, T., and Mohammed, F., J. of Mat. Sci., Vol. 25, pp. 1137-1158, (1990). 5. Kumar, K. S., Brown, S. A., and Pickens. J. R., Acta Metar., Vol. 44, No. 5, pp. 1899, (1996). 6. Langdon, T. J., and Pickens, J. R., In Aluminum-Lithium Alloys V, VA, March 27-31, pp.691-700., (1989). 7. Segal, V., "In 1st International Conference on-Proc. Mat. For Prop." eds., Henein, H. and Oki, T., TMS, Honolulu, pp. 611-613, (1993). 8. Ferrasse, S., Segal, V., Goforth, R., and Hartwig, T., "Development of SubmicrometerGraind (SMG) Microstructure in Aluminum 6061 Using Equal Channel Angular Extrusion", (in press). 9. Goodhew, P. J., Mater. Sci., pp. 153, (1973). 10. Zaidi, M., and Wert, J., "Treatise on Materials Science and Technology", eds., G. Mahon and R. Ricks, PA, pp. 139 (Personal collection, Salem, H.)., (1989) 11. Humphreys, F., and Hatherly, M., "Recrystallization and Related Annealing Phenomena", British Library Cataloguing Publication Data, Pergamon, pp. 167, (1995). 12. Skorlzki, B., Shiflet, G., and Starke, E. A., J. Metallurgical and Materials Transactions A, Vol. 27A, pp. 3431-3444, (1996). 13. Ahmed, M., and Ericsson, T., Scripta Metall., Vol. 19, pp. 457264, (1985). 14. Gayle, F., Heubaum, F., and Pickens, J., In Aluminum-Lithium V, eds., T. H. Sanders and E. A. Starke. Jr., Materials and Component Engineering Publication Ltd., Birmingham UK, pp. 701-711, (1989). 15. Ashton, R. F., Thompson, D. S., Starke, E. A., Jr., and Lin, F. S., Aluminum-Lithium III, eds., C. Baker, P.J. Gregson, S. J. Harris, and C. J. Peel, Institute of Metals, London, pp. 57-65, (1986). 16. Kumar, S., Mcshane, H. B., and Shepherds, T., Materials Science and Technology, vol. 10, pp. 162-170, (1994).
Current Advances in Mechanical Design and Production, MDP. 7
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Current Advances in Mechanical Design and Production, MDP-7
Fl&
5 TEM Photomicmvphs for ECAE and Rolled alloys, (a) and @) Low and high magnifications for 2095 ECAE pmctsSett alloys via Mute A with elongatcd (marked A a B) and cquiaved (marked C) structural regions, s h m n g
dislocation aclivlty (marked B). And a SAD pattern carresponding to zone aurs. (C) L (d) Via mute C, wth extinction contours along gram boundaries And a SAD pattern corresponding to [ 112]~lzone axis. and (e) 2095 shccts pmcssed via rolling where some distmtion actiww show around p’ dark spherical precipitates. h d a S A D paitern aonespondinR to [ 1 ionc axis.
Current Advances in Mechanical Design and Production, MDP-7
Flpl. 6 TEM Photomicrographs for 2095 ECAE billets processed via m i t e C M e r SHT at 5 10°C for l h followed hy WQ. (a) SMF situcturc < t pm. Large B ~ O I Y Sshow cxtinaion contours along some o l the grain boundanes mdicahve of energetjc structure. and small amrvs show the dark p-phase pfnning grain horindanes. &I) shows a SAD pattern corrcsponding to II12]hl 7onc axis with riipcrlnnicc rcflections for fil andlor P’-phnse<
F~E.7 TEM Phatomicropphs for 2095 processed via rolling aRer SHT at 510°C for I h followed by WQ. (a)
shows fine grain structure -4pm with httle dislocation actiwty within p n s , mow show thc dark $-phase pinning pain boundants. (b) shows a S A D pancrn corresponding to [ O I I ] M zone axis wlth supcrfaibce reflecrms for fif andlor F’-phases
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Fig. 8 TEM Photomicrographs for 2195 ECAE billets processed via route C in the "I"4 temper condition. (a) low magnification of a structure g~th high dislocation activity within the grains interacting ~1 precipitates, and extinction contours along some of the grain boundaries, (b) higher magnification of the grain marked by. the arrow with a SAD pattern corresponding to [100]~ zone axis where ~ l a t t i c e reflections for 8/ and/or [~-phases (marked by large arrows), and streaks representing O/(Ainu) phase are revealed.
Fig. 9 TEM Photomicrographs for 2195 ECAE billets processed via route C in the T6 temper condition. (a) low magnification of a well defmed and relaxed grain structure ~l.2pm in size, (b) higher magnification showing the Tt(AI2CuLi) precipitation within the grains and along some grain boundaries marked by arrows. A SAD pattern corresponding to [I 10]~azone axis showing spots and streaks for 8/,15l, TI, and SI.
Fig."lO'---~M Photomicrographs for 2095 processed via rolling in the T6 temper condition. (a) low magnification of a well defmed and relaxed grain structure ~4.1pm in size, and an evidence for PFZs formation (marked by arrows), (b) higher magnification with a SAD pattern corresponding to [011 ]~ zone axis showing spots and streaks for 8/, 15/,and T~ phases, while no evidence for S/ phase precipitation.
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
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STRUCTURE AND ENGINIEERING PROPERTIES OF SOME DUCTILE IRONS Refaey, A. *, Hafiz, M.** and Fatahalla, N.***
*Postgraduate student (Mechanical Engineer, Geisum Oil Company, No. 10, 250, st. Sarayat, Maadi, P.O. Box. 282 Maadi, Cairo-Egypt.) * * Assistant Professor, ***Professor of Materials Science, Mechanical Dept., Faculty of Engineering, AI Azhar University, 11371 Nasr City,Cairo-Egypt.
ABSTRACT Microstructure, hardness and wear characteristics were studied for: (1) Austempered SG-iron (ADI); and (2) low alloyed SG-iron. Comparison has been made between the properties of these two types and that of conventional SG-iron. The present ADI samples were produced by austenitising at 1203K, for 3.6 ks and austempered at 773K for 1.8 ks. The second type was prepared by adding low percentages ofNi, Cr, Mn and Mo to SG-iron. The microstructure of ADI consists of upper bainite and retained austenite, while the low alloyed SG-iron showed the graphite nodules to be surrounded by ferrite, pearlite, and iron carbides. Macrohardness study showed a maximum value for low alloyed SG-iron, and a minimum for conventional SG-iron, and hardness of ADI is intermediate between the two irons. The wear properties were determined using pin on ring machine, under dry sliding conditions at room temperature. The variation of wear rate with sliding distance, at variable loads and speeds were studied. The wear mechanisms were investigated by means of subsurface observations with an optical microscope. Microhardness test was used to study the change in the matrix strength with distance from the wom surface due to plastic deformation. Mild wear was observed for a wide range of sliding distances in the ADI. On the other hand, its range in low alloyed SG-iron was very narrow and most of the curves of mass loss revealed severe wear. KEYWORDS Conventional SG-iron, ADI, Low alloyed SG-iron, Austempering, Microstructure, Hardness, Wear Characteristics. I. INTRODUCTION Spheroidal graphite (SG) irons have significantly good combination of tensile strength, ductility and toughness, along with good wear resistance, and hardnability [1]. The most recent development is to obtain a variety ofbainitic structures [2]. Bainitic matrix in SG-irons can be obtained in two ways: (i) In the as-cast form where bainitic transformation is achieved by judicious alloying elements such as Ni, Mn, Mo and Cr. (ii) By subjecting conventional SG-iron castings to a specialised heat treatment known as austempering 12]. The remarkable combination of properties attainable from ADI has made them emerge lately as a new class of SG-iron [3]. ADI is present now as an attractive alternative to steel [3]. It has been reported that ADI is about twice as strong as standard SG-irons grades at the same level of toughness
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[3]. The resulting ADI have combinations of tensile strength and ductility, high fatigue strength, high toughness, greater vibration damping-capacity, excellent wear resistance, and cost less than that of steel casting. Because of their superior properties and low production cost, ADI have been increasingly used for industrial parts such as hydraulic valve bodies, exhaust systems, automotive gears, crankshafts, cylinder heads and many others [3]. In some of these applications, wear is an important problem and one of the major ways by which materials ceases to be useful [41. From the tribological point of view the commercial significance of ADI together with the lack of scientific investigations of their friction and wear behaviour prompted us to pursue this research work. At present very limited information on the wear characteristics of ADI is available [4]. This study was, therefore, to investigate the microstructure, hardness, and wear characteristics of ADI and low alloyed SG-iron compared with the conventional SG-iron. Also to fill in gaps in the previous studies and to produce wear properties at various parameters of loading, sliding distance, and speeds. 2. EXPERIMENTAL PROCEDURE
2.1 Materials, Casting and Melting Conventional SG-iron was produced in a high frequency induction furnace, in a commercial foundry. The charge was made up of 80% pig iron with 3.9 mass% C, 0.9 mass% Si, and 20% SG-iron return scrap, with 3.6 mass% C, 2.4 mass% Si and 0.05% Mg. The liquid metal was then treated (1.6% by weight of charge) with Fe-Si-Mg alloy (45, 50 and 5 mass% respectively). The melt was then inoculated (0.6% by weight of charge), using ferro-silicon alloy (20 and 80% by weight, respectively), having a grain size from 0.2-3 mm. The Sandwich technique [51 was used in producing SG-iron. Addition of the following master alloys produced the low alloyed SG-iron: FeNigs, FeMoT0, FeMnT0, and FeCrT0, in the as cast condition. Table I shows the chemical analysis of conventional SG-iron before and after Mgtreatment and low alloyed SG-iron. To maintain the same cooling rate and other casting conditions, the melts of conventional as well as low alloyed SG-irons were poured into test block sand moulds with dimensions of (120 x 100 x 75 mm3).
2.2 Tempering and Austempering Tempering heat treatment was carded out for the low alloyed SG-iron block at 873K for 7.2 ks in a heating furnace. Austempering was conducted using small cubes; 20 mm side, taken from the block of conventional SG-iron in the following steps [1]: (i) Heating the SG-iron to austenitising temperature of 1203K for 3.6 ks in a muffle furnace with accuracy • 5K. (ii) Quenching in a fiuidised salt bath containing "Die Gusse" As 140, at a temperature of 723K. (iii) Holding the SG-iron in a salt bath for 1.8 ks. (iv) Removing from the salt path and cooling in atmospheric air to room temperature (300K).
2.3 Microstructure Observations The Reichert optical microscope "Me F2" with optical bench was used in all metallographic examinations. Standard metallographic techniques [6.~ were employed to reveal the different microconstituents of the structure.
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2.4 Hardness Test Macrohardness study was performed using three indentation hardness methods being Vickers, Brinell, andRockwell. In the Vickers test a square diamond-pyramid indenture having an angle of 136 ~ and a load of 62.5 kg was applied on the sample for a period of 15 s. On the other hand, a steel ball of 5mm diameter was used in the Brinell test under a load of 750 kg applied for 15 s. Thirdly, the Rockwell tests were carried out on three scales; A-scale using a steel ball of 1.6 mm diameter under a load of 60 kg. The B-scale using a steel ball of 1.6 mm diameter under a load of 100 kg. The C-scale using a diamond cone indenture with 0.2 mm diameter under a load of 150 kg load applied for 15 s. All the hardness tests were carried out at about 300K using Otto Wolpert-Werke, hardness testing machine. Standard microhardness test was applied on the specimens of each type to obtain the hardness of its matrix constituents. It was also carried out on longitudinal cross section perpendicular to the worn surface at different distances from the worn surface. To reveal any changes in hardness of the subsurface region of the pin due to plastic deformation. All microhardness tests were carded out at 300K using Shimadzu microhardness tester type M with 200g load and 15 s time of loading.
2.5 Sliding Wear Tests The wear resistance test for the 3 types of SG iron, were carded out with pin-on-ring apparatus using the TNO-Tribometer, under dry sliding conditions in ambient air at 300K. The specimens of SG-iron in the form of cylindrical pin of 8 mm diameter and 12 mm height were used. Hardened steel ring of HV 8100 MPa of 70 mm diameter and 10 mm thickness was used as a counter body. During the tests the pin was pressed against the ring with variable loads of 180 N, 265 N and 445 N. Three sliding speeds of the disc were chosen i.e. 0.57, 0.95 and 1.53 m/s. Five sliding distances were selected starting from 0.5 km with interval of I km up to 4.5 km. The specimen was removed from the testing machine after each interval. The variation of wear rate with sliding distance, at variable loads and speeds were studied. After testing the worn surface of the pin was examined by optical microscope. Three tests were performed for each set of conditions and the average was taken. 3. RESULTS AND DISCUSSION
3.1 Metallography Figure 1 (a, b and c) illustrates the matrix constituents for conventional SG-iron, ADI and low alloyed SG-iron respectively. Figure I (a) reveals, generally, graphite nodules embedded in a ferritic-pearlitic matrix, known as bull's-eye structure (cf. arrow in photo (a)). Figure 1 (b) shows the matrix of ADI generally consisting of bainite (B) (acicular) and retained austenite (non-transformed white regions (cf. arrow in photo (b)). No evidence of intercellular martensite or austenite decomposition products (carbides) was seen in ADI. During austempring at 723K and after austenitising at 1203K the austenite transformed first to an upper bainite. This structure agree with Korichi, et al [7l who indicated that during austempering at 713K after austenitising at 1173K the austenite transformed first to upper bainite that was coarser than that obtained at 653K. Grech, et al 181 concluded that, at higher austempering temperatures, particularly at 673K, larger, more blocky austenite volumes occur between the ferrite plates. These are randomly distributed and relatively non-uniform in carbon content. On cooling to room temperature, martensite is likely to form in their centres.
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This last phenomenon is contradicting with our investigation due to short austempering time, because further holding in austempering bath may however, lead to the decomposition of this metastable austenite to ferrite and cementite and to general deterioration in properties. Figure 1 (e) shows the graphite nodules surrounded by ferrite (F), pearlite (P), and iron carbides occurring in the inter-nodular regions (cf. arrow in photo (c)). This structure is in good agreement with Muthukumarasamy, S., [21 who indicated that the low alloy addition lead to the formation of intercellular carbides. Moreover, he indicated that repetition of production of bainitic ductile iron in the as-cast condition based on Ni and Mo can be achieved. Table 2 shows the volume fraction (Vf) of the matrix-phases. It can be seen that the Vf of ferrite has a higher value than that of pearlite in the matrix of conventional SG-iron. This may refer to the chemical composition of the matrix and high carbon equivalent of 4.6% [9]. Doubrava [10] concluded that the chemical composition of the iron has been established as one of the significant factors determining the matrix structure. Silicon favours ferrite, so increased amount of ferrite can be obtained by increasing the silicon content in SG-iron. Carbon also is expected to favour ferrite over pearlite. Fatahalla, et al [9] showed that the Vf of ferrite increased with increasing %CE up to 4.61. The Vfofupper bainite has a higher value than that of retained austenite in the matrix of ADI. This refers to a higher austempering temperature, which leads to faster rate of carbon diffusion and, consequently, the growth rate of these ferritic platelets is rather rapid. This result in a lower Vf of ferrite and more austenite in the matrix, but these ferritic platelets will be rather large or coarse in nature. This matrix agree with Susil K, et al [11] who concluded that the Vfofaustenite in the matrix increases with increasing the austempering temperature, whereas the Vf of ferrite decreases as the austempering temperature increases. The Vf of pearlite has a higher value than that of ferrite, and iron carbide in the low alloyed SG-iron matrix. This may refer to presence of elements that favour pearlite and carbides in the composition. These elements include manganese, nickel, molybdenum, and chromium. This may also stem from the high solidification cooling rate combined with relatively low carbon equivalent [91. Doubrava [101 indicated that, higher pouring temperatures somewhat increase the amount of pearlite.
It can be noted from Table 2 that, the Vf of low alloyed SG-iron showed lower values than that of conventional SG-iron and ADI. Although the theoretical determination of the Vfof graphite in such SG-iron results in a value around 14% [12],however, Table 2 revealed higher values for all types investigated. This phenomenon refers to the high solidification cooling rate (SCR) on the industrial scale which in turn results in the movement of the eutectic point [12]. 3.2 Hardness Table 3 shows the macro-and micro-hardness values of low alloyed SG-iron, ADI, and conventional SG-iron. The lowest values of hardness refers to conventional SG-iron, this may refer to the presence of ferritic-pearlitic matrix which showed the Vickers hardness (HV) values ranging from 1560 to 2170 MPa. The present hardness values are in consistence with literature of SG-iron according to their matrix and nodule characteristics [13]. The intermediate values of hardness refer to ADI, this may refer to an increase in austempering temperature which leads to the formation of upper bainite and reduction in austenite content. This may cause a reduction in strength and hardness if compared with low austempering temperature 573K which showed lower bainite and retained austenite structure having a HV values of 5300 MPa. Stenfors, S., et al [14] concluded that, the hardness decreased when the austempering temperature is increased and a minimum is observed at about 673K. The results
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of the present ADI are in consistence with Ref. [7] who showed that during austempering at 713K after austenitising at 1173K, retained austenite peaked at approximately 20% after 15 minutes, and the HV fell to a plateau level of approximately 3150 MPa. It also agrees with standards of ASTM A 897 M: 1990, which showed the HV values ranging from 3020 to 3630 MPa. The high values of the hardness corresponds to low alloyed SG-iron, this may be due to the presence of alloying elements (Mn, Ni, Mo and Cr) which cause the formation of harder phases such as carbide and pearlite as shown in the microstructure given in Fig. 1 (e). These results agree well with Ref. [2] who indicated that the highest values of HV recorded was 4200 MPa at 2 mass% Ni. The Vickers microhardness results of ferrite matrix constituents showed little lower values when compared with Ref. [151, in which the ferrite hardness reported 2000 MPa. The microhardness results of pearlite matrix constituents showed a value of 2600 MPa. This value agrees well with Ref. 115] in which the pearlite ranges from 2230 to 3020 MPa. The bainitic constituents showed higher value when compared with the grades of ASTM standard A897 M: 1990 in which the hardness of the bainite ranges from 3020 to 3630 MPa. This maybe due to the variations of chemical compositions and increase of the volume fraction of the harder phase (upper bainite). It may also refer to the different procedures used in the two researches, and additionally, due to the complexity of defining the hardness property and the factors that affects it [9]. The retained austenite of ADI, showed a consistence value with Janowak, [16] who stated that the hardness of austinite ranges from 1000 to 4000 MPa. The ferrite, pearlite and iron carbides showed higher values when compared with the values of Ref. [17]. This may refer to the same reasoning mentioned above, and may also stem from the different carbon contents and the amount of residual stresses of the alloys [121. Additionally may refer to the increase of volume fraction of the harder phases such as pearlite and iron carbide 19]. 4 WEAR CHARACTERISTICS 4.1 Variation of Wear Rate Versus Sliding Distance at Constant Load
Figure 2 shows the variation of wear rate with sliding distance for the 3-types of SG-iron. It is interesting to note that the wear characteristics for the 3-types of SG-iron follow the normal behaviour of metals and alloys 13]. That is to say the low alloyed SG-iron, showed that; the running-in period starts from zero to 3.5 km followed by steady state period from 3.5 km to 4.5 km. The running-in periods of conventional SG-iron and ADI, starts from zero and continues up to 4.5 km. After 4.5 km of sliding the wear rate of conventional SG-iron is 5 times greater than that of ADI and 33 times than that of low alloyed SG-iron. Consequently the wear rate of ADI is 6.5 times greater than that of low alloyed SG-iron. The conventional SG-iron has by far the highest wear rate which maybe associated with its relatively low hardness, this maybe due to the presence of ferritic-pearlitc matrix. The values of wear rate of ADI are intermediate between the other two types, this may refer to the change in hardness which is obviously related to the amount of retained austenite. The increase in austenitising and austempering temperatures decreases the bulk hardness because of the increased amount of retained austenite, resulting in an increase in wear rate of ADI greater than low alloyed SGiron. This result agrees with Chang et al [4]. They concluded that the wear resistance absolutely depends on the amount of retained austenite and the hardness of the matrix structure. The large amount of retained austenite results in lower wear rate. The low alloyed SG-iron showed superior wear resistance compared to the other two types, this may refer to the high hardness which is a good indication of the low wear rate. The nature of wear characteristics during the tests shows that the process is essentially one of abrasion of the
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surface-treated layer. Hence the rate of wear would be expected related to hardness, as indeed was demonstrated by the mild wear result. These results agree with those published previously [18]. They concluded that wear resistance absolutely depends on the hardness of the matrix structure. The hardness versus wear rate for the 3-types of SG-iron is illustrated in Fig. 3. It can be seen that the wear rate decreases with increasing the hardness. 4.2. Variations of Wear Rate with Sliding Distance at Different Loads
Figures 4 and 5 illustrate the variation of wear rate with sliding distance at different loads for low alloyed SG-iron and ADI respectively compared with the conventional SG-iron. As shown in the two figures, the effect of increasing the applied load and sliding distance is to increase the wear rate. Figure 4 shows that, at low and medium loads (180N and 265N) the wear rate shows approximately linear relationship with sliding distance with no indication of an initial transient stage of mild wear to severe wear from the beginning to the end of the curves. This phenomenon refers to the surface of the pin which was relatively smooth with only minor surface grooves. For relatively high loads of (445N) testing, the running-in region start from zero to 0.5 km of sliding distance. Thereafter, a linear increase is observed in transient region up to 3.5 km of sliding distance and finally a drastic increase (severe wear) up to 4.5 km of sliding distance. This phenomenon refer to the surface of the pin which is irregular and heavily damaged giving further indication that a basic change in wear behaviour has occurred. These features are characteristic of a mild-to-severe wear transition. Figure 5 shows that at low load (180N) the wear rate was relatively small and it shows curvilinear relationship from the beginning to the end of the curve. For medium and high loads (265N and 445N), it is interesting to note that the wear rate increased drastically from the beginning to the end of the curves. It is also noted that there is no indication of transition period in the curves, and most of the curves revealed severe wear. For low loads the mild wear was observed for a wide range of sliding distances in the ADI. On the other hand, its range in the low alloyed SG-iron was very narrow. These two phenomena may refer to the same reasoning mentioned above. Additionally, they may refer to the increase in volume fraction of retained austenite and upper bainite which are sorer than carbides. For conventional SG-iron, it is noticed that, the wear rate shows approximately linear relationship with sliding distance, and applied loads. The running-in region starts from zero and continues up to 4.5 km with no indication of an initial transient stage from running-in region to a steady state region for both types of SG-iron. Investigations of these two figures delineated that the maximum values of wear rate (minimum wear resistance), from the beginning of the curves up to their ends at a sliding distance of 4.5 Ion, refer to conventional SG-iron at 265 N load. On the other hand, the minimum values of wear rate (maximum wear resistance) referred to the low alloyed SG-iron. This phenomenon is due to the presence of alloying elements in the low alloyed SG-iron such as the molybdenum which is a mild carbide former and has a very modest effect on chill. In Ref. [19] they concluded that, adding molybdenum at levels of 0.5 % or more, results in appearance of some grain boundary carbides. This carbide stabilising effect will be accentuated by the presence of other carbide-stabilising element, such as Cr, Mn, and V. The structure of the matrix, as well as the hardness and wear resistance is markedly affected by the presence of carbides. Shaikh et al {20] showed that the carbide size, type, and distribution are all related in some way to wear resistance. Eyre et al [21] studied the wear characteristics of flake and nodular graphite cast irons. They observed a transition from mild to severe wear with increasing load in both cases.
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4.3 Variation of Wear Rate With Sliding Distance at Different Speeds Figures 6 and 7 illustrate the variation of wear rate with sliding distance at different speeds of low alloyed SG-iron and ADI compared with conventional SG-iron. As shown in the two figures, the effect of increasing the sliding speed is to increase the wear rate. Figure 6 shows that, at low speeds (0.57 m/s, 0.95 m/s) the wear rate is somewhat regular with low values (mild wear), and the running-in period started from zero to 2.5 km of sliding distance. After 2.5 km of sliding distance the wear rate is transformed to steady state condition until 4.5 km. For relatively high speeds (1.53 m/s) the curve showed normal behaviour up to 2.5 km of sliding distance. Thereafter, a drastic linear increase (severe wear) is observed up to 3.5 km and finally it tended to stabilise. Figure 7 shows that; at low speeds (0.57 m/s, 0.95 m/s) the running-in region starts from zero and continued up to 4.5 km of sliding distance, and the mild wear has a wider range than that of the low alloyed SG-iron. At relatively high speed (1.53 m/s) the wear rate increased drastically from zero to 0.5 krn of sliding distance. Thereafter, a monotonic gradual increase occurred with sliding distance up to 4.5 km. It is clear that the conventional SG-iron shows maximum values of wear rate in both figures. The drastic increase in wear rate at higher speeds can be attributed to more irregular surfaces. The increase of sliding speed make the sliding surface rough when wear particle interfere between the two sliding surfaces. It may also be due to the high temperature and vibration generated during the test at high speeds. Terheci, et al [22] concluded that the increase in wear rate observed at higher speed is a result of vibrations which in turn are mainly the result of higher impact shocks exerted by the two sliding surfaces, one against another. 4.4 Study of Wear Track Figure 8 (a, b and e) shows optical micrographs of longitudinal sections perpendicular to the worn surface, of conventional SG-iron, ADI and low alloyed SG-iron, tested in wear. It can be seen that the edges of conventional SG-iron and ADI samples are irregular, as in Fig. 8 (a,b) whereas that for the low alloyed SG-iron sample it is slant as in Fig. 8 (c). The worn surfaces of the 3-materials show that, the mode of particle deformation is mainly by surface adhesion and plastic deformation. These wear characteristics are strongly related to the microstructure of the three types of SG-iron, which indicated that the low alloyed SG-iron is higher wear resistance. Under low load mild condition, the three samples exhibit what appears to be work hardened layer. This layer developed increased in thickness with increasing load on the surface in the mild region. Increasing the load seems to increase the surface temperature, further metallographic examination and hardness testing indicated that, this layer was essentially carbide in the low alloyed SG-iron sample. 4.5 Subsurface Microhardness Investigation Figure 9 shows the Vickers microhardness values of the low alloyed SG-iron, ADI, and conventional SG-iron. Clearly they are strongly related to the microstructures in the present results. For instance, the matrix hardness increase as the distance from the contact surface decreases in the three types of SG-iron, indicating that plastic strain hardening occurred during the wear test [3]. For low alloyed SG-iron the results shows that, the wear test produced a hard surface layer in mild wear, and very high increase in hardness can be observed in only a very narrow layer, less than 160 ~tm apart from the surface. The hardness increase from 6000 MPa to 7800 MPa on the worn surface of the low alloyed SG-iron and from 3300 MPa to 4800 MPa for ADI, and from 1650 MPa to 3000 MPa for conventional SG-iron, due to the subsurface deformation produced in sliding under a load of 265 N. With
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the increase of load to 445 N and during severe wear, this hard surface layer appeared to have softened in the immediate vicinity of the surface, the hardness increased to a still higher value for the three types of SG-iron. The above observations of microstructure are strongly related to the microhardness curves of the three types of SG-iron. The worn surface of conventional SG-iron shows that, the deformed layer has a wider range than that of ADI and low alloyed SG-iron. We can observe that, the hardened layer of conventional SG-iron has a wide range of 260 ~tm, while the ADI range was 200 ~tm, and for low alloyed SG-iron has a narrower range of 160 ~tm. Generally speaking, the Vickers microhardness decreasing with increasing the depth into the bulk matrix hardness and then steady state condition for the 3-types of SG-iron was attained. CONCLUSIONS 1.
Austempering at 723K, after austenitising at 1203K, revealed a microstructure of upper bainite with ferrite needles somewhat shorter and more widely spaced. This structure containing 20% retained austenite results an excellent combination of hardness and wear characteristics. 2. The relative amounts of matrix constituents for low alloyed SG-iron are ferrite, pearlite, and iron carbides, were directly related to the alloy additions ofNi, Cr, Mo and Mn. 3. Hardness of low alloyed SG-iron showed higher values than ADI and conventional SGiron. 4. Wear resistance depends on the combined effect of hardness. The low alloyed SG-iron and ADI generally have higher wear resistance than that of conventional SG-iron. 5. The wear rate of conventional SG-iron is greater 5 times than that of ADI and 33 times than that of low alloyed SG-iron. Consequently the wear rate of ADI is greater 6.5 times than that of low alloyed SG-iron. 6. The production process of ADI is easy to implement and also results in large economic benefits and a comparable properties than that of low alloyed SG-iron [due to expensive alloying additions (Ni, Cr, and Mo)]. ACKNOWLEDGEMENT The authors wish to express their gratitude to Prof. Dr. Adel Nofal the Head of the Central Metallurgical Research and Development Institute (CMRDI) in Tebbin-Helwan, for the help in studying the wear characteristics on the TNO-Tribometer machine. The first author wishes to express his gratitude to Eng. Osama Hussein, the Manager of The Information Centre ELNasr Company for Castings, for kindly supplying of raw materials and facilities allowed in the foundry-shop. REFERENCES 1. 2.
3.
Cast Metals Development Ltd. "Austempered-Ductile Iron Castings, Advantages, Production, Properties and Specifications", Mat. and Design, Vol. 5, pp. 285-297 (1992). Muthukumarasamy, S., Sabu, A.J.S., and Seshah, S.," High Strength Ductile Irons-As Cast Bainitic Ductile Iron and Austempered Ductile Iron", Indian Foundry Journal, August, pp. 23-29, (1992). Ping, L., Bahadur, and Verhoeven, D.,"Friction and Wear Behaviour of High Silicon Bainitic Structures in Austempered Cast Iron and Steel", Wear ,Vol. 138, pp. 269-284, (1990).
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4. Jeng, M.C., "Abrasive Wear Study of Bainitic Nodular Cast Iron", Journal of Materials Science, Vol. 28, pp.6555-6561, (1993). 5. Fatahalla, N., Bahi, S., and Hussein, O., "Metallurgical Parameters, Mechanical Properties, and Machinability of Ductile Cast Iron", Journal of Materials Science, Vol. 3 l, pp. 5765-5772, (1996). 6. ASTM Designation: A 247-67 (Reapproved 1984) "Standard Method for: Evaluating the Microstructure of Graphite in Iron Castings, Annual Book of ASTM Standards Vol. 03.03, (1996). 7. Korichi, S., Priestner, R., "High Temperature Decomposition of Austemperd Microstructures in Spheroidal Graphite Cast Iron", Materials Science and Technology, Vol. I l, pp.901-907, (1995). 8. Grech, M., and Young, J, M., "Influence of Austempring Temperature on the Characteristics of Austempered Ductile Iron Alloyed with Cu and Ni", AFS Trans., Vol. 98, pp. 345-352, (1990). 9. Fatahalla, N., Hakim, H., Abo Elezz, A., Mohamed, M., "Effect of Carbon Equivalent on the Microstructure and Mechanical Properties of Ductile Cast Iron" ,Z. Metallkd, Vol. 89, pp.507-513, 7 (1998). 10. Doubrava J.H., S.F. Carter and J.F. Wallance (The Influence of Processing Variables on the Matrix Structure and Nodularity of Ductile Iron), AFS Transactions, Vol. 89, pp. 229250. (1981), 11. Putatunda, S. K., and Singh, I., "Fracture Toughness of Unalloyed Austempered Ductile Cast Iron (ADI)" Journal of Testing and Evaluation, (1995), pp. 325-332.[By the American Society for Testing and Materials] 12. Fatahalla, N., Gomaa, T., Bahi, S., Negro, M., " Microstructure, Plastic Deformation Characteristics, and Mechanical Properties of AS-Cast and Ferritic Heat-treated SG-Iron in a Wide Range of Solidification Cooling Rate", Z. Metallkd, Vol. 89, pp.554-561, 8 (1998). 13. ASM; Metals Handbook (Casting), Vol.15, 9th Edition, American Society For Metals, Metals Park, Ohio, USA, (1992). 14. Stenfors, S.E. , Storestm, J . , and Sandstorm. R., "Influence of Heat Treatment and Composition on the Mechanical Properties and Machinability of ADI", 2"d International Conference of ADI (Proceedings Conference), Ann. Arbor, Michigan, USA, 17-19 March, pp. 227-237, American Society of Mechanical Engineers, (1986). 15. S. Avner; (Introduction to Physical Metallurgy), 2nd Edition, McGraw-Hill bkogakusha, Ltd., Tokyo- New Delhi- Sydney, (1974). 16. Janowak J.F. (Cast Iron Metallurgy for Improved Machinability), Journal of Applied Metalworking, Vol. 4, pp. 223-229,(1986). 17. Y. Lakhtin (Engineering Physical Metallurgy), (1971) MIR Publishers. Moscow. 18. Shimizu, K., Noguchi, T., Kamada, T., and Takasaki, H., (Progress of Erosive Wear in Spheriodal Graphite Cast Iron) Wear, Vol. 198, pp. 150-155, (1996). 19. Angus, H., (Cast Iron, Physical and Engineering Properties), 2n~Edition. British Cast Iron Research Association, London-Boston-By Butterworths, (1978). 20. Shaikh, Q.A. (Wear and Microstructural Studies of Alloy Sintered Steel) Materials science and technology, Vol.7, pp. 728-734, August (1991). 21. Eyre, T.S., Iles, R. F. and Gasson, D.W., "Wear Characteristics of Flake and Nodular Graphite Cast Iron", wear, Vol. 13, pp. 229-245, (1969). 22. Terheci, M., Manory, R, R., and Hensler, J, H., "The Friction and Wear of Automotive Grey Cast Iron Under Dry Sliding Conditions" Part l- Relationships Between Wear Loss and Testing Parameters, Wear, Vol. 180, pp73-78, (1995).
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Current Advances in Mechanical Design and Production, MDP-7
Table I Chemical analysis of the conventional SG-iron before and after magnesium treatment and low alloyed SG-iron |
C
Si
S
Mn
P
Mg
Cr
Ni
3.75
1.3
0.01
0.75
0.04
0.003
0.06
0.02
4.33
After Mg Treatment
3.45
2.4
0.009
0.089
0.042
0.06
0.061
0.024
4.6
Low alloyed SG-iron
3.53
1.42
0.006
0.6
0.037
0.045
0.75
2.2
Mo
%C.E
Conditi'Om~ Before Mg Treatment .
,
4.18
0.52
,.
Table 2 Volume fraction of matrix constituents for 3 types of SG-iron Constituent Phases Volume Fraction,(% ) Alloy type
Graphite
Ferrite
Pearlite
Conventional SG-iron
2O
50
3O
ADI
20
Low alloyed SG-iron
15
25
40
Retained Austenite
Bainite
20
60
Iron Carbide
.................
20
Table 3 Macro and microhardness values for constituents of the 3 types of SG-iron Vickers Microhardness (MPa)
Rockwell
HB MPa
HV MPa
Conventional SG-iron
1820
1850
51
ADI
3180
3200
---
59
33
Low alloyed SG-iron
4420
4600
---
61
45
,.
B
Type
C
Ferrite
Pearlite
1650
2600
Austenite
3300 2830
4000
Bainite Carbide
3730 6000
379
Current Advances in Mechanical Design and Production, MDP-7
180
160
160
o
Conventional SG-iron
o 120
40
140
.-.x 120
_E
~ ~o 9
.... Conventional SG-iron
~
80
~
6o ADI
ADI M
20 o
. . . . . .
~ o
1ooo
2000
3000
1850
4000
Sliding Distance (m)
16
160 .-~
m
14
m 140 -~
o
12
120 q.t
10
100 o U
.j "
3200
4600
Macrohardness HV (MPa)
Fig. 3. R e l a t i o n s h i p Between H a r d n e s s and
Fig.2. Relationship between w e a r rate and sliding distance for the 3 types of SG-iron o
Low alloyed SG-iron ~ _
Wear Rate for the 3 TvDes of SG-iron ~8o
50 _ O r
14o
.~
r
8
8o
~
6
60
~
"1"
160 ~
~
120
o ,i-
x E 30
O
100 o
920
8o
-~
60
'~
o
40 ~ 2
ly
0
20
0
~0oo 2 o o o 3 o o 0 4o0o Sliding distance (m)
0
9
n,,
0 0
1000 2000 3000 Sliding distance (m)
20
v0
o
r
4000
Fig. 4 Relationship between w e a r rate and sliding distance at different loads of l o w alloyed SG-iron
Fig. 5 R e l a t i o n s h i p between wear rate
c o m p a r e d with c o n v e n t i o n a l SG-iron
ADI c o m p a r e d with c o n v e n t i o n a l SG-iron
and s l i d i n g d i s t a n c e at different loads of
Current Advances in Mechanical Design and Production, MDP- 7
380
"r (n
0 m
14
160 ~
140 (n
12
0
120 i
10
x E
1~1.53
140
~0.57
'toil'm0.95 m/s
~ 120
L_ "I'
m/s :
8
=..
6
40
2O ~=_
20
0
0
1000 2000 3000 Sliding distance (m)
0
4000
m O
Fig.6 Relationship between wear rate and sliding distance at different speeds of low alloyed SG-iron compared with conventional SG-iron
1,53 m/s
Jm : O
1000 2000 3000 Sliding distance (m)
Fig.7 Relationship between wear rate and slidig distance at different speeds of ADI compared with conventional SG-iron
Conventional SG-iron
40
60
I~
4000
ADI
20
20 o
0
3000
.u :E 1500
~
0,57 m l s A.D.I
0
L o w alloyed S G - i r o n
4500
6o
O
A.D.I
900o
v u)
k,.
0,95 m / s
Fig. 8 Optical microgrphs showing longitudinal cross section near the worn surface for the (a) Convention SG-iron, (b) ADI, and (c) Low alloyed SG-iron
m 75OO A. :S A > 6OOO -r
| C
100 o o 0 v.,
60
4o'[
~=
120
lu I= 0
8O ~
2
O m
1 4 0 ~m
conventional SG-iron
100~> ~'1oo ,~
.J t._
~
160
"11"
80 100 120 140 160 180 200 220 240 260 280 300 Depth from the worn surface (mm)
Fig. 9. Variation of microhardness with distan from the worn surface for the 3 types of SG-iron
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
381
CONVENTIONAL VERSUS THIN SLAB CASTING: A NUMERICAL SIMULATION APPROACH FOR THE COMPARISON OF MICROSTRUCTURAL PROPERTIES
Youssef, Y.M.', Megahed, G.M.'" and Lee, P.D."" Alexandria National Iron and Steel Co. (ANSDK), Alexandria Tabbin Institute for Metallurgical Studies, Cairo. Dept. of Materials, Imperial College of Science, Technology and Medicine, UK **
ABSTRACT Due to improved efficiency and reduced environmental impact, steel producers are exploring the feasibility of thin slab casting to replace conventional continuous casting. A numerical simulation of the solidification process, including thermal and microstruetural predictions, was developed and applied to compare the quality of the microstructures in these competitive processes. Through appropriate assumptions, the model was optimised to run quickly on a PC, allowing fast and inexpensive comparisons to be made. The predicted temperature time history was used to calculate the microstructural features of both secondary Dendrite Arm Spacing and carbon microsegregation across dendrite arms. The model was validated by comparing the model results with published thermal data for conventional slab casting, together with calculated and measured data of secondary Dendrite Arm Spacing for conventional and thin slabs. Experiments were also performed to physically simulate the thin slab casting process, providing data for the heat transfer coefficients and a means for further validation of both the thermal and microstructural predictions. The validated model was used to compare the conventional and thin slab casting processes for the production of two steel grades, AISI 1008 low carbon and AISI 1026 medium carbon steels. It was found that a more refined structure was obtained rising the thin slab process, leading to improved quality in the final product. KEYWORDS Continuous Casting, Thin Slab Casting, Modelling, Microstructural Simulation, Steel. I. INTRODUCTION Steel companies are under continuous pressure to reduce operating costs whilst improving product quality. To achieve this goal a number of new manufacturing methods have been developed that provide an excellent product whilst benefiting from the economic savings of a reduced number of production steps compared to conventional processes. Thin slab and strip casting are two examples of technologies that both reduce the number of processing steps whilst combining others into a continuous line operation. Therefore, both methods produce cast material nearer to the dimensions of the final product with reduced material loss, energy consumption and manpower. Such technologies have become known as near net shape
382 casting
Current Advances in Mechanical Design and Production, MDP-7
[1,2].
Near net shape casting of steel is currently a subject of world-wide activity
because of its expected advantages concerning investment costs, energy consumption and saving of numerous costly process steps in the production of flat products [3]. From the viewpoint of material properties, the refinement of the microstructure in the as-cast state, namely: secondary Dendrite Arm Spacing (DAS) has one of the most beneficial effects [4]. This effect cannot be ascribed to the reduction in casting thickness only, which decreases in the sequence thin slab casting, strip casting and thin strip casting, but also depends on other process parameters such as metal-mould heat transfer and casting temperature. To determine the significance of these parameters upon the as cast structure, simulations of heat transfer and solidification structure can be applied. The mathematical simulation of conventional continuous casting is now well established using either finite difference/volume [5-7] or finite element [8] methods. Recently, several authors have also simulated near net shape processes [9-11], however, one of the most significant parameters, the heat transfer coefficient at the mould-metal interface, has not been as well characterised. Furthermore, the coupling to the prediction of microstructural properties has not been widely applied. The present work is aimed at producing a combined model that can provide a simulation tool that can predict the microstructural features for a range of continuous casting processes for quick comparison, even on a personal computer. The assumptions required to produce an effective fast tool for process comparison together with the model implementation are presented first. An instrumented mould developed to both determine the boundary conditions and validate the model is then outlined. Finally, the application of the programme for the comparison of thin slab processes (cast thickness ranging from 20 to 80 mm) to conventional slabs (normally 200 to 270 mm thickness) is described. Although two microstructural features were compared, secondary Dendrite Arm Spacing and microsegregation, only the DAS values will be discussed in this paper. 2. EXPERIMENTAL METHODS A static mould with an adjustable thickness and the ability to apply varying pressures was designed and constructed in order to measure the heat transfer coefficients over a range of conditions and to produce samples for model validation. The experimental apparatus consisted of a static water cooled copper mould that simulates the mould cooling zone for different thin slab thicknesses by varying the distance between the two mould sides. The slab to mould heat transfer coefficient was determined using thermocouples embedded in the copper mould. Castings of different steel grades were made, and optical emission spectrometry was used to verify the composition whilst the microstructure was examined using optical microscopy. Further details of the experimental apparatus and methods are given by Youssef [12]. 2. MATHEMATICAL MODELLING 3.1 Solidification Model Theory The high heat capacity and low thermal conductivity of steel combined with the high casting speeds in thin slab casting (5 to 6 m/min), make it a reasonable assumption that the heat transferred by conduction in the axial (z) direction is negligible compared to the heat
Current Advances tn Mechanical Design and Production, MDP-7
383
transferred by the bulk motion of the strand. Furthermore, in thin slab casting the thickness of the slab is very small compared to the width. Therefore, by using a moving frame of reference where a slice is followed along the caster, only the heat transfer through the thickness need be simulated. Using these assumptions, the transient heat transfer and solidification in continuous casting of steel slab can be simulated by the Fourier equation in one dimension: 8 k(T) 8T
~x
OH
-~-- = p ( r ) - ~ -
(I)
where T is the temperature, k is the thermal conductivity, is the density, t is the time, x is the through thickness dimension and H is the enthalpy. In order to include the latent heat of solidification, the enthalpy-temperature relationship takes the following form: H = Cpr + L , r >_ 7'1
(2)
H = CpT + L ( I - f~), T~ < T < T~
(3)
H = C p r , r <_ 7",
(4)
where L is the latent heat, Tt is the liquidus temperature, Ts is the solidus temperature and fs is the solid fraction. Due to the non-linearity of the associated boundary conditions, as well as the temperature dependence of both thermal conductivity and specific heat capacity of steel, an analytic solution to equation (1 was not possible; therefore, it was solved using an explicit finite difference method as described by many authors, for example Croft and Lilley [13].
3.2 Initial and Boundary Conditions and Thermophysical Properties Appropriate initial and boundary conditions must be determined to solve equation (1 and the thermophysical properties known. The initial conditions were fixed by assuming the initial temperature was constant through the thickness and equal to the temperature of the metal poured into the mould, Tpou,. Assuming that the heat transfer is equal on both faces of the mould, symmetry around the slab centre was used, fixing the boundary condition at the centre line as:
~(x,t) c~
=0atx
0, t > 0
(5)
At the slab-mould interface, the effects of convection anradiation were lumped into one bulk heat flux and the boundary condition was set to: c~(x,t)
k- c~
+qb = 0 t>0
(6)
where qb is the heat flux in W/m 2, which was set equal to the effective heat transfer coefficient multiplied by the difference in temperature between the billet and the mould. The quantification of the heat transfer coefficient between the billet and the mould in the casting processes is one of the most important factors affecting the accuracy of solidification models. The precise measurement of this factor is essential to refine the solidification models in order to accurately predict the microstructure properties. For the purpose of comparison, the values of the heat transfer coefficient presented in the literature for conventional and thin slab casting were used. For final application of the model, the heat transfer coefficients measured using the static mould were used.
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Current Advances in Mechanical Design and Production, MDP-7
The thermal conductivity and specific heat of steel were calculated using the equations given by Pehlke et al. [14]. The density and latent heat of solidification were taken as constant and equal to 7800 kg/m3 and 277000 J/kg respectively. 3.3 Microstructural Modelling Two aspects of the microstructure were modelled, the secondary Dendrite Arm Spacing and the microsegregation of carbon. The DAS is governed by a coarsening process, and hence is given by the following equation [in microns]: D A S = 143R-~
(7)
where R is the cooling rate in ~ The carbon microsegregation was calculated using the approach developed by Clyne and Kurz [15] in which the variation of the solid composition at the interface (C, ~in wt%) is calculated by equation (8, assuming a binary Fe-C system with complete diffusion in the liquid and incomplete back-diffusion in the solid: (k-t)
C~ = k C o {1 - (1 - 2 n k ) f , }0-2nt)
where Co is the initial bulk concentration (wt%), k is the partition coefficient and modified diffusion parameter. For further details refer to Clyne and Kurz's work [ 15].
(8) is the
3.4 Other Parameters Calculated by the Model Other important parameters estimated by the model that are useful for the design of the continuous thin slab casting machine are: shell thickness, S; metallurgical length (the distance from the meniscus to the location of complete solidification), Lm; and the solidification constant, K. The solidification constant was calculated using:
S = K,~
(9)
3. RESULTS AND DISCUSSION 4.1 Model Validation The model was validated by comparing its predictions with the thermal histories and DAS values published in prior literature.
4.1.1 Thermal history The thermal history predictions were validated by comparison to the work of Mizikar [16], who modelled a 152 mm thick AISI 1010 steel slab cast with a teeming stream temperature of 1555~ The mould heat flux was calculated using the following equation from Mizikar: Q = 850,000- l O0,O00d7
(I0)
where Q is the heat flux into the mould (Btu/fd/hr), and t is the time in seconds. The spray cooling heat transfer coefficient was taken as constant and equal to 75 Btu/~/hr/~ (426 W/m2/~ Although convection was not simulated directly, its affect on enhancing the heat transfer was accounted for by increasing the thermal conductivity in the liquid region by a factor of seven.
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Current Advances in Mechanical Design and Production, MDP-7
1600 ...-- Surface Temperature C entre Tempera ture 9Tsurf - Mi zikar Tcentre - Mizikar
.~. 1400 0
1200 1000 800 0
200
400
600
Time [s] Fig. 1 Comparison of cooling curves from the current work (solid lines) to that of Mizikar (dotted lines) [16] for a 152 mm thick steel slab for a constant spray heat transfer coefficient of 426 W/m2fC (75 Btu/ft2/hr/~ Figure 1 shows a comparison of the solidification time (590 s) predicted by the current model to that obtained by Mizikar (584 s). The correlation is good, however the predicted final surface temperature, 973~ is higher than that presented by Mizikar, 871 ~ The value of the spray heat transfer coefficient and the method of incorporating convection were not clearly stated, which would easily account for the discrepancy. 4.1.2 Secondary dendrite arm spacing LOser et al. [10] presented both measured and calculated DAS values for conventional and thin slab castings of AISI 1008 steel. Using their published heat transfer coefficients, the DAS was calculated using equation (7. The results obtained by the current work are compared to the DAS values measured experimentally by LOser et al. [ 10] in Figure 2. The current model is in excellent agreement for both conventional and thin slab casting.
W
300
eo 200
i m
<
....
100
Thin Slab [ 5 0 m m ] Conventional [230mm] 9 Thin S l a b - Expt. 9 Conventional - Expt.
50
100
Distance from Surface [mm] Fig. 2 Comparison of calculated DAS for thin slab (50 ram) and conventional slab (230 ram) to the measured values given by L6ser et al. [10]. 4.2 Application of the Model for Comparison of Conventional and Thin Slab Castings The predictions of the fast PC based model presented in this work are in good agreement with prior published results for both conventional and thin slab castings. Its application for the
386
Current Advances in Mechanical Design and Production, MDP-7
comparison of the predicted microstructures in AISI 1008 steel for a range of billet thickness is discussed below.The initial and boundary conditions used were a pouting temperature of 1577~ with a heat transfer coefficient in the mould of 1500 W/m2/~ (mould zone is 0.7 m) and 350 W/m2/~ during secondary cooling. Convection in the liquid was simulated by enhancing the thermal conductivity by a factor of 7, and the liquidus and solidus temperature were 1535~ and 1495~ respectively. A constant production rate ofl.56 tonnes/min(26 kg/s) was assumed and the appropriate casting speed was calculated for each slab thickness. The above conditions were applied for five different slab thicknesses ranging from a conventional 200 mm slab and down to a 25 mm thin slab. The results of the model runs are listed in Tablel. Table 1. Predicted macroscopic process responses for five different slab thicknesses. The values for shell thicknes s,and surface temperature are those at the mou!d exit. Run # Slab Casting Shell Ts,,/ Lm Tsu,f K Thickness Speed Thickness at Lm [mm] [m/min] [ram] [~ [ram] [~ [ram/rain ~ 1 200 1.0 22.0 981.7 16.73 955.9 24.45 2 150 1.33 18.0 1040.7 14.14 1044.6 23.03 3 lO0 2.0 13.0 1126.5 I 1.70 1159.1 20.67 4 50 4.0 7.0 1254.2 8.73 1304.7 16.92 5 25 8.0 3.0 1365.5 7.03 1400.6 13.33 As shown in Table 1, decreasing the slab thickness fTom the conventional 200 rnm slab to the 25 mm thin slab leads to a decrease in metallurgical length and shell thickness at the mould exit. Because of the need to maintain a minimum shell thickness at the mould exit to prevent break outs, either the casting speed must be reduced or the heat transfer rate between the slab and mould must be increased. In thin sections, pressure can be applied to increase the heat transfer rate, in addition to increasing the cooling water flow rate. The beneficial effect of increasing the heat transfer coefficient on increasing the shell thickness is shown by the sensitivity study results given in Figure 3. 15
. . . . . .
13
11
~
~
'
'
~
9 7 1500
1750
2~
I 2250
I , 2~
,I 27~
3OOO
Wm2,K Fig. 3 Effect of heat transfer coefficient on the shell thickness, S, at mould exit. Using the thermal history, the microstructural features can be calculated. Figure 4 shows the thermal history at the surface, quarter thickness, and centre of the slab for 200 mm and 25 mm thicknesses. The cooling curves shown in Figure 4 illustrate the range of difference in metallurgical length and surface temperature which was quantified in Table 1.
Current Advances in Mechanical Design and Production, MDP-7
1500
Centre
G' ?..
Quarter t h i c k n e s s
Q t,,,
0
Q.
387
1000
Surface
E Q
--.---
I-
25 mm Slab 200 mm Slab
500 0
5
10 Tim
15
20
9[m in]
Fig. 4 Cooling curves for 25 mm and 200 mm thick sla0bs. The effect of slab thickness on the secondary Dendrite Arm Spacing is shown in Figure 5. The DAS was found to be as large as 360 m for a 20ram slab whilst for a 25 mm slab the maximum was approximately 145 m. This reduction in DAS is a significant refinement in the microstructure obtained for thin slab casting. 400
,-- 300 - o - S l a b 25 S
< ~
200
Slab 50 - - o - S lab 100 -o--Slab 150 --4t- Slab 200
100
0
25
50
75
100
D i s t a n c e from Surface [mm] Fig. 5 Secondary Dendrite Arm Spacing as a function of distance from slab surface for different slab thicknesses ranging from 25 to 200 mm. 5. CONCLUSIONS A model for the comparison of thin slab casting to conventional slab casting was developed for use on a personal computer. The model was in good agreement with prior experimental and computational results. A sensitivity study was performed which determined that the heat transfer coefficient was a critical factor in controlling the Dendrite Arm Spacing and the shell thickness. The model was used to compare a variety of slab thicknesses ranging from conventional to thin slab casting. The thin slab process leads to a refining of the microstructure which in turn improves the mechanical properties of the resulting product.
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Current Advances in Mechanical Design and Production, MDP-7
ACKNOWLEDGEMENTS One of the authors (YMY) gratefully acknowledges ANSDK for the financial support provided to him which made this study possible. REFERENCES 1. Birat, J.P., "Manufacture of Flat Products for 21st Century", Ironmaking and Steelmaking, 14 (2), pp. 84-92, (1987). 2. Shibuya, K. and Ozawa, M., "Strip Casting Techniques for Steel", ISIJ Int., 31 (7), pp. 661-668, (1991). 3. Agarawal, J.C., Loreth, M.J., "Economic Analysis of Thin Sections Casting", Iron and Steel Maker, Nov. (1990). 4. Flemings, M.C., Solidification Processing, McGraw-Hill, Inc., USA, (1974). 5. Choudhary, S.K. and Mazumdar, D., "Mathematical Modelling of Transport Phenomena in Continuous Casting of Steel", ISIJ Int., 34 (7), pp. 584-592, (1994). 6. Lally, B., Biegler, L. and Henein, H., "Finite Difference Heat Transfer Modelling for Continuous Casting", Met. Trans., 21 B, pp. 761-770, (1990). 7. Reza Aboutalebi, M., Hasan, M., and Guthrie, R.I.L., "Coupled Turbulent Flow, Heat, and Solute Transport in Continuous Casting Processes", Met. Trans. 26B, pp. 731-744, (1995). 8. Storkman, W.R. and Thomas, B.G., "Mathematical Modelling of Continuous Casting of Stainless Steel Slabs to Optimize Mold Taper", Sol. Proc., Sheffield, Sep.(1987). 9. Megahed, G.M., Development Of A Production Technique For Continuous Thin Slab Steel Casting, Ph.D. thesis, Cairo University, (1994). 10. L6ser, W., Thiem, S. and Jurisch, M., "Solidification Modelling of Microstructures in Near-Net-Shape Casting of Steels", Mat. Sci. and Eng. A173, pp. 323-326, (1993). 11. Yang, B.J., Liu, W.T., Su, J.Y., "Numerical Modelling On The Temperature Fluctuation And Thermostress Development Of The Continuously Cast Steel Thin-Slab", Mod. Cast., Weld. and Adv. Solid. Proc. VII, pp. 265-273, (1995). 12. Youssef, Y.M., Mathematical Modelling of Thin Slab Casting Technology, MSc Thesis, ICSTM, Univ. of London, (1997). 13. Croft, D.R. and Lille),, D.G., Heat Transfer Calculation Using Finite Difference Equations, Applied Science Publishers Ltd, U.K, (1977). 14. Pehlke, R.D., Jeyarajan, A. and Wada, H., Summary Of Thermal Properties For Casting Alloys And Mould Materials, Dec. (1982). 15. Clyne, T.W., Kurz, W., "Solute Redistribution During Solidification With Rapid Solid State Diffusion", Met. Trans., 12A, pp. 965-971, (1981). 16. Mizikar, E.A., "Mathematical Heat Transfer Model For Solidification Of Continuously Cast Steel Slabs", Trans. TMS-AIME, 239, pp. 1747-1753, (1967).
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
389
SIMULATION AND CONTROL OF THE COOLING OF HOT ROLLED STEEL WIRE ROD
Labib, H.F. ~ Megahed, G.M. "~ EI-Mahallawi, 1.!, Dashwood, R.J. ~ and Lee, P.D. ~ It
Alexandria National Iron and Steel Co. (ANSDK), Alexandria '" Tabbin Institute for Metallurgical Studies, Cairo * Department of Metallurgy, Cairo University, Cairo : Dept. of Materials, Imperial College of Science, Technology and Medicine, London, UK e-mail: [email protected], [email protected] and [email protected]
ABSRACT A combined thermal and metallurgical finite difference based model has been developed to simulate the Stelmor process for controlled cooling of steel wire rod. The model predicts the cross-sectional temperature and microstructure distribution in the rod from exiting the last rolling stand until it reaches room temperature. Plant trials were conducted to evaluate heat transfer coefficients for the water cooling and forced air cooling stages of the process in terms of the actual process parameters, namely: wire diameter, rolling speed, number of active water boxes, and the operation ratio of the Stelmor fans. The model has been validated for three carbon manganese steel grades ranging from 0.07wt.% to 0.67wt.% C (AISI 1008, 1019 and 1060) with wire diameters of 5.5 to 14 ram. A good agreement between the predicted and measured thermal behaviour of the wire was achieved. The model was used to predict the effect of individual process parameters including their extreme settings on wire temperature distribution, forming the basis for developing the most efficient controlled cooling regimes. KEYWORDS Stelmor Process, Heat Optimisation, Steel
Treatment,
Modelling,
Microstructural
Simulation, Process
I. INTRODUCTION The Stelmor process has become a popular technique for controlling the cooling regime and microstrucmral properties of wire rod throughout the world, due to the fact that a wide range of microstructural properties can be achieved. However, time consuming and expensive inplant experimental trials are required to optimise the Stelmor line parameters in order to obtain the required microstructures and properties. Mathematical modelling has become a successful approach for the prediction, both qualitatively and quantitatively, oft he phase transformations during many heat treatment processes. Mathematical models describing heat transfer and phase transformation of metals during cooling processes have been developed [14] to predict the temperature distribution and fraction of phase transformed during cooling of steel bars, plates, wire rod and extruded products. Such models can provide a fast and
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Current Advances in Mechanical Design and Production, MDP-7
inexpensive technique to check the effect of the Stelmor process parameters on wire properties. In order to implement such models it is necessary to obtain accurate heat transfer coefficients for the water cooling and forced air-cooling stages of the process. In particular, the relationship between heat transfer coefficients and Stelmor operational parameters are required. A procedure that has been developed for in-line assessment of heat transfer coefficients using non-invasive techniques will be presented, and their implementation into a mathematical model outlined. The results of validation experiments will be discussed and the application of such models for optimisation of the Stelmor process briefly introduced. 2. EXPERIMENTAL Heat transfer measurements and validation trials were conducted on the Stelmor cooling system attached to the rod rolling line at Alexandria National Iron and Steel (ANSDK). There are two stages of cooling involved in a Stelmor cooling line. The first stage of the cooling process is effected via relatively straightforward direct water cooling, whilst the second stage of cooling is effected by fan assisted forced convection (see Fig. 1). Three different grades of carbon manganese steel were studied, ranging from 0.07wt.% to 0.67wt.% C (AISI 1008, 1019 and 1060). Metallographic and mechanical property samples were taken for each plant trial.
~ m impo mommom an,m~n
,Rollinl~ 9 direction.~
Fig. 1. Schematic of the Stelmor Cooling System
Fig. 2. Schematic diagram of the selected Stelmor fan and positions of measuring points.
2.1. Assessment of Heat Transfer Coefficients during the Water Cooling Stage For the purposes of this study, the settings of the water cooling boxes were taken as being fixed, such that the water cooling boxes were treated as on/offdevices. If the water boxes were inactive, a standard radiation heat transfer coefficient was employed. The three water cooling boxes were assumed to be identical and in all boxes only three water valves (out of total of four valves per box) were opened. In order to estimate the actual heat transfer coefficient during water cooling, the wire surface temperatures entering and leaving the cooling boxes were assessed, and an iterative scheme (using a finite difference solution to solve for the rod surface temperature) was employed [5]. A difference of 0.05 ~ was taken as a limiting value for each series of iterations.
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2.2. Assessment of Heat Transfer Coefficients during the Forced Air Cooling Stage During cooling on the Stelmor conveyor, the heat transfer from the surface of the wire is effected via a number of mechanisms: radiation (hR), natural convection (hNc), forced convection due to movement of the conveyor (hrcw), and forced convection due to air flow from the Stelmor fans (hFcF). Of the four components, the contribution from forced convection via the fans (hFCF), is the key thermal variable that can be controlled during the Stelmor process. The following experimental approach was followed to relate hFCFto the Stelmor fan operation ratio. The first fan in the line was selected for performing the plant trials as the temperature in this zone is the highest, giving the best accuracy for the infrared pyrometer. For each setting of the Stelmor fan operation ratio, the wire surface temperature was measured by means of an infrared pyrometer at six points distributed along the fan operating distance as shown in Fig. 2. Measurements were performed for four fan operation ratios, namely: 0%, 20%, 50% and 100%. These experiments were conducted on 10 mm diameter rod of steel grade AISI 1008. The rolling speed was 41 m.s "1, with a conveyor speed of 0.45 m.s "! and an inter-loop distance of 36.7 mm. The overall heat transfer coefficient (hov) was calculated between each pair of successive points using the equation given below:
h~176
]
2
(1)
TTs-TAMB tO - Tau8
where p is the density of steel, ro is the wire radius, Cp is the specific heat capacity, t is the time, Ts is the surface temperature, TAMS is the ambient temperature, and To is the initial temperature at zero time. The heat flow at the wire surface, (q) in given by:
hov (Ts - Tau B) hov = hFCF+ hrcw + hNc + hR
q =
(2) (3)
This technique is considered to be suitable for the heat transfer conditions pertaining to the Stelmor process [6,7,8]. In order to calculate the contribution of the forced convection due to the Stelmor fans (hFcr), the heat transfer coefficients due to radiation and natural convection were calculated using the following equations:
hR = 4.53
3[ft' f-t41 - -~
/(Ts - T ~ ) [2]
(4)
!
huc = 1.3 l(Ts - T~n )~ [41
(5)
These were then subtracted from the measured overall heat transfer coefficient (hov) to leave the contribution to heat transfer from just the movement of the Stelmor conveyor (hrcw) and the forced convection from the Stelmor fans (hrcr). The heat transfer coefficient as a result of movement of the Stelmor conveyor was taken as the residual heat transfer measured when the fan operational ratio was zero, and this was then subtracted to find hrcr.
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Current Advances in Mechanical Design and Production, MDP-7
3. THERMAL AND METALLURGICAL MATHEMATICAL MODEL A computer program was developed to implement the thermal, microstructural and property predictions using FORTRAN 77, based on the work of Campbell et al. [5], the precise details of which can be found elsewhere [5, 9]. A finite difference scheme was employed for the thermal calculations, including internal heat generation to account for the latent heat of transformation during decomposition of austenite into ferrite and pearlite. For the purposes of the calculations, the density of the steel was assumed to be constant; however, the thermal conductivity and the specific heat were both assumed to be dependent on temperature and calculated at each node for each time step using a second order polynomial function of temperature. The kinetics of the phase transformations were calculated using a modified Avrami equation and the coefficients established by Campbell et al. [10] in combination with the additivity principle. The model includes metallurgical and property predictions based on equations developed by Campbell et al. [10] which estimate microstructural dimensions using the transformation temperature and cooling rate, and then uses these dimensions to calculate mechanical properties. 4. RESULTS AND DISCUSSION 4.1. Evaluation of Heat Transfer Coefficients (Ifl'Cs) in Water Cooling Boxes Table I lists the average wire temperature prior to entering the water cooling boxes and the surface temperature of the wire on exit for a range of rolling speeds. The heat transfer coefficients, calculated using an iterative method, are also included. The results display the expected trends with increasing rolling speed leading to increased heat transfer coefficients as a result of a decrease in boundary layer thickness.
Table I. Calculated HTCs for water cooling box for different wire diameters. do (mm)
VR, (m.s "~)
5.5 6 8 10 12 14
90 80 60 40 30 26.4
Average temperature before cooling box . . ~C) 1200 1200 1170 1130 1120 1100
Surfa 9 temperature after cooling box ,, (~ .... 920 900 850 780 780 750
Calculated heat transfer coefficient OV-m~Z-K'l) , 20500 19000 16500 14000 6800 7500
4.2Evaluation of HTCs on the Stelmor Conveyor The results obtained from the overall heat transfer calculations (hov) for the fan-assisted cooling stage of the Stelmor cooling process are represented graphically in Fig. 3, as a function of the fan operation ratio.
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tSO
.A
:,i,o ~., 160
"r ~
u E
S 15o
!
/
J
=.
0 0
:
~
10
20
~30 40 Fp Opa'atioe
;-
;
'
SO 60 70 Ratio - FOR (%)
,,. :
;
'
80
90
IGO
Fig. 3. Effect of fan operation ratio on overall heat transfer coefficient for ANSDK's Stelmor line.
o0
I0
20
310
FR
40
Operation
50
60
70
!10
90
100
Rado - FOR I%1
Fig. 4. Effect of fan operation ratio on heat transfer coefficient due to forced convection by Stelmor fans.
The fan operation ratio indirectly controls the overall heat transfer coefficient through forced convection exerted by fans (hFcr). In order to obtain the relation between hrcr and the fan operation ratio, the contribution to heat transfer via radiation and natural convection were evaluated using equations 4 and 5 respectively, and then subtracted from the overall heat transfer coefficient to yield the contribution from the fan assisted convection. It was found that the average forced convection heat transfer coefficient calculated for a fan operation ratio of 0% was not equal to zero but had a value of 30.4 W.m'2.K"l. This value is attributed to the forced convection as a result of wire movement due to the Stelmor conveyor system. This value was then also subtracted from the calculated overall heat transfer coefficient to obtain a true relationship between the fan operation ratio and the heat transfer coefficient solely as a result of the fan-assisted convection (Fig. 4). Although this relationship is specific for ANSDK's Stelmor line, this relationship was found to be comparable to that experimentally obtained by Anelli [8] for a different Stelmor line, suggesting that it could be applied to any Stelmor line with minor modifications of the coefficients. 4.3. Model Validation
Before using the model developed for the prediction of wire properties, the output of the model was validated against actual in-plant measurements. Figs. 5 and 6 show examples of the predicted temperature profiles (along with predicted ferritr and pearlite evolution)for 8mm AISII019 (0.16 wt.%C, 0.7 wt.% Mn) and 5.Smm AISI1060 (0.67 wt.%C, 0.7 wt.% Mn) respectively. The actual temperature profiles, measured on-line during experimental work, are plotted on each graph for comparison. As can be seen from these figures the model predictions for thermal response showed excellent agreement with the measured temperature for both low carbon and eutectoid steels. In addition, the microstructural and property predictions are in good agreement with those measured from plant trials. Taking all the validation experiments conducted, it was found that the level of confidence in the predicted yield stress was over 90%.
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Current Advances in Mechanical Design and Production, MDP-7
1200
1.2 Predicted Surface Tcmpentturc
1000
Fertttl FiIctkm
8O0
R~rrite
g
Fen~ Grin OWneW (l~n) Pelttte Spedng (pro) o, (MPI)
o~m(MPa)
Predicted
Measured
0.78 18 0.152 283 489
0.79 22 0.209 292 488
1.0 0.8 E
=1
~ oO0 g ~E 4oo
0.6
g
0.4 Pcarlite
2OO
0.2
9Sud'ace-rn~sur~!
