Structural Dynamics of Electronic and Photonic Systems

Structural Dynamics of Electronic and Photonic Systems

Structural Dynamics of Electronic and Photonic Systems Copyright © 2011 John Wiley & Sons, Inc.

Edited by Ephraim Suhir, David S. Steinberg and T. X. Yu

Structural Dynamics of Electronic and Photonic Systems

Edited by

Ephraim Suhir David S. Steinberg T. X. Yu

JOHN WILEY & SONS, INC.

This book is printed on acid-free paper. Copyright © 2011 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at www.wiley.com/go/permissions. Limit of Liability/Disclaimer of Warranty: While the publisher and the author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor the author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information about our other products and services, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Structural dynamics of electronic and photonic systems / edited by Ephraim Suhir, David S. Steinberg, T. X. Yu p. cm. Includes index. ISBN 978-0-470-25002-0 (hardback); ISBN 978-0-470-88665-6 (ebk); ISBN 978-0-470-88678-6 (ebk); ISBN 978-0-470-88679-3 (ebk); ISBN 978-0-470-95001-2 (ebk); ISBN 978-0-470-95162-0 (ebk); ISBN 978-0-470-95179-8 (ebk) 1. Electronic apparatus and appliances—Reliability. 2. Optoelectronic devices—Reliability. 3. Fault tolerance (Engineering) 4. Microstructure. 5. Structural dynamics. I. Suhir, Ephraim. II. Steinberg, David S. III. Yu, T. X. (Tongxi) TK7870.23.S77 2010 621.382—dc22 2010031072 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

Contents

Preface vii Contributors ix 1 Some Major Structural Dynamics-Related Failure Modes and Mechanisms in Micro- and Opto-Electronic Systems and Dynamic Stability of These Systems 1

David S. Steinberg 2

Linear Response to Shocks and Vibrations

19

Ephraim Suhir 3

Linear and Nonlinear Vibrations Caused by Periodic Impulses Ephraim Suhir

35

4

Random Vibrations of Structural Elements in Electronic and Photonic Systems Ephraim Suhir

5

Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures Subjected to Sinusoidal or Random Vibrations 75

53

David S. Steinberg 6

Drop/Impact of Typical Portable Electronic Devices: Experimentation and Modeling 135

T. X. Yu and C. Y. Zhou 7

Shock Test Methods and Test Standards for Portable Electronic Devices C. Y. Zhou, T. X. Yu, S. W. Ricky Lee, and Ephraim Suhir

159

8

Dynamic Response of Solder Joint Interconnections to Vibration and Shock David S. Steinberg

9

Test Equipment, Test Methods, Test Fixtures, and Test Sensors for Evaluating Electronic Equipment 183

175

David S. Steinberg 10 Correlation between Package-Level High-Speed Solder Ball Shear/Pull and Board-Level Mechanical Drop Tests with Brittle Fracture Failure Mode, Strength, and Energy 195

Fubin Song, S. W. Ricky Lee, Keith Newman, Bob Sykes, and Stephen Clark 11 Dynamic Mechanical Properties and Microstructural Studies of Lead-Free Solders in Electronic Packaging 255

V. B. C. Tan, K. C. Ong, C. T. Lim, and J. E. Field 12 Fatigue Damage Evaluation for Microelectronic Components Subjected to Vibration 277

T. E. Wong

v

vi Contents 13 Vibration Considerations for Sensitive Research and Production Facilities

309

E. E. Ungar, H. Amick, and J. A. Zapfe 14 Applications of Finite Element Analysis: Attributes and Challenges

327

Metin Ozen 15 Shock Simulation of Drop Test of Hard Disk Drives

337

D. W. Shu, B. J. Shi, and J. Luo 16 Shock Protection of Portable Electronic Devices Using a “Cushion” of an Array of Wires (AOW) 357

Ephraim Suhir 17 Board-Level Reliability of Lead-Free Solder under Mechanical Shock and Vibration Loads 371

Toni T. Matilla, Pekka Marjamaki, and Jorma Kivilahti 18 Dynamic Response of PCB Structures to Shock Loading in Reliability Tests

415

Milena Vujosevic and Ephraim Suhir 19 Linear Response of Single-Degree-of-Freedom System to Impact Load: Could Shock Tests Adequately Mimic Drop Test Conditions? 435

Ephraim Suhir 20 Shock Isolation of Micromachined Device for High-g Applications

449

Sang-Hee Yoon, Jin-Eep Roh, and Ki Lyug Kim 21 Reliability Assessment of Microelectronics Packages Using Dynamic Testing Methods 485

X. Q. Shi, G. Y. Li, and Q. J. Yang 22 Thermal Cycle and Vibration/Drop Reliability of Area Array Package Assemblies 519

Reza Ghaffarian 23 Could an Impact Load of Finite Duration Be Substituted with an Instantaneous Impulse? 575

Ephraim Suhir and Luciano Arruda Index

589

Preface

Electronic, optoelectronic, and photonic components and systems often experience dynamic loading. In commercial electronics, such loading can take place during handling or transportation of the equipment. In military, avionic, space, automotive, and marine electronics, dynamic loading, whether deterministic or random, is expected to occur even during normal operation of the system. On the other hand, random vibrations are sometimes applied (in addition to, or even instead of, thermal cycling or environmental testing) as an effective and fast means to detect and weed out infant mortalities. In addition, the necessity to protect portable electronics from shock loading (typically, because of an accidental drop) resulted in an elevated interest in the development of theoretical and experimental techniques for the prediction of the consequences of an accidental shock, as well as for an adequate shock protection of portable products. Development of new shock absorbing materials is regarded equally important. Finally, owing to numerous optoelectronic and photonic technologies emerged during the last decade or so, the ability to evaluate and possibly optimize the dynamic response of various photonic devices to shocks and vibrations is becoming increasingly important. The following objectives are pursued in this book: • familiarize the readers with the major problems related to the dynamic behavior of

electronic and photonic components, devices, and systems; • examine typical failure modes and mechanisms in electronic and photonic structures

experiencing dynamic loading; • address the basic concepts and fundamentals of dynamics and vibration analysis, includ-

ing analytical, computer-aided, and experimental methods, and demonstrate how these methods can be effectively used to adequately approach the above problems; • discuss and solve particular problems of the dynamic response of electronic and photonic systems to shocks and vibrations, and • suggest how to choose the appropriate mechanical design and materials to create a viable and reliable product. The reader of the book will become familiar with the mechanical, materials, and reliability related problems encountered in systems experiencing shocks and/or vibrations and will learn about the theoretical and experimental methods, approaches, and techniques which are used to solve these problems. This will enable those in the field to enhance their knowledge and skills in their profession and will teach those not in the field yet how to apply their background in mechanics, materials, and structures to this exciting and rapidly developing area of “high-tech” engineering. The book is unique: it is the first time that a book of such a broad scope is written. The content of the book covers some of the most important mechanical, materials, and reliability aspects of the dynamic response, stability, and optimal design of electronic and photonic components, devices, and structural elements experiencing dynamic loading. The book contains 23 chapters written by leading specialists in the field. After getting familiar with the book’s chapters, readers will better understand the reliability problems in, and mechanical behavior

vii

viii

Preface of, typical microelectronic, optoelectronic, and photonic structures subjected to dynamic loading, as well as be able to select the most appropriate materials for, and geometries of, such structures. Some of the design decision could be made based on simple and easy-to-apply formulas which will be provided in the book. These formulas indicate the role of different materials and geometrical factors affecting the mechanical behavior and reliability of a structure and can be effectively used prior to, and quite often even instead of, computer-aided modeling or experimental analyses. The technical emphasis of the book is on the application of the basic principles of the dynamic structural analysis to understand, analyze, and improve the dynamic behavior and reliability of microelectronic and photonic structures operating in dynamic environments. The book will enable a design and reliability engineer, who did not work before in the field of electronics and photonics, to apply his/her knowledge in dynamical analysis to this new and exciting field. At the same time, physicists, materials scientists, chemical or reliability engineers who deal with “high-technology” components and devices for many years will learn how methods and approaches of mechanical and structural engineering can be effectively used to design a viable and reliable product. The book is written with the emphasis on the physics of the phenomena. No in-depth knowledge of the mechanical, materials, or structural engineering is required. The needed information is given in the book chapters, when appropriate. Nonetheless, some knowledge of the basic calculus, strength of materials, and theory of vibrations is desirable to better understand the contents of the book.

Contributors

Hal Amick Colin Gordon & Associates Brisbane, CA 94005

C. T. Lim National University of Singapore Singapore 117576

Luciano Arruda Instituto Nokia de Technologia Terra Nova Manaus-AM, 69048-660, Brazil

J. Luo Nanyang Technological University Singapore 639798

Stephen Clark Dage Precision Industries Rabans Lane, Aylesbury Bucks HP19 8RG United Kingdom J. E. Field University of Cambridge Cambridge, CB30HE United Kingdom Reza Ghaffarian Jet Propulsion Laboratory California Institute of Technology Pasadena, CA 91125 Ki-Lyug Kim Agency for Defense Development Yuseong, Daejeon 305-600 Republic of Korea Jorma K. Kivilahti Helsinki University of Technology 02015 TKK, Finland S. W. Ricky Lee Hong Kong University of Science and Technology Kowloon, Hong Kong People’s Republic of China G. Y. Li South China University of Technology Guangzhou 516040 People’s Republic of China

Pekka Marjam¨aki Helsinki University of Technology 02015 TKK, Finland Toni T. Mattila Helsinki University of Technology 02015 TKK, Finland Keith Newman Oracle Corporation Santa Clara, CA 95054 K. C. Ong National University of Singapore Singapore 117576 Metin Ozen Ozen Engineering, Inc. Sunnyvale, CA 94085 Jin-Eep Rho Agency for Defense Development Yuseong, Daejeon 305-600 Republic of Korea B. J. Shi Nanyang Technological University Singapore 639798 Daniel X.Q. Shi Applied Science & Technologies Research Institute (ASTRI) Shatin, Hong Kong People’s Republic of China D. W. Shu Nanyang Technological University Singapore 637331

ix

x

Contributors Fubin Sung Hong Kong University of Science and Technology Kowloon, Hong Kong People’s Republic of China

T. Eric Wong Raytheon Company El Segundo, CA 90245

Dave S. Steinberg (retired) Steinbergelectronics, Inc. Westlake Village, CA 91361

Q. J. Yang Bosch Chassis Systems Asia-Pacific Ltd. East Bentleigh VIC 3165 Australia

Ephraim Suhir University of California Santa Cruz, CA 95064

Sang-Hee Yoon University of California Berkeley, CA 94720

Bob Sykes Dage Precision Industries Rabans Lane, Aylesbury Bucks HP19 8RG United Kingdom

T. X. Yu Hong Kong University of Science and Technology Kowloon, Hong Kong People’s Republic of China

V. B. C. Tan National University of Singapore Singapore 117576 Eric E. Ungar Acentech, Incorporated Cambridge, MA 02138-1118 Milena Vujosevic Intel Corporation Folsom, CA 95630

Jeffrey A. Zapfe Acentech, Incorporated Cambridge, MA 02138-1118 C. Y. Zhou Hong Kong University of Science and Technology Kowloon, Hong Kong People’s Republic of China

CHAPTER

1

SOME MAJOR STRUCTURAL DYNAMICSRELATED FAILURE MODES AND MECHANISMS IN MICRO- AND OPTO-ELECTRONIC SYSTEMS AND DYNAMIC STABILITY OF THESE SYSTEMS David S. Steinberg Steinbergelectronics, Inc. Westlake Village, California