9 0
10
0.0
20
30
40
50 T i m e (s)
60
70
80
90
100
Fig. 5. Predicted temperature profile and fraction of austenite transformed into ferrite and pearlite compared to measured values. (8.0 mm diameter AISI 1019) 1200
1.2 Predicted Surface Temperature
I000
/ [
A ~ o
"9~
/.----
P~.ti,~
F,,~. F,,~o~
Fent~ Gnlin Diameter(1~) ~ ~ (~) ""--" ~ (MPII)
~
~
Momur~
0.o2
0.02
"" o.m 630 1
1.0 a
*" o.17o
0.8
651 1
:1 Z~o
0.6 1
g E
i
400
0.4 E [--
200
0.2
............... 0
I . . . . . . . . . . . . . . . . . . . . 10
20
30
40
, ....... 50 T i m e (s)
60
.... 70
80
90
0.0 100
Fig. 6. Predicted temperature profile and fraction of austenite transformed into pearlite compared to measured values. (5.5 mm diameter AISI 1060) 4.4. Model Predictions for the Effect of Process Parameters The validated model can be used to study the effect of different operational variables on the cooling rate and metallurgical transformations. A good demonstration of this is how the fan operation ratio can control the cooling rate and hence the final properties. The strong effect of the fan operation ratio on the cooling rate and metallurgical transformations can be clearly seen in Fig. 7. Increasing the fan operation ratio significantly increases the cooling rate, hence decreasing the ferrite fraction, grain size and pearlite spacing, giving rise to an increase in yield stress and tensile strength. The model can be used in a number of w a y s - assessing a new process set up, alloy, or product size, are only a few possible applications. However, by evaluating the effect of all controllable line parameters on the metallurgical and mechanical properties of the wire, it is possible to derive a control algorithm for these parameters, the purpose of which is to be able to alter process variables to achieve certain properties required by customers for different wire diameters and steel grades. Another benefit of such a model is that by running a sensitivity analysis, it is possible to assess the most critical line parameters,
Current Advances in Mechanical Design and Production, MDP-7
395
thus identifying aspects of the process that require the most control. Figures 8 and 9 display the results of such a sensitivity analysis for both a hypo eutectoid and eutectoid steel. The effect of process parameters such as heat transfer coefficient, wire diameter, carbon content and prior austenite grain size, on the final yield strength of the product are all illustrated in these figures. This study clearly reinforces the importance of accurate measurement of heat transfer coefficients, as, of all the parameters studied, it is this variable that has the greatest influence on the final properties. Another point highlighted is the fact that for different grades of steel the key parameters vary. For eutectoid steels, the strength is derived from the pearlite spacing. Therefore cooling rate dominates. The prior austenite grain size has very little effect on yield stress for the eutectoid steel since this parameter has no influence on the pearlite spacing, whilst it does have a strong effect for the hypo eutectoid AISI 1008. 1200 ~
1.2
iooo
-
i
'-
6O0
u
0.8
E
% = 270 M P I
0.6 _
e~ ["
la
.2 FOR = 0%
t
4 0 0 -~
_
FOR
% : 290 MPo
av :
--
0.4 ~
300 MPa
9.
0.2
= 50%
. . . .
0
~
~ ~ . ~ 0 0 %
FOR - 0%
2 0 0 -~
~
20
+ . . . .
4o
60
;
. . . .
: . . . .
ioo Time (s)
80
; . . . . 120
; . . . .
;
140
. . . .
160
: '
'
'
180
'
0 200
Fig. 7. Predicted wire temperature profiles and ferrite ratios under different fan operation ratios - FORs (10.0 mm diameter AISI 1019).
IPtrcmlalr Cklinlr In Ykld SllrlU
rm~aav omqe k VkM Stnm io .v
'~ t
/. / J i
"""p~-p +-v,,,,+k
.
.,o
.~. ~
.
. +
-+J;
~
.~.+--+~-~4
.
m
+ I
ImVm'tJk
.so
" +i /
+.
ii I '"'-e- HClll Tmmfcr CoelTlcient ! --41- WimDiluneter
I
--4b-%C Comenl
t
--4- AusTnile Gram Size
'
-e-tl-~-~-
Hint TnmhCmmcAm WkeDimmm, '~C Cnmem m O J n Sire
.io +'
Fig. 8. Influence of variation in input variables on the predicted yield stress for AISI 1008 (0.08 wt.% C).
Fig. 9. Influence of variation in input variables on the predicted yield stress for AISI 1060 (0.67 wt.% C).
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Current Advances in Mechanical Design and Production, MDP-7
5. CONCLUSIONS A mathematical model was developed to simulate the controlled cooling (Stelmor process) of steel wire rod after rolling. The simulation integrates three models (thermal, metallurgical and mechanical) to allow the processing of the rod to be related to its final properties, predicting the thermal history, the kinetics of austenite transformation into ferrite or pearlite, final ferrite ratio, ferrite grain diameter, pearlite inter lamellae spacing, and the yield stress and tensile strength of the wire under variable process parameters. The model was validated on an industrial scale for a number of steel grades and for a range of wire diameters and processing conditions, and showed good agreement with experimentally measured thermal history and final properties. A sensitivity study was conducted to determine the dependency of the final properties upon the process parameters: wire diameter, prior austenite grain size, carbon content and the heat transfer coefficients. The heat transfer coefficient was found to be a critical process parameter and a method for measuring these coefficients non-invasively was presented. The potential application of the model with respect to Stelmor process optimisation was demonstrated. ACKNOWLEDGEMENTS The authors would like to thank ANSDK for supplying the material and facilities to allow the experiments to be conducted, and for financially supporting one of the authors (HFL). REFERENCES 1. Morales, R.D., Lopez, A. and Oliveres, I. M., "Mathematical Simulation of Stelmor Process", Iron Making and Steel Making, 18 (2), pp. 128-138, (1991). Ozisik, M. N., Heat Transfer- Basic Approach, USA, McGraw Hill, (1985). 3. Prakash, K., Agarwal and Brimacombe, J. K., "Mathematical Modelling of Heat Flow and Austenite Pearlite Transformation in Eutectoid Carbon Steel Rods for Wire", Met. Trans., 12B, pp. 121-133, (1981). 4. Munira, M., Dhindaw, B.K., Biiswas, A. and Roy, A., "Modelling of Eutectoid Transformation in Plain Carbon Steel", ISIJ Int., 34 (4), pp. 355-358, (1994). 5. Campbell, P.C., Hawbolt, E.B. and Brimacombe, J.K., "Microstructural Engineering Applied to the Controlled Cooling of Steel Wire Rod- Part III. Mathematical Model Formulation and Predictions", Met. Trans., 22A, pp. 2791-2805, (1991). 6. Kruse, M. and Maul P.J., "Controlled Rolling and Cooling for Wire Rod Mills", Germany, SMS Schloe.-Siemag Aktiengesellschafl-Dussedorf and Hilchenbach, (1995). 7. Campbell, P.C., Hawbolt, E.B. and Brimacombe, J.K., "Microstructural Engineering Applied to the Controlled Cooling of Steel Wire Rod- Part I. Experimental Design and Heat Transfer", Met. Trans., 22A, pp. 2769-2778, (1991). 8. Anelli, E., "Application of Mathematical Modelling to Hot Rolling and Controlled Cooling of Wire Rods and Bars", ISIJ Int., 32(3), pp. 440-449, (1992). 9. Labib, H.F., Modelling of Microstructure Evolution and Properties During Cooling of Hot Rolled Steel Wire Rod, M.Sc. Thesis, ICSTM, University of London, (1997). 10. Campbell, P.C., Hawbolt, E.B. and Brimacombe, J.K., "Microstructural Engineering Applied to the Controlled Cooling of Steel Wire Rod- Part II. Microstructural Evolution and Mechanical Properties Correlations", Met. Trans., 22A, pp. 2779-2790, (1991). .
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
397
EFFECT OF ALUMINA ADDITIONS ON THE MECHANICAL BEHAVIOR OF PM MMC WITH LOW STRENGTH MATRIX Mazen, A.A.* and Ahmed, A.Y.** *Assistant Professor, Faculty of Engineering, Menia University, Menia, Egypt. ** Graduate Student, the American University in Cairo, Cairo, Egypt.
ABSTRACT Metal matrix composites (MMC) were manufactured using the powder metallurgy (PM) technique of hot pressing followed by hot extrusion. Low strength matrix of pure aluminum was used to which different weight fractions of alumina particles were added. Mechanical testing, microstructural examinations, and scanning electron microscopy (SEM) were used for characterization of mechanical behavior and fracture surface analysis. It is shown that improvement in strength was achieved for all reinforcement weight fractions except for AI-2.5 wt% A1203. It is also shown that a minimum reinforcement weight fraction of 3.4 wt% A1203 is required to induce strengthening in the composite. The unreinforced aluminum's yield strength was improved by 106% and its tensile strength achieved 88% improvement compared to the corresponding values of ingot pure aluminum. The yield and tensile strengths of AI-10 wt% A1203 composite showed improvements of 30% and 24%, respectively, compared to the unreinforced matrix's strength (i.e. 160% and 132.6% compared to ingot pure aluminum). Microstructural examinations revealed the development of fine grain size structure with some residual porosity and clusters of alumina particles. It is shown that as more A1203 particles were added to the matrix, the grain size became finer. No dynamic recrystallization effects were observed in the microstructures. Also, no direct relationship seems to exist between the number of grains and reinforcement particles. KEYWORDS Metal Matrix Composites, Powder Metallurgy, Mechanical Behavior, Microstructure. I. INTRODUCTION The most commonly used methods for metal matrix composite (MMC)manufacturing are casting techniques[l,2,3,4], and powder metallurgy techniques[4]. PM processing proved to be effective in avoiding the limitations and defects of casting techniques, and in producing materials with superior properties inherited from the powder characteristics usually related to rapid solidification. Most studies on PM MMC's concentrated on a composite with high strength aluminum matrix[5,6,7,8]. Lai and Chung[9] presented one of the very few papers on MMC using pure aluminum as a matrix. In their work, the MMC's were manufactured by vacuum infiltration of liquid aluminum into a porous particulate perform under inert gas. Several studies on aluminum alloy matrix composites reported a decrease in mechanical strength of the composite compared to the unreinforced matrix[8,10,11 ]. This was attributed
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Current Advances in Mechanical Design and Production, MDP-7
to the overloading of the brittle reinforcement particles during the load transfer process. Thus, these particles fractured causing deterioration of mechanical properties of the composite. Manoharan and Lawandowski [10] explained that low strength matrices may benefit more from brittle reinforcement particulates than high strength matrices. The goal of the present work is to present the results of an attempt to manufacture MMC's using low strength pure aluminum matrix with alumina particles as reinforcement. The effects of different alumina weight fractions on the mechanical behavior of the composite are investigated. The manufacturing technique consisted of simple hot pressing followed by hot extrusion without use of complicated techniques of degassing or vacuum pressing. The processing conditions consisted of a compaction stress of 157 MPa, a compaction temperature of 873K, applied for 3 hrs. at maximum load followed by hot extrusion at 823K. This work represents part II of an on going research effort on AI- A1203 PM MMC. Part I of the work presented the experimental results of the mechanical behavior of AI-AI203 PM MMC manufactured under a different set of processing parameters, i.e. compaction stress of 74 MPa, compaction temperature of 723K, and compaction time of 144 ks under maximum load, followed by hot extrusion, see ref. [12]. 2. EXPERIMENTAL PROCEDURES Preweighed mounts of pure aluminum (AI), and alumina (A1203) powders were mixed thoroughly in a mechanical mixer. The mixture was then hot pressed in well lubricated dies using single-acting press. Four different reinforcement weight fractions were used, i.e. AI-0 wt% A1203, AI-2.5 wt% A1203, AI-5 wt% Al203 and AI-I 0 wt% Al203. Table I shows the particle size distribution for both powders. Using the present technique of manufacturing, it was found that the maximum possible alumina weight fraction is 10 wt%. Adding more alumina caused the specimens to be fragile and difficult to be machined. Tension tests were carried out on subsize specimen, Fig. 1. The tests were conducted at room temperature (300 k) using an average strain rate across the gage length of 10-3 s"t. Optical microscopy was used to study the different microstructmal features. 3. EXPERIMENTAL RESULTS Figure. 2 shows the stress-strain curves for the different material compositions used in the present investigation. Each point represents the average of at least three tests. Table 2 gives a summary of all mechanical parameters determined from testing. As shown in Fig. 2, the unreinforced aluminum (AI-0 wt% A1203) processed from powder using hot pressing followed by hot extrusion achieved a yield strength of 62 MPa. This value represents a significant improvement if compared to the yield strength of ingot pure aluminum processed by casting techniques. In the latter technique, a yield strength of 25 MPa to 35 MPa is reported[l-3]. Table 1. Particle size distribution for A! and A!203 powders .Size
Aluminum Alumina
125 90 63 ijjn Retained wt% in each mesh average 12.8
30.7
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Current Advances in Mechanical Design and Production, MDP-7
r
8R
I.
[
399
83
Dims. in mm
~J
r~"
Fig. 1 Tension test specimen. 200 o 0. 160 ~E
Fig. 2 Stress-strain curves for AI-AI2Oa MMC.
I-
120 , AI- 0 w t % A I 2 3 x AI-2.5 w t % .. o AI-5 wt% .. a AI -10 wt % ..
~-- 40 3
6
9 12 15 18 True s t r a i n , %
160 -
~"
~---
o
Fig. 3 Variation of the yield and tensile strengths with alumina weight fraction
100-
2{,
~-------~
140-
z
21
Y -%
~-o
I
/-
wt'/.A~O3M~
. ~ i~o~ p,,, AI
.-8o~/ 11...~I..~-.--o 60~-~- ~
- ~
-~/AI-0 wt%At2C)3MMC
Effective yields l r ~ 40~",2(-
-o~ ingot pure AI o Yield strength
9Tensile ,.
0
0
I 1 2
I ~ 6 Almino wt %
8
10
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Current Advances In Mechanical Design and Production, MDP-7
Table 2 summary of tensile parameters of AI-A1203 MMC
Composition
O'y
O'u
Omax
N
Ou/O'y
30.0
0.20
2.3
%El , ,
Ingot AI*
30
69
AI-0 wt% A1203
62
129.6
153
18.0
0.32
2.09
AI-2.5 wt% A1203
60
112.4
136
21.5
0.32
1.87
74
151.7
176
16.2
0.34
2.05
80
160.5
183
14.4
0.37
2.00
,
-
AI-5.0 wt%
A!203
AI-10 wt% A1203 *
,
....
.,
Ref. il21.
Key: (Yy O'u O'max
n %El P Ao
9yield strength, determined by proportional limit method. :tensile strength, defined by ou= Pmax/Ao. :max. strength, MPa. : work-hardening exponent. : Elongation percent to fracture. : Load, N. 9Initial cross sectional area, m 2.
Figure. 2 also indicates that the mechanical properties of the aluminum matrix improved as a function of A1203 weight fractions, except for the AI-2.5 wt% A1203 where the composite showed a decrease in its mechanical properties. The improvement in yield strength for AI-10 wt% A1203 compared to the yield strength of the unreinforced matrix, is about 30%. Compared to ingot pure aluminum, an improvement of about 160% is obtained (based on average ingot yield strength of 30 MPa). The tensile strength of the unreinforced matrix was found to be 129.6 MPa a (based on the conventional definition: ou= Pmax/Ao where Pmax is the maximum load during the test, and Ao is the initial cross-sectional area of the specimen before testing). The tensile strength of AI-2.5 wt% A1203 MMC decreased to 112.4 MPa, a reduction of 13.3% is exhibited. The AI-5 wt% A1203 and Al-10 wt% A1203 showed improvements of 17% and 23.8%, respectively, compared to the tensile strength of the unreinforced matrix of the present work. Compared to ingot pure aluminum (ou=69 MPa[12]), the improvements are 87.8%, 62.8%, 119.8%and 132.6%, respectively. The ductility of the different material compositions was evaluated using the elongation present to fracture. No necking was observed in any of the tested specimens. Such behavior supports the use of elongation percent to evaluate the ductility without concern of the complications of necking on the calculated values. The ductility decreased as a function of the alumina weight fraction. A range of 21% to 14% was achieved which represents considerable ductility, as compared to MMC manufactured by casting methods[l,2,3]. The work-hardening potential was evaluated using the work-hardening exponent in the power law; a=Kc n, where o, and c are the true stress and true strain, K is a constant, and n is the workhardening exponent. For ingot pure aluminum, n=0.15 to 0.25[13]. In the present work, n ranged from 0.32 to 0.37 such increase in n is reasonable in view of the strengthening due to processing and the presence of reinforcement particles.
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4. ANALYSIS AND DISCUSSION OF MECHANICAL TESTING RESULTS The addition of small weight fractions of alumina particles (up to 2.5 wt% A1203) caused the yield and tensile strengths of the composite to decrease compared to the unreinforced matrix. As more alumina was added to the matrix, the yield and tensile strengths of the composite increased above those values of the unreinforced matrix. The strength of particulate PM MMC is the resultant of two competing and opposite microstructural factors: A negative factor that reduces the strength, i.e. stress concentration effect due to hard particles being embedded in the ductile matrix, retained residual porosity, and clusters of particles. A positive factor which adds to the composite strength comes from the role of reinforcement particles in generating more dislocations in the matrix. Miller and Humphreys[15], summarized the different strengthening mechanisms in particulate MMC as: i) strengthening due to thermal processing: the dislocation density generated is a function of reinforcement size and volume fraction, and the product of the thermal mismatch, and the temperature change, ii) Orowan strengthening by particle bypassing by dislocations. This source is inversely proportional to interparticle spacing, however, it is negligible except for very mall particles (less than 1 micron), iii) grain size strengthening, this is the Hall [16], Perch [17] and Li [18] suggested mechanism. Arsenault et al.[19], found that in aluminum alloys strengthened by SiC particles, the dislocation density increases and the subgrain size decreases as the volume fraction of SiC increases, iv) work-hardening mechanism which is reinforcement volume fraction dependent. From this discussion, it is clear that at small reinforcement volume fractions the negative factor overrides the positive one causing the composite strength to be lower than the strength of unreinforced matrix. Manoharan and Lewandowski [ 10], investigated the effect of reinforcement size and matrix structure condition on the fracture behavior of a 7091 aluminum alloy reinforced with SiC particles. They found that in both the underaged (UA) and the overaged (OA) conditions, the yield strength of the composite decreases invariably as compared to the unreinforced matrix. Thy attributed that to the load transfer process which caused the reinforcement particles to fracture. This could be true in the case of MMC with high strength matrices. However, in the present investigation, a low strength matrix is used, and besides, optical as well as SEM examinations did not reveal any particle cracking. In the present case, the effect of load transfer is to weaken the matrix/particle interfacial bonds. Such bond is purely thermomechanical bond which results from the severe friction and plastic deformation during hot pressing and hot extrusion. No interfacial reactions were observed at the particle/matrix interfaces. The fact that improvement in composite strength depends on the inherent strength of the matrix was also reported by Corbin and Wilkinson[8], and Friend[l 1]. According to the results of the present investigation, it can be seen from Fig. 3, that the minimum alumina weight fractions required for improvement in yield and tensile strengths to exceed those values of unreinforced matrix are 2.9 wt% A1203 and 3.4 wt% A1203, respectively. Aradhya and Surrapa[20] conducted a finite element analysis (FE) to predict the mechanical properties of 6061 AI-SiC particulate composites. They also conducted several experiments on a cast-extruded 6061 AI-SiC MMC containing 5, 10, and 15 vol % SiC. The FE analysis showed that a minimum of 25 vol% SiC is required to improve the local yielding stress in the matrix of the composite compared to the unreinforced matrix. However, their experimental results showed that the overall yield strength of the composite is higher than that of the
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unreinforced matrix at all SiC volume fractions. They attributed this discrepancy to the use of the 0.2% proof stress in calculating the yield strength in their experimental work, which does not indicate the first yielding in the matrix of the composite. Nardone and Prewo [21 ] showed that the yield strength determined by proportional limit is consistently lower than that determined by 0.2% proof stress. Once the above mentioned volume fractions are exceeded, improvement in yield and tensile strengths of the composite could be achieved easily. In the present work, at 10 wt% AIzO3 the yield strength improved by 30% while the tensile strength improved by 24%, compared to those values of the unreinforced matrix. The strengthening mechanisms responsible for this improvement are mechanisms i), iii), and iv), given above. Such mechanisms achieve more strengthening in MMC's with soft matrices than in MMC's with hard ones, as discussed earlier. On the other hand, the presence of residual porosity, characteristic of PM processing, and clusters of reinforcement particles, which is common in particulate MMC's[21 ], limited the improvement in strength obtained in the present study. These limitations could by avoided by incorporating a degassing step in the processing. However, such technique was not used in the present work to keep the processing sequence simple and the set-up uncomplicated. 5. ANALYSIS AND DISCUSSION OF OPTICAL AND SEM MICROSTRUCTURAL RESULTS 5.1. Microstructurai Features
The common observation in the microstructures of unreinforced matrix, Fig. 4-a as well as the aluminum composites with different AI203 weight fractions, Fig. 4b, 4e and 4d is that small-size grains and subgrain structures could be observed. These fine grains resulted from powder metallurgy processing in the case of unreinforced specimens, and from PM processing together with particles stimulated nucleation in AI-AI203 MMC. In the latter case, the grain size refinement increased as the weight fraction of A1203 increased. However, no direct relationship was clear of how many grains are due to each reinforcement particle. Figure 4e indicates that the alumina particles are surrounded by fine porosity. This explains why the bond at particle/matrix interfaces is not so strong. When the ratio of matrix (AI) particle size to reinforcement (A1203) particle size is high, each aluminum powder particle will be uniformly coated with fine alumina particles during mixing. Consequently, during pressing and extrusion, the presence of these A1203 particles prevents metal to metal contact, resulting - in unfilled volume or voids[23]. This effect increases in proportional to reinforcement weight fraction. One way to get rid of such porosity is by incorporating a degassing step in the processing sequence[24]. Such step was not used in the present work as mentioned above. The distribution of reinforcement particles was investigated using optical microscopy, Fig. 5a, 5b and 5r as well as SEM micrographs, Fig. 5d, 5e, 5t'. From both sets of figures, it is clear that a homogeneous distribution of particles was achieved in the matrix. Figure 5d through 5f show the presence of clusters of A1203 particles in the microstructure. These clusters also result from the smaller size of reinforcing particles as compared to that of matrix Al-particles. Lewandowski et a1.[25] found that when the ratio of reinforcing particle size to matrix particle size rp/r= was 3:10 many clusters appeared in the microstructure compared to the case where the ratio was 7:10 at which minimum clustering was observed. Thus, to avoid clustering, powders with a ratio rp/rm close to unity should be used. For the present study the ratio was about 0.8:10. However, the observed clustering was not extensive which indicates that the ratio of rp/rm is not the only factor controlling the
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Fig. 4: Microstructural features of AI-A1203 MMC (1520x). a. AI-0 wt% AI203 b. AI-2.5 wt% A!203 c. AI-5 wt% Al203 d. AI-10 wt% A!203 e. Porosity associated with AI203 particles
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Fig. 5: Distribution of A!2o3 particles in the AI matrix (a,d) AI-2.5 wt% A!203. (b,e) AI-5 wt% A!203. (c,f) AI-10 wt% A!203
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clustering of reinforcing particles in the matrix. Other factors seem to include the accuracy of mixing which is affected by the mixing time as well as the condition of particles prior to mixing and exposure to atmospheric conditions. Another effective way to disperse clusters that form during PM processing is by applying a secondary processing step, e.g. extrusion, forging, rolling, etc. Hong and coworkers[26] obtained better reinforcement alignment and higher mechanical properties for composites extruded at an extrusion ratio of 17:1than for composites extruded at 10:1 Kalu and McNelley [27] observed clusters after the use of extrusion with a 17:1 ratio (a strain of 2.8). However, after applying a warm rolling step at further strain of 2.3 (total strain of 5.1), clusters were homogeneously distributed in the aluminum alloy matrix. In the present study, an extrusion ratio of 5:1 was used, thus, it was not effective in breaking-up or redistributing the clusters in the matrix. It was used merely to density the microstructure by closing some porosity due to the compressive stresses applied in the process. 6. CONCLUSIONS 1. Appreciable strength improvement could be achieved in low strength aluminum matrices by addition of alumina reinforcement particles coupled with PM processing using hot pressing followed by hot extrusion. 2. Two competing and opposite factors determine the strength of this PM composite. Microstructural problems like retained porosity and particle clustering, lead to strength reduction. Processing induced grain size refinement as well as particle induced one lead to strength improvement in strength. 3. A minimum reinforcement weight fraction must be exceeded to achieve strength improvement. Below this minimum, the strength of the composite is lower than that of the matrix. For the present work, this minimum was shown to be 3.4 wt% A1203. Above this minimum an improvement of up to 30% in mechanical strength compared to unreinforced matrix (and up to 160% compared to ingot aluminum) could be achieved. 4. The present manufacturing technique produces MMC's with considerable ductility compared to casting techniques. 5. Microstructural examinations revealed grain size refinement as a result of processing and more refinement as a result of reinforcement additions. It also revealed the presence of some clustering of reinforcement particles. Porosity was shown to dominate the areas surrounding reinforcement particles. Both clustering and peripheral porosity are the result of the large size difference between reinforcement particles and matrix powder particles. REFERENCES 1. Harris, S.J. "Cast Metal Matrix Composites", Mat. Sci. Tech, Vol. 4, p. 231, (1988). 2. AI-Latif, N.I., Khedr, A.I., Goal, S.K. "Microstructure and Mechanical Mroperties of AIA1203 mg O Cast Particulate Composites", J. Mat. Sci., 22, p. 466, (1987). 3. Arsenault, R.J., Wu, S.B., "A Comparison of PM vs. Melted SiC/AI Composites", Scripta. Metall, Vol. 22, p. 767, (1988). 4. Strivatsan, T.W., Ibrahim, I.A., Mohamed, F.A., and Lavemia, E.J. "Processing Techniques for Particulate Reinforced AI Composites", J. Mat. Sci., Vol.26, p. 5965, (1991) 5. Sircar, S., and Humphreys, F.H. "Modelling Microstructural Formation in Two Phase Aluminum Alloys After Hot Deformation", Mat. Sci. Tech, Vol. 12, p. 158, (1996). 6. Singh, J., Goel, S.K., Mathur, V.N.S. and Kapoor, L.M. "Elevated Temperature Tensile Properties of Squeeze Cast AI-AI203-MgO Particulate MMC'S" J.Mat. Sci., 26, p.2750, (1991).
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7. Pestes, R.H., Kamat, S.V., and Hirth, J.P. "Effect of Microstructure Parameters on the Yield Strength of AI-45% Mg/AI203 Composites", Scripta Metall., Vol. 30, 8, p. 963, (1994). 8. Corbin, S.F., and Wilkinson, D.S. "Effects of Temp on the Mechanical Behavior of a SiC Particulate AI-Alloy" in Proceedings of International Symposium on Processing of Ceramics and MMC's, ed. H. Mostaghaci, Halix, p. 401, 0989). 9. Lai, S.W. and Chung, D.L. "Superior High Temperature Resistance of Aluminum Nitride Particle-Reinforced Aluminum Compared to SiC or A1203 Particle-Reinforced aluminum", J. Mat. Sci., 29, p. 6181, (1994). 10. Manoharan, M., and Lewandowski, J.J. "Effect of Reinforcement Size and Matrix Microstructure on the Fract. Properties of AI-MMC" Mater. Sci. Eng. A 150, p. 179, (1992). 11. Friend, C.M. "Effect of Matrix Properties in Short Alumina Fiber Aluminum MMC", J. Mat. Sci., Vol. 22, p. 3005, (1987). 12. Mazen, A.A., and Ahmed, A.Y. "Mechanical Behavior of AI- A1203 MMC Manufactured by PM Techniques, Part I: Scheme I Processing Parameters", J. Mater. Eng. Perf., Vol. 7 (3), p. 393, (1998). 13. Callister, W.D. Jr., "Materials Science and Engineering", John-Wiley & Son, p. 95, (1989). 14. Hertzberg, R.W. "Deformation and Fracture Mechanics of Engineering Materilas", 4th ed., John-Wiley & Sons, p. 18, (1996). 15. Miller, W.S., Humphreys, F.J., "Strengthening Mechanisms in Particulate Metal Matrix Composites", Scripta Metall. Mater, Vol. 25, p. 33,199. 16. Hall, E.O., Proc. Phys. Soc. London, Vol. 643, p. 747, (1951). 17. Petch, N.J., J. Iron Steel Inst. London, V 173, p. 25, (1953). 18. Li, J.C.M., Trans. Metall Soc. AIME, Vol. 227, p. 239, (1963). 19. Arsenault, R.J., Wang, L., and Feng, C.R. "Strengthening of Composites Due to Microstructure Changes in the Matrix", Acta Metall. Mater, Vol. 39, p. 47, (1991). 20. Aradhya, K.S.S., and Surappa, M.K. "Estimation of Mechanical Properties of 6061 AISiCp Using Finite Element Method", Scripta Metall. Master Vol. 25, p. 817, (1991). 21. Nardone, V.C., and Prewo, K.M. "On the Strength of Discontinuous SiC Reinforced AIComposites", Scripta Metall., 20, p. 43, (1986). 22. Raghunathan, R., Joannidis, E.K., and Sheppared, T. "Fabrication, Properties, and Structure of a High Temperature Light Weight Alloy Composite", J. Mater, Sci., Vol. 26, p. 985, (1991). 23. Daehn, G.S., and Doncel, G. "Deformation of Whisker-Reinforced MMC Under Changing Temperature Conditions", Metall. Trans., Vol. 20A, p. 2355, (1989). 24. Ibrahim, I.A., Mohamed, F.A., and Lavemia, E.J. "Particulate Reinforced MMC-a Review", J. Mater. Sci., Vol. 26, p. 1137, (1991). 25. Lewandowski, J.J., Lui, C., and Hunt, W.H., "Microstructural Effects on the Fracture Microstructures in 7xxx AI PM-SiC MMC", In processing, Properties and Microstructures of SiCw 2124 AI Composites", Mat.Sci. Eng., A206, p. 225, (1996). 26. Hong, S.H., Chungs, K.H., and Lee, C.H. "Effects of Hot Extrusion Parameters on the Tensile Properites and Microstructures of SiCw 2124 AI Composites", Mat. Sci. Eng., A 206, p. 225, (1996). 27. Kalu, P.N., and McNelley, T.R., "Microstructure Refinement by Thermomechanical Treatment of a Cast and Extruded 6061 AI- A1203 Composites", Scripta MetaU. Mater, Vol. 25, p. 853, (1991). 28. Williams, P., Cannon, S., and Ralph, B. "Investigation into Fracture Mechanisms and Effect of Stretch Straightening on a Extruded MMC", J. Mat Sci., Vol. 29, p. 4906, (1994). 29. Whitehouse, A.F., and Clyne, T.W. "Cavity Formation During the Tensile Straining of Particulate and Short Fiber MMC", Acta Metall, Vol. 41, 6, p. 1701, (1993).
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
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EVALUATION OF DAMPING BEHAVIOR OF SPRAY DEPOSITED SiC PARTICULATES REINFORCED A! COMPOSITES Abo EI-Nasr, A.A.
Department of Production Engineering and Mechanical Design Faculty of Engineering, Menoufia University, Shebin EI-Kom, Egypt
ABSTRACT The damping capacity of silicon carbide particulates (SiCp) reinforced 6061 AI metal matrix composites is studied. These composites were processed by spray atomization and deposition. The effect of SiC particulate volume fraction on the damping behavior is investigated. A dynamic mechanical thermal analyzer is used to measure the damping capacity and elastic modulus over a fairly wide temperature range (30-300 ~ The results of the present work show that the damping capacity can be improved by the addition of the SiC particulates through spray deposition processing. The damping capacity is observed to increase slightly with increasing the particulates volume fraction. The largest gains were made at high temperature (above 100 ~ and low frequency. However, associated with the damping gains of the materials there is a loss in elastic modulus. The operative damping mechanisms are analyzed and discussed in the light of the data obtained. KEY WORDS Damping behavior, Damping capacity, Spray deposition, Metal Matrix Composites. I. INTRODUCTION The damping capacity (DC) of a material refers to its dissipation of energy to the surrounding environment by a reversible microstructural movement or an irreversible thermoelastic process inside the material during mechanical vibration [1]. When utilized effectively in a structural application, this property allows undesirable noise and vibration to be passively attenuated and removed to the surroundings as heat. The application of high damping materials may eliminate the need for special energy absorbers or dampers. Recently, metallurgists have begun to use different alloying, processing and heat treatment techniques to improve the damping behavior of metallic materials. Metal matrix composites (MMCs) are attractive as high capability materials. They have a potential to meet the requirements imposed on the next generation materials. On the other hand, Spray Atomization and Deposition (SAD) is of interest as a processing technique that combines near-net-shape capabilities with the structural control available through rapid solidification [2]. This technique was developed as an alternative to Powder Metallurgy, Ingot Metallurgy and Liquid Infiltration techniques. In the past ten years, previous experimental and theoretical results have revealed the potential advantages of spray deposition for the synthesis of MMCs[2-6]. These include: (a) reducing interfacial reactions between metal matrices and ceramic
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reinforcement, (b) improving the dispersion of ceramic reinforcements, (c) refining and modifying the microstruc-ture of matrices, and (d) formation of non-equilibrium phases that in some cases enhance mechanical properties. SiC particulates have been used as a secondary phase to improve the mechanical properties of AI alloys with the objective of developing high damping materials. Much effort has been devoted to the improvement and optimization of these properties in various structure materials [2,5], since the light weight composite materials are primarily used in dynamic load environments, for example; aerospace structures and jet engine rotor components. Many applications of MMCs under dynamic load environments require not only high strengths and high stiffness but also good vibration damping properties as well. The present study was undertaken with the following two primary objectives. First, the present work sought to characterize the damping behavior of spray atomized and deposited 6061 Al and SiCp-6061 Al MMCs. Second, the study was completed to provide insight into the operative damping mechanisms that were active in these types of materials. 2. EXPERIMENTAL PROCEDURES 2.1 Material Synthesis Both 6061 AI alloy and SiCp-6061 AI composites were synthesized using spray atomization and deposition conducted in an environmental chamber filled with a nitrogen atmosphere as shown in Fig. 1. The fabrication procedures are described as follows. First, the alloy is charged in a ceramic crucible. Then, the atomization chamber is evacuated. Once this step is complete, the alloy is induction melted to a certain degree of superheat which is determined for each individual experiment, but usually is about 100-200 ~ above the liquidus of the alloy. The melt is then delivered through a ceramic delivery tube to an atomizer, where it is energetically disintegrated into very fine droplets using inert gas, such as Ar or N2. The droplets then travel through the evacuated chamber and impinge on a water-cooled Cu substrate to form a solid deposit. In case of composite deposition, the SiC particulates carried by a separate flow of N2 gas, were co-injected via nozzles directed normal to the outline of the
Fig. 1 Schematic drawing of the spray atomization and deposition technique used in the present study.
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atomization cone. The partially solidified droplets and entrained particulates then continued to travel under the effect of gravity and drag forces to impact and collected on a water-cooled Cu substrate. The composites were fabricated with two distinct volume fractions of SiC particulates (5.4% and 11.8%). In order to avoid extensive oxidation of the material during processing, the chamber was evacuated to a pressure of 0.1 MPa. A summary of the processing variables used in the present study is given in Table 1. Chemical analysis revealed that the nominal composition of the matrix alloy is of 1.0%Mg, 0.6%Si, 0.2%Cr, 0.15%Mn, 0.70%Fe, 0.28%Cu, 0.25%Zn, 0.15%Ti and balance is AI (in wt.%). The matrix alloy was supplied by Alcoa, Pittsburgh, Pa, USA. The 1200 SiC particulates were employed as reinforcement phase. Table 1. Processing variables used in the present spray deposition process. i
ii
i
i
. . . .
. . . . .
J
,
L,,
,
Processing Variables Unreinforced Reinforced Atomization pressure 1.14 MPa 1.14 MPa Atomization gas N2 N2 Atomized droplet flight distance 40.6 cm 40.6 cm Pouring temperature 800 ~ 800 *C Ratio of metal/gas mass flow rates 2.29 2.29 Number of Injectors 2 Injector distance to atomizer 25.4 cm ' Particulate !njection angle * 90 deg. Particulate injector press.ure 0131 MPa * Injection an~ie..measurementwith respect to outline of atomization cone,.. . . . . . . . . . i
i
i
ii ii
~IRI
i
i
i
,
.
.
.
.
.
.
,,,
,,,i
,
i
Table 2 lists the relevant data of the matrix and SiC particulates at ambient temperature. In this table, d denotes the average particulate size, p the density, E the modulus, r/the loss factor (damping capacity) and CTE the coefficient of thermal expansion.. Table 2. Material data used in the present study. i,
,i
Material
d 0am)
p(g/cm 3)
E (GPa)
rl,('l 0 -3)
cTE' (I0-6)C-I
6061Al 1200 SiC
Bulk 13
2.73 3.20
70 300-400
' 4 1-2. 5
'" 23.4 4.3.,5.6
i[i
Ref,
t71 l
[8]
2.2 Hot Extrusion A 2.54 cm diameter extrusion billet was removed from the central portion of the deposited materials and was subsequently hot extruded at 400 ~ using a reduction ratio of4:l. This procedure was used to close the porosity that is normally present in the spray deposited materials [9]. The damping specimens were cut from the extrusion billet. The dimensions of the specimens were 101.6 mm long, 12.7 mm wide and 2.54 mm thick. 2.3 SiC Particulate Volume Fraction Measurement The volume fraction of SiC particulates in the bulk MMCs was determined using acid analysis. The acid analysis was performed as follows. First, MMC samples were placed in a solution of 37 wt.% HCI to dissolve the AI matrix. After dissolution, the residue was collected using vacuum filtration, dried and weighed at room temperature. As control
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experiments, AI matrix with the same composition but without reinforcements were also dissolved and filtered under identical experimental conditions. The resultant dry SiC particulate mass, combined with the dry and submerged MMC weights and densities of Al and SiC particulate, was sufficient to calculate SiC particulate volume fraction, AI volume, and porosity using the equations
Vp
Mp/pp Mc / Pc
=
Vm = ( g c - Mp) / Pm
(l)
(2)
Mc / Pc P = ( 1 - V p -I'm)
(3)
where Vp, Mpand p p denote the volume fraction, mass and density of SiC particulates, respectively; while Vm , Pro, Mc and, Pc represent like values for Al matrix and the MMC. The term P represents porosity. 2.4 Damping Measurements The cantilever beam technique was used in the present study to characterize the damping behavior of the deposited materials. In this technique, one end of a rectangular specimen was fixed while the opposite end was allowed to move freely in response to a mechanically induced displacement on the dynamic mechanical thermal analyzer (DMTA). The damping capacity of the materials was then determined from the resulting displacement specmun, by utilizing the logarithmic decrement and the half-power bandwidth analysis methodologies. In this method, a history of amplitude versus time during free vibration of the cantilever beam specimen was recorded by an oscilloscope through an optical displacement transducer. Fig. 2 shows a typical example of the response obtained in time domain. By measuring the free amplitude decay after excitation, the logarithmic decrement (8) can be calculated from [10]
~-lln( .
A' ~
kA,+.)
(4)
where A ~ and A (i+,) are the amplitudes of the i th cycle and the (i+n) th cycle, respectively, separated by "n" periods of oscillation (Fig. 2). Finally, experimental data for the logarithmic decrement 8 and the loss factor ( Q-' ) can be checked by using the relationship [4,10] 8=~Q-'
(5)
The damping capacity (tan ~) and loss factor (r/) are related to the real and imaginary components (denoted E' and E", respectively) of the complex dynamic modulus by tan + = ,7= Q-'= E"/E'
(6)
The force and displacement curves were used to calculate the magnitude of E'. This value is referred to as the storage modulus and corresponds to Young's modulus for the nondynamic case. All the samples were displaced to a m a x i m u m strainof2.6xl0 -4 and were placed in a furnace to investigate the effect of temperature on damping capacity. The temperature was increased at a rate of -3 ~ from 30 to 300 ~ and was monitored through a platinum resistor adjacent to the test specimen. During the temperature cycle, the sample was oscillated at three different frequencies of 0. I, I, 30 Hz.
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Fig. 2 A typical free decay response of a 12% SiCp-A! MMC sample 3. RESULTS AND DISCUSSIONS
3.1 Microstructure Fig. 3 shows an SEM micrograph of SiCp reinforced 6061 AI composite. The SiC particulates display a sharply faceted geometry with a clean surface. Also, uniform distribution of the particulates can be seen. 3.2 Characterization of Damping Capacity Figs. 4 and 5 show the damping capacity (tan ~) and dynamic modulus (E) versus the temperature. As shown in the figures, the damping capacity increases with increasing temperature for each frequency recorded. Up to 100 ~ no indications of frequency or temperature effect on the damping capacity were observed. Above 100 ~ a significant difference in the damping capacity can be seen. The dynamic modulus, as displayed in the figure, decreases with increasing the temperature. This behavior is expected for the modulus as a result of the matrix softening under high temperatures. Direct comparison of damping capacity data for the composite at 1 Hz is made in Fig. 6. Damping capacity of the as-received and as-extruded alloy were included for reference. As can be seen, the hot extrusion is appeared to enhance the damping capacity of the matrix alloy and large increases are made with the addition of SiC particulate, particularly, at high temperature. On the basis of the present experimental findings, it is evident that the presence of the SiC particulates leads to an improvement in damping capacity. It is believed that the intrinsic damping of SiC exhibits a constant value with increasing temperature upto 300 ~ [8]. Analysis of the experimental data on the basis of the rule of mixtures (ROM) in Eq. 7 suggested that the observed damping behavior of the present composites may not be rationalized solely on the basis of the intrinsic damping capacity of the composite constituents. Therefore, the observed improvement in damping response may be attributed, not only to the intrinsic damping capacity of SiC particulates, but also to the modifications in the microstruc-ture of the matrix derived from the presence of the SiC particulates and processing technique.
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Fig. 3 SEM micrograph of a polished section of 12% SiCp-AI MMC sample showing the fairly regular particulates distribution (x200). 76
--.
9 ,
9
,
72 A D.. c3
,
jl~O.O8
I).06
80
0.08
,,e.
0.06
..e
o
r
68
"r @
o.~ E
-~ 9
~. ~o
~
_~
g9 m
.~ 0.04
65
~ "K
Off/
0.02 60
60 - 0
100
200
0.00 300
Temperature, C F i g . 4 Damping capacity and modulus versus temperature for 5% SiCp-AI composite,
55 0
100
200
0.00 300
Temperature, C Fig. 5 Damping capacity and modulus versus temperature for 12% 3 i C p - A I composite.
Expected operative damping mechanisms in the present composites may include intrinsic damping of particulate, particulate/matrix interface damping, crystallographic defect damping and thermoelastic damping in the matrix.[7,8]. Thermoelastic contributions in the present composites are neglected due to the relatively high frequency (100 Hz) at which the mechanism is reported to become significant in particulate reinforcement MMCs [11]. It is well established that dislocation density is increased in the MMCs as a result of thermal misfit strains at the particulate/matrix interface associated with the presence of SiC particulates [1,7,12]. The dislocations are generated to accommodate the residual thermal misfit strains associated with the difference between the coefficient of thermal expansions (CTEs) of the matrix and the particulates (Table 2). These dislocations are located primarily near the interface and decrease with increasing distance from the interface. Consequently, these dislocations become sources of damping because of internal friction to the motion of the dislocations under elastic waves or cyclic loading. Based on the dislocation theory of Granato-L(icke [13] the dislocation damping is expected to decrease with increasing temperature because of the annihilation of dislocations and the damping capacity is proportional to the dislocation density and strain amplitude dependent.
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413
The effect of particulate/matrix interface may be considered by the Schoeck theory,[14] in which the internal friction of an alloy containing precipitates was increased by relaxation at the semicoherent or incoherent precipitate/matrix and the anelastie strain induced by dislocations in the matrix in the vicinity of the interface. Therefore, the resultant internal friction was proportional to the volume friction of the precipitated particulates, as well as to the magnitude of the developed shear stress at the interfaces. For the present composites, the addition of the SiC particulates into the metal matrix generated a large amount of interface area and the difference in CTEs between particulates and matrix was likely to result in stress concentration at the incoherent interfaces, thereby increasing internal friction through the Schoeck interface effect. At low temperature this effect is minor. As the temperature increases, the interface effect becomes significant because the matrix softens relative to the SiC particulates and a reversible movement at the interface is likely to occur. Consequently, the internal friction at the interface is thought to be thermally activated, suggesting that interface damping becomes dominant at high temperatures. The damping capacity of the present composite can be predicted analytically in terms of damping properties and volume fractions of the used particulates (r/p) and matrix (r/m). The rule of mixture gives a simple means of explaining the increase of damping capacity with increasing volume fraction of silicon carbide particulates. The overall damping capacity, r/c, ~7 c = 17 p . V p + rl m " Vm
(7)
A comparison of the theoretical data with the corresponding results from the DMTA (Fig. 7) reveals the ROM predictions to be lower than the experimental measurements. This agrees with the basic premise that the ROM should account for only the intrinsic damping mechanisms existent in the matrix and reinforcement, thereby neglecting additional damping mechanisms produced as a result of interfaces and interactions between the two. There has been some experimental evidence suggesting that the presence of pores or porosity (P) in deposited materials enhances their intrinsic damping response [2,12]. Therefore, a correction factor can be used to consider the change in the microstructure. The value of this correction factor is estimated from the experimental data in Fig. 6 based on the amount of improvement made in the as-extruded matrix material compared with the as-received one. 0.009 0.06 .@.
i
o D 9 o
As-received(AI Matrix) As-exu'uded(AI MalAx) 5% SiCp-AI Composite 12 % SiCp-A! Composite
:a~ r~c~1-J p-ss
;s
"- 0.04
0.008
|
i
............ co~,~,o~o ~
.e-
._~ 0,007t
Damping o 0.1Hz .~ I Hz o 30Hz
-~. 0.0O6 ~
0.02 0.005
0.00
0
I00 200 Tempcratrue, C
j
300
Fig. 6 Comparison of damping capacity of matrix material and the composites at I Hz.
ROM 0.004
~ 0.0
I 0.2
,
a 0.4
Volume Fraction, V p
,
,
o.(
Fig. 7 A comparison between the experimental results and predicted data using rule of mixture and the corrected one.
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CONCLUSIONS 1. Spray atomization and deposition of SiCp-6061 AI MMCs exhibit significant damping gains comparing with the as received 6061 AI. 2. The addition of SiC particulates enhances the damping capacity of the matrix metal, meanwhile, the damping capacity was relatively independent of the volume fraction of the SiC particulates. However, associated with the damping gains of the materials, there was a loss in elastic modulus. 3. At low temperature (below 100 ~ damping capacity was noted to be independent, relatively, on the applied frequency and temperature. At high temperature (above 100 ~ the damping capacity is appeared to be frequency and temperature sensitive with the lowest frequency exhibiting the highest damping. This suggests that, the various defect mechanisms are believed to be operative in dissipating energy. 4. The damping mechanism associated with SiC-AI MMCs are attributed to dislocation damping, interface damping at low testing temperatures, and to interface damping at high temperatures. Hot extrusion appeared to be an additional factor for the increase of damping capacity. ACKNOWLEDGMENT I would like to acknowledge the materials group at the School of Engineering, University of California, Irvine, for their help during the processing of the materials, especially, Dr. Ya Wu and Dr. Qyunh Doung. REFERENCES 1. Lazan, B.J., "Damping of materials and Members in Structural Mechanics", Pergamon Press, pp. 38-78, (1968). 2. Lavemia, E.J., and Wu, Y., "Spray Atomization and Deposition", Wiley, New York (1996). 3. Cai, W.D., Smugeresky, J. and Lavernia, E.J., "Low-Pressure Spray Forming of 2024 AI Alloy", Mater. Sci. and Eng. A241, pp. 60-71, (1998). 4. Ucok, I., Duan, X. and Grant, N.J.; "Processing and Structure of Rapidly Solidified Mg. 8.5 wt.% Li Alloy Via LDC", Presentation at NATO workshop, West Point, NY, July, (1994). 5. Dalai, R.P., Prichard, P.D. and Tom, T., "Inter. Gas Turbine and Aeroengine Congress and Exposition", Cincinnati, OH, May, pp. 24-27 ASME, New York, NY 10017, (1993). 6. Wren, G.G. and Kinra, V.K., "Damping in Metal Matrix Composites: Theory and Experiment", Technical Report, Texas A&M University, pp. 19,140, (1990). 7. Zhang, J., Perez, R.J., Gungor, M.N. and Lavernia, E.J., "Developments in Ceramic and Metal-Matrix Composites", (edited by K. Upadhya), TMS, Warrendale, Pa 203, (1992). 8. Nishiyama, K., Yamanaka, M., and Umekawa, S.; J Mat. Sci. Lea., Vol. 9, p. 526, (1990). 9. Lavernia, E.J., Ayers, J.D., and Srivatsan, T.S.; Int. Mater. Rev. Vol. 37, pp. 1-44, (1992). 10. Nowwick, A.S. and Berry, B.S., "Anelastic relaxation in Crystalline Solids" (Academic, New York), pp. 18, 21,436, (1972). 11. Bishop, J.E. and Kinra V.K., "A Second Law Analysis of Thermoelastic Damping in Elementary MMCs", Technical Report 91-15, Texas A&M University, (1991) 12. Zhang, J., Perez, R.J., and Lavemia, E.J.; J. Mater. Sci. 28, 2397, (1993). 13. Granato, A. and LOcke, K.; J. Appl. Phys., 27, 583, (1956). 14. Schoeck, G. ; Phys. Stat. Sol., 32, 651, (1969).
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
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BONDING AND PROPERTIES OF EXPLOSIVELY COMPACTED COPPER POWDER AND POLYPROPYLENE GRANULES Hegazy, A. Abousree
Assoc. Prof., Prod. Eng. Dept., Faculty of Engineering, Helwan University, Helwan, Cairo, Egypt
ABSTRACT Metal and non-metal powders can be consolidated by making use of the high shock wave pressures obtained from detonating explosives. The shock wave front propagates at a high velocity compacts the powder instantaneously. Local deformation at the particle surfaces leads to adiabatic heating and surface melting. The microstructures of copper powder and polypropylene granules compacts obtained by using explosive compaction technique have been examined to elucidate the process leading to welding between individual particles. Several experiments were carded out to declare the effect of wave proceeding distance on density, hardness and microstructure through the whole length of the compacts. The results observed have shown that the great effect of the traveling distance of the shock wave on the properties studied. Regions of jetting and interparticle bonding have been identified. KEYWORDS Powder Metallurgy; Explosive Compaction System: Copper Powder; Polypropylene; Microstructure. 1. INTRODUCTION Most of powder compaction techniques lead to green compacts require sintering in order to gain high density and strength of the product. By using dynamic compaction systems, it is possible to attain a density very close to the theoretical density of the material without needs to sintering process [1-4]. Explosive powder compaction has been employed for producing shaped parts from powders of metals, ceramics, polymers and composite materials [ 1-9]. Explosive powder compaction methods are based on the application of very high shock wave pressures developed over very short period of time, few microseconds [6], while the explosive charge is detonated. There are two compacting systems used for explosive powder consolidation. These are designated as direct and indirect techniques. The direct compaction system is by far the most often used. In such method a cylindrical tube containing the powder is surrounded by an explosive charge of uniform thickness. The detonation oft he charge develops a radial compression wave as the detonation front travels along the container. As these waves proceed inside the powders, individual particles have been accelerated toward the tube center, where they compact simultaneously.
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As a matter of fact, in dynamic compaction process the energy needed for powder consolidation may be deposited very rapidly, one second or few microseconds, so that very high heating and cooling rates (106 C/ sec) can be reached[10]. This energy is essentially deposited at the interfaces of particles and the mean bulk temperature remains low enough for the initial structure of the powder to be retained [ 11]. It has been previously reported by the author and others, independently, that a direct explosive compaction is suitable for producing a good compact of metallic and non-metallic powders [ 1,12-14]. In fact, all these works based on evaluating the physical and mechanical properties on the compact cross section, i.e. perpendicular to the wave propagation direction. The present work aims to characterize the jetting and detect the effect of shock wave proceeding distance on the properties of the obtained compacts. 2. EXPERIMENTAL WORK Copper powder and polypropylene granules are the raw materials used in this investigation. The size of the spherical granules is about 3mm. The chemical composition and sieve analysis of the copper powders are given in the following tables.
Table 1: Chemical composition of copper powders J
Iron[ [ 0.05
Lead 0.03
sulpher 0.01
Acid insoluble 0.07
H2 Loss 0.02
Copper Reminder
Table 2: Sieve analysis of powder used Sieve Coarse % Fine %
+325 29.4
-325 70
+250 Trace
+150 10
-150 +106 50 o.6
-106 +75 40
Direct explosive compacting system is employed in the investigation. In this system a cylindrical mild steel tubes were used as containers for containing the powder. The container was made of a mild steel with an inner diameter of 26 mm, a wall thickness of 2 mm and a length of 140 mm. Each container was surrounded by an explosive charge of uniform thickness. The explosive, which was detonated from one end, was a mixture of ammonium nitrate and TNT with a density of 1 g/cm 3 and detonation velocity of about 2600 m/sec. After consolidation process, samples were recovered by splitting the tube with milling machine. Most of the samples were consolidated and they were cracks free. Green density was measured by an Archimedean technique using water and vaseline to coat the sample. Two sides along the axes of the compacted samples were ground and polished with AI2 03 powder. The micro-Vicker's hardness was measured at 100 grams load for 15 sec. At least six indentations were made for each quoted value. Hardness measurements were carried out at the particles surfaces and at the interparticle boundaries. These measurements have been carried out at different depths from the front face in as consolidated specimens free of cracks. Fracture surfaces of the samples were observed by a scanning electron microscope.
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3. RESULTS AND DISCUSSION For the purpose of identification of melting zones, jetting, and compact properties versus its length, two batches of copper powders and polypropylene granules have been investigated using direct explosive compaction technique. The effect of shock wave proceeding distance on the compact density, hardness and microstructure is presented. These properties are summed up in Figs. 1-4. As we know, the main feature of the direct system is that a shock wave is transmitted from a detonating explosive through the wall of the container to the powder. Depending on the shape of the shock wave and the generated heat under- correct or overcompaction may result [2,4]. Shock consolidated powders are characterized by a great part of the energy supplied is consumed at the surface of the powder particles, leading to high temperatures and causing melting [1,9].
1 ~
o
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Fig. 1. Micrographs of explosively compacted coarse copper powder at 5.6 GPa for three different positions
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Figure 1 shows three micrographs of an explosively compacted copper powder at 5.6 GPa for three positions along the compact height. These microstructures reveal the differences in their morphology along the axis of the specimen. Fig. I a shows rearrangement of the particles with little plastic deformation. Heat generated in the particle surfaces and the compressive effect of the shock wave result in neck formation and interparticle bonding. The neck formation is very clear in Fig. lb. Fig. l c clarifies the melting zones around the particle surfaces lead to welding. The melting mechanism could be explained as follows. Small areas of contact between particles causes thermal build-up, thus raising the temperature until local melting begins in region of intimate contact. The liquid metals expand outward and fill spaces between the solid particles. A necked region of molten material form with a concave surface due to the surface tension between the solid particles. This is clearly seen in Fig. 1 b and c. These results are in consistent with one of the mechanisms proposed by Morris [3]. Figure 2 supports this phenomenon by using polypropylene granules. It can be seen from the figure that jetting between two granules has been achieved. This jetting means that, melting at the granules surfaces and inter-granular bonding takes place.
Fig.2. Optical micrograph for polypropylene compact Figures 3&4 show the variations of density and micro- Vicker's hardness with the compact length at shock wave pressures of 5.6 GPa and 3.2GPa for coarse and fine copper powders respectively. These properties have been measured for sound compacts on their outer surfaces. The expected increase in the compact hardness and density with increasing shock wave pressure was found. It is clear from figures 3a & 4a that the as-consolidated density ,green density, is affected by the height of the compact. It can be seen that the relative density, i.e. the ratio of the green density of the compact to the density of solid material, increases from top to bottom along the compact axis. These results may be due to the build-up of the shock wave pressure during the wave propagation after detonation [2,4]. It can be noticed also that, at lower pressure, 3.2 GPa, the maximum relative density achieved equal to 94%, while
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for higher pressure, 5.6 GPa, has been 98.7%. Under the higher initial shock wave pressure, 5.6 GPa, a decrease of the relative density has been observed at the lower end portion, (see Fig. 1). This may be due to the debonding effect of the strong reflected tensile shock wave at high pressures. 2 2 0 ~,
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Fig. 4. Variations of density and hardness with the compact positions for fine copper powder The micro-Vicker's hardness measurements shown in the figs. 3b& 4b show that the hardness of the compacts increases with compaction pressure and proceeding wave distance (through positions 1-5) from an initial value of about 170 VHN to about 210 VHN. This set of data, however, shows leveling off at lower shock wave pressure ( 3.2 GPa) but under higher pressure (5.6 GPa) the hardness values tend to decrease as the compact approaches its limiting value of density (which is less than the theoretical density). This can be correlated with the pressure build-up and increasing the particles interior temperature. Since the particle surface temperature and consequently the mean temperature of the compact depend upon the induced pressure and particle deformation. Mammals et al [9] reported that, a considerable softening of material takes place under shock compaction at very high pressure used, these results are in good agreement with those which have been reported under different shock wave pressures. Therefore, to get nearly constant density and hardness along the whole compact length it
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should take into account the effect of pressure build-up during the wave propagation. For this, further investigations should be carried out to overcome this problem. Figure 5 shows an optical micrograph and hardness variations for the portion no. 4 (see Fig. 1) which has higher mean values of density and micro-hardness (see Figs. 3 and 4). Microhardness measurements performed on the central region of the particles, revealed a microhardness of 208 + 5 HV, where that performed on the interpanicle boundaries revealed a microhardness of 223+ 3 HV. This means that the microhardness of the interparticle boundaries is always higher than that of the particles surfaces. The increase of the microhardness at the interparticle boundaries is probably due to the melting of these boundaries which is rapidly solidified. This may be another fact supporting the occurrence of interparticle bonding.
Fig. 5. Optical micrograph showing the microstructure of compact in portion no. 4 of the copper powder compact and microhardness on the particle surfaces (Ps) and on interparticle boundaries(l b) On the other hand, it can be noticed from the figures that the compacts obtained from coarse copper show higher values of relative density (98.7 %) than that achieved for fine powder (96%) under same compaction pressure. This may be due to the ability of coarse powder to deform plastically and good bonding between particles obtained consequently. CONCLUSIONS The purpose of the present work was to detect the effect of shock wave proceeding distance on the properties of the obtained compact. Strain localization at the particle surfaces leads to adiabatic heating and melting zones, jetting and interparticle bonding have been identified. Micro-Vickers hardness, density and morphology of the compacts produced indicate that the great effect of the proceeding wave distance. This correlated to the pressure build-up through the propagating shock wave distance. The results also show that the particle size has significant influence in the compact density. The product homogeneity along its length is needed to further investigation to overcome the pressure build up effect.
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REFERENCES 1. Torsten, R., Prummer, R., and Roll, W. "Explosive Compaction of Nanosized TiNPowders", Material Sci. Forum Vols. 235-238 pp. 285-290 (1997). 2. Hegazy, A. Abousree, "Some Aspects Of Implosive Powder Consolidation", 5th Int. Conf. On Prod. Eng. and Design For Development FEDD5, April 28-30, pp. 409-417(1998). 3. .Morris, D.G., "Bonding Process During Dynamic Compaction of Metallic Powders", Materials Sci. and Engineering, 57 pp. 187-195 (1983). 4. Staudhamer, K.P. and Jhonson, K.A., "Controlled Powder Morphology Experiments in Megabar 304 L Stainless steel Compaction", In Metallurgical Applications of Shock Wave and High Strain Rate Phenomena, ed. By Murr, L.E. et al., Marcel Dekker, INC, New York and Basel, pp. 149- 166 (1986). 5. Dogan, C.P., Rawers, J.C., Gorier, R.D, and Korth, G., "Mechanical Processing, Compaction, and Thermal Processing of Alpha- Fe Powder", Nanostructural Materials V4 n 6 pp. 631-644 (1994). 6. Kochsiek, D., Prummer, R., and Bnmold, A.,"Synthesis of Intermetallic Aluminides by Explosive Reaction Pressing", Metall Vol.49, Jahrgang, No. 3 pp. 168-172 (1995). 7. Sivakumar, K., Prasad, K.S., Bhat, T., Balakrishana, and Ramakrislman, P., "Microstructural Characteristics Of Shock Consolidated 2124 AI Alloy Compacts", J. of Mat. Sci. V32 n 19 pp. 5271-5278 (1997). 8. Rawers, J., Gorier, D., and Korth, G., "Consolidation, Mechanical Properties, and Phase Stability Of Mechanically Alloyed Fe-N Powder Compositions", ISIJ International VOL.36 n 7 pp. 947-950 (1996). 9. Mamalis, A.G., Gioftsidis, G.N., and Prohaszka, J., "Manufacturing of Thin Rectangular Plates By Explosive Compaction Of Copper Powder", J. of Eng. Manufacture, 204 pp. 237-247 (1990). 10. Thomas, T., Bensussan, P., Chartagnac, P., and Bienvenu, Y.,"Dynamic Compaction of Copper Powder: Experimental Results and 2D Numerical Simulation" In "Shock Wave and High Strain Rate Phenomena in Materials" ed. Meyers, M.A., Murr, L.E., and Staudhammer, K.P., Marcel Dekker, Inc., New York, pp. 433-441 (1992). 11. Schwar'z, R.B., Kasiraj, P., Vreeland, T.V. and Ahrens, T.J., "The Theory for the Shock Wave Consolidation of Powders", Acta Metall. Vol. 32 n 8 pp. 1243-1252 (1984). 12. Jaramillo, D., Hinojosa, G., Hallen, J.M., Balmor, N., Inal, O.T., "Mechanical and Microstructural Characterisation Of AI-SiC Composite Material Obtained By Dynamic Compaction", In 1st Int. Conf. On Ceramic and Metal Matrix Composites, San Sebastian, Spain (1996). 13. Hegazy, A. Abousree, and Hegazy, A.A., "Explosive Compaction of Metallic and Polymeric Powder Mixture", In Proc. Of 4th Cairo University, MDP Conf., Cairo, pp. 7986 (1988). 14. Bystrzycki, J., Paszula, J., Trebinsk, R., and Varin, R.A., "Microstructure and Interface Behaviour ofNi/Ni AI Composites Produced By The Explosive Compaction of Powders", J. Materials Sci. Vol. 29 n 23 pp. 6221-6226 (1994).
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Section IV
MANUFACTURING PROCESSES
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
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NEW TRENDS IN CIM: VIRTUAL MANUFACTURING SYSTEMS FOR NEXT GENERATION MANUFACTURING
Choi, B.K." and Kim, B. H. * Director of the VMS Lab Department of Industrial Engineering, KAIST, Taejon, South Korea e-mail: [email protected]
ABSTRACT The paper provides a structured review on recent developments and new trends in CIMrelated technologies and business strategies, and presents a new human-centered CIM framework called Human-Centered Virtual Manufacturing Systems which is suitable for meeting the requirements of Next Generation Manufacturing. KEYWORDS CIM, Computer and Human Integrated Manufacturing (CHIM), Intelligent Manufacturing Systems (IMS), Holonic Manufacturing Systems (HMS), Virtual Manufacturing System (VMS), Agile Manufacturing (AM), Next Generation Manufacturing (NGM), HumanCentered Systems (HCS). I. INTRODUCTION The acronym CIM is one of the most pervasive, but somewhat illusive, concepts in modem manufacturing. Since early seventies, a large variety of CIM architectures (or frameworks) have been proposed, and several academic journals beating the term "Computer Integrated Manufacturing" have been published including the Int. Jr. of CIM (IJCIM). A typical CIM architecture [1] is shown in Figure 1. In order to explore new trends in CIM, it is first necessary to establish a common view of CIM. In Figure 1, a CIM system is viewed as a structure consisting of three areas - Product and Process Design (PPD), Manufacturing Planning and Control (MP&C), and Production Process - that are linked together via an Information Technology (IT) infrastructure. CAD/CAM, MRP (Manufacturing Resource Planning), and FMS (Flexible Manufacturing System), respectively, were regarded as the representative technology in each of the three areas. Thus, CIM is not regarded as a "tangible" technology by itself, but is viewed as a competitive strategy or as a manufacturing paradigm. The two views of CIM are complementary, however, since the former defines the goals of CIM while the latter prescribes means to achieve them. In the paradigm view, the main theme is integration (and computerization) of its subsystems [2]. An implication of the paradigm-view of CIM is that the whole manufacturing system, from order picking to selling, is regarded as a complex machine requiting little human interference
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[3]. Along this direction of the CIM paradigm, the concepts of Intelligent Manufacturing System (IMS) were evolved. Especially, the Japanese initiative on the IMS Program [4] had fueled worldwide interests in IMS. By pursuing this direction to its extreme, there have been proposed various "new manufacturing paradigms" such as Bionic and Genetic Manufacturing, Fractal Manufacturing System, and Holonic Manufacturing System (HMS) [5]. A common theme of these paradigms is manufacturing with minimal human intervention, which may be characterized as Computer-Centered CIM.
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Fig. 1. A Typical CIM Architecture [1] In the mean time, failure with CIM and advanced manufacturing technology (AMT)had widely been reported, and it was recognized that the failure was mainly due to the neglect of human and organizational factors. As a result, different kinds of new manufacturing paradigms, such as Human Integrated Manufacturing and Computer and Human Integrated Manufacturing (CHIM), were emerged [3]. A common theme here is "manufacturing with human involvement", which is often called Human-Centered CIM. More recently, general concepts of Human-Centered Systems had been elaborated [6]. Another line of development related to the technological aspect of CIM is the concepts of Virtual Manufacturing (VM). The idea of a VM-system is proposed in order to integrate various kinds of virtual manufacturing [7], and the VM-concept is employed to achieve a total manufacturing integration [8]. The VM-concept is a "neutral" CIM-concept because it may be used as either a skill supporting or skill substituting means (or both). A CIM-concept is called "human-centered" if it's purpose is to support human skills, and is called "computer-centered" if it aims to substitute human skills. A line of development related to the strategy-view of CIM is the concepts of Agile Manufacturing (AM) and Next Generation Manufacturing (NGM). The concept of AM is built around the synthesis of a number of enterprises that each has some core skills which they bring to a joint venturing operation [9]. The term "a number of enterprises" in the above sentence is often called a Virtual Enterprise [ 10] or Extended Enterprise [ 11]. The concept of NGM is similar to that of AM, but it provides a comprehensive strategic framework as well as a set of requirements or goals (that has to be satisfied by a CIM system).
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The purpose of this paper is to (1) provide a brief but structured review of the recent trends in CIM-related concepts and technologies and (2) propose a new CIM framework, which we call a "Human-Centered VMS" framework, to meet the manufacturing requirements in the 21st Century. 2. OVERALL TRENDS IN THE PARADIGM-VIEW OF CIM 2.1. Trends in Computer-Centered CIM The paradigm-view of CIM, in which the whole manufacturing system is regarded as a complex machine requiring little human interference, had led to the concepts of IMS. Even though the original vision of IMS was to better harmonize human beings and intelligent machines, most of the IMS research activities have been focused on computer-centered systems [4, 12].