1

PHYSICS OF ELECTRONIC FAILURES IN VIBRATION AND SHOCK Modern electronic equipment is being used in a very large number of different areas that range from simple applications, such as automobile keys and temperature control devices, to very complex applications, such as airplanes, space exploration vehicles, and optical scanning medical devices. It is probably safe to say that virtually all electronic systems will be exposed to some form of vibration or shock during their lifetime. The vibration and shock exposure may be due to the operating environment experienced by an airplane or an automobile. The vibration and shock exposure may also be due to shipping the product across the country by truck or train. Electronic systems that are required to operate in a harsh shock or vibration environment will often fail. If a failure occurs in an automobile temperature-sensing device or a fuel gage, it may be inconvenient for the owner, but the chances of someone being injured or killed are small. If the electronic failure happens to occur in the flight control system or navigation control system of an airplane or a missile, several hundred people could be injured or killed. Many different types of materials, for example, metals, ceramics, plastics, glass, and adhesives, are being used today to fabricate and assemble a wide variety of electronic systems for commercial, industrial, and military applications. Many types of sophisticated electronic component parts are now available from a large number of manufacturers for specific applications and functions that did not exist only a decade ago. These components are often soldered to multilayer printed circuit boards (PCBs). PCBs may have from 6 to 12 internal layers of thin copper ground planes and voltage planes to remove heat and to provide electrical interconnections. The PCBs make it easy to assemble, remove, and maintain complex sophisticated electronic equipment at reduced costs. The PCBs must also protect the electronic components in storage, shipping, and operation in severe vibration, shock, thermal, and high-humidity environments. A wide variety of special plastics and metals are available to fabricate cost-effective and reliable electronic systems for special conditions. The soldering process must be carefully controlled because solder is often the major source of early failures in the field. Good solder joints usually require the use of paste and flux to obtain a reliable connection. The paste and flux must be carefully removed from the PCB to prevent electrical malfunctions in sensitive systems after several months of operation Structural Dynamics of Electronic and Photonic Systems Copyright © 2011 John Wiley & Sons, Inc.

Edited by Ephraim Suhir, David S. Steinberg and T. X. Yu

1

2

Dynamics-Related Failure Modes and Mechanisms in harsh environments. The normal procedure is to mount the electronic components slightly above the surface of the PCB, so that there is a small gap under the components. This makes it easy to flush out any paste and flux accumulated under the components. A thin protective coating should be applied to the PCB after cleaning to avoid the growth of dendrites, which can degrade the electrical performance of sensitive electronic systems operating in humid conditions. Dendrites are thin semitransparent plastic whiskers with a high electrical impedance that will often grow between electrical conductors in the presence of a chemical residue such as paste and flux and moisture exposed to an electrical current. Extended exposure for periods of several months can produce such a large mass of these whiskers that it will change their electrical resistance to an extent sufficient to cause malfunctions and even short circuits in the electrical system. Several thin protective coatings are available that can effectively prevent the growth of dendrites on PCBs. Materials such as paralyne, solder mask, polyurethane, epoxy, and acrylics can be applied to the clean PCB surfaces using different methods such as spray, brush, dip, and even vacuum processes. One of the best materials for protection is paralyne, which can be applied as a vapor. However, it is expensive and very tough. It is difficult to remove from the PCB if repairs have to be made and can often create new failures while trying to repair the old failures. PCBs are often enclosed within a box or housing for easy transportation and handling. The housing can also protect sensitive electronic components from hostile external environments, such as sand, dust, sun, humidity, rain, insects, mice, and birds. Very small insects often make nests inside the warm interior of the electrically operating system. Their residue can build up inside the housing and cause bridging across multiple pin connectors, resulting in short circuits with early electrical failures. Electronic systems that will be required to operate in open outside areas must be fabricated and assembled to prevent small insects from entering the housing and making nests. Removable covers must have a very close fit or gaskets must be used to provide a good seal. The PCBs are normally attached to the inside walls of the housing to help conduct away excessive internal heat to the outside ambient, where it is carried away. This also prevents the PCBs from impacting against each other and causing damage in vibration and shock conditions or the PCBs may have a multiple pin or socket connector added to one end of each PCB with a mate on the housing, so that each PCB can be plugged into the housing for electrical operation. Side wedge clamps can be used on the PCBs or the housing to improve the internal conduction heat flow path to the outside of the housing to reduce internal hot-spot temperatures. Reducing the internal hot-spot temperatures will usually increase the fatigue life of the electronics. The use of wedge clamps also helps to support the sides of the PCBs, which increases the PCB stiffness and natural frequency. A higher PCB natural frequency substantially reduces the PCB dynamic displacements in vibration and shock conditions. This reduces the stresses in the PCBs, in the components, in external lead wires, and in external solder joints. This increases their fatigue life. Reducing the PCB dynamic displacements also increases the fatigue life of the die bond wires and the ball bonds inside the electronic components. Therefore, by increasing the PCB natural frequency, one could increase substantially the fatigue life of the lead wires, the solder joints, and the ball bonds on the semiconductor components mounted on the PCBs. The natural (or resonant) frequency of the outer housing must be well separated from the natural frequency of the internal PCBs to avoid severe dynamic coupling and rapid structural failures of the PCBs in the housing during sine vibration. When the natural frequency of the outer housing is excited during exposure to a sine wave, the housing can sharply amplify the magnitude of the input acceleration (g) level, depending upon the damping in the system. As is known from the theory of damping in linear vibration systems, when the structural damping in the system is zero and the system is being vibrated at its natural frequency using

2

Case History for Design, Analysis, and Testing of Electronic Chassis

3

a sine wave, the transmissibility (amplitudes) of that structure will be infinite. This condition is impossible, however, because every real structural system has damping.

2

CASE HISTORY FOR DESIGN, ANALYSIS, AND TESTING OF ELECTRONIC CHASSIS REQUIRED TO OPERATE IN SEVERE SINE VIBRATION ENVIRONMENT AND EFFECTS OF USING VISCOELASTIC DAMPING MATERIAL ON PCBs TO INCREASE FATIGUE LIFE Failures in electronic systems can occur in many ways, often due to carelessness and lack of experience in handling a new environment or a new material. A large company with extensive electronics experience was awarded a multi-million-dollar contract for a program with a very severe vibration environment. Several other companies no-bid this contract because of the potential problems with the environment and low weight requirements for a system about the size of a shoe box. The company put a team of its top engineers together to solve the problem. This team spent several months investigating different proposals involving different exotic materials and the use of prototype test models to prove their capability to withstand the severe environment. The standard approach for providing reliable operation in severe vibration is to use vibration isolators. However, in this case the equipment had to be hard mounted so that isolators would not work. The team of experts finally selected a fabrication method for the PCBs inside the electronic enclosure that used a viscoelastic material. When this material was bonded to the PCBs, it provided high damping. The prototype vibration test models showed excellent results. The viscoelastic material was very effective in reducing the vibration “G response” levels enough to assure reliable operation in the severe vibration conditions. Reports were written and presentations were made to upper management. The experts’ team was given a green light to proceed with the fabrication and assembly for a large number of production units. Everyone was happy and sure that they had a reliable lightweight design that would survive the required severe qualification vibration tests. Their successful prototype test data were proof their new design would not fail. Their customer was invited to witness the qualification tests. The day for the qualification tests arrived. One of the customers selected one chassis assembly to start the vibration tests. The selection was made from a group of about 40 production units, ready for delivery to the customer. The first part of the vibration sine sweep tests went well, with no problems, so that everyone was happy. The next phase was the nasty 60-min dwell at the primary resonant frequency of the chassis. It was a mess. There were very loud cracking noises and a complete electrical failure of the electronic unit after a few minutes into the resonant dwell. The chassis was removed and the top cover was opened to inspect the condition of the PCBs inside of the chassis. Dozens of electronic components that had been soldered to the PCBs had broken off and were now lying at the bottom of the chassis. It was a major disaster. It was obvious that the design of the chassis did not meet the contract environment requirements. It was also obvious that the other 39 chassis assemblies sitting on the shelf ready to be delivered to the customer were also unacceptable. An investigation of the failures showed that all the prototype models were tested at room temperature. No one in the expert team had any experience with the properties of viscoelastic damping materials. All the data they had from various sources showed the general material had excellent damping properties. No one in the expert team thought of calling or talking to the various viscoelastic suppliers to get more information on these materials for their severe environments. After all, the expert group had their own test data that showed the viscoelastic material was acceptable for their environment. What else did they have to know? What they

4

Dynamics-Related Failure Modes and Mechanisms did not know was that most viscoelastic materials are extremely sensitive to temperatures. At room temperatures and lower, these materials work very well and can provide good damping. However, at higher temperatures these materials can lose almost all their damping properties and are almost useless. Since all the prototype fabricated test models were tested at room temperatures, the loss of damping at higher temperatures was not observed. No one in the expert group thought of the heat that can be generated when the electronic system was in operation. All the existing finished assembled units ready for delivery were now scrap, a whole new system had to be developed and put into production as quickly as possible, and the new system would still have to survive the severe environment. The company had a contract to deliver a large number of systems at a predetermined price. They had two choices. Go bankrupt or go ahead with a new design and production to meet the contract with their cost of about $25 million. They chose to redesign the unit and to meet the contract with their own money. The company searched for other engineers that had the knowledge and experience to meet the contract requirements. The new engineers were asked if they had ever used viscoelastic damping to improve the reliability of electronics in severe vibration. Their answer was that their experience showed that damping does not work very well, so they prefer not to use those techniques. Instead, the new engineers recommended the use of snubbers to reduce the dynamic displacements of the PCBs in severe vibration and shock conditions. Snubbers are usually made from epoxy fiberglass dowel rods, 0.25 in. in diameter. When several plug-in types of PCBs of the same size are inserted in parallel groups, the dowel rods should be epoxy bonded at the center of every PCB on both sides. The dowel rods on adjacent parallel PCBs should be facing each other with a small gap about 0.005–0.010 in. between facing dowel rods. These snubbers can reduce the dynamic displacements of the PCBs in severe vibration and shock conditions. This then reduces the forces and stresses acting on the electronic components mounted on the PCBs, which substantially increases the fatigue life of the equipment (see [1], Fig. 7.7, p. 160). Snubbers are successfully used in PCBs for air dropping electronic sensors in the ocean at an altitude of 1000 ft. Shock levels of 1000g are produced under these conditions with very few failures.

3

WHAT HAPPENS WHEN 20 PLUG-IN PCBs ARE TIED TOGETHER, THEN INSTALLED IN A CHASSIS THAT IS SUBJECTED TO A 5G PEAK SINE VIBRATION INPUT LEVEL? Another case history that demonstrates types of electronic failures that can be hard to predict involves an electronic enclosure about the size of a shoe-box. It operates in a 5g peak sine vibration environment. Several different types of prototype models were fabricated using different methods for mounting plug-in types of PCBs supported on four sides of their perimeter. Rapid failures occurred with every system that was tried. A single PCB with a bolt added through the center of the plug-in PCB was tested and that worked. The fundamental resonant frequency of this PCB with the center bolt was 425 Hz. The problem was how you clamp the center of 20 plug-in PCBs. The best idea was to drill a quarter-inch-diameter hole through the center of each PCB. A hole was also drilled at both ends of the enclosure. A long steel rod with a slightly smaller diameter could then be inserted through each PCB, so that the rod extended through the holes at each end of the enclosure. The clearance between each PCB was measured, so that a spacer the length of each measured clearance could be cut from an aluminum tube with a hole slightly larger than the steel rod. Each spacer has to be located between each pair of the PCBs. Spacers were also required between both inside end walls

4 Using Snubbers to Increase Fatigue Life of Electronic Systems

5

and the end PCBs. The steel rod can be inserted through each spacer, each hole in the PCB, and the holes at each end of the enclosure. Both ends of the steel rod have screw threads cut so that nuts can be added and tightened. The centers of all the PCBs and the two end-housing walls are now locked together to form a very stiff structure. There are a total of 20 PCBs in the housing, each with a weight of about 1 lb, for a PCB weight of about 20 lb. The enclosure housing was fabricated by using aluminum plates 0.090 in. thick that were dip brazed to form the outer enclosure. The 5g peak sine vibration environment with a resonant dwell at the natural frequency was considered to be severe. Therefore, an extra stiff belly band was brazed to the outer housing that extended 1.0 in. around the perimeter of the housing with 0.38 in. thickness (see [1], Fig. 14.29, p. 374). Vibration tests were run with the system bolted to an oil film slider plate. Before the tests were run, several engineers had questions regarding the safety characteristics of the PCBs all clamped together with spacers. There is a lot of vibration data on PCBs that were free to vibrate at their own individual natural frequencies. There was very little data, however, on vibration of many PCBs with spacers all bolted together. Very few people have experience with the large amounts of damage that can be generated by large masses all vibrating at the natural frequency. Accelerometers were located on PCBs inside the housing, on the outside surfaces of the housing, and on the oil film slider plate. The vibration tests were run in a direction perpendicular to the plane of the PCBs in the housing. The 5g peak sine vibration test was started using a slow sweep and increasing the frequency. Everything appeared normal as the frequency was slowly increased to 100 Hz, then to 200 Hz, then to 300 Hz. At this point the noise level was starting to increase slightly. As the frequency was increased more, the noise level also increased, but at a much higher level, and the people standing near the vibration machine began to back away. As the forcing frequency approached the 425 Hz resonant frequency, it sounded like a railroad train running through the environment test area followed by an explosion as pieces of the aluminum housing split apart. It should have been obvious that when a large number of heavy components are bolted together they will all have the same natural frequency. A large mass vibrating at its natural frequency will have a high kinetic energy that can cause a lot of damage even to a strong structure. A very massive structure has to be designed and fabricated to prevent this type of failure. This type of system should not be used on structures subjected to high vibration and shock because they can have a high failure rate. Properly designed snubbers can be very effective for high vibration and shock levels when used properly.

4

CONSIDER USING SNUBBERS TO INCREASE FATIGUE LIFE OF ELECTRONIC SYSTEMS REQUIRED TO OPERATE IN SEVERE VIBRATION AND SHOCK ENVIRONMENTS One type of electronic system that works very well for severe vibration and shock is to bond snubbers to plug-in types of PCBs. These devices reduce the forces and dynamic displacements of the individual PCBs when they are properly used. This decreases the PCB stress, increases the PCB’s fatigue life, and improves its reliability. Snubbers allow each PCB to vibrate at its own natural frequency. Every PCB will now have a different natural frequency. This prevents the individual PCBs from adding all their kinetic energy together and sharply increasing their failure rates. Snubbers usually use epoxy fiberglass dowel rods about 0.25 in. in diameter epoxy bonded at the center of each plug-in PCB on both faces of every PCB. The snubbers on adjacent PCBs facing each other must have a very small space between them, about 0.008 in., and be aligned so that they impact against each other during vibration, thereby reducing the displacement for improved fatigue life.

6

Dynamics-Related Failure Modes and Mechanisms

5

SAMPLE PROBLEM: CALCULATING FORCES, STRESSES, AND FATIGUE LIFE OF THE END ALUMINUM PLATES IN PREVIOUS HOUSING ENCLOSURE High vibration acceleration “G forces” will often produce high stresses that can reduce the fatigue life of poorly designed structures (see [1], Fig. 14.29, p. 374). In this sample problem the dynamic forces F acting on the end plates are calculated first. The resulting stresses are calculated next. The expected fatigue life is calculated last: F = mass × acceleration = ma, where Mass = weigh/gravity. So w m= g Hence,

and G =

(1) acceleration a = . gravity g

(2)



   w a Force = ma = (a) = (w ) = wG. (3) g g During resonant conditions the transmissibility Q amplification factor for the 20 internal PCBs must be included to obtain the total dynamic force F acting on the full structure. Test data showed that a good approximation for the PCB Q factor for the assembly with an input acceleration level of 5g sine could be approximated by the square root of the expected natural frequency. The natural frequency for the PCBs in this dip-brazed aluminum assembly was about 425 Hz: √ Approximate transmissibility Q = 425 = about 20 (dimensionless) (4) F = wGIN Q = (weight)(G)(Q) = (20)(5)(20) = 2000 pounds on 2 ribs,

(5)

Dynamic force acting on one rib = 1000 lb, Chassis width = 8 in., Half of Chassis width = 8/2 = 4.0 in. The approximate dynamic bending stress on the aluminum end plates of the housing can be obtained using the equation MC Sb . (6) I Here the Bending moment M for one reinforcing rib is given as 1000 (4) = 2000 lb in., (7) M = 2 C = distance from end plate to neutral axis = 0.50 in., (8) I = approximate moment of inertia, one rib = 0.0316 in.4 ,   (2000)(0.50) (9) Sb = = 31, 645 lb/in.2 0.0316 The approximate vibration fatigue life of the 6061 T-4 aluminum dip-brazed chassis can be obtained from [2], as shown below:  b S2 N1 = N2 S1

6 How Displacements Are Related to Frequency and Acceleration so

 N1 = N2

S2 S1

b

 = (1000)

36, 000 31, 645

7

6,4 = 2282 cycles to fail.

(10)

The value b of the exponent was determined for the aluminum structure by using the typical physical properties of the aluminum alloy where the endurance stress is typically one-third of the ultimate tensile stress. A stress concentration factor of 2 was used to compensate for manufacturing tolerances and material properties, to ensure a good fatigue life in vibration and shock environments, without a large increase in the size, weight, and cost. The exponent (b = 6.4) represents the slope of the vibration fatigue curve on a log–log plot. The value of this function was derived in [1] (p. 168, Eq. 8.3). The estimated time to fail can be obtained from the chassis natural frequency of 425 Hz, as shown below:   2282 cycles to fail Time to fail = = 0.089 min (425 cycles/sec)(60 sec/min) = 5.4 sec. (11) The above time to fail is very close to the time of failure from the vibration test.

6

HOW DISPLACEMENTS ARE RELATED TO FREQUENCY AND ACCELERATION A rotating vector is often used to describe the simple harmonic motion of a single spring–mass system. The projection of the rotating vector can be used to describe the vertical displacement Y of the mass at any time compared to the maximum displacement of the mass, Y0 , as the mass moves up and down, as shown in the following equation: Y = Y0 sin t.

(12)

Here the rotation of the vector is  = 2π f , where t is time and f is frequency. The velocity V is the first derivative as shown below: dY (13) = .Y0 cos t. V = dt The acceleration A is the second derivative: d 2Y (14) A = 2 = −2 Y0 sin t. dt The maximum acceleration will occur when sin t is 1: AMAX = 2 Y0 .

(15)

The negative sign indicates the acceleration acts opposite to the displacement direction. The acceleration can be obtained in terms of the gravity units (g) by dividing the maximum acceleration by the acceleration of gravity g = 386 in./sec2 using inches for English units and the acceleration of gravity g = 980 cm/sec2 using centimeters for metric units. The radians are changed into cycles per second as  = 2π f and substituted into Eq. (15): ⎧ 2 Y0 4π 2 f 2 Y0 f 2 Y0 AMAX ⎪ ⎪ ⎪ = = = (English units, in.), (16) ⎨ g g 386 9.8 G= ⎪ AMAX 2 Y0 4π 2 f 2 Y0 f 2 Y0 ⎪ ⎪ ⎩ = = = (metric units, cm). (17) g g 980 24.9 It is convenient to change the displacement reference Y0 to ZSA for the single-amplitude displacement. Then, solving for the maximum single-angle displacement for the English and

8

Dynamics-Related Failure Modes and Mechanisms the metric units gives the values

ZSA

⎧ 9.8G ⎪ ⎪ ⎨ f2 = ⎪ 24.9G ⎪ ⎩ f2

(English units, in.), (18) (metric units, cm).

When the input G level is used, the above equations will give the input displacement Z . When the response (or output) G level is used, the above equations will give the output displacement Z usinsg the proper displacements in inches or in centimeters. The amplification factor, which is called the transmissibility or the Q factor, can often magnify the input acceleration G level to the outer housing and also to any electronics that are mounted inside the housing. These Q factors can magnify the input acceleration G levels to the outer housing and the internal electronics by values as high as 10 or 20 or more; this can cause extensive damage and rapid structural failures. The transmissibility Q is defined as the ratio of the output divided by the input. This ratio can be in terms of the acceleration G levels, or displacements, or velocities, or forces. One of the most important equations in dynamics is ⎧ 9.8GIN Q ⎪ ⎪ (English units), ⎪ ⎨ fN2 (19) ZSA = ⎪ 24.9GIN Q ⎪ ⎪ (metric units) ⎩ fN2 where: Z SA = displacement, single amplitude, zero to peak, English units = inches, metric units = centimeters G IN = input acceleration G level, dimensionless f N = frequency or natural frequency, Hz Q = transmissibility, dimensionless ratio of output/input The above equations can be written in another way to demonstrate the effect of using a very low input G level for a vibration test in order to prevent damaging the hardware. In English units, f 2 ZSA . (20) Q= N 9.8GIN Equation (20) shows that if a very low input acceleration G is used for a sine vibration test (usually to prevent damage) on any nonrigid structure such as a PCB, the transmissibility Q can become very high. For example, test data show that, if an input acceleration level of 0.2g peak is used on a PCB with a natural frequency of 250 Hz, the response Q level of the PCB will be about 70, depending upon the damping available in the construction of the PCB. Transmissibility Q values of 70 will frighten many people who are not familiar with vibration. They will multiply the response Q of 70 × their 5G peak input required for the qualification test. This will give them a value of 350g peak. They are shocked because acceleration G levels this high will destroy the system very quickly. These people do not know that if an input acceleration level of 5g peak is used, test data shows the response Q level of the PCB will only be about 16, depending upon the damping. The higher 5g sine input acceleration level increases the force, stress, and displacement in the PCB. This will increase the PCB damping, which will reduce the transmissibility Q level. The new PCB response is now about 16 × 5 = 80g peak. This is still a high acceleration level that may require the use of an isolation system or some structural reinforcement, or perhaps snubbers, to survive lengthy qualification tests. This also shows that the input acceleration G level, even for preliminary sine vibration tests, must be based upon

8

Resonance Coupling of Outer Housing Enclosure with Internal PCBs

9

the acceleration G levels expected in the qualification test program. This will assure the reliability of the electronic assembly so it will be able to pass the qualification test program without any failures. Equation (20) also shows that a higher frequency will always produce a higher transmissibility Q because the displacement Z and the acceleration G level are linear functions. However, the frequency is a square function so it increases at a more rapid rate. Equation (18) shows that the displacement, acceleration G, and frequency cannot be separated. They are locked together. Any two parameters determine the third parameter. Equation (18) also shows that increasing the natural frequency will result in a substantial reduction in the dynamic displacement because the frequency is a square function. Equation (18) shows that an increase in the acceleration G level will also increase the dynamic displacement if the frequency is held constant.