As a result, some researchers regard Holonic Manufacturing System (HMS) as the enabling technology for implementing IMS. For example, [ 13] states that "IMS in general and HMS in particular are new concepts in the Manufacturing arena. They differ from conventional Manufacturing Systems - even advanced ones - in their inherent capability to adapt to changes without external intervention". HMS aims to develop a holonic organization as a means to realize the IMS vision. The research in HMS has resulted in some prototype systems in the areas of process sequencing and scheduling and autonomous co-operative control [5, 14]. 2.2. Trends in Human-Centered CIM As discussed earlier, failure with CIM and advanced manufacturing technology (AMT) had widely been reported, and it was recognized that the failure was mainly due to the neglect of human and organizational factors. Furthermore, in the case of Flexible Manufacturing System (FMS), it was realized that running the system required close attention from qualified people. And much of the responsibility for the efficient daily operation of the FMS was shown to depend less on the technology and more on the persons attending it [15]. It was further recognized that human skill and its application be optimized in harmony with AMT [16] and that system productivity depended directly on human skill and the human element in AMT was responsible for vital flexibility [17]. These observations, together with social science influences, had led to the concepts of Human-Centered Design (of CIM components)and Computer and Human Integrated Manufacturing (CHIM) [3].
Human-Centered CIM-components should 1) provide the user with the opportunity to learn and develop skills and knowledge, 2) facilitate the maximization of operator choice and control, 3) integrate the planning, execution and monitoring components of the work, 4) encourage social communication and interaction between its users. And empirical evidence shows that companies tend to move toward the CIM paradigm first and experience low performance, and then shift from CIM to the CHIM paradigm [3]. 2.3. Trends in Neutral CIM The concept of Virtual Manufacturing (VM) represents a technological aspect of CIM, and is defined as the concept of "executing manufacturing processes in computers as well as in the real world" [l 8]. The purpose of VM is to achieve a total manufacturing integration [8] by integrating various kinds of manufacturing in virtual as well as real domain.
In this ' view [ 18], a manufacturing system is decomposed into a real physical system (RPS)
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and a real information system (RIS). An RPS is composed of substantial entities that exist in the real world. An RIS involves activities of information processing and decision making in manufacturing. Then, it is defined that: 1) Virtual Physical System (VPS): A computer system that simulates the responses of the RPS, 2) Virtual Information System (VIS): A computer system that simulates the RIS and generates control commands for the RPS, 3) Virtual Manufacturing System (VMS) consists of a VIS and a VPS. The concept of VMS was proposed to achieve a total manufacturing integration by integrating various kinds of manufacturing in virtual as well as real domain [ 19]. 3. OVERALL TRENDS IN THE STRATEGY-VIEW OF CIM With ever-increasing global competition, sustained technological gains, and the need for rapid response to customer demands, manufacturing industries are constantly seeking for new business strategies as a way of survival. As a result, we have witnessed such buzzwords as Just-in-time (JIT), Flexible Manufacturing, Lean Production, Agile Manufacturing (AM), and Next Generation Manufacturing (NGM). In order to define strategic requirements of future CIM, it is necessary to understand these "paradigm shifts in manufacturing".
3.1. Agile Manufacturing The concept of Agile Manufacturing (AM) is built around the synthesis of a number of companies, each of which has some core skills or competence brought to a joint venturing operation, based on the each partner's facilities and resources [9]. The "synthesis of a number of enterprises" in the above sentence is often called a Virtual Enterprise [10] or Extended Enterprise [ 11 ]. Using the virtual enterprise concept, agility is about rapidly reconfiguring the whole supply chain, if necessary, to meet the current and emerging market needs. The concept of AM is more characteristic of project shops and job shops (where high variety, high uncertainty of "what the next specification will be" are the operations orders of the day [20]) and is about rapidly reconfiguring the whole supply chain to meet the market needs. But it places a particular emphasis on the participation of people [21 ].
3.2. Next Generation Manufacturing The concept of Next Generation Manufacturing (NGM) is similar to that of Agile Manufacturing, but it provides a more comprehensive strategic framework. Key characteristics of NGMS (or NGM System) were first developed at an NGMS Program Definition Workshop conducted by CAM-I t in February 1994 for twenty large manufacturing companies from Europe, Japan, and the United States [10]. The key characteristics of NGMS identified during the workshop are 1) support for virtual enterprise, 2) customer focus and business centered, 3) Re-configurable, adaptable, and flexible in response to customers, 4) Global support for design and production, 5) information and knowledge based, human intelligence oriented, 6) modular to support distribution/autonomy, but cooperative to achieve enterprise goals, 7) environmentally aware. To meet the above requirements, an NGMSarchitecture called "The NGMS Conceptual Pyramid" is proposed However, a more comprehensive effort was launched in 1995 involving a team of nearly 500 experts to sculpt a vision of future manufacturing in the United States, and an NGM Report of over 600 pages was prepared in early 1997 [ l l ]. The purpose of the project is to develop a t Consortiumfor AdvancedMfg. Integration(CAM-I): 1250E CopelandRoad, Arlington,TX 76011,U.S.A.
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broadly accepted model of future manufacturing enterprises, and recommend actions that manufacturers can use to attain world-class status. The concept of NGM is based on the concept of Extended Enterprise. It is a group of companies that collaborate to create and support timely and cost-effective services and/or products, which requires (1) sharing of knowledge and resources, (2) collaboration to create products and services, and (3) seamless integration. The two key words of the NGM attributes are responsiveness and teaming.
3.3. Human-Centered Systems Traditionally, the theme of technology development has been "Science Finds, Industry Applies, Man Conforms". But, in February 1997, participants at an NSF Workshop discussed a very different theme: "Humans Live, Science Finds Out How, Technology Conforms" [22]. The topic of the workshop was "Human-Centered Systems: Information, Interactivity, and Intelligence", and the goal was to define this emerging interdisciplinary field and articulate research, education, and infrastructure needs to support work in this area [ 11 ]2. The paradigm change, from the technology-driven approach to the human-centered approach, is depicted in Figure 2
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Fig. 2. The Paradigm Change in HCS [1 l l In the HCS concept, the human's creativity can not be developed and utilized unless he 1) feels of "mastery" over the problem while using the tool, 2) is able to "achieve" a result that satisfies him, and 3) feels that "he" has solved the problem. In addition, there is a strong motivation for collaboration because of the augmentative, integrative, and debative nature of the work. And there are strong social incentives that knowing that others depend on you may motivate you to do your part and that working in a team is more fun and offers possibilities for friendship. Thus, a HCS should be designed to support human creativity and collaboration. 4. HUMAN-CENTERED VMS FOR NGM In this section, we will propose a new Virtual Manufacturing Systems Framework to meet the requirements of NGM in a human-centered way. The new framework is termed a HumanCentered VMS.
4.1. Scope of the Human-Centered VMS Since the term "manufacturing" has different meanings depending on the context, so does the VMS. Thus, it is necessary first to define the scope (or domain) of the VMS to be proposed in this paper. As depicted in Figure 3, the top-level model of the Virtual Enterprise (or Extended Enterprise) is called a NGME (Next-Generation Manufacturing Enterprise). The "Product Related Processes" portion of the NGME is called a NGMS, which consists of 1) Product & Process Development, 2) Production, 3) Logistics, and 4) Post-Sales Services. -"The document is available at http://www.ifp.uiuc.edu/nsfhcs/
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We call the NGMS in Figure 3 a "wide" manufacturing system. The four sub-functions in the NGMS are a generic grouping. However, we have developed a more structured model of the same manufacturing system, as depicted in Figure 4. In our model, there are five subfunctions: Order Control System (OCS), Manufacturing Execution System (MES), Product Design System (PDS), Manufacturing Preparation System (MPS), and Factory. In this model, the Logistics function and the Process Development function in the NGMS model may be handled by the MES, and the Post-Sales Service function can be handled by the Order Control System.
Fig. 4. A Structured Model of Wide Manufacturing System The entire "wide-manufacturing" function of Figure 4 may be handled either by one company or by a "group of company" forming an extended enterprise. In the latter case, which is becoming more common these days, the Product Design function may be handled by more than one Design Company and the rest of the functions by another group of companies, as shown in Figure 5.
Fig. 5. An Extended Enterprise Model for the Wide Manufacturing System
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In this "extended enterprise" model, a Design Company consists of three sub-systems: Order Control System, Design Execution System (DES), and Product Design System (PDS). And, a Manufacturing Company has four sub-systems - Order Control System, Manufacturing Execution System (MES), Manufacturing Preparation System (MPS), and Factory-, which constitute a "narrow manufacturing system". From now on, we will focus on the narrow manufacturing system, the Manufacturing Company in the extended enterprise model of Figure 5, which becomes the domain of the Human-Centered VMS to be described shortly. 4.2. Architecture of Human-Centered VMS
Our goal is to design and operate a manufacturing company so that it can serve both external customers (or the society) and internal customers (i.e., its employees). To meet this goal, we propose a Human-Centered Virtual Manufacturing System that will (1) expand human capabilities, (2) support human creativity, and (3) support collaboration among the people so that the responsiveness and teaming of the NGM Company are maximized. We define a VMS as a "cyberspace structure of interface, planning, and control mechanisms to support human decision-making via monitoring & simulation of actual manufacturing situations through modeling of all activities and resources in a physical manufacturing system". Furthermore, we regard a (narrow) manufacturing system to be composed of 1) a hardware system including tools, machines, facility, materials, parts, and products, 2) people including organizational structures, 3) a VMS, and 4) others including paper documents, company policies, etc. In this sense, the VMS is also a"real" part (as a decision support system) of the manufacturing system. ....
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Shown in Figure 6 is a reference model of the Manufacturing Company in Figure 5, which we called a narrow manufacturing system. In this model,
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OCS (Order Control System) consists of Marketing, Bidding & Negotiation, Delivery Control, and Post-Sales Services functions. MES (Manufacturing Execution System) consists of Shop-level Process Planning, Loading Scheduling, Progress Control, and Logistics functions. MPS (Manufacturing Preparation System) consists of Process Planning, Tooling, P/G, and Exception Handling functions. The Exception Handling function is responsible for facility maintenance, ECO (engineering change orders), trouble shooting, etc. The Factory has machining, inspection and assembly stations as well as transportation and storage facilities. The proposed VMS framework supports two types of VMS: Prototypic and Operational VMSs. As depicted in Figure 7, the Prototypic VMS is used as a virtual prototyping tool for designing a new manufacturing system. This concept is used widely in designing anew product, which is often called Digital Manufacturing. An Operational VMS may be obtained from the Prototypic VMS (for a new manufacturing system) or constructed from the scratch for an existing manufacturing system.
The Human-Centered VMS concept is geared to the Operational VMS. The Architecture of our Human-Centered VMS is presented in Figure 8. The Humans in the Physical Manufacturing Domain are hooked up to the VMS, and there are "high- speed" Data Transfer Lines connecting the VMS to the physical Factory and to the outside world (via Internet). The VMS shown at the bottom of Figure 8 has four "virtual" sub-systems - Virtual MES, Virtual MPS, Virtual Factory, and Common DB - as well as User Interface Control and Data Handler. The overall structure of the VMS looks similar to the CIM Architecture of Figure 1, but its concept is quite different. In the case of CIM architecture, integration is achieved in a physical domain (with or without human involvement). In the VMS concept, integration is achieved in a virtual domain to serve the humans in the physical domain where teaming and human creativity are more important. That is, the role of VMS is to expand the capabilities of humans and support their creativity and collaboration efforts. 5. DISCUSSIONS AND CONCLUSIONS Proposed in the paper is a Human-Centered Virtual Manufacturing System that is compatible with current trends in CIM and is suitable for meeting the requirements of the NGM (Next Generation Manufacturing) and HCS (Human-Centered Systems) concepts. Even though there remain some technical and theoretic issues in modeling and simulation (of products, machines, manufacturing processes, planning and control activities, etc.), there are available various kinds of commercial VM technologies.
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However, in order to realize the proposed VMS concept, a number of technical and social issues have to be resolved. One major issue in designing a Human-Centered VMS is how to allocate functions among humans and computer systems. Another key issue is how to design a human-oriented software system to capitalize on and accommodate human skills in perception, attention, and cognition, while minimizing the opportunities for and effects of human error [23]. Also the Clean Interface Concept [24] and the Machinery Station Concept [21] have to be developed further, and the general perception that "human operators are sources of error and unpredictability" is likely to be a huddle. The proposed Human-Centered VMS concept is a new one and has to be validated and seasoned under real manufacturing environments. The VMS Lab3 at KAIST (Korea Advanced Institute of Science and Technology) has been focusing on developing "component technologies" and sub-systems of VMS as well as integration methods, and is fully devoted to continuous research efforts along these directions. It is one of the National Research Laboratories endowed by Korean Government.
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In the Virtual MPS area (of Figure 8), a virtual machining software system has been developed based on the unified CAM system architecture proposed by the first author of this paper [25]. In the Virtual MES, a Gantt Chart based MES system has been developed [26]. These two software systems, named Z-Master| and mold-MES | respectively, have been commercialized. We have also been developing a Virtual Factory software, named VFS| based on the modeling scheme proposed by the first author and his former students [27, 28]. Shown in Figures 9, 10, and 11 are screen images of these software systems.
ACKNOWLEDGEMENT The research was supported by The Ministry of Science and Technology of Korea through its National Research Lab Program. REFERENCES 1. Gunn, T.G., Mfg. Competitive Advantage - Becoming a World Class Manufacturer, Ballinger Publish (1987). 2. Choi, B.K., "CIM in the Four Dragons", IJCIM, Vol. 5(4), pp. 199-200, (1992).
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3. 4. 5. 6. 7. 8. 9. 10.
11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.
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Sun, H. and Frick, J., "A Shift from CIM to CHIM", IJCIM, Vol. 12(5), pp. 461-469, (1999). Hayashi, H., "Intelligent Manufacturing", IEEE Spectrum, Vol. 30(9), pp. 82-84, (1993) Brussel, H.V., et al, "Ref. Architecture for HMS", Computers in Industry, Vol. 37(3), pp. 255-274, (1998). NSF, "Reports from Break-out Groups (BOGs)", NSF Workshop Final Report, pp. 21-91, (1997). Iwata, K., et al., "A Modeling and Simulation Architecture for Virtual Manufacturing Systems", Annals of CIRP, Vol. 44, pp. 399-402, (1995). Kimura, F., "Product and Process Modeling as a Kernel for Virtual Manufacturing Environment", Annals of the CIRP, Vol. 42, pp. 147-150, (1993). Kidd, P.T., "Agile mfg.: a Strategy for the 21st Century", lEE Colloquium on Agile mfg., pp. 1/1-1/6, (1995). Jordan, J.A. and Bunce, P.M., "Requirements for Next Generation Manufacturing Systems", IEEE/CPMT Int"l Electronics Manufacturing Technology Symposium, pp. 247252, (1994). NGM Project Report, Agility Forum, (1997). Valckenaers, P., "Intelligent Mfg. Systems", Computers In Industry, Vol. 37(3), pp. 169170, (1998). Sousa, P., and Ramos, C., "A Distributed Architecture and Negotiation Protocol for Scheduling in Manufacturing Systems", Computers in Industry, Vol. 38(2), pp. 103-113, (1999). Leeuwen, E.H. and Norrie, D., "Holons and Holarchies", Mfg. Eng., Vol. 76(2), pp. 8688, (1997). Martensson, N., Martensson, L. and Stahre, J., "Human-Centered Flexible Manufacturing System in Machining and Assembly", Human-Intelligent Based manufacturing, SpringerVerlag, pp. 67-82, (1993). Murphy, S., "Human-Centered CIM Systems" ESPRIT 88, Putting the Technology to use, Proceedings of the 5th Annual ESPRIT Conference, Brussels(BE), pp. 1615-1629, (1998). Slatter, R.R., et al, "Human-Centered Approach to the Design of Advanced Manufacturing Systems", Annals of the CIRP, Vol. 38, pp. 461-464, (1989). Onosato, M., et al., "Development of a Virtual Manufacturing System by Integrating Product Models and Factory Models", Annals of the CIRP, Vol. 42, pp. 475-478, (1993). Iwata, K. et al., "Virtual Manufacturing Systems as Advanced Information Infrastructure for Integrating Manufacturing Resources and Activities", Annals of CIRP, Vol. 46, pp. 335-338, (1997). Harrison, A., "From Lean to Agile Manufacturing", lEE Colloquium on Agile manufacturing, pp. 1-7, (1997). Tu, Y., "Production Planning and Control in a Virtual One-of-a-Kind Production Company", Computers in Industry, Vol. 34, pp. 271-283, (1997). Talbert, N., "Toward Human-Centered Systems", IEEE Computer Graphics and Applications, Vol. 17(4), pp. 21-28, (1997). Chappell, S.L., "Certification of Usability: A Process for Creating a Human-centered System", Proc. of the digital avionics systems conference - 17th DASC, pp. E1 l/l-El 1/8, (1998). Jerard, R.B., "FACILE: A Clean Interface for Design and Fabrication Intemet-Aided Design, Manufacturing and Commerce", 18th Computers in Engineering Conf., Atlanta, Sep., (1998). Choi, B.K., et al, "Unified CAM System Architecture for Die and Mold Manufacturing", CAD, Vol. 26(6), pp. 235-243, (1994).
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26. Choi, B.K., Kim, D.H. and Hwang, H., "Gantt Chart Based MES for Die and Mold Manufacturing", Proc. IFIP WG 5.7 Conf. On Managing Concurrent Manufacturing, Seattle, USA, pp. 105-114, (1995). 27. Choi, B.K., Han, K.H. and Park, T.Y., "Object-oriented Graphical Modeling of FMS", Int. Jr. ofFMS, Vol. 8(2), pp..159-182, (1996). 28. Park, T.Y, et al, "An OOM framework for AMS", IJCIM, Vol. 10(5), pp. 324-334, (1997).
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
437
INTELLIGENT MACHINING SYSTEMS - CHALLENGES AND OPPORTUTINTIES
Teltz, R. ~ and Elbestawi, M.A.** ** Professor and Chair, Department of Mechanical Engineering Research Engineer, Intelligent Machines and Manufacturing Research Center (IMMRC) Faculty of Engineering, McMaster University, Hamilton, Ontario, Canada Email: elbestaw~mcmaster.ca and n_'[email protected] w
ABSTRACT Machine tool control systems typically involve servo motion and discrete input/output control. In this manner they only indirectly control the metal cutting process and consequently the economic and efficient production of discrete parts. Intelligent Machining Systems attempt to more directly control machining processes by incorporating models of the cutting process and real time sensor data. Operational parameters such as cutting speeds and feeds are formulated to directly address goals of economy and product quality. Intelligent Machining Systems integrate technologies related to sensing, modeling, control and monitoring. The process variables involved include tool condition, process cost, rate of production, and part quality. More recently, the use of Open Architecture concepts in CNC machine controllers has allowed a greater realization of complex Intelligent Machining Systems; particularly with regards to systems which involve the control or monitoring of multiple process conditions. This paper describes research performed in the IMMRC labs relating to modeling, monitoring and control for Intelligent Machining Systems (IMS). Applications in milling and turning are described. Key points regarding the challenges and opportunities for future research are discussed. KEYWORDS Intelligent machining systems, machine tools, metal cutting, sensing, modeling, controls 1. MODELING FOR THREE AXES MILLING OF DIES AND MOLDS
The Role of Process Modeling in Intelligent Machining Systems Accurate models and simulation systems of metal cutting processes are used to evaluate operational parameters (ie: feeds, speeds, depth of cut) with respect to critical process constraints such as cutting loads, cutting stability, produced part features, etc. In this regard, process modeling is a major component of Intelligent Machining Systems for optimization algorithms and for the generation of base line data for actual process evaluation. There are several aspects of metal cutting that make process simulation complex. The first involves the accurate calculation of the geometry of interaction between the tool and workpiece - an issue compounded by the use of tooling and work pieces with complicated profiles. Moreover, for milling of dies and molds in particular, tool paths can involve continuously varying immersion geometries, with rough, semi finish and finish passes. This introduces the need to manage very large amounts of data. Added to all of this is the requirement for inclusion of the effects of machine tool dynamics due to their critical effect on chip geometry. Other issues include model calibration where for example, variation in cutting velocities along the tool edge profile require a nonlinear calibration.
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Research at the IMMRC has addressed several aspects of these problems, with the particular goal of making simulation systems more accurate in their predictions and more applicable to the needs of Intelligent Machining Systems. The follow sub sections discusses some of this research.
Cutting Edge Representation The primary type of tooling used today to manufacture dies and molds includes ball and flat end mills. However, the specific cutting edge geometry varies significantly within this class of tooling. While the majority of published research has dealt with simple cutting edge designs (ie. fixed diameter, zero helix, constant helix, constant pitch), the problem remains in having a generalized modeling procedure to represent a varied cross-section of tooling, including form tools. Recent work at the IMMRC has attempted to generalize the representation of the cutting edge. Imani [ 1] and Abrari [2] use a cubic piecewise NURBS curve [3] to represent the tool cutting edge profile for a constant lead/pitch ball end mill. A key advantage this representation is that there are standard techniques available for manipulating the location and orientation of a NURBS curve. Thus, virtually any cutting edge profile can be used by the simulation allowing for example, the tool profile itself to be a manipulated variable in an optimization strategy of the Intelligent Machining System. A p'" degree NURBS curve is defined by: I!
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~ i _ o Ni.v ( u )o)i where the Pi are the control points, the t.o, are the weights and the Ni.p(U) are the pth B-spline basis functions. The set of three dimensional control points P~ along the cutting edge profile are typically determined from either of 1) an analytical expression for the edge profile, 2) a tooling drawing, or 3) discrete three dimensional points measured by a coordinate measurement machine.
Part Local Surface Topology Representation The surface topology of a die or mold is typically composed of many complex surfaces. This complexity is a result of both the desired part form in addition to any "marks" left on the surface from previous machining operations (eg: roughing, semi finishing). To simulate the machining of even moderately sized parts, the required storage of surface data can quickly become unwieldy. To overcome this problem, the developed simulation systems uses only the local topology of the part in the vicinity of the tool. Figure I depicts this procedure. The "Control Space" shown in the figure is chosen to consider all surfaces within possible contact of the tool. A simple bounding box intersection algorithm is used to determine which tool path segments contribute to the local surface topology of the part in the vicinity of the current tool position. In the case of secondary operations (ie: semi finishing or finishing) all previous operations are reconstructed to provide the surface data seen by the current pass. In this manner the need to store vast amounts of surface data is avoided.
Mechanistic Force Model The mechanistic force model has been formulated following those models presented by Imani [ l ] and Yucesan [4]. The modeling approach taken was to first develop the cutting edge geometry, then to develop the cutting force expressions. The cutting edge geometry for a ball end mill is shown in figure 2. The coordinate values of any point P~on the cutting edge can be determined from the NURBS curve representation, if given the corresponding parameter u,. The parameter u~, normalizes points at equal spacing along the rotational axis (z axis) of the tool. Given features of the geometry in the locale of point Pi, the instantaneous
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Figure 1. Concept of local surface topology representation. \x.
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+ K,, . [ - ~ , ( O. z ) + Kre . ~r,( O. z ) ] . dS } Calibration The coefficients K=, Kfr and K=, Kf, are functions of cutting edge geometry, cutting speed, and feed per tooth. Various researchers have addressed the problem of calibrating these coefficients to experimental data; some use averaged values over a range of cutting conditions [4], while others have considered only the variation with cutting speed for example [ 1]. It should be noted that the subject of calibration is an ongoing research issue for metal cutting process modeling. The method proposed by the work presented here expresses the variation of pressure and friction coefficients as an exponential function of the cutting speed, the helix angle and the feed: K~ = e "'§ (10) where ~ is the instantaneous cutting coefficient, and V~, 13~are the instantaneous cutting speed and helix angle respectively. The coefficients a l, a2, a3 are functions of the instantaneous feed per tooth. Figure 3 shows the comparison of some experimental results with the calibrated force model. The calibration has been shown to provide good correlation with experimental data over a wide range of cutting conditions. Force X fl~:
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2. PROCESS M O N I T O R I N G USING FUZZY NEURAL N E T W O R K S
Machining Process Monitoring Modem techniques for automated machining process monitoring include expert systems, neural networks, and fuzzy pattern classification. Indirect measurements such as cutting force and vibration signals are typically used however, the measured alone signals do not often give sufficient information to discriminate between the different tool conditions. Accordingly, further processing stages is required to yield useful features which are more sensitive to tool conditions, but insensitive to noise. The problem of decision making relies on criteria relating the variables in the feature space to the classification space. Typically. parameters (weights) in a matrix form are used for this mapping, such as the weights in a linear classifier or in a back-Propagation neural network. In more sophisticated implementations, multi-layered neural networks are employed, which consists of nonlinear connections between the inputs and the outputs.
Implementation of the Fuzzy Neural Network As an alternative to typical neural networks, the work discussed here [5] proposes a fuzzy neural network approach for tool condition monitoring. Using this combination of technologies, it is possible to exploit the parallel nature of the classification and to provide a fuzzy reasoning approach. Different from matrix-type decision making, a tree structure is used for reducing unnecessary interconnections within the networks. This approach results in simpler and faster training and classification. An example neural network that utilizes fuzzy logic is shown in Figure 4. The input layer, FA, has processing elements, for each of the n dimensions of the input pattern Xk(feature vector). The parameters of the probability distributions represent the connections to the hidden layer. The hidden layer FB, consists of neurons that use fuzzy classification to address subsets of the original data set and any necessary information from other neurons. A membership function is used for the classifications at neurons, the interconnections within the hidden layer and the connections to the output layer. The neurons of the output layer, Fc, represent the degrees to which the input pattern Xk, fits within the each class (tool conditions). The output decision may be either "soft" or "hard" depending on the requirements.
Figure 4. Structure of the Fuzzy Neural Network.
Figure 5. Example of Maximum Partition.
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Learning and Classification
In the learning phase, the hidden layer is constructed based on available learning samples. A key operation here is determining the "the Maximum Partition", Xp = Ap + Bp, where Ap is the set which contains the samples that belong to certain tool conditions and Bp is the set of all the remaining samples. The partition is determined from probability distributions of the data, using a technique referred to "Multiple Principal Components"[5]. The monitoring index associated with a resulting partition is defined as "the pivot index" and is applied as a major input of a neuron in the hidden layer. Figure 5 gives an example of the Maximum Partition of a 2-index, 3-class problem. By studying the distribution of the samples from classes A, B, and C, we notice that 12gives the maximum partition of the learning set for class C with respect to the other classes. Here, 12 is chosen as "the pivot index" at this neuron for the maximum partition. The maximum partition in multiple directions at neuron p generates two new neurons: one with the learning set Xp and the other with the learning set Bp = X~ - Ap. In this manner the distribution of learning samples is used to measure the supporting strength of the partition. Implementation of this procedure in algorithmic form results in an automated learning procedure - also referred to as unsupervised learning capability. Classification is performed by searching a path of neurons through the network, leading to an output decision. At each neuron in the hidden layer, the search is directed by the fuzzy membership grade at that neuron, using the membership function. This information propagates through the hidden and/or the input layer to the output layer where the final decision is given. Experimental Verification
Cutting tests were performed using 10 Hp NC lathe, instrumented with a cutting force dynamometer mounted under the tool holder (Fx, Fy, Fz), two accelerometers located on the tool holder (Ax, Ay), and a spindle current sensor for cutting power, Pw. All signals were low pass filtered at 1 kHz, and then were sampled at 2 kHz. The work pieces were AISI 1014 steel bars and the cutting tools were ISO K21 and K6 grade carbide inserts. Fifty two cutting conditions were tested as folows: cutting speeds 96 to 322 m/min, feeds 0.024 to 0.246 mm/rev., and depths of cut 1.2 to 3.5 mm. Five different tool conditions were considered:, sharp tool (no chipping, flank and crater wear <0. I mm), slight flank wear (0.1-0.16mm), medium flank wear (0.16-0.3mm), severe flank wear (greater than 0.3mm) and tool breakage (chipping area > 0.04mm2). Table I summarizes the results obtained from those tests. For comparison, the results obtained using a well-known feed-forward neural network trained by back-propagation (BPNN) [6]. For the various parameters of the BPNN algorithm, those chosen for comparison represent the best results obtained. The Fuzzy Neural Network was found to outperform the BPNN approach. Test
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3. INTELLIGENT, OPEN A R C H I T E C T U R E C O N T R O L IN TURNING Open architecture machine tool control is recognised as an enabling technology for advanced machining systems. The full realization of process control in machining depends on flexible and open interfaces to critical machining functions, In this manner, Open Architecture Control (OAC) facilitates the implementation of complex Intelligent Machining Systems. The IMMRC OAC [7]is based on a distributed, peer to peer functional model of the entire machine control system. Industry standards are used for hardware, software, and communication media. Machine motion and I/O functionality is provided by a module referred to as the 'Machine Server'. Generalized services support command, data and timing mechanisms between the Machine Server and 'client' applications (ie: IMS modules). Figure 6 depicts the relationship between IMS modules and the OAC Machine Server. FEED, SPEED, CLIENT APPLICATION
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A comprehensive Intelligent Machining System has been developed and demonstrated on the OAC. Figure 7 depicts the system for optimization and control of multi pass turning. The temporal aspects of the multi pass problem are addressed by the three feedback loops depicted. In pass control addresses time critical events related to the safety and integrity of the current pass. Tool breakage, chatter, and force regulation are addressed. Per pass control provides a plan of tool paths and feeds to provide constant cutting forces for the next pass. The model used for planning and prediction of cutting forces incorporates tool wear measurements acquired between passes by an on machine vision system. Multi pass control records and models longer term process behaviors. Lower levels of control are compensated; models are recalibrated and parameters are adjusted. In pass control tasks are performed by C language routines implemented at the real time level of the OAC structure. The time critical nature of these activities is supported here by a real time operating system and low latency access to sensor data. The per pass control tasks include tool wear measurement and image data processing cycles, and feed schedule calculations. These algorithms execute at the non real time level of the OAC and, in the case of the feed scheduling tasks, use routines
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from a commercial solid modeling package (ACIS, by Spatial Tech. Inc.). Multipass control is realized using CLIPS (C Language Integrated Production System), an open source expert system. The system described here is detailed in the references given. The discussion here is provided to underscore the advanced and seemingly limitless possibilities for optimization of machining processes offered by the open architecture control platform. The concept of Intelligent Machining systems, and moreover the adoption of such technologies in industry clearly stands to benefit from further development and proliferation of the OAC systems. 4. DISCUSSION AND CONCLUSIONS This paper has attempted to give an overview of research related to the development of Intelligent Machining Systems at the IMMRC. Some advancements in process modeling was described for die and mold milling applications, tool condition monitoring using hybrid fuzzy logic and neural network technologies, and open architecture machine tool controls. These three areas emphasize the broad scope of focused research required to further the larger realization of Intelligent Machining Systems. Many challenges remain before such systems become commonplace in industry. Slow, but deliberate movements within the machine controls sector is advancing the proliferation of Open Architecture machine control. These platforms are the practical basis of Intelligent Machining Systems in that they allow the other technologies - modeling, monitoring, and controls to be integrated into the production machining environment. In this regard, there can be expected an increasing number of opportunities for these Intelligent Machining technologies to manifest tangible benefits for manufacturers in the future. Several challenges remain however, before such benefits are truly seen by practicing manufacturers. Process modeling methods must become more accepted as part of commercial CAM installations. Future engineers and manufacturers need to appreciate the potential of such tools, and be ready to rely on their predictions for process development. This underscores the need for further research towards more accurate modeling strategies, more comprehensive models (eg: the inclusion of fixture and part dynamics), and the development of simpler and better calibration schemes. Monitoring systems must also progress forward to provide higher reliability and sensitivity to process conditions. False alarms cannot be tolerated in high volume machining systems. Perhaps a greater understanding of their limitations by manufacturers would promote their use today in proper applications. In this regard, the mind set of end users of Intelligent Machining Technology remains perhaps the greatest challenge to overcome. -
REFERENCES 1 - lmani, B., "Model Based Die Cavity Machining Simulation Methodology", Phd Thesis, McMaster University, (1998). 2 - Abrari, F., "MultiAxis Milling of Flexible Parts", Phd Thesis, McMaster University, (1998). 3 - Piegl, L., Tiller, W., "The NURBS Book", Springer, 2"a Edition, (1997). 4 - Yucesan, G., Altintas, Y., "Mechanics of Bali End Milling Process", ASME Journal of Manufacturing Science and Engineering", Voi 64, pp. 543-55 I, (1993). 5 - Li, S., Elbestawi, M.A., "Tool Condition Monitoring in Machining by Fuzzy Neural Networks", ASME Journal of Dynamic Systems, Measurement and Control, Dec. 1996, Vol.55, pp 1019-1034, (1994).
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6 - Lippmann, R.P., "Pattern Classification using Neural Networks", IEEE Communication Magazine, pp 47-64, Nov., (1989). 7 - Teltz, R., Elbestawi, M.A., "Design Basis and Implementation of an Open Architecture Machine Tool Controller", Proc. of the 25th NAMRC, Lincoln Nebraska, (I 997).
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Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
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CAM SYSTEM FOR EFFICIENT GENERATION OF PART-PROGRAMS FOR WIRE-EDM EI-Midany, T.T. ~ EI-Keran, A.A. ~ and Radwan, H.T. ~176
'Professor, *'Assist. Prof., Prod. Eng. and Mech. Design Dept. '"Production Engineer. Faculty of Eng. P.O.B. 2 Mansoura University 35516, Egypt
ABSTRACT In the recent years wire-EDM became a necessity for many engineering applications, particularly in the dies making. Design of a good CAM system as an aid for making dies or other wire-EDM applications, became a challenging job with increased use of complex part geometry. In this paper a proposed CAM system is designed to overcome the limitations of the existing CAM systems. The proposed CAM provides automatic correction for maximum taper angle, and automatic creation of the programmed fixation stop. Programming time has been reduced using the automatic tool path creation, postprocessing, trace checking feature, and simulation check. Enhanced two-axis programming capabilities include, multiple operation automatic lead-in and lead-out generation, automatic comer filleting for internal, external, or all sharp intersections. One directional trim cut can be written as a main program which calls a separate sub-program NC file for each trim cut to reduce the NC program size. No-Core cut capabilities are also added to avoid the machine damage. Great saving in programming time, user effort, and the precision for all steps have been achieved. KEYWORDS Wire EDM Programming, CAM System, No-Core Cut, Taper Cutting, Trim Cut, NC. 1. INTRODUCTION Wire cutting process is widely used for making stamping dies, turning form tools, templates, extrusion dies, progressive dies, and prototype production. The wire-EDM programming start with a manual method [ 1]. With the increased use of NC systems and growth in complexity of production the manual programming became a tedious work and time consuming. The manual method depends to a great extent on the experience of the operator, which leads to many human errors. So, the part programmer was no longer able to calculate efficiently the required tool path, and the use of CAM systems as an aid to part programming became a necessity [2,3]. Automatic generation of standard radius, sequence of wire path, fixation stability, and No-Core cut are examples for developing wire-EDM programming. When developing CAM software, one must consider how the interaction between the user and the program will take place. The portion of the software that provides this link is referred to as the user interface. The interface should provide a convenient means of moving through the various options of the program. In addition to the selection of program options, the interface must provide error
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detection and recovery as well as the ability to cancel any operation midway through the process. Other desirable attributes of the user interface include making it easy to enter the information, providing graphical feedback to user inputs (such as blinking of selected entities), and preventing unpleasant surprises such as program crashes. Certainly, the items listed above make the construction of good user interfaces a challenging job. Thus, the objective of this work is to design a CAM system to sequence the wire path in a way to minimize the machining time, and generating the standard radius, maintaining fixation stability and optimizing the wire curing process. Some of the wire-EDM techniques such as, making the conical curing, trim cuts, and calculating the optimum location for the threading/breaking point of the wire, are also achieved through the proposed system. Through different applications of the proposed system, the results show a great saving in programming time, user effort, high precision for all steps of the curing process, particularly in the compound dies, prototype, and form tools. 2. THE INFORMATION FLOW THROUGH THE PROPOSED SYSTEM The information flow through the proposed system is illustrated in Fig. 1.
3. THE PROPOSED SYSTEM FUNCTIONS The proposed system was designed to satisfy the following factors:
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a. Sequencing the wire cutting motion in a way to minimize the machining cutting time. b. Considering the fixation stability to avoid the drop of the work piece. c. Analyzing the part-geometry and adding special features to the contour to improve the quality of the cutting process. d. Optimizing the wire cutting parameters for individual part-elements according to the part-features and technology data. e. Graphically simulating the cutting process of the spark erosion to verify the cutting sequence. f. Generating NC part-programs directly from CAD geometry database. 4. THE PROPOSED CAM SYSTEM The proposed system contains two processors for CAD/CAM calculations. Geometry processor for CAD database calculations and Technology processor for CAM and technology aspects calculations 4.1 Geometry Processor
The geometry processor analyses the extracted part-geometrical elements, and decides an optimized cutting sequence. The criteria for an optimized sequence are the fixation stability for the remaining sheet and the cutting time. During this step, part geometrical elements are processed according to the following steps: 1 Filter part-geometrical elements by removing ~emoveoverlapped ~ g ~" = E "the overlapped elements and joining together any )o]nCl--ose'Elements ) ~ ~ ) close vertices if the distance between them is less ......... than kerf width, as shown in Fig. 2. Fig. 2.Geometry Filter 2. Convert the individual geometrical elements to polygons by sorting and joining together any two identical vertices. Then identifying all generated polygons as closed or open polygons. 3. Generate a vertex searching space 0 sTART and END vertices of all open ~ I olygons. list that contains start and end vertex All vertices of all closed polygons. coordinates of all open polygons as well as all vertex coordinates of all closed .... Fig: 3"Sear.rh Lis t Generation . ~ , ~ J polygons, Fig. 3. 4. Sequencing the cutting motion by selecting from search space list the polygon that has the closest vertex to the previous tool (wire) position, taking into consideration that if the selected polygon includes other polygons, then skip this selection. If the selected polygon includes other polygons, then all inner polygons must be machined before the outer polygon. Remove the selected polygon from search space list and repeat the step no. 4 until the end of all nesting parts. Insert standard radius for the sharp comers, as shown in Fig. 4. 5. Read the standard radius value from cutting libraries; the option default value equal to 2 mm. Searching for sharp comers, and making grips around them, and then filtering. 4.2 Technology Processor
The technology processor analyses the data stored in the wire cutting libraries, which is available for modifying from the "Wire Option" dialogue box, and decides the optimum cutting conditions. During this step, cutting conditions are processed according to the following steps:
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_~
C]
f:
Select from search list the polygon, which
L
[
has the closest vertex to origin point, and make this
vertex, the START vertex. ,,
polygon was closed
'~
START
Set Origin Point = END Vertex
Sequencing _ ~ Yes Remove this polygon temporarily fiom search list
if this polygon includes another
N~ I Set origin point = START vertex
l
No
iq-oo,o,nd 1
Fig. 4.Sequencing Calculate for the polygon area, if the polygon area < 3 mm, the technology processor automatically adds the codes for No-Core cut in the NC file (*.cnc). 1. Add trim cut, correction, and the taper angle in the technology and NC files [9,10,11]. 2. Make the multiple cuts Fig. 5 and employing the 9 P o l a r arra~" ,'+ dimensions (mm/inch) in the job file. 666 3. Determine the best location of the 9 RectanrularArrav: ~ ~ threading/breaking point of the wire. Store all cutting files, NC file, position, technology file, and Fi~. 5.Makinl[ Multiple Cuts job file, as shown in Fig. 6. 4 44
4 4~
5. THE PROPOSED SYSTEM PART-PROGRAMS
5.1 Position File Position file contains the command for positioning the machine head for x-y and z-axis without wire and cutting. 5.2 Numerical Control File This file contains the elements of the drawing in the ISO code format. The basic structure of this file contains lines ~ G01, curves C C W ~ G03, curves in C W ~ G02, stop ~ MOO, [12] etc. 5.3 Technology Data File The cutting libraries use this file to define the technology functions.
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............ ~ Determine ThreadingPoints ~ ~
r
1
451
~ "Point
-'-~nd Point
Punch
i,l Select the Outer Point
Select the Inner [ Point
Fig. 6.Determination Threading/Breaking point 5.4 Job File In order to execute the erosion of a workpiece the machine needs the job file. This file is used to call the three files, position, NC and technology file.
6. HIGH-PRECISION TAPERING In many cutting dies, the taper is only included to allow guidance and ejection of the cut part (slug) away from the machine. A medium level of taper accuracy may be often enough to ensure this function. However, a number of applications require precise tapers. These include, dies for powder materials, dies having small taper angle, fine-blanking dies, resharpening of chipped tools and milling cutters, and electrode geometries for die-sinking EDM of die and mold cavities. For thin sheet material, higher taper accuracy is needed to ensure that the cut parts are not bent during ejection. To obtain high accuracy on the cut parts, it is necessary for the wire to have the best possible mechanical properties when it enters the machining area [ 13]. The wire must be selected and treated in such a way that it does not plastically deform during its passage through the wire drive system and through the guides [ 14,15]. 7. SIMULATION CHECK Simulation aims to make the best use of resources by helping the user to decide what is needed, and where is the best position to locate it. The proposed system uses the simulation check to save hours of machining through few minutes of verifying check, and to save the machine time for the actual cutting. 8. PROPOSED SYSTEM APPLICATION 8.1 manufacturing of Compound Dies The proposed system is able to machine compound dies in one path, instead of two programs (one for blanking punch and the other for piercing die), as shown in Fig.7. (Refer to appendix
A for the list ofoutputfiles)
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Fig. 7.Manufacturing of Compound Dies 8.2 Prototype Production Prototype production is one of the main applications of the wire-EDM. The required number of the prototypes in this example is five. These are fired by rivets and machined by stack machining, Fig. 8. (Refer to appendix B for the list of output files) 8.3 Manufacturing of Form Tools Manufacturing of turning form tools is very important, particularly for automatic and semiautomatic lathes in mass production, using a suitable fixture equipped to the machine to produce clearance, Fig 9.
9. CONCLUSION The application affirms that the time saved through the proposed CAM system starting from the initial calculation to the generation of the NC file is about 50 % or more compared with EasyCut CAM software. By using the proposed CAM system, sequencing the wire cutting motion in a way to minimize the machining time, can be accurately and simply performed. Suitable wire cutting techniques are selected based on the geometry specifications. The type of polygon (punch/die) is selected by an automatic method based on the fixation stability and the geometry specification. The location for the threading and the breaking point of the wire is selected by an automatic method to attain the optimum loeation._Adding taper angle, trim cut, and wire-offset are achieved by an automatic method. Programming time has been reduced using postprocessor. Making NC programs from minimal information. The proposed CAM system has been used to produce several practical workpieees at private sector in the 10th of Ramadan City / Egypt. REFERENCES Herbert, Y.W., "Manufacturing Process", pp. 303-323 Prentice-Hall, Inc. Englewood Cliffs, New Jersey, USA, (1997).
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2. Jain, K.S., "Manufacturing Engineering Technology", 2nd Ed., Addison Wesley Publishing Co. Inc., (1991). 3. Fuller, J.E., "Electrical Discharge Machining", Vol. 16, Metal Handbook, 9th Ed., (1988). 4. User Manual, "AutoLISP", for AutoCAD R. 10-14, USA, (1996). 5. User Manual, "A-CAD R. 13 for Microsoft Win 4.0 (x86)", Autodesk, Inc., USA, (1996). 6. Beltrami, I. and et al, "Simplified Post Processor for Wire EDM", Vol. 58 No. 4, pp. 385389, Journal of Materials Processing Technology, AGIE Ltd., Losone, Switzerland, April 15th (1996). 7. AGIECUT Sprint Operating Manual; Published by AGIE Industrial Electronics Ltd., Losone, Locamo, Switzerland, April (1995). 8. "EasyCut CAD/CAM System" Software Package, AGIE Industrial Electronics Ltd., Losone, Locamo, Switzerland, Feb. (1991). 9. Noaker, P., "EDM Gets with the Program", Manufacturing Engineering Vol. 110, No. 6, pp. 55-57, June (1993). 10. Kozak, J., and etal, "Material Removal In WEDM of PCD Blanks" Journal of Engineering For Industry ASME Vol. 116, No. 3, See. B, pp. 363-369, August (1994). 11. Luo, Y.F., and Chen, C.G., "Effect of a Pulsed Electromagnetic Field on the Surface Roughness in Super Finishing EDM", Precision Engineering Vol. 12, No. 2, pp. 97-100, April (1990). 12. EI-Midany T.T., Elkeran A.; and Radwan H.T., "CNC Automatic Programming System for Wire-EDM"; The Twelfth International Conference on Industrial and Engineering Applications of Artificial Intelligence and Expert System (lEA / AIE-99); Cairo; Egypt; May 31 st-June 3 rd (1999). 13. David, T., "EDM Software", Manufac. Engineering Vol. 112, No. 6, pp. 57, June (1995). 14. Bally, F. and Buemi, C., "High Precision Tapering Made Easy", American Machinist Vol. 140, pp. 33-35, June (1996). 15. Michael, T., "Leasing Lets EDM Machines Go To College", American Machinist Vol. 138, July (1994). APPENDIX A. List of proposed system output files for compound dies A.I Sample List of NC file (examplel.cnc) $Proposed system Ver 1.01 cod(iso) ldr(6) lit(%) N001 G01 X+056678 Y+ 104772 M22 :00002 M63 . o ,
N043 M02 A.2 Sample List of Technology data file (example l.ted) % I 13 12 11 00 00 00 14 (AEGDIR_3) TOO 15 05 19 002 02 02 02 002 002 02 02 . . .
M02 A.3 Sample List of position file (example l.pof) cod( POSITION File )
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lit(!) Pos O1 PAX+056678 PaY+I05348 !
$$ Position Without Wire
= 168.0207
A.4 Sample List of job file (example l.job) Dimensions COMPLETE MM POF CAProgram Files~Vlicrosoft Visual Basic\pof~xample(1).pof TEC C:LProgram Files~4icrosofl Visual Basic\ted~Example(l).ted CNC sav RETRY FROM : 9 TIMES WITH TOO TECH SKIP STP IF NOT THREAD STOP CNC CAProgram FilesLMicrosoR Visual Basic\CNC\Example(l).cnc EXEC STP EXEC WIE EXEC TO1 EXEC SO1 FROM 001 TO 999 FIN
APPENDIX B. List of proposed system output files for prototype production B.I Sample List of NC file (example2.cnc) $Proposed system Ver 1.01 cod(iso) ldr(6) lit(%) N001 GO1 X+028000 Y-015074 M22 N043 M02 B.2 Sample List of Technology data file (example2.ted) % I 13 12 11 00 00 01 14 (ASGDIR_7) D03-178 D04-178 M02 B.3 Sample List of position file (example2.pof) cod( POSITION File ) lit(!) Pos 02 PAX-001050 !
$$ Position Without Wire
-- 127.6244
B.4 Sample List of job file (example2.job) Dimensions COMPLETE MM POF CAProgram Files~d~1icrosoR Visual Basic\pof~Example(2).pof Fin
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, Februat3, 15-17, 2000
455
ON THE PREDICTION OF SURFACE ROUGHNESS IN TURNING USING ARTIFICIAL NEURAL NETWORKS
El-Sonbaty, I." and Megahed, A.A.*" * Assoc. Prof. and ** Graduate Student Mechanical Design and production Engineering Department, Faculty of Engineering, Zagazig University, Zagazig, Egypt.
ABSTRACT The aim of the present work is to use the technique of artificial neural networks, with backpropagation routine, for the prediction of surface roughness of workpieces produced by turning process. To provide the data required to train the networks, specimens were turned at different cutting conditions using tools having different pre-machining tool-flank wear. The vibration level during machining, and the after-machining tool-flank wear were measured and the surface roughness parameters of turned specimens were assessed. Several neural networks were trained with changing both of the network structure and the number of training samples. The best of these networks were selected to be used for the prediction of surface roughness parameters, vibration level and tool wear. From the results obtained it may be concluded that the developed neural networks permit acceptable outputs for the prediction of the investigated surface roughness parameters, vibration level and tool-flank wear. Also, based on these networks, an integrated system for continuous prediction of surface roughness was developed, and a computer program was designed for selecting the appropriate machining conditions required to achieve desired values of surface roughness parameters. KEYWORDS Surface Roughness, Vibration Level, Tool-flank Wear, Turning, and Neural Network. I. INTRODUCTION Surface roughness is a factor of great importance in the evaluation of workshop production and considerable attention is now being focused on its measurement as a means of quality control [l]. Several attempts were performed to predict the surface roughness in turning operation by using either the mathematical models or the artificial neural network technique. As an example of the mathematical models, Mital and Mehta [2] generated surface finish prediction models. All proposed prediction models take into account cutting speed, feed, cutting tool nose radius, and their interactions. Fei and Jawahir [3] presented a methodology for predicting surface roughness in turning through a fuzzy knowledge-based system for a given set of cutting conditions, work material, tool insert type and tool geometries (including chip groove geometries). EI-Baradie [4] developed a mathematical model which uses cutting speed, feed rate and nose radius as inputs for predicting surface roughness of machined components in turning gray cast iron. Arsecularatne, Fowle and Mathew [5] described an
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analytical approach to predict the chip flow direction, cutting forces and surface roughness under finishing conditions. The major weakness of the mathematical model systems lies in the random errors contained in the machining response data. It has usually been found that metal cutting data are susceptible to an appreciable amount of scatter. This is due to the non-homogeneity of the material being cut and its effect on the measurement of the machining response data, as well as due to the errors in the measuring technique itself [4]. So, another method was used based on using the artificial neural network technique. The primary advantage of the artificial neural networks approach over the mathematical models is that the neural networks can model very complex relationships and yet are relatively insensitive to noise included in the machining data [6]. Chryssolouris and Guillot [6] developed neural networks to predict surface roughness (Re parameter), spindle power, noise level, and chip merit mark in orthogonal turning operation using process parameters (feed rate, cutting speed, rake angle, width of cut, and flank wear) as inputs. Matsumura, et al [7] proposed an autonomous turning operation planning, to minimize machining cost by predicting tool-flank wear analytically and surface roughness with a neural network. Darenfed, Miao and Boudreau [8] developed a multi-layered learning network for predicting surface roughness using cutting parameters (cutting speed, feed, depth of cut, and nose radius) as inputs and one surface roughness parameter as an output, as a performance criterion of learning networks. Aboul-Nour, et al [9] used the artificial neural networks to predict the surface roughness (1~, Rt and Rz parameters) of products as a representator for the machining performance in milling operation. Fang and Yao [10] developed neural networks for on-line prediction of machining performance (characterized by surface roughness, cutting forces, chip breakability, and dimensional deviation due to tool wear) during tool wear progression using a single sensor. The tool wear influences the surface roughness of the workpiece and the value of surface roughness is one of the main parameters used to establish the moment to change the tool in finish turning [11]. Tool-flank wear was predicted either analytically [12-14], or with the neural networks [9,15]. The present work is an approach to predict the surface roughness, in turning operation, using the feed-forward artificial neural networks (ANN) technique trained with the backpropagation routine. The inputs to the neural network of surface roughness prediction are cutting speed, feed, depth of cut, nose radius, pre-tool-flank wear, and vibration level. The tool-flank wear and the vibration level are predicted using other neural networks. An integrated system, comprising the three developed neural nets, is aimed to be made for continuous prediction of surface roughness. Also, a computer program is designed for the selection of the most suitable machining conditions, which are required to achieve a desired value of surface roughness parameters. 2. EXPERIMENTAL WORK In order to train the neural network, input-output pairs must be collected. In the present work input-output pairs required to train the developed neural networks are collected through the experimental work. The results of this experimental work have been used to train, evaluate, and test the developed neural networks. Sixty workpieces were prepared, each with 80 ram. diameter and 260 mm. length, from steel bars having the composition; 0.39% C, 1.5% Mn, 0.12% S, 0.025% P, 0.21% Si and 0.07% Cu, with a hardness value of 170 HB. The used length of the workpiece in machining and
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measurements (effective length) was 200 ram. and the remainder 60 ram. was used for clamping. In the machining process, each effective length was divided into five equal regions (40 mm. each) which represent five different samples. The experimental results were gained by dry finish turning of the sixty workpieces (i.e. 300 samples) using coated cemented carbide throw-away square tips (grade TNI50) with three different nose radii (r = 0.4, 0.8 and 1.2 ram.). The machining was implemented on a Colchester 7.5 HP lathe. Four different cutting speeds (V = 130, 180, 250 and 350 m/min.), four feeds (f = 0.06, 0.08, 0.10 and 0.12 mm/rev.) and two depths of cut (d = 0.5 and 0.8 ram.) were selected as machining conditions. Each workpiece was machined with fresh (new) tip. For each sample the maximum values of tool-flank wear, before machining (Vbt) and after machining (Vb2), were determined using an optical microscope (type 2158) with vernier resolution of 0.01 mm. During machining of each sample the vibration level of the cutting tool (V.L.) was measured using an integrated vibration meter (type 2513). After machining each workpiece, the surface roughness of each sample was assessed by using "Taylor Hobson Surtronic 3+" surface roughness measuring instrument with its software (version 1992) taking a cut-off value of 0.8 mm. and a traversing length of 4 mm. Thus, after the finishing of the experimental work, the gained results were; 300 values for vibration level, 300 values for tool-flank wear (for each of Vbl and Vb2), and 300 values for each of seven surface roughness parameters (six vertical parameters; Ra, Rq, Rt, R3z, Rp, Rv and one horizontal parameter S). For the purpose of testing the developed neural networks, other four samples were machined with machining conditions which are different from that on which the neural networks were trained. Beside these four samples, six samples were selected from the above 300 samples to permit test sets used to check the ability of the developed neural networks to be considered as a generalized one. 3. RESULTS AND DISCUSSION 3.1. The Developed Neural Networks In order to develop the required neural networks, from all of the samples that were machined in the experimental work, samples were selected randomly as; 240 samples for training patterns, 54 samples for evaluation patterns (which were used as a training stoppage criterion [16]), and 10 samples were used as test patterns. Table 1 shows a sample of the results of the experimental work that were used to train, evaluate and test the developed neural networks. Actually, each row in the Table presents the results of one sample of a machined workpiece. Table 1. Sample of the results of the experimental work i
Machining conditions No. l 2 3 4 5 6 7 8 9 10
V --
f
d
r
m/min,
mm~-
mm
350 250 130 180 130 250 180 350 350 130
():'06 0.08 0.10 0.12 0.10 0.06 0.08 0.12 0.06 0.06
0.~ 0,4 0.5 0.4 0.5 0.4 0.5 0.8 0.5 0.8 0.5 1.2 0.5 1.2 0,5 1.2 0.8 0.4 0.8 0.8
mm
q
Surface roughness parameters 0tm.)
Vbl
V.L.
Vb2
ram.
dB
ram.
Ra
0.09 0.13 0.12 0.14 0.09 0.11 0.09 0.11 0.13 0.20
0.9'7 1.40 !.79 0.77 1.69 0.73 0.77 0.7! 0.99 0.99
Q.O0 0.11 0.11 0.14 0.00 0.11 0.09 0.10 0.11 0.18
109,0 ll0.0 108.5 109.0 109.0 109.0 108.0 109.5 !11.0 111.5
Rq
Rt
1.21 .6,70 1.71 10.1 2.15 10.4 1.03 5.50 2.10 12.9 0.95 7.60 1.00 7.20 0.90 5.10 1.23 6.20 1.30 9..00
Raz
P',r
4.50 3.90 5.40 5.50 7.60 ~6.20 4.10 ~3.30 6.80 6.40 3.20 4.80 3.70 3.20 3.10 .3"10 4.90 3.60 4.30 3.50
Rv
S
2.80 27.58 4.60 20.64 4120 22.09 2.20 29.22 6.60 26.35 2.80 28.18 4.00 21.10 2.00 26.65 2.60 20.67 5.50 27.24
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Fig. 1 shows the schematic diagram of the neural net for surface roughness prediction. The output is the surface roughness which is assessed by seven different parameters; Ra, Rq, Rt, R3z, Rp, Rv and S. For each parameter, ten neural networks were developed using two network structures (5 and 10 hidden units) with five different numbers of training samples (70, 100, 150, 200 and 240). Fig. 2 illustrates the neural net for vibration level prediction. Fifteen neural networks were developed using three network structures (5, 10 and 15 hidden units) with five different numbers of training samples (70, 100, 150, 200 and 240). Fig. 3 shows the neural net for tool-flank wear prediction. Fifteen neural networks were developed using three network structures (5, 10 and 15 hidden units) with five different numbers of training samples (70, 100, 150, 200 and 240). From the all developed neural networks, nine networks were selected as the best nets developed for prediction of 1~, Rq, Rt, R3z, Rp, Rv, S, V.L. and Vb2. The criterion of selecting the best net is the minimum root mean square of the differences between the measured and the predicted values obtained from the net (Es), applied for the test data sets. Table 2 shows the best neural network of every predicted output, each associated with the number of training samples, the number of evaluation sets, the number of training iterations and the test sets error (Es). From this Table, with respect to surface roughness parameters the prediction of which is the main goal of this work, it is found that the prediction of R, gives the smallest test sets error (0.023 ttm.). So, the P~ parameter can be considered as the best parameter for surface roughness prediction in the present work. Figs. 4 to 12 present the measured and predicted values of the nine best developed neural networks. From these figures it can be seen that, for the most of the developed nets, there is a small error between the measured and predicted values for both training and evaluation sets, while for the test sets there is a relatively larger error (from-54.59% to 57.52%). For the surface roughness patterns, these errors may be attributed to the randomness in the obtained results which usually comprise the random portion of the surface roughness assessment. However, it can be said that the best developed neural networks permit acceptable results for the prediction of the studied outputs irrespective of the existence of some errors which can be minimized by increasing the number of inputs to the networks.
Input Information
Input Layer Units
Hidden Layer Units I
Cutting speed Feed of cut Nose radius
Output Layer Unit
Output Information Workpiece surface roughness
Pre-tool-flank
Fig.l. Schematic diagram of the neural network for surface roughness prediction.
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Input Information
Input Layer Units
Hidden Layer Units
Cutting speed
Output Layer Unit
Feed
Output Information
Depth of cut [
Nose radius Pre-tool-flank wear
Fig.2. Schematic diagram of the neural network for vibration level prediction.
Input Information
Input Layer Units
Hidden Layer Units
I
Cutting speed
Feed
O,
._...._.__._.__..]_~~ Depth of cut [
Layer Unit
Output Information After-machining tool-flank wear
Nose radius
[ Machining time
[ Pre-tool-flank wear Vibration level
Fig.3. Schematic diagram of the neural network for tool-flank wear prediction. Table 2. The best developed neural networks
Output
Net structure
R. R~ R3z ..
R~ S V.L. Vb2
6-5-i 6-5-1 6-10-1 6-5'1 6-5-1 6-5-1 6-5-1 5-15-1 7-5-1 .,
,
. .
Number of training samples 1o0
Number of evaluation sets 25
~oo
25
150 150 100 100 70 240 70
35 35 25 25 15 54 15 ,..
Number of training iterations 8070 8620 1759 87790 133159 ..... 72119 1970 11538 98895 i
Test sets error (E,)
ii
. ,
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Fig. 5. M e a s u r e d vs P r e d i c t e d values of Rq.
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Fig. 6. M e a s u r e d vs P r e d i c t e d values of Rt.
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Fig. 7. M e a s u r e d vs P r e d i c t e d values of R3z.
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Fig. 8. M e a s u r e d vs P r e d i c t e d values of Rp.
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Fig. 9. M e a s u r e d vs P r e d i c t e d values of Rv.
12
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Current Advances in Mechanical Design and Production, MDP-7
70 ~. 60 E "-"50
116
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Fig. 10. Measured vs Predicted values of S.
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Fig. 11. Measured vs Predicted values of V.L.
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0.2
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Fig. 12. Measured vs Predicted values of Vbz. 3.2. The Continuous Prediction of Surface Roughness With the aid of the best obtained developed neural networks the surface roughness, the toolflank wear, and the vibration level, resulting from machining of a workpiece at certain machining conditions, can be predicted. Each one of the outputs of these networks may be obtained separately by preparing the inputs for the networks and getting the predicted results (direct prediction). One of the inputs of both the surface roughness and tool-flank wear networks is the vibration level. Therefore, before using both of these networks, the vibration level should be predicted. If the same cutting tool will be used to perform another operation, after the operation under study, the after-machining tool-flank wear must be predicted to complete the inputs for the prediction of the surface roughness of that operation. Therefore, the integration between the networks of the prediction of vibration level, surface roughness and tool-flank wear is deemed to be urgent in order to establish a system for continuous prediction of surface roughness. Fig. 13 shows the flowchart of such integrated system which is capable to predict the surface roughness continuously without the need to measure the vibration level or the tool-flank wear.
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Start ] 1~...
Yes
.
.
.
Pre-tool-flank wear=0
Enter the pre- / tool-flank wear "-T
.
h.~.
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[ Choose the machining conditions for finish turning ] Get the vibration level during the operation' (using the neural net of Fig. ,2.) Get the surface roughness (using the neural net of Fig. 1.) Get the alter-machining tool-flank wear (u (using the neural net of Fig. 3.) .
/
.
.
.
Print the surface / roughness
Yes
No
Yes
[ Set the pre-tool-flank wear--Vb2 I ,,
]
Fig. 13. Flowchart of integrated system for continuous prediction of surface roughness. A comparison between the direct and continuous systems for predicting the surface roughness parameters was made using the results of 30 samples selected randomly from the samples of the experimental work. For each case the error, which is the difference between predicted (obtained) value and the measured (nominal) values, was determined. Then, the mean value and the standard deviation (S.D.) for the samples were estimated and presented in Table 3. From this Table, it can be noticed that most of the mean values of errors for continuous prediction system are smaller than that of direct prediction system, and most of the values of standard deviation of errors are higher for continuous prediction. This may be attributed to the accumulation of the dispersed errors in the pre-predicted values of vibration level and toolflank wear. It is clear that, the continuous prediction, however, can offer a reasonable mean for prediction of surface roughness of any number of continuous machining operations without the need to measure the vibration level or the tool-flank wear.
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Table 3. Comparison between the errors resulting from direct and continuous predictions of surface roughness parameters
Roughness Parameter
!
R~
........
P~ R3z
P~ Rv
Direct prediction Error =predicted-measured Mean (~tm.) s. D. (jam.) 0.109 0.284 0.038 0.289 -I.105 2.716 0.126 1.046 -0.390 1.609 -0.207 1.004 0.841 4.22 i ,
. . . .
i
Continuous prediction' Error--predicted-measured Mean (~m.) ,S.D. (gm.) 0.105 0.297 0.030 0.293 -I.033 2.746 0.169 1.150 -0.316 !.654 -0.203 1.042 1.254 4.211 ....
i
3.3. Selection of the Appropriate Machining Conditions After the neural networks of surface roughness prediction have been constructed, it is desirable to perform the reverse process, i.e. selecting the appropriate machining conditions (cutting speed, feed, depth of cut and tool-nose radius) that lead to a desired value of workpiece surface roughness. Based on the best developed neural networks of the different outputs, the steps of the selection process are as follows (Fig. 14): - Enter the workpiece length, workpiece diameter, pre-tool-flank wear and the desired surface roughness parameter with its value (R). - Choose the machining conditions that are suitable for finish turning process on the used machine tool (from machining literature) to get the input patterns (N). - Present these chosen machining conditions and the pr-tool flank wear as inputs to the vibration level neural network, and get the vibration level. - Take the value of the predicted vibration level and feed it as the sixth input to the neural network for surface roughness prediction, and get the workpiece surface roughness (S.R.). - Determine the machining time and feed it to the neural network of tool-flank wear prediction, and get the after-machining tool-flank wear. - Choose the patterns that fulfill the condition; S.R. _
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(Start) ..._|
~!
nter the workpiece length, workpiece iameter, pre-tool-flank wear and the d e s i r e d ~ ufface roughness parameter with its value ( R ) . ~
I Cet tho number of input pattern, (N) ] .~-~_'-~ For (N) pattern [ Get the vibration level ]
i 8~t the ~ , ~ e roug~e~ (S.R.) ] 1'(~al'culate the machining time [ I Get the tool-flank wear ] No
[ Difference (R S.R.) [
,
[ Exclude this pattern [
Save the pattern and its results Yes No Choose the pattern(s) with min. machining time or rnin. tool wear. ~ s p l a y
the pattern which has the rnin. differencq.~ ~ ~ Yes
Fig. 14. Flowchart summarizing the selection procedure of machining conditions. From Table 4, it can be seen that there are differences (errors) between the obtained (measured) and the desired values of surface roughness parameters. These errors may be attributed to many interacted factors such as the errors existing in the networks used in the selection system of machining conditions, the random nature of surface roughness itself and the uncertainty of roughness measurement process.
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Table 4. The resulting errors due to the selected m a c h i n i n g conditions
Parameter Ra Rq R~
Desired value (lam) '0.80 1.50 2.50 0.90 2.00 3.00 5.00 10.00 13.00 .
R3z Rr Rv
3.50 7.00 9.00 3.00 5.00 7.00 2.00 4.50 7.00 24.00 29.50 55.00
.
.
.
.
.
.
.
.
.
.
.
.
,
.,,
.
S
The selected machining conditions .... Obtained Speed Feed Depth of Noseradius value ,,,,(rpm), (mm/rev) cut (ram) (mm) (lam) 1120 . 0.'12 .... 0 . 5 1.2' 0.96 1500 . 0.12 0.8 0.8 1.72 1500 0.12 0.5 0.4 2.39 !120 0.10 0.5 1.2 0.73 ....!.500 0.12 0.8 0.8 2.15 1500 0.12 0.5 0.4 2.87 1500 0.12 0.8 1.2 5.40 1500 0.12 0.5 0.8 9..5.0 ! 500 0.12 0.8 0.4 | 1.9 1500 0.12 0.5 1.2 3.20 150.0 0.12 0.5 0.8 7.10 1500 0.12 0.8 0.8 8.00 1500 0.12 0.8 1.2 3.30 1500 0.12 0.8 0.8 5.50 1500 0.12 0.5 0.4 8.00 1500 0.10 0.5 1.2 2.50 1500 0.12 0.5 0.8 4.80 1500 0.12 0.8 0.8 5.40 1500 0.12 0.8 1.2 23.06 1500 0.12 0.8 0.8 33.91 1500 0.12 0.5 0.4 66.67 .
.
.
.
.
.
.
.
.
.
.
%Relative error (obtaineddesired/desired) 20.00
_
14.67 i
. . . . . .
-4.40 -18.89 7.50 -4.33 8.00
-5.00 -8.46 -8.57 1.43
'-11.11 10.00 10.00 14.29 25.00 6.67 -22.86 -3.92 14.95 21.22
,
In case of precision dimensions in engineering the tolerance in 95% of all cases is less than 1% of the nominal (desired) dimension, but in the case of surface roughness a parameter tolerance of this order is absurd owing to the much less precise nature of the assessment. So, for example, the standard system of assigning the Ra values (N-system) established a built-in tolerance (the allowed relative error) o f - 2 5 % to +50% [17]. However, it can be shown from Table 4 that the relative errors of the obtained values of Ra don't exceed the range o f - 4 . 4 % to +20%. Moreover, the relative errors of the obtained values of the all investigated roughness parameters are within the range-22.86% to +25%. Therefore, it can be said that the proposed selection program offers, with acceptable results, a reliable system for selecting the most appropriate machining conditions needed to achieve the desired surface roughness. 4. C O N C L U S I O N S In this work the artificial neural networks technique, with back-propagation training routine, was used for the prediction of surface roughness, as well as the prediction of vibration level and the tool-flank wear in turning operation. The experimental results obtained at different machining conditions were used to train, evaluate and test the developed neural networks. Several attempts were made to obtain the best form of these networks. The results obtained lead to the following conclusions; The developed neural networks, with the best form, permit acceptable outputs for the prediction of the investigated surface roughness parameters, the vibration level and the tool-flank wear. The best developed neural networks have been integrated by a developed system which is capable to predict the surface roughness continuously without the need to measure the vibration level or the tool-flank wear. Based on the best developed neural networks a computer program has been designed, and considered as a reliable approach, for selecting the most appropriate machining conditions according to the desired surface roughness parameter and its value. .