7

SAMPLE PROBLEM: FIND DYNAMIC DISPLACEMENT OF PCB EXPOSED TO SINE VIBRATION USING ENGLISH UNITS AND METRIC UNITS A PCB supported on two opposite sides has a natural frequency of 225 Hz and a transmissibility Q of about 15. Find the single-amplitude displacement at its natural frequency using a sine wave with an input level of 5g peak using Eq. (18): ⎧ (9.8)(5)(15) ⎪ ⎪ : 0.0145 in. (English units), (21) ⎨ (225)2 ZSA = ⎪ (24.9)(5)(15) ⎪ ⎩ = 0.0368 cm (metric units). (22) (225)2

8

OCTAVE RULE CAN AVOID VIBRATION FAILURES DUE TO RESONANCE COUPLING OF OUTER HOUSING ENCLOSURE WITH INTERNAL PCBs Many electronic control systems are required to operate in severe vibration, shock, and humidity conditions, so sensitive PCBs are often mounted inside an enclosed housing for protection. Housings must be strong enough to withstand severe environments. Consider a relatively simple system where several plug-in types of PCBs are in an enclosure to provide protection from harsh outside environments. Vibration and shock tests are to be run so small accelerometers are mounted in several areas on the outside of the housing and on several PCBs inside the housing. These will record the acceleration levels observed at various points outside the housing and on the components inside the housing. The housing will have a fundamental (first) natural frequency which can be recorded and the various PCBs will typically have their individual fundamental natural frequencies. It is very important to avoid conditions where the housing natural frequency is very close to the natural frequency of any of the PCBs mounted inside the housing. This can cause a resonance coupling effect where the housing resonance multiplies the PCB resonance that can produce failures in some of the PCBs. When the peak sine vibration input acceleration level is 5g and the housing Q value is about 4, the response of the housing at its natural frequency will be 5 × 4, or 20g peak. Acceleration levels this high may cause some failures in the housing structure and possible failures to some PCBs, depending upon the natural frequency of the PCBs. The same conditions must also be examined for the PCBs inside the outer housing when the assembly is exposed to a sine vibration input level of 5g peak. The internal PCBs have their own individual natural frequencies that can be excited during exposure to the sine vibration imposed on the outer housing because the PCBs are attached to the inside of the housing. The PCBs can amplify the

10

Dynamics-Related Failure Modes and Mechanisms housing acceleration G levels when the PCB natural frequencies are excited. Amplification Q factors for each of the PCBs will also depend upon the damping of the various PCBs. Vibration test data show that the PCBs inside the housing can further amplify the acceleration levels they receive from the outer housing. When the natural frequency of the housing is close to the natural frequency of the PCBs, their Q’s can couple. This will increase the acceleration levels on the PCBs. Test data show the transmissibility Q values of the PCBs and the housings are multiplied. They are not added. For example, when the peak input sine level to the housing is 5g, and the transmissibility Q for the housing at its natural frequency is 4, the transmissibility Q value for the housing at its natural frequency will be about 5 × 4, or 20. When the transmissibility Q of the typical PCB inside the housing is about 9, and the natural frequency of the PCBs are very close to the natural frequency of the housing, the resulting transmissibility Q of the housing will couple with the Q of the PCBs. The typical PCB Q will then be about 5 × 4 × 9 = 180g peak. The actual peak acceleration levels will be somewhat lower because the high-acceleration G levels will increase the dynamic displacements of the PCBs. This will increase the stress levels in the PCBs and convert more strain energy into heat, which increases the damping in the PCBs. This will slightly reduce the transmissibility Q acting on the PCBs, which will reduce the forces acting on the PCBs slightly. However, the reduced G forces in this general range can still do a lot of damage very quickly, even to rugged electronic systems. Sine vibration-induced failures in electronic systems can be reduced by using the “octave rule.” “Octave” means to double. In this case it means that the natural frequency of the outer housing structure must be an octave (factor-of-2) away from the natural frequency of the PCBs mounted inside the housing. Test data show that when the natural frequencies of the outer housing and the internal PCBs are separated by a factor of at least 2, the magnitude of the coupling forces between the housing and the PCBs are sharply reduced. This substantially increases the fatigue life of the electronic system by reducing the internal forces and stresses. The outer housing and the internal PCBs will have several different natural frequencies. The first natural frequency usually has the highest Q value. Vibration tests on prototype models have to be made to ensure the most critical combinations of housing frequencies and PCB frequencies are used to follow the octave rule for the minimum forces acting on the internal PCBs. The dynamic force path through the structural assembly should be examined closely to make sure that the octave rule is being followed properly. For example, when the fully assembled electronic system is being vibrated, the exciting force first acts on the outer housing. The outer housing then becomes the first degree of freedom. The force path then goes from the outer housing to the internal PCBs that are attached to the housing. The internal PCBs then become the second degree of freedom. The natural frequency of the first-degree-of-freedom structure (the housing) must be at least one octave (a-factor-of-2) or more away from the natural frequency of the second degree of freedom (the PCBs). This will avoid severe coupling that can generate high forces and stresses that can reduce the fatigue life of the housing and the PCBs. This means that if the lowest natural frequency of the typical internal PCB is about 200 Hz, the lowest natural frequency of the external housing should be about 100 Hz (which is half of 200 Hz on the PCB), or the natural frequency of the housing should be about 400 Hz (which is 2 times the 200 Hz on the PCB) to follow the octave rule. In the case just described, the housing natural frequency should be either slightly below 100 Hz or slightly higher than 400 Hz to follow the octave rule. Housing natural frequencies between 100 and 400 Hz for this specific condition should be avoided, if possible, to ensure a good fatigue life for the PCBs mounted inside the housing. This means that the internal PCB natural frequency design must be carefully separated from the housing natural frequency. This will avoid close resonances between the PCBs and the housing that can produce rapid failures in the PCBs during vibration conditions. It should be noted that the natural frequency of the PCBs and the

10

Vibration Failures due to Connector Fretting Corrosion in Random Vibration

11

natural frequency of the outer housing refers to their first resonant mode, which is usually the lowest natural frequency of the PCBs and the lowest natural frequency of the housing. Shock is not as important in this application because the shock environment very seldom experiences as many stress reversal cycles as vibration in most applications. Shocks usually produce much higher acceleration G levels than sine vibration, which can also produce rapid structural failures in the outer housing and the internal PCBs.

9

ANOTHER APPLICATION WHERE SNUBBERS CAN BE USED TO IMPROVE FATIGUE LIFE OF PCBs IN SEVERE VIBRATION AND SHOCK ENVIRONMENTS The octave rule is often difficult to implement because of possible size and weight limitations and lack of information relating to the transmissibility Q values of the structural elements or the natural frequencies and these structural members. Under these conditions the use of snubbers is highly recommended. The best snubbers are made from epoxy fiberglass dowel rods about 0.25 in. in diameter. They should be epoxy bonded near the center of every circuit board on both opposite faces of the circuit board. These snubbers must be positioned on each circuit board and aligned so the snubbers on adjacent PCBs impact against each other, which will reduce their dynamic displacements. The spacing between the snubbers on adjacent PCBs must be small, typically less than about 0.010 in., so that they will impact against each other at each PCB resonant frequency. This also reduces the PCB dynamic stresses in the electronic components and their electrical lead wires and solder joints, which improves the reliability of the electronic system. Some sine vibration testing may have to be done with the snubbers to make sure the spacing forces snubbers to impact against each other. Another method for reducing large dynamic displacements and stresses in the PCBs is to just fill all the empty spaces in the enclosure, between all of the PCBs, after their final assembly using small, lightweight spheres. These look like small hollow ping pong balls that add very little weight to the electronic assembly. These small spheres can be added at the final assembly while it is being vibrated at a very low frequency to allow the spheres to settle in place.

10

VIBRATION FAILURES DUE TO CONNECTOR FRETTING CORROSION IN RANDOM VIBRATION A wide variety of electrical connectors are being used in a very large number of different applications, in many different types of industries, and in many countries all over the world. These applications vary from the very simple devices to the very complex devices. Some applications monitor sensitive stationary conditions and some monitor rapidly changing conditions. Most connector applications involve the transfer of some type of analog or digital data from one point to another point. Sometimes the data being transferred can be very critical and sometimes not really important. When the data being carried are very important, then it is necessary to understand the characteristics of the connector that will be used to transfer the critical data, and the operating environment, in order to be sure the data will be accurate. Extensive test data on many different types of connectors operating in different types of vibration and shock conditions have shown some strange results. A series of tests were run with PCBs that had multiple plug-in pin types of NAFI flat-blade and tuning fork connectors. Each connector blade had two electrical contacts, one on each face of the blade. All the electrical brass contact faces were protected with 30 millionths of gold over 150 millionths of nickel over the brass connectors. The tests were run using a 5g root-mean-square (rms) random vibration input with a PCB resonant frequency of about 150 Hz. This showed what appeared to be open circuits (called “glitches”). Each glitch lasted about 1 μsec. The first

12

Dynamics-Related Failure Modes and Mechanisms glitch was observed after 15 or 20 min of testing. No one was worried. All went well for the next 10 min, then another glitch was observed. Now the engineers began to worry. In a high-speed digital system a lot of information can be lost if there is an open circuit. Again all went well for another 5 min, then another glitch was observed. As the tests continued, the glitches were observed to occur more often at closer intervals. The tests were stopped and the housing was opened so the PCB solder joints, the connectors, and the cables could be inspected for problem areas. Everything looked normal so the housing was assembled and the vibration started again. The system looked normal for a few minutes, then the glitching started again. Every time the electrical system was tested without the imposed vibration, the system worked perfectly normal. Every time the tests were run with the imposed random vibration, the glitching occurred. A mechanical engineer from another group walked up to the test engineers and asked them if they checked the plug-in connectors on the PCBs to see if the protective layers of gold over nickel over brass were worn away. The test engineers were very upset by this outside engineer asking questions and they wanted him to leave. The test engineers said they checked the gold coatings and they looked good. The outside engineer walked over to the housing and wiped his finger across the inside surface and showed a finger with a fine black powder to the test engineers. He said his black finger was from the black copper oxide generated by the heat of a very rapid weld shear action because the nickel and the gold protection on the contact pins was worn away. They did not believe him when he said polished brass looks like gold. He reached into a box and removed a small bottle that contained a 5% solution of sodium sulfide. He brushed this solution on every connector on every PCB. A small black spot appeared on every connector contact in about 7 sec. This showed there was copper present and that the gold-over-nickel protection on the contacts had been worn away by the rapid 150-Hz oscillations generated by the vibration. Some care must be used when purchases are made of the sodium sulfide solution for testing the presence of any copper alloys. Many companies sell a 3% solution of sodium sulfide. This solution is not strong enough to detect the presence of any copper-based alloy. The minimum solution strength of 5% sodium sulfide must be used to detect the presence of a copper-based alloy. The test engineers did not understand why the glitches were occurring. It was pointed out that each connector pin had two points of contact for improved reliability. When one contact point on the pin failed, there was always a second contact point that could pick up and transmit the same signal to ensure the reliability of the signal and the system. The fine black powder is an electrical insulator. During vibration, the fine black powder can be deposited on both faces of one flat connector pin. The black powder can creep in between some of the pins and the sockets on the plug-in PCBs, producing an open circuit for a 1-μs glitch. Long vibration periods can accumulate such a large amount of black powder that it coats the entire inside surface of the housing. This problem can be solved by using connectors with pins that have more than two points of contact. Tests with connectors that had four or more contact points on each pin for the electrical contacts did not show any glitching when the vibration was imposed. The normal fretting corrosion still occurred and the fine black powder was still there. However, the probability of all four points on a pin having the black powder get under all four points at the same time and causing an open circuit was shown by the tests to be almost impossible. No open-circuit glitches were observed after several hours of random vibration using an input level of 5g rms. Many of the latest Air Force airplanes have had similar glitching problems with oldtype connectors on new electronic equipment. Changes were made using the new Bendix Bristle Brush (B cube) connectors with redundant pins that have about 12 points of electrical contact to solve the problem. The Smith Hypertronix connector is also popular for solving the glitching problem. This connector has about eight points of electrical contact for each pin. The fine black powder will still be created with these connectors, but it will not cause any glitching with open circuits with the use of random vibration, since these connectors have

11 Why Some Failures May Be Difficult to Solve or May Never Be Solved

13

more than four points of electrical contacts on each pin. They have been used to avoid the glitching open-circuit problems during vibration when the resonant frequency is excited. This can impose millions of stress cycles that can wear through the gold-over-nickel protection for the copper-based alloy connectors, but this will not affect the reliability of these connectors, even in harsh environments. The fretting corrosion and open circuits only appear to occur with random vibration during extensive exposure to long periods of high-vibration G rms levels. This can produce many million stress cycles that can cause extensive wear on the gold and nickel protective coatings over the connector pins. Test data using sine vibration with a greater equivalent energy input acceleration peak level did not cause any problems with glitching or open circuits or the fine black powder being developed. The rapidly changing random vibration frequency and amplitude, over the very wide bandwidth, sharply increases the number of critical stress cycles imposed at different frequencies. This increases the wear through the gold and nickel connector protective coating and decreases the fatigue life of the electronic assembly.

11

WHY SOME FAILURES MAY BE DIFFICULT TO SOLVE OR MAY NEVER BE SOLVED When groups of people work together on the same project, there is usually a feeling of friendship among the workers. There are times when failures will occur during the testing of an important new prototype electronic assembly. People with the most knowledge and experience will usually be assigned to examine the assembly to try to find the source of the failure and to make recommendations for some corrective action. Everyone in the group appears to be willing to help solve the problem as quickly as possible. Time is money, so that a quick solution to the problem is desired. Under these conditions no one at this time would believe that the failure could have been caused by one of their own friends. An investigation of the failure might show something out of the ordinary, like a bent pin on a connector, or a loose wire, or a cracked solder joint in a strange place. These do not look like natural failures. However, after repairs have been made and tests are resumed, more failures in different areas appear. Some people may begin to wonder if the failures are natural or man made. It can be very difficult to separate real failures from deliberate man-made failures, depending upon the experience and skills of the people involved. Several years ago an engineer was working on a program where the vibration levels were quite high, with a requirement for a lightweight system. A qualification test was set up using several prototype models to evaluate proposed designs. Natural frequencies with calculations of the forces, stresses, and expected fatigue life of critical structures were made before the vibration test program to ensure a reliable design. The vibration tests on some of the models were very good. Some models showed a rapid fatigue failure in one of the structural elements that was used on all of the prototype models. This was very strange because the calculations showed this structural element was expected to be very reliable. The structural element involved was a steel shaft about an inch long with a diameter of 0.50 in. One end was machined down to a diameter of about 0.25 in. over a length of about 0.50 in. to fit a bearing. Several failures occurred at the step in the shaft where the shaft diameter changed from 0.50 to 0.25 in. High stresses were expected at the step because of high stress concentrations, so that a generous fillet was used at the step to reduce these stresses. One of the engineers suspected there might be a problem with cutting the step in the shaft using a single cutter on a lathe instead of three cutters at 120◦ apart. A single cutter would apply a high concentrated force at the end of the shaft which could produce cracks at the step and generate more rapid failures. He went to the machine shop and spoke to the manager of the department about a single cutter. The manager became very angry at the idea and insisted that he personally directed every machinist to use three cutters on the shafts. The manager

14

Dynamics-Related Failure Modes and Mechanisms then insisted that the engineer leave his shop immediately, because the failures were due to the engineer’s poor design. The engineer went back to his calculations that showed the shafts should not fail. He went to the machine shop again and waited until the manager left the area. The engineer then took a walk through the small machine shop to see for himself how the shafts were being cut. He saw one of the shafts being cut with a single cutter. The manager saw the engineer and demanded that he leave the machine shop immediately. The engineer did not move. Instead he pointed to the operation that was using a single cutter. The manager looked at the single-cutter operation and his face turned white. He yelled at the machinist, asking him why he was using a single cutter when he was told to use three cutters. The answer came back that he could not make his quota using three cutters because it took much more time to make the adjustments. The single cutter was much faster and the results were just as good. The engineer was lucky because the machine shop was operating two shifts. With a little bad luck he would have completely missed seeing the operations using one cutter so he would never know the real reason for the failures. He would have less confidence in the accuracy of his calculations. This also points out the poor practice of some managers. They give instructions to people but never explain why the instructions are important and they never take the time to follow up to see that these people are really following his instructions.

12

COMPANIES WITH FINANCIAL PROBLEMS MAY REDUCE QUALITY TO SAVE MONEY Companies often try to find ways to improve their profit margins when their business slows down. An example of this involved the temper of aluminum wedge clamps that were specified as 6061-T6 for plug-in PCBs to operate in severe environments. These conditions involved vibration, shock, and thermal cycling. Many electronic systems had early failures due to loose wedge clamps that were blamed on the manufacturing group. The manufacturing group said they always inserted the PCBs in the guides at the sides of the housing; then they tightened the screws using the proper torque until the PCBs were locked in position. There must be another reason for the failures. The engineering group looked and looked for other possible reasons for the failures, but they could not find any. They were about to go back to the manufacturing group again when one engineer took another look at the three-piece wedge clamps on both sides of the PCB. These centerpieces were very long and they were attached to the sides of the PCB. The centerpiece had a 45◦ wedge at each end. The top and bottom end pieces were very short. They each had one 45◦ wedge on one face. The 45◦ wedges on the two short pieces interfaced with the 45◦ wedges on the long centerpiece. The small bottom wedges had threads for a long screw. The center wedge and the top wedge had clearance holes for the long screw. When the screw is tightened at the top wedge, the bottom wedge rides up the 45◦ angle on the centerpiece and the top wedge rides down the 45◦ angle on the centerpiece. This forces the top and bottom wedges to expand and lock the PCB in position against the side guides on the PCB. The engineer noticed a small notch near the bottom of the center wedge and a similar small notch near the top of the bottom wedge. These two notches appeared to be very close to each other. The engineer turned the screw at the top of the wedge and watched the bottom wedge rotate and start to ride up the centerpiece. He continued to turn the screw to the required torque and noticed that the bottom wedge did not fully rotate and did not properly ride up the centerpiece. The small notch on the half-rotated 45◦ wedge on the bottom part was now locked on the small notch on the centerpiece. The proper torque had been applied to the screw for locking the PCB to the housing, but the PCB was still loose. During normal operation the heat from the PCB must be carried away to prevent the PCB from overheating and failing. The heat must be conducted across the interface from the wedge clamps on the PCB to the housing metal side walls. This carries the heat from the PCB to the local ambient

13

Why Some People Will Ship Electronic Equipment They Know Will Fail

15

outside the housing so the PCB stays cool. If the PCB wedge clamps are loose, there will be air gaps between the PCB and the housing. Air does not conduct heat very well so the PCB temperature rises and the PCB fails. Wedge clamps have been used for many years without any failures. Small notches in the wedge clamps mean the aluminum metal must be softer than 6061-T6. The engineer calls the wedge clamp company to verify if the wedges are using the proper heat-treated aluminum. The company insists his wedge clamps have the proper hardness. The wedge clamps are then sent to three outside testing groups to verify the temper of the aluminum. The reports show a temper of only T-4, which shows the aluminum is too soft and is causing the failures. The engineer calls the wedge clamp company and reports the outside tests show a temper of only T-4 which is causing the failures. The company again insists his temper is T-6 and the other outside testing groups do not know how to run proper tests. The engineer goes to his purchasing department and tries to cancel the contract and go to a different company to purchase the wedge clamps with the proper T-6 hardness. He is told the contract cannot be canceled because this owner of the wedge clamp company is a relative of the president of the engineer’s company. The engineer must find a way to use the T-4 temper aluminum wedge clamp so they do not fail in harsh environments. Every idea he has is not practical because it requires a redesign and a large increase in the cost with a long time delay in the delivery of the hardware. He finally remembers that aluminum can be anodized with a thin coating that is very hard because it is a ceramic, with a modulus of elasticity that is three times higher than the aluminum. He also finds the process will only cost about one dollar for each wedge clamp. This solves the problem with no more loose PCBs and no more failures.

13

WHY SOME PEOPLE WILL SHIP ELECTRONIC EQUIPMENT THEY KNOW WILL FAIL JUST TO GET THEIR SHIPPING BONUS Many companies often reward their employees with extra bonus money if they ship some equipment ahead of schedule. These companies always assume their employees are very honest so they would never ship any equipment they know is bad and will fail quickly. In this particular investigation, a large well-known Midwest company always performed an environmental stress screening (ESS) on its electronic equipment before shipping to ensure a reliable product. The purpose of the ESS is to stimulate the electronic equipment for a short period of time in some way that will force the most critical hidden flaws into hard failures. The equipment can then be repaired before it is shipped so the customer will receive a very reliable product with a long fatigue life. The stimulus is often called a bake-and-shake test. This covers about a dozen thermal cycles over a broad temperature range and several minutes of a broadband random vibration test, each performed separately while the equipment is electrically operating. The problem in this case was that a 50-pin ceramic microprocessor about 2 in. long kept failing after only a few minutes. The solder joints were failing in the thermal cycling tests and the lead wires were failing in the random vibration tests. Surface-mounted lead wires were tried and through-hole lead wires were tried with the microprocessor with no success. An engineer made an analysis of the solder joint and lead wire forces and stresses that showed the failures would not occur if the wires were made more flexible and more compliant. This could be done by forming an S shape in the lead wires with a special forming die. The production people said that could not be done. Instead they wanted to epoxy bond a nickel steel shim under the component to stiffen the PCB near the area of the lead wires. This might work if the shim could be attached very rigidly to the PCB. The production people wanted to simply apply the epoxy adhesive to the shim and place it on the PCB under the component. The engineer said that will not work because the nickel steel shim is very smooth so the epoxy bond will not hold. The engineer wanted to drill about 10 small holes through the shim so the epoxy

16

Dynamics-Related Failure Modes and Mechanisms adhesive will flow through the holes and act like rivets. This arrangement can carry very high shear forces without failing. This idea fell on deaf ears as the production people went ahead with their method. The epoxy bond failed very rapidly. The production people decided to solve the problem another way, by simply leaving the microprocessor off of the PCB and sealing the assembly. They then simply ran reduced ESS thermal and vibration tests without the microprocessor. The seals were then broken and the microprocessor was hand soldered back on the PCB. The new systems were then shipped. The upper management people were never aware of what was being done in the production area. The engineer overheard some production people talking and laughing about the new production shipments. The engineer ran to the people in the production area demanding to know why they shipped defective hardware. They said, “If we do not ship, we do not get our bonus.” The engineer was shocked. He did not know how many different organizations were buying this defective equipment. He was going to blow the whistle and inform upper management what was going on without their knowledge. He spoke to a lawyer who advised him to keep his mouth closed. The lawyer said that whistle blowers always lose. You are only one against a dozen other people who will claim you are trying to get even with them because they did not listen to your advice. You will probably be the one that gets fired. Do not say anything to anyone. Do not write anything about this. The engineer kept quiet. The sad part of this situation was that some of this electronic equipment was for military use, which could affect the lives of soldiers. The good part of this situation was when the company decided to outsource the production of this piece of electronics to a job shop. A short time later all of the production people that were involved with the shipping of the defective electronics were laid off.