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REFERENCES 1. Shiraishi, M. and Sato, S., "Dimensional and Surface Roughness Control in a Turning Operation", Trans. of ASME, J. of Eng. for Industry, Vol. 112, pp. 78-83, (1990). Mital, A. and Mehta, M., "Surface Finish Prediction Models For Fine Turning", Int. J. of Product. Research, vol. 26, No. 12, pp. 1861-1876, (1988). Fei, J. and Jawahir, I.S., "A Fuzzy Knowledge Based System for Predicting Surface Roughness in Finish Turning", 92 IEEE Int. conf. Fuzzy Sys. FUZZ-IEEE. Publ. by IEEE, (IEEE Cat n92Ch3073-4), PP. 899-902, (1992). EI-Baradie, M.A., "Surface Roughness Model for Turning Grey Cast Iron (154 BHN)", Proc. of the Institution of Mech. Engineers, Part B: J. of Eng. Manufact., Vol. 207, nB1, pp. 43-54, (1993). 5. Arsecularatne, J. A.; Fowle, R. F. and Mathew, P., "Prediction of Chip Flow Direction, Cutting Forces and Surface Roughness in Finish Turning", Trans. of ASME, J. of Manufact. Science and Eng., Vol. 120, pp. 1-12, (1998). 6. Chryssolouris, G. and Guillot, M., "A Comparison of Statistical and AI Approaches to the Selection of Process Parameters in Intelligent Machining", Trans. of ASME, J. of Eng. for Indust., Vol. 112, pp. 112-131, (1990). 7. Matsumura, T.; Obikawa, T.; Shirakashi, T. and Usui, E., "Autonomous Turning Operation Planning with Adaptive Prediction of Tool Wear and Surface Roughness", Journal of Manufact. Systems, Vol. 12/No. 3, pp. 253-262, (1993). 8. Darenfed, S.; Miao, S. and Boudreau, R., "Performance Criteria of Learning Networks: Application to Surface Roughness Modeling in Turning", Trans. of the Canadian Society for Mech. Eng., Vol. 18, No. 4, pp. 303-315, (1994). 9. Aboul-Nour, A.A.; S h o u t , S.N.; EI-Sonbaty, I. and Ata, M.M., "Prediction of Machining Performance For Milling Operation With Neural Network", Proc. of 6th Int. Conf. of Prod. Eng., Design and Control, pp. 427-435, (1997). 10. Fang, X.D.; and Yao, Y.L., "In-process Evaluation of the Overall Machining Performance in Finish Turning via Single Data Source", Trans. of ASME, J. of Manufact. Science and Eng., vol. 119, pp. 444-447, (1997). 11. Bonif~cio, M.E.R. and Diniz, A.E., "Correlating tool wear, tool life, surface roughness and tool vibration in finish turning with coated carbide tools", Wear, Vol. 173, n 1-2, pp. 137-144,(1994). 12. Park, J. and Ulsoy, A.G., "On-Line Flank Wear Estimation Using an Adaptive Observer and Computer Vision, Part 1: Theory", Trans. of ASME, J. of Eng. for Indust., Vol. 115, pp. 30-36, (1993). 13. Park, J. and Ulsoy, A.O.,"On-Line Flank Wear Estimation Using an Adaptive Observer and Computer Vision, Part 2: Experiment", Trans. of ASME, J. of Eng. for Indust., Vol. 115, pp. 37-43, (1993). 14. Rao, S.B., "Tool Wear Monitoring through the Dynamics of Stable Turning", Trans. of ASME, J. of Eng. for Indust., Vol. 108, pp. 183-190, (1986). 15. Teshima, T.; Shibasaka, T.; Takuma, M. and Yamamoto, A., "Estimation of CuRing Tool Life by Processing Tool Image Data with Neural Network", Annals of CIRP, Vol. 42/1, pp. 59-62, (1993). 16. Swingler, K., "Applying Neural Networks, a Practical Guide", Academic Press Inc., London, (1996). 17. Collett, C.V. and Hope, A.D., "Engineering Measurement", Pitman Books Limited, London, 2nd Edition, (1983). .
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ON THE ADJUSTMENT AND VALIDATION OF FINITE ELEMENT MODELS FOR HEMISPHERICAL CUP FORMING Wifi, A.S., t Ragab, M.S. 2, Hussein, A.A. 3 and Abdel-Hamid, A. 4
1'2professor, and 3Graduate Student, Mechanical Design and Production Dept, Facultyof Engineering, Cairo University, Giza 12316, Egypt. tProfessor, Currently on leave to Mechanical Engineering Dept., KSU, Riyadh, Saudi Arabia 4professor, The American University in Cairo, Egypt.
ABSTRACT The general purpose finite element code ABAQUS is used to analyze the process of deep drawing of hemispherical cups. Solid element models for axisymmetric elastoplastic large strain formulation with friction and contact boundary conditions are used. An experimental/computer-aided simplified inverse technique is developed to optimize the model parameters, namely, number of elements, element type, number of layers and generalized coefficient of friction, such that the difference between the numerically predicted and experimentally measured maximum drawing force is minimum. To validate the optimized model, a comparison between the results obtained from the adjusted model and those found experimentally is carded out with changing the blankholder force. Good agreements are obtained between numerical and experimental load-stroke relation and strain distributions at different blankholder force. KEYWORDS Finite Elements, Cup Forming, Model Optimization, Simplified Inverse Technique. 1. INTRODUCTION The last two decades have witnessed rapid development of computers and computational techniques. This motivated many research workers to attempt the direct simulation of the sheet metal stamping process especially by the finite element method. Today, the degree of advancement of this research indicates that this kind of simulation has become an efficient tool in the design of many industrial stamping operations. The objective of the simulation of deformation process is to determine forces and power required to complete the operation, as well as the displacements, strains and stresses at every point in the deformation region during processing such that the final shape of the product and its properties, residual stresses and formability limit can be predicted. A complete analysis of sheet metal forming problems requires solutions in both elastic and plastic regions. However, classical methods of solution, such as the uniform deformation energy method, slip line field and slab methods [1 ], usually consider rigid-plastic material and
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make many simplifying assumptions that limit their applicability. Accurate determination of the effects of various parameters involved in sheet metal forming process on the detailed metal flow became possible when the finite-element method (FEM) was introduced into the analysis of metal forming process. This is due to the ability of the FEM to deal with complicated geometry in one, two and three dimensions, sophisticated constitutive laws of the material, and to treat the boundary conditions effectively. By the FEM, it is possible to follow the deformation path and get complete information about stresses and strains, plastic zones and plastic flow during the deformation process [2,3]. The first finite element complete solutions of hemispherical stretch forming and deep drawing problems, taking into account the contact problem at blankholder, die and punch, were obtained by Wifi [2] in 1976, using elasto-plastic large strain formulation for axisymmetric solid elements. This was followed by Wang and Budiansky [4], who have used a nonlinear membrane theory and simulated the axisymmetric hemispherical punch stretching of sheet metals. A systematic survey of the developments and applications of the FEM in the field of material processing was given by Wifi [5], and a nice survey of the process modeling and simulation for sheet metal forming including stretch forming and deep drawing, was given by Lee et al. [6]. Recently, many analyses of these processes were carded out with emphasis on the investigation of the influence of important parameters such as blankholder control schemes, use of draw bead and the frictional contact problems encountered [7-11 ]. For any finite element model, there are many controlling variables that would affect the model performance, such as element type, number of elements, and frictional contact conditions. The output results of the finite element model are very sensitive for such variables. If the selection and/or adjustment of the model variables is correct, the obtained results will be realistic, which is the main concern of any successful simulation process. Surprisingly, this crucial issue was overlooked in many finite element analyses of sheet metal forming processes. Some researchers compared the output results of the finite element models with some existing experimental, analytical or numerical data, to validate their models; see for example, Saran et al. [ 10], Lee et al. [11 ], and Wifi et al. [9]. However, no systematic procedure was given for the adjustment of the models in these works. In the present study a computer-aided/experimental simplified inverse technique is developed to adjust the finite element model parameters, such that the difference AF between numerical and experimental maximum drawing force is minimized, where AF = [(Fcxp - Fnum)/Fexp]X100, F,um is the maximum drawing force obtained numerically for a given drawing conditions and Fexp is the maximum drawing force obtained experimentally for the same conditions [12]. The minimization process follows a simplified inverse technique where the results of various systematic simulations are compared to the experimental value, F~p, which is considered as a reference value. If AF is less than or equal to 5%. the adjusted value of the model variables are considered to be optimum [13]. The model is validated by comparing the numerically predicted drawing force and strain distributions with those obtained experimentally in ref [12]. 2. SIMULATION OF THE HEMISPHERICAL CUP FORMING PROCESS Figure 1 shows the main features of the process of deep drawing of a hemispherical cup, with an edge drawbead, as modeled by the finite element method (FEM). The following dimensions are used in the finite element analysis: Initial blank diameter, Db = 150 mm, Blank thickness, to = 1 mm, Punch radius = 50 mm, Die diameter = 102.25 mm, Drawbead
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Current Advances in Mechanical Design and Production, MDP. 7
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profile radius = 4mm, Blankholder edge radius = 4 mm, Drawbead height = 4 ram, and Clearance between blankholder and drawbead = 1 ram. At the start of the analysis, the blank is positioned precisely on top of the die, and the blankholder is precisely in touch with the top surface of the blank. The drawbead is positioned 0.18 mm below the bottom surface of the blank while the punch is positioned 100 mm above the top surface of the blank, Fig. 2. The problem is handled as an axisymmetric problem, therefore the nodes lying along the axis of symmetry move only along the axis. The die is fixed in all directions. The blankholder moves only in the vertical direction as well as the punch and the drawbead. The simulation is carried out in four steps. In the first step the blankholder is pushed into the blank with a prescribed displacement of-7.25 E-5 mm. This value is chosen by trial to obtain a reaction force approximately equal to the applied force on the blankholder. In the second step, the prescribed displacement boundary condition at the blankholder is relaxed and replaced by an applied force on the blankholder. This force is kept constant throughout the following steps. This means that the blankholder is allowed to move freely vertically under the applied blankholder force. During the third step the drawbead is moved towards the blank through a total distance of 4.18 mm. This step models the first drawing process which produces a dish. In the fourth step the punch is moved towards the blank through a total distance of 146 mm. The other rigid surfaces (die and drawbead) are fixed in position during this step. The step models the actual drawing process which produces a hemispherical cup, Fig. 2. Various types of axisymmetric elements were tried, and the number of elements as well as the number of layers (elements across sheet thickness) were changed to determine a proper finite element model of the blank. The following elements were considered, [ 14], 1- CAX4 4-noded bilinear axisymmetric solid element. 2- CAX4R 4-node bilinear, reduced integration axisymmetric solid element. 3- CAX4I 4-node bilinear, incompatible axisymmetric solid element. 4- CAX8 S-node biquadratic, axisymmetric solid element. 5- CAX8R 8-node biquadratic, reduced integration solid element. To study the effect of the number of elements on the results, the blank was modeled with different number of elements varying between 40 and 80 axisymmetric elements, arranged in one or two layers. Two materials were considered, namely, steel St37 and steel X5CrNi 189. The materials were in sheet metal form with a nominal initial thickness of 1 mm. The mechanical properties and anisotropy ratio for these materials were taken from reference [12] and are summarized in Table 1.
Table 1. Mechanical properties of the used materials (Fouad, [12]). Material
O'y ,
steel St37 Steel X5CrNi189
(3' u
MPa 271 245 ,
MPa 368 645
elongation 25.7 41.7
K MPa 644.6 1394.5
n-value 0.225 0.405
r-value 1.34 ' 1.21
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Current Advances in Mechanical Design and Production, MDP-7
3. DETERMINATION OF THE CONTROLLING VARIABLES OF THE FINITE ELEMENT MODEL BY A SIMPLIFIED INVERSE TECHNIQUE In all adjusting runs data from section 2 are used with the blankholder force of 134 kN for steel X5CrNi 189 and 69 kN for steel St37 which are the experimental values used by Fouad [12]. In the present study a total of 41 runs were carried out in order to adjust the model variables by the inverse technique, Figure 3.
Table 2. Effect of element type on the calculated maximum drawing force.
Element Type
Fnum
AF=[(Fexp-Fnum)/Fexp]x 100
CAX4 CAX4R CAX4I CAX8 CAX8R
217.508 182.610 219.680 225.200 217.000
5.4 20.6 4.4 2.08 5.65
,
,,
For steel XSCrNi189, Table 2 shows that the axisymmetric element CAX8 with 8 nodes and full integration seems to be the most suitable element for modeling of the present problem. This element shows the minimum difference between Fnum and Fexp, AF = 2.08%. Similar results were obtained for Steel St37. Figure 4 shows the effect of the coefficient of friction ~t on the percentage differences in drawing force, AF. The figure indicates that the optimal generalized coefficients of friction suitable for this model are ix = 0.22 for Steel St37 and la = 0.25 for steel X5CrNi189. These values of ~ minimize AF in the simulation process. It is also worthnoting that a value of ~t > 0.15 was commonly used in finite element calculations for sheet metal forming [9,10,15]. It should be emphasized that the coefficient of friction obtained by the above mentioned inverse approach is an average generalized coefficient of friction, that describes the overall process. In fact, there are experimental evidences I16-17], that the value of ~t changes from one deformation zone to the other and could be affected by the strain, strain rate and contact pressure which change during the process. careful analysis revealed that there is no significant differences between the predicted drawing forces when using one layer or two layers. Hence, one layer of elements is suitable to model the blank properly. No much difference in the predicted punch force when either 40 element or 80 elements are used. However, the 40 element model shows some oscillations towards the end of stroke. Nevertheless, the trend of this model seems to be correct, as seen from Fig. 5. Such oscillations are usually acceptable in finite element analysis [15]. Thus to keep the CPU time and hard disk storage within practical limits, it was decided to utilize the one layer, 40 elements CAX8 element model throughout the present study.
Current Advances In Mechanical Design and Production, MDP-7
472
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Figure 3 THE PLAN OF THE MODEL VARIABLES ADJUSTING RUNS.
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4. VALIDATION OF THE MODEL To validate the finite element model and the simplified inverse technique developed in the previous sections complete deep drawing processes for hemispherical cups made of steel St37 and steel X5CrNi 189 are analyzed. Figure 5 indicates that the predicted punch force agrees very well with the experimental values [12] throughout the whole drawing process, and not just at maximum punch force (which is the criterion used for the model adjustment). Note that in the example of steel XSCrNi189 a blankholder force of 90.8 kN is used, which is different from the one used in the previous section. This suggests that ~t = 0.25 is also valid under this blankholder force. Figure 6 shows the effect of changing the BHF on the maximum drawing force. The agreement between the finite element prediction and experimental results is obvious. As shown in Fig. 7, more thinning is obtained at higher BHF. However for St37 steel the maximum thinning is predicted relatively nearer to the drawbead, whereas for X5CrNil89 steel the maximum thinning is observed near to the punch zone, Figs. 8 and 9. For the values of the BHF considered the thinning of XSCrNi189 steel seems to be less affected by the change in BHF in comparison with St37 steel, because of the higher n-value of X5CrNil89 steel. Good agreement between experimental and numerical predictions can be seen in Figs. 8 and 9. In fact, the finite element results, with the adjusted values of the variables, could represent the complexity of the strain distributions at different zones of the cups, which proves the validity of the simplified inverse technique developed in the study. 5. CONCLUSIONS A simplified inverse technique has been developed to adjust the finite element model parameters, namely number of elements, number of layers, element type and generalized coefficient of friction. In this technique, the minimized difference between the numerical and experimental maximum drawing force was obtained through an experimental/computer-aided optimization by simulation approach. The optimized model was validated by comparing the predicted drawing force at different punch stroke, and the strain distributions with those obtained experimentally under different blank holder forces. The simplified inverse technique has shown that the adjusted model with one layer of 40 elements of 8 noded axisymmetric solid element (CAX8) was good enough to simulate the blank during the cup forming process. The global generalized coefficient of friction has been found to be 0.25 and 0.22 for steel X5CrNi 189 and St37 respectively in case of conventionally lubricated blank and tool. It should be emphasized that the optimal values of the model parameters could be affected by the finite element formulation and the numerical techniques used to handle contact boundary conditions. However, the presented approach is general and the obtained results could be used as guidance.
Current Advances in Mechanical Design and Production, MDP- 7
474
250
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i Mat. X5CrNi 189 2110 84 - B H F = 90.8 K N Coefficient of Friction 0.25
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Current Advances in Mechanical Design and Production, MDP-7
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REFERENCES 1- Hosford, W.F. and Caddell, R.M., "Metal Forming Mechanics and Metallurgy", 2nd edition, Prentice-Hall, Inc., N.J., USA, (1993). 2- Wifi, A.S., "An Incremental Complete Solution of the Stretch Forming and Deep Drawing of A Circular Blank Using A Hemispherical Punch", Int. J. Mech. Science, Pergamon Press, Vol. 18, pp. 23-31, (1976). 3- Kobayashi, S., "A Review on the Finite Element Method and Metal Forming Process Modelling", J. Applied Metal Working, ASM, Vol. 2, No. 3, pp. 163-169, (1982). 4- Wang, N.M., and Budiansky, B., "Analysis of Sheet Metal Stamping by the Finite Element Method", J. Applied Mechanics, Vol. 45, pp. 73-81, March, (1978). 5- Wifi, A.S., "Finite Element Correction Matrices in Metal Forming Analysis (with Application to Hydrostatic Bulging of Circular Sheet)", Int. J. Mech. Sci., Vol. 24, pp. 393-406, (1982). 6- Lee, D., Majless, S.A. and Vogel, J.M., "Process Modelling and Simulation for Sheet Forming", Metals Handbook, 9th edition, ASM, Vol. 14, pp. 911-927, (1988). 7- Karima, M., and Tse, W., "Effect of Process and Material Windows on Formability of Drawbead Controlled Geometries", Ontario Centre for Advanced Manufacturing Report, Cambridge, Ontario, pp. 153-168, (1988). 8- Chabrand, P. and Pinto, Y., "Numerical Model for Blankholders with Drawbead", MECAMAT 91, Eds. Teodosio et al., Balkema, Rotterdam, pp. 379-385, (1993). 9- Wifi, A.S., Halim, S.P. and Abdel-Hamid, A., "A Finite Element Analysis of the Deep Drawing Process Using a Vibrating Blankholder", AMPT 97, Portugal, (I 997). 10- Saran, M.J., Schedin, E., Simuelsson, A., Melander, A., and Gustatsson, C., "Numerical and Experimental Investigation of Deep Drawing of Metal Sheets", J. Engng. Ind., Trans. ASME, Vol. 112, pp. 272-277, August, (1990). 11- Lee, J.K., Cho, U.Y., Hambreeht, J. and Choudhry, S., "Recent Advances in Sheet Metal Forming Analysis", Advances in Finite Deformation Problems in Materials Processing and Structures, ASME, Vol. 125, pp. 119-130, (1991). 12- Fouad, M.A., "A Study on the Deep Drawing of Hemispherical Parts", MSc. Thesis, Cairo University, (1996). 13- Zabaras, N. and Badrinarayanan, S., "Inverse Problems and Techniques in Metal Forming Processes", Inverse Problems in Engineering: Theory and Practice", ASME, pp. 65-76, (1993). 14- ABAQUS Version 5.5: User's Examples and Theory Manual, Hibbit, Karlsson and Sorensen, Inc. Rhode Island, USA (1995). 15- Marques, B.J.M. and Baptista, R.M.J.M., "Theoretical and Experimental Analysis of Axisymmetric Deep Drawing", J. Mater. Proc. Tech., Vol. 24, pp. 53-63, (1990). 16- Nine, H.D., "The Applicability of Coulomb's Friction Law to Drawbeads in Sheet Metal Forming", J. Appl. Met. Work., ASM, Vol. 2, No. 3,200-210, (1982). 17- Wilson, W.R.D., "Tribology in Cold Metal Forming", J. Manufacturing Science and Engng., Vol. 119, pp. 695-698, November, (1997).
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
477
DEVELOPMENT OF SPOT-WELD BONDED LOW CARBON STEEL DAMPING SHEETS
Darwish, S.M." and Ghanya, A.'" 'Mech. Eng. Dept., King Saud University, Saudi Arabia *'Mining and Metal. Dept., Assiut University, Assiut, Egypt
ABSTRACT It is well established that weld-bonded joints enhance the fatigue as well as the corrosion resistance of spot welded joints. In the present work, both techniques of weldbonding, namely, Weld-through and Flow-in have been evaluated, during the development of low carbon steel damping sheets. Metallic fillers, at optimum percentages were also tried in order to enhance the electrical conductivity of the adhesive material, while preserving the joint mechanical s,rength. The present work demonstrated that the Flow-in technique is far more better when compared with the Weld-through technique, from the technological and economical view points. KEY WORDS Spot Weld, Weldbond, Photomicrograph.
Weld-through,
Flow-in,
Micro-hardness,
Metallic
fillers,
I. INTRODUCTION Bonding of metals is becoming increasingly important, both in absolute terms and relative to mechanical fastening. Applications of adhesive bonding are found in the assembly of many products including aircraft, cars, trucks, and office furniture. This is because adhesive bonding offers more uniform distribution of stresses, increased fatigue life, weight savings, prevention or reduction of corrosion between dissimilar materials, added to the ability to join small and delicate parts [1-3]. In machine tool manufacturing, adhesives are used for cementing gear wheels, joining elements of hydrostatic bearings and transmissions with load beating surfaces [ 1-3]. In electrical and electronic industry adhesives are used in a variety of ways. These range from holding microcircuits to bonding coils in mammoth electrical generators. In addition to mechanical fastening adhesives are required in electrical applications to seal and protect substrate and conduct or insulate heat and electricity [4]. Bonded structures can be of two types based on either purely adhesive or on adhesive/ mechanical connections. The purely adhesive connections include shaft- pinion joints, laminated metal-metal joints and three layer honeycomb structures. The bonded mechanical types include bonded-welded, bonded- riveted and bonded screwed connections [3-10]. The combined connections (bonded-welded, bonded-riveted and bonded-screwed) ensure high
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Current Advances in Mechanical Design and Production, MDP- 7
fatigue strength of the structures. It has been demonstrated that the application of the adhesive material has improved the fatigue life from 911 cycles for spot-welded, to 15360 for weld bonded specimen [ 11] and are extremely economical, because they do not require any fixtures for use during the cementing process. Damping sheets (weld-bonding) composed of two metallic sheets with a polymer resin layer between them, have been developed in order to prevent noise and vibrations from automobile transmissions, railway carriages, etc. Weld-bonded joints were first developed and used in the USSR in airplanes of the type AN-24 [12]. The weld-bondingprocess, is essentially spotresistance welding of parts that subsequently have their overlapping areas adhesive-bonded. The Soviet Union initially perfected this technology which is known there as "glue welding". The approach was a "flow-in" method (Fig. 1(a)), whereby parts were welded together first then the adhesive was flowed into the joint. A low-viscosity adhesive is used which penetrated the overlap joint by capillary action and was subsequently cured. The technique used in the United States is the weld-through method (Fig. l(b)), whereby the adhesive is applied to the parts to be cured, spot welded and subsequently cured. In comparison with mechanical fasteners, weld-bonding offers the following benefits [ 10-12]: 1- Reduced manufacturing costs and adaptability to mechanization.
234567-
High static strength. Improved fatigue strength. Improved corrosion resistance. Eliminates sealing operations. Eliminates shop noises rivets. High peel strength.
Adhesives normally function as good insulators, so improving the electrical properties for weld bonding application is a must. Metallic fillers have been used for improving the electrical properties of structural epoxy resin adhesives, where nine filler materials having different particle sizes were used [6]. These fillers were classified into three groups. l.Powders: graphite (5-10 I~m), iron (147 ~tm) cast iron (-~ 100 ~tm), aluminium (-~ 160 ~tm), bronze (~- 100 ~tm), nickel (-~ 45 pro) and copper (-~ 45 ~tm). 2.Fibers: chopped copper wires having a diameter of 0.077 mm and lengths ranging from 0.04 - 0.08 ram. 3. Networks: brass (60/40) having a mesh number of 0.123. The mechanical strength of the different mixtures was assessed using a single lap specimen according to BS 5350 [13]. The mixtures giving the maximum shear strengths were chosen for enhancing the electrical resistivity [6]. The degree of acceptance of weldbond applications has been increasing, as the process has been understood and its mechanical properties developed [10-12]. So, the aim of the present work is to evaluate both techniques of weldbonding; namely flow-in and weld-through from both economical and technological view points.
Current Advances in Mechanical Design and Production, MDP- 7
2. IMPROVING THE RESIN ADHESIVE
479
ELECTRICAL PROPERTIES OF STRUCTURAL EPOXY
Effective improvements in the electrical conductivity of structural epoxy resin adhesives were found to be achieved when fillers (powder) were mixed with the adhesive [6]. Additions of 10% grapfhites 15% aluminium or 20% copper powder by weight markedly improve electrical conductivity [6]. Luckely, the shear strengths of the epoxy resin adhesive was found to attain its maximum at these percentage additives (10% graphite, 15% aluminium and 20% copper) [61. 3. RESISTANCE SPOT WELDING Resistance spot welding, is a process in which faying surfaces are joined in one or more spots due to the heat generated by resistance to the flow of electric current, through workpieces that are hold together. The contacting surfaces in the region of current concentration are heated by a short time pulse of low voltage, high amperage current to form a fused nugget of weld metal [ 14-26]. The size and shape of the individually formed welds are limited primarily by the size and contour of the electrode forces. In the present work, the welding electrodes were made of a special copper alloy and were sized to give 6 mm nugget. In the present work, the welding current, electrode force, as well as the weld time were adjusted to the level that ensures test specimens free of surface fusion, deep electrode indentation, electrode deposits, and abnormal discoloration around the weld (welding current = 6.6 K.A., welding force = 920 Newtons, welding time - 52 cycles (The cycle is one sixtieth of a second in 60 Hz power system)). 4. EXPERIMENTAL WORK
4.1 Weld Bond Test Specimens The configuration and dimensions of the weldbonding specimens used (B.S. 5350) throughout the present work are shown in Fig. 2. The specimens were cut from a 1.5 mm thick low carbon steel sheets. After cutting they were subjected to a stress relief anneal at 650~ for 30 minutes. The interface of which was prepared by grinding with silicon carbide paper up to 220 grit, then degreased, dried and kept in a desiccator, ready for weld-bonding. The structural epoxy resin adhesive (Araldite 2004) used in the present work was a room temperature cure, two-part epoxy resin with a mix ratio of 100 by weight of resin to 40 parts by weight of hardener. This adhesive had a pot life of 30 min. at 23 ~ and a cure cycle of 2428 hours at 24 ~ For weldbond specimens (flow-in technique) the adhesive was spread over the overlap length, with the thickness recommended by the manufacturer (0.5 ram). Afterwards the specimens were immediately spot welded. As for the weld-through weldbond specimens, the adhesive material plain or filled (with 10% graphite, 15% aluminium or 20% copper) was applied over the overlap length, then the specimens were cured at 24 ~ for 48 hours, and were spot-welded thereafter. It is worth mentioning that none of the specimens (having plain or filled adhesive) could be welded after the adhesive material has been cured. However, only the specimens containing 15% AI can be welded, after being heated up to 100 ~
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Current Advances in Mechanical Design and Production, MDP-7
4.2 Metallurgical Examination After welding is completed, the weld-bonded specimens were cut into two halves, and were prepared for metalographic examination (according to standard procedures) [11]. The photomicrographs were conducted at (37.5 X) for the welding nugget, and at (300 X) for the micro-structure of each zone.
4.3 Micro-hardness Measurements Vickers micro-hardness measurements were conducted on a Buehler Micromet microhardness tester. The micro-hardness measurements were taken along the major and minor axes of the weld nugget (from center and outwards). 5. RESULTS AND DISCUSSION
5.1 Microstructure of Reference Spot Welded Joint Fig. 3, shows a polished and etched cross section of a spot welded 1.5 mm low carbon steel sheets. The figure shows different structural regions as unaffected, transition, refined, coarsened, and fused zones. The unaffected zone, represents the typical grain structure of the parent low carbon steel sheet. This has not been heated enough to reach the critical zone. The transient zone was subjected to a temperature between AI and A3 transformation temperatures, where incomplete transformation takes place. The refined zone is heated to just above A3, where the fine austenite transforms to fine ferrite and pearlite. At higher temperatures (above A3), and at adjacent zone to melting, coarse austenitr is produced and consequently transformed to coarse structure which yields coarsened zone. At temperatures above the solidus, at the laying surface actual melting of the parent metal has taken place and resolidified rapidly epitaxially which lead to perfect coalescence. In summary, the structure of spot welded low carbon steel has the following features: 1. Columnar dendritic structure in the fusion zone. 2. Heat affected zone showing gradual transition from a coarse overheated structure through a normalized region (refined), to the original structure of the unaffected base metal. 3.Narrow ferritic zone in the interface between the overheated and unaffected zones, which is not always well defined. The above principles are demonstrated in Fig. 3.
5.2 Effect of Welding Parameters on Weld Quality Nugget formation depends on many variables, but welding heat input is probably the most effective parameter. The available spot welding machine has three levels of current density along with two voltage levels (220 V, 380 V). It is worth mentioning that with 380 V almost all of the specimens demonstrated expulsion (surface fusion), so this voltage was discarded. As for the 220 V with different current densities, the weld nugget quality appeared to improve with increasing current density (almost all specimens produced were free of surface fusion, deep electrode indentation, and electrode deposits). As a result of these preliminary tests it was decided to work with 220 V and the maximum available current density. Table 1, shows the microhardness measurements associated with each condition.
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Current Advances in Mechanical Design and Production, MDP-7
5.3 Effect of Welding Technique on Weld Bond Quality The two techniques of weld-bonding, namely, weld-through and flow-in have been conducted and studied. In the flow-in technique the adhesive material was uniformly spread over the overlap length, afterwards they were spot welded immediately (before the adhesive layer has been cured). In the weld through technique the adhesive material plain or filled [6] (10% graphite, 15% aluminium or 20% copper) was spread on the overlap length then the bonded specimens were left for curing at 24 ~ for 48 hours, and were subjected to spot welding after curing. It is worth noting that none of the specimens (having plain or filled adhesive) could be spot welded due to the higher resistivity of the adhesive material. In order to reduce the electrical resistivity of the bonded specimens, these specimens were heated-up in an oven until they reached the temperature of 100 ~ (the maximum working temperature of the Expoxy resin used is 120 ~ However, only the specimens containing 15% aluminium could be spot welded. Figs. 4, 5 show the micrograph of a two weld bond specimens manufactured using the different techniques of weld-bonding namely, weld-through and flow-in. The figures show that with weld through technique the weld nugget pronounced an irregular shape, especially at the ends, also the heat affected zone is not well defined. The flow-in weld bond specimens, shows a uniform elliptic shape, clear and distinguished micro-structure and a more defined heat affected zone. Also, a more complete dissolution of thin film of adhesive absorbed by the weld. As for the micro-hardness of the weld nugget, the weld-through weldbond (15% aluminium) specimens demonstrated 42% reduction in the maximum microhardness, when compared with spot weld specimen. Table 1 Microhardness measurements m
Condition of Spot Weld
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Fig. 6, shows different structural regions of a polished and etched weldbond (flow-in) having 10% carbon powder mixed with adhesive. In order to study the effect of filler type on weldbond quality, two flow-in weldbond specimens filled with metallic powder were compared. The former contains 10% by weight carbon powder [6], while the latter contains 20% by weight Copper [6]. Both specimens demonstrate a well defined nugget with pronounced heat affected zone. This indicates that the thin film of adhesive material in both specimens was dissolved in molten metal, so welding is complete. The only difference that can be realized that the heat affected zone is more wider in specimens having 10% carbon. This may be attributed to the lower conductivity of the first specimen (10% C), when compared with the second specimen (20% C).
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Current Advances in Mechanical Design and Production, MDP-7
6. CONCLUSIONS The flow-in weldbond technique is much easier to implement when compared with the weldthrough technique. As it better preserves the microstructure and hardness of the produced weldments, when compared with the weld-through. From the economical view point, the flow-in weld bond technology is much cheaper when compared with weld through technology, as the weld-through technology the welded specimens (only filled with 15% Al metallic powder) must be heated in an oven to lower its electrical resistivity, besides it should be left to cure for 48 hours before welding. REFERENCES 1. Karashor, D.A., "The use of adhesives in aircraft construction", Vestnik Mashinostroeniya, Vol. 58, pp. 50-53, 1987. 2. Brewis, D.M., "Critical assessment of factors affecting bonding of metals", Mat. Sc. and Tech., August, Vol. 2, pp. 761-767, 1986. 3. Apartseva, E.L., "The use of adhesives in mechanical engineering", Vestnik Mashinstroeniya, Vol. 58, pp. 47-50, 1978. 4. Darwish, S.M., Niazi, A., Ghania, A. and Kassem, M.E., "Formulation effects on some properties of structural epoxy resin adhesives", 3rd Applied Mechanical Engineering Conf., Military Technical College, pp. 147-155, Egypt, 1988. 5. Kilik, S., Davies, R. and Darwish, S.M., "Thermal conductivity of epoxy resin adhesives", Int. J. of Adhesion and Adhesives, Vol. 9, No. 4, pp. 219-223, 1989. 6. Darwish, S.M., Niazi, A., Gahanya, A. and Kassem, M.E., "Improving electrical conductivity of structural epoxy resin adhesives", Int. J. of Adhesion and Adhesives, England, Vol. 1I, No. 1, pp. 37-41, 1990. 7. Darwish, S.M., Azaym, K.M. and Sadek, M.M., "Design philosophy of a bonded gear box", Int. J. of Mach. Tools and Manufact, England, Vol. 31, No.4, pp. 625-631, 1991. 8. Darwish, S.M., Azaym, K.M. and Sadek, M.M., "Dynamic characteristics of industrial double containment joints", Int. Conf. on CAPE, Edinburgh, pp. 45-49, 1989. 9. Darwish, S.M., AI Abbas, R.K. and Sadek, M.M., "Design rationale of industrial containment joints", Int. J. of Adhesion and Adhesives, Vol. 11, No. 2, pp. 65-70, 1991. 10. Darwish, S.M., Niazi, A. and Ghanya, A., "Phase stability of Duralumin machined with bonded and brazed metal curing tools", Accepted for publication in Int. J. of Mach. Tools and Manufact., Vol. 32, No. 4, pp. 593-600, England, 1992. 11. Darwish, S.M., Soliman, M.S., AI-Fahead, A.M., "Manufacturing and characteristics of brass damping sheets", J. of Materials Processing Technology, Vol. 79, pp. 66-71, 1998. 12. Schwartz, M.M., "Metals joining manual book", McGraw-Hill, pp. 8-1, 8-32, 1979. 13. "British standard of testing adhesives" B.S. 5350, 1978. 14. Wu, K.C., "Resistance spot welding of high contact resistance for weld-bonding", Weldbonding Journal, pp. 436-443, Dec, 1975. 15. Roest, C.A., Rager, D.D., "Resistance welding parameter profile for spot welding aluminium", Welding Journal, pp. 529-538, Dec., 1974. 16. Andrews, D.R., and Broomhead, H., "Quality assurance for resistance spot welding", Welding Journal, pp. 431-435, 1975. 17. Sawhill, J.M., Watanabe, H. and Metchell, J.W., "Spot weldability of Mn-Mo-Cb, V-N and SAE 1008 steels", Welding Journal, pp. 217-224, 1977. 18. Sawhill, J.M. and Baker, J.C., "Spot weldability of High-strength sheet steels", Welding Journal, Jan. 1990.
Current Advances in Mechanical Design and Production, MDP-7
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19. Diccknson, D.W. et al., "Characterization of spot welding behaviour by dynamic electrical parameter monitoring", Welding Journal, pp. 170-176, 1980. 20. Kaiseret, J.G., "The effect of electrical resistance on nugget formation during spot welding", Welding Journal, pp. 167-174, June 1982. 21. Kimchi, M., "Spot weld properties when welding with expulsion, A comparative study", Welding Journal, pp. 58-63, Feb., 1984. 22. Nied, H.A., "The finite element modeling of the resistance spot welding process", Welding Journal, pp. 123-132, April, 1984. 23. Cohen, R.L. and West, K.W., "Aluminium spot weld strength determined from electrical measurements", Welding Journal, pp. 37-41, June 1985. 24. Aidun, D.K., and Bennett, R.W., "Effect of resistance welding variables on the strength of spot welded 6061 T6 Aluminium alloy", Welding Journal, pp. 15-26, Dec. 1985. 25. Gould, J.E., "An examination of nugget development during spot welding, using both experimental and analytical techniques", Welding Journal, June, 1987. 26. Alcini, W.V., "A measurement window into spot welding", Welding Journal, Feb., 1990.
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Current Advances in Mechanical Design and Production, MDP- 7
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Current Advances in Mechanical Design and Production, MDP-7
485
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Current Advances in Mechanical Design and Production, MDP- 7
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488 Current Advances in Mechanical Design and Production, MDP-7
Section V DESIGN AND T R I B O L O G Y
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Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
491
CONTRIBUTION OF CAD-CAM AND REVERSE ENGINEERING TECHNOLOGY TO THE BIOMEDICAL FIELD Hosni, Y.A.
Martin Marietta Distinguished Professor of Engineering Industrial Engineering and Management Systems University of Central Florida Orlando, FL 32816 - USA e-mail: yhosni@mail,ucf.edu
ABSTRACT Through a combination of CAD/CAM, Reverse engineering (RE) technologies such as Computed Tomography (CT) scan or Magnetic Resonance Imaging (MRI), and Stereolithography; physical models of the human anatomy can be produced directly from the scan data. The models are used not only in designing and producing surgical tools and equipment, but also in diagnosis, aiding in pre-surgical planning and rehearsing of complex surgical procedures. In this paper we provide an overview of the engineering technologies which make these applications feasible as well as recent advances in this field. We describe the different methods of integrating CT or MRI with image processing and rapid prototyping technologies in producing stereolithographic models. We provide results of ongoing research that promises to have a great impact on the Biomedical field. KEYWORDS Computer Tomography, CT. MRI, CAD/CAM Stereolithography; Medical Modeling; Biomedical Engineering
Rapid Prototyping;
1. INTRODUCTION Physicians in general, and surgeons in particular often seek better methods to optimize treatment of their patients. From better diagnostics to optimal surgery, doctors use all kind of "engineering" methods and tools. Such tools may include X-ray pictures for better diagnostics to surgery planning. One such method traditionally used by surgeons, who perform complex operations, is through rehearsing and practice operations on physical models. Typically, models are constructed to visualize anatomy, design custom implants, and to use as a preoperative planning and practice tool. Through the models as visual and tactical aids, the probability of costly mistakes in the operating room is minimized. By providing surgeons with a physical model of the anatomy of the patient on whom they will operate, a more accurate planning of the surgical procedure and better assessment of equipment requirements are possible than with imaging alone. In a survey conducted among 25 surgeons to assess the value of physical models to physicians in supporting operations 94% of the surgeon participants assessed the use of physical models as
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either essential, very useful or useful as a diagnostic or therapeutic method for surgery [1 ]. Beyond their use in visualizing complex geometry, advantages lay in the ability to manipulate the models with surgical instruments. Advances in the last few years in the field of rapid prototyping (RP) enabled the physical modeling of complex designs. A logical extension of RP is in the field of medical modeling. With the invention and the improvements in the medical imaging, it is now feasible more than ever to produce a physical model "directly" from Computed Tomography (CT) scan or Magnetic Resonance Image (MRI)with great accuracy. While the technology is still in its developing stage, compared to other improvements in the medical environment, the technical challenges and the pace by which such technology is advancing proves very promising. In this paper we provide an overview of the engineering technologies which make these applications feasible, specifically, the integration of CAD/CAM with RE, image processing and Rapid Prototyping (RP) in the development of accurate anatomical physical models. We introduce the results of two experiments conducted at the University of Central Florida Design and Rapid Prototyping Lab (UCF-DRPL). The first experiment was aimed at building a customized implants - Knee- directly from scanned data. The second experiment was aimed at modeling "deformed" spines of two patients. The models were used for diagnosis and to help the surgeon planning procedures to correct the spinal deformities. We start with brief summary of the technology of RP, and that of Medical imaging. We then describe the techniques used in converting the "image to a solid". We conclude with an assessment of the technology and its potential. Due to the limitation of this paper, we limit ourselves to the techniques and methods behind the technology specifically when applied to the building of human parts from scanned data. The results of the two experiments demonstrate the application of the integrated technology. 2. THE TECHNOLOGY
2.1 Rapid Prototyping (RP) RP is the process of converting 3D CAD files into physical models "rapidly". There are several techniques for the conversion process and all are based on layered manufacturing. The idea behind RP is to convert the 3D CAD file into 2 dimensional cross-sections, by slicing the solid model of the object being designed/built. To enable the slicing procedure, a triangulated file (refer to as STL file) has to be generated of the 3D-model. A surface in the STL file is a mesh created by triangular elements. This process is done automatically via software and the STL-format is a standard accepted by all RP techniques. Depending on the object design, its orientation, and the RP technology; support structure is added to support overhang and to avoid sagging during the build up. Depending on the technology used, the 2D files are used to guide a laser beam (or depositing/ curing head)which either"cure/ solidify" liquid photopolymer resin or deposit material/cut sheets into the equivalent solid layers. By building layer by layer on top of each other, a physical model for the 3D object is obtained (grown). The main advantage of the Stereolithography technique (SLA) regarding its application in the medical field, is that any object regardless of its complexity, including free form human body pans, can be built automatically from its "CAD" file without the need for tools or manual interference. In addition the accuracy obtained can be very high. The drawback of using stereolithography, is that the part has to be built from a special medical graded resin, to be brought in to the operating theater. Even though medical graded resin is used, medical models can only be used as a reference during the surgery and can not be used as implants. Fig. 1. shows the procedure of the SLA machine, which uses resin in building the
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objects. The figure shows a schematic of building a skull in which layers are cured on top of each other to form the "solids" of the skull. The laser beam receives its path instructions for each slice/layer from the pre-processed (sliced) object file. Layers are built bottom up, where a perforated tray is dipped one layer at a time allowing a new layer of resin on top of a previous layer, as shown in Fig. 2. At the end of the process the entire skull will be under resin in a semi-cured status. The base tray then will be elevated exposing the object and the curing is completed using a curing oven with ultraviolet light.
Fig. 2. Building the object
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2.2 Medical Imaging (MI)
MI has been used for years for medical analysis and diagnosis. Several methods can be used. The most commonly used techniques are the Computed Tomography (CT) and the Magnetic Resonance Imaging (MRI). Either of the techniques provides cross sectional images of a scanned part of the human body. The main difference is that the CT scanner uses radiation in the process while MRI does not. The quality of the finished model totally depends on the accuracy of the scanning machine and the resolution of the data. Resolution can be increased by decreasing the scan distance, which produces more slices along the same scanned region. The longer scanning period required for a high-resolution scan, however, must be weighted against increasing the patient's exposure to radiation, scan time and cost, and patient discomfort. The new spiral CT-scan technology allows faster acquisition of smaller scan distances compared to traditional scanners that must translate the patient for each transverse section. [2]. In either of the techniques, the output of the scanning process is a set of crosssectional data images. CT-data is most suitable for modeling bone structures and MRI-data is best suited for modeling of soft tissues. Fig. 3. is a typical CT scan process while Fig. 4. Shows the outcome of the process. .
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Fig. 4. Typical outcome of the scan process.
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3. CONVERSION AND INTERFACE There are five steps for interfacing between CT/ MRI equipment and rapid prototyping equipment. (1)Reading the scanned data from the appropriate medium. Typically, the CT/MRI equipment can supply scanned data on magnetic tape or optical disk. (2) Converting the data into a manipulative format. In this stage, the CT/MRI format is translated into an image format specific for the conversion software. Since CT scanners' produce a 12-bit gray scale image, and most computer system has a standard 8 bit displays, truncation of data normally occurs. Image digitization, or segmentation is the process in which artifacts are removed and only useful portions of the scan are retained. To isolate the parts to be built, thresholding of the grayscale is used, since the data is represented in pixels of grayscale density representing the tissues or the bone. The resolution of CT-data is usually 512x512, so planar interpolation is used to smoothen the selected areas to better represent the actual geometry. The output can be produced as 2 dimensional cross sections with a thickness equal to the scan distance or as a contour file. Fig. 5. Shows an example of such process.
Fig. 5. Contour identification.
Fig. 6. The solid model of a hip joint.
(3)Creation of 3D-model. To generate a 3D-model from the 2 dimensional scan images, region growing is used to combine all cross sections of the selected object. This procedure will search all images and select the area of each picture, belonging to that particular bone or tissue. Several iterations of this procedure might be conducted with different thresholds, to obtain desired accuracy of the model. By repeating the region growing process, several parts can be chosen and will be represented by different colors as shown in Fig. 6. Since each cross section has a certain thickness determined by the scan distance, interpolation in the Z-direction is necessary to smoothen the parts and prevent stair stepping surfaces. If it is desired to build several part as one unit, support structure can be added where no physical connection exist. When the 3D-model or solid model is obtained, an stl-file used by the RP-machine will be exported. Fig. 6 shows the solid model of a hip joint. In certain cases, it may be of advantage to convert the CT data contours directly into an stl-file.
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(4) Depending on the part being built, various amount of support structure has to be generated for the part. Typically the structure is composed of thin crossing webs, perforated enough to allow drainage of the resin, and strong enough to support the part during the build. In addition, the structure must be distinguishable from other elements of the body part being built to allow its removal upon the resin curing. (5) Once an stl-file is ready (or contour data), it can be transferred directly to the RP machine where it can be built as a regular part. Recent advances in RP materials enable coloring of selected potion of the physical model, through the application of multiple curing for parts that need coloring. Such tool is of great value for surgeons who may be able to distinguish between parts requiring removal and other parts. Fig. 7. are typical outcomes of the RP process in medical applications.
Fig. 7. Typical outcomes of the RIP process for medical applications. 4. EXPERIMENTS CONDUCTED AT UCF 4.1 The Knee Implant Experiment The procedure described above was applied as an experiment aimed at studying the feasibility of customizing a knee implant from CT data and to carry the necessary biomechanical analysis, which would enable prototyping of a customized knee implant. Fig. 8. shows the elements of the Knee implants.
Fig. 8. Elements of the Knee implant
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The reason for customizing a knee implant is that the current procedure uses standard sizes of implants, which cause the implant to loosen over time due to lack of perfect fitting. Recent studies proves that a customized implant can last for as long as 20 years, which is twice the time for standard implants. The experiment proved the feasibility of using 3D-models created from CT-data to design a customized knee implant with perfect fit. Once the CT are converted to solid model, the model was imported to a standard CAD/CAM software package, where the knee implant components were" custom" designed. Fig. 9. shows the solid model for the femur and the tibia.
Fig. 11. Customization of the Femur and Tibia
Fig. 12. The designed components
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Stress analysis of the knee components was conducted at different angles, positions, and load distribution that can vary due to different activities such as walking or running. For example, Fig. 10 . Shows Stress Distribution on the Femoral Component at 3 Times Body Weight (partial results). Once the design was completed, the CT data for the subject was used to prototype the femur, tibia and the corresponding customized implants. Using a special software and Boolean operations, the exact cuts needed for the Femur and Tibia could be determined. Fig. 11. shows the customized femur and tibia, and Fig. 12. Shows the associated designed components. With advances in the RP material, it is now feasible to use the prototypes directly for investment casting. This would enable the manufacturing of customized implant components with high precision.
4.2. The Spine Project One of the applications of the rapid prototyping technology is its use for medical diagnostics. The spine project involves cases of two patients suffering from spinal deformations. The doctor requested physical models of the portions of the spines that are deformed, as opposed to the conventional 2D slides, to aid him in determining the future of his patients. The first patient is a 2-years old girl suffering from torticollis and the second is a 14-year old boy suffering from scoliosis. Tonic or intermittent and usually painful spasms of the neck muscles cause torticollis. Obvious signs of torticollis are a forward, backward or lateral leaning of the head at the neck. The most successful approach for treating torticollis is a selective denervation surgical process. As with all surgeries there are risks associated with it, especially for children. Scoliosis is defined as a structural lateral curvature of the spine. Sixty to eighty percent of cases occur in females. Scoliosis can be first detected when the shoulders are not the same height. The proper reading of the x-ray images is key in detecting the onset and severity of scoliosis. The prognosis depends on the site and the severity of the disease. The correction of the existing deformity requires surgery. Because of the critical age associated with typical scoliosis patients, extra care should be exercised when determining treatment options.
Fig. 12. The Scoliosis Case
Fig. 13. The Torticollis Case
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The experiment was aimed at providing the doctor with a physical model for the spine of the two patients for diagnosis and surgery planning. The experiment included compiling and processing the CT data before it would be feasible for physical model creation. After a number of iterations, the solid model was generated. Fig. 12. and Fig. 13. are the physical models for the two patients - the outcome of the experiment. The physical models created using the stereolithography were invaluable tool for the physician.. The physician was able to identify the root cause of the spine of deformity. Without going in the details of the surgery, one can easily determine how valuable such model would be for the physician in case he has to perform a corrective surgery. 5. CONCLUSION The applications of CAD/CAM, RE and RP techniques in the medical field, is an invaluable contribution of engineering technology to the medical field. The ability to produce physical models directly from the scanned data, promises to be the way of the future in medical surgery. In the upcoming years it is expected that parts/ inserts will be "designed" to complement nonfunctional human parts of the body. Advances in RP material will undoubtedly enable producing "clean" parts that can be used directly as implants. It is the ability of such engineering techniques to produce complex "designs" coupled with the advances in surgical procedures, which will enable replacement of human parts, reconstruction of others, and the performance of operations with great precision. With the ability of coloring and the development of new material with its low toxicity rate and increased strength, it is very possible that in not too distant future, implants and replacement parts will be directly fabricated and used in patients with great precision.
REFERENCES 1. Smet, M.H., "Medical Model Requirements and Validation of Medical Models." PHIDIAS Project. Consortium Partners: Materialise NV, Belgium; Katholieke Universiteit Leuven, Belgium; Siemens AG, Germany; Zeneca Specialties, U.K., (1996). 2. Sam M.W., III, Quyen Tong, Adrian Dupre, and Ronald E. Barr; "Proceedings of the 7th International Conference on Rapid Prototyping, San Francisco, CA, March/April (1997). 3. STEREOCOLI, Part Building Guidelines for Stereolithography Machines, Zeneca Specialties, Manchester, England. 4. MATERIALISE, USA, MIMICS/MAGICS Presentation, Detroit, MI, USA 5. Medical Applications of Stereolithography: Case of Design and Creation of Customized Knee Implants, Thesis, by Zadhya R. Mohamed, UCF, Orlando, (1998). 6. Swaelens, B., and Vancraen, W., "Laser Photopolymerisation Models Based on Medical Imaging: A Development Improving the Accuracy of Surgery"; Proceedings of the 7 th International Conference on Rapid Prototyping, San Francisco, CA, March/April (1997)
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Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
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DESIGN FOR MANUFACTURE AND ASSEMBLY (DFMA): CONCEPTS, BENEFITS AND APPLICATIONS
Bayoumi, A.M.E. Professor and Chair, Department of Mechanical Engineering University of South Carolina, Columbia, SC, USA 29208
ABSTRACT Design for manufacture and assembly (DFMA) is the practice of designing products with manufacturing in mind so they can be designed in the least time with the least development cost; make the quickest and smoothest transition into production; be assembled and tested with the minimum cost in the minimum amount of time; have the desired level of quality and reliability; and satisfy customers needs and compete well in the marketplace. DFMA considers manufacturing issues early to shorten product development time and ensure smooth transitions to manufacturing, thus, accelerating time-to-market. DFMA reduces costs since products can be quickly assembled from fewer standard parts. Parts are designed for ease of fabrication and commonality with other designs. This, in tum, means a broader product line can be created by assembling common "building blocks" modules into new products. I. INTRODUCTION One of the earliest efforts in DFMA was specifying tolerance close enough to enable interchangeability to parts. The important results were not only greater manufacturing efficiencies but also uncomplicated field repairs. A major prerequisite of Henry Ford's "mass production" (perhaps more of an innovation than the moving assembly line itself) was tightening tolerances so that a car could be assembled with any part in the bin without the usual "forge and file to fit", a common practice of the time. DFMA is often used as an optional technique primarily by individual designers who personally feel strong enough about manufacturability to design for it. This may be based on the designer's intuition which, in turn, is based on experience. Thus, the designers who do the best DFMA designs are those who have had manufacturing experience. Today, DFMA is no longer optional, but, in many cases, required for being competitive. To determine a product's competitive vulnerability, ask the question: "How much better could your product line be designed for the following?" The same function, but with lower cost, better quality & reliability, better delivery, more responsive to customer needs, and new products developed sooner. "It does little good to design state-of-the-art products, if within a short time, our foreign competition can manufacture them more cheaply." John Young, President, Hewlett-Packard
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In addition, products "fail" and go out of production because the cost are too high, quality is too low, introduction was too late, or production couldn't keep up with demand. These are all manufacturability issues and therefore are very much affected by DFMA. Designs that do not take manufacturability into account cause the following manufacturing problems: Time to market: Non-DFMA products take longer to get into production because they are not designed for the existing processes and special arrangements must be made to be able to build them at all. Equipment: Non-DFMA products require more specialized equipment, which results in additional cost. In addition, delays in obtaining special production equipment can be substantial and can have a major negative impact on product introduction schedules. Delivery: Non-DFMA products will take more time to build and deliver because they may require extra steps or manual operations which, in turn, result in poor quality and more rework. Quality: Non-DFMA products have more quality problems because they have more parts from more vendors, require more manual assembly, and may not take full advantage of factory quality control procedures which are set up for the typical processes. Cost: Quality problems and extra rework translate into higher manufacturing cost, especially if any defects get out to the customers. The extra effort for "fire fighting" problem introductions also increase overhead costs.
2. MANUFACTURING IMPROVEMENTS Automation: Companies that automate their plants without DFMA find the job is more difficult than anticipated because products have too many parts of the wrong shape that don't go together easily. IBM Corporation wanted to build an automated plant to build the Selectric typewriter but, after looking at the complexity of the original product, realized that automation was impossible. So the product was redesigned for automation and their showplace factory in Lexington, Kentucky was built around the new design. Just-in-Time (JIT): JIT programs depend on parts standardization. The most obvious results of DFMA are being able to order parts in high enough quantity to encourage frequent deliveries. Fewer part types greatly simplify the flow of parts within the factory, allowing products to be built with minimum of Work-in-Process (WIP) inventory. Flexible Manufacturing: Flexible Manufacturing depends on parts commonality programs, which strive to minimize the number of part types and maximize commonality throughout the product line. In order for a factory to be flexible, each workstation must be able to work on a variety of products. Standardization of design features is also a prerequisite for flexible manufacturing. Different products will be traveling along the same material handling system and will be built by the same tools and equipment. Computer Integrated Manufacturing: Product design simplification and other simplifications (like JIT) should be completed before large-scale integration is attempted. This is why DFMA is usually the first step in Computer Integrated Manufacturing (CIM) programs, since DFMA greatly simplifies designs, reduces the number of part types and, in turn, streamlines the flow of parts in a factory. The method of design will determine how well the design fits into CIM environment. Computer Aided Designs (CAD)flow easily into Computer Aided Manufacturing (CAM). Once the factory is dependent on CAD input, it will not accept manual designs. CAD and DFMA should be implemented early in CIM programs so that computer-based simplified designs will be prevalent when the CIM program is implemented. 3. THE EARLY EFFECTS OF DESIGN By the time a product has been designed, only about 8% of the total product budget has been spent. But by that point, the design has determined 80% of the cost of the product! The
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design determines the manufacturability and that determines a significant part of the introduction and production cost (the 80%) of the product. Once this cost is locked in, it is very hard for manufacturing to remove it.
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4. DO IT RIGHT THE FIRST TIME Everyone that practices DFMA should adopt the motto: Do it right the first time. This advice seems so simple because no designer would begin a design expecting to redesign it later. However, it is distressing how many companies routinely tolerate redesigns, maybe because they always have. Some companies camouflage the process by euphemistically calling them "revisions" or ever "updates." But, regardless of the label, redesigns can have a severe impact on a product's cost and time schedule, not to mention company morale. The results of doing it right the first time are: no need for costly redesign or changes, quick & easy product introduction, trouble-free production, and good product cost, quality and delivery On the other hand, if designers use these DFMA techniques to "do it right the first time," the product will sail through product introduction into stable, trouble-free production with the best cost, delivery and quality to provide a competitive advantage. In many companies, the DFMA program evolves out of a "do it right the first time" thrust including the following methods: Design for Manufacturability: making changes for manufacturability is one of the most common reasons for redesigning a product. Computer Aided Design (CAD): CAD can greatly reduce many types of errors, i.e. since dimensions are entered numerically or computed, they are more accurate than the scaled dimensions prevalent in manual drafting; since CAD drawings are all drawn in the same scale (real units), the drawings of all parts and assemblies can be easily superimposed to check for clearances and interference, in addition, drawing features from previous designs can be easily and accurately copied for use in current designs; Product Simulation: computer simulation programs have revolutionized the evaluation phase of design. The tedious prototypes and breadboards of the past can, in many cases, be dispensed with in favor of functional verification on the computer. This is not only much quicker and more accurate, but will allow simulating functions that might have been impossible any other way; Factory Simulation: manufacturing flow can also be optimized by factory simulation. If this is done in the early design stages, it may reveal processing
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difficulties that the design itself could correct. This is especially true when there are many parts that need to be assembled on slow and/or expensive machines. Computer simulation of factory flow can spot bottlenecks on new products long before they reach the factory floor. With this knowledge, designers might be able to design the product for smoother processing. 5. BENEFITS OF DFM
The benefits of DFMA range from the obvious cost, quality, and delivery to some important subtle benefits. The OBVIOUS BENEFITS include Lower production cost: lower assembly costs results from easy assembly motions with fewer parts leading to the minimum m o u n t of manual labor. Lower "cost of quality" results from fewer parts and from foolproof assembly. Smoother product introduction means less time spent on costly "fire fighting" to deal with product introduction problems. Higher quality: higher quality results from fewer parts, foolproof assembly, easy to inspect features, the use of more standardized parts with known good quality, and a good utilization of stable factory processes. Quicker time to market: DFMA products fit fight into existing processes and do not require special equipment and procedures. The use of standard parts means most will be on hand or be easy to obtain. The SUBTLE BENEFITS include Lower capital equipment cost: designs that assemble easily need less time on assembly machinery. Less need for special equipment saves equipment capital. The use of standardized parts results in fewer set-ups to change to non-standard parts and this results in greater machinery utilization. Greater use of automation: designing for automatic assembly allows for greater use of automation with all its cost and quality advantages. Production up to speed sooner: fewer introduction problems and less need for special equipment or procedures mean production is up to speed sooner. Less chance of redesign: if the design satisfies all the goals and constraints, it will not have to be redesigned for manufacturability or any other reason. Fewer parts to purchase from fewer vendors: fewer parts to purchase saves purchasing expense especially for standard parts. Dealing with fewer vendors strengthens relations with those vendors and results in less cost and effort to qualify parts and deal with quality problems. Factory availability: fewer production problems makes factory more available for other products. DFMA alone may make the difference between being competitive and not succeeding in the market place. Most markets are highly competitive, so slight competitive advantages (or disadvantages) can have significant impact. DFMA may make the difference between a competitive product line and, in the extreme, products that are not manufacturable at all. Products "fail" and go out of production because costs are too high, quality is too low, introduction was too late, or production couldn't keep up with demand. These are all manufacturability issues and therefore very much affected by DFMA. 6. DESIGN PHILOSOPHY: "CONSIDER ALL GOALS AND CONSTRAI NTS EARL Y" 6.1. Traditional Design Consideration Function: the product has to work properly, but it must be kept in mind that, although function is the most obvious consideration, it is far from being the only one. A redesign to correct a purely functional problem will result in another product introduction and that can be an unexpected drain on manufacturing resources. Cost: cost has been the battleground of competition for two decades now. But the lowest product cost does not result from "cost reduction" measures, per se. Design determines well over three fourths of a product's cost. For example, one high tech company appointed a "cost reduction manager" for a critical new
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product line, which managed to reduce the projected cost within the goal by buying the cheapest parts. However, the parts came from 16 different countries and took 9 months to deliver. And, this was on a leading edge product! Furthermore, when production began, the part quality was so poor that the plant actually ground to a halt, thus delaying delivery even further.
6.2. Factory Design Considerations Delivery: delivery is greatly affected by the design because the design determines how difficult the product is to be built and assembled. Design standardization and commonality will affect the effectiveness of JIT programs, which are the key to a fast factory throughput. Quality and Reliability: like cost, quality and reliability are determined more by the design than is commonly realized. Designers determine the number of parts and so determine the cumulative effect of part quality on the product, an important consideration on complex products. Designers are responsible for the tolerance sensitivity. Designers are responsible for ensuring that parts are designed so that they cannot be assembled wrong. These are very much manufacturability issues since quality problems must be consistently corrected in the plant before a product can be shipped. Ease of Assemble: this is what comes to mind when most people think of DFMA because much attention has been focused on Design for Assembly (DFA). Ability to Test: test strategy is very much affected by the company quality culture. " At companies using Total Quality Control (TQC), quality is everyone's responsibility. The TQC philosophy is that, instead of being tested in, quality should be built in using control of the process. Designers of these products are responsible for devising a way to not only test the product, but also to diagnose it if that is needed by the factory. In complex products, test development cost can exceed product development costs and can even take more calendar time. Ease of Service and Repair: being able to repair a defective product is a manufacturability issue because any product failing any test will have to be repaired. Shipping: shipping considerations should not be left until the first manufactured product reaches the shipping dock. Every effort should be made to use standard packaging used for many products.
6.3. Social Design Considerations Human factors: human factors or ergonomics are social considerations that should be considered at the very beginning, since ergonomic changes would be difficult to implement after the design is complete. Appearance/style: appearance and style should be considered an integral part of the design, not something that is added later. All factors in a design need to be considered simultaneously throughout the design. Also, building an attractive product will be good for morale in the factory. Safety: safety should not be considered after the first lawsuit. When a major safety issue surfaces, production ceases until all products in the factory are corrected and all affected parts in inventory and in the field are reworked. Designers should make every effort to design safe products as a moral and legal obligation.
6.4. Marketing Design Considerations Customers' needs: the ultimate goal in designing a product is to satisfy customers' needs. The designers must have up to date marketing input and even talk to the customers directly to ensure that the real customers' needs will be satisfied by the product design. Breadth of product line: using the group technology principles, products can be designed with common, standard parts and be produced on flexible manufacturing lines. Product customization: if the designers use CAD and incorporate group technology principles, it may be possible to customize products to better satisfy customer needs. Time-to-Market: time-to-market is becoming a major source of competitive advantage. Expansion/upgrading: designers should
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design products so that they are easy to expand or upgrade by the plant or by the customer. Future designs" similarly, current products should be designed so subsequent products can be based largely on the current design. 6.5. Environmental Design Considerations Product pollution: designers should anticipate environmental trends and design products "clean" enough for future environmental standards. Anticipating forthcoming pollution standards will minimize the chances of having to redesign the products to incorporate them. Processing pollution: product designers specify the process whether they realize it or not. The 3M Corporation has a new environmental thrust called the "3P" program: "Pollution Prevention Pays." The theme is prevention of pollution at its source. The three elements of the program are recycling, redesign products and equipment for less pollution and creating products that do not pollute in the first place. Ease of recycling product: similarly, everyone has a moral obligation to be concerned about what happens to the product after its useful life is over. Can it be recycled into new materials? Can it be rebuilt for extended life7 This leads to one of the most important design principles for design in general, not just for DFMA: The further into a design, the harder it is to start satisfying additional needs.
7. DFMA VERSUS DESIGN FREEDOM AND DESIGN TIME Designers may be tempted to think that fewer constraints means more design freedom and many may resist DFMA on those grounds. But, in reality, too few constraints may lead to the design equivalent of "writers block." If every design decision has many open choices, the whole design will represent an overwhelming array of choices that can lead to design paralysis. So the designer breaks the impasse by making arbitrary decisions. Not considering all the goals and constraints at the beginning results in arbitrary decisions that eliminate solutions downstream. Designers may be also tempted to think that considering all these constraints will take more time to complete the design. But it really takes no more time (maybe even less time) because thinking about all the constraints at once will steer the designer more quickly to the optimal design. In order to make such a design manufacturable; it may be necessary to make changes in the design. The cost of changes rises drastically as the product progresses toward production. 8. THE SYSTEMS APPROACH TO DFMA The systems approach is a way of thinking that looks at the whole problem as a system and clearly defines its components: Objectives: determine the real objectives. The stated objectives may not include all the design objectives. Objectives are goals to achieve either by meeting specifications or by maximizing or minimizing some variable. Constraints: constraints are not under the control of the designer. Cost would be a constraint if some commitment were made regarding production cost. But cost would be an objective if the goal were to minimize cost. Resources: The resources are the people and money available to develop the product. 9. CREATIVITY AND BRAINSTORMING The better the design is (the better it meets its objectives within constraints and resources); the better will be the chance of incorporating DFMA principles. Creativity may be needed to
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accomplish all the objectives and constraints including manufacturability. The prerequisite for creativity is an open mind. Trigger your mind with questions like, "What if .... ?" A fivestep model of the creative process is outlined here to include the systems approach for DFMA: Steps 1: Preparation, the real objectives are defined; the constraints are summarized; and the resources are identified. Step 2: Concentrated Effort, work hard to find solutions to the objectives subject to the constraints within the resources available. Look for multiple solutions. Use brainstorming techniques. Step 3: Incubation, give the subconscious mind a chance to work on the problem. You might even think of a solution for one project while working on another project. Even taking a walk can "clear the mind" and overcome hurdles. Step 4" Insight, the idea or reorganization of thoughts which is the solution could leap into your mind or it could be looking at the problem in a different way. Step 5: Follow-through, steady, thorough follow-through is essential for a good design. Simply revise the idea or think of a new one using the same steps. Brainstorming is a technique, normally used in-groups, which can generate very many ideas. The leader can organize a formal session or just steer a spontaneous discussion into brainstorming using the following rules. The leader should make the rules clear and record everything said. 1. Criticism and judgement are not allowed: criticism discourages the generation of new ideas and inhibits everyone's responses. 2. Praise all ideas: all ideas should be praised because the point of a brainstorming session is simply to generate ideas. 3. Generate many ideas: the secret to getting good ideas is to have a lot of ideas and some criterion for choosing. Plus, one idea can lead to another. 4. Think wild: what may seem to be a ridiculous idea might trigger the thought process that ultimately leads to an original solution. Think of the mathematical process of using imaginary numbers (square root of negative one) to derive real answers. There are many calculations using these imaginary numbers, but in the end a real number answer appears. After the Brainstorming Session: 5. Sort out all the candidate ideas: this is where judgement is applied. Try to prioritize the ideas to a list of leading candidates. Some ideas may seem promising but need more investigation. 6. Choose the final solution: choose the best solution based on the objectives, constraints and resources. 10. IMPLEMENT DESIGNS After design ideas have been chosen and survived the initial follow-through analysis, they must be given a thorough follow-through effort to reduce the idea to practice. This must be pursued ambitiously until the design meets all its design goals. If the tentative design falls short, the designers should iterate the above process until the product meets all the goals. Dropping the manufacturability goal because the design is having problems causes many DFMA problems.
Paul Cook, founder and CEO of Raychem Corporation, said: "What separates winners from losers in innovation is who masters the drudgery." By drudgery, he means reducing ideas to practice. 11. DESIGN FOR ASSEMBLY The way the product will be assembled must be considered early in the design because of demands each type of assembly places on the design. The optimal method of assembly can be chosen early in the design process if the product and process are designed simultaneously. Assembly systems include manual assembly, semi-automatic assembly, adaptive assembly, automatic assembly, and flexible assembly.
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Designing for Manual Assembly: human assemblers can use skill, dexterity and judgement. However, the quality of manual assembly is rarely consistent. Therefore, parts and assembly should be designed to minimize the chance and consequences of errors. Some examples of minimizing errors include parts are symmetrical so they don't need to be oriented, assemblies are designed so parts can't go in wrong, assemblies are designed so omissions are obvious, test/inspection is designed to catch all manual errors. Establishing and maintaining workmanship standards is a massive undertakings, if done properly. Designing for Adaptive Assembly: adaptive equipment can adapt to predictable variations through sensors and computer logic. This may ease design restrictions on tolerances, orientation, and size ranges. Features, however, may have to be added to parts for visual orientation and inspection criteria. Designing for Automatic Assembly: designers need to realize that most automation machinery and robots are "deaf, dumb and blind." Parts must be designed for consistency and predictability, to be oriented and singulated, to be transported and presented, with shapes that can be gripped, and within the capabilities of the equipment. Designing for Flexible Assembly: in flexible assembly, group technology families must be similar enough to be able to be assembled by the same equipment. Either the set-up is fixed for the family or the equipment must be able to identify different parts and quickly and automatically change the set-up. 12. DESIGN FOR REPAIR AND MAINTENANCE
The need for ease of repair is proportional to the need for the product to be repaired in the factory or the field. A repair strategy may be to simply replace parts or modules, which are themselves not repaired. Some parts are literally "throw-away" (or recycle) parts that are not intended to be repaired at all. However, part cost might preclude this practice. Easy repair may depend on diagnostics which tests would supply. The need for ease of maintenance depends on the reliability of the product and demands on "uptime" (how much time the product needs to be available for use). Design for Ease of Repair: provide ability for tests to diagnose problems. Make sure the most likely repair tasks are easy to perform. Ensure repair tasks use the fewest tools. Use quick discount features. Ensure that failure or wear prone parts are easy to replace with, disposable replacements. Provide inexpensive spare parts in the product. Ensure availability of spare parts. Sensitive adjustments should be protected from accidental change. Protect parts with fuses and overloads. Design for Ease of Maintenance: maintenance can either be performed after something fails (unscheduled maintenance) or at scheduled intervals to replace parts before they are likely to fail (preventive maintenance). Unscheduled maintenance: restoring operation after a failure. Preventive or scheduled maintenance can be scheduled to replace parts before they are expected to fail. Ease of maintenance should be designed into the product as one of the early design goals. Reliability studies should predict part failure frequencies and, of course, be a criterion for part selection. 13. CLOSING REMARKS DFMA provides a systematic procedure for analyzing a proposed design form the point of view of assembly and manufacture. This procedure results in simpler and more reliable products, which are less expensive to assemble and manufacture. In addition, any reduction in the number of parts in an assembly produces a snowball effect on cost reduction because of the drawings and specifications that are no longer needed, the vendors that are no longer needed, and the inventory that is eliminated. All of these
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factors have an important effect on overheads, which, in many cases, form the largest proportion of the total cost of the product. DFMA tools encourage dialogue between designers and the manufacturing engineers and any other individuals who play a part in determining final product costs during the early stages of design. This means that teamwork is encouraged and the benefits of simultaneous or concurrent engineering can be achieved. The savings in manufacturing costs obtained by many companies who have implemented DFMA are astounding. For example, Ford Motor Company has reported savings in the billions of dollars as a result of applying DFMA to the Ford Taurus line of automobiles. NCR anticipates savings in the millions of dollars as a result in applying DFMA to they new point-of-sales terminals. These are high volume products. At the other end of the spectrum, where production quantities are low, Brown & Sharpe have been able, through DFMA, to introduce their revolutionary coordinate measuring machine, the MicroVal, at half the cost of their competitors, resulting in a multimillion-dollar business for the company. These are but a few examples that show that DFMA really works. REFERENCES 1. American Society of Metals, "Mat.DB User's Manual," ASM International, Cleveland, Ohio, (1990). 2. Bilotta, A.J., Connections in Electronic Assemblies, Marcel Dekker, Inc., N.Y., (1985. 3. Boothroyd, G., "Design for Economic Manufacture," Annals of the CIRP, Vol. 28/1, p 345, (1979). 4. Boothroyd, G., "Design for Manual Handling and Assembly," Report No.4, Dept. of Mechanical Engineering, U. of Mass., Sept. (1979). 5. Boothroyd, G. and Dew Hurst, P., "Product Design for Assembly, "Boothroyd Dew Hurst, Inc., Wakefield, R.I., (1990). 6. Crane, F.A.A. and Charles, J.A., "Selection and Use of Engineering Materials," Butterworths, London, (1984). 7. Farag, M.M., "Materials and Process Selection in Engineering," Applied Science Publishers, Barting, England, (1979). 8. Funk, J.L. et al., Programmable Automation and Design for Manufacturing Economic Analysis, National Science Foundation, (1989). 9. Ham, I. And Lu, C.U., "Computer Aided Process Planning: The Present and the Future," Annals ofCIRP, Vol. 37 (2), p 591, (1988). 10. Kalpakjian, S., "Manufacturing Processes for Engineering Materials," 1st Edition, Addison-Wesley, Reading, Mass., (1984). 11. Matisofi, B.S., Handbook of Electronics Manufacturing Engineering, Van Nostrand Reinhold Co., (1986). 12. P.E.R.A., "Survey of Machining Requirements," PERA, Melton Mowbray, U.K. 13. Quick, Joseph H., "Work Factor Time Standards," McGraw-Hill Co., New York, (1962). 14. Seth, B. and Boothroyd, G., "Design for Manual Handling," Report No.9, Dept. of Mechanical Engineering, U. of Mass., Jan. (1979). 15. Shea, C. and Dew Hurst, P., "Computer-Aided Materials and Process Selection," Proc. 4th Int. Conf. on Product Design for Manufacture and Assembly, Newport, R.I., June, (1989). 16. Taylor, T., Handbook of Electronic Industry Cost Estimating Data, John Wiley & Sons, (1985).
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INVERSION OF FRUSTA AS IMPACT ENERGY ABSORBERS
Aljawi, A.A.N. and Alghamdi, A.A. Department of Mechanical Engineering, King Abdulaziz University, P.O. Box 9027, Jeddah 21413, Saudi Arabia
ABSTRACT
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In this paper a noval crushing mode of frusta is presented for the first time. The details of the plastic inversion of frusta as energy absorbers are given. The deformation modes of capped frustum are investigated both experimentally and analytically. An Explicit version of ABAQUS 5.7-3 finite element (FE) code is used for computing and describing the proposed deformation mode. Good agreement is obtained between the experimental results and the FE predictions. KEYWORDS Energy Absorber, Frusta Inversion, Finite Element. 1. INTRODUCTION Energy absorbers are systems that convert kinetic energy into other forms of energy, such as elastic strain energy in solids and plastic deformation energy in deformable solids. The converted energy may be reversible, as in pressure energy in compressible fluids, and elastic strain energy in solids, or irreversible, as in plastic deformation. The process of conversion for plastic deformation depends, among other factors, on the magnitude and method of application of loads, transmission rates, deformation displacement patterns and material properties [1]. The predominant domain of applications of collapsible energy absorbers is that ofcrash protection. Such systems are installed in high-risk environments with potential injury to humans or damage to property. The aim is to minimize the risk of injury or damageby controlling the deceleration pulse during impact. This is achieved by extending the period of dissipation of the kinetic energy of the system over a finite period of time. Cushioning devices on vehicle bumpers, crash retards in emergency systems of lifts and crash barriers used as roadblocks are everyday examples. Familiar plastic deformable energy absorber units include cylindrical shells [2], wood-filled tubes [3], foam-filled columns [4], sand-filled tubes [5], PVC shells [6], tube inversions [7] and tubular elements [5]. The active absorbing element of an energy absorption system can assume several common shapes such as circular tubes [8], square tubes [9], multicomer metal columns [10], frusta [11] and rods [12]. Axisymmetrical and circular shapes provide perhaps the widest range of all choices for use as absorbing elements because of their favorable plastic behavior under axial forces, as well as their common occurrence as structural elements.
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In this paper the selected absorber has a truncated capped frustum shape. Frusta are employed over a wide range of applications, especially in the domains of aerospace and armaments. Common examples occur in the nose cones of missiles and aircraft. 2. AXIAL LOADING OF TUBULAR COMPONENTS The study of deformation of tubular energy absorbers in general falls into two main categories, lateral, and axial loading. Investigations often lead to accounting for geometrical changes, interactions between modes of collapse, as well as strain hardening and strain rate effects. Johnson and Reid [1] identified the dominant modes of deformation in simple structural elements in the form of circular and hexagonal cross-section tubes when these elements were subjected to various forms of quasi-static loading. They described the loaddeformation characteristics of a number of these elements. Thin-walled absorbers having symmetrical cross sections may collapse in concertina or diamond mode when subjected to axial loads. The collapsing of such components by splitting or by inversion is also reported [9]. The behavior of thin tubes (large diameter D/ thickness t), with circular and square cross sections, when subjected to axial loads, has been of particular interest since the pioneering works of Alexander [2]. In fact circular tubes under axial compression are reported to be the most prevalent components in energy absorber systems. This is because the circular tube provides a reasonably constant operating force. Furthermore, circular tubes have comparatively high energy absorbing capacities, and stroke length per unit mass. In comparing lateral with axial compression, the axial buckling mode has a specific energy absorbing capacity, which is approximately ten times that of the same tube when compressed laterally between flat plates. This is due to the fact that all wall material in a tube can be made to participate in the absorption of energy by plastic work in axial loading. 2.1 Thin-Walled Frusta
Frusta are tnmcated circular cones, see Fig. 1. Literature on the utilization of frusta for dissipation of energy is meager. Postlethwaite and Mills [11 ] first studied the frustum in this context in 1970. In their study of axial crushing of conical shells they used Alexander's extensible collapse analysis [2] to predict the mean crushing force for the concertina mode of deformation for frusta made of mild steel. Mamalis et al. [ 13] investigated experimentally the crumbling of aluminum frusta when subjected to axial compression load under quasi-static conditions. They proposed empirical relationships for both the concertina and the diamond modes of deformation. Mamalis et al. [14] extended their experimental study to include mild steel at elevated strain rates. They concluded that the deformation modes of frusta could be classified as a) concertina, b) concertina-diamond, and c) diamond. Mamalis and associates [15] refined the work of Postlethwaite and Mills [11 ] in using the extensible collapse analysis for predicting the mean crushing load, and fair agreement with the experimental results were reported. In another paper, Mamalis and his group [16] modeled the progressive extensible collapse of frusta and gave a theoretical model that depicts the changes in peaks and troughs of the experimental load-displacement curves. The comparison with the experimental results gave a fair degree of accuracy. The above studies deal with axial crushing (or crumbling) of frusta between two parallel plates. However, an innovative mode of axial deformation is presented in this paper. This mode is inward (free or direct) inversion. In what follows, results of experimental work as
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well as finite element modeling conducted on the inversions of capped spun aluminum frusta are presented. 3. FINITE ELEMENT MODELING The finite element method (FEM) has been used extensively to simulate many applications in structural dynamics [8,17-19]. In the present study, ABAQUS Explicit FEM code (version 5.7-3) is employed to investigate the modes of deformation of frusta under quasi-static loading. Fig. 2 shows the finite element models used in this study for the inversion. An axisymmetric four-nodded element, CAX4R, is used for modeling the frustum shown in Fig. 2. About 300 elements are used for the model. Material properties of the model were taken as rigid perfectly plastic with yield strength Sy=I25MPa, and density 0=2800 Kg/m 3. All nodes at the centerline of symmetry were selected to move only in the vertical direction. Both upper and lower surface were set in contact with rigid body surfaces. These rigid surfaces were modeled using two nodal axisymmetric rigid elements, RAX2. A coefficient of friction of ~t=0.15 was incorporated between the contact surfaces. A reference node was introduced at the top end surface of the model. This node was set to move at a velocity of0.01m/s representing quasi-static case. The upper small capped end of the frustum was in contact with a rigid body moving at a constant velocity. The lower end was restrained from moving in vertical and horizontal directions as shown in Fig. 2. 4. EXPERIMENTAL A large number of frusta, featuring different thicknesses and apex angles were subjected to various loading conditions. The program involved the use of twelve different sizes of aluminum frusta (3 different apex angles and 4 different thicknesses) for inversion. Additional tests were conducted to investigate the effect of impact speed on the inversion process. Tests were conducted by the use of a 50-ton Instron Universal Testing machine (UTM) as well as a falling weight hammer (FWH) of 7 m/s striking speed. Special jig for inversion was manufactured and utilized. The jig consisted of an inversion rod and a base cylinder, as shown in Fig. 1. The upper jaw of the UTM clamped the rod, and the base rested on the lower jaw. The same jig was utilized also with the FWH, in which case the inversion rod was simply attached to the falling weight. 5. RESULTS AND DISCUSSION
5.1 Static Testing In this section details of the experimental load-displacement curves and finite element results for inversion are presented in details for quasi-static loading. Results of dynamic loading at high impact velocity are summarized in Section 5.2. A spun frustum (Specimen No. 23) was inverted at quasi-static Condition using the UTM moving at a cross-head speed of 10 mm/min. The specimen has an angle ct=60~ large diameter D=73 mm, small diameter d=22.5 mm, thickness t= 1.25ram, height h=44 ram, and mass m=25.72 grams. Figure 3 shows experimental and finite element (FE) load-displacement curves for the spun aluminum frustum. It can be observed that good agreement is obtained between the experimental results and FE predictions. It can be observed from the figure that the frustum passes through a number of stages. The load rises quasi-linearly from the origin to point (a). The force at point (a) represents the load of instability. Up to this point the deformation is recoverable, i.e., elastic and beyond which plastic behavior sets in. The zone between (a) and
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(b) is a zone of incubation, within which the cap of the frustum is deformed in such a manner as to facilitate the inversion type of deformation. Three localized plastic hinges developed from point a to point b and extensible mode of deformation was observed. Point (b) signals completion of the development of the inversion zone. Inversion then proceeds towards the larger (lower) end of the frustum, until point (c) is reached, see photograph in Figure 3. The increase in the inversion force from (b) to (c) is attributed to the progressive increase in the volume of the deformation one ith he ncreasing D/t ratio. Point (c) in Fig. 3 signals the termination of the inversion zone, the bending front having reached the vicinity of the free large end of the frustum. From point (c) to (d) inversion mode changes into flattening mode and the undeformed part of the frustum has the shape of BeUeville spring, see photograph in Figure 3. The free end of the frustum is flattened parallel to the shoulder of the jig base. The energy absorbed recorded experimentally by the frustum through this inward inversion is 9.73 J/gm, whereas the predicted energy by FE is 9.47 J/gm. The FE details of the inversion process can be seen in Fig. 4 that gives the inversion mode of deformation in 9 stages. These stages were captured at the following axial intervals: 0.3, 10, 20, 25, 50, 60, 77, 83, 85mm. The second and the eighth stages show the initiation and termination of the inversion process, respectively. Figure 5 shows half of the frustum (Specimen 23) before and after the inversion as predicted by the FE and the true photograph of the frustum. Excellent agreement between the two deformed shapes is obtained. Frusta made of mild steel, galvanized sheet steel, low carbon steel and PVC were tested successfully. Also, frusta made by riveting, welding, machining and spenning were also tested successfully.
5.2 Dynamic Testing In order to assess the effect of speed on the process of inversion, identical frusta were tested using UTM at cross-head speeds of 2, 20 and 200 mm/min. Additional tests were conducted on the FWH facility using different falling masses. Impact velocities up to 7m/s were used in these tests. As all specimens in these tests behaved as in quasi-static tests. It was concluded that inversion is not affected by strain rate for low impact velocities. Shapes of the inverted frusta at quasi static condition are very similar to those inverted at dynamic case. Figure 6 shows a photograph of inverted frusta at dynamic conditions using FWH. The possibility of re-using the inverted frusta was investigated, everal ests were conducted for inversion and then re-inversion of inverted frusta. Figure 7 shows the load displacement curves of the first, second, third and fourth inversions. Results from such experiment show that it is possible to invert and re-invert the frustum. All specimens failed, however, during the fourth inversion. 6. CONCLUSIONS New mode of axial deformation of frusta is presented. The proposed mode is repeatable and predictable. Although the energy density of this axial mode of deformation is less than that of tube inversion, the inversion of frusta required simpler test fig and no die is required. In fact it was found that a frustum might be inverted several times, indicating that it is possible re-use the same absorber. Since all specimens in the impact tests behaved s n he uasi-static ests, it is concluded that within he xperimental ange f mpact peeds 0-7m/s), he rocess o inversion is not affected by the speed of deformation. Finally, good agreement was achieved between the experimental results and the predictions by the FE model at the condition investigated.
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REFERENCES
1. Johnson, W. and Reid, S. R., 'qVletallicEnergy Dissipating Systems, "Appl. Mech. Rev., Vol. 31, pp. 277-288, (1978). 2. Alexander, J.M., "An Approximate Analysis of the Collapse of Thin Cylindrical Shells Under Axial Loading," Quart. J. Mech. Appl. Math., Vol. 13, pp. 10-15, (1960). 3. Reddy T.Y. and AI-Hassani, S.T.S., "Axial Crushing of Wood-Filled Square Metal Tubes," Int. J. Mech. Sci., Vol. 35, pp. 231-246, (1993). 4. Abramowicz, W. and Wierzbicki, T., "Axial Crushing of Foam-Filled Columns," Int. J. Mech. Sci., Vol. 30, pp. 263-271, (1988). 5. Reid, S.R., 'Metal Tubes as Impact Energy Absorbers," Metal Forming and Impact Mechanics, ed. S. R. Reid, Pergamon, New York, pp. 249-269, (1985). 6. Mamalis, A.G., Manolakos, D.E., Viegelahn, G.L., Vaxevanidis, N.M. and Johnson, W., "On the Inextensional Axial Collapse of Thin PVC Conical Shells," Int. J. Mech. Sci., Vol. 28, pp. 323-335, (1986). 7. AI-Hassani, S.T.S., Johnson, W. and Lowe, W.T., "Characteristics of Inversion Tubes Under Axial Loading," J. Mech. Eng. Sci., Vol. 14, pp. 370-381, (1972). 8. Reid, S.R., "Plastic Deformation Mechanisms in Axially Compressed Metal Tubes Used as Impact Energy Absorber," Int. J. Mech. Sci., Vol. 35, pp. 1035-1052, (1993). 9. Lu, G., Ong, L.S., Wang, B. and Ng, H.W., "An Experimental Study on Tearing Energy in Splitting Square Metal Tubes," Int. J. Mech. Sci., Vol. 36, pp. 1087-1097, (1994). 10. Abramowicz, W. and Wierzbicki, T., "Axial Crushing Of Multicomer Sheet Metal Columns," J. Appl. Mech., Vol. 56, pp. 113-120, (1989). 11. Postlethwaite, H.E. and Mills, B., "Use of Collapsible Structural Elements as Impact Isolators with Special Reference to Automotive Applications," J. Strain Anal., Vol. 5, pp. 58-73, (1970). 12. Alghamdi, A.A. and Aljawi, A.A.N., "Cubic Steel Rod Cells as Energy Absorbers," Proceedings of the Complas 2000, Barcelona, Spain, September 11-14, 2000, submitted. 13. Mamalis, A.G. and Johnson, W., "The Quasi-Static Crumpling of Thin-Walled Circular Cylinders and Frusta Under Axial Compression," Int. J. Mech. Sci., Vol. 25, pp. 713-732, 1983. 14. Mamalis, A.G., Johnson, W. and Viegelahn, G.L., "The Crumbling of Steel Thin-Walled Tubes and Frusta Under Axial Compression at Elevated Strain-Rate: Some Experimental Results," Int. J. Mech. Sci., Vol. 26, pp. 537-547, (1984). 15. Mamalis, A.G., Manolakos, D.E., Saigal, S., Viegelahn, G. and Johnson, W., "Extensible Plastic Collapse of Thin-Wall Frusta as Energy Absorbers," Int. J. Mech. Sci., Vo128, pp. 219-229, (1986). 16. Mamalis, A.G., Manolakos, D.E., Viegelahn, G.L. and Johnson, W., "The Modeling of the Progressive Extensible Plastic Collapse of Thin-Wall Shells," Int. J. Mech. Sci., Vol. 30, pp. 249-261, (1988). 17. Bammann, D. J., Chjiesa, M. L., Horstemeeyer, M. F. and Weigaten, L. T., "Failure in Ductile Materials Using Finite Element Methods," Structural Crashworthiness and Failure, ed. N. Jones and T. Wierzbick, Elsevier, London, pp. 1-54,(1993). 18. Kormi, K., Shaghouei, E. and Duddell, D.A., "Finite Element Examination of Dynamic Response of Clamped Beam Grillages Impacted Transversely at Their Center by a Rigid Mass," Int. J. Impact Eng., Vol. 15, pp. 687-697, (1994). 19. HKS, Inc, ABAQUS/Explicit User's Manual, Theory and Examples Manual and Post Manual, Version 5.7, Explicit, (1997).
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Inversion Rod Initial Frustum o~ f....
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Cylinder Figure 1. Direct inward inversion of frusta.
Figure 2. FE model for direct inversion.
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Figure 3. Experimental and FE load-displacement curves for quasi-static inward inversion of cappedaluminum frustum.
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Figure 5. Comparison between the expe~memai and the FE prediction of a frustum before an~l aner inversion.
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Figure 6. Photograph shows spun aluminum frusta inverted using falling weight hammer. 3500
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Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
521
EFFECT OF LASER SURFACE TREATMENT AND WORK HARDENING ON THE FRETTING WEAR RESISTANCE OF Zr-2.5Nb ALLOY AT HIGH TEMPERATURE
Attia, M.H. Adjunct Professor, Mechanical Engineering Department, McMaster University, Hamilton, Ontario, Canada
ABSTRACT In this study, the effect of laser surface treatment (by heating and glazing) and work hardening (by shot peening and cold working) on the fretting wear resistance of Zr-2.5Nb alloy at 265~ is investigated. Results showed that while laser surface heating provides the best fretting wear resistance, cold working ensures the optimum performance of the tribosystem, when the wear of the mating Zr-4 material is considered. The results also showed a good correlation between the wear resistance and the increase in the material hardness by surface treatment, with the exception of laser surface heating and glazing treatments. The former shows remarkable improvement in wear resistance in spite of its relatively low hardness. This reversed effect is attributed to the self-induced changes and plastic deformation during fretting. KEYWORDS Fretting Wear, Zirconium Alloys, Laser Surface Treatment, Work Hardening 1. INTRODUCTION Fretting wear occurs between two contacting surfaces having oscillatory relative motion of small amplitude [1]. This type of damage has great economical and safety impact on the operation of nuclear reactors, since a large number of their components are subjected to flowinduced vibrations [2-4]. Zirconium alloys are highly desirable for nuclear applications due to their high corrosion resistance in combination with a transparency to thermal energy neutrons. Their wear resistance is, however, much lower than other nuclear-grade materials, e.g., nickel alloys and stainless steels. The early work on fretting wear of Zircaloy-2 [5,6] indicated that while the fretting wear process is self-limiting at relatively small contact pressures, it continues with time in a linear fashion at high contact pressures. The results also indicated that surface pre-oxidation has insignificant effect on the fretting wear mechanism, while Lupoli [7] showed that the passivation of Zr-2 has a negative effect on fretting wear rate. Trowse and Moore [8] showed that the relationship between the wear volume and the mechanical work input to the tribo-system, is nearly linear. Contradictory conclusion was, however, drawn by Spinsa et al. [9]. To compare the wear resistance of different reactor materials, Ko [4] conducted a series of tests with Zircaloy-2 fretted against palladium, tantalum, niobium, and austentie stainless steel.
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The measures used to minimize the fretting wear damage are generally based on using lubricants, surface coatings, and interfacial layers, as well as modifying the surface by thermomechanical treatment and cold working [ 1,10,11 ]. In recent years, researchers have focused on ceramic coatings applied by physical vapor deposition PVD, chemical vapor deposition CVD, and laser plasma generation. Schouterden et al.[12] showed that PVD- TiN and CVD-DLC coatings failed by contact fatigue under fretting conditions. Although TiN coatings are limited to small thickness (< 5~tm) to ensure good adhesion, De Bruyn et al.[13] showed that the fretting wear resistance of thick (25~tm), well adherent Ti2N coating is as good as the conventional TiN coating. The fretting wear resistance of multilayered TiN, silicon nitride and some selected surface coatings and plasma surface modified materials were also investigated by many researchers [14-16]. Laser surface treatment LST was only investigated for corrosion applications [17] and for conventional sliding tribo-systems [18,19]. The wear testing results indicated that the LST produced improved tribological properties over conventional heat treatments. The results reported in [ 18] also showed that while laser surface heating increases the wear resistance of high-chromium cast iron by a factor of two, the laser glazing effect was found to decrease the wear resistance by a factor of five. For nuclear applications, the use of surface coatings is restricted by some basic requirements to ensure that the coating material does n o t absorb neutoms, reduce reactor activity, or include elements that produce long-lived radioisotopes. Therefore, the objective of this investigation is to explore the potential of increasing the fretting wear resistance of Zr-2.5Nb alloy by only modifying the material surfaces through cold working and laser surface heating and glazing. 2. EXPERIMENTAL SET-UP
2.1 Test Specimens The oscillatory specimen is made of Zr-2.5 Nb alloy, which has the following composition: 2.6 wt% Nb, 0.14 wt% 02, and the balance are Zr and impurities. The dimensions of the specimen are: 4.0 mm thick, 6.5 mm wide and 45 mm long. The fretted surface is cylindrical with an inner radius of 51.7 mm. The stationary specimen is made of Zircaloy-4, which has the following composition: 1.35 wt% Sn, 0.23 wt% Fe, 0.08 wt% Cr, 0.12 wt% 02, and the balance are Zr and impurities. The dimensions of the specimen are: 1.1 mm thick, 2 mm wide and 24.1 mm long. The fretted surface is cylindrical with an outer radius of 38.1 mm. Microhardness measurements near the surface, on a number of specimens, showed that Hv =185+ 30. 2.2 Fretting Wear Tribometer, Measurement and Control System The design of the fretting tribometer ensures that the following conditions are kept constant during the test: (a) the frequency and amplitude of the oscillatory relative motion, (b) the normal applied load, and (c) environmental conditions and temperature. A schematic of the fretting test set-up is shown in Fig. 1. It consists of two test chambers, which allows two separate, but similar, tests to be conducted simultaneously to assess the repeatability of the test results. The holder (1) of the stationary specimen is attached to the chamber (2) by a taper joint to ensure rigidity. Since temperature has a significant effect on the fretting process [20] the temperature is controlled at T = 265 ~ in each chamber through control heaters HI, I-I2 and II3. The lower specimen holder (9) is made of structural ceramic material to achieve the required rigidity and thermal characteristics. The oscillating specimens are driven by an electromagnetic vibrator, which is connected to a vibration controller to maintain the displacement and the frequency of oscillation constant during the test. The feedback signal is
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provided by a piezoelectric accelerometer, which is connected to a charge amplifier to convert the acceleration signal into displacement. The axis of the vibrator and force transducer are positioned to be collinear with fretting interface I-I. The contact pressure between the fixed and moving specimens is maintained constant by using a dead weight system, while allowing the fixed specimen to move vertically. The movement is guided by a water-cooled linear bearing (6), which is pre-loaded to minimize movement in the horizontal direction. A cylinder pivot (4) is used to ensure a proper initial alignment between the moving and fixed specimens.
Normalcontactpressure
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(dead weight)
~-]
IF] .[ 6, ....r ~ -
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l I i~,- ;_ 1~ 1
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.
.
.
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.
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.
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.
.
.
.
.
.
.
.
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.
.
I PowerAmplifier 1 '; cimlent& ,
Fre,quencyController DataAcquisiti~ i
Fig. 1 Schematic of the fretting tribometer, and the measurement and control system 2.3 Test Conditions The fretting wear tests were conducted under the following conditions: normal load FN = 121.4 N, displacement amplitude 8= • ~tm, frequency f = 15 Hz, number of cycles N=6.16x 106 (equivalent to a test duration of 144 h). The tests were conducted in air at 265 • 2
~
3. SURFACE MODIFICATION AND TREATMENT The following types of surface treatment were considered in this investigation: 1. Oxidized zirconium ZrO2-autoclaved to approximately 1 ~tm in thickness on as-received and heat treated (475 ~ for 6 hours) Zr-2.5Nb alloy. These specimens are designated as HT. 2. Oxidized zirconium ZrO2-autoclaved to approximately 1 ~tm in thickness on shot-peened Zr-2.5Nb alloy. The shot peening of the specimens was performed at Almen intensity A15 and 100% coverage. These specimens are designated as SP. 3. Cold worked Zr-2.5Nb specimens. These specimens, which are designated as CW, were cut from tubes that were manufactured through the following steps: forging, beta treatment, extrusion at 1 I" 1 and 1100 K, cold drawn to approximately 27% and then autoclaved at 673 K for 24 h to grow an oxide film of approximately 1 ~tm in thickness.
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4. Laser surface heating LSH. In this process, the specimens were moved under the laser
5.
beam at a speed of 12.7, 16.9 or 21.2 mm/s. A 1.4 kW continuous wave CO2 laser was used in the Gaussian mode to perform the LSH process in a vacuum chamber with a NaCI window. Laser surface glazing LSG. This process, in which the surface melting is followed by fast quenching to the cold and solid substrate, was carried out by rapidly scanning the surface of the specimens with a beam from a 1.4 kW continuous wave, CO2 laser. The transverse speed of the specimens was 42, 60 or 76 mm/s. The LSG process was also performed in a vacuum.
Figure 2 shows the effect of the transverse speed in the LSH process on the micro-hardness in the subsurface layers. These measurements were made along three lines, 0.45 mm apart, on cross sections of metallographieally prepared specimens using Vieker indenter with 200 g load. The figure shows that high hardness (-= 265-280 Hv) were obtained at a depth of approximately 240-250 ~tm from the surface. This is followed by a sharp decrease in the hardness between 250 lira to 1 ram. For depths greater than 1 ram, no significant change in hardness is observed. In the following wear tests, a single pass process with transverse speed of 21 mm/s was selected. Another series of tests was also performed to examine the dependance of the micro-hardness of the surface layer on the LSG process parameters, namely, the number of beam passes and the percentage of overlap. The measurements were made along a number of lines, 0.28 mm apart. Figure 3 shows that with a single pass PI, the maximum hardness at a depth of 50 ttm varies between Hv --- 325 and 350, for 42
300 250
+ 'a', 17 mm/s - x - 'b', 17 mm/s
200 t_.
+
etl .m
6 t_ t#
150
'c', 17 mm/s
---O-'c', 21 mm/s ---o-'c', 13 mm/s
100 0
0.5
1 1.5 2 Depth below the surface, mm
2.5
Fig. 2 Effect of the LSH process parameters on the micro-hardness profile
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500 t ~" 450 o-
>:4oo
i
;
o 'd'. 60 mm/s, P2, 7 5 %
~300 .~ 250 r
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.
.
.
200 150 100
. . . . . . . . . . . . 0.5 1 1.5 2 Depth below the surface z, mm
0
o 'd', 42 mm/s,P1 A 'd'. 60 mm/s, P 1 o 'd', 76 mm/s, P1 + 'a', 42 mm/s, P3, 25% • 'd', 42 mm/s, P3,25% x 'g', 42 mm/s, P3, 25% E 'a', 76 mm/s, P4, 25% 2.5 o 'd', 76 mm/s, P4, 25% # 'g', 76 mm/s, P4,25%
Fig. 3 Effect of the LSG process parameters on the micro-hardness profile
Fig. 4 Macro-structure of the LSG specimen, showing the heat-affected zone (4 passes) The micro-hardness profiles in the HT, SP and CW specimens are shown in Fig. 5 (identified as 'initial' condition). While the shot-peening and heat treatment result in hardness values of Hv ~- 205-210 near the surface, the cold working process causes the micro-hardness to rise to about 225 Hv at a depth of 400 lam. The depth of hardening in these treatments is approximately 1.5 ram. & LsH, initial
350
9 LSit, fretted
~:
O
300
O I.SG. initial 9 LSG. fretted
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~
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o
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;" tiT. initial
0
1 2 3 4 Depth below the surface z, mm
5
+ HT, fretted
Fig. 5 Effect of the SP, HT and CW processes on the micro-hardness profile
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4. TEST RESULTS AND DISCUSSION The fretting wear volume was measured by mapping the topography of the worn surface, using an automated scanning and analysis system, which incorporates a stylus-type surface roughness measuring insmunent. Examples of fretted surfaces are given in Fig. 6 for the CW specimen and the Zr-4 counter part. Examination of the fretted surfaces showed a visible white color oxide layer (Fig. 7, for specimen LSH). This oxide film was also observed on all Zr-4 stationary specimens. The fretting wear resistance of the Zr-2.5Nb and Zr-4 materials are expressed in terms of the wear rate coefficient k, which is defined as the ratio between the total wear volume V and energy input to the system W through the normal load wear [21 ] k = V / W = V ! ( 4 8 N FN), where 8 is the amplitude of oscillation, N is the number of cycles and FN is the normal load.
Fig. 7 Optical microscopy photograph (magnification 100x) of the cross-section of LSH specimen near the fretting zone Figure 8 shows the effect of the surface treatment on the wear coefficient for both types of zirconium alloys. Analysis of the results showed the excellent repeatability of the tests; two
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tests conducted under similar conditions vary within + 17% of the average values. In comparison with the reference case of conventional heat treatment HT, the figure shows that the fretting wear resistance of Zr-2.5Nb alloy is significantly improved by the LSH process (by approximately factor of 3.1) and to a lesser extent with the LSG process (by approximately factor of 1.7). Cold-worked specimens showed also some improvement in the fretting wear resistance (by approximately factor of 1.4). Autoclave oxidation of shot-peened specimens exhibit, however, less wear resistance by a factor of 0.9. The poor performance of the HT and SP surfaces in comparison to the cold-worked specimens can be attributed to the relatively low hardness of the surface layers (Figs. 2, 3 and 5). The results shown in Fig. 8 also show that the improvement in the wear resistance of Zr-2.SNb alloy by laser surface modification is associated with reduction of the wear resistance of the mating Zircaloy-4. Cold-worked specimens, on the other hand, seem to provide the optimum condition for the Zr-2.5Nb/Zr-4 tribo-system. It is worth noting that the LSH results in better wear resistance than the LSG process in spite of its relatively low hardness (Figs. 2 and 3). This result is consistent with the conclusion drawn in [18].
[] Wear rate coefficient of Zr-2.5Nb [] Wear rate coefficient of Zr-4 t -~
6E- 14
ex..
5E-14 4E-14 ca
3E-14 2E-14
1E-14 LSH
LSG
CW
HT (Ref)
SP
Fig. 8 Effect of surface treatment on the wear coefficient of Zr-2.5Nb and Zr-4 alloys
In attempting to explain this effect, the fretted specimens were sectioned and the microhardness profiles were measured outside and below the center of the fret marks. This provides a partial answer to the self-induced changes in the material due to the plastic deformation associated with the fretting wear process. As Fig. 5 shows, the fretting wear process has nearly no effect on the hardness distribution in the subsurface layer of the LSG, CW, SP, and HT specimens. On the other hand, there is clear indication that the hardness of the LSH specimen has significantly been increased (for depths up to 2 mm) above the initial level and that of the LSG treatment. 5. CONCLUSIONS The following conclusions can be drawn form this investigation: I. In comparison with the conventional heat treatment process, the results show that the fretting wear resistance of Zr-2.5Nb alloy is significantly improved by the LSH process (by a factor of ~ 3.1) and to a lesser extent with the LSG process (by a factor of--- 1.7). Cold-worked specimens showed also some improvement in the fretting wear resistance. Autoclave oxidation of shot-peened specimens exhibit, however, less wear resistance by a factor of 0.9.
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2.
The improvement in the wear resistance of Zr-2.SNb alloy by laser surface modification is associated with reduction of the wear resistance of the mating Zr-4. Cold-worked specimens, on the other hand, seem to provide the optimum condition for the Zr-2.SNb/Zr4 tribo-system. 3. The plastic deformation associated with the fretting process has nearly no effect on the hardness distribution in the subsurface layer of the LSG, CW, SP, and HT specimens. On the other hand, the hardness of the LSH specimen has significantly been increased above the initial level and that of the LSG treatment. ACKNOWLEDGMENT The work presented in this paper was conducted under the support of the Natural Sciences and Engineering Research Council of Canada, which the author greatly appreciates. REFERENCES 1. Waterhouse, R.B., "Fretting Corrosion", Pergamon Press, Oxford, (1972). 2. Alicino, F., and Zampini, G., "Integrity Assessment of Zircaloy-2 Pressure Tube With Fretting Corrosion Defects", 5th Int. Conf. on Pressure Vessel Tech., pp. 577-598, (1984). 3. Schmugar, K.L.,"Vibration Wear of Fuel Rods", ASME, Paper No.75-WA/HT-79, (1979). 4. Ko, P.L., "Wear of Zirconium Alloys Due to Fretting and Periodic Impacting", Wear, Vol. 55, pp. 369-385, (1997). 5. Lobsinger, R.J., "Evaluation of fretting Corrosion of Zr-2", Hanford Atomic Products Operation, Washington, HW-61915, UC-25, Feb. (1960). 6. Davis, S.M., and Mercer, W.L., "Fretting and Corrosion of Zircaloy-2 in Steam", Atomic Power Construction Ltd, UK, TRG Report 1094, April (1965). 7. Lupoli, P., "Experimental Results of Some Fretting Corrosion Tests of Metallic Materials", National Committee on Nuclear Energy, Italy, AEC-TR-7450, (1973). 8. Trowse, F.W., and Moore, D.A., "Fretting of Zr-2 against Zr-2 and Stainless Steel in High Temperature", Atomic Power Construction Ltd, UK, TRG Memorandum 3051 (5), (1965). 9. Espinosa, P.D., Lelli, G., and Possa, G., "Fuel String Dynamics and Pressure Tube Fretting Corrosion in the CIRENE Power Channel" Energia Nucleare, pp. 406-424, (1978). 10. Beard, J., "The Rational Selection of Palliatives for Avoidance of Fretting", Proc., Int. Conf. on Tribology- Fifty Years on, Instn. of Mech. Eng., London, Vol. 1, pp. 311319,(1987). 11. Zhou, Z.R., and Vincent, L., "Lubrication in Fretting-A Review", Wear, Vols. 225-229, Part II, pp. 962-967, (1999). 12. Schouterden, K., Blanpain, B., Celis, J.P., and Vingsbo, O., "Fretting of Titanium Nitride and Diamond-like Carbon Coatings at High Frequencies and Low Amplitude", Wear, Vol. 181-183, pp. 86-93, (1995). 13. De Bruyn, K., Celis, J.P., Roos, J.R., Stals, L.M., and Van Stappen, M., "Case StudyFretting Behaviour and Porosity of Ti2N Coatings", Wear, Vol. 181-183, pp. 356-861, (1995). 14. Sandstrom, P.W., Sridharan, K., and Conrad, J.R., "A Machine for Fretting Wear Testing of Plasma Surface Modified Materials", Wear, Vol. 166, pp. 163-168, (1993). 15. Feng Zhong-Chao, Dong Xiang-Lin, Li Mei, Zhang Bing-Chun, and Wang Ya-Qing, "Fretting Wear Resistance of Ti(C,N) Films Produced by Laser Chemical Vapor Reaction", Proc., Int. Symposium on Fretting, Chengdu, P.R., China, pp. 188-194, (1997).
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16. Novak, S., Drazic, G., Kalin, M., and Vizintin, J, "Interactions in Silicon Nitride Ceramics vs. Steel Contact Under Fretting Conditions",Wear, Vols. 225-229, pp. 1276-1283, (1999). 17. Amouzouvi, K.F., Clegg, L.J., and Styles, R.C., "Surface Modification of Zirconium Alloys by Laser Glazing," Int. Conf. on Surface Engineering, Toronto, Canada, pp. 270279, (1990). 18. Gureyev, D.M., Mednikov, S.I., Yamtchikov, S.V., "Laser Radiation Influence on the Surface Wear of Machine Parts Made of High-Chromium Cast Iron", Proc., 5m Int. Congress on Tribology, Finland, pp. 335-340, (1989). 19. Sagaro, R., Ceballos, J.S., Blanco, A., and Mascarell, "Tribological Behaviour of Line Hardening of steel Ul3A with Nd'YAG Laser",Wear, Vols. 225-229, pp. 575-580, (1999). 20. Attia, M.H., "A Thermally Controlled Fretting Wear Tribometer-A Step Towards Standardization of test Equipment and Methods", Wear, Vol. 135, pp. 423-440, (1990). 21. Stowers, I.F. and Rabinowicz, E., "The Mechanism of Fretting Wear", Trans ASME, J. Lubrication Technology, pp. 65-70, (1973).
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Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
531
IN-VITRO MODEL TO EVALUATE THE EFFECT OF ATTACHMENT DESIGNS ON STRESSES TRANSFERRED TO SURROUNDINGS IN IMPLANT-RETAINED OVERDENTURES
El-Wakad, M.T. Assistant Professor, Production Engineering Department Faculty of Engineering, Helwan University, Helwan, Egypt. e-mail" [email protected]
ABSTRACT Overdentures in full edentulous patients have been developed using dental implants as supports instead of using natural roots in partial edentulous patients. The overdentures are connected to the implants via several common attachment designs such as the ball and socket, straight bar, and magnets. The function of the attachment is to retain the denture and to lower the stresses transmitted to the bone surrounding the implants. The aim of this study is to find out the design that transmits the least stresses to the surrounding bone. Therefore, an in-vitro strain-gauged model of a lower mandible with two implants was devised. The results obtained after loading showed that magnet attachment transmits the least stresses followed by the bar, then, by the ball and socket. It, also, showed that both, the ball and socket and the bar designs expose the implant on the balancing side to a pulling effect, which increases the risk of implant failure. KEYWORDS Biomechanics, Dental Implants, Strain Gages, Bridge Attachments I. INTRODUCTION Complete overdentures supported and retained by few natural roots proved to have functional benefits. Therefore, the same concept has been developed for edentulous patients using osseointegrated implants instead of natural roots. This new concept was used with high success rate [ 1]. This success encouraged many investigators to use it in practice. Documented results of these investigators showed a great success [2-4]. In any implantation operation the main concern is to preserve the marginal bone against overloading. This concern is not valid in the case of the natural teeth due to the presence of the ligament [5]. Therefore, research in the implantaion area was directed towards minimizing the harmful effects of the overloading due to the absence of the natural ligament [6]. This could be done using various procedures including design of implants and their attachments [7]. Understanding the biomechanical principles involved in implant supported denture helps in minimizing and/or preventing the overloading of the prosthesis [8].
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Different attachment designs were used to improve the retention between the implant and the overdenture. The most used attachment designs in the literature are the ball and socket, straight bar, and magnets [9-11 ]. Reports about failure rates of implants that employ different attachment designs were a matter of controversy. Some investigators reported no difference between the three attachment designs mentioned above [9]. On the other hand, other studies reported that marginal bone resorption around implants inter-connected with bars is less than those unconnected [7, 11-13]. Therefore, it is the aim of this study to compare the effect of the currently used attachment designs on the stresses transmitted to the surrounding medium around implants supporting complete overdentures. 2. MATERIALS AND METHODS
2.1. Model preparation A model of a correct edentulous mandible was made using alginate impression material (Kromopan, Lascod S.P.A. Lab., Fiorrentino, Italy). The crest was covered with a light body of rubber base (Silicon Rubber, Bayer Co., Leverkysin, Germany) to simulate the soft tissue. The model was placed on a surveyor table. The straight hand piece of the surveyor was used to drill two parallel holes in place of the canines. Two brass analogues of screw implants (Impladrill Swiss) were fixed in the two holes using acrylic resin. 2.2. Strain gage application The model was thinned on both buccal (towards the cheek) and lingual (towards the tongue) sides of each analogue abutment. Four axial gages (TML, FLA5-IL-II-120f~, gage size 10 mm x 3 mm); one on each prepared site; were glued. A dummy gage was glued on an unloaded piece of the model material. The gages were connected in a half bridge configuration to a strain measuring device (Philips-Carrier frequency bridge, model 9307, Germany). 2.3. Experiment procedure A surveyor was modified to act as a loading device (Fig. 1). A load of 20 N was used to produce measurable readings on all gages. Two ball and socket attachments (Fig. 2) were screwed in the implant analogues (design 1). The denture was, then, adapted [14] to the sockets. The load was applied to one side of the mandible and readings were recorded from the two strain gages on this loaded side as well as from the two gages on the unloaded (balancing) side. The ball and socket attachments were, then, replaced by the bar attachment (Fig. 3) which was modeled to fit on the analogues (design 2). Finally, the magnet attachment (Dyna magnet 300 gin., 2.5 mm, Dyna Dental Eng., 4600AB, Bergen, Holland)was used (design 3 in Fig. 4). The loading and measuring procedures were repeated five times for each gage. The obtained data were tabulated and analyzed using ANOVA. 3. RESULTS The average of the measured readings from each of the four gages on the loaded and unloaded sides under a load of 20 N are shown in table 1 for the three attachment designs. Generally, the recorded readings showed both compressive and tensile values. This means that some of the forces were extrusive while others were intrusive. The readings on the loading side gages were more compressive than their corresponding ones on the balancing side. On the loaded side the buccal readings were higher than the lingual readings in all designs. On the balancing side, the buccal readings were opposite in direction to lingual readings in all designs.
Current Advances in Mechanical Design and Production, MDP-7
Fig.2. Ball and socket attachment in place. Arrows show axial buccai gages
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Fig.4. The magnet attachment
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The largest compressive and tensile readings recorded were in the ball and socket design from the buccal gage on the loaded side (-36.4 ttstrain) and the lingual gage on the unloaded side (35.2 ~tstrain). The smallest compressive and tensile readings recorded were in the bar attachment from the lingual gage on the balancing side (-1.2 ttstrain) and from the buccal gage on the balancing side (9.0 lastrain). TABLE 1: Readings of strain gages in ttstrain at a Load of 20N
1"~~ GAGE DESIGN ~ BALL AND SOCKET BAR MAG'NET' SIGNIFICANCE (P),
LOADEDSIDE BUCCAL 'LINGUAL -36.4 • 2.2 "14.4• i 1 -22.0 ~: 2.0 ' -5.0~ 0.7 -16.8 • 2.6' -8.6 + 413
-
BALANCINGSIDE .......... BuccAE LINGUAL :17.8 • 2.3 35:2 ~ 3.0 9.0 + 1.0 -1.2 • 1.79 .7.4 • 1.1 918 • 1.5 <0.001 <0.001:"'
Looking at the results of the different attachment designs from all the gages on both the loaded and unloaded sides it is seen that it can be ranked in a descending order. It was observed that the highest rank was for the readings recorded from the ball and socket design followed by the bar and magnet attachments. The readings on the loaded side were compressive in nature. It showed a bending towards the buccal gage. On the unloaded side the readings were both compressive and tensile. It showed a bending towards the buccal gage in both the magnet design and the ball and socket design. In the bar attachment the bending was towards the lingual gage. DISCUSSION The aim of this study was to determine the attachment design that transmits less stresses to the surrounding material. This assures the protection of the surrounding bone from failure. Lower stresses are preferable on both loaded and unloaded sides. Therefore, the measurements were carried out on both sides to determine whether or not the design that lowers the stresses on loading side is able to lower them on the balancing side. First the causes behind these results will be discussed. Then, the contribution of these results to choosing one design will be discussed. Usually, in the case where only two supporting implants are used, the magnitudes of the moments are expected to be large [8]. The transfer of these moments to the bone surrounding the implant may cause overloading. This will risk the integrity of the implant and consequently the failure of the prosthesis. Therefore, it is recommended to use more than two implants to reduce the stress transferred into the bone. However, in cases where only two implants can be used then other techniques should be employed to reduce the effect of the bending moment [8, 18]. The technique available, now, is the use of attachments. Figure 5 shows the model with the load (P) acting at a point on an overhang. Studies have shown [15-17] that such a load causes two perpendicular bending eomponents, MI in the mesio-distal direction (towards the front of the body) and M2 in the bucco-lingual direction (towards the cheek).
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On the loading side it can be seen that the buccal gages to be subjected to higher compressive stresses (-36.4, -22.0 and -16.8 ~tstrains for designs 1,2 and 3 respectively) than lingual gages (-14.4, -5, -8.6 ~tstrains for designs 1,2 and 3 respectively). According to the loading situation described in the figure and from the gage readings this is attributed to the effect of the bending moments added to the effect of the load (P).
Buccal strain gages
M2
Fig.5. Drawing showing moments resulting from the applied load P On the balancing side readings tend to be generally lower than the loaded side. This is because its gages are further from the load. In addition the compressive effect of the load (P) and of the moments M1 and M2 are not augmenting each other. The highest readings recorded were in the case of the ball and socket attachment. It may be expected that this design should give lower readings since ball and socket does allow transmission of moments. However, the case here is that two attachments are used for the purpose of retaining the overdenture in place during mastication. This causes the overdenture to be completely locked. This lock allows the transfer of all applied loads and moments on the denture to the implant and then to the surrounding medium. Thus, the risk of failure of the implant and the denture is higher [19]. The risk is even greater on the implant on the balancing side. This is due to the pulling effect from the intrusive force (35.2 ~tstrains) [20]. The observed readings (bending while pulling) on the balancing side mimic the force applied by the dentist to extract a tooth. In the case of the bar attachment the situation has some similarity to the ball and socket. The bar attaches the two implants to each other. All the loads are transferred from the overdenture and distributed to both the bar and the implants. Therefore, the recorded readings in this case are lower than those in the ball and socket attachment. On the balancing side, there is a pulling effect on the implant (9 ~trains) similar to the ball and socket case but to a lower extent. The lowest readings were recorded from the magnet attachment, on the loading side -16.8 and -8.6 ~tstrains while on the balancing side -7.4, 9.8 ~tstrains. This could be due to the fact that this attachment permits free horizontal movement of the overdenture while maintaining the vertical retention [11]. This is a favorable loading because it reduces the effect of the moments on the supporting bone. Thus, longer functional life of the implant and prosthesis is expected [ 19]. The readings on the balancing side are close to each other with different signs.
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This means that the net effect is almost bending with very little pulling. Thus, the pulling effect on the balancing side is not as great as those observed in the ball and socket designs. CONCLUSION Based on the study of the three attachments and the previous discussion the following could be concluded: The magnet seems to be the one that transmits the least stresses while the ball and socket attachment is the one to transmit the most stresses to the surrounding materials. Therefore, the risk of failure is lowest in the magnet attachment. .
On the balancing side the ball and socket attachment expose the implant to relatively large pulling effect which increases the risk of failure. From a biomechanical point of view the development and/or selection of attachment design should consider the designs that permits free movement while maintaining retention and avoid designs that provide complete lock.
REFERNCES 1. Ettinger R.L., Taylor, T.D., and Scandrett, F.R., "Treatment Needs for Overdenture Patients in a Longitudinal Study: A Five Year Results", J Prosthetic Dentistry, Vol. 52, pp. 532, (1984). 2. Engquist, B., Bergendal, T., Kallus, T., and Linden, U., "A Retrospective Multicentre Evaluation of Osseointegrated Implants Supported Overdenture", Int. J. Maxillofacial Implants, Vol. 3, pp. 129, (1988). 3. Mericske-Stern, R., "Clinical Evaluation of Overdenture Restorations Supported by Osseointegrated Titanium Implants: A Retrospective Study", Int. J. MaxiUofacial Implants, Vol. 5, pp. 375, (1990). 4. Naert, I., DeClerc, M., Theuniers, G., and Schepers, E., "Overdentures Supported by Osseointegrated Fixtures for the Edentulous Mandible: A 21/2 Year Report", Int. J. Maxillofacial Implants, Vol. 3, pp. 191, (1988). 5. Renenr, PR., "The Overdenture Concept", Dental Clinics of North Am., Vol. 34, No.4, pp. 595, (1990). 6. Branemark, P.I., Zarb, G., and Albrektsson, T., "Tissue Integrated Prosthesis" in "Osseointegration in Clinical Dentistry", 2"0 Ed., Quinstessence Pub. Co., pp. 89-116, (1985). 7. Quireynen, M., Naert, I., Van Steenbergh, D., and Nys, L., "A Study of 589 Consecutive Implants Supporting Complete Fixed Prosthesis. Part I: Periodontal Aspects", J. Prosthetic Dentistry, Vol. 68, pp. 655-663, (1992). 8. Rangert, B. and Sullivan, R., "Biomechanical Principles: Preventing Prosthetic Overload Induced by Bending", Nobelpharma News: Vol; 7, No. 3, pp. 4, (1993). 9. Naert, I., Quirynen, M., Hooghe, M., and Van Steenberghe, D., "A Comparative Prospective Study of Splinted and Unsplinted Branemark Implants in the Mandibular Overdenture Therapy: A Preliminary Report", J. Prosthetic Dentistry, Vol. 72, pp. 486, 1994. 10. Nieport, H.M. and Plooij, J., "The Treatment of Severely Resorbed Edentulous Mandible with an Implant Supported Magnetic Overdenture. A Clinical Study in a 100 Patients:
538
11. 12.
13.
14.
15.
16.
17. 18. 19. 20.
Current Advances in Mechanical Design and Production, MDP-7
Preliminary Results", Abstract represented in the I st International Implantology Conference for ESSDI, Cairo, March (1995). Plooij, J. and Mieuport, H.M., "Magnetic Retention for Implant Supported Magnetic Overdenture", Abstract represented in the 1st International Implantology Conference for ESSDI, Cairo, March (1995). Quirynen M, Naert I, and Theuniers G, "Prosthetic Aspects of Osseointegrated Fixtures Supporting Overdentures: A 4 Year Report", J. Prosthetic Dentistry, Vol. 65, pp. 671, 1991. Quirynen M, Naert I, Van Steenbergh D, Teerlink J, Dekeyser C, and Theuniers G, "Periodontal Aspects in Osseointegrated Fixtures Supporting an Overdenture: A Four Year Prospective Study", J. Clinical Periodontology, Vol. 18, pp. 719, (1991). Kolodney, H., Holder, R., and Gray, W.C., "A Reliable Index for Correct Positioning of Precision Attachments into an Existing Overdenture", J. Prosthetic Dentistry, Vol. 67, pp. 335-338, (1992). EI-Charkawi, H.G., EI-Wakad, M.T., "Effect of Splinting on Load Distribution of Extracoronal Attachments with Distal Extension Prosthesis In Vitro", J Prosthetic Dentistry, Vol. 76, pp. 315-320, (1996). Skalak, R., "Aspects of Biomechanical Considerations", in Branemark PI, Zarb GA, Albrektsson T (eds.): "Tissue Integrated Prostheses", Chicago, Quintessence Pub. Co., pp. 117-128, (1985). Skalak, R., "Biomechanical Considerations in Osseointegrated Prostheses", J. Prosthetic Dentistry, Vol. 49, pp. 843-848, (1983). Sullivan, D., "Prosthetic Considerations for the Utilization of Osseointegrated Fixtures in Partially Edentulous Arch", Int. J. Maxillofacial Implants, Vol. 1, pp. 39-, (I 986). Rangert, B., "Lab Research at Rensselaer Bodes Well for Clinical Progress", The Nobel Biocare Global Forum, Vol. 10, No. 2, pp. 6, (1996). Sheets, C.G., Earthman, J.C., "Tooth Intrusion in Implant-Assisted Prosthesis", J. Prosthetic Dentistry, Vol. 77, No. 1, pp. 39-45, (1997).
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference
539
Cairo, February 15-17, 2000
ASSESSMENT OF RELIABILITY PARAMETERS FOR MAINTENANCE-BASED UNITS USING LINEARIZED WEIBULL MODEL
Mostafa, A.A-F." and Khattab, A.A." "Professor, Dept. of Mechanical Design and Production, Faculty of Engineering, Cairo University, Giza, Egypt.
ABSTRACT Assessment of reliability parameters for maintenance based units involves evaluation of: reliability function, maintainability function, availability function, failure and repair frequency functions, mean time to failure and mean time to repair, as well as the effect of preventive maintenance action on reliability and its relation to the cost-effectiveness function. The problem relating reliability with maintainability is solved using Markov's model. The Markov's model is well suited to exponential frequency distributions for both reliability and maintainability functions. The exponential distribution assumes constant hazard and repair rates. However, the Weibull model is the most used model to describe frequency distributions in reliability and failure problems. A linearizing scheme is proposed so that the Weibull distribution could be approximated to the exponential distributions, and therefore the Markov's model can be applied. The problem is solved for units connected in a system. Three types of units are considered; a single unit to be connected in series to other units, two units in parallel with one unit as active redundant and two units with one unit as stand-by redundant. KEYWORDS Reliability, Maintainability, Weibull Distribution, Markov's Model. I. INTRODUCTION The stochastic scheme of natural hazards or accidental loads and variation in material properties dictate a probabilistic approach for rational assessment of system safety and performance. In conventional reliability evaluation, the usual 'useful life period' of equipment is considered. In this case, time homogeneous Markov processes are used to evaluate reliability. However, during the 'wear out' period, where the failure rate is not constant, the use of Markov's model is not valid [1]. In this case the process is Non-Markovian [2]. One method for handling Non-Markovian processes is the device of stages. This is a method of representing a non-exponential state by a combination of stages, each of which is exponentially distributed [3]. Parameters that are usually listed to evaluate reliability condition are: failure probability density function, hazard (instantaneous failure) rate, cumulative failure probability function, reliability function and mean time to failure MTTF,[4]. The trend in
540
Current Advances In Mechanical Design and Production, MDP-7
modem technology is to intensify the use of plant and machinery. The occurrence of failure can be reduced by design or material change. However, some causes of failure will remain which are known as random fatTures. Different policies are suggested for reducing the inservice failure by replacing components before they fail. These policies are known as preventive maintenance PM schemes. Block and age-replacement policies have been treated by many investigators [5] and [6]. It was suggested that the best policy is to replace the component if its age exceeds a control limit [7]. The best preventive maintenance policy is based on economic consideration. Optimum policies could provide cheaper way for replacing components to block policies. The criterion used in search for optimal policy is the average cost per unit time. This concept was illustrated with trade-off equations that combines cost of preventive maintenance and cost of failure [8]. The result is the cost-effectiveness equation. The assumption of an increasing failure rate corresponds to the intuitive idea of a component or a unit deteriorating in service. The Weibull distribution could describe cases of increasing failure rates, if suitable values are assigned to its parameters. The three-parameter Weibull model contains three parameters: shape parameter, scale parameter and location parameter. In many cases, the location parameter is ignored and is dropped from the equation yielding a two-parameter density function. Weibull parameters could be evaluated using different methods [9]. 2. PERTINENT FORMULAE Reliability and maintainability parameters are evaluated with equations derived on basis of obtaining correct factors that fit with the assumed case. 2.1 The Weibull Distribution
The three-parameter Weibull probability density function is given as;
(1) where a, b and c are the scale, shape and location parameters respectively. In many cases, c is equal to zero, yielding two parameters Weibull function for the frequency distributionj~t); tb-~ ex f (t) = bai
- -~T
(2)
2.2 The Hazard Rate
For the two parameter Weibull model, the hazard rate z is; z = a
tp
where ; ct = b/a b and ~ = b-1
(3)
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2.3 Reliability and Failure Probability Functions
Based on the assumption of Weibull distribution, the reliability R(t) and the failure probability F(t) can be evaluated ;
and;
j
(4) (5)
F(t) = 1- R(t)
2.4 Maintainability Function
Based on the assumption of Weibull frequency distribution, the time to repair (0 is; I r/t r ] g(t) = rI t r exp - r + 1
(6)
The instantaneous repair rate(y) can be assumed similarly as; (7)
Y = rl t r
and the maintainability function M(t) can be obtained as ; M(t)= l-ex
p~ r/tr
--~.~j
/
(8)
2.5 Mean Time To Failure (MTTF) and Mean Time To Repair (MTTR) Based on the assumption of Weibull distributions, the mean time to failure MTTF and the mean time to repair MTTR can be derived; |
MTTF= o
where-
F is the gamma function;
1
F
1
(ft+i)
(9)
and
!
MTTR =
(~0)
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2.6 The Linearized Model
The two parameters Weibull model could be approximated, within a short period of time, to the exponential distribution, if 9 - For the failure frequency distribution; fl = 0, and A, = a - For the repair frequency distribution;
( = 0, and r/= ,u
Values of 2 and/~ are evaluated from eqns. (9) and (10) as; 1
A =-----=MTTF
(ll)
and; 1
/~= MTTR
(12)
2.7 Availability F u n c t i o n The Markov model could be solved for the hazard rate 2 and the repair rate/z as evaluated by equations (11) and (12) to give the availability function for one unit model in a series system as;
Aft) =
/~
/~+2
+
2
p+2
exp[-(2 +/~)t]
(13)
For steady state availability, eqn. (13) becomes; A(oo) .
/1 . . .u+2
UT . UT+DT
(14)
where ; UT = (1/~,) is the up-time and DT = (1/tt) is the down time. 2.8 Reliability and Availability of Two-units Parallel System with Repair The use of redundant units in mechanical systems is usually imposed by condition of safety and reliability. The use of more than one redundant unit in mechanical systems requires usually considerable expenditure for small return. Therefore, the two-units parallel system with one unit as being redundant is considered .The redundant unit could be connected as active-redundant or stand-by-redundant, as the case may require. Using the Markov's reliability model for two-units parallel system with repair, the equation for reliability can be assessed [10];
R(t) =
sl $1 - $2
erZ'
s2
en'
(15)
$1 - $2
where; (16)
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Eqns. (15) and (16) could be used to asses the system reliability, where Xo, Xt and ~tl are the hazard rates for no-failure (both units working, subscript 0), for one failure (one unit working and the other is faulty, subscript 1) and repair rate when one unit is working and the other is being repaired. For two-units with one unit as active-redundant: X0=2~., Xl=X and ~tt=la For two-units with one unit as stand-by-redundant: Xo=X, Xt=Z. and ~tl=~t X and ~t are estimated using Eqns (l 1) and (12). The availability can also be estimated using the same model [ 10].
s~s2j
s,-s2
s,
s2)
(17)
where;
(18) where the subscripts indicates the following states of the two units as: 0 : indicates state of no failures, 1 : indicates the state of one unit has failed and one is still active and 2 : indicates the state that both units have failed. For the case of one unit as active redundant: L0=2;~, Lt=L, lal=~t and ~t2=2~t For the case of one unit as stand-by redundant: L0=L, Z.I=TL,~tm=~tand l.t2=2~t.
2.9 Reliability with Preventive Maintenance PM Action If (T) indicates the time interval for PM action, the reliability of the system at time (t>T) is; R(t/T) = R(t-T) R(T)
(19)
where; R(tlF) is conditional probability for successful operation. After maintenance has been carried out, we may assume that the system becomes as good as new. The reliability of the system after repair has taken place is, therefore, equal to the probability of the system being operational at time (T) in the Markov's model. Therefore; R(T) = Po(T) - A(T)
(20)
and eqn. (19) afterj times PM services becomes; R(t) = A(/T) R(t-jT)
, forjT < t<(/'+ 1)T
(21)
2.10 Mean Time to System Failure (MTSF) with PM Action Eqns. (9) and (I0) were derived assuming that no maintenance action has taken place to modify the state of the unit. If maintenance action has taken place, the mean time to failure is called the Mean Time To System Failure, MTSF. Assuming that PM action are taking place at regular time interval (T); therefore
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Current Advances in Mechanical Design and Production, MDP-7
MTSF =
pj+l)T
JR(t- jT)dt =
d~TR
(x)dx
(22)
For a unit connected in series in a system, the MTSF aRerj times PM actions is;
(23) For two units connected to a system with one unit as active redundant;
(MTSF)j
A(jT)(3_4e-ar +e -2~r) 22
=
(24)
For two units connected to a system with one unit as stand-by redundant;
A(J~[2-e-'ir(2+,l,T)] 2
(MTSF)j =
(25)
2.11 Preventive Maintenance and Cost-Effectiveness Theoretically, there exists a point where the total cost of failure and cost of preventive maintenance action per unit time of service is at minimum. The time interval for this minimum is considered the optimum time for preventive maintenance action. The equation for this condition (cost effectiveness) is given by Dodson [8] as; oo
C~
T
oo
T
T
0
(26)
T ~f(t)dt + ftf(t)dt T
0
where ; Cp is the cost of PM actions C!is the cost of failure. T is the time interval for PM action. Eqn. (26) can be applied to the different cases of a unit in a system. For the case of a single unit connected in a system, the failure frequency equation is;
f(t) = ge -~r exp[-2(t - jT)] and the cost-effectiveness equation is ;
CurrentAdvancesin MechanicalDesignandProduction,MDP-7
545
Cpe -xr +Cf(1-e -at) C=
(l_e -at)
(27)
2 For two units with one unit as active redundant connected in a system, the failure frequency equation is ;
and the cost-effectiveness equation is ; C = A/B where;
(28)
A= Cp[l-(l-e-ar~]S[2-eXr(s-')~xr('-')+ C,[I - (1- e-Xr)~l' •
{ear(s-l)[e'lr(S-l>-2]-e'Vr(e asr-2)} B= T[I- (1- e-ar)2]s[2- ear(s-')]ear(S-')+[1-(1- e-ar)~} • e 2a(s-l)r T
- 2e a(s-I) T + 22
+---e 2
For a new system that is first installed,j = 0 For two units system with one unit as stand-by redundant connected in a system , the frequency equation is;
f
C=A/B
(29)
where ; ,4 = C p [ 1 - A T ( y - I ) ~
s-')r
B= T[I- 2T(j-1)]e (j-l'ar
+ Ci [(2j'T -
A T - l)e 'j-')ar - ( 2 j T
-1)e *r ]
+(jT+ jT22-2T 2 -2T-.2)ea(S-I'r - [ j T - 2 ) e ~r
Similarly, if the system is first installed, j = 0
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Current Advances tn Mechanical Design and Production, MDP-7
3. ILLUSTRATING EXAMPLES The above equations estimate most of relevant reliability and maintainability parameters needed to describe the performance of a unit in a system. To demonstrate the applicability of these equations, the following examples are solved : a- A Weibull prabability density distribution for failure is assumed with the following parameters' values : - modified scale parameter" ot = X =10 .7 - modified shape parameter : 13= 0 , 0 . 5 , 1 , 1 . 5 and 2. b- An Exponential distribution is assumed for the probability density distribution for repair, or a Weibull distribution with the special Weibull values as" - modified scale parameter: 11 = ~t ---0.01 modified shape parameter: ~ = 0 The parameters for Cf, and Cf are assumed to vary in proportion to (W~,) with Cp=l. -
5. RESULTS AND DISCUSSION Figure 1 shows curves for reliability of a single unit and for different Weibull shape factors, over a period of 1000 hours. The effect of the value of 13(the shape factor) can be evidently noted. The curve for Maintainability indicated in Fig. 1, as M(t) is seen to follow the exponential distribution. Figure 2 is for the probability density distributions for the cases shown in Fig. 1. The curves for 13<1.5, can be hardly noted in the figure, on account of their small magnitudes, compared with other curves. The curve ofj(0 for 13= 2 is similar to that of the normal curve. The curve for g(t) for the maintainability function is seen to follow the exponential curve. Based on the Weibull assumption the MTTF is calculated and is shown in Fig. 3, where it can be seen the great difference of values for the MTTF for different values of 13. Values of MTTF are used to estimate the mean value of the hazard rate X for an approximating exponential distribution. The difference between the approximated exponential distribution and the actual Weibull distribution for the probability density is calculated and reported as error ratio:
fw-f, Error ratio =
fw
where, fw is the value of frequency distribution for Weibull distribution, andfe is the value of frequency distribution for exponential distribution. The result is shown in Fig. 4, where it can be seen that there are periods of time where the approximation may be acceptable. For other time periods, the approximation is not as good. This suggests the solution of [3] by taking the approximation for different time period-steps rather than considering the whole life span. Based on the assumption of approximating the Weibull distribution, the availability function could be evaluated and examples are shown in Fig. 5. The repair is assumed to follow the exponential distribution with instant repair rate p = 0.01. The steady state availability is seen to be affected with the value of the shape parameter [3.