14

CAN VIBRATION ALONE PRODUCE A V-SHAPED DEEP HOLE IN A 65-POUND ALUMINUM CASTING 12 INCHES IN DIAMETER THAT IS 0.25 INCH THICK? Three graduate engineers wheeled a cart to the office of an engineering manager who had extensive testing experience. The cart contained a 65-lb cast aluminum sphere about 12 in. in diameter with a wall thickness of 0.25 in. and full of electronics. The three visiting engineers wanted to know how vibration produced a deep V-shaped hole in the aluminum casting 0.25 in. thick. The manager took one look at the hole in the casting and said the hole was not caused by vibration. It was produced by a high impact, probably caused by dripping it on the floor, in an area like a machine shop where there are often some loose bolts and nuts lying on the floor. The visiting engineers said they had proof that the hole in the aluminum sphere was caused by vibration and not by being dropped. When they were asked for the proof, they said they questioned the test people who maintained there was no hole in the casting before the vibration test started. The hole in the casting was noticed after the vibration test was completed. Therefore, the hole in the casting has to somehow be related to the vibration. The manager laughed and said that the only other way the hole could have been caused was by someone hitting the casting with a great deal of force using a ball peen hammer. Anyone with a little testing experience associated with vibration would come to the same conclusion. It was obvious that the testing people were lying about the hole in the casting to protect their friends who dropped the casting. There was the fear that they might have to pay for the damage to the casting assembly, which cost about $150,000. The most disturbing part of this investigation was that three graduate engineers, each with several years of experience, could not tell the difference between a vibration-induced crack in a casting and a hole in the thick wall of the casting that could only be caused by a single heavy impact.

16 Structural Dynamics and Dynamic Stability of Systems 17

15

FAILURE MODES IN OPTO-ELECTRONIC FIBER-OPTIC SYSTEMS RELATED TO STRUCTURAL DYNAMICS Optical fibers are being used in fiber-optic communication systems because they are very efficient for transmitting data over long distances. They permit higher data rates than the wired or wireless forms of communication with cables. Two common types of systems are the multimode fiber optics and the single-mode fiber optics, often called the monomode fiber. The glass optical fibers are usually made from silica. Plastic optical fibers (pof) are also used, but they have a much higher attenuation than the glass fibers so they are not used for long-distance communications. Protective plastic jacket layers of a tough resin are applied to the outer glass fibers. This adds strength to the brittle glass fiber cable which allows it to have a long fatigue life so it can be used in severe environments such as vibration and shock. These reinforced cables are also used extensively in areas where extra protection is needed from animals that chew through the cables. Underwater applications also work very well for protection against sharks. Some heavy-duty cables can have as many as 1000 fibers assembled in a single cable. These optical fibers are often connected to networks by splicing or fusion. Here the fiber ends are melted and welded together with an electric arc. The ends must be very closely aligned to get good performance. Opto-electronic systems make extensive use of optical fibers that are made of very thin glass, which is brittle. When these materials are used in dynamic environments, their displacements must be kept very small to avoid cracking the glass. The most critical areas are at the termination points at the beginning and at the end of the fiber-optic cable. These areas must be reinforced so they are stiff enough to reduce the bending and the failures in the glass fibers where the glass fibers are joined to the operating system using plastic jackets or fusion methods. These connecting joints must be supported to prevent relative motion in the thin glass fibers. The individual glass fibers can be fastened to an epoxy fiberglass pad about 0.12 in. thick that is wide enough and long enough to support all of the glass fibers. The fibers can be bonded to the epoxy pad using a semirigid room temperature vulcanized (RTV) material for support, so repairs can be made if necessary. The epoxy fiberglass pad itself may have to be bonded to a rigid support to prevent excessive motion if the fiber optics will be exposed to severe environments that can produce these effects. The safe displacement levels can be established by testing the cables in environments similar to the expected operating conditions to ensure the reliability of the cables.

16

HOW ELECTRONICS ARE BEING USED TO CONTROL STRUCTURAL DYNAMICS AND DYNAMIC STABILITY OF SYSTEMS FOR IMPROVED RELIABILITY AND SAFETY New electronic components are rapidly shrinking in size while they are expanding their functional capabilities at the same time. This has resulted in the development of many new applications for control systems that were not available before. This new technology has been given many different names. One of the most popular names is electronic stability control (ESC). This new technology is being applied to many different industries for use in industrial, commercial, and military applications such as homes, hospitals, construction, manufacturing, and many other areas. Improvements associated with this new technology are reduced costs, improved safety, performance, reliability, stability, and control of many different types of mechanical devices. This includes automobiles, airplanes, helicopters, trains, trucks, and missiles, just to name a few. The automobile area is very interesting in the use of special sensors that can prevent accidents and injuries by controlling skids, yaw, steering,

18

Dynamics-Related Failure Modes and Mechanisms poor tires, improper air pressure in a tire, direct fuel to different engine cylinders, traction, rollover, automatic reduction in speed in unsafe conditions, and many other safety functions. Airplanes are also interesting. The B2 flying wing cannot be flown manually because there is no tail to control stability, which is controlled by sensors. The latest fighter aircraft, such as the Lockheed F-22 and F-35, are designed to be unstable and cannot be flown by hand. The sensors control all the flight functions so the airplane can instantly change its flight maneuvers for a snap roll or dive or a quick turn. A stable airplane will lose a couple of seconds, which can mean the difference between life and death in combat.

REFERENCES 1. Steinberg, D. S., Vibration Analysis for Electronic Equipment, 3rd ed., Wiley, New York, 2000. 2. Crandall, S., Random Vibration, Wiley, New York.

CHAPTER

2

LINEAR RESPONSE TO SHOCKS AND VIBRATIONS Ephraim Suhir University of California, Santa Cruz, California University of Maryland, College Park, Maryland ERS Co., Los Altos, California

In the analyses that follow we address some simple and basic problems of the linear response of structural elements encountered in electronic and photonic engineering to shocks and vibrations. See also Chapters 4, 16, 18, 19.

1

SINGLE-DEGREE-OF-FREEDOM (SDOF) SYSTEM Consider the simplest single-degree-of-freedom (SDOF) system m y(t) ¨ + c y(t) ˙ + ky(t) = F (t)

(1)

subjected to an arbitrary excitation F(t). The system consists of the concentrated mass m, a spring with the spring constant k, and a source of viscous damping c (Fig. 1). Let y(t) be the deviation of the mass m from the position of static equilibrium. This position is defined as such, where the weight P = mg of the mass m is equilibrated by the reaction ky(t) of the spring (“restoring force”) and the external (“excitation”) force F(t) is equal to zero. When the deviation y(t) is not zero, the system becomes subjected to the following forces: (1) inertia force m y(t), ¨ whose direction is always opposite to the direction of the acceleration y(t); ¨ (2) damping force c y(t), ˙ whose direction is always opposite to the direction of the velocity y(t); ˙ (3) restoring force ky(t), whose direction is always opposite to the direction of the displacement y(t); and (4) excitation force F (t). The equation of equilibrium (1) is written in accordance with d’Alembert’s principle in dynamics: The equilibrium condition can be obtained by equating to zero all the acting forces, including the force of inertia. Equation (1) can be rewritten as F (t) y¨ (t) + 2r y˙ (t) + ω02 y(t) = , (2) m where the following notation is used:  k c , ω0 = . (3) r= 2m m Equation (2) is an inhomogeneous ordinary differential equation of the second order. In Chapter 4 we obtain a solution to such an equation using theory of spectra. In this chapter we use a much simpler approach based on the more or less elementary theory of differential equations and seek the general solution to the inhomogeneous equation (2) as the sum of the general solution to the homogeneous equation, y(t) ¨ + 2r y(t) ˙ + ω02 y(t) = 0,

(4)

and the particular solution yp.s. (t) to the inhomogeneous equation (2). Equation (4) can be obtained from Eq. (2) by simply putting its right part (excitation force) equal to zero. Equation (4) Structural Dynamics of Electronic and Photonic Systems Copyright © 2011 John Wiley & Sons, Inc.

Edited by Ephraim Suhir, David S. Steinberg and T. X. Yu

19

20

Linear Response to Shocks and Vibrations F k

c

y m

Figure 1 A single-degree-of-freedom system subjected to external excitation. describes free vibrations of the system. These vibrations can be triggered by either displacing the system from the equilibrium position by introducing the initial displacement y(0) = y0 and/or by applying to the system an instantaneous impulse s = m y(0) ˙ = m y˙0 , that is, by applying an initial velocity y˙0 . The biquadratic characteristic equation z 2 + 2rz + ω02 = 0

(5)

of the differential equation (4) has the following two roots: where i =

√

z = −r ± i ω∗ ,

(6)

−1 is the “imaginary unity” and  ω∗ = ω02 − r 2

(7)

is the frequency of free vibrations, with consideration of the effect of damping. If this frequency is positive, that is, if damping is small, then vibrations take place. If not, the solution to Eq. (4) is expressed through exponential functions, and the corresponding solution describes nonperiodic motion. The solution to Eq. (4) is y(t) = e −rt (C0 cos ω∗ t + C1 sin ω∗ t),

(8)

where C0 and C1 are constants of integration. These constants can be found from the initial conditions y(0) = y0 , y(0) ˙ = y˙0 , (9) so that C0 = y0 ,

C1 =

y˙0 + ry0 . ω∗

(10)

Then solution (8) yields   y˙0 + ry0 sin ω∗ t = A0 e −rt cos(ω∗ t + ϕ0 ), (11) y(t) = e −rt y0 cos ω∗ t + ω∗ where    y˙0 + ry0 2 2 (12) A0 = y0 + ω∗ is the initial amplitude of damped vibrations, ϕ = ω∗ t + ϕ0 is the phase of vibrations, and ϕ0 is the initial phase of vibrations. As to the particular solution to the inhomogeneous equation (2) it can be obtained using the following Duhamel integral (see also Chapter 4)  t 1 e −r(t−τ ) F (τ ) sin[ω∗ (t − τ )] d τ. (13) yp.s. (t) = mω∗ 0

2 SDOF System Subjected to Harmonic Excitation

21

The rationale underlying formula (13) can be found in Section 4, Chapter 4. Solution (13) corresponds to zero initial conditions. The general solution to Eq. (2) is as follows:    t y˙0 + ry0 1 yp.s. (t) = e −rt y0 cos ω∗ t + sin ω∗ t + e −r(t−τ ) F (τ ) sin[ω∗ (t − τ )] d τ. ω∗ mω∗ 0 (14) This solution determines the response of a SDOF system with arbitrary initial conditions y0 and y˙0 to an arbitrary excitation force F (t).

2

SDOF SYSTEM SUBJECTED TO HARMONIC EXCITATION In the analysis that follows we address the response of a SDOF dynamic system to a harmonic excitation with an arbitrary frequency ω. A similar situation was examined in Chapter 4 based on the theory of spectra. In this section we use more or less elementary considerations to solve the problem in question. Thus we examine the following system: F0 sin(ωt + δ) y(t) ¨ + 2r y(t) ˙ + ω02 y(t) = , (15) m where F0 is the amplitude of the excitation force and δ is the initial phase of this force. As follows from solution (14), the initial conditions y0 and y˙0 need not be considered if damping is appreciable and/or when the time t is significant. Generally speaking, the solution to Eq. (15) in this case can be obtained on the basis of formula (13). We will use, however, a simpler way, using the fact that the forced vibrations y(t) have the same frequency ω as the excitation force: y(t) = B0 cos(ωt + ε) + B1 sin(ωt + ε).

(16)

Introducing the sought solution (16) into Eq. (15), we obtain the following equations for the constants B0 and B1 :  2



  F0 ω0 − ω2 cos ε − 2rω sin ε B0 + ω02 − ω2 sin ε + 2rω cos ε B1 = sin δ, m



   F0 − ω02 − ω2 sin ε + 2rω cos ε B0 + ω02 − ω2 cos ε − 2rω sin ε B1 = cos δ. m (17) These equations yield



F0 ω02 − ω2 sin(δ − ε) − 2rω cos(δ − ε) , B0 (ω) = 2

2 m ω0 − ω2 + (2rω)2

F0 ω02 − ω2 cos(δ − ε) + 2rω sin(δ − ε) B1 (ω) = . 2

2 m ω − ω2 + (2rω)2

(18)

0

The amplitude of the induced vibrations can be found as  1 F0 a(ω) = B02 + B12 = . 2   2  2 mω0 2 ω 2r ω  1− + ω0 ω0 ω02

(19)

Let us revisit Eq. (15). If the force F0 were applied statically, then neither inertia forces, nor damping forces could possibly occur, and the (static) displacement yst would have been expressed as F0 A0 yst = = 2. (20) mω02 ω0

22

Linear Response to Shocks and Vibrations Z0

d1

d2 A0/w02

d3 d4 w0

0

Ω

Figure 2 Dynamic factor (“resonance curves”): ratios of “output” amplitudes to static response for different damping coefficient values. While damping is significant, no vibrations could possibly occur. Resonance condition occurs when the frequency of the harmonic excitation becomes equal to the natural frequency of the damped vibrations. When damping is zero, the resonance amplitudes are infinitely high. Comparing formulas (19) and (20), we conclude that the dynamic factor of the system is expressed as a(ω) 1 Z0 = =  (21) 2  2 .

ast 2

2rω  1− ω + ω02 ω02 This formula is the same as formula (58) in Chapter 4, where it was obtained based on the theory of the Fourier integral. The relationship (21) is plotted in Fig. 2, where δ = r/ω0 .

3

SYSTEMS WITH MULTIPLE DEGREES OF FREEDOM (MDOF) While linear vibrations of a system with a single degree of freedom are described by a single differential equation of the type (1), linear vibrations of a system with n degrees of freedom is described by a system of differential equations of the second order: n  (agk T¨ k + bgk T˙ k + cgk Tk ) = Qg (t), g = 1, 2, . . . . (22) k =1

Here Tk are the generalized (particularly, principal) coordinates of the system (linear, angular, curvilinear, etc.), Qg (t) are the generalized forces (forces, moments, etc.), and the coefficients agk , bgk , and cgk depends on the structure of the particular elastic system and the way that the differential equations (22) are obtained. It can be shown that system (22) will indeed describe low-amplitude vibrations in the vicinity of the state of its stable equilibrium if the determinant of this system obtained by substituting the Euler relationship Tk = Ak e zt into system (22) is equal to zero:    agk z 2 + bgk z + cgk  = 0.

(23) (24)

In this equation g is the number of the line and k is the number of the column in the determinant (24). It is noteworthy that the generalized coordinates Tk can be chosen in different ways, and such a choice will affect the system (22). This choice, however, will not (should not) affect the roots (natural frequencies) of the system (22) since the physics of the vibrations

4 Forced Vibrations of Elongated PCB due to Harmonic Oscillations of Its Support Contour

23

should not be dependent on the choice of the coordinates. The most attractive choice of the generalized coordinates is such when the system (22) can be substituted by a system of uncoupled equations, but this is not always possible, even in the case of linear systems. The following Lagrange equations are often used in the theory of low-amplitude vibrations of MDOF elastic systems:   d ∂K ∂V = Qk . (25) + ˙ dt ∂ Tk ∂Tk Here K is the kinetic energy of the system, V is its potential (strain) energy, and Qk is the generalized force that corresponds to the k th generalized coordinate. The kinetic energy, K, and the potential (strain) energy, V , assumed in this theory are homogeneous functions of the second order of the generalized velocities and generalized coordinates, respectively: n n n n 1  1  agk T˙ g T˙ k , V = cgk Tg Tk . (26) K = 2 2 g=1 k =1

g=1 k =1

When damping is considered, this could be accounted for by considering the generalized force Q k expressed as ∂R Qg = − , (27) ∂ T˙ g where

1  bgk T˙ g T˙ k 2 n

R=

n

(28)

g=1 k =1

is the so-called dissipation function. When the generalized coordinates Tk are principal coordinates, formulas (26) can be simplified as n n 1 1 Mk T˙ k2 , V = Ck Tk2 , (29) K = 2 2 k =1

k =1



where ωk =

Ck Mk

(30)

are the natural frequencies of the k th mode of vibrations.

4

FORCED VIBRATIONS OF ELONGATED PCB DUE TO HARMONIC OSCILLATIONS OF ITS SUPPORT CONTOUR Let an elongated PCB be subjected to steady-state vibrations caused by the lateral oscillations of its support contour. The induced PCB displacements, w (x , t), can be sought in the form ∞  w (x , t) = w0 (t) − Tk (t)Xk (x ), (31) k =1

where w0 (t) is the displacement of the support contour, X k (t) is the coordinate function of the k th mode of the free vibrations, and T k (t) is the corresponding principal coordinate. The kinetic, K , and the strain, V , energy of the board can be expressed, considering (31), as follows:   a/2  ∞ ∞   1 1 ∂w 2 ˜ k T˙ k (t) + 1 K = m M dx = maw˙ 02 (t) − w˙ 0 (t) Mk T˙ k2 , 2 ∂t 2 2 −a/2 k =1 k =1 (32)  a/2  2 2 ∞ 1 2 1 ∂ w 2 dx = ωk Mk Tk . V = D 2 ∂t 2 2 −a/2 k =1

24

Linear Response to Shocks and Vibrations where m is the PCB mass per unit area, a is the PCB width,  a/2  a/2 Mk = m Xk2 (x ) dx, M˜ k = m Xk (x ) dx −a/2

−a/2

(33)

are the k th generalized masses associated with the inertia force and the excitation force, respectively,  a/2   2 D 2 X (x ) dx (34) ωk = Mk −a/2 k is the square of the vibration frequency of the k th mode, Eh3 (35) 12(1 − ν 2 ) is the flexural rigidity of the board, h is its thickness, and E and ν are Young’s modulus and Poisson’s ratio, respectively, of the PCB material in the x –y plane. The origin, O, of the coordinate x is in the midpoint of the PCB width. It is assumed that the electronic components mounted on the PCB are small in size, so that the flexural rigidity D of the board is not affected by the flexural rigidities of the components. The board’s distributed mass, m, however, should be determined with consideration of the components’ masses. This could be done, for instance, by “spreading out” these masses over the PCB surface. In formulas (31), it is taken into account that the coordinate functions, X k (x ), possess the property of mutual orthogonality, that is,  a/2 Xk (x )Xj (x ) dx = 0 for i = j . (36) D=

−a/2

Introducing (32) into the Lagrange equations   d ∂K ∂V = Qk (37) + dt ∂ T˙ k ∂Tk and considering viscous damping, so that the generalized force Q k can be expressed as (38) Qk = −2rk Mk T˙ k , we obtain the following equation for the k th principal coordinate: T¨ k (t) + 2rk T˙ k (t) + ωk2 Tk (t) = γk w¨ 0 (t). Here r k is the damping coefficient of the k th mode, and the factor  a/2 Xk (x ) dx ˜k M −a/2 =  a/2 γk = Mk Xk2 (x ) dx

(39)

(40)

−a/2

considers the effect of the k th coordinate function, X k (x ), on the excitation force. Let the PCB be subjected to harmonic oscillations, so that w0 (t) = A0 e i ωt .

(41)

Then the solution to Eq. (39) can be sought in the form Tk (t) = Tk0 e i ωt .

(42)

Introducing (41) and (42) into Eq. (39), we obtain the following expression for the complex amplitude, Tk0 , of the principal coordinate, T k (t): Tk0 =

1−

γk2 A0 . 2 ωk /ω2 + 2i (rk /ω)

(43)

4 Forced Vibrations of Elongated PCB due to Harmonic Oscillations of Its Support Contour From (31) and (43) we find w  (x , t) = −A0

∞ 

γk2 Xk (x )

k =1

1 − ωk2 /ω2 + 2i (rk /ω)

e −i ωt .

25 (44)

If the electronic component is located at the point x = x 0 , then the complex amplitude of the angle, ψ 0 (t), of rotation of the PCB’s cross-section at this point can be found as ∞  γk2 Xk (x0 ) . (45) ψ0 = −A0 1 − ωk2 /ω2 + 2i (rk /ω) k =1 The real amplitude of the rotation angle is expressed as ∞  γk2 Xk (x0 ) . ψ0 = −A0

2 2 2 + [2(r /ω)]2 k k =1 1 − ωk /ω

(46)

In this formula, A0 is the real amplitude of the vibrations of the PCB support contour. If, for instance, the PCB is simply supported along its edges, that is, if kπ Xk (x ) = cos , (47) a then we obtain     ma ˜ k = 2 ma sin k π  = 2 ma , , M Mk =  2 kπ 2  kπ  4 (48) D 4 k π ωk2 = , γk = , k = 1, 3, 5, . . . , m a kπ and the formula (46) yields ∞ 16 A0  1 sin(k π x0 /a) . (49) ψ0 =  2   π a k  r 2 kπ 4 D k =1,3,5,... k 1− + 2 a mω2 ω This series is rapidly converging, and therefore one can assume sin(π x0 /a) 16 A0 (50) ψ0 ∼ =   π 4 D 2  r 2 . π a 1 1− + 2 a mω2 ω For nonresonant conditions, the dissipation term in the denominator can be neglected: ∼ 16 A0  sin(π x0 /a)  . (51) ψ0 =  π 4 D 2 π a 1− a mω2 In the case of low excitation frequencies,  2 16 mω2 π x0 ψ0 ∼ sin . (52) = 9 A0 a 7 π D a This formula indicates that the rotation angles of the component’s attachments to the board are very small, if the frequency ω is low, and the board is narrow and thick. In the case of very high frequencies, we have ∼ 16 A0 sin π x0 . ψ0 = (53) π a a Hence, in such a case the rotation angles become frequency independent, and, for the given location of the component are determined by the ratio x0 /a and the amplitude-to-width ratio A0 /a only.