Current Advances in Mechanical Design and Production, MDP-7
547
Fig. 6 shows the effect of preventive maintenance action PM on reliability of single unit. The PM interval is taken as 250 hours, and it can be seen how the reliability is greatly affected with the PM action. For example for [I = 2, the reliability of the unit as shown in Fig. 1 is nearly zero for working time exceeding 500 hours, while in Fig. 6, reliability is greater than 0.65 for the same shape factor over a working time of 750 hours. This demonstrates the beneficial effect of carrying out a preventive maintenance scheme on units specially in the deteriorating stage (worn-out) period. If the unit is deteriorating in service and is backed-up with a redundant unit , the situation is further improved. This can be seen in Fig. 7, for two units with one unit acting as active redundant unit. Though the PM time interval is increased to 500 hours, reliability is higher. With maximum value for [~ = 2, it is greater than 0.8. The higher reliability values and the increased PM time interval can justify the economy of using a redundant unit despite the expected increase of installation cost . The situation is further improved, if the redundant unit is connected as stand-by unit. Fig. 8 shows that for two units, with one unit acting as stand-by redundant, and for [3 = 2, reliability is better than 0.88, even for 1000 working hours. To complete the picture, the cost effectiveness factor is estimated. Fig. 9 gives an example for a single unit in a system. The effect of the shape factor [3, is not seen to affect the value significantly, and the cost effectiveness ratio is seen to diminish with increasing PM time-interval. The same observation is noted by Dodson [8], that for exponential distribution, the optimum mean value between maintenance action is infinite, and "the optimal policy is no preventive maintenance". This statement is based, however, purely on economic bases, with no regard to safety or other obliging commitments. However, the cost effectiveness is estimated for values of Cp and Cf that were assumed arbitrarily regardless of actual values which could modify this observation. Dodson [8], moreover, noted that "for Weibull shape parameter, less than 1, preventive maintenance restores the system to a less reliable state than before preventive maintenance action was taken place ". This is a case of reducing the hazard rate (,8<0) with working time. This is a case similar to what happens in the running-in period, and such case is not considered in the investigation. 5. CONCLUSIONS Equations are derived to asses values of different reliability parameters. The Weibull distributions could be approximated to exponential distributions, over limited periods of time. Reliability can be greatly improved by PM actions and/or by using redundant units. Stand-by redundancy has an advantage over active redundancy. If the performance of the unit has deteriorated, and its reliability is reduced, the use of redundant units can improve not only the reliability of the system but also the economy of operation, as it allows PM action to be taken at greater time interval. Preventive Maintenance, if carried out properly, could prolong the service life almost indefinitely. The mathematical equations derived, herewith, seem to certain these conclusions. REFERENCES 1.
2.
Asgarpoor, S., Mathine, M.J., "Reliability Evaluation of Distribution Systems with NonExponential Down Times", IEEE transactions on Power Systems, Vol. 12, No., pp. 579 584, May (1997) . Singh, C., Billinton, R., " System Reliability Modeling and Evaluation", Hutchinson Educational, London, (1977).
Current Advances in Mechanical Design and Production, MDP-7
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3. 4. 5. 6. 7. 8. 9. 10.
Billinton, R., Lian, G., "Monte-Carlo Approach to Substitution to Reliability Evaluation", IEEE proceedings - C, Vol. 140, No 2 March , pp. 147-152,(1993). Leemis, L.M., "Life Time Distribution Identities", IEEE transactions on Reliability, Vol. R-35, No 2, pp. 170-174., (1986) Flehinger, B.J., "A Great Model for the Reliability Analysis of Systems under Various Preventive Maintenance Policies", Annuals Math. Statistics, Vol. 33 , pp. 137, (1962). Crook P.C.I., "Replacement Strategies", OR, QI4, Vol. lc, p. 167, (1963). Woodman R.C., "Replacement Policies for Components that deteriorate", Operation Research Quarterly, Vol. 8, No 3., p 267, (1967). Dodson, B., "Determining the Optimum Schedule for Preventive Maintenance", Quality Engineering, Vol. 6, No 4 , pp. 667-679, (1994). Sinha, S.K., Sloan, J.A., "Bays Estimation of the Parameters and Reliability Function of the 3-Parameter Weibull Distribution", IEEE trans, on Reliability, Vol. 37, No 4, (1988). Balagurusamy, E., "Engineering Reliability", Tara McGraw-Hill Pub. Co., pp. 118-138, (1984). , Rr
o., 0.7 I "~ ~
0.6
~
./ 1
o.a
o~
o.~
t
0
.
, '
L M
0.4
........ R ( , . IS.o.s
/
\
I
..
I=Kt]-~ -
"~J ,,
\
1
--,a,.-n(t)-~= 9 t .s l
--x-- ~t)- D -2 - - , o - - M(1)-./,l=.O'~
.
.
.
""
200
0
400
8OO
eO0
I
1OOO
(am,m)
F I G . 1. R E L I A B I L I T Y
& MAINTAINABIUTY
4.00E-03 r
3.ooE-o3
- - . - - x - - - - I~,,l.o
"~- 7_oos-o3
------o..--- r ..-...o-~ F. 2
! .00 -E,-~3
,--9 -dE - - p,,O.Ol O.C~E+~
0
200
400
600
800
I0OO
lrwlz 0mum) FIG.
2 . FAILURE
AND
REPAIR
FREQUENCY
DISTRIBUTIONS
1.00E*O6
!
.--."'
m •
|_ | : ....
|
:i
1.00E,~2
0
0.2
o.,
o.6
o.s
w~euu.
FIG.3.
MEAN
,.2,
t SHA~
TIME
TO
~Ac'ron
L4
,.~
(p)
FAILURE
(MTTF)
t.s
2
549
Current Advances in Mechanical Design and Production, MDP- 7
'IX ,~, ""i
o I-..
0
--
-
&i.-~ .~ ~
-5 -10
o w
-15-
!
.
.
-20
.
, . %
I
'f
.
/
-;
""
~ _i ,i T
~-----,,-_~ ,,, ~-
~
,
9
~...,
-~_ ...=' " - G -_~_=_~. "~ - , .....
!. . . . . . . . . . . a~
:x; ~
'
13-2 !
.
9
9
\-
\
X
-25 ~
8 ,i-=
TIME ( h o u r s ) FIG. 4. ERROR RATIO OF FREQUENCY DISTRIBUTION FOR LINEARIZING EFFECT
0.98
1
~
I
,
-
~k~
,,_
,
p=0
0.96 0.94
I~=U.01
....
1~=0.5
....
13=1
-
~2.5
0.92 9
0.9
200
,. . . .
|
.' . . . . .
400
600
= ~ ,, ' " , 800
1000
TIME (hours)
FIG. 5. AVAILABILITY FOR DIFFERENT WEIBULL SHAPE FACTORS
1
w
-
, - - - - . - - - . -
0.95
i
0.9 0.85 ~"
t
0.8
- . m - 13=o.~
--a--p-~.5
O.75
--o--13=2
0.7 0.65 .
I
13=1.5
1
O
"rIME ( h o u r s )
FIG. 6. RELIABILITY AND MAINTAINANCE (single unit)
Current Advances In Mechanical Design and Production, MDP-7
550
~ ' 13=0
0.g5
--x-- 13--0.5
~. 0.9 w
--e~
p=l
---o-- !~=~.51
0.85 0.8
TIME (hours)
FIG. 7. R E L I A B I L I T Y WITH ONE A C T I V E - R E D U N D A N T AND P.M. TIME INTERVAL 500 hrs ! .02
~.o.s
I 0.98 0.96
m 0.94
-o-13-2
0.92 0.9
0.88 0
0
0
0
0
0
0
0
0
0
0
TIME (bourn)
FIG. 8. R E L I A B I L I T Y WITH ONE S T A N D - B Y UNIT AND P.M. TIME I N T E R V A L 500 hours 2.5OE~2
:9
Cp=l .Cf: 1E+05 :Cp= 1 ,C :f=g.32
o Imffl o u
- - - - - - 13=2
2.~E-t~
1.SOE-02,
75
100
125 P.M. TIME (hours)
150
175
FIG. 9. C O S T R A T I O FOR DIFFERENT VALUES OF P.M. TIME INTERVALS
Section VI
INDUSTRIAL E N G I N E E R I N G
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Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
553
OPTIMAL REPLACEMENT OF COMPONENTS SUBJECT TO DEGRADATION
Elsayed, E.A. Department of Industrial Engineering, Rutgers University Piscataway, NJ 08854-8018 email: [email protected]
ABSTRACT Many of civil engineering structures and constructed facilities throughout the world are in a condition of serious deterioration. This deterioration causes significant degradation in their mechanical properties and their ability to perform the intended functions without failure. Indeed, the degradation is accelerated by exposure to environmental conditions that contain sulfur dioxide and other oxidation agents. Moreover, infrequent actions such as earthquake, fire and improper use or maintenance result in acceleration of the structure degradation. Therefore, it is important to develop reliability models, based on degradation data, which accurately determine the residual life of structures. In this paper, we propose a reliability model that utilizes the degradation data obtained from a structural component. We then determine the residual life of such component. 1. INTRODUCTION
1.1 Reliability and Degradation Reliability is one of the most important characteristics of products, equipment, structures and service. Since reliability is a time-dependent characteristic, it is impossible to measure it at the time of the product release, it can only be estimated. Therefore, researchers have focused their effort on developing methodologies for reliability estimation. One of the most widely used approach for estimating reliability is testing. This is usually accomplished by using operational life testing, reliability demonstration testing, burn-in testing and accelerated life testing. In all cases, failure data is gathered and used in a reliability prediction model to estimate the expected life of the product. Highly reliable products or components may not experience failures at realistic test conditions. Moreover, causing a failure of some products is undesirable such as in the case of one --of-a-kind products; e.g., concrete beams of bridges, satellites and electric power generators. In such cases, some measures of the system performance are used as an indicator of its reliability. Drift in the value of a resistor of an electric circuit and change in the light intensity of light emitting diodes (LED) are examples of such measures. We refer to this measure as a degradation measure, which is observed over time, and its values are recorded to be used later in reliability estimation. When the change in the degradation measure is slow, we utilize accelerated degradation testing so that we can observe the change of the degradation measure in a short time. We refer to this type of testing as accelerated degradation testing.
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Degradation is the most appropriate measure for reliability estimate of structures. Many researchers have investigated the structural reliability problem. It has gained special importance due to the increased attention to the serious deteriorating state of the civil infrastructure that includes bridges, tunnels, highways and large structures. When an engineering structure is loaded in some way it will respond in a manner which depends on the type and magnitude of the load and the strength and stiffness of the structure. Whether the response is considered satisfactory depends on the requirements which must be satisfied [Melchers, 1987]. Such requirements might include safety of structure against collapse, limitation on damage, deflection or any other measures. Not meeting such requirements is considered a limiting state violation. Hence, the reliability of a structure is defined as the probability of occurrence of the limit state violation at any stage during its lifetime. This probability is can be obtained from measurements of the long-term occurrence of the event on similar structures, subjective estimation, or by using small-scale prototypes and testing them at different conditions. Alternatively, a measure of the structure degradation with time such as rate of corrosion can be used to determine the time at which the structure will seize to function as desired. The level of degradation at that time is referred to threshold degradation
level. 1.2 Classification of Degradation Models
Elsayed [1996] classifies degradation models as physics-based and statistics-based models. The physics-based degradation models are those in which the degradation phenomenon is described by a physics-based relationship such as the Arrhenius law and corrosion initiation equation or experimentally-based results such as crack propagation or crack growth models. The statistics-based degradation models are those in which the degradation phenomenon is described by a statistical model such as regression. Relevant research on physics-based degradation models is described below. Nelson (1981) develops a life time distribution based on the performance-degradation relationship. Tydeman and Kirkwood (1984) design an accelerated degradation model to analyze the stability of biological standards. Carey and Koenig (1991) estimate the reliability of an Integrated Logic Family (ILF) using a degradation model. Ioannides et al. (1993) utilize a degradation model to predict the life time of ball and roller bearings. They focus on the importance of the cleanliness for bearing performance and relate both the size and the hardness of the particles in lubricant to the bearing performance. Meeker and Hamada (1995) discuss a genetic model for a tmivariate degradation process. Tseng et al. (1995) present an experimental design for improving system reliability. Chick and Mendel (1996)develop a statistical lifetime model for curing tools based on the tool-wear curve. The tool wear rate is directly proportional to the degradation rate of the tool. Feinberg and Widom (1996) use the Arrhenius law and present a thermally activated time-dependent model that relates aging to catastrophic failure. They show that the Arrhenius degradation behavior can lead to a lognormal distribution for the corresponding time to failure distribution. Qureshi and Sheikh (1997) use a physics-based relationship to model adhesive wear. Chuang et al. (1997) present a kinetic model for the degradation of LEDs. Recently, Meeker et al. (1998) present a physics-based model for accelerated degradation tests. The limitations of physics-based degradation models are: There is no universal physics-based or experimental-based relationship that describes the degradation phenomenon of all products, which makes it impossible to develop a general degradation model.
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It is time consuming to develop a physics-based (or experimental-based) relationship for new products. Physics-based (or experimental-based) degradation models may not be suitable for the development of closed form reliability functions. This is due to the fact that some of the parameters are random variables; the degree of difficulty of deriving a reliability function depends on the distribution of these random variables. Except for simple eases, the reliability function can not be easily derived.
Statistics-based models are more general than the physics-based models which has lead to the development of several statistical approaches for modeling degradation data. Lu and Meeker (1993) use Monte Carlo simulation to obtain point estimates and confidence intervals for reliability estimation based on degradation data. They present a general path model for degradation that includes both fixed-effect parameters, common for all units, and randomeffect parameters representing individual unit characteristics. Stock et al. (1994)classify component performance into four states: operational state, degraded state, failed state, and maintenance state. They assume that the amount of time spent in a particular state is an exponentially distributed random variable. They then utilize continuous time Markov models to estimate the time to failure and other measures of product performance. Yang (1994) categorizes the source of variation in the product performance as piece to piece initial variation (caused by manufacturing), piece to piece degradation rate variation (caused by disparities) and measurement error or non-degradable stationery environmental variations. Lu et al. (1997) present a linear degradation model with random coefficients and time-dependent standard deviation for the error term. It is assumed the random coefficients follow a bivariate parametric distribution. These statistics-based models assume linear degradation paths and/or constant standard deviation. Furthermore, they are developed for a given stress level so that reliability can only be estimated at the stress conditions at which the degradation data are collected. Recently, Eghbali [1999] developed a Proportional Degradation Hazards Model (PDHM), a statisticsbased model, at which the degradation paths can be considered nonlinear functions of time and the standard deviation is time dependent. In this paper, we develop a statistics-based approach that relaxes the linearity of the degradation paths and constant standard deviation assumptions. The proposed approach can be extended to provide reliability estimates at different stress conditions. Researchers have utilized the approach of level crossing of random processes to determine the time for a random process (such as degradation) to cross, i.e., to exceed some specified event [ Diltlevsen, 1983]. Blake and Lindsey (1973) discuss the level crossing problem, in general, and present a brief review of their solution techniques. Domine (1995, 1996) presents the first passage time probability density function and its moments when the random process is a Wiener process (or Brownian motion) with drift. Pieper et al. (1997) apply the first passage time theory to determine the life time distribution assuming that a Wiener process can model the degradation phenomenon. He obtains the probability density function of the first exit time for a degradation process with mean /~ and standard deviation tr and lower and upper degradation boundaries/11 and h2 respectively as
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f,(t) = ~-' ~ ,-, (/h _/~)2 exp
k21r2o'2t
2(/h _/~)2
/~2t
/z
20.2 ~ 5 x
x [ e x p ( - ~ ) ( - l ) '§ + e x p ( - ~ , ] sin(k~r ~ _ ~ , where x is the starting point. Doksum and Hoyland (1992) develop accelerated life testing models based on the Wiener processes. The assumption that a Wiener process (or Brownian motion) governs the degradation phenomenon (or cumulative damage){B(t):O<_t<_oo} with driR g > 0 a n d a diffusion constant tr ~ implies that for any 0 ~ s < t, the distribution of the cumulative damage at [ t,s ], or B(t) - B(s), follows a normal distribution with a mean of g ( t - s) and variance of o.2 ( t - s). The proposed approach does not require the normal distribution assumption for the cumulative damage, i.e., the Wiener process assumption is relaxed. We model the random process (or degradation phenomenon) by a family of distributions in which at least one parameter is a function of time. Details are presented next. 2. MODEL DEVELOPMENT It is assumed that the effect of the degradation phenomenon on the product performance can be expressed by a random variable called degradation measure. Typical measures include the amount of wear of mechanical parts such as shafts and bearings, the drift of a resistor, output power drop of light emitting diodes, the gradual corrosion of a reinforcing bar, the propagation delay of an electronic chip and the loss of stiffness of elastomeric mounts. It is clear that units with the same age would have different degradation measure levels; the degradation measure is a time-dependent random variable that can follow different distributions at different distinct times. In general, the degradation measure, X, may follow a distribution which changes with time in the type of the distribution family and its parameters. This paper assumes the degradation measure follows the same distribution family but its parameters may change with time. Furthermore, it is assumed that the degradation paths are monotonic functions of time; they are either Monotonically Increasing Degradation Paths (MIDP) or Monotonically Decreasing Degradation Paths (MDDP). X T
f(x;O F(x;t) r(x;t )
degradation measure (r.v.), x > 0 time probability density function (pdf) of the degradation x at given time t complement of the cumulative distribution function (CDF) hazard function off(x;O, referred to as the degradation hazard
function g(x), q(t) a,b,3,,;~, 0
positive functions constants
Rx(t) = P(Tx > t) reliability at time t and threshold degradation level x Hx (t) = - I n Rx (t) We assume that the degradation hazard function r(x;t ) can be represented by two functions as
Current Advances in Mechanical Design and Production, MDP-7 r(x; t) = g(x)q(t),
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q(t) > O, g(x) > 0
where g(x) and q(t) are positive functions of the degradation measure and time, respectively. This assumption ensures that the distribution family of the degradation measure does not change with time. where q(t)=
bexp(-at)
g(x) = ?x r-t and b> >0
The parametric form of q(t) is determined based on graphing the degradation data versus time. Moreover, the distribution of the degradation data is studied in order to determine g(x). It is shown that g(x) = )Ixr-~ is a good fit for most of the degradation data. The corresponding degradation measure distribution for this degradation hazard function is a Weibull distribution with a time-dependent scale parameter f (x; t)= v(t) ?-~x y-i exp(o--~), - xy t > 0
where O(t)= be -~ is the scale parameter, x is the degradation measure at which a failure occurs and the corresponding reliability function can then be determined as --X r
Rx(t) = P(X > x;t)= exP[bexp(_at)]
(1)
3. APPLICATIONS Corrosion in reinforced concrete bridges is a major concern to professional engineers because of both public safety and cost that associated with needed repairs and replacement. Prediction of bridge functional degradation due to corrosion conditions will be investigated utilizing the proposed model. The two main corrosion parameters which affect the reinforcing bars in reinforced concrete bridges are the corrosion rate, rr and the time it takes to initialize corrosion, TI. Enright and Frangopal (1998) present several mean and variance test measurements for both rco,, and TI. In a typical case, they show that the mean and variance of rco,, will be taken as 0.005 in/year and 3XI0 "6 in/year, respectively. The mean and variance ofT1 will be taken as 10 years and 0.4 years, respectively. In order to estimate the time-variant strength of a reinforced concrete corroded beam, the corrosion effects on the diameter of the reinforcing bars should be evaluated first. After corrosion initiation time, TI, the diameter of a reinforcing bar, D(t), can be evaluated as D(t)= D, - rco,, ( t - T1)
(2)
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Where Di = 1.41 in, is the initial reinforcing bar diameter and t is the elapsed time. Note that t > TI and D(t) > O. For more details of Equation (2) the reader is referred to Enright and Frangopal (1998). The time-variant reinforced concrete strength, Mp(t), can now be evaluated using the conventional design equations in [3].
and
a=(nAs fr)/~.85 fc b )
(4)
Note that As= xD(t)2/4. The reinforcing steel and the concrete strengths are fy and fc, respectively. The number of reinforcing bars is n. The effective depth and the width of the beam are d and b, respectively. For the current example, the values of different parameters were chosen as fy = 40 ksi, fc =3 ksi, d = 27 in. and b = 16. Using Equations 2 through 4 the random time-variant strength, Mp(t), can be estimated. Using the previously mentioned statistical values of rco,~ and 7'1 and a Monte Carlo simulation technique, different strength values for different reinforced concrete beams can be simulated. Thus, a discrete time-variant reinforced concrete strength, x# can be evaluated from the Monte Carlo simulation of the continuous strength Mp(t). Figure 1 shows the time-variant strength, xo -- M p(t) for different reinforced concrete units. It is assumed that the strength/degradation data of 20 beams are monitored for 40 years. Note that these data are assumed to be independent as discussed in Ettouney and Elsayed [ 1999]. The Maximum Likelihood method was utilized to estimate the parameters of the model as given by Eq(1 ) wI
L(y,a,b,t)=I" m
[ ( Y )., ,-i b exp( - at ~)
n~
1-I I-I
,=~ j=~
u
r-I exp(
xo.
x~ b e x ,,( .. -at5 )
(5)
-
where m is the number of years, n, is the total number of degradation data in a year i and x# is the strength of unitj in year i. Taking the logarithm of Eq.(5) we obtain: m
•
m
i=l
t=1
i=!
lnL=~n, lny-~'n, lnb+~'n,at,+ (6) (y' - 1)In x o i=1
j=l
"=
x~ j-t b e x p ( - a t , )
Equating the partial derivatives of Eq.(6) with respect to 7', a and b to zeros and solving the resulting equations using a modified Powell hybrid algorithm and a finite difference approximation to the Jacobian yields: a=0.12, b-11346x 107 and y=1.49.
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Substituting the values of q(t) and g(t) into Eq.(l), the reliability function can then be determined as _x r
Rx(t) = P ( X > x; t ) = e x P [ b e x p ( _ a t ) | _
or
R~(t) = exp[
X 1-49
11346598exp(-0.12t) ]
(7)
The reliability for different threshold values of the strength is shown in Fig. 2. The time to failure for threshold values of 4800, 4000, 3500, 3000, and 2500 are 25.04, 27.25, 28.88, 30.76, and 33.0 years The mean residual life of the beams can be obtained as follows: The mean residual life (MRL) function e(t) of a non-negative random variable T with reliability function Rx(t) is
e(t) = E(T - t / T > t) = R~ (t)-' ~R x (u)du t
The mean residual lives for different degradation threshold values are shown in Fig.s 3. REFERENCES: 1. Blake, I.F., and Lindsey W.C., "Level Crossing Problems for Random Processes," IEEE Transactions on Information Technology, Vol. IT-19, No.3, pp. 295-314, (1973). 2. Carey, M.B. and Koenig R.H., "Reliability Assessment Based on Accelerated Degradation: A Case Study," IEEE Transactions on reliability, Vol. 40, No.5, pp. 499506, (1991). 3. Chick, S. E. and Mendel, M.B., "An Engineering Basis for Statistical Lifetime Models with an Application to Tribology," IEEE Transactions on Reliability, Vol. 45, No. 2, pp. 208-214, (1996). 4. Chuang, S. L., Ishibashi, A., Kijima, S., Nakayama, N., Ukita M., and Taniguchi S., "Kinetic Model for Degradation of Light Emitting Diodes," IEEE Journal of Quantum Electronics, Vol. 33, No. 6, pp. 970-979, (1997). 5. Ditlevsen, O. and Madsen, H.O., Structural Reliability Methods, John Wiley, (1996). 6. Domine M., "Moments of the first passage time of a Wiener process with drift between two elastic barriers," J. Appl. Prob., Vol. 32, pp. 1007-l 013, (1995). 7. Domine M., "First passage time distribution of a Wiener process with drift concerning two elastic barriers," J. Appl. Prob., Vol. 33, pp. 164-175, (1996). 8. Doksum, K. A. and Hoyland, A., "Models for Variable-Stress Accelerated Life Testing Experiments Based on Wiener Processes and the Inverse Gaussian Distribution," Technometrics, Vol. 34, No. l, pp. 74-82, (1992). 9. Eghbali, G., "Reliability Estimate Using Accelerated Degradation Data," Ph.D. Dissertation, Department of Industrial Engineering, Rutgers University, Piscataway, NJ, (1999). 10. Elsayed, E. A., Reliability Engineering, Addison-Wesley, (1996). 1I. Enright, M. P. and Frangopol, D. M., "Probabilistic Analysis of Resistance Degradation of Reinforced Concrete Bridge Beams Under Corrosion," Engineering Structures, Vol. 20 No. I l, pp. 960-971, (1998).
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12. Ettouney, M., Elsayed, E. A., "Reliability Estimation of Degraded Structural Components Subject to Corrosion," 5~ ISSAT International Conference on Reliability and Quality in Design, August 11-13, Las Vegas, Nevada, pp. 291-295, (1999). 13. Feinberg, A.A. and Widom, A., "Connecting Parametric Aging to Catastrophic Failure Through Thermodynamics," IEEE Transactions on Reliability, Vol. 45, No. 1, pp. 28-33, (1996). 14. Ioannides, E., Beghini E., Bergling G., Goodall J. and Jacobson B., "Cleanliness and its importance for bearing performance," Ball Bearing Jouma1242, pp. 8-15, (1993). 15. Lu, J. C. and Meeker W. Q., "Using Degradation Measures to Estimate a Time-to-Failure Distribution," Technometrics, Vol. 35, No.2, pp. 161 - 173, (1993). 16. Lu, J., Park J. and Yang Q., "Statistical Inference of a Time-to-Failure Distribution Derived From Linear Degradation Data," Technometrics, Vol. 39, No.4, pp. 391-400, (1997). 17. Meeker, W. Q. and Hamada, M., "Statistical Tools for the Rapid Development & Evaluation of High-Reliability Products," IEEE Transactions on reliability, Vol. 44, No.2, pp. 187-198, (1995). 18. Meeker, W.Q. and Escobar L. A. and Lu C.J., "Accelerated Degradation Tests: Modeling and Analysis," Technometrics, Vol. 40, No. 2, pp. 89-99, (1998). 19. Melchers, R. E., Structural Reliability, Ellis Horwood Limited, (1987). 20. Nelson, W., "Analysis of performance-degradation Data from Accelerated Tests," IEEE Transactions on reliability, Vol. R-30, No.2, pp. 149-155, (1981). 21. Pieper, V., Domine, M., Kurth, P., "Level Crossing Problem and Drift Reliability," Mathematical Methods of Operation Research, Vol. 45, 355-375, (I 997). 22. Qureshi, F. S. and Sheikh A. K., "A Probabilistic Characterization of Adhesive wear in Metals," IEEE Transactions on Reliability, Vol. 46, No. 1, pp. 38-43, (1997). 23. Stock, D., Vesely W. E. and Amanta P.K., "Development and Application of Degradation Modeling to Define Maintenance Practices," NUREG/CR-5967, (1994). 24. Tydeman, M. S. and Kirkwood T. B. L., "Design and analysis of accelerated degradation tests for the stability of biological standards I. Properties of maximum likelihood estimators," Journal of Biological Standardization No. 12, pp. 195-206, (1984). 25. Tseng, S. T., Hamada M. and Chiao C. H., "Using Degradation Data to Improve Fluorescent Lamp Reliability," Journal of Quality Technology, Vol. 27, No. 4, pp. 363369, (1995). 26. Yang, K., "Robust Design and Reliability Engineering, An Integrated Approach," Proceedings of Ford 2000 Conference on Integration of Quality Methods, Reliability, and Robust Design, Nov. 17-18, 1994, Fairlane Club, Dearborn, Michigan, pp. 141-185, (1994).
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s=2500
._
0,8
-
0.6
-
s=3000
.
-, ,, , \
.4 ~ 0,2
/
%
/
s=3500 s=4000 s=4800
'.~
lib
0 0
10
20
30
40
Time (Years)
Fig 2. Reliability vs. time for different strengths
50
60
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Mean Residual Life
I ~=4~~
40 35
9
s=4ooo
I
.... ] I
30 25 20 15 10 5 0
10
20 Time 30
Fig 3. Mean residual life
40
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Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
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FEATURE RECOGNITION ALGORITHM FOR PROCESS SELECTION McCormack, A.D. and Ibrahim, R.N.
Dept of Mechanical Engineering, Monash University GPO Box 197 Caulfield East, 3149 Vie, Australia
ABSTRACT The function modeller presented in this paper was designed to extract product features for automatic process selection in an independent application. This modeller works on principles of Boundary Representation (B-Rep), specifically Euler formula and Attributed Adjacency Graphs and Constructive/De-Constructive Solid Geometry (CSG) modelling. These models are reviewed and presented as introductory material to this paper. The product CAD data that is used within these principles is obtained from a neutral STEP file format, ensuring its compatibility amongst existing CAD programs. A recursive checking method, utilising volume decomposition of a CSG model, is employed to ensure that a valid product model is developed. KEYWORDS Feature Recognition; Process Planning; Attributed Adjacency; Feature Extraction 1. INTRODUCTION Recent progresses in CADCAM research, particularly in three dimensional product representation, have created new opportunities for the previously difficult task of automatic generation of manufacturing process plans. The generative approach to process planning is an example of these relatively new manufacturing philosophies. This approach allows for new process plans to be generated for each component using decision logic and precoded algorithms. To enable automatic selection of process plans for an individual product, the Computer Aided Process Planning (CAPP) system must be able to extract detailed information from the product description. These components include geometry, dimensions, tolerances and surface conditions. Included in CAPP systems is the solid modeller, which consists of a interface for the extraction of shape information, data structures to represent the CAD file, and an algorithm to generate the shape information for the specified application. The data representation scheme is an integral part of the solid modeller and there are generally two recognised models [l]; boundary representation (B-REP) and constructive solid geometry (CSG). The boundary representation scheme is based on the concept that a solid object can be considered bounded by a series of faces. These faces are then made up of a series of edges, which consists of a series of vertices or points. This representation model basically breaks down the solid model in a series of cartesian points and their connectivity with each other.
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The relationships between the individual faces of the product are stored and make up an integral part of the boundary representation model. The topological relationships between the various features of a product can often directly determine the feasible process sequences [12]. There are typically four types of feature relations defined: intersections between protrusion, adjacency between protrusion or normal adjacency, adjacency otherwise, and intersection between protrusion or depression. Representing a product under the CSG method results in a collection of basic three dimensional solids, such as cubes or prisms. Their orientation inside a 3-D space is utilised to combine or subtract the primitive solids from each other to form a completed solid model. CSG works with a forward planning approach [6], but the steps can be worked in a reverse order. This requires the use of simulated stock work piece where basic three dimensional solids are removed.This process can be linked to a feature-based design technique known as volume-decomposition, where the solids removed correspond to a machine process. There are four major phases to a volume decomposition process [5]" 1. Determine the expected volume that will be removed. This should be seen as the raw stock minus the finished product. 2. Partitioning the removed volume into individual and basic three dimensional solids. 3. Combine adjacent cells for typical machining features. 4. Match machining features to typical operations. The algorithm being presented in this paper follows these steps when creating a secondary product model. The extraction of, the previously mentioned, machining features that are characteristic of a certain application is the key to an efficient process planning task. This can dictate the actual process selection and consequently the process sequencing, which is essentially the body of a process planning task. 2. REVIEW OF RELEVANT WORK There have been many documented cases of automated process planning systems in the past. The GT and CAPP system [2], utilises a group technology code to identify the features. Upon identification of the features, the processes are then selected and sequenced according to a typical algorithm of decisions, or rules. The disadvantages of a system based on a GT code are that the coding system must go into great detail, and involve a very lengthy searching process, to ensure the process plan will be valid for all products. Another notable example is the EXCAP system [7], which was designed to generate process plans for rotational components using both group-technology codes and specific generative language approaches. This was expanded further into the CADEXCAP system [8,9], which integrated EXCAP with CAD systems using a neutral file exchange system. Neutral file exchanges are extremely important in CAPP systems as they ensure compatibility amongst various CAD systems. Solid modeling is a concept that is not oRen considered along side rotational product modeling as this is often a two dimensional field but the CADEXCAP system utilises the product profile to extract features. It is now in three dimensional areas, which are not often required in rotational products, that the new solid modeling techniques benefit the process planner the most. The process selection algorithm being presented has been developed from a generative point of view, meaning that it creates a new process plan for each product. As was previously
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discussed, there are generally two recognised forms of product representation, B-Rep and CSG, and this algorithm seeks to create a solid model based on both techniques to utilise the advantages of both, while simultaneously compensating for their failings. It is important to note that the algorithm does not create two product models. The secondary CSG solid model is merely a way of verifying the proposed process plan. The main disadvantage of solid model constructed under CSG is there is no explicit information about faces, edges or vertices of the object, which makes it less suitable for application such as assembly. It is for this reason that the product model used in the algorithm being presented here is developed as a B-Rep model. A solid model also created under a boundary representation model can be easily recognised as unique if its geometric data and topological relationships are identical, therefore it is an excellent way of comparing models. 3. SELECTION ALGORITHM In this algorithm, data is obtained from a CAD file in STEP format via an interface. A B-Rep product solid model is developed to extract features using topological relationships and geometric data. Machining processes are linked to the features, via a machine dependent database and a CSG model, created using volume decomposition methods, tests the features as they are removed from a raw-stock model. The details of this algorithm are discussed below. This initial model that the algorithm creates is in the B-Rep form. It needs to read the topological and geometric data of the product from the CAD file and create a simple array for storing the data. As it is not possible for two different shaped objects to have the same topological relationships and geometry data, this is a very simple way of interpreting product data. When examining a product representation in B-Rep form, there will be various data arrays existing for each individual feature. For example, each face of the product will join another face at an edge on some point of the model. This relationship can be recorded in the form of a topological graph [1,3,10]. However, a more convenient fashion is a simple matrix with each face listing and showing the faces to which it is connected. If a search is done through the CAD file, we can compare the various topological matrices and geometric data to a database containing features typical to the application and machining environment. One important aspect of the geometrical data that will have a large influence on the recognition of features is the angles at which joining faces make with each other, also known as convexity or concavity. 4. MODIFIED ATTRIBUTED ADJACENCY Under a graph representation, or matrix structure, the face adjacency and relationships are stored, but the information regarding the type of relationship is also of great importance when feature recognition is required. The Attributed Adjacency Graph [13,14], which is also representable under matrix notation, is a way of storing the type of relationship. Often the relationship is given an attribute of 0, 1 or 9, which can represent data such as convex or concave relationship, parallel relationship or perpendicular relationships. The algorithm being presented in this paper utilises a modification on the Attributed Adjacency Graph/Matrix by further classifying the attribute of the relationship. For example, in one half of the matrix a convex relationship between two faces may be stored, but in its reflected element the actual angle is stored. Other data regarding the relationship may include noting that the interaction is between a cylindrical opening and a planar face. The data proves to be extremely useful when
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identifying features. An example would be the ability to immediately distinguish between a square slot, a dovetail slot, and a V slot. All of these slots would have otherwise been recognised as a sequence of three planar faces with concave relationships. Joshi and Chang [15] developed a heuristic for extracting sub-graphs/matrices from the Attributed Adjacency Matrix that is based on the statement: a face that is adjacent to all its neighbouring faces with a convex angle does not form part of a feature. This allows the features specific to the application to be extracted so that each individual machining operation can be identified. To separate an individual sub-graph/matrix from the modified attributed adjacency matrix we classify the end of a depression feature by the presence of a convex edge. The end of a protrusion feature is thus found via the presence of a concave edge. This extraction process is simplified by using the Euler equation [4] to determine whether the current feature is a passage or non-passage feature. The Euler formula validates all vertices, edges and faces against various criteria dependent on the class of the object. For example, in its simplest form the Euler equation determines if an object is valid, in that all faces are simply connected, and that is has no holes through it. This decision is shown in Figure 1, which depicts the decision flow of the algorithm.
Renrer,entntirm
o
Required from File - Material - Process Machine - Tool Fixture - End Effector
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J
T
Hybrid/Non-Hybrid f
. . . . . . . . . . . ,
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,,
!
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A passage feature is then given a genus rating [3] to determining the level of feature complexity. Genus, or genera, are other criteria with which the Euler equation must be compared to when dealing with non-simple objects, such as spheres or cylindrical through holes. Passage features are often in the form of holes, blind or open, so the process selection phase is quite simple. This genera data originates from the geometric data, which can indicate a relationship between a cylindrical opening and a planar face in a very simple manner. When non-passage features are found, they must typically be examined based on their
convexity or concavity, which can be determined from their individual topological relationships with other features. This is basic classification for non-hybrid features in that they are seen as either protrusions or depression. Simple features, such as protrusions or depressions, are stored in a features database, which the algorithm accesses in part with selecting the required machining process. Hybrid features of the non-passage type, which are not automatically recognised and attributed to a common machining process, must be broken down into individual features and then re-compiled to develop a small process sequence. An individual feature may be in the form of single face that requires only one machine pass. If the process sequence is deemed successful then the hybrid feature needs to be stored, in the form of a topological relationship and geometric data, to ensure future recognition. Once the individual features sub-graphs have all been identified, they must be indexed based on complexity. An obvious example would be that a counter-sunk thread could not be tapped until an initial basic drilling operation was performed. This sequencing is based on data stored in the features database and forms the foundations for the sequencing of the individual processes. The precedence of a feature by another indicates that it should be produced first in order to satisfy the algorithm. The selection of each machining process for each individual feature is made from data available in a database that is machine specific. This process entails patternmatching techniques between the individual feature sub-graphs and the process/feature database. The volume removed by each machining process that exists in the case-specific machining environment is extremely important to the validation of the product model and is contained in this machine tool database but is not used here. Once all features have been indexed, the dimensions, symmetry and orientation of each individual feature are examined simultaneously and compared to the current tool data to ensure machining compatibility. Symmetry of the features is examined only to simplify the process path selection. However, non-symmetrical feature can still be used. The accessibility should be viewed as the starting point for the individual feature's process path and will update the sequencing of each process. The determination of the approach direction was investigated by Aldak,hilallah and Ramesh [ l l ]. They proposed that a system should develop a set of all the directions from which a tool can approach a face. For example {+x,-x,+y,-y,+z,-z} would be a set possible for most three dimensional products. From this set, the system needs to interpret a set of feasible directions from which a face can be machined. A further refining process involves examining the following and previous processes approach path so that the loading and unloading of the workpiece is minimised.
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5. PROCESS SELECTION The machining data is now obtained and a product model is developed based on raw stock data and material removal of the current tool. An extension on the constructive solid geometry technique is utilised to form a trial of the product. This process is known as volume decomposition and is an excellent way of creating a secondary solid model as it refers to the volume removed by each process as an individual 3D solid. Each specific volume removed is constructed based on geometric constraints from the B-Rep model and from data that is unique to the required machine and tool. This produces the physical characteristics of each primitive and the orientation of the primitive has already been determined by the calculation of the tool approach direction. Since the original boundary representation model contains explicit geometric location data, a reference point for the three-dimensional primitive is then copied directly from the B-Rep model to the new CSG model. This reference point then positions the primitive solid according to original B-Rep model and the orientation is determined from the tool approach direction. Each solid is then subtracted from a user defined stock piece. Interactions between solids are irrelevant as all solids are being subtracted from the basic stock solid. This new CSG model has now essentially been created by an internal "design by features" function. Upon removal of each three-dimensional solid, a check is performed on the current feature and it is converted into a boundary representation model. This model is updated with each new process and is the basis for the recursive checking system. The conversion to B-Rep depends greatly on the geometric data of the machining processes stored in the feature/process database. What this means is that the algorithm can compare the topology matrix and geometry data of the newly removed feature and compare it to the corresponding feature in the original solid model. If the feature that is being proposed does not match the required original feature, the algorithm loops back to the initial feature indexing and evaluates the feature according a variant on the Euler formula. The Euler formula can be presented in various forms and these are stored to ensure that the feature checking process is as accurate as possible. If a problem arises in feature identification, this secondary CSG solid model can be quite useful for user prompting as this is an area which B-Rep often lacks [13]. Now that all processes have been synthesised on this trial, the geometric and topological data of the completed model are compared to the actual product model and the machining database to give an approximation of the final tolerance on the model. Appendix A depicts a simple example of the algorithm at work on a simple slotted workpiece. It shows the logical steps that the algorithm presented in Figure l would follow to produce a complete process plan. 6. CONCLUSION The presented algorithm developed using B-Rep and CSG techniques was used to determine the features for different products. In the algorithm, these techniques were employed to utilise the advantages of both in that the complexity of the product being modeled can be increased and the time taken to recognise the feature can be reduced. This ability is derived from the modified Attributed Adjacency Graph/Matrix. With a simple method for storing data regarding product features and their related machining processes, the process sequencing can be performed at a quicker speed also.
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REFERENCES 1. Bronsvoort, W.F. and Jansen, F.W. "Feature Modelling and Conversion- Key Concepts to Concurrent Engineering", Computers in Industry, 21, pp. 61-86, Elsevier, (1993). 2. Joshi, S.B. et al, "Design, Development and Implementation of an Integrated Group Technology and Computer Aided Process Planning System" Volume 26, Number 4, IIE Transactions, July, (1994). 3. Rembold, U., Nnaji, B.O. and Storr, A. "Computer Integrated Manufacturing and Engineering", Addison-Wesly, (1994) 4. Mortenson, M.E., "Geometric Modelling", John Wiley & Sons, (1996) 5. Shirur, A., Shah, J.J., and Hirode, K., "Machining Algebra for Mapping Volumes to Machining Operations for Developing Extensible Generative CAPP", Journal of Manufacturing Systems, Vol. 17/No. 3, (1998). 6. Chang, T.C., Wysk, R.A. and Wang, H.P., "Computer-Aided Manufacturing 2nd Ed.", Prentice Hall, (1998). 7. Kalta, M., Davies, B.J., "Product Representation for an Expert Process Planning System for Rotational Components'; International Journal of Advanced Manufacturing Technology, Vol. 9, pp. 180-187, (1994). 8. Kalta, M., Davies, B.J., "CADEXCAP: Integration of 2D CAD Models of Turned Components with CAPP" International Journal of Advanced Manufacturing Technology, Vol. 8, pp. 145-159, (1993). 9. Kalta, M., Davies, B.J., "Guidelines for building 2D CAD Models of Turned Components in CAD-CAPP Integration", International Journal of Advanced Manufacturing Technology, Vol. 8, pp. 285-296, (1993). 10. Shah, J.J., Mantyla, M. and Nau, D.S., "Advances in Feature Based Manufacturing", Elsevier, (1994). 11. Aldakhilallah, K.A. and Ramesh, R., "An integrated framework for automated process planning: design and analysis", International Journal of Production Research, Vol. 36, No. 4, pp. 939-956, (1998). 12. McMahon, C.A. et al, "Representation and reasoning in computer aided process planning ", Proceedings from the Institute of Mechanical Engineers, Vol. 211, Part B 13. Singh, N., "Systems Approach to Computer-Integrated Design and Manufacturing", John Wiley & Sons, Inc., (1996). 14. Zhang, C., Chan, K.W. and Chen, Y.H., "A Method for Recognising Feature Interactions and Components within the Interactions" International Journal of Advanced Manufacturing Technology, Vol. 13, pp 713-722, (1997). 15. Joshi, S. and Chang, T.C., "Graph-based Heuristics for Recognition of Machined Features from a 3D Solid Model", Computer-Aided Design, Vol. 20, Number 2, pp. 58-66, March, (1988).
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APPENDIX A
#1
#2
9 9 9
9
The above part geometry is employed as an example of the algorithm presented. CAD file interfaced Geometric and topological data extracted Euler verification: V-E+F=2*(S-G)+R Therefore 16-24+ I 1=2"(14))+ I and thus the object is valid under boundary representation The Attributed Adjacency Graph/Matrix stores the type of relationship between faces.
Feature Identification A Convex relationship between three faces of equal length indicates a slot. Slot immediately recognised as a V-slot. The modified AAG and AAM for this slot is shown below. In matrix form, the 1350 between two faces would be stored in the symmetrical unit of the matrix.
.
r, Ft 9
.
9 Connectivity between faces examined using AAG. 9 Based on attribute, further data regarding the relationship is stored, creating the modified Attributed Adjacency Matrix. 9 An edge created by two planar faces requires the angle between the two faces to be stored. A Blind hole requires reference face to the hole opening (ring) relationship to be stored.
0
9 The Blind hole has a reference face that is recognised as face 02 of the V-slot. 9 V-slot feature is indexed first. Therefore machining of this feature will occur before machining of blind hole. 9 Blind hole is indexed 9 Geometric data compared with process/feature relationship database to determine accessibility.
G 135 o 9
F2 0
9
135 o
F39
0
9
Q
0('35~
Tool Path Generation Access to feature determined to be at trapezoidal opening of slot. This produces a starting point for tool path generation. The width of the slot at top and bottom, is compared to machine tool database and determines number of passes. Hole diameter and depth extracted.
.
Q
0('35~
0
Verification Tool path is essentially generated from slot length, opening, and blind hole centre point. Volume Decomposition: solids subtracted from cubic primitive.
8
9 Topological relationship of feature with primitive examined via AAM comparison with original 9 Completed CSG model confirms process plan 9 Therefore, tool path will be created using endmill of diameter d.
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Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
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DEVELOPMENT OF A GENETIC ALGORITHM BASED ON FUZZY LOGIC SETS FOR SOLVING FACILITY LAYOUT PROBLEMS
Ramadan, M.Z.* and Abou EI-Ez, S.R.S.** *Associate Professor, **M. Sc. Candidate Prod. Eng. Dept., Faculty of Engineering, Helwan University, Helwan, Cairo, Egypt. Email: [email protected]
ABSTRACT Facilities layout is concerned with how to arrange block areas within the facility. Traditionally, this problem has been approached through the use of analytical tools ranging from scaled templates to computer-aided layout algorithms. The latter algorithms are characterized by using the relationship charts or preference charts. The analyst is typically uncertain about what these inputs should be. In fact, one of the real difficulties in developing such models for layout design is the natural vagueness associated with the model inputs. The use of fuzzy logic relationships is one approach for handling such kind of inexact vague data. Therefore, this paper presents a genetic algorithm based on fuzzy logic sets for solving facility layout problems. Better solutions that satisfy multiple objectives are produced by implementing genetic operators to the proposed gene structures. Significant better quality layouts that are obtained by employing this approach are compared to test problems available in the literature. A numerical example is given to illustrate the proposed approach and to show its capability to deal with multi-criteria problem effectively. KEYWORDS Facility Planning/ Layout; Multiple Criteria Programming, Fuzzy Rule-based System; Genetic Algorithms; Search Strategies. 1. INTRODUCTION Facility layout deals with the selection of the most efficient layout of physical departments based on their inter-relationships in production and service organizations in order to operate costs effectively. The design of the facility layout of such organizations is of tremendous importance for its effective utilization. This fact was emphasized in Tompkins and White [1], with the authors pointing out that 20-50% of the total operating expenses in manufacturing are attributed to material handling and layout related costs. The use of effective methods for facilities planning can reduce these costs by at least 30%. They listed several layout design objectives such as: 1) improve materials handling; 2) utilize people, equipment, space, and energy; 3) minimize capital investment; 4) provide employee safety and job satisfaction; 5) be flexible and promote ease of maintenance.
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Traditionally, this problem has been dealt with the use of analytical tools ranging in sophistication from scaled templates to computer-aided layout algorithms. According to Islier [2], facility layout algorithms can be classified into two main categories: optimal (like total enumeration, branch and bound, or curing plane)and suboptimal methods. Optimal methods are not computationally feasible for even moderate size problems. This fact has led researchers to focus on developing heuristic algorithms. These algorithms are classified as: 1) construction methods, 2) improvement methods, 3) hybrid methods, and 4) graph theoretic methods. These layout approaches start with a collection of data that are either quantitative such as from-to-charts, or qualitative as relationship-charts. Regardless of this type of data, there is an element of vagueness in it [3,4]. The traditional layout techniques treat these inputs as exact. The analyst is typically uncertain about what these inputs should be. The use of Fuzzy methodology is one approach for handling inexact vague data and yet to work in a mathematically strict and rigorous way [5]. On the other hand, Elwany et al. [6] pointed out that the facility layout problem was treated traditionally as a single objective, single criterion problem. It could be solved by applying any simple management science technique. In the real life, problems contain a considerable number of variables, and a number of conflicting constraints that make the optimization of the problem a difficult task. John Holland [7] introduced genetic algorithms (GAs) that have received a great deal of attention in the recent literature, because of its highest capability to deal with multi-criteria combinatorial optimization problem and that they do not rely on the analytical properties of the function to be optimized. Moghaddam & Shayan [8] used GAs to solve Quadratic Assignment Problem formulation of equal and unequal sized facility layout problems. Islier [3] presented a GAs based model for dealing withmulti-criterea facility layout problem. Rajasekharan [9] proposed a GAs based strategy for solving facility layout problem in flexible manufacturing system. The previous literature indicated that better layout solutions can be obtained using GAs for solving the facility layout problem. No one tried to have both techniques in one model. Therefore, in this work, GA is used to optimize facility layout problems based on fuzzy rule-based outputs. 2. PROPOSED APPROACH: 2.1. The fuzzy sets representation: The fuzzy logic, or fuzzy set theory, aims to represent fuzzy concepts in an understandable form. It links natural language with reasoning computing system (e.g., soft computing system) through the use of linguistic variables and quantifiers. Linguistic variables can represent words such as "age", "tall", "hard", "street length", or "beauty". In addition linguistic quantifiers such as "young", "some", "many", "less than", or "average" are quantifiers as fuzzy subsets of the real line which correspond to imprecise values of an amount. The variables and quantifiers are mapped to fuzzy membership functions. Suppose a grade of membership measure fA(u) -- 0.6 means that u is an element of set A to a degree of 0.6 on a scale where zero is no membership at all, and one is a complete membership.
The main components of the fuzzy decision-making model are shown in Fig. 1. Numerical (crisp) values or fuzzy expressions (results of subjective evaluation or estimation) are allowed to be the process input. The main part of the fuzzy model is a knowledge base with a set of linguistic rules and definitions of fuzzy sets describing the system to be modeled. A fuzzy output value is computed on the basis of these rules using a certain inference method.
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Then this fuzzy output value can be transformed into a numerical value (so-called defuzzification) or approximated to one of the fuzzy, or linguistic values that have been defined for the output variable. This approximation (defuzzification process) can be accomplished by means of calculating the center of area between fuzzy sets. The following steps are description of how to obtain a fuzzy model, as described above, in a systematic way: Step 0. Select a set of input linguistic variables that are available. Step 1. Select a set of output linguistic variables whose crisp values are needed. Step2. Determine the membership functions for all linguistic labels of the linguistic variables. Step 3. Develop a knowledge base of fuzzy rules. Step 4. Select fuzzification technique in which crisp inputs are converted into fuzzy representations (i.e., suitable linguistic of fuzzy sets). Step 5. Develop a knowledge base of fuzzy rules. Step 6. Develop a fuzzy inference strategy applying compositional rules of inference (CRI) using IF-THEN rules. Step 7. Select defuzzification technique in which the propagated fuzzy representation is converted to a set of crisp output values. i
Fuzzy Database !
Crisp i ~ . ~
Rules
.~ Fuzmficat~on Interface
Fuzzy inputs
~ Fuzzy Inference] Inference ,~ 9
,,,[
Crisp output
l i
Fuzzy output l llll ii
Engine
f
Fig. 1. Architecture of typical fuzzy system. 2.2. The genetic algorithm representation Genetic algorithms are basically search techniques. They imitate the natural process of evolution by processing towards the optimum. In a genetic algorithm, an individual chromosome (any feasible solution to the problem) is the basic element of the population. These chromosomes are combinations of facilities numbers, known as genes. The best chromosomes are selected by the roulette wheel principle to be parents. Three genetic operators known as crossover, mutation, and inversion are applied to those parents to generate new offsprings. In this model, population size is five times the number of the facilities (length of the chromosome) in the layout. The crossover, mutation, and inversion probabilities are: 0.9, 0.01, and 0.05 respectively. The maximum number of generated population is 120. The proposed genetic algorithm is established as follows"
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Step 0. Initialize randomly a population size of chromosomes equal to five times the number of the facilities (n). In addition, each chromosome (as a potential solution) is represented by a decimal string of length equal to the number of the facilities. Step 1. Calculate the fitness value for each chromosome fv i (i-l,..., population size). Step 2. Find the total fitness of the population using the sum of the distances times the pop-size
relationships for each pair of facilities, F - ~ fv i ill
Step 3. Calculate the selection probability for each chromosome, Pi = fvi / F. i
Step 4. Calculate the cumulative probability for each chromosome, q = Z pj i
Jffil
Step 5. Develop the new population by spinning the roulette wheel population-size times for selecting the next chromosomes in the following way: a) Generate a random number r from the range { 0, ..., 1}. b) If r < q l then the first chromosome is selected; otherwise the ith chromosome is selected such that qi-I < r _
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T a b l e 1. M i n i m u m cost o f the facility for different algorithms based on five s t a r t i n g solutions.
Number of Departments 12
HillierConners 304
CRAF T 289
COL
Proposed Genetic Algorithm (G.A.) 291
FRAT . . . . . Biased , Sampling 295 289*
293
20
1319
1324
i312
i 300
30
3161
3148
3152
3129
1304 '
1285'
b
.
.
.
.
.
3093* .
3093*
.
* Best solution over all methods.
4. SIMPLE ILLUSTRATED EXAMPLE: Consider the layout design of a small metal fabrication shop with five facilities. Suppose that managers, employees, and consultants have indicated that the material flow, information flow, personal flow, equipment flow, supervision of personnel, employee satisfaction, and safety consideration are shown in Fig. 2. Facility
Area
Name
(m 2) ii
A. Shipping & Receiving B. Storage C. Fabrication Area D. Inspection & Assembly E. Offices i
i
i
i
mnn him
i
75 150 500 100 75
j
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Fig. 2. Desired relationships among different facilities
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The factors that affect the decision of the designer are: 1.
Material Flow (MF): represents the number of products in terms of raw materials, semifinished, or finished products moved among different facilities, and fuzzified in Fig. 3. 2. Information Flow (IF): represents the number of oral or written communications among different facilities, and fuzzified in Fig. 4. 3. Personal Flow (PF): represents the number of moved employees among different facilities for doing their required jobs, and fuzzified in Fig. 5. 4. Equipment Flow (EF): represents the number of material handling equipment used for handling different products among different facilities, and fuzzified in Fig. 6. 5. Personnel Supervision (PS): represents the number of supervisors moved among different facilities for doing supervision jobs, and fuzzified in Fig. 7. 6. Employee Satisfaction (ES): represents the employees' ratings of satisfaction of closing some facilities to the others, and fuzzified in Fig. 8. 7. Flexibility Rating (FR): represents the ease by which facilities earl be arranged and rearranged; it can be expressed in terms of a closeness relationship among the facilities, and fuzzified in Fig. 8. 8. Safety Consideration (SC): represents the conditions of closeness among facilities in a way of a noisy places should be far as possible, etc., and fuzzified in Fig. 8. V.L.
L.
M.
H.
V.L.
V.H.
L.
M.
H.
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1.0 IXg(IF) 0.5
VVV
0 20 40 60 80 0 8 16 24 32 Fig. 3 Membership function of material flow. Fig. 4 Membership function of information flow. V.L.
L.
M.
H.
V.L.
V.H.
L.
M.
H.
V.H.
1.0 ~tg (EF) 0.5
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0 7 14 21 28 0 25 50 75 I00 Fig. 5 Membership function of personal flow. Fig. 6 Membership function of equipment flow. V.L.
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I
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o.5--~V 0 2 4 6 8 Fig. 7. Membership function of Personnel Supervision.
'0
V.L.
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0.5 X U O I E A Fig. 8. Membership functions for Employee Satisfaction, Flexibility, and Safety.
For the simplicity of the demonstration and the limitation of the paper pages, the eight factors affecting the decision are assumed to have the same weight effect in the designer decision. Then, all factors are fuzzified based on maximum rule of the fuzzifieation process. The results of the fuzzification process are presented in Table 2.
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T a b l e 2. I n f o r m a t i o n collected f r o m the fuzzification process. Activity between facilities
.g (M F) V.H.* 0.625+
pg (IF) M. 0.85
V.H. 0.60
A-C
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L.
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"'L.
A-D
0.375 L. 0.625
0.60 L. 0.85 L. 0.85 H. 0.75 M. 0.70 L. 0.60
0.76 L. 0.68 L. 0.68 M. 0.88 L. 0.60 L. 0.88
0.57 M. 0.57
A-B
)~g (PF)
pg (EF)
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. g (FR)
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1.00 V.L. 1.00 M. 1.00 V.L. 1.00 M. 1.00 V.L. ! .00 V.L. 1.00 V.L. 1.00 V.L. 1.00
1.00 V.L. 1.00 V.L. 1.00 V.H. 1.00 M. 1.00 V.L. 1.00 L. 1.00 V.L. 0.00 V.L. 0.00
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.
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.
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.
.
.
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.
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.
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.
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.
.
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.
.
.
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.
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.
Fuzzy subset type (function): V.H.= Very High, H. = High, M.= Meduim, L.=Low, V.L.= Very Low. + Grade of membership value (e.g., 1 = complete membership, 0 = no membership at all). The relation between facilities A and B, for example, for factor 1 (MF) has a value of 29. It belongs to the fuzzy subset Very High (VH) with a membership value of 0.625, see Fig. 3. Similarly, this process was done for the rest of the factors and all pairs of facilities in the layout. The second step is to apply the IF-THEN rules that were developed by the layout planner in the fuzzification inference of the fuzzy decision-making stage. These rules are: Rule Rule Rule Rule Rule
1. If a 2. If a 3. If a 4. If a 5. If a
factor belongs factor belongs factor belongs factor belongs factor belongs
to the to the to the to the to the
fuzzy fuzzy fuzzy fuzzy fuzzy
subset subset subset subset subset
Very High (VH) Then the rating is 5. High (H) Then the rating is 4. Medium (M) Then the rating is 3. Low (L) Then the rating is 2. Very Low (VL) Then the rating is 1.
The last step of the fuzzy decision-making process is the defuzzification interface. This step implements the center of area equation. For example, the activity relationship crisp value (ARCV) between facility A and facility B is calculated below: ARCV_[5 (0.625) § 3 (0.85) + 5 (0.6) + 5 (0.57) +5 (0.5) + 4 (1.0) +4 (1.0) + 5 (1.0)]= 4.4 [ 0.625 + 0.85 + 0.6 + 0.57 + 0.5 + 1 + 1 + 1 ] This process is repeated for the other relationships between facilities as shown in Fig. 9. The final step of the model is to feed the genetic algorithm with the fuzzy activity relationship chart. The output of the model is presented in Fig. 10 with a final score of 80.28.
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Dept. Code From A
B
C D
i
To B
C
D
E
4.40 1.98 1.79 2.22 3.54 2.85 1.33 2.88 1.21 1.26
CCCCCCBEE CCCCCCBBE C CCCDDBBA C CCCDDBAA Fig. 10. The model output of the layout.
i
Fig. 9. Crisp outputs of the fuzzy model 5. CONCLUSION: The proposed model has been extended to have a weight difference for each factor relative to the others. In addition, the final layout is evaluated based on the fuzzy rules using the fuzzification process for both distance and layout score. The developed GA is tested on hundred problems ranged in size from five to fiRy facilities developed in random fashion. The results compared with well known algorithms published in the literature. The tests proved that the model has a capability to handle large problem size with different weighted variables in a reasonable time effectively. 6. REFERENCES
1. Tompkins, J.A. and White, J.A., "Facility Planning", Wiley, New York, (1984). 2. Islier, A.A," A Genetic Algorithm Approach for Multiple Criteria Facility Layout Design", Int. J. of Prod. Res., Vol. 36, pp. 1549-1569, (1998). 3. Dweiri, F. and Meier, F.A., "Application of Fuzzy Decision Making in Facilities Layout Planning", Int. J. of Prod. Res., Vol. 34, pp. 3207-3225, (1996). 4. Evans, G.W., Wilhelm, M.R. and Karwowski, W., "A Layout Design Heuristic Employing the Theory of Fuzzy Sets", Int. J. of Prod. Res., Vol. 25, pp. 1431-1450, (1987). 5. Zimmerman, H.J., "Fuzzy Sets: Decision Making Expert Systems", (Boston, MA: Kluwer Academic Publications)(1987). 6. Elwany, M.H., Abou-Ali, M.G., and Harraz, N.A., "The Development of a DecisionSupport Expert System for Solving Facilities Layout Problems", The 6th Int. Conf.on Prod. Eng., Design and Control Proceedings, Alex.: Egypt, Feb., pp. 703-713 (1997). 7. Holland, J.H., "Adaptation in Natural and Artificial Systems : An introductory Analysis with Applications to Biology, Control, and Artificial Intelligence", MIT Press (1992). 8. Moghaddam, R.T., and Shayan, E, "Facilities Layout Design by Genetic Algorithms", The 22 nd Int. Conf. on Computers and Ind. Eng. Proceedings, Dec., Cairo:Egypt, pp. 415-418 (1997). 9. Rajasekharan, M., Peters, B.A., and Yang, T., "A Genetic Algorithm for Facility Layout Design in Flexible Manufacturing Systems", Int. J. of Prod. Res., Vol. 36, No. 1, pp. 95110, (1998). 10. Nugent, C.E., Vollman, R.E., and Ruml, J., "An Experimental Comparison of Techniques for the Assignment of Facilities to Locations", J. O.R., Vol. 16, p 150 (1968).
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-I 7, 2000
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MANUFACTURING CELL FORMATION PROBLEM: A GRAPH PARTITIONING APPROACH
Selim, H.M. Assistant Professor of MIS - United Arab Emirates University and Assistant Professor of Industrial Engineering - Helwan University E-mail: [email protected]
ABSTRACT The design of a cellular manufacturing system requires that a machine population be partitioned into machine cells. A new graph partitioning heuristic is proposed to solve the manufacturing cell formation problem (MCFP). The proposed heuristic represents the MCFP as a graph whose node set represents the machine cluster and edge set represents the machinepair association weights. A graph partitioning approach is used in forming the manufacturing cells. This approach offers improved design flexibility by allowing a variety of design parameters to be controlled during cell formation. The effectiveness of the heuristic is demonstrated on several problems from the literature. KEYWORDS Graph Partitioning, Manufacturing Systems, Cellular Manufacturing, Group Technology, and Sequential Algorithms. I. INTRODUCTION The Graph Partitioning (GP) problem is defined by Fj~illstr6mas [ 10] and by Kemighan and Lin [1 1] as follows: Given a graph G=(N, E) (where N is a set of weighted nodes and E is a set of weighted edges) and a positive integer p, find p subsets NI, 312..... Np of N such that: 1. O~_-tN, = Nand N, n N j =0 fori , j , 2. W(i) ~ W/p, i =1,2,...,p, where W(i) and W are the sums of the node weights in N~ and N, respectively, and 3. The cut size, i.e., the sum of weights of edges crossing between subsets is minimized. Any set {N, c_ N :l _
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for each part are known, which machines and their associated parts should be grouped together to form cells ?" Wu and Salvendy [17]. MCFP solution approaches have been surveyed and classified by Selim et al [14]. GP methods applied on the MCFP treat the machines and/or parts as vertices and the processing of parts as arcs connecting these nodes. These methods aim at obtaining disconnected subgraphs from a machine-machine or machine-part graph to identify manufacturing cells. Rajagopalan and Batra [13] suggested the use of Jaccard's similarity coefficients and graph theory to form machine groups. Faber and Carter [9] developed a graph theoretic algorithm for grouping machines and parts into manufacturing cells by converting the machine similarity matrix into a cluster network. The cluster network is partitioned into cells by solving a minimum cost flow problem. Kumar et al [12] developed a 0-1 quadratic programming with linear constraints to solve the part grouping problem. Askin and Chiu [ 1] proposed a cost-based mathematical formulation and heuristic solution for MCFP. Vohra et al [16] proposed a network-based algorithm to minimize the amount of machine times performed outside the part primary cells. Wu and Salvendy [ 17] developed a network model to partition the machine-machine graph into cells by considering operation sequences. This paper proposes a cell formation heuristic method involves clustering individual machines to identify manufacturing cells. The heuristic is used to create manufacturing cells by partitioning the individual-machine set into a number of desired cells. The machine set is represented by a graph with individual machines as nodes with weights represented by machine potential utilization or usage and machine-pair desirability measures as arc weights. The objective is to create manufacturing cells with minimum intercell movements. The remainder of this paper is organized as follows. Section 2 describes the proposed heuristic. Section 3 presents an illustration of the methodology. Section 4 evaluates and compares the proposed heuristic to data sets from the literature. Finally the implications and conclusions of the research are discussed in section 5. 2. CELL FORMATION BY GRAPH PARTITIONING METHOD An overview of the proposed Cell Formation by Graph Partitioning method (CFGP) is shown in Fig. 1. The CFGP method consists of three stages, the machine-machine (mc-mc) graph is constructed in the first stage, an initial assignments of individual machines to cells is generated in the second stage, and a local improvement heuristic is implemented in the last stage in order to develop the final cellular manufacturing cells. ii
Construct mc-mc graph with arc costs = desirability for machine pairs to be in the same cell
Get initial assignments of individual machines to cells
Improve on the initial assignments of individual machines to cells
Fig. 1 CFGP Method Overview 2.1 CFGP-stage 1: Construction of the MC-MC graph
A graph is constructed in which each node represents an individual machine. Each machine node on the mc-mc graph has an associated weight, which is measured as the potential
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machine utilization or usage. The machine nodes are connected by an arc with an associated weight defined in terms of a desirability measure. Let d,/.,,2 represents the desirability measure between machine nodes n l and n2 (each representing an individual machine of type m l and m2 respectively) which is proposed by Askin et al [3].The desirability index is defined as a convex combination of two parts:
d.z,. 2 =aW~,.2 +(l-a)W.~.. 2
0 < d.)..2 < 1 and 0 < a _< 1
Where W:~,,2 =max {W.,..2,W.2.. , },
W~,,,2 = Proportion of machine nl usage associated with parts that also use machine n2 s
W~.,2
~--
s
s
Yml,j + " Ym2, j -
-
Yml, J Ym2, J
J
s-I x
s
f
Xc
jl = l j2 = jl + I
s-,
s
E
E
X~lj2
j l = 1.12 = j l + 1
fj,j2 = Proportion of current parts that require operations j 1 and j2 j, j l , j2 - Indexes of operation types with values of 1,2 .... ,J X~tj2 =max {Y.,.j,,Y.zj, }.max {YmI.y2'YmLi2}-- max {Y.,.j,'Ym,j2,Y.2.j,'Y.2.j2} y.,j = 1 if machine type m can perform operation j; 0 otherwise.
m, ml, m2 = Indexes of machine types. W~.,2 is computed as the maximum proportion of machine n l usage or machine n2 usage that involves parts that also visit the other machine given the current part-operation assignments. W~.,2 is a function of the combined processing capability of machine pair nl (of type ml) and n2 (of type m2). W~.,2 is the product of two terms. The first term is the proportion of part operations that can be performed by the combined set of machine pair nl of type ml and n2 of type m2. The second term in W~.n2 takes a weighted average over those operation pair that requires both machines of the likelihood that an arbitrary future part will require both r operations. Xjtj2 takes on the value one only if neither machine type ml nor m2 can perform both operation j 1 and j2, but this pair of operations can be completed by the union of the two machines. The weights fj,j2 then indicate the probability that a future part will require this pair of operations. The parameter a is specified by the user. A high value of a indicates a user preference for designing cells based on the current part set and demand requirements. A lower value of a indicates focusing on the possible set of operations that could be performed by both machines nl of type ml and n2 of type m2 rather than the current set that have been assigned. Assuming that the part operations have been assigned to machine types. The potential machine utilization or usage can be calculated using the forecasted part demands and part processing needs. We assume that the individual machine potential utilization or usage are given as part of the input to the CFGP heuristic, which can be used in caleulating the
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desirability measures as explained before. The input data to the CFGP heuristic consists of the arc weights (machine pair desirability index) between all individual machine node pairs, the available number of individual machines of each type, the individual machine utilization or usage. The user is required to interactively specify the number of cells (C). The cell/partition size is calculated as [N/C'I, where N is the number of individual machines. 2.2 CFGP-stage 2: Initial Assignments of Machines to Cells On the basis of the mc-mc desirability measures (computed in stage 1) and the number of individual machines available of each machine type, the procedure for initial partitioning (initial cellular configuration) of the mc-mc graph is used. A newly proposed heuristic (HEUIP) has been developed in order to generate an initial cellular configuration. The proposed HEU-IP can be used as a quick stand-alone method of generating cellular configuration. The HEU-IP's pseudo code is given below (cell and partition are used interchangeably): HEU-IP Pseudo Code: Begin
Until each cell contains exactly one machine, Do Identify machines nl and n2 such that dn1~2 i$ the minimum. Assign n l and n2 to two different and empty cells Discard machines n l and n2 from the unassigned machines set. I f only one cell is remaining then Assign nl to this cell Discard machine nl from the unassigned machines set. End Until Until the unassigned machines set becomes empty, Do Identify machines nl and n2 such that dnl,n 2 is the maximum. Assign nl and n2 to the same cell End Until Read V (* interactively from the user *) Add V% dummy individual machines to each cell, such that the C cell sizes are equal. End
The result of this stage is an initial cell configuration. Each individual machine is assigned to a cell and the mc-mc graph is partitioned into C cells. At the end of this stage, the cell sizes are equalized by complementing the unbalanced cells (cells which contain less [N/C] machines). The equal cell sizes are obtained by adding enough number of dummy machines to each unbalanced cell. The dummy machine is a machine with zero desirability with all the other machine nodes and has a usage of zero. The equal cell sizes are necessary to implement the local refinement heuristic. Also allowed is the inclusion of a fixed additional percentage of dummy machines for all cells (V%). This facilitates switching a machine from cell to cell without corresponding to switch real machine from cell to cell. Thus, note that the greater the number of dummy machines introduced, the more likely that the number of final machine assignments is not balanced across cells.
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2.3 CFGP-stage 3: Locally Refining the Initial Cellular Configuration In this stage, a modification of the graph-partitioning heuristic originally due to Kernighan and Lin (KL) [ 11] is applied as a local refinement scheme. KL is the most widely used local refinement algorithm for the GP problem. It is an iterative method based on inner and outer iterations. The fundamental idea behind KL heuristic is the concept of gain associated with moving a machine into a different partition or cell. The gain is simply the net reduction in the weight of cut arcs (intercell moves) that would result from switching a machine to a different cell. More formally, if machine n l in cell c I were to be interchanged with machine n2 in cell c2, the corresponding gain ( gel,c2 .~..2) can be expressed as:
I. nec 2
Where
mecl
I raecl,ma*nl
nec 2,n~n 2
n and m are individual machine indexes.
The first term of g~l.,z .c2 represents the total external desirability of machine nl and n2 that becomes internal after interchanging the two machines (hence, the value is positive). The second term represents the total internal desirability between machine n l and all machines in cell c I; and machine n2 and all machines in cell c2 that become external after the interchange is implemented (hence, this term is negative). The n l and n2 desirability measure is counted twice, then it is subtracted twice. Clearly, if the gain associated with interchanging nl and n2 is positive, then making the interchange will reduce the total desirability measures on the cut arcs which represent the intercell moves. The gains are calculated for all the possible pairwise interchanges between cell c I and cell c2. The machine-pair with the maximum gain is set aside temporarily in an "interchange list". The gains are recalculated for the machines in cell [cl-{nl }] and for cell [c2-{n2}]as ifnl and n2 have been actually interchanged. The process of determining the machine pair with the maximum gain, from the remainder set of machines in cells cl and c2, is repeated. Another pair of machines is set aside in the "interchange list". This process is repeated until all machines in cells c I and c2 have been exhausted. K is chosen to maximize the partial sum K
g, , g, ~ (Interchange List) denoted by G. Now, if G>0, a reduction in intercell moves of t-I
value G can be made by interchanging the K sequence of machine pairs from the "interchange list". After this is done, the procedure is repeated with another pair of cells till all the cell pairs are considered. The coverage of all the possible cell pairs is called a "pass". The passes are repeated until the pass gain G = zero. The original KL heuristic partitions a set of even number of nodes into two equal partitions minimizing the cost of arcs cut. The KL heuristic has also indicated an approach for the case where the number of partitions is not fixed but is to be equal to or greater than a specified value. In the case of cell formation, there is no constraint on the number of partitions (cells) and the inequality of cell sizes is permitted. The KL heuristic is based on swapping equal sets of nodes. In the present work, the KL heuristic has been modified as follows: 1. The number of partitions is not fixed and can be interactively controlled by the user. 2. The partition sizes have been manipulated to be unequal, by introducing dummy machines. 3. The modified KL heuristic is not restricted to even number of nodes.
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4.
Unequal sets of nodes can be swapped by including dummy machines in the swapped sets. 5. Further improvement is introduced to the KL final cellular configuration as follows: 9 An attempt is made to reduce exceptional part-operations by reassigning them to the primary cell to which the associated part is assigned. This attempt is limited by the individual machine utilization or usage constraint; and the availability of a machine that can perform the exceptional part-operation to be reassigned to the primary cell. 9 An attempt is made to reallocate the machines that process part-operations on parts for which the primary cell is not the one in which the machine is currently allocated. The pseudo code of the modified KL heuristic is shown below. Modified KL Pseudo Code:
Begin Until Pass Gain=O Do (* each outer loop iteration is called a "Pass" *) For c l = l to C-1 Do (* c I & c2 are cell indexes *) For c2=cl + 1 to C Do Compute g.lg~ - ~ ~2 for each machine pair. Select nl e cl and n2 E c2 such that g.l..2 _c~.c2 is maximum. Append nl & n2 to the exchange list. Update the gain values g
Choose K to MAX G =
~ g, ~=l,g,e( Interchange List )
Implement the sequence o f K exchanges. Update gains of all neighbors of switched machines End Until Implement the part-operations reallocations. Implement the bottleneck machine reallocations. End
The CFGP method can be used as a generator of alternative cellular designs. Several initial partitions using other heuristics could be used. The user-controlled parameters such as , V, and C can be used to generate alternative cellular configurations. This can assist the decision maker in selecting the suitable alternative. 3. ILLUSTRATION A hypothetical example is used to illustrate the CFGP heuristic. There are 9 machine types consist of 16 individual machines. The individual machine utilization is calculated based on their processing capabilities and the current operation assignments. Table 1 shows the individual machine utilizations and the machine types. Table 2 shows the desirability measure values for each machine pair (the results of stage 1). The entries of this matrix represent the arc weights of the mc-mc graph. The machine utilization or usage is used as a node size in the local refinement KL heuristic. Stage 2 results are given in Table 3. The final partitions after applying stage 3are given in Table 4.
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Table 1. Individual Machines Data Individual MC i
2 3 4 5 6 7 8 9 10 II 12 13 14 15 16
MC Type
Utilization
!
o.588
! 2 2 3 3 4 5. . . . . 5 6 6 7 7 8 9 9
0.214 0.712 0.654 0.200 0.591 0.226 0.317 0.590 0.225 0.217 0.273 0.180 0.378 0.521 0.097 .....
Table 2. Desirability Measures of Individual Machines 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0.0() 0.00 0.46 0.00 0.78 1.00 0.65 0.00 0.00 0.00 0.00 0.28 1.00 0.33 0.00
2 3 0.00 0.00 0.00 0.00 0.00 0.00 0.73 0.00 0.00 0.00 0.27 0.00 0.38 0.00 0.36 0.45 0.00 1.00 0.00 0.26 0.73 0.47 0.27 0.00 0.38 0.00 0.27 0.00 1.00 0.00
4 5 0.46 0.00 ~0.00 0.73 0.00 0.00 0.00 0.00 ' 0.74 0.00 1.00 0.00 0.30 0.35 0.00 0.70 0.00 0.00 0.00 0.00 0.00 1.00 0.86 0.00 1.00 0.30 0.80 0.00 0.00 1.00
6 0.78 0.00 0.00 0.74 0:00 0.73 0.65 0.00 0.0() 0.74 0.00 0.57 0.73 0.53 0.00
7 1.00 0.27 0.00 1.00 0.00 0.73 0.65 0.00 0.00 0.00 0.00 1.00 0.92 1.00 0.00
Table 3. Initial Cellular Configuration Partition / Cell I . 2
Individual Machines 1'3'4'5'7'10'14,!.6 t__ ..2,6,8,9, I l, ! 2,.13,15
. 8 0.65 0.38 0.0(i 0.30 0.35 0.65 0.65 0.00 0.00 0.00 0.35 0.00 1.00 0.00 0.00
9 0.00 0.36 0.45 0.00 0.70 0.00 0.00 0.00 0.10 0.00 0.80 0.00 0.00 0.00 1.00
10 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.10 0.00 0.33 0.00 0.00 0.00 0.00
II 0.00 0.00 0.26 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.26 0.00 0.00 0.00 0.00
12 0.00 .73 0.47 0.00 1.00 0.00 0.00 0.35 0.80 0.33 0.26
13 0.28 0.27 0.00 0.86 0.00 1.00 1.00 0.00 0.00 0.00 0.00 0.00
14 1.00 ().38 0.00 1.00 0.30 0.73 0.92 1.00 0.00 0.00 0.00 0.14 0.86
15 0.33 0.27 0.00 0.80 0.00 0.53 1.00 0.00 0.00 0.00 0.00 0.00 1.00 0.86
0.00 0.14 0.86 0.00 1.00 0.86 1.00 0.00 0.00 0.00
16 0100 i.00 0.00 0.00 1.00 0.00 0.00 0.00 1100 0.00 0.00 1.00 0.00 0.00 0.00
Table 4. Final Cellular Configuration Partition /Cell
1 2
....Individual Machines
1,4,6,7,8,11,13,14,15 ...... 2,3,5,9,10,12,16
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Note that the initial partitions have equal cell sizes, 8 machines in each cell. Using V=25%, the final two cells contain 9 and 7 machines respectively. Table 5. Initial Part-Machine Incidence Matrix MC No. MC Type I 2 3 4 5 6 7
Z
lO
~.,
12 13 14 15 16 17
1 I 2 ! 3 ! 4 15
i 6
I I 112121313
7 i 8 i 9 I I O ! I1 I i-2 i 13 1 1 4 1 1 5 l l6 1415151616l 71718 [ 9 [ 9
I
1 i 1
1 1 ! I
I I I 1
I
1 I
1 ! ! 1
I
I
I I
I 1 1 I I !
I I 1 1
! 1 1
1 I 1
I 1 I 1
I I I I I
I i I 1 1 1
I I
I I 1
I I
1 I 1
I 1
Table 6. Final Part-Machine Incidence Matrix MCNo.
,Z
1 1 2 _ 3 ' 1 4 ~ 7 I 8 1 9 1 I0 1
5 6
~-i~
[ 13 114 [
1 [ 4 I 6 I 7
MC rypel I i 2 1 3 I 1
1
14151 I
I I 1 1
1 ! I i i 1 I I
1 I 1
i8 19
1 I I
I I I I
2 [ 3 I 5 19-1
lo112116
1121315161719
11
1 I I 1
I I
FT]
ill
13 14 15 16
17
15
6 I 7 I 8 I 9
II I
I I
1 I I I 1 1 1 I
1
1 1 1
I 1
1 I 1 1 I
1
1 1
1 1
1 1 1
To visualize the effect of CFGP heuristic, the initial part-machine incidence matrix is given in Table 5 (this is not part of the CFGP input). The rows represent parts and the columns represent machines. An entry of 1 at the intersection of row r and column s indicates that part
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r requires an operation on machine s. The final part-machine incidence matrix is given in Table 6 (this is not part of the CFGP output). The two partitions that represent the two cells are clearly visible. This concludes the CFGP illustration. The proposed heuristic generates machine assignments to manufacturing cells, given the part-operations assignments to machines. 4. EVALUATION AND COMPARISON The CFGP heuristic, described in section 2, has been implemented in the C programming language and can partition a 1000-node graph. Fast data structures have been used in the C program to speed the search and looping operations such as single linked lists and double linked lists. In identifying the previous CF approaches for comparison, the Hamiltonian Path Heuristic (HPH) proposed by Askin et al [2] and the method proposed by Co and Araar [7] (CA) have been selected. The reasons for selecting these two methods are: (1)detailed solutions generated by these methods to one or more data sets are available and hence, there is no need to develop codes for operationalizing these procedures; and (2) The HPH method has shown to dominate previous matrix restructuring methods for the CF problem and thus, is a valid choice for comparison. CA method considers information similar to that of the CFGP method, which makes it a reasonable choice for comparison. Given that both the HPH and CA methods focus on the objective of minimizing intercell moves, Grouping Efficiency (GE) measure developed by Chandrasekharan and Rajagopalan [6] has been used as the basis of comparing the solutions generated by the HPH, CA, and CFGP methods. The GE measure is defined as follows: G E = qO I + (1 - q)O 2 0 <_ q <_ 1 where; 0t - The ratio of the number of ' 1' entries in the diagonal blocks of the rearranged partmachine incidence matrix to the total number of possible' 1' entries in all the diagonal blocks. Thus 0 < 0 ! < 1. This measure focuses on the within cell 'utilization'. 02 = The ratio of the number o f ' 0 ' entries in the off-diagonal blocks of the rearranged partmachine incidence matrix to the total number of possible '0' entries in off-diagonal blocks. Thus 0 _<02 _<1. This measure focuses on the intercell moves. q is assumed to be 0.5 which means that both intercell and intracell linkages are equally weighted.
Co and Araar demonstrated their method on a 15-part, 63-machine problem. Their method produced a 6-cell solution with a GE of 0.68. Setting the maximum allowable machine usage to be 100%, C=6 and =0.50, an improved GE of 0.72 is obtained. The CFGP's Cell density was improved and the number of exceptional part operations was reduced. The CFGP method is compared with the HPH method developed by Askin et al [2]. Six published data sets have been used for comparison purposes. For each data set we assume that one machine of each type is available and =0.5. The results of applying the CFGP heuristic and the HPH to the six data sets are presented in Table 7. The GE measure has been calculated at p=0.5. For data sets 1,3, and 6 both methods yielded the same solutions. For data sets 2,4, and 5 the CFGP heuristic dominated the HPH method.
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Although the CFGP input has been reduced to match the two other methods, the CFGP has shown its superiority over the HPH and CA methods. It took almost 10 seconds to solve the CA data set on a Pentium II - 200 MHz processor. The CFGP solves the problem of assigning machines to cells given the part-operation assignments to machines and it considers more than one machine of each machine type. This is a major step in solving the cellular manufacturing system design, and hence, a good solution obtained at this stage could lead to improved solutions at later design stages. Table 7. Comparison of CFGP and HPH methods Problem Number 1
Reference
Chan & Milner ,[5] De Witte [8] Chand. & Rajag. [6] Stanfel [l 5] Burbidge [4] Burbidge [4]
C 3 3 3 4
5 5
Size Part, M/C 15,10 19,12 20,8 24,14 43,16 43,14
CFGP
HPH
GE 0.9150 0.8487 0.9583 0.9920 0.8014 0.8302
GE 0.9150 0.7664 0.9580 0.8390 0.7986 0.8302
5. CONCLUSIONS In this paper a cell formation problem solution approach has been proposed. The proposed technique is unique in many aspects. First, the CFGP method allows the user to design several alternative cellular configurations by setting the number of cells and the cell size desired. Secondly, The CFGP approach is capable of solving problems of realistic sizes while including consideration of important factors such as alternate machine types and usages. The CFGP method consists of three stages. The first stage evaluates the individual machine desirability index for each machine pair. Stage 2 generates an initial cellular configuration. Finally, stage 3 locally refines the initial cellular configuration in order to reach the best possible solution given the initial partitioning. The CFGP has been compared with 2 published methods and has shown dominating or equivalent solutions. REFERENCES 1. Askin, R.G. and Chiu, K.S., "A Graph Partitioning Procedure for Machine Assignment and Cell Formation in Group Technology," International Journal of Production Research, Vol. 28, pp. 1555-1572, (1990). 2. Askin, R.G., Cresswell, S.H., Goldberg, J.B., and Vakharia, A.J., "A Hamiltonian Path Approach to Reordering the Part-Machine Matrix for Cellular Manufacturing," International Journal of Production Research, Vol. 29, pp. 1081- 1100, (1991). 3. Askin, R.G., Selim, H.M., and Vakharia, A.J., "A Methodology for Designing Flexible Cellular Manufacturing Systems'" lie Trans., Vol. 29, pp. 599-610, (1997). 4. Burbidge, J. L., "The Introduction Of Group Technology," John Wiley & Sons, New York, (1975). 5. Chart, H.M., and Milner, D.A., "Direct Clustering Algorithm for Group Formation in Cellular Manufacturing", J. of Manufacturing Systems, Vol. 1, pp. 65-74, (1982). 6. Chandrasekharan, M.P., and Rajagopalan, R., "An Ideal Seed Non-hierarchical Clustering Algorithm for Cellular Manufacturing," International Journal of Production Research, Vol. 24, pp. 451-464, (1986).
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7. Co, H.C., and Araar, A., "Configuring Cellular Manufacturing Systems," International Journal of Production Research, Vol. 26, pp. 1511-1522, (1988). 8. DE Witte, J., "The Use Of Similarity Coefficients In Production Flow Analysis," International Journal of Production Research, Vol. 18, pp. 503-514, (1980). 9. Faber, Z. and Carter, M.W., "A New Graph Theory Approach for Forming Machine Cells in Cellular Production Systems," Flexible Manufacturing Systems: Methods and Studies, ed. Kusiak, A., New York: North Holland, pp. 301-318, (1986). 10. Fj~illstr6mas, P., "Algorithms for Graph Partitioning: A Survey," Link6ping Electronic Articles in Computer and Information Science (ISSN 1401-9841), Vol. 3, No. 010, "http// www.ep.liu.se/ea/cis/1998/010", (1998). 11. Kemighan, B.W., Lin, S., "An Efficient Heuristic Procedure for Partitioning Graphs," AT&T Bell Labs. Technical journal, Vol. 49, pp. 291-307, (1970). 12. Kumar, K.R., Kusiak, A., and Vannelli, A., "Grouping of Parts and Components in Flexible Manufacturing Systems", European Journal of Operational Research, Vol. 20, pp. 387-397, (1986). 13. Rajagopalan, R. and Batra, J., "Design of Cellular Production Systems- A Graph Theoretic Approach", International Journal of Production Research, Vol. 13, pp. 56-68, (1975). 14. Selim, H.M., Askin, R.G., and Vakharia, A.J., "Cell Formation in group technology: Review, Evaluation and Directions for Future Research", International Journal of Computers and Industrial Engineering, Vol. 34, pp. 3-20, (1998) 15. Stanfel, L.E., "Machine Clustering for Economic Production", Engineering Costs and Production Economics, Vol. 9, 73-81, (1985). 16. Vohra, T. and Chen, D.S., Chang, J.C. and Chen, H.C., "A Network Approach to Cell Formation in Cellular Manufacturing", International Journal of Production Research, Vol. 28, pp. 2075-2084, (1990). 17. Wu, N and Salvendy G., "A Modified Network Approach for the Design of Cellular Manufacturing Systems", International Journal of production Research, Vol. 31, pp. 14091421, (1993).
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AN INVESTIGATION OF THE GROUP SCHEDULING HEURISTICS IN A FLOW-LINE CELL Huzayyin, A.S.*, Badr, M.A.** and Helal, M.E. ***
* Professor and Dean, Benha High Inst. of Technology ** Researcher, Mechanical Eng. Dept., The National Research Centre- Cairo *** Demonstrator; Mechanical Eng. Dept., Benha High Inst. of Technology
ABSTRACT A comparative study of group scheduling (GS) in a flow line cell is presented. Three simple scheduling heuristics are compared with two iterative improvement heuristics. The objective is minimizing makespan and total flow time separately. A number of modifications are proposed in order to explore the performance of the heuristics and to investigate the characteristics of the GS model. In addition, a timetabling procedure for multi-family cells is proposed, considering the presence of the zero processing times. Results showed that the proposed modification could improve the performance of the heuristics under study. The iterative improvement techniques were found appropriate for GS not only because of their superiority over the simple methods but because they can handle the phases' interaction in GS as well. The tabu search heuristic is found preferable to the simulated annealing heuristic. KEYWORDS Group Scheduling, Heuristics, Flow-Line Cell, Tabu Search, Simulated Annealing. I. INTRODUCTION Group scheduling (GS) is applied when parts are classified into different part families based on setup and processing requirements. The machines are assumed grouped into a manufacturing cell. The creation of part families leads to the creation of a two-phase scheduling model: scheduling part families (family phase) and scheduling jobs within families (job phase). A typical application of GS is the scheduling of a static flow line manufacturing cell [ 1-4]. The scheduling task is greatly simplified with GS in addition to the reduction of the setup times. Results generally indicated that the GS approach yields superior performance over the corresponding traditional (single-phase) models [5,6]. GS problem is Non-Polynomial complete. An optimal solution based on permutation schedules with respect to makespan was obtained by Hitomi and Ham in 1976 using branchand-bound. Their work is a two-stage application of the branch and bound model of Ignall and Scharage developed in 1965 for the general flow shop problem [3,4,7]. The main feature of the methodology is that schedules in both phases must be determined simultaneously in order to achieve optimality [1]. However, as optimization techniques are computationally burdensome, heuristic techniques are used instead to obtain near optimal solutions. The traditional flow line heuristics can be modified for the flow line GS problems where heuristics
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are executed in two stages, family phase and job phase. Grasso et-al. [4] proposed a general framework to modify the traditional heuristics to GS applications. The problem according to them is simplified into the separated determination of job sequences within families, and the determination of family sequences. This does not take into account the possible interaction between the two phases and hence leads to suboptimal results. Still, the simplification is useful in order to derive rapid and efficient GS heuristics from the traditional heuristics. Makespan is the criterion in consideration. Modified versions of the known traditional flow line scheduling heuristics were developed in a way similar to Grasso's framework. Hitomi [8] developed a GS version of Petrov's method. Wemmerlov and Vakharia [7] developed GS versions for Campbell, Dudek and Smith's heuristic (CDS) and for Nawaz, Enscore and Ham' s heuristic (NEH). A drawback of the above simple GS heuristics is that they do not consider the phase's interaction. In addition, the number of generated solutions is small. With the increase of computer capabilities, researchers developed iterative improvement techniques to solve the GS problem. The iterative methods seem able to consider the phase's interaction, while enumerating a much larger number of feasible solutions. Two of the generic techniques applied are the simulated annealing (SA) and the tabu search (TS) approaches. SA is a randomized iterative improvement technique that was originally developed as a simulation model for a physical annealing process. Basic concepts for the algorithm were developed by Kirkpatrick et-al, in [9]. SA starts off from an arbitrary initial configuration associated with a cost given by a relevant cost function. In each iteration, by slightly perturbing the current configuration a new configuration is generated. Difference in cost between the two configurations is compared with an acceptance criterion that tends to accept improvements but also admits, in a limited way, deteriorations in cost in order to avoid being traped in a local optimum. Initially, the acceptance criterion is taken such that deteriorations can be accepted at a high probability. As the process proceeds, the probability of accepting deteriorations decreases until zero [10]. TS is a meta-strategy that has its origins in combinatorial optimization procedures applied to some non-linear problems in the late 1970s. Strategic principles of TS in a broader sense have been laid out in [11,12]. It starts from a random initial solution and performs a set of alterations (moves) on it to move to neighbour solutions in a search for a better result. The basic components of TS are defined as follows. A tabu list is comprised of moves that are not allowed (tabu moves) to be performed at the current iteration. Function of the list is to prevent returning to previously visited solutions to avoid cycling. The aspiration level function whose role is to provide added flexibility to choose good moves, by allowing the tabu status of a move to be overridden if this aspiration level is fullfilled. A Long term memory (LTM) is employed to achive regional intensification and global divesivication of the search. LTM records and compares features of the best trial solutions generated during a particular period of search. Features that are common are taken to be regional attributes of a good solution. The method then seeks new solutions that exhibit these features (intensification). Or LTM may be used to guide the process to regions that contrast with those examined so far (diversification). Vakharia and Chang [2] proposed a SA heuristic to minimize makespan in multi-family cells. They used an acceptance probability that is independent of the change in the objective function value. The heuristic spends 90% of the search efforts in the job phase and 10% in the family phase. It starts from an initial random solution and performs a pair-wise interchange of
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jobs or families. Makespan is checked and the acceptance probability is then used to accept or reject an inferior solution if reached. Initial value of the acceptance probability is 50% and it is reduced by a constant amount each iteration until reaching zero. Kapov and Vakharia [ 13] proposed a TS heuristic to minimize makespan in the multi-family cells. An initial random schedule is generated and moves are checked in a pair-wise interchange of jobs or families. A best move is identified and performed at each iteration given it is not a tabu move. The tabu list is a record of a predefined number of recent moves. It is updated each iteration by appending the last performed move and deleting the oldest one. A LTM is used to rerun the procedures five times by generating new initial solutions. LTM is FxF frequency matrix that contains records about the number of times a family occupied a certain position in the trial schedules during the search process, where F is the number of families. For a new start, a new family' schedule is generated by letting each family be in the position in the schedule it occupied for the largest number of times (intensification). Only a new schedule for part families is generated while jobs schedules are randomly reset. The aspiration level function allows a tabu move to be performed if an overall improvement is got. 2. OBJECTIVES OF THE CURRENT STUDY
2.1. Proposed timetabling procedure Sridhar and Rajendran [14-16] notified that a feature of cells is the presence of some zero processing times for some jobst. They showed that disregarding this yields erroneous jobs start and finish dates, incorrect information about machine availability, and overestimates of makespan and total flow time. They presented a modified timetabling formulation that accounts for the zero times. Their work considers a single-family cell. For the multi-family situation, Hitomi [8] and Kapov and Vakharia [13] proposed two different timetabling procedures. Both neglected the zero times. The procedure proposed by Kapov and Vakharia could not be implemented. It contained errors such that some valuse calculated in some latest seteps of calculations are needed in eariler steps [ 18]. The following timetabling procedure is proposed for the multi-family cells considering the presence of the zero times. Assuming a flow cell of M machines for the processing of F part families each contains ni jobs, let the family index be i (i =l ,2. . . F) and jobs in family i indexed by j (j =l,2...ni). The setup time of family i on machine k (k =I,2...M) is Sik and the processing time of.job j in family i on machine k is Pijk. Let Startij,k be the start time for.job j in family i on machine k and SetStarkk setup time start for family i on machine k. Fori---1,2 .... F , F o r j - l , t Start,.i. k = Where:
kk jj
Fork = 1,2 ..... M IStart,.~.~ + Pi.t.~ max [Start, ,~.k +. P,, . , .k + S, k x Z~
If P~,t.k > 0
0 Otherwise Last machine that.job 1 in family i was processed on. The job preceeds job 1 in family i on machine k.
2In 1974, Baker [17] mentioned the possibility that a job may not be processed on some machines in the general flow shop and he assigned zero processing times for such cases.
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ii
Family containing job jj.
ZI
Binary variable such that = tl0
If PU.k > 0
If Pi,t,k = 0
For i = 1, 2,.. F , For j = 2,3,...,ni , For k = 1,..., M
Starts.j,k =
Start
+ P~,~,k + S+,k• Z2
IfPij k
>0
Otherwise
SetStarti,k = Where
9
kkk JJJ
Startidiij,k-
Si,k
J~jJ
Last machine job j in family i visited. The job preceeds job j in family i on machine k. Family containing job jjj. First job in family i having a non-zero time on machine k.
Z2
Binary variable such that = {10
~ 1 7 6 1 7 6
**~
111
Makespan =
otherwise If iii < i
max { Start v.,.k + Pvt,k }
kffil,2 .....M
Total Flow Time = ~ ~ Finishi,j.. i=t ~(here
9
t V
lk
j-t
Last job processed on machine k Family containing job t Last machine in the cell that job Jij was processed on.
2.2. M o d i f y i n g the GS heuristics
The three simple GS heuristics of Hitomi [8], CDS and NEH [7], and the two iterative improvement techniques of SA [2] and TS [4] are selected for the study. Modified versions of each of them are proposed. The details of the proposed modifications and description of the modified versions are found in [18]. Basically, the modifications to the simple methods were to test Rajendran's modification [ 15,16] that suggests dividing the scheduling indices by the number of the non-zero times for the job. This is to account for the zero processing time but no explanation is provided in [ 15,16]. This was tested and found ineffective [ 18]. In addition, an iterative version of CDS was proposed. It was found to be remarkably superior to the original CDS, which indicates the importance of taking the two-phase nature of GS in consideration. However, its superiority is limited by the finite number of solutions enumerated by CDS [ 18]. The iterative improvement techniques were found superior to the simple methods. They are considered in the following analysis. Three modifications to SA and two to the TS are proposed. The modified heuristics are described below.
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In SA-M-I, a change-dependent acceptance probability version of the original SA [2] is proposed. The standard form of the acceptance probability [9,10] is employed. The advantage of using change-dependent acceptance probability as defined in the listing below, is that solutions which cause drastic changes in the objective function value are avoided as iterations proceed [14]. In SA-M-2, slight modifications in Steps 6 and 7 in the original SA is suggested to increase the efficiency of the search process. The idea is to prevent the use of two successive random numbers of the same value so that to avoid reversing (cancellation) of the last performed perturbation operation during the current operation, hence to avoid wasting efforts. In SA-M-3, the control on the random number behaviour in SA-M-2 is added to Steps 6 and 7 in SA-M-I. Being the best SA version (as shown later) SA-M-3 is listed below [18].
Step 1. Step 2.
Step 3. Step 4. Step 5. Step 6.
Step 7.
Set X = 25, Y = 50, APo = 0.5, r = 0.9, and set the GP value. Generate a random initial schedule. This includes a complete sequence for all jobs (fio), a family sequence (x) and a sequence for jobs in each family (~i); i = 1, 2 . . . . , F. Let this be current solution with total flow time Flow ~ Let fi~ be the incumbent solution with total flow time Flow*. Set fi ~ = f~o and Flow* - Flow ~ Let X = rX. If X < 1.62 then stop, else set y = 0 and continue. Set y = y + 1. If y > Y then go to Step 3, else go to Step 5. Generate a random number v (0 <_.v < 1). If v >_.GP go to Step 7, else go to Step 6 Generate a random number v l (1 __< v l _< F). If v l = last v l and previous perturbation was a families interchange, then regenerate v 1, Else if v 1 = last v 1 and previous perturbation was a jobs interchange and no change in current sequence has occurred in that perturbation, then regenerate vl, Else interchange the families in positions vl and v l + l (if vl - F, interchange the families in positions F and 1) and get a new family sequence x I. Based on x t specify a new complete job sequence D t and calculate its total flow time Flow t. (a) If Flow I > Flow" then go to (b). Else let fi* = fit, Flow' = Flow I, go to (b). (b) If Flow I > Flow ~ then let A = Flow I - Flow ~ calculate the acceptance probability APx - EXP( - A / X ) and go to (c). Else let x = x ~ and fio = f~l in the current solution, and set Flow ~ = Flow l and go to Step 4. (c) Generate a random number v2 (0 < v2 < 1). If v2 >_ A P go to Step 4, Else let D ~ = fii, x = x I in current solution, set Flow ~ = Flow I and go to step 4. Generate a random number v3 (1 _
(c)
If Flow ! >__Flow' then go to (b), Else let fi* = f2 ! in the incumbent solution, set Flow' = Flow I and go to (b). If Flow j _> Flow ~ then let A = Flow t - F l o w ~ calculate the acceptance probability APx = EXP( - A / X ) and go to (c). Else let ~tfI = ~ttrt ,fio = fi! in the current solution, and set Flow ~ = Flow I and go to Step 4. Generate a random number v2 (0 < v2 < 1). If v2 >_. A P go to Step 4, Else ~trI = ~tt fl ,D~ = fl t in current solution, set Flow ~ = Flow I and go to Step 4.
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Tne GP parameter controls the amount of search efforts given to each scheduling phase. Its value of 0.1 in [2] leads to spending 10% of efforts to the family phase and 90% to the job phase. Alison [3] stated that the family phase is more worthy. To investigate this GP will be given values of 0.1, 0.3, 0.5, 0.7, and 0.9. This is used for all SA versions. In TS-M-I, when generating the new restart schedule, the current jobs sequences within families are kept instead of being randomly regenerated. The concept is to make use of the search efforts in the job phase. LTM is used for the same purpose in the family phase. In TSM-2, long term memories for jobs within the families are developed and used to complete the new restart schedule by generating restart jobs sequences within families as done in the family phase. Actually, original TS is completed rather than being modified. Being best TS version (as shown later) TS-M-I is described below. Variable list sizes are used. Two types of tabu list are needed for the two phases. In the family phase, the initial f-tabu list size is Int(F/2), decreased size is Int(F/3) and the increased size lnt(F/0.5). In the job phase the initial j-tabu list size is Int(N/F), decreased size is Int(N/2F) and the increased size Int(N/0.5F). N is the total number of jobs.
Step 1.
Initialize the f-tabu-size (tabu list for family phase) and j-tabu-size (tabu list for job phase). Set the number of LTM restarts = 5. Set LTM matrix = 0. Step 2. If LTM - 0, generate a random families sequence, Else generate a families sequence using LTM, and for each family, generate a jobs sequence using LTMi. Let this be current solution f~o with a total flow times Flow ~ Let f~' represent the incumbent solution with total flow time Flow'. Set f~" = fl ~ , Flow" -- Flow ~ . Set LTM = LTM + 1 and LTMi = LTMi +1 for all i = 1,2,..., F. Step 3. Start counting iterations for family exchange. Set f-iter - 0. Step 4. Stopping criterion for family exchanges: (a) If no improvment in the last 5F itaerations with the initial f-list size then decrease the f-list size and go to (b). (b) If no improvment in the last 2F itaerations with the decreased f-list size then increase the f-list size and go to (c). (e) If no improvment in the last 3F itaerations with the increased Size then set the list size to its initial value and go to Step 6. If at any poine therse an improvement then go to Step 5 Step 5. Family exchange phase of search. Evaluate completely the neighbourhood Ni (f~o) and select the best exchange of families. Denote the new complete sequence by f~l and its makespan by Flow I. If Flow I < Flow' then set f~'--f~m and Flow' = Flow*. Set f~o = f~l, Flow o = Flow* and go to Step 4. Step 6. Start counting iterations for job exchanges. Set j-iter = 0 and go to Step 7. Step 7. Stopping criteria for job exchanges. Set j-iter = j-iter + 1. (a) If no improvment in the last lnt(N/3)iterations with the initial j-list size then decrease the j-list size and go to (b). (b) If no improvment in the last Int(N/3) itaerations with the decreased j-list size then increase the j-list size and go to (c). (c) If no improvment in last Int(N/3) itaerations with the increased size then set the list size its initial value and go to Step 9. If at any point there is an improvement then go to Step (8) Step 8. Job exchange phase of search. Evaluate completely the neighbourhood Nj (fl ~ and select the best exchange of jobs. Denote the new complete sequence by fl~ and its makespan by Flow I. If Flow* < Flow" then set f ~ ' - f ~ , Flow" - Flow I. Set fl ~ - fl I, Flow ~ = Flow i ang go to step 7.
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If the incumbent solution was changed during job phase, then go to Step 3. If the required number of LTM restarts was performed then stop, Else go to Step 2.
3. COMPARISON OF THE SCHDULING HEURISTICS For Carrying out the comparison of the GS heuristics, GS problems of various sizes are randomly generated. Data configuration is similar to that used in [13,2]. 30 problems for each of 8 problem sizes were generated. Written as (F x M x ni), problem sizes are (3x3x3), (3x4xS), (4x4x4), (6x5x4), (5x5x5), (6x6x6), (5x6xS), and (Sx8xS). Jobs processing times are integer random variables uniformly distributed in U(1,10). To generate the zero times a uniform random number is sampled, if it is less than or equal to 0.2 a zero time is used. Hence 20% of job processing times are set to zero. This percentage is used in [ 14,15,16]. The family setup times are integer random variables uniformly distributed in the three sets U(1,20), U(1,50) and U(1,100) so that to study the impact of the different values for the family setup time to job processing time ratios (S/R)of approximately 2, 5 and 10 respectively. Scheduling objectives are minimizing total flow time (sum of completion times of jobs [14,15]) and minimizing makespan separately. Makespan is commonly used however, Rajendran and Sridhar [ 14,15,16] notified that total flow time is more relevant criterion in cells. Better reduction in total scheduling costs is achieved by minimizing total flow time than minimizing makespan [ 16,17]. A measure of performance is established as follows. The total flow time and makespan obtained by the original Hitomi's heuristic are standardized to be 100. Then the average total flow time or the average makespan for each heuristic is compared with respect to that of Hitomi. For instance, let Faitmoi and Fx represent the average total flow times obtained by Hitomi and the X heuristic, respectively, then the relative total flow time for X is denoted by RELFx (similary relative makespan is RELMx ) and is given by: RELFx- (Fx / Fsitomi ) x 100. A value below 100 will indicate that X outperforms Hitomi and is preferred to it. And generally lower values are for better performance. Heuristics are coded in Quick Basic 4.5 and expermental calculations are performed on a Pentium 100 MHz PC. 4. RESULTS The complete set of results of the comparison is found in [18]. Results showed that the iterative improvement methods are superior to the simple methods for all conditions. The iterative methods investigate a much larger number of solutions and are able to account for the phases' interaction. The simulated annealing heuristics. Figs.1 and 2 show the performance of SA-M-2 at the different values of GP, at S/R = 5 for total flow time and makespan respectively. It is observed there are no significant differences for the different GP values. However GP of 0.5 and 0.7 gives relatively better results in most cases. GP of 0.9 is then preferred to 0.3. This means that giving the majority of the search efforts to the family phase is preferable. This is true for all S/R and is true for the other SA versions at less differences for SA-M-1 and SAM-3. This is also true for minimizing makespan at less important differences than for total flow time in general. In all, GP of 0.1 used in [2] gives the least performance.
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Figures.3 and 4 compare the four versions of SA for total flow time and makespan respectively at GP = 0.7 for all S/R. It can be observed that SA-M-1 and SA-M-3 are better than the original SA and SA-M-2 for the two objectives. This shows that the changedependent acceptance probability is more efficient than the change-independent probability used in [2]. It is found also that SA-M-2 is slightly better than the original SA for about 58% of cases (for all GP)[18]. Similarly SA-M-3 is better than SA-M-1 for about 60% of cases
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[18]. That is some control on the effect of the random numbers behaviour in SA is recommended. The proposed SA-M-3 is thus the preferred SA version. Nevertheless, tile performance of the SA versions is generally tending to be inferior at the larger problem sizes. This is clearer for total flow time and at lower S/R values. The tabu search heuristics. No much difference is observed among the TS versions. Still, the proposed TS-M-1 is generally better than the original TS and TS-M-2 for about 66% of cases. At S/R of 10 TS-M-1 is better all the time. This can be observed in Figs. 5 and 6. The TS methods are relatively more robust than SA when increasing problem sizes.
By counting the number of times a TS version is better than the others it was found that [18], the original TS ranks second after TS-M-1, and TS-M-2 is the last. Using a relatively good initial solution, TS-M-2 could outperform the original TS. Still, TS-M-l is better than both of them for most of time [ 18]. Hence, it is the complete LTM; in the two phases in TS-M-2; is the reason that it came forward to the original TS when using a relatively good initial solution. Nevertheless, TS-M-l is always better than TS and TS-M-2, while using a partial LTM in the family phase and making use of the search efforts in the job phase by simply keeping the current jobs sequences during restarts. It thus possible to conclude that LTM is needed in both scheduling phases, however, including frequency information only in LTM as in [11 ] is not
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sufficiently effective. Other search-based information may be concerning phases' interaction, should be included in the LTM. Thus the SA-M-3 for GP of 0.7 (SA-M-3.7) and TS-M-1 are identified to be preferable to the other heuristic versions for the current study. Figs.7 and 8 presents a comparison between the two methods. A better level of improvement over the reference value is observed for total flow time than for makespan for both methods. No significant difference is observed between the two methods. However it can be seen that SA-M-3.7 is slightly better for total flow time. For makespan, the two methods are approximately equivalent. Nevertheless, TS-M-1 is generally better than SA-M-3.7 at the larger problem sizes for the two objectives. This is true for all S/R values [18].
TS-M-1 seems to be more robust to increasing problem sizes. It offers the ability to redefine its components (LTM, tabu lists and the aspiration level function) so as to improve its performance incorporating search-based information, which is not available for SA-M-3. Consequently as TS-M-I is more robust and flexible, it is generally preferred to the SA-M-3.7 in this study. SA-M-3.7 may be used for scheduling with respect to total flow time at the relatively smaller problems. 5. CONCLUSIONS GS model was investigated while studying the relative performance of selected simple and iterative improvement GS heuristics in a static flow line cell. Besides, a time-tabling procedure that can account for the presence of zero processing times in a multi-family cell, is proposed. The major conclusions of the study are: 9 The two-phase nature of GS should be considered in the heuristic methods in order to compensate for the possible phase's interaction. 9 The iterative improvement techniques were found preferable to the simple methods, not only because of their superior performance but because they can consider the phases' interaction in GS as well. 9 The proposed modifications to the iterative GS heuristics studied provided slight, yet observable improvements over the original formulations of SA, and TS. 9 SA-M-3.7 and TS-M-1 are relatively better than the other methods under study. TS is found to be more robust than SA while offering the possibility to redefine its components to include more relevant search-based information thus to increase its efficiency. ~ The change-dependent acceptance probability in the SA is more efficient than the changeindependent acceptance probability. The change-dependent acceptance probability can avoid solutions that results in drastic changes in the objective function value thus to force
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the procedure to converge, while avoiding local optimality. The probability of accepting such solutions (non-improving) would be very low due to the large value of the change in the objective function value. 9 The possibility of the zero processing times in cells should be in consideration during time tabling calculations. Otherwise, erroneous time tabling information and overestimation of makespan and total flow time would be obtained. The proposed time tabling procedure for the multi-family cell was found effective in accounting for the zero times and removing their effects. REFERENCES 1. Hitomi, Ham, Yoshida, "Group Technology", Kluwer-Nijhoff, Hingham, MA, (1985). 2. Vakharia A. J., Chang Y., "A Simulated Annealing Approach to Scheduling a Manufacturing Cell", Naval Research Logistics, Vol. 37, pp. 559-577, (1990). 3. Allison J. D., "Combining Petrov's Heuristic and The CDS Heuristic in a Group Scheduling Problems", Computers and Industrial Engineering, Vol. 23, pp. 457461,(1990). 4. Grasso, Masanta, Piacentini, "Heuristic Procedures for Group Scheduling"; 4~ Cairo Univ. MDP Conf., Dec. pp. 629-636, (1988). 5. Mahmoodi, Dooley K. J., "A Comparison of Exhaustive and Non-exhaustive Group Scheduling Heuristics in a Manufacturing Cell", Int. J. of Prod. Res., Vol. 29, No. 9, (1991). 6. Ruben R. A., Mosier C. T., Mahmoodi F., "A Comprehensive Analysis of Group Scheduling Heuristics in a Job Shop Cell", Int. J. of Prod. Res., Vol. 31, No. 6, (1993). 7. Wemmerlov U., Vakharia A. J., "Job and Family Scheduling of a Flow-Line Manufacturing Cell: A Simulation Study", IIE Transactions, Vol. 23, No. 4, (1991). 8. Hitomi K., "Group Scheduling with Interactive Computer Graphics", Memories of the Faculty of Eng., Kyoto Univ., Vol. 50, No. 3, (1988). 9. Kirkpatrick S., Gelatt C. D., Vecchi M. P., "Optimization by Simulated Annealing", Science, Vol. 220, No. 4598, 13 May (1983). 10. Schuur P. C., "Classification of Acceptance Criteria for the Simulated Annealing Algorithm", Mathematics of Operations Research, Vol. 22, No. 2, May (1997). I I. Glover F., "Tabu Search: Part I", ORSA J. of Computing, Vol. 1, No. 3, pp. 190-206, (1989). 12. Glover F., "Tabu Search: Part II", ORSA J. of Computing, Vol. 2, No. 1, pp 4-32, (1990). 13. Kapov, Vakharia A. J., "Scheduling a Flow-Line Manufacturing Cell'a Tabu Search Approach", Int. J. of Prod. Res., Vol. 31, No. 7, (1993). 14. Sridhar J., Rajendran C., "Scheduling in a Cellular Manufacturing System: a Simulated Annealing Approach", Int. J. of Prod. Res., Vol. 31, No. 3, (1993). 15. Sridhar J., Rajendran C., "A Genetic Algorithm for Family and Job Scheduling in a Flowline-Based Manufacturing Cell", Computers and Industrial Engineering, Vol. 27, No. I-4, pp. 469-472, (1994). 16. Rajendran C., "A Heuristic for Scheduling in Flow Shop and Flowline-Based Manufacturing Cell with Multi Criteria", Int. J. of Prod. Res., Vol. 32, No. 11, (1994). 17. Baker, K. R., "Introduction to Sequencing and Scheduling", John Wiely and Sons, New York, (1974). 18. Helal, M., "Investigating Group Scheduling in a Flow Line Based Manufacturing Cell", Unpublished Master thesis, Benha High Inst. of Technology, (1999).
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SCHEDULING APPROACH FOR MIXED NETWORKED BATCH PROCESSES Soltan, H.A.M.
Prod. and Mech. Design Eng. Dept., Faculty of Engineering, Mansoura University, Mansoura 35516, Egypt
ABSTRACT The nature of flow in chemical industries stimulates the authors to develop special scheduling techniques for batch processing. In the networked processes, one or more products are completed through multistage-multiunit facility in forward paths. The problem may be more complicated if a facility assigned more than one product simultaneously which could be named mixed networked production. This paper formulates the mixed batch processing problem as a Mixed Integer Nonlinear Programming model in its Lagrangean relaxation. It is mainly based on the idea of balancing the facility at the bottleneck capacity without finite intermediate storage. The considered performance criteria are tardiness and earliness penalty costs and setup costs. Due to complexity, it is proposed to heuristically obtain a near optimal rate and sequence of processing batches along a periodical time horizon. Aided by computer experiments, an iterative heuristic procedure is developed to solve the model. It periodically builds up an ordered list for batches fathoming the most earlier batches until one or more batches bubble up. The approach demonstrates a satisfactory schedule for a case. More, it could be extended to other applications. KEYWORDS Multi-Product/Stage; Programming
Due Dates; Batches; Scheduling; Lagrangean relaxation; Nonlinear
I. BACKGROUND AND PROBLEM DEFINITION
Batch scheduling resolves the processing order of batches from one or more products using one or more units of a plant. Intuitively, the problem is characterized by the nature of the process. Several of the general scheduling methods can be used for solving such problems in chemical industries. But in chemical industry, the processes have some peculiarities (Fortemps et al.[ 1]), specially the material transfer and storage, which stimulate the authors to develop special methods. Kim et al. [3] mentioned major types of chemical batch plants, multi-product flowshop and multi-purpose jobshop. Studies adopted different performance criteria and intermediate storage policies. Musier and Evans [4] reported from their survey that most of industries adopted due date customer satisfaction. The storage policy is a reflect of capacity constraints which affect the parameter setting (Kim et al. [3], Rodrigues et al. [5], and Graells et al. [2]). Networked flowshop batch process is performed through a facility exploiting the physical properties of serial flowshop. The facility consists of stages and machines (units). The
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multiple units at each stage are identical in operation and may be different in capacity. So, intermediate storage containers are installed after each stage to accommodate the difference in production rates between stages. Moreover, delivery containers are installed after the last stage. The machines and containers may be connected or not, but in the most of cases they are found connected especially for fluids. On the other hand, the direct connection between machines may exist. R is presumable to assign more than one path for the same product batch and those paths may be changed for another batch of such product according to the schedule set. Actually, the networked process is an enlargement to the simple serial multi-product batch process. Thus making it possible to get some benefits of multi-purpose jobshop batch process (Kim et al. [3]). However, contributions were focused on multi-product and multi-purpose batch scheduling (Kirn et al. [3]). Any way, a robust relation exists between the three types of batch processing. In general, different algorithms are used to solve chemical scheduling problems such as Mixed Integer Linear/Nonlinear Programming (MILP/M1NLP), Simulated Annealing (SA), Tabu Search (TS) and Genetic Algorithm (GA). The latter three algorithms belong to a class called metaheuristics. For an overview, refer to Rodrigues et al. [5], Musier and Evans [4], and Kim et al. [3]. More, Tandon et al. [9] applied SA with due date criterion to schedule multiple products on a network of a single stage of unrelated parallel units. They characterized the problem explicitly and classified such networks according to processing time dependency. Graells et al. [2] presented a SA technique in two phases for multi-purpose scheduling based on due date satisfaction. Gantt-Chart is incorporated as a monitor for allowed shifts to enhance the initial schedule. In terms of MILP, Rodrigues et al. [5] developed a multi-purpose technique with a formulation based on cruciality weighting and no finite intermediate storage policy. This weighting parameter is used to reflect the relative batch importance. Kim et al. [3] analyzed the networked flowshop batch processes and applied a modified GA aided by Gantt-Chart to set an initial optimal schedule which minimizes due date earliness and tardiness penalties. Fortemps et al. [1 ] proposed a heuristic procedure involved in two stages which was applied to schedule a real life piping networked facility. The first assigns a fixed order to the jobs while the second applies SAandTS to optimize the order. 2. APPROACH FOR MIXED NETWORKED BATCH PROCESSES
Scheduling of networked flowshop batch processes is adopted in a new algorithm allows more than one product to join the production facility concurrently. The purpose is to sequence several batches in the queue and set the facility at a satisfying production rate for each batch. The schedule appears in a finite time horizon consists of a number of shared equal or unequal periods. The algorithm presented in this paper is characterized by the following: (1)all machines at each stage are identical and unrelated; (2) production rates may be stage dependent but not product; (3) setup and production costs are product and stage dependent; (4) switching may be allowed if a new batch is placed but under nonpreemptive processing (it is appropriate for the chemical process industry to continue with the current joined batch); (5) the problem is static deterministic; (6) setup costs and holding costs may be set to zero if they are not competitive with the other criteria; (7) earliness penalties may be explained as holding costs or early receiving penalties imposed by the customer; and (8) the products transfer continuously between stages. Moreover, to write the model, several nomenclatures are stated at the end of
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the paper supposing that i, j, k, and p be product index, machine index, stage index, and period index respectively.
2.1 Problem Formulation The problem is described by a MINLP model. The objective cost function (1)is mainly constructed with two terms. First includes tardiness and earliness penalty costs while the second includes production and setup costs of facility units for products. Although, in this formulation, production costs is not a decision parameter, it is included to complete the total cost evaluation.
Min Z(Q,,t,)=Z{[Y,L,a, Di+(Y,-I)(L,)flIDj]+h, ~}+ Z X ~=1
Xep{caDi+QJSa}
(l)
per ~p k-i Rk
Where y~ is an integer variable indicates if a batch is early or late. Let X~pbe an integer variable indicates the product and period relationship. Substitute by the value of Le and rearrange to
+2;X
(2)
pET i~p
To meet the due date as possible, it is intuitive that the processing start time ti should be a direct proportional function of both El and Q~. Therefore, it could be heuristically set as co~pQ~E, where COepis proposed as a proportion parameter (constant for each period)and named the batch priority multiplier. The optimistic value of this multiplier, at the start of a period p, occurs when Le=0 and Qj =G while (Ertr) replaces Ej in the numerator or (.O,p = [(E~ - tp )G - D, ]/E,G 2
(3)
This parameter may be slightly altered but, here, it is experimented numerically and found parallel to what is called Lagrangean multiplier. The numerator of (3) explains by how much the batch is currently in lead (positive) or lag (negative). In other words, it indicates the current penalty conditions. Substitute by a~ep in (2), rearrange the function, and set the constraints according to the problem characteristics. Hence, the MINLP is constructed as
Min Z(Q~,coe)= Z
Zx, e/~ {((~
p~T t~p
M
+ Z X X ;x"
E~)[Yi(ae+~)-fl~]+O, Lv,Ca, +
+e, s,,}
Q,
+
h,} (4)
peT iEp k=l a X k
subject to
iEp
Y"%Q~ <_G
(5)
Q, > 0
(6)
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% =
10if product i is scheduled in period p otherwise
(7) (s)
~ x ep = 1V i __ pet N
~" ~" xt. = N
(9)
peT i=l
xi. < min.(mk )
(1 O)
iep
Y~ =
10if (t e - Ej + D i / Qt) > 0 if (t~ - E~ + D~ / Q~) < 0
(ll)
~1 V k if a machine j assigned a product i
(12)
x#k(P) = [0 otherwise
(13)
~"~x#k(p)=lVj, k tep
t~ = tp
ifxr =1
and
tp =tp_ I
(14)
+ max.(~), i e p - I
Let ~, be a vector of Lagrangean multipliers for the capacity constraints (5) (see, e.g., Taha [8], Thizy and Wassenhove [10]). The model Lagrangean relaxation, subject to (6)-(14), is given by Min Z~.,~p,~)= Z
Zx.cDe{(~E,~.
p~T iep
+
Z 5: :--"
p e t i~p k=l "~'k
- E~)[ye(ae+ ~)-~]-o
p~T
qfy,(~ +fl,)-fl,]+~} Q.
iep
Different methods solve the Lagrangean relaxation in similar problems (Thizy and Wassenhove [10], Sox and Muckstadt [7], Soltan [6]). Because of the complexity, the solution will be heuristically proceeded. The problem is partitioned periodically to determine the products that could share the facility in each period. Each point of solution is similar to the multi-item inventory problem. Since, for each period, Z is a convex function of Q~, the optimal facility sharing occurs where 0Z / OQ~ = 0 or
Current Advances In Mechanical Design and Production, MDP-7
D~[y,(a, + fl~) - fli ] + D,h, M S,~ co,E,D,[y,(ot, + f l , ) - fl,] + 3.p + ~ Rk
Q, =I"
607
(16)
k=!
For late products, y~'s= 1, Qj will be as (17) while for early products, yt' s=O, Qi will be as (18).
Q: = I
D~ot, + D,h,u S__L
(17)
k=!
(18)
Q7 = I Ap -W,pE~D~fl, + ~M s~__ RkL k=l
Since it is not known in advance if a batch will be finished late or early, we subject both cases to the experiment. The two equations are used for finding the production rates under facility sharing principle. The batches sequence appears after periodically checking all the model constraints. The next procedure uses the quantities of eqs. (17) and (18) as principals for differentiation. 2.2 Solution Procedure and a Case The computer experimentation is conducted to reach a heuristic solution procedure based on a ranking mechanism. The following steps are proposed.
Step 0:Start a new period and calculate its new parameters, tp and co~p. Step l:Test the sign of ~ , using a suitable increment, to find for Q's the range which is real valued and the range which is imaginary valued; eqs. (17) and (18).
Step 2:Rank the batches according to cojv;the minimum is ranked I and so on. Step 3:Rank the batches according to the precedence of real appearance of Q's; for first real Q+'s, rank I while for first real Q"s, rank maximum.
Step 4:Rank the batches according to the values of Q§
rank I for maximum value.
Step 5:Rearrange the current list in ascending order according to the total rank of each batch. Step 6:Check if the number of current batches is less than or equal to the bottleneck number of machines (here two machines in stage two). If not go to next; otherwise go to step 8.
Step 7:Exclude the batch in the bottom of the list. This batch is still earlier and not competitive for the current assignment. Go to step 2.
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Step 8:For each batch, calculate the number of machines required at each stage. If any stage
needs a number more than the currently available, exclude the batch in the bottom of the list and go to step 2; otherwise go to next. Step 9:Settle the sharing production rates of the current batches using eq. (17) and go to step 0. The steps from 0 to 9 are repeated until the queue becomes zero. Ranking process needs to some sense taking in considerations the significance difference in values and the precedence of real appearance. In other words, the step of ranking may be variable. Also, the sharing between batches may be optional to accommodate a current policy of management. An academic instance is used to well display the steps of the solution procedure. The facility consists of three stages to produce different five batches as depicted in Table 1. The batch sizes, due dates, production rates, and costs are declared in units, days, units per day, and dollars. The start is assumed at time zero with empty facility. Note that the holding cost is proportional to the batch size. Table 1. Multi-Product multi-unit processing information.
B~ch A B C D E R,
1
.
De 16000 8500 5000 20000 4000 .
.
.
.
he 64 34 20 80 12 .
.
.
ae 0.015 0.015 0.020 ~010 0.020 .
~e , 0.010 0.010 0.015 0.005 0.020 .
.
Ee 6 5 10 '15 10
~t Sa 100 120 75 50 200 80 125 125 150 100 1500. 2500
Sn 110 I00 180 125 100 1000
De/G 4~000 2.125 1.250 5.000 1.000 13.375"'
** the minimum time required to finish the five batches.
* the bottleneck stage.
The problem is solved in four periods. First period, 0.000-4.000, is settled in five iterations and assigned batch A with rate 4000. Second period, 4.000-6.125, is settled in four iterations and assigned batch B with rate 4000. Third period, 6.125-10.264, is settled in two iterations and assigned batches C and E with rates 1208 and 1112 respectively. Finally, fourth period, 10.264-15.264, is assigned bath D with rate 4000. Note that batch D is delayed 0.264 days. This difference can be reduced by expediting period three to consume the facility bottleneck capacity as Qc=-4000[1208/(1208+1112)]=2083 and QE=4000[I 112/(1208+1112)]=1917. But it is impossible to do that because stage three has only four machines at rate 1000 each, therefore Qc may be set at 2000. This modification makes period three ends at 8.625 days and period four ends at 13.625 days as shown in Fig. 1. Moreover, the difference could be set at zero by consuming capacity less than 4000; this depends on the schedule conditions especially when another batch is placed. These results are just the final summary of a constructed PC program output. 0
I
4.000 A
._
I
6.125
....
8.625
13.625 days
c
E
D 8.211
Fig. 1. Time chart for the modified schedule.
.....
I
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It is obvious that the procedure postponed batch D until reaches period four and assigned batches C and E to share the third period. Hence, the competition arises between batch A and B which have earlier due dates and set in the schedule top. This proves that the procedure is able to accommodate the due dates as possible and evaluate the production rates when more than one batch share the facility. The idle capacity shown in Fig. 1 (hatched) can be managed in one of three ways. To finish batch D before 13.625 days, assign this capacity to batch D or batch C or delay the start of batch E. However, using those alternatives depends on the change of the cost function. Many more enhancements could be submitted by scanning the program output. 3. CONCLUSIONS The common types of batch processes used in the chemical industry are multi-product flowshop batch processes and multi-purpose jobshop batch processes. Networked flowshop batch processes are the frequently used class. In general, the problem of scheduling multiproduct on multi-stage multi-unit facility don't receive a considerable attention in the literature. The common industrial scheduling methods can be used to solve such problems with less efficiency. In this paper, a new approach is developed for networked flowshop batch processes with mixed nature accommodates the nature of chemical flow. It is implemented on a PC to solve the problem as a sequence of multi-item inventory sub-problems. In spite of the simplification assumptions, the approach has an attractive property that it is capable of determining the sequence of processing batches and their production rates using the same procedure. Moreover, it introduces different alternatives for sharing the facility and it can meet additional requirements such as cleaning and setup times. Those requirements don't affect the main formulation and could be inserted after obtaining the initial schedule. Mini modifications may be imposed under recommendation of the practitioners. The method is applicable in a straightforward manner to the chemical industries in which the unit transfer between stages has a continuos nature with small times. An example is the pharmaceutical industry. It may be extended to cover the cases which need to significant stay times in stages such as refineries; but this subject the formulation to major modifications. Moreover, it can be extended to other industries paying attention to different intermediate storage policies. Finally, the approach is considered a competitive to the metaheuristics because it is not highly affected by the change of the problem parameters under the same conditions. Also, the solution may be enhanced by conducting an algorithm such as GA to the existing formulation. ACKNOWLEDGEMENT I warmly thank Professor Salah E. Elmaghraby, NC State University, USA, who stimulates me to touch such type of problems. REFERENCES 1.
2.
Fortemps, P.; Ost, C.; Pirlot, M.; Teghem, J. and Tuyttens, D., "Using Metaheuristics for Solving a Production Scheduling Problem in a Chemical Firm. A Case Study," Int. J. Prod. Econ., Vol. 46-47, pp. 13-26, (1996). Graells, M.; Espul~a, A. and Puigjaner, L., "Sequencing Intermediate Products: A Practical Solution for Multipurpose Production Scheduling," Comp. Chem. Engng., Vol. 20, pp. 1137-1142, (1996).
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3. 4.
5.
6. 7. 8. 9.
10.
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Kim, M.; Jung, J.H. and Lee, I., "Intelligent Scheduling and Monitoring for Multi-Product Networked Batch Processes," Comp. Chem. Engng., Vol. 20, pp. 1149-1154, (1996). Musier, R.F.H. and Evans, L.B., "An Approximation Method for the Production Scheduling of Industrial Batch Processes with Parallel Units," Comp. Chem. Engng., Vol. 13, pp. 229-238, (1989). Rodrigues, M.T.M.; Gimeno, L.; Passos, C.A.S. and Campos, M.D., "Reactive Scheduling Approach for Multipurpose Chemical Batch Plants," Comp. Chem. Engng., Vol. 20, pp. 1215-1220, (1996). Soltan, H.A., "Experimental Analysis of Multi-Item Inventory Systems," Mansoura Engng. J., Vol. 23, pp. 1-7, (1998). Sox, C.R. and Muckstadt, J.A., "Multi-Item, Multi-Period Production Planning with Uncertain Demand," IIE Trans., Vol. 28, pp. 891-900, (1996). Taha, H.A., "Operations Research: An Introduction," Macmillan Co., New York, (1992). Tandon, M.; Cummings, P.T. and LeVan, M.D., "Scheduling of Multiple Products on Parallel Units with Tardiness Penalties Using Simulated Annealing," Comp. Chem. Engng., Vol. 19, pp. 1069-1076, (1995). Thizy, J.M. and Wassenhove, L.N.V., "Lagrangean Relaxation for the Multi-Item Capacitated Lot-Sizing Problem: A heuristic Implementation," IIE Trans., Vol. 17, pp. 308-313, (1985).
NOMENCLATURES ai Ck cik D~ Dt/G
Ei G hi Le M mk
N Rk
Q, Qi §
Q; s,k T ti
tp cor
9tardiness penalty cost imposed on batch i unit per unit time; : earliness penalty cost imposed on batch i unit per unit time; : production rate of stage k, (mt, Rt,); : unit time cost of operating a machine at stage k to produce batch i; 9demand (batch size) of batch i; 9processing time of batch i if it join the facility alone; : due date of batch i; : bottleneck production rate of the system, ( min.(Ck )); 9cost of holding or maintaining a tank for batch i per unit time; : a vector of unrestricted Lagrangean multipliers; : tardiness of batch i, (t i + D~ /Qt - E~). It refers to earliness if it is negative; : number of (operations) stages; : number of machines in stage k, : number of batches in the queue; 9 production rate of a machine at stage k, 9planned delivery rate of batch i; : production rate of tardy batches i's; : production rate of early batches i's; : setup cost of a machine at stage k to produce batch i; : set of equal or unequal production periods; 9time at which batch i joins the system; 9start time of period p; 9priority multiplier of batch i at the start of period p.
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
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A FUZZY REACTIVE APPROACH FOR THE CREW ROSTERING PROBLEM El Moudani, W . * + , Brochado, M.R. # , Handou, M. 4 and Mora-Camino, F. * +
* LAAS du CNRS, 7 Av. du Colonel Roche, 31077 Toulouse, France. + Air Transportation Dept., ENAC/DGAC, 7 Av. Edouard Belin, 31055 Toulouse, France. Centro Federal de Educa~;ao Tecnica, CEFETfRJ, Av.Maracana 229, Rio de Janeiro, Brazil. 4Division Op6rations A6riennes, EAMAC/ASECNA BP 745, Niamey, Niger.
ABSTRACT In this paper the airline Crew Rostering Problem is considered from a dynamic point of view. The study tackles disturbances which lead to modifications for the monthly personalized assigmnent of the crew staff to flights. An adaptive solution approach is proposed through the asynchronous resolution of an updated optimization problem whose objective is to minimize changes from the previous crew staff assignment while coveting the flights of a sliding time period. A mathematical formulation, including penalties which are applied to the assignment changes, is developed. Since the discomfort caused by these changes is often difficult to quantify, a fuzzy logic approach is developed to realize this evaluation. Then, a numerical solution approach based on a forward dynamic programming technique is described. The proposed solution strategy is applied to a medium size problem. KEYWORDS Combinatorial Optimization, Crew Scheduling, Fuzzy Logic, Air Transportation Operations. 1. INTRODUCTION The airlines crew scheduling is a problem that has retained the attention of the Management and Operations Research community for more than three decades now. Crew costs in air transport are extremely high, amounting 15-20% of total airlines operations costs. Therefore, airlines consider that the efficient management of their crew staff is a question of the highest economic relevance. Unfortunately, the numerical solutions of the associated large scale and combinatorial optimization problems, are very difficult to obtain. Early rules of thumb have been quickly overrun by the size of the practical problems encountered (hundreds or thousands of crew members to be assigned to at least as many flights) and by the complexity of the set of constraints to be satisfied, leading to poor performance solutions. Then, following the development of Computer Science, many optimization approaches have been proposed to solve this problem : first mathematical programming methods (large scale linear programming and integer programming techniques) and more recently artificial intelligence methods (logical programming, simulated annealing and genetic algorithms) as well as heuristic approaches and their respective combinations [ 1-9].
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It appears that the Crew Scheduling Problem (CSP) has always been treated as a static decision problem, in general on a monthly basis. In this case, the input data is constituted of the monthly table of flights, then, using one of the different available solution approaches, an assignment is provided for the month. The disturbances present in daily operations imply frequent changes in the monthly personalized planning. In general these modifications are performed manually using practical rules. This corrective approach is sub optimal and feasible solutions can be difficult to obtain in this way for medium or large problems. So, in this paper a new systematic approach, based on the repetitive resolution of the assignment problem is proposed. Since equity is supposed to be satisfactorily treated with the first assignment solution computed for a nominal period, a possible goal for the corrective assignment can be to minimize assignment modifications while covering all the remaining rotations. These modifications have not the same impact on the quality of the solution, so a penalization of the criterion appears necessary. In this communication, it is shown how this task can be achieved using fuzzy logic techniques. This paper is organized as follows : first the Dynamic Crew Scheduling Problem (DCSP) is described, then the strategy solution approach is presented, mathematical formulations for the nominal and corrective Crew Rostering Problems in a dynamic context are developed, a Dynamic Programming solution method is described and a fuzzy logic approach is developed to provide relevant penalties for the additive elements of the criterion function, finally a numerical application of the proposed solution strategy to a medium scale Crew Rostering Problem (CRP) is displayed and discussed. 2. DESCRIPTION OF THE DYNAMIC CREW SCHEDULING PROBLEM The CSP is a classical problem that consists of an airline which assigns the crew staff to the programmed flights. In general, the CSP is considered once the schedule of the flights has been established for the next month and once the available fleet has been assigned to the scheduled flights. National regulations, internal company rules and agreements with unions regarding the crew's working conditions and remuneration provide elements for the formulation of a complex set of operational constraints which take into account aspects such as :crew qualifications, medical examination, training requirements, office duties, holidays and other operational or personal constraints. While some of these conditions are common to most airlines, others are of interest for some classes of airlines and some few are specific to each airline. Until today, a sub optimal, but widely accepted approach, is to decompose the CSP, which is of the NP-hard computational complexity class, in two sub problems of lower difficulty. The first problem, called the Crew Pairing Problem (CPP), involves the construction of rotations composed of a sequence of flights that start and return to the same airline base, and with rotations which may last one or a few days. These flights which compose a rotation, are separated by rest periods in accordance with legal working rules. The solution oft he CPP provides a set of rotations which are the demand side of the CRP which is in charge of assigning nominal crews to rotations in order to build the "line of work" for the rostering period. In this paper the rostering period is the month as it is adopted by a majority of airlines. Many airlines continue to produce crew schedules using heuristic computer methods, among which Rubin's algorithm has probably been the most widely accepted. However, a number of airlines have developed or are developing •217 techniques to solve the CSP.
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Bianco, et al. [2] describe an heuristic procedure to solve the CRP. Column generation techniques have been used to solve set partitioning problems associated with the CSP [6,7,8]. In the literature, the research tackling with the Crew Rostering Problem has considered it as a static assignment problem whose input data is exactly known beforehand and is not supposed to suffer any modification. However, a large range of perturbations can turn the resulting optimized solution unfeasible : canceled or delayed flights, aircraft failure, strikes, unexpected unavailability of crew members, etc. Since in this approach, the monthly optimized assignment is communicated in advance to the crew members, about ten days in general, they organize, in accordance, their private life and their other working activities. So, when a crude restart of the optimization process over the remaining part of the planning rostering period occurs, uncomfortable and awkward situations can be generated for the crew staff. The frequent occurrence of such events will lead inevitably to a reduction of the levels of service and security as well as to a degradation of the relations between the crew staff and the management of the airline. So, it appears of first importance to cope with the CRP in a dynamic framework. 3. A SOLUTION STRATEGY FOR THE DYNAMIC CREW ROSTERING PROBLEM In this study, the solution approach to the Crew Rostering Problem is separated in two phases. In the first one, considering the normal planning period, the crew assignment is performed on a nominal basis using a dynamic programming technique. In the second phase, the dynamic dimension of the CRP is considered. This leads to introduce a new optimization criterion to penalize the possible changes on the previously computed monthly personalized assignments when a significant disturbance occurs and when no local standard corrective procedure can be done at hand. Then the optimization process must be restarted on these new grounds. Since, the nominal monthly personalized planning is communicated with some antecedence (one or two weeks before the beginning of the next rostering period) to the crew staff, it is proposed here to distinguish two cases to determine the range of the optimization period" when the operation is subject to a disturbance before the issue of the new nominal assignment for the next rostering period, the optimization horizon of the corrective problem goes to the end of the current rostering period, in the other case, the optimization horizon goes to the end of the next rostering period (Fig. 1). In doing this, the late corrective actions on the crew assignment will not introduce cost penalizing constraints for the next rostering period (RP). Perturbation before establishing the new schedule
l
Perturbation after establishing the new schedule
Establishingnew schedule
'
,,
]
'
to+d..~ to~RP I
Rostering Period (l) (Re)
1
II
. . . . . . .
to+RP+dm,x to~2*RP
. . . . . . . . . .
I
Rostering Period (2) (Re)
Fig. 1 : Definition of the optimization horizon.
Then, the optimization horizon H t = [t, Tmtax] can be chosen as follows:
II
illl
iin
:
I
t~ I
,
Rostering Period (3) (Re)
,
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Current Advances in Mechanical Design and Production, MDP- 7 t = t o + K x R P + A t with 0 ~ ; A t < R P , K r n
Ttax = t o + ( k + l ) x R P / f A t ~;dmx and T~ax = t o +(k+2) x R P ifAt>-dmx where At is the time elapsed since the beginning of the month. Depending on the nature of the disturbance, before to seek corrections for the CRP solution, the Crew Pairing Problem should be reconsidered to provide a new efficient and feasible set of rotations. For example, if a flight is canceled, the corresponding rotation is either broken or reduced. So, it is necessary to re-optimize the set of selected rotations to avoid if possible operations like dry flight or deadheading. Once the new set of rotations is established, the nominal reassignment of the crew staff can be performed efficiently. 4. MATHEMATICAL FORMULATION OF THE DYNAMIC CREW ROSTERING PROBLEM In the specialized literature, the Crew Rostering Problem is most often treated as a zero-one integer mathematical programming problem whose formulation is such as: m
Min ~ ~ ccxiy with the following set of constraints: jffilieAj
E'u = ~ ieAj
v1 e J
(1)
iffil m
Y~dsx u
vi e I
(3)
Vi EI
(4)
j=l m
Z t j x i j ~LT y=l
where ! is the set of the n pilots, J is the set of the m rotations to be flown in the nominal planning period, Aj is the set of pilots able to fly rotation j, Osl is the set of rotations overlapping with rotation Jl, xt/ is a binary decision variable such that xr = I if pilot i is assigned to rotation j, xr = 0 otherwise. Constraint (1) insures that, each rotation is assigned to a unique crew. Constraint (2) insures that the same crew is not assigned to two overlapping rotations. Constraint (3) insures that the number of hours flown by each pilot over the rostering period does not exceed the common upper limit LH, and constraint (4) insures, that the total number of takeoffs realized by each crew over the rostering period does not exceed the upper limit L T (here ti is the number of segments composing rotation j). The upper limits LH and LT, may vary from one airline to
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another. If necessary, other operational constraints can be added to the discrete formulation of the mathematical programming problem. The objective function of the static CRP considers the individual costs, c/j, of assigning pilots "i" to rotations "j". These costs are established by the airlines considering different factors such as seniority of the crew, availability of the crew, equity with respect to workload assignment and others. As mentioned above, when a disturbance occurs the CRP is reformulated with the definition of a new criterion to optimize over a new period. So, the objective consists now in minimizing the sum of penalties associated to the changes on the previously computed personalized planning over the remaining rostering period and eventually (see section 3) the next rostering period. The following expression has been adopted for the modified criterion in this study : Min
~ ~,po.(xu -x0.) 2 , jeJt i~Pt
~ = It n
Aj
The CRP consists now in minimizing this criterion subject to constraints (1), (2), (3) and (4) where LH and L T are updated, It is the set of pilots and,It is the set of rotations supposed available for the planning period starting at t and xr represents the new binary decision variable associated with the reassignment of pilot i to rotation j. Here also, Dynamic Programming can be used to obtain a current solution for the reassignment problem. The p#. represent weights used for penalizing the assignment changes of the pilots to the rotations, they can be evaluated using a fuzzy logic technique as described in the next section. The combinatorial nature of both problems (the static and the dynamic CRP) appears clearly in the proposed formulation given above. The dynamic programming technique has been adopted in this study to solve both of them since it is recognized to be an efficient technique to treat combinatorial problems in a dynamic environment. This technique is appropriate here since the chronological succession of the rotations provides the problem with a dynamic nature while the separability structure of the constraints insures its direct applicability. The key point to use Dynamic Programming in an efficient way in this particular application seems to be the identification of each rotation, after J has been ordered chronologically, with a stage of the search process to which is attached the set of possible states identified with the set of pilots L The choice is judicious since it reduces drastically the number of states at each stage (its upper limit is then equal to n, the number of crew members) while limiting excessive data storage and processing difficulties. 5. A FUZZY APPROACH FOR THE DEFINITION OF PENALTIES In this section, the process of computation of the weights present in the criterion function used in the dynamic formulation is detailed. Since it is very difficult to evaluate the a priori importance of each possible modification of the rostering plan, the proposed approach takes advantage of the ability of fuzzy logic to manipulate specialized and mainly qualitative knowledge of experts (airlines operations engineers in general) by the construction of a set of fuzzy inference rules. Current experts' opinion is that the actual amount of flown hours (WL~) by a pilot (i) and the duration (DRj) of a rotation (j) are the main factors which should contribute to the definition of the penalty p/j. This leads to the definition of a set of fuzzy inference rules with two inputs, the fuzzy evaluations of WL and DR, and one output, the
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fuzzy evaluation of a weight rl. The following fuzzy sets are defined to qualify adequately the different variables" For WL" Very Light (VL), Light (LT), Moderate (ME), Heavy (HY), Very Heavy (VH). For D R " Very Short (VS), Short (S), Moderate (ME), Long (L), Very Long (VL). For l'I" Very Small (VS), Small (S), Medium (M), High (L), Very High (VL). with their corresponding graphical representation in Fig. 2.a, 2.b and 2.c.
VL
L ME HY
VH
VS
houri 16
32
48
64
80
M L
16
24
VL
.':.ours
0 8
96
2.a. Qualification of WL
VS
S
32
40
48
2.b. Qualification of DR
S
M
L
VL
2.c. Qualification of
Fig. 2.a, 2.b, 2.c. Fuzzy membership functions for WL, DR and FI. The Mamdani's defio2ification process has been adopted here to get easily the weights FI/j over the interval [0, 1]. Then the evaluation of the final penalties p# is done as follows" 1. if xr = 0 and x0. = 1 then pc = FIe 2.
ifxr
vi
Aj,vj
andxr162
Some of the 25 generated elementary fuzzy rules are displayed here" 1. If (WL is VL .or. WL is L .or. WL is ME) and DR is VS then FI is VS 2. If WL is HY and DR is VS then FI is M 3. If WL is VH and DR is VS then l-I is L 4. If (WL is VL .or. WL is L) and DR is S then FI is VS 5. If (WL is ME .or. WL is HY) and DR is S then Fl is M
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The defuzzification process is displayed through an example in the following figure (Fig.3) :
6. CASE STUDY An application of the proposed approach to a medium scale problem has been developed to evaluate its efficiency as well with respect to the quality of the results, as with respect to computational considerations. The application has been developed with Turbo C++ and has been run on a last generation microcomputer. To each crew member has been attached the set of rotations, organized chronologically, that he can technically perform during the roster period, then personal constraints have been introduced to reduce the dimensions of these sets. The application has considered 24 crew members for 100 rotations. The total CPU-time needed to produce a solution for the static as well as the dynamic CRP ranges between 5 and 8 seconds. The resulting workload for the static CRP is displayed in Fig. 4, there, a high degree of equity is observed among the crew staff.
Figure 5 displays the redistribution of the workload through the dynamic CRP after a major disturbance (unavailability of one crew member). It appears that the proposed approach
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provides improved solutions with respect to previous manual solutions while maintaining a good level of equity.
Fig. 5 Redistribution of the workload
7. CONCLUSIONS In this paper, one of the main operations problem facing airlines has been dealt with. The proposed approach, does not produce an exact solution in pure mathematical terms but appears to be quite adapted to the operational context of airlines and provides, through a comprehensive process for the decision-makers, an improved solution. Some characteristic features of the proposed approach deserve to be pointed out" - The approach developed provides systematically and in real time a solution for the DCRP. - The computer time needed to treat either the nominal or the revised CRP remains small. - The quality of the solutions obtained is satisfactory. Finally it is important to note that this solution strategy can be also useful to establish, according to the expected characteristics of the operations disturbances, the safe size of the reserve crew and its localization. REFERENCES 1. Ball, M. and Roberts A.," A Graph Partitionning Approach to Airline Crew Scheduling", Transportation Science, Vol. 19, N ~ 2, pp. 107-126, (1985). Bianco, L. and Bielli M., " A Heuristic Procedure for the Crew Rostering Problem", European Journal of Operational Research, Vol. 58, pp. 272-283, (1992). 3. Desaulniers, G., Desrosiers, J., et al., "Crew pairing at Air France", European Journal of Operational Research, Vol. 97, pp. 245-259, (1997). 4. Desaulniers, G., et al., "Crew Pairing for a Regional Carrier". Les Cahiers du GERAD, G97-33, Ecole des Hautes Etudes Commerciales, Montrral, Canada, H3T 1V6, (1997). 5. EL Moudani, W., Mora-Camino, F., Handou M. and Brochado, M.R., "Airlines Fleet Operations Management 9An Integrated Solution Approach", XV lEE on CAD/CAM, Aguas de Lindoia, SP, Brazil, (1999). 6. Gamache, M., Soumis, F., Marquis, G. and Desrosiers, J., "A Column Generation Approach for Large Scale Aircrew Rostering Problems". Les Cahiers du GERAD, G9420, Ecole des Hautes Etudes Commerciales, Montrral, Canada, H3T 1V6, (1994). 7. Grainic, T.G. and Rousseau, J.M., "The Column Generation Principle and the Airline Crew Scheduling Problem", INFOR 25, pp. 136-151, (1987). 8. Lavoie, S., Minoux, M. and Odier, E., "A New Approach for Crew Pairing Problems by Column Generation with an Application to Air Transportation", European Journal of .
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Operational Research, Vol. 35, pp. 45-58, (1988). 9. Rubin, J., "A Technique for the Solution of Massive Set Coveting Problems, with Application to Airline Crew Scheduling", Transportation Science, Vol. 7, pp. 34-48, (1973). 10. Campello, R.E., Maculan, N., "Algoritmos e heuristicas", Ed. Universidade federal Fluminese, Niter6i, (1994).
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Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
621
EFFECTS OF STATIC AND DYNAMIC MEAN VARIATION ON THE PROCESS CAPABILITY
Mohammed, H.H. Assistant Professor, Production Engineering Department Faculty of Engineering, Helwan University, Helwan, Cairo, Egypt e-mail: [email protected]
ABSTRACT Measures of process capability are used widely to assess a process's ability to meet the quality specifications. The capability is quantified by dimensionless indices that provide common and easy principles for evaluating the performance of a process. In calculating a process's capability index(s), the process must be in control statistically, i.e., the mean and standard deviation parameters are consistent all the time. However, some processes are characterized by having unavoidable and repeated shifts and drifts in their mean values. These are called processes with mean variation. Such mean variation could be static, dynamic, or simultaneous static and dynamic. The reason(s) of that variation may be tool wear, differences in raw material, or change of suppliers. In this research, processes with dynamic mean variation (DMV) and those with simultaneous static and dynamic mean variation are analyzed. Their capability indices are developed and the corresponding production yields are calculated. Other useful practical utilities are investigated. The analyses and the developed capability indices are verified through simulation. An application to the results of this research will be conducted on an assembly process in Helwan Company for Metallic Appliances. KEY WORDS Dynamic Mean Variation, Inflated Standard Deviation, Process Capability, Process Tolerance, Production Yield, Robustness. I. INTRODUCTION In production management, the production processes must be characterized, i.e., their means, variances, capabilities, etc. should be determined and recorded. Such process characteristics are usually determined by monitoring and observing data when the process is in statistical control and having consistent and sustained performance. However, it is rare to have such consistency either because of natural or assignable variability causes. As mentioned by Gunter [1 ], if the data could be collected again under the same process conditions that existed the first-time around, we would get somewhat different data, and thus different means, variances, and capability indices. The phenomenon of unstable process mean parameter will be analyzed in this paper. When a process has its mean parameter changes continuously and randomly for unavoidable reasons; and the expected value of the mean over many production intervals has a strong bias toward
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the target value, such process is considered to have DMV. On the other hand, the process is called to have a static and dynamic mean variation when the expected value of the mean has a deviation from the target value. The study of these processes is important in finding their capability and, accordingly, their production yields and robustness. Analysis of shifts and drifts of a process's meat,, parameter is also essential in tolerance allocation analysis. Woo [9] and Balakrishnan [ 10] introduced different tolerance allocation models that can be applied to processes having repeated shifts in their means. Also, Cheng [11 ] presented a Taguchi loss function-based assembly tolerance model that considers a process with continuous shifts in its mean parameter. Usually, it is difficult to predict quantitatively the continuous changes in a process parameter's mean value, but the knowledge that they will occur helps to cope with the situation. Although the use of this type of processes bothers designers and manufacturing engineers because of their high variability, it may be better economically to utilize them especially when the precision of the characteristic is not sensitive to the product function. The more typical example of the processes with mean variation are those of the plastic industry in which the material composition, the cooling rate, and the product size affect strongly the output dimensions of the molded parts. Also, the machine could be the reason of instability of the mean parameter because of its low precision especially when it approaches, or already has passed, its economic life. 2. THE PROCESS INFLATION BEHAVIOR The phenomenon of dynamic mean variation have been investigated by several researchers. Harry [2] found that, on the long nm, the DMV translates a normally distributed process to another normally distributed one that coincides with the original process in its mean parameter (the location parameter) but has a wider standard deviation. Bender [3] analyzed the effect of DMV on the process's instantaneous capability and concluded that the dynamic process mean tends to mask the instantaneous capability of the process. AI-Sultan [12] investigated the effect of the process with random linear drift on the optimal production cycle and on the total cost per good item. Hussein [4] simulated the DMV behavior and found the values of the inflation factor for different ranges of a mean's dynamic variation. As exhibited in Fig.(l-a,b), the population's mean movement takes place from time to time. The normal distribution holds true for this particular characteristic. Also, the assumption of equal variances must prevail. The overall effect across many such manufacturing cycles is a tail area probability that is greater than any instantaneous estimate. Evan [5] suggested that the inflated standard deviation (OLT)can be found from OLr = COST
(1)
where C is a constant, called the inflation factor, and OST is the short-term standard deviation which is the instantaneous standard deviation of the process. The use of C allows for deriving first-order estimates of long-term capability and, in turn, production yield as will be introduced in the following sections. Coupling these estimates of manufacturing capability, a designer or a manufacturing engineer can assess the robustness of the parameter of interest to random and systematic influences of a dynamic nature. The value of C is also important in calculating the out-of-tolerance percentage. For such importance of the C parameter, a MonteCarlo simulation study (a reinforcing study to that in [4]) is carried out and detailed in Appendix A. The purpose of the simulation study is to find out the values of that parameter
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with sufficient confidence and validation. Table (1) includes the conducted simulation output average values and 99.5% confidence intervals of the C parameter. Table 1: The inflation factor average value with confidence level for different shift ranges
_+015o
Range C Value Confidence Interval
_+ 1.0o 1.15 1.144 1.156
1.11 1.103 1.117
,
......
,,
+ 1.5o' 125 1.241 1.259 .
.
.
.
.
.
.
.
.
+_2.00 1.39 1.3801.400 ,,,
.......
.
:1:2.50
1.51 1.501 1.519
3. ADDRESSING THE PROCESS'S MEAN VARIATION Processes with continuous shifts in their mean parameter have two important and independent key factors to be found. The first is the range in which the mean moves (that is used to find the value of the inflation factor C), and the second is the expected value of the process mean in the long run. When the mean's expected value (~t) has a strong bias to the process target value (T) such that the mean's offset, Z~, is zero then the process has only a dynamic mean variation (Fig. l-a). Otherwise, if Z~t >0, then the process has a simultaneous static and dynamic variation as illustrated in Fig. (l-b). Historical data and their statistical analysis are the means to find these key factors. In collecting historical data, screening suspended values or observations obtained in unusual circumstances, like measurement errors, is a must. Statistical control charts, if used, can be a good tool to find the range of the mean movement by noticing the changes in the center limit and find out its maximum and minimum values. Also, averaging these center limits gives the expected value of the global or long-term process mean; and consequently the process mean offset, Z~t, from its target value. Z~t
T= IJ
z
Fig. l-a. Process with DMV
[4
Z
Fig. l-b. Process with static and dynamic mean variation
4. PROCESSES WITH DYNAMIC MEAN VARIATION To find the capability index of processes with DMV, an equivalent static mean shiR is to be obtained. An equivalent mean shift is nothing more than a compensatory static offset in the mean which directly corresponds to dynamic inflation of the standard deviation such that the probability of nonconformance is the same in both cases. This provides us with comparable
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tail area when considering both types of occurrences. Based upon that notation of equivalent tail area, it would be reasonable to postulate equivalent capability ratio: CPK ---- CpD
(2)
where Cpg is the equivalent capability ratio of a process with static mean offset, and CPD is the dynamic capability ratio. More details about Cpg and other process capability indices can be found in [8]. Since the process with DMV has its mean bias toward the target value, the CpD capability ratio may be expressed as:
(3)
CPD IT - SL l/3 c osr --'--
where T is the process target mean, and SL is the closest specification limit to that target. Substituting about CPD and Cpg in Eq. (2),
I I T- SLI/3 osrl(1-K)
--
IT- SLI/3 c osr
(4)
Eq.(4) can be reduced to: K-
1 - I/C
(5)
Eq.(5) expresses the relationship between the process mean's static offset "K" (in units of oST) and the inflation factor C. The K parameter represents the tolerance ratio consumed by the process mean offset from the target value. The value of K is given by [6]: K= IT-~[/[T-SLI
(6)
Substituting Eq.(6) in Eq.(5) and dividing both sides by OST one can get the Z-transform that relates C to the process's mean shift from its target value:
I z~l
-- I Zsrl (1-1/c)
(7)
Eq.(7) translates C to an equivalent sustained mean shift (Zp) expressed in Z units of measure. According to the values of C and the process's instantaneous capability, the equivalent shift of the mean (Z~t) from its target value can be found. In Table (2), the equivalent mean shift, Z~t, is calculated for different values of short-term capabilities Cp ( where C p - ZST/3) at some values of the inflation factor C. At this point of analyses, the process with DMV can be calculated its capability value. Given the process's short-term capability and its equivalent mean offset Z~t, then the long term tolerance band in Z-form, ZLT, is defined as: ZLT ffi ZST- Zp = ZsT (l-K)
(8)
The capability index CpD call be found from: CPD = Z~T (I-K)/3
(9)
or
CrD = 7-,sT/3C
(10)
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Table 2: Equivalent mean offset (Z~t) at different values of C and
LCo
i r ->
1.2
~
0.501 1.002 1.25
1 2 3
"
1.4
1.6
0.857 . . . . 1.714 2.i43
...
Cp
l.s
1.125 2.25 2.813
2.0
1.332 2.664 .... 3.33
1.5 3.0 3.75
The practical utilities of the analysis above are important. A designer and/or a production manager can decide whether to use a process with DMV based on its expected production yield for certain tolerance bands. To illustrate, suppose that the required target production yield, in a part per million (ppm), is 1000, i.e., the number of out-of-specifications parts in a lot of million parts is 1000 (500 ppm at each side of the specifications). Then the corresponding ZLT (from the Normal Distribution Tables)should be 3.29 and, accordingly, CpD = 1.1. Now, if the process is historically characterized by having a DMV with, for example, C=1.5 (K=0.333), the specification limits should be set at least at ZST = 4.94 (from Eq. 10) to achieve the target value of the production yield (ppm = 1000). This means an equivalent potential capability of Cp = 1.65. The designer can decide whether he can allow for that value of tolerance, and if not, the production manager should look for a more precise (but usually expensive) process. On the other hand, if the designer insist on the tolerance he decided, the production manager can expect the production yield, and, if satisfied, he can go ahead and use the process. Table (3) introduces the production yield for processes with DMV (at some C values) for some selected values of tolerances (expressed in Z-form, ZST). The Table can help a production manager to determine the production yield given the process tolerance and the inflation factor of its standard deviation. Table 3: The expected ppm for processes with DMV at different tolerance bands
C 1.2 1.4 1.6 1.8 2.0
I ZsT " 9
3.0 32420 33360 60100 94920 133620
~
4.s 176 1326 4954 12420 24440
6.0 "
19 176 868 2700
7.0 2.8 30 176
Another practical utility is to find out the process robustness (CR) for the inflation parameter. As given by Richards [7], robustness is defined as the ability to resist or respond to perturbations without loss of performance as well as the ability to grow and expand. Since ZLr is the response of the process due to the dynamic mean behavior and Zsr is the original response without perturbations, one can apply the above definition, that is, the process robustness CR may be given by: CR- ZLT / ZST-- ZsT (l-K) / ZST = (l-K) -- I / C
(ii)
Expressed as a ratio, the process robustness may be given as Ca percent robust to perturbing influences of the mean variation, e.g., a process with C = 1.5 is 66.7 % robust due to that mean variation causing the inflation.
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5. VERIFICATION OF THE ANALYSIS A Monte-Carlo simulation program is developed to check out the derived capability index (in Eq. 10) associated with the processes with DMV. The simulation program, indeed, tests both the inflation factor and the process's expected production yield. The same procedure of generating normally distributed random variates with specified DMV, explained in Appendix A, is utilized. The out-of-specifications number of units for batches of 5000 (1000 data sets times 5-unit set) production units is counted. Ten replications are carried out for each one of the nine selected cases exhibited in Table (4) below. In each case, the calculated and simulation-resulted ppm are compared. The calculated ppm values are found through the same procedure used to develop Table (3).The simulation resulted ppm are found by counting the average out-of-specification number per batch and then applying the chi-square distribution method for best estimate detailed in [2]. Table 4: The expected ppm from calculations and simulation for some selected cases
Cv C Calculated ppm Simulatedppm
1'.4 33360 33506
1.0 1.6 60100 60424
1.8 94920 95234
1.4 1326 1384 .
.
1.5 1.6 4945 4880 .
.
.
1.8 1.4 12420111~' 20 12102 21
2.0 1.6 176 181
1.8 868 842
The analysis verification is based on comparing the calculated versus the simulation-resulted ppm considering the simulation results as the reference. As can be observed from the table, differences between the calculated and simulation-resulted ppm, in general, are not significant. The maximum difference is (324 / 106 %, at Cp = 1.0 and C = 1.6 in the Table). Also, there is no bias in the calculated ppm, and in turn in the developed capability index, since non of the calculated and simulation-based ppm values dominate the other. 6. PROCESSES WITH SIMULTANEOUS STATIC & DYNAMIC MEAN BEHAVIOR In some situations, both forms of static and dynamic mean behavior may come into play. In such a situation, the estimate of process capability and the corresponding production yield is more complicated because it should consider the process mean offset (static mean offset) as well as the dynamic influences that result over time. In this research, two approaches are used to evaluate a process's capability when it has a simultaneous static and dynamic mean behavior. The first approach, by Harry [2], computes the degrade in the process capability as the summation of the static and dynamic mean variation effects. The second approach, proposed by the author, bases the calculation of the capability indices for a process with a static mean shift on the inflated standard deviation to compensate for the dynamic mean behavior. Harry [2] suggested to add the effects of both the static and dynamic mean behavior. He proposed that the portion of mean variation due to dynamic variations (K') is given by: K" = 1 - I(ZsT + Z~ ) / Zs-r !
(12)
Since K is given by {1 - (ZLT / ZST )}, from Eq. (8), the effect of the static mean offset is simply given by ( K - K" ). To illustrate, suppose that ZST =5.0 and ZLT = 3.75. The K value will be given by 1- (3.75/5.0) = 0.25. Suppose that the mean offset Z~ = 1.0, then by applying Eq. (12), K* will be 0.05. The conclusion is that the dynamic mean behavior
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consumes 5.0 percent of the semi-tolerance zone while (0.25 - 0.05) 100 = 20.0 percent of the same zone would be consumed by the static mean offset. The long-term capability index (C*pK) is given by [2]: CpK = Cp(ZLT / ZST) =Cp[l - (K + KST)I
(13)
The author here introduces another approach for dealing with processes with static and dynamic mean behavior. This approach makes use of the inflation factor given in Table (1) and the well known process capability indices. As given in [8], the most popular capability indices for processes with a static mean shift from its target value are Cpr and Cpm where: CpK = Cp ( l - K ) ,
and
Cpm = Cp / 3/1 +(It -oT)2
In terms of the notations used in this research:
CpK -- TZST
(1 - K )
(14)
Applying the inflated standard deviation, the capability indices for a process with simultaneous static and dynamic mean behavior will be:
Cpm---~ ,~Jlq-Z~
(15)
CpmD
= ZST / 3/1 + Z 2 3C ~t
(17)
Ceua~
= zsT (I - K) 3C
(16)
7. COMPARISON BETWEEN DIFFERENT APPROACHES In this section, a Monte-Carlo simulation program is developed to evaluate the two approaches introduced in the previous section. The Harry's approach to compute the capability index for processes with simultaneous static and dynamic mean variation (Eq. 13) and the author's approach for the same purpose (Eqs. 16,17) are used to compute the expected out-of-specifications percentage for some selected combinations of static and dynamic mean variations (presented in Table (5) below). The simulation program is used to generate normally distributed random variates with a specified offset of their mean from its target value. The same technique applied in section (5) is used to cause the effect of the dynamic variation; and the specified mean's offset from the target value is added to the generated variates to reflect the static variation of the mean. As in section (5), ten replications are conducted for every case, each case includes a batch of 5000 variates. The average
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percentages of out-of-specifications for the ten replications (shown in the last row of Table (5)) are used to compare their corresponding values calculated using the two approaches. Table 5: The expected out-of-specifications percentages for processes with simultaneous static and dynamic mean variation ZsT C
C'PK 0.5
Cpr,D CPmD simulation
C'pK l.O
Cpgn
CPmD Simulation
1.5
C*PK CPKD CPmD Simulation
....
6'0
4.5
3.0
1.4 3.14 3.67 2.74 3.61 5.81 7.6 6.4 7.45 10.02 i4.2 11.7 14.31
,,
1.8 6.92 8.23 6.81 8.39 10.37 13.3 11.9 12.95 15.8 20.3 17.9 20.35
1.4 .16 0.21 0.21 0.19
0.54 0.62 1.16
6.71 1.13 1.62 3.75 1.85
1.8 1.19 1.32 1.25 1.35 2.2 2.6 3.8 2.9 4.42 4.7 8.2 5.61
1.4 3.9e-3 4.4e-3 6.4e-3 4.2e-3 1.5e-2 1.8e-2 0.12 3.1e-2 5.7e-2 6.6e-2 0.86 0.09
1.8 .09 0.11 0.14 0.12 0.23 0.28 0.91 0.37 0.55 0.62 3.2 0.95
As can be observed, Harry's approach always underestimates the out of specification percentage or, in other words, overestimates the capability of the process. The author's approach of using the well known indices and basing their calculations on the inflated standard deviation seems more reasonable; and the simulation results comply to a high extent with their corresponding values of either CpKD and CpmD. Comparing the results of Cp~ and CpmD, one can observe that the CpmD -- based values are generally greater than those of the simulation results especially in high values of Ztt and ZST. However, the CpmDis better than C'pr as an index of process capability but is not as good as CpKD. The CpgD-based out of specification values are the closest to the simulation results, with no bias, such that it is recommended to be used as a capability index in characterizing processes with simultaneous static and dynamic variation. Table (5) can also help a production manager deciding whether to use the process based on the expected out of specification percentage. 8. APPLICATION The process of "Assembling Water Housing and Water Cover" in Helwan Company for Metallic Appliances is selected to demonstrate how to find out capability and other performance measures of processes with mean variation. In this application, the focus will be on the boring process that is used to produce the water housing cover. This process, as claimed by the quality control manager in the company, is experienced to have repeated mean shift from time to time. The inner diameter of the cover has a nominal size of 88.8 mm and a tolerance zone of :1:0.4 mm around the nominal size. Measurements of the inner diameter over about one year have been collected. Gauge and operator errors have been neglected. A 180 samples, five-unit each, have been plotted their means (and also their individuals) on a chart where the horizontal axis represented the sample number in a time sequence. The chart included also the process's target value (88.8 mm) and specification limits. Samples seem to have high variation are excluded. The plotted samples represented clusters of different means and run lengths, and that proves that the process has a
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dynamic mean variation. Although the exact timing of process mean shift is hard to be accurately determined, the clusters of measurements guided us to visually figure out these shifts. The process analysis has been conducted through the following steps: 1. The range and the mean of each sample are calculated. The average range is calculated and then the short-term standard deviation is computed and found to be 0.08 mm. The process's grand average (long-term average) is calculated by averaging the samples' means (after excluding samples with abnormal variations) and found to be 88.68 mm. 2. The maximum and minimum cluster means are addressed and found to be 88.85 and 88.53 mm respectively, i.e., about • 2 Csr around the mean value. The corresponding value of the inflation parameter (Table 1) is C=1.4. 3. The process's short-term, or potential, capability index is calculated as: Cr
= Z s x / 3 = ( S L - T ) / 3 o s r = ( 8 9 . 2 - 8 8 . 8 ) / 3 x 0 . 0 8 --- 1 . 6 7
4.
Since the process's grand average (88.68 mm) deviates from the target value (T = 88.8 mm) and the mean has random shifts (from the samples' plotting), then the process is considered to have a simultaneous static and dynamic mean variations, and the Corn should be calculated. 5. The degree of mean offset (K) can be calculated as: K = IT - 6 1 / I S L -
6.
TI = 0 . 3
Applying Eq. (16), the Cp~a) can be obtained as: Cer~ = (Zsr / 3C )(1 - K) = 0.83
The corresponding production yield (from the Normal Dist Tables) is expected to be 99.38 percent. 9. CONCLUSIONs Some manufacturing processes have natural shifts and drifts of their mean that happen randomly across their specification limits. This phenomenon is called dynamic mean variation if the process's long-term average coincides with its target value. Otherwise, the process is considered to have simultaneous static and dynamic mean variation. A process with DMV can be translated into an equivalent process with the same mean but having an inflated standard deviation. The inflation factor depends on the range of the random process's mean shift. This inflation can be related to a process static mean offset and, accordingly, the process's degraded capability can be calculated. In case of simultaneous static and dynamic mean variation, the problem gets more complicated. In this research, two approaches are introduced, one by Harry [2] and the other by the author, to compute the capability index. The two approaches are assessed by comparing their results of the expected out-of-specification percentages against other percentages resulted from a simulation program developed for this purpose. It was found that, within the tested range of parameters, the results of the author's approach comply to high extent with the simulation results; while those of Harry's approach are always less than the simulation results. REFERENCES 1. Gunter, B.H., "The Use and Abuse of CPK, Part 2.", Quality Progress, pp. 108, (1989). 2. Harry, M. J., and Lawson, J. R., "Six Sigma Producibility Analysis and Process Characterization", Motorola Inc., (1992).
630
3. 4. 5. 6. 7. 8. 9. 10. 11.
12.
Current Advances in Mechanical Design and Production, MDP-7
Bender, A., "Statistical Tolerances as it Relates to Quality Control and the Designer", Society of Automotive Engineers, SAE, Paper No. 75049, pp. 412-419, (1975). Hussein, M., "Analysis of Quality Characteristics for Processes with Dynamic Mean Variation", Eng. Research Journal, Helwan University, pp. 46-52, Dec. (1998). Evans, D. H., "Statistical Tolerancing: The State of the Art, Part II: Methods for Estimating Moments", Journal of Quality Technology, Vol.7, No. 1, pp. 1-12, (1975). Montgomery, D. C., "Introduction to Statistical Quality Control", 2"~ John Wiley & sons Inc., (1991 ). Richards, L.D., "Analysis of Robustness in the Formulation of Technology Strategy", ASEM Proceedings, pp. 7-16, (1996). Sweeny, P. J., and Mohammed, H. M., "Capability Indices and Process Evaluation", ASME Conference Proceedings, pp. 157-164, (1996). Woo et al, "Tolerance Synthesis for Nonlinear Systems Based on Nonlinear Programming", IIE Transactions, Vol. 25, No. 1, pp. 51-61, (1993). Balakrishnan, N., "A Multiple-Choice Knapsack Model for Tolerance Allocation in Mechanical Assemblies", IIE transactions, Vol. 25, No. 4, pp. 13-14, (1993). Cheng, B.W., et al, "Optimization of Mechanical Assembly Tolerances Incorporating Taguchi's Quality Loss Function", Journal of Manufacturing Systems, Vol. 14, No. 4, pp. 264-276, (1995). AI-Sultan, K.S., and AI-Fawzan, M.A., "Variance reduction in a process with random linear drift", Int. J. Prod. Res., Vol. 35, No. 6, pp. 1523-1533, (1997).
APPENDIX A ESTIMATING THE INFLATION FACTOR USING SIMULATION A computer program, encoded in the C-language, is developed to estimate the values of the inflation factor (C) at different ranges of mean variation. The selected ranges are exhibited in Table 1. For each of these ranges, the program is run 250 times (250 replications). The average values and their corresponding confidence intervals of C are given in Table 1. In each run, the program is initiated by generating N=I000 data set, each contains 5 measurements distributed normally N(0, l). The mean and standard deviation of each set are computed; and the cumulative standard deviation is progressively calculated for the 1000 set. Nonrandom perturbations in the mean parameter are calculated as follows. A uniform random distribution is used to create a series of mean shift vectors, one for each data set. A positive vector called for positive shift and vice versa. The two vectors are similar for each range and have values range from zero to the specific mean variation range (five equidistant va!ues).The extent of the mean shift is determined by a random uniform number. Each of these numbers is multiplied by the corresponding vector. To test the inflation in the process standard deviation, the cumulative standard deviation is calculated progressively across all the 1000 data sets. A ratio is calculated of the cumulative standard deviation for each shifted subgroup to that of the respective unshifted group; resulting in the value of C for that replication. The expected value and confidence intervals for the inflation parameter C are based on the 250 replication-resulted C values.
Current 4dvances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
631
SHORT TERM MANAGEMENT OF WATER RESOURCE SYSTEMS
Faye, R. M. 3, Sawadogo, S. 3, Gonzalez-Rojo,
S. 1'4 and
Mora-Camino, F. t'2
ILAAS du CNRS, 7 Av. du Colonel Roche 31077 Toulouse France 2Ecole Nationale de l'Aviation Civile 7, Av. Edouard Belin 31055 Toulouse - France 3Ecole Sup6rieure Polytechnique B.P. 10 Thins- S6n6gal 4Instituto Tecnologico de Chihuahua, Av. Tecnologico N~ C.P 31310, Mexico
ABSTRACT This paper addresses the problem of short term management of Water Resource Systems (WRS). This problem is coped through an adaptive generation of a recurrent optimization problem which is solved by a specialized primal linear programming technique. A fuzzy approach is used in each instance of the recurrent optimization problem to express accurately the demand constraints and the terminal level stock constraints. The proposed approach, applied to a WRS of medium complexity, shows that it can be adapted to the management of particular situations like flood management, management in presence of failed devices. KEYWORDS Water Resource Systems, Hybrid Dynamics, Intelligent Management Systems, Fuzzy Logic, Linear Programming. 1. INTRODUCTION WRS can be viewed as hybrid dynamics systems subject to continuous operations broken by discrete events when new goals must be defined on the short term either to guarantee an efficient use of the upstream water reserves or to enforce security [ 1]. Until very recently little has been developed on methodological grounds to face at the tactical level the large uncertainties that impair the operations of such systems. So in this communication, this problem is coped with two levels 9first, through an adaptive optimization approach where initial conditions and constraint levels are updated according to the last available information [3] and second, through a reactive approach when the occurrence of significant events leads to the definition of new short term goals and constraints for the current optimization problem. Fuzzy Logic is used in each instance of the recurrent optimization problem to express accurately the demand constraints and the level of the terminal stock constraints, then a Linear Programming solution is developed. The proposed approach is illustrated by its application to a water resource system of medium complexity. 2. MODELLING OF WRS FOR SHORT TERM MANAGEMENT The structure of WRS can be considered as ordered graphs composed of upstream reservoirs, sequences of interconnected reaches, end users pumping stations and outlet sections (Fig. 1). It
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Current Advances in Mechanical Design and Production, MDP. 7
appears that to cope efficiently with the management of WRS on the short term, stock and transfer dynamics must be taken into account. This leads to the following unified dynamic representation [2]" Z i ( t + A t ) = Z i ( t ) + [ ~ y,.hiJ .Qi(x)_ E Q ~ ( t ) - ~ P ~ ( t ) + a i ( t ) - d i ( t ) ] A t / o i (1) jEA~
't
'-'
J
je~
jePi
where Ai is the set of convergent canals towards reach i, Si is the set of canals originated at reach i, Pi is the set of pumping stations at reach i, 9is the transfer delay, Zi(0 is the downstream water level in reach or dam i at time t, Q~(z) is the mean inflow to reach or dam i during period [~, ~+Az], ai(t), Q~(t), di(t) and P)(t) are the respective mean values of natural inflows, controlled outflows, overflows and pumpings from, or to, reach i or dam i during period [t, t+At], m are shape parameters for the storage components and hiJt-~ are transfer coefficients representative of the flow dynamics along the reaches.
Overflow Inflows
--~ Losses
v~
~-~[ v-] Pumpings /
intermediateOutlet ",~jDam v-
Fig. 1. Components and structure of a WRS
3. THE PROPOSED APPROACH FOR SHORT TERM MANAGEMENT OF WRS Like in the case of many other large scale distributed systems, the management of water resource systems can be organized in three levels according to the span of time considered [1 ]: long term, short term and very short tenn. At the long term level, WRS is managed in order to gumantee availability of the resource and efficiency in its use over the whole annual cycle. At the short term level, reference values and control parameters are optimized to adequate long term goals with the actual operational state of the system. At this level, discrete decisions must also be taken to enforce security of operations and equity in deliveries. Following an adaptive approach, it is proposed in this communication to identify on a hourly basis the current operational situation and to reprogrmn the pumpings and releases over a chosen period of time [4] (Fig. 2). When no terminal event is expected for this period, its span should be taken longer than the maximum transfer delay present in the system. However, since the short term demand forecasts present a decaying accuracy with an increasing horizon of prediction, this period of time should be restricted to a few days. The planned hourly releases and pumpings are then obtained by the resolution of a recurrent optimization problem (ROPh) taking into account the current conditions of the system (water level in the reaches and dams, operational state of the actuators) and short term demand forecasts.
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Current Advances tn Mechanical Design and Production, MDP-7
Fig. 2. Proposed short term WRS management structure
4. ADAPTIVE GENERATION OF THE RECURRENT OPTIMIZATION PROBLEM At first, a general linear formulation is considered for the ROPh problem : h+H-I
max
Y'.)".
~
P~(t),Qij (t)i r162
(PJ(t)-Di(t))
(2)
t=h
with the dynamics constraints 9 ,.
Zi(t+l)_Zi(t)_[E
~ h 0 .Qi(x)_ ~ Q](t)- ~ P j ( t ) + a i ( t ) - d i ( t ) ] A t / c i=O x_
i~B, 1~
max {zmin,zi min h+H } ~ Zi ( h + H ) < Z max h+H-!
E [)".Q](t)+ ZP](t)]_< Li(h,H )
t-h je~i
jd:i
(5) (6) (7)
p~min ~ P2(t)~ pjmax i~B, jcPi, h~.t<_h+H-I i Dijb (t) i~I3, jEP~, h.g.t<_h+H-1 Pj(t)_< 0 _
(3) (4)
(8) (9) iCB, i~R
(10)
i~R
(11)
icR
(12)
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Current Advances in Mechanical Design and Production, MDP.7
where D~(t) is the predicted demand, 13 is the set of dams and reaches, R is the set of dams, Zim(h) is the initial measured level of reserve for reach i at the beginning of the h th hour. Since the behavior of users with respect to water demand is not well known, it is difficult to anticipate needs on the short term. In fact, many users needs are relative to a given quantity of water to be delivered over a given time span, so constraints (8) should be rewritten as" P~(t)~ P~max
)t
t~"+T~i'
t ~[t~0,t~ i) + T~ ')
0=t~i)
(13)
Dj (t
P~(t)= 0 ift ~[tJ(i),t j(i) + T~ i) ] where j(i) is the index ofuserj attached to reach i and where [t[i),t~ i) + T~i)] is the n th period of water demand for user j(i). In other situations when two (or more)different end uses can be distinguished at the same pumping station, constraints (8) and (13) must be introduced simultaneously with the additional condition: p~ i(t)+Pj2(t)
icB,
t~[t~ i) t~ i) +T~(i)]
(14)
If water demand is known only in qualitative terms (Fig. 3), qualitative scales based on fuzzy sets can be used to introduce the effect of intensity and flexibility of demand in these two constraints. I
12
Very
small
small
medium
high
I
Veryhigh
Ah hours
Fig. 3. Linguistic evaluation of demand flexibility
The long term management of the system provides for each storage dam the quantity of water which can be released per week. When constraint (11) is not saturated in the case of a storage darn by the solution of ROPh, it appears necessary to update the formulation of constraints (11) in problem ROPh+I. This is done considering the following performance indeces: 1. the demand flexibility with respect to dam i, is given by: Ah=
E A h j . dij / E d ji jcSuc(i) [jcSuc(i)
(15)
where Suc(i) is the set of direct users of dam i, Ahj is the temporal interval of demand dI from storage i at pumping station j. 2.
the demand trend for this storage dam which is given by:
Current Advances
in M e c h a n i c a l
Design and Production,
635
MDP-7
ii ~i i ~ii~ !
ADih = Di(h + H ) - Di(h + H - 1) i~R,
i
(16)
i~ iii ~I
h+H-I
where,
Di(h+H)=
E
)".
j ~ S u c ( i ) gr
3.
E
D~h(t)
~~
~ii I
t=h
the current release variation with respect to the nominal release estimated at time h over the period [h, h+H- 1], Ri(h+H- 1).
These three performance indeces are evaluated in a qualitative way using an adequate definition of scaled fuzzy sets 9Then the updated limit for constraint (11) to be adopted for thenext planning period [h, h+H] is given by: L i (h, H)
= L i (h,
H) + ~,~. R i (h + H - 1)
....
(17)
where X~ is an adjustment coefficient (Fig. 4) obtained as output from a base offuzzy rules whose entries are the fuzzy evaluations of "demand flexibility", "demand trend" and "release variation".
. . . . .
!/, I
NVL
NL
NM
Ns
NVs
!
i
ii i L v
......
Z
PVs
Ps
PM
PL
PVL
ii~
!i
k
Fig. 4. Defuzzification scale for parameter X i iii~i
5. MANAGEMENT OF PARTICULAR SITUATIONS According to the operational situation, the general formulation of the ROPh will be modified. 5.1 Flood management In the case of an excess of resource in the system, the demand plays a secondary role and it will be rather a question of managing the capacity in order to diminish the risk of flood while building up reserves. Then the ROPh can be formulated as ' min
~
P~(t),Qitt),di(t)idl
h+H-I
~di(t )
t=h
(18)
with constraints (3), (4), (5), (5), while constraints (8), (9) and (10) can be eliminated for a subset of end u s e r s ( j ~ , 7icPi, ieB). Here constraints (ll) become " ( I - E : i ) . Z max _
~
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Current Advances in Mechanical Design and Production, MDP-7
Given the solution of ROPh-I at (h-l)" Z:(t+l), Q~i(t), j~Si,
Pj'i(t), jCPi, d~(t), ir
l~t
e It
3i
Z[(,+ 1) ~ (1-6i ).Zmax
[d,(~)~di
such as "~and *
(19)
rain
with : h+H
E E ah(t)>(l-~,i)[E(EP~ max+ EQ~max). H]
(20)
where a~(t) is the forecast realized at time of inflows for dam i during period t which is h, ~i is a positive real number small w.r.t, unity. 5.2 Management with failed devices When pumping station jo(io) goes down during period [ho-l,ho] with a predicted repair time iho of some hours (iho
max
Z
Pi(t) +
t=h o
ho+H-I
.
Z_
P~~
(21)
t=ho+th o
j*jo(Io) with constraints (3),(4),(5),(6) and :
p~ rain ~ p~ (t) ~ P~max i r B, j r Pi, j;~jo(io), hog
(22)
p~min< p~o(t)< p~max ho +ih
(23)
P~(t)< Djin(t) icB, j~Pi, j~jo(io), h ~ + H - 1 9
P;o~(t)< D~~
h o +ih < t < h o + H - I
(24)
(25)
while constraints (9),(10),(11) and (12) remain unchanged. This formulation and the subsequent solution will be maintained until the failed device is repaired. Other special situations such as the ones resulting from programmed maintenance operations and from the presence of contamination have been analyzed [4]. 6. APPLICATION The proposed approach has been applied to different WRS. Here its application to a dual WRS dedicated to irrigation (see Fig. 5) is displayed to illustrate its practicability.
637
Current Advances In Mechanical Design and Production, MDP-7
a3 ~
2 /N
1~
Di '/~' '-'
P2
D2
~4
Fig.5. The W R S of application
The WRS is characterized by the following parameters' min=3m ' zmax=16.5m 7mi"=2m ' zmax=14.5m, - the boundary levels in the dams "ZR, R, , "JR, R2 - the boundary levels in the reaches" Z~'in =l.5m, Z~"i" =l.5m, Z~'in =l.5m, Z["~"=3m, Z~'~" =3m, Z~"X=3m, - the boundary flows in the canals0 9_
D~ (k)=0.37 106 m 3, - the contribution supply" at(h)=0.5 103 m3/h, a2(h)=0.3 10a m3/h, a3(h)=l.2 103 m3/h, -the estimated transfer coefficient" hgl=0.1, hll=0.7, h~l=0.15, h[l=0.05, h~l=0, hg3=0, h13=0.2, h~3=0.6, h[3=0.15, h~3=0.05, h~3=0, h =0. l, h,2'=0.7, h =0.15, h -0.05, h =0, h23=0, h23=0.1, h23=0.5, h23=0.2, h23=0.15, h23=0.05, h~3=0, - the surface coefficient of the dams and reaches" a R, =6 105 m 2, 6 R2 =3.5 105 m 2, a ! =0.3 5 105 m 2, a 2 =0.3 3 105 m 2, o 3=0.25 105 m 2. The demands over a period of three days are taken such as" i#
io'
m~4~our
,
..-._--_
,
, . , ,,,
mJ/how
,
,
;,
24
12
,
._._.,.____ I~
12
2a
12 houri
Fig. 6.1. Hourly predicted demand at station 1
1~
12
24 horn
Fig. 6.2. Hourly predicted demand at station 2
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Current Advances in Mechanical Design and Production, MDP-7
Here it is supposed that at the current planning period the supply/demand balance is such that" Rt(h+H-I)/D~(k) = 0.02, R2(h + H-l)/D~(k)=-0.01 AD~/Dt(h + H-I) = -0.02, AD~/D2(h + H-l) = -0.01 In the case where demand flexibility is medium, the fuzzy logic approach proposed in section 4 leads to the definition of the following parameters and constraints" X~=-0.12 and ~.h=-0.05 h+H-I
X j~Q~(t) + jZiPl (t)-<0.238 106 m3
t=h
(26)
" "
h+H-I
)-, Q2 (t) + j~P2 (t) <0.659 106 m 3
(27)
t=h j ~
with 9 ZR, (h + H)> Z~ in and ZR2 (h + H)> Z~ in
(28)
The resolution of the ROPh provides the following short term management plan"
't
10s
*2
~
12
24
12
.t 9.
24kom
Fig. 6.3. Planned water level at reach I to' m~thouT
'
12
2
24
L
12
J
)4
)_
12
horn
Fig. 6.4. Planned water level at reach 2 I#
41
o
t
kom
Fig. 6.5. Planned pumpings at station I
12
24
12
24
12
24hom
Fig. 6.6. Planned pumpings at station 2
It is shown that with a medium flexibility of the demand, all the authorized release at station 2 is used. This is not the case at station 1, because of the contribution supply a3 and the capacity Pmma~. However, for another degree of flexibility (high), all the authorized release is used.
Current Advances in Mechanical Design and Production, MDP-7
639
7. CONCLUSION In this communication the short term management problem of WRS has been considered. An adaptive management approach has been proposed to tackle the large uncertainties which impair the operations of such systems, a Linear Programming formulation has been adopted for the resulting recurrent optimization problem. It has been shown that this formalism is able to cope efficiently with a large variety of operational situations, while Fuzzy Logic techniques can be used to tune the current level of constraints related with the management of the water resource. REFERENCES 1. Carpentier, P., and G. Cohen, Applied Mathematics in Water Supply Network Management. Automatica, Volume 29, N ~ 5, pp. 1215-1250, (1993). 2. Faye, R.M., F. Mora-Camino and A.K. Achaibou, Adaptive Optimization Approach for the Supervision of Irrigation Systems. IFAC/IFIP Conference MCPL '97, Volume 1, pp. 175-180, 31 Aug.-3 Sept., Campinas, Brazil, (1997). 3. Faye, R.M., S. Sawadogo, F. Mora-Camino and A. Niang, An Intelligent Decision Support System for Irrigation System Management. IEEE Conference on Systems, Man and Cybernetics, SMC'98, Vol. 4, pp. 3908-3913, 11-14 Oct., San Diego, USA, (1998). 4. Faye, R.M., Une Approche Int6gr6e pour la Gestion des Ressources en Eau faisant appel aux Techniques Floues et Neuronales, Ph.D Thesis, University Paul Sabatier Toulouse, 14 June (1999).
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AUTHOR INDEX Ababneh, M.A. Abd EI-Ghany, K.M. Abdel-Hamid, A. AbdeI-Hamid, A.A. Abdel-kader, M.S. Abdel-karim, M. Abdel Motelb, M.S. AbdeI-Shafi, A.A.A. Abduljabbar, Z.S. Abo EI-Naser, A.A. Abo-EI-Ezz, A.E. Abo-Elkhier, M. Abou EI-Ez, S.R.S. Ahmed, A.K.W. Ahmed, A.Y. AI-Bastaki, N.M. Alghamdi, A.A. AI-Haddid, T.N. Ali-Eldin, S.S. Aijawi, A.A.N. Alkhoja, J. Almakhdoub, S.A. Aiy, M.F. Aref, N.A. Attia, M.H. Attia, M.S. Badr, M.A. Bahei-EI-Din, Y.A. Bahgat, A. Bahr, M.K. Bakhiet, E. Bayle, B. Bayoumi, A.M.E. Behnam, W.M. Bravo, R.R. Brochado, M.R. Choi, B.K. Darwish, S.M. Dashwood, R.J. Dokainish, M. EI-Keran, A.A. EI-Arabi, M.E. EI-Beheiry, E.M. Elbestawi, M.A. EI-Koussy, M.R. EIMadany, M.M. EI-Maddah, M.M. EI-Mahallawi, I. EI-Midany, T.T. E! Moudani, W. Elrafei, A.M. Elsawy, A.H.
339 261 467 331 195 185 75 303 95 407 253 293 571 105 397 243 511 339 205 511 233 347 205 195 521 185 591 271 75 139 281 57 501 233 37 611 425 477 389 37, 131 45, 447 85 85, 113 437 261 95 195 261,389 447 611 271 313
Elsayed, E.A. El-Shaer, Y.I. El-Sonbaty, I. El-Wakad, M.T. EI-Zoghby, A.A. Era, H. Fahmy, M.F. Fanni, M. Fatahalla, N. Faye, R.M. Fouda, N. Fourquet, J.-Y. Ghanya, A. Gharieb, W. Goforth, R.E. Gonzalez-Rojo, S. Hafiz, M. Hammouda, M.M.I. Hamza, K.T. Handou, M. Hassan, M. Hedia, H.S. Hegazy, A. A. Helal, M.E. Ho, K. Hosni, Y.A. Hozayyin, A.S. Hussein, A.A. Ibraheem, A.A. ibrahim, R.N. Jamshidi, M. Karnopp, D. Kassem, S.A. Khalil, M. Khattab, A.A. Kim, B.H. Kishitake, K. Krempl, E. Labib, H.F. Lee, P.D. Liu, P. Mahmoud, F.F. Masoud, M.I. Mazen, A.A. McCormack, A.D. Megahed, A.A. Megahed, G.M. Megahed, M.M. Megahed, S.M. Meguid, S.A. Mohammed, H.H. Mora-Camino, F.
553 195 455 531 243,281 323 313 45 369 631 303 57 477 67 357 631 369 215 27 611 131 303 415 591 151 491 591 467 75 563 3 85 139 281 539 425 323 151 389 381,389 105 205 323 397 563 455 381,389 185, 195 27 161 621 123, 611,631
642
Mostafa, A.A-F. Nagib, G. Naito, K. Noda, N. Radwan, H.T. Ragab, M.S. Ragab, A.R. Rakheja, S. Ramadan, M.Z. Recho, N. Refaey, A. Renaud, M. Richard, M.J. Saleh, Ch. A.R. Salem, H.A. Sawadogo, S. Seleem, M.H.
Current Advances In Mechanical Design and Production, MDP-7
539 67 323 223 447 467 173 105 571 233 369 15, 57 105 173 357 631 215
Selim, H.M. Shabana, Y. Shahzad, M. Shehat& I.H. Slama, J.G. Soliman, M.S. Soltan, H.A.M. Teltz, R. Tohgo, K. Varzavand, S. Weaver, D. Wifi, A.S. Youssef, Y.M. Zaid, A.I.O. Zeyada, Y. Zhang, X.B.
579 223 123 313 123 347 603 437 223 313 131 467 381 331,339 85 233
643
SUBJECT INDEX
Acoustic emission Aeroengine Air transportation Aircraft, Guidance Al-alloys, 2024-T3 , AL-Cu-Li , AI-Mg-Zn , Copper-Berllium , Creep , Dynamic strain aging , ECAE processing , Fatigue stren~h , Poisoning , Solidification , Welded , Welding AM Autofrettage Batch processes Biaxial tension Biomechanics CAD CAM Casting, Continuous , Microstructural , Steel , Thin slab Cellular manufacturing CIM CNC Collision avoidance Combinatorial optimization Composites, Microstructure , Cementitious , Dampin~ , Fibrous , Laminates , Metal matrix , Powder Compressibility Compressor Control, Preview , Vehicle , Robust , Bilinear , Complex systems , Fuzzy , Guidance systems Non-linear , Variable structure , Vibration ,
253 16 I 61 ! 123 339 357 347 323 347 347 357 339 331 331 339 357 425 195 603 185 531 491 491,447 381 381 381 381 579 425 437, 447 85 611 397 313 407 271 281 397, 407 397 151 161 95 95 75 113 3 3, 67, 75 67 37, 67 37 113
Copper, Microstructure , Polypropylene , Powder Crack, Detection , Polymer , Propagation Crew scheduling Cup forming Dental implants DFMA Driver modeling Ductile iron, Austempering , Hardness
, Microstructure Dynamic Contact Viscoelasticity Dynamics, Nonlinear EDM Elastic, Stress Aging Elastic, Plastic Facility, Layout Planning Feature Recognition Finite element ,
,
Flexible manipulators Flow line Frusta inversion Fuzzy logic Generation, Trajectory Genetic algorithms Glass, Recycling Group scheduling Grouping technology Gun barrels Hot rolled, Steel IMS Iterative elastic techniques Laminates, Angle-ply , Composites , Cross-ply Lane following Maintainability Manufacturing system Markov's model Mechanics, Fracture Mixed networked Mobile manipulators
415 415 415 253 253 253 611 467 531 501 85 369 369 369 205 205 131 447 323 323 215,223,243 571 571 563 27, 45, 131,185, 195, 215, 233,261, 271,293,467, 511 27, 37, 45 591 511 3, 75, 85, 611,571, 631 123 571 313 591 579 195 389 425,437, 631 185 281 281 281 85 539 579 539 233,243, 271,281 603 57
644
Current Advances in Mechanical Design and Production, MDP-7
Multi-body dynamics 27 Multiple criteria 571 NC 447 Neural networks 123, 455 NGM 425 Nonlinear programming 603 Plastic instability 173 Plastic,Elastic 215, 223,243 ,Notch 215 , Stress concentration 215 Plasticity 151 Plasticity,Necking 173 Polyvinylochloride 243 Process, Capability 621 , Planning 563 , Tolerance 621 , DMV 621 Pump, Axial piston 139 , Swash plate 139 Ratcheting 15 i Reliability 539, 553 Replacement, Optimal 553 Residual stresses 195 Reverse engineering 491 Robot, Autonomous control 3 Robot, Configuration 15 , IKM 15 , Nonholonomy 57 , Optimization 45 , Redundancy 57 Robotic applications 3 Robotics 67 Roll stability 105 Rolling, Shape 293 Rolling, Square-to-oval 293 Scheduling 603 Sequential algorithms 579 Simplified inverse technique 467
Simulated annealing Steel, Casting , Heat treatment , Hot rolled , Optimization , Welding Strain gages Stressconcentration Stress, Fatigue , Optimization , Shielding , Thermal , von-Mises Structural damping Surface roughness Suspension, Design , Anti-roll , Interconnected , Vehicle Tabu search Vehicle Vehicle, Control , Dynamics Vehicle, Ride Vibration, Flow induced Viscoplasticity VMS Void growth Water resource Wear, Fretting Weibull distribution Welding Welding, AI-alloys , Bonding , Elongation defects , Steel Work hardening Zirconium alloys
591 381 389 389 389 477 531 261 303 303 303 223 303 27 455 95 105 105 105, 113 591 113 95 85 105 131 151 425 173 631 521 539 233 339,357 477 261 477 521 521
645
LIST OF PARTICIPANTS
Ababneh, M.A. Abbas, A.T. Abd EI-Aal, U.M. Abd EI-Azim, A.N. Abd EI-Ghany, K. Abd EI-Magied, R. Abdel Azim, A.N. AbdeI-Alim, H. AbdeI-Hamid, A. AbdeI-Hamid, A.A. AbdeI-Karim, M. Abdel Motelb, M.S. AbdeI-Shafi, A.A. Abduljabbar, Z.S. Abo EI-Naser, A.A. Abo-EI-Ezz, A.E. Abo-Elkhier, M. Abo-Ismail, A. Abou-Ali, M.G. Abou EI-Ez, S.R.S. Abou EIKhair, M. Abou-Hamda, M. AbuI-Naga, A.M. Abutaleb, H.S.A. Achaibou, A.K. Aggag, G. Ahmed, A.K.W. Ahmed, A.Y. Ahmed, M.H.M. AI-Bastaki, N.M. Alghamdi, A.A. AI-Haddid, T.N. AI-Hozwany, W. Ali-Eldin, S.S. Aljawi, A.A. Alkhoja, J. AImakhdoub, S.A. AIswailem, S.I. Aly, A.A. Aly, M. F. Aref, N.A. ARia, M.H. ARia, M.S. Ayad, M. Badr, M.A. Bahei-EI-Din, Y.A. Bahgat, A. Bahr, M.K. Bailey, J.A Bakhiet, E. Barakat, A.F.M.
Jordan Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Jordan Egypt Egypt Egypt S. Arabia Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt France Japan Canada Egypt S. Arabia Bahrain S. Arabia Jordan Syria Egypt S. Arabia France S. Arabia S. Arabia Japan Egypt Egypt Canada Egypt Egypt Egypt Egypt Egypt Egypt USA Egypt Egypt
Basily, B.B. Bayle, B. Bayoumi, A.M.E. Bedewy, M.K. Behnam, W.M. Bhat, R.B. Bo Zhang, X.Y. Bravo, R.R. Brochado, M.R. Chakraverty, S. Chen, J. Choi, B.K. Darwish, F. Darwish, S.M. Dashwood, R.J. Dawood, O.M. Dokainish, M. Dossoky, S. EI-Arabi, M.E. EI-Axir, M.H. EI-Baradie, Z.M. EI-Batahgy, A. EI-Beheiry, E.M. Elbestawi, M.A. EI-Gamil, M.A. EI-Habak, A.M. EI-Hebeary, M.R. EI-Kady, M. Aly EI-Keran, A..A. EI-Khabeery, M.M. EI-koussy, M.R. EIMadany, M.M. Ei-Maddah, M.M. EI-Mahallawi, I. EI-Mahallawi, N.M. EI-Meniawi, M.A.
EI-Midany, T.T. El Moudani, W. EI-Mounayri, H. EI-Mously, H.I. EIrafei, A.M El-Said, R. A. EIsawy, A.H. Elsayed, E.A. EI-Shaer, Y.I. EI-Shammari, F.D. EI-Sonbaty, I. EI-Wakad, M.T. Elwany, M.H. EI-Zoghby, A.A. Era, H.
Egypt France USA Egypt Egypt Canada China Canada Brazil India USA S. korea Egypt S. Arabia Egypt Egypt Canada Egypt Egypt Egypt Egypt Egypt Egypt Canada Egypt Egypt Egypt Egypt Egypt Egypt Egypt S. Arabia Egypt Egypt Egypt Egypt Egypt France USA Egypt Egypt Egypt USA USA Egypt Egypt Egypt Egypt Egypt Egypt Japan
646
Current Advances in Mechanical Design and Production, MDP-7
Fahmy, M.F. Fanni, M. Fatahalla, N. Faye, R.M. Foda, M.A. Fouda, N. Fourquet, J.-Y. Gadalla, M.A.E. Ghanem, M. Ghanya, A. Gharieb, W. Goforth, R.E Gomaa, A.H. Gonzalez-Rojo, S. Hafiz, M. Hammouda, M.M.I. Hamza, K.T. Handou, M. Harraz, N.A. Hasheem, N.E. Hassan, M.F. Hassan, M. Hedia, H.S. Hegazy, A. A. Helal, M.E. Henriksen, S. Ho, K. Hosni, Y.A. Hozayen, A.S. Hussein, A.A. Ibraheem, A.A. Ibrahim, I.A. Ibrahim, R.N. Jamshidi, M. Kandil, A. Kanki, H. Karnopp, D. Kassem, S.A. Khalil, M. Khalil, M.A. Khashaba, U.A. Khattab, A.A. Kim, B.H. Kishitake, K. Krempl, E. Kridli, Gh. Labib, H.F. Lee, P.D. Lener, Y.S. Liu, P. Maeda, T. Maes, J. Mahmoud, F.F.
USA Egypt Egypt Senegal S. Arabia Egypt France Egypt Syria Egypt Egypt USA Egypt Mexico Egypt Egypt Egypt Niger Egypt Egypt Egypt Canada Egypt Egypt Egypt Australia USA USA Egypt Egypt Egypt Egypt Australia USA Egypt Japan USA Egypt Egypt Egypt Egypt Egypt S. korea Japan USA USA Egypt UK USA USA Japan Belgium Egypt
Masoud, M.I. Mawsouf, N.M. Mazen, A.A. McCormack, A.D. Megahed, A.A. Megahed, G.M. Megahed, M.M. Megahed, M.M. Megahed, S.M. Meguid, S.A. Mohammed, H.H. Mora-Camino, F. Mostafa, A.A-F. Muhe Guo, K. H. Naga, S.A.R. Nagib, G. Naito, K. Noda, N. Ohuchi, H. Oraby, E.A. Patterson, R.L. Radwan, H.T. Ragab, A.R. Ragab, M.S. Rakha, M.M. Rakheja, S. Ramadan, M.Z. Rashad, R.M. Recho, N. Refaey, A. Reif, W. Renaud, M. Riabov, M.V. Richard, M.J. Salah, M.F. Saleh, Ch.A.R. Salem, H.A. Sanad, F.H. Sawadogo, S. Seleem, M.H. Selim, H.M. Shabana, Y. Shahzad, M. Shash, Y.M.S. Shehata, I.H. Shehata, T. Slama, J.G. Sol, H. Soliman, M.S. Soltan, H.A. Stiharu, I. Takahashi, K. Teltz, R.
Japan Egypt Egypt Australia Egypt Egypt Egypt Egypt Egypt Canada Egypt France Egypt China Egypt Egypt Japan Japan Japan USA Australia Egypt Egypt Egypt Egypt Canada Egypt Egypt France Egypt Germany France USA Canada Egypt Egypt Egypt Egypt Senegal Egypt UAE Japan France Egypt USA Australia Brazil Belgium S. Arabia Egypt Canada Japan Canada
Current Advances in Mechanical Design and Production, MDP-7
Tohgo, K. Varzavand, S. Waly, M.A. Weaver, D. Wifi, A.S. Wusthof, P. Yoon, D.
Japan USA Egypt Canada S. Arabia Germany USA
Youssef, Y.M. Zaghloul, B. Zaid, A.I.O. Zeyada, Y. Zhang, J. Zhang, X.B.
Egypt Egypt Jordan Egypt USA France
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