26 5

Linear Response to Shocks and Vibrations

FORCED VIBRATIONS OF A HEAVY ELECTRONIC COMPONENT SUBJECTED TO HARMONIC EXCITATION External electric leads soldered into plated-through-holes (PTHs) of a PCB provide, in addition to electrical interconnection, also mechanical support for heavy (high-mass) electronic components mounted on the boards. The leads can possibly break if the PCB support contour is subjected to extensive vibrations transmitted to the heavy component. In this analysis we examine the dynamic response (forced vibrations) of a heavy component to harmonic (sine) excitations applied to its external electric leads. The vibrations are due to the angular oscillations of the clamped ends of the leads. These oscillations in turn are due to the vibrations of the PCB support contour in the PCB through-thickness direction. The obtained results can be used to evaluate the stresses occurring in the electric leads because of the harmonic oscillations of the PCB support contour. These results can be used also for the assessment of the “output” spectrum of the dynamic stresses when the oscillations of the PCB support contour are random and are described by an appropriate “input” power spectrum. We assume that the heavy electronic component is supported by just two electrical leads that experience vibrations in the vertical plane that is perpendicular to the plane, in which the leads are located. This situation reflects, of course, the worst-case scenario. The objective of the analysis is to determine the response function of the dynamic system in question, that is, to evaluate the vibration amplitudes of the component for the given frequency of vibrations of the leads’ clamped ends. Although the problem in question is a rather specific one, the way its solution is obtained is typical for a number of practically important problems of the dynamic response of PCBs to the vibrations of their contours. The equations of motion of an electronic component mounted on a PCB (Fig. 3) can be obtained in the following form (see Appendix): 6EI −3(1 + 2 g )M¨ 0 (t) + (1 + 3 g ) N¨ 0 (t) − 2λ20 N0 (t) = (1 + g )ψ¨ 0 (t),  2  2 ¨ 0 (t) − 2 λ0 M0 (t) + 3 N¨ 0 (t) + 2 λ0 (1 + g ) N0 (t) = 6EI ψ¨ 0 (t). −6M ρ ρ (54) In these equations, M 0 (t) and N 0 (t) are the bending moment and the lateral force, respectively, at the clamped end of the lead; g = g / is the ratio of the distance, g , of the component’s

z Ψ0(t) lg z1(t)

1 l0

G 1

El 2

O

3

x

w (x,t )

x0 a

Figure 3 Heavy electronic component that experiences forced vibrations because of the oscillations of the PCB.

5 Forced Vibrations of a Heavy Electronic Component Subjected to Harmonic Excitation

27

center of gravity, G, from the outer (“free”) end of the lead to the lead’s length, ;  3EI (55) λ0 = M 3 is the fundamental frequency of the component’s vibrations evaluated for the hypothetical situation when the entire mass M is concentrated at the lead’s “free” end, z = ; EI is the flexural rigidity of the lead; ρ = ρ/ is the ratio of the radius of inertia, ρ, of the component’s mass, M , to the lead’s length, ; and ψ0 (t) is the angle of rotation of the clamped end of the lead. It is clear that the hypothetical, but easy to evaluate, frequency λ0 of vibrations, when the entire mass M is concentrated at the component’s free end, should be considerably higher than the actual frequency λ of free vibrations when the mass M of the component is distributed over its length. Excluding the lateral force function, N 0 (t), from Eq. (54) and adding the viscous damping force λ2 R = 2r 4 M0 (t) (56) η to the inertia and the restoring forces, we obtain the following differential equation for the bending moment, M 0 (t), at the clamped end of the lead:  λ2  4EI  IV M0 (t) + βλ40 M0 (t) = − ψ0 (t) + λ2∗ ψ0 (t) . 4 η In this equation, r is the coefficient of damping, M0IV (t) + 2αλ20 M0 (t) + 2r

λ = ηλ0

(57)

(58)

is the actual fundamental (lowest) frequency of free (natural) vibrations of the component,1   

β  η < 1, (59) η = α 1− 1− 2 , α is the factor that considers the effect of the finite g and ρ values on the natural frequency λ, the parameters α and β are expressed as   4 2 β α = 1 + 1 + 3 g + 3 g , (60) , β= 2 3ρ 2 and the frequency λ* is expressed as  2

1 + g  λ∗ = λ0 1 + . (61) ρ The physical meaning of this frequency will be shown below in the “Response Function” section. The primes in Eq. (57) indicate time derivatives. The dissipation term (56) is introduced in such a way that the damping coefficient, r, has a unit of frequency, as it is usually accepted in the theory of damped elastic vibrations (see, e,g., [2]). In addition, damping is considered low, and therefore its effect on the vibration frequency need not be accounted for. Damping should be accounted for, of course, when the amplitudes of forced vibrations are evaluated, especially in the close-to-resonance conditions. Note that formula (59) can be obtained directly from Eq. (57). Indeed, in the case of free [ψ¨ 0 (t) ≡ 0] and undamped (r ≡ 0) vibrations, this equation yields M0IV (t) + 2αλ20 M0 (t) + βλ40 M0 (t) = 0.

28

Linear Response to Shocks and Vibrations The characteristic equation

λ4 − 2αλ20 λ2 + βλ40 = 0

of this differential equation results in formula (50). Let us determine the response function. If the excitation ψ0 (t) is harmonic, that is, if it can be represented as √ (62) ψ0 (t) = ψ0 e i ωt , i = −1, then the steady-state response M 0 (t) is harmonic as well and has the same frequency ω, so that the response can be sought as (63) M0 (t) = M0 e i ωt . In formulas (62) and (63), ψ0 and M0 are the complex amplitudes of the angle of rotation, ψ 0 (t), and the bending moment, M 0 (t), respectively. The complex frequency characteristic, M (i ω), of the induced process M 0 (t) of bending moments can be found as the ratio of the complex amplitudes, M 0 and ψ 0 , of the “output” and the “input” processes, M 0 (t) and ψ 0 (t) (see also Chapter 4):  2   ω 2 λ∗ −   2 M0 4EI 2 ω λ λ = , 0 ≤ ω ≤ λ∗ . (64) η M (i ω) =    ω 2   ψ0 λ ω2 2rω 2 β− η 1− 2 +i 2 λ λ λ The response function (complex-frequency characteristic) for the maximum bending moment (occurring at the lead’s clamped end) is aM (ω) = |M (i ω)|2 = M (i ω)M (−i ω)   2 λ∗ 2  ω 2 −   λ λ 4EI 2  2 ω 4 η =    2   , 2   λ ω 2 ω2 2rω 2 2 β− η 1− 2 + λ λ λ2

0 ≤ ω ≤ λ∗ . (65)

The units for this function are the units of the bending moment squared. As evident from expressions (64) and (65), forced vibrations of the component will not occur if the frequency ω of the excitation is zero or when this frequency exceeds the λ* value. If the excitation frequency is equal or higher than the λ* value, the vibrations of the component simply cannot follow the high-frequency vibrations of the board and are not initiated. This conclusion can be useful, particularly when planning random vibration tests for detecting infant mortality failures: The excitation frequencies should be within the range of 0 ≤ ω ≤ λ* in order to generate the component vibrations and to produce meaningful results. On the other hand, if the excitation frequencies in actual operation conditions exceed the λ* value, the dynamic stability of the system is ensured: Such high excitation frequencies are simply unable to induce forced vibrations of appreciable amplitude. At the resonant conditions (ω = λ), formula (65) yields 2      2 4EI 2 4 λ 2 λ∗ −1 . (66) η aM (ω) = aM (λ) = 2r λ The dynamic (bending) stress can be found from the calculated bending moment, M , as σb =

M , S

(67)

5 Forced Vibrations of a Heavy Electronic Component Subjected to Harmonic Excitation

29

where S is the section modulus. For a circular cross section, π (68) S = r03 , 4 and therefore √ 4M 4 aM = ψa , (69) σb = π r03 π r03 where ψ a is the (real) amplitude of the angular displacement of the lead’s clamped end.

5.1 Numerical Example Let a heavy electronic component whose weight G = 200 gf = 1.96 N (the corresponding mass M = G/g = 2.0408 × 10−5 kg × sec2 × mm−1 = 200 g) is supported by two electric leads whose diameters are d = 1.0 mm (r 0 = 0.5 mm). The moment of inertia of the lead’s cross section is I = 14 π r04 = 0.04909 mm4 . Let the Young’s modulus of the lead’s material be E = 2.0 × 104 kg/mm2 = 196 GPa. Then the flexural rigidity of the leads is EI = 2 × 2.0 × 104 × 0.04909 = 1963.5 kgf × mm2 = 0.01924 N × m2 . Let the leads length be = 15 mm, the distance of the component’s center of gravity from the lead’s end be g = 12 mm, and the radius of inertia of the component’s mass be ρ = 6 mm. Then we obtain λ0 = 292.441 sec−1 , g = g / = 0.8, ρ = ρ/ = 0.4, β = 8.3333, α = 23.1666, η = 0.4249, λ = 124 sec−1 , and λ* = 1348 sec−1 . Let the dissipation be such that the ratio of the coefficient of dissipation to the actual frequency of free vibrations is r/λ = 0.01. Then expression (12) results in the following formula for the response function:   2 2  ω 4 ω 1 − 0.09217(ω/λ) aM = aM = 5.8108 10 

2   2 . λ λ 1 − 0.00391(ω2 /λ2 ) 1 − ω2 /λ2 + 0.0024(ω/λ) (70) The calculated aM values are shown in Table 1 and plotted in Fig. 4. As evident from the obtained data, the response function has a strongly pronounced peak at the resonant frequency ω ∼ = λ and, in the below-resonance region, rapidly decreases with the decrease in the excitation frequency. As to the above-resonance region, the response function decreases relatively slowly with an increase in the excitation frequency. Therefore the most effective means to ensure high dynamic stability (low vibration amplitudes) of the system is to make the frequency ratio ω/λ as low as possible, well below the resonant value, that is, to stiffen dramatically the structure. This could be done, for instance, through stiffening the lead structure (by employing shorter leads of larger diameter, by introducing surrogate leads, not needed from the viewpoint of the electrical performance and located in a different plane than the electric leads, etc.) or, if possible, by keeping the excitation frequency well below the natural frequency of free vibrations. If, for one reason or another, the structure is intended to be operated in the frequency range, exceeding the resonant frequency, then it is advisable to operate the structure in the ω/λ > 11 region (ω > λ* ). Let us now evaluate the dynamic stress for the amplitude of the angle of rotation, ψ 0 (t), of ψ 0 = 0.5◦ = 0.00873, and for the frequency of excitation of ω = 497.0 sec−1 . With the Table 1 ω/λ

0

0.5

0.75

1.0

1.5

2.0

4.0

8.0

10.0

a M ×10−3 , 0 6.4416 95.5600 9,917,518.6 184.3713 70.9290 56.1706 22.2111 3.6193 kgf2 × mm2

10.85 —

Linear Response to Shocks and Vibrations 1.8 1.6 Response Function, aM × 10−5, (kg × mm)2

30

w = frequency of excitation l = natural frequency

1.4 1.2 1.0 0.8 0.6 0.4 0.2 0

0

1

2

3

4 5 6 7 Frequency Ratio, w/l

8

9

10

11

Figure 4 Response function for maximum bending moment in external electric leads that support an electronic component that experiences forced vibrations because of oscillations of the PCB. input data from the previous example, we have ω/λ = 4.0, and Table 1 yields a M = 56170 (kgf × mm)2 = 5.3946 (N × m) 2 . Then formula (16) results in the following bending stress: σ b = 2.634 kgf/mm2 = 25.813 MPa = 3,746 psi. At the resonance conditions, σ b = 8855.5 kgf/mm2 = 86.784 GPa = 12,593 kpsi. The leads will most likely break when subjected to such a high stress. Thus, in the example in question, as long as one could avoid resonance conditions, the induced stress is low, and no special design measures seem to be necessary. In practical applications, it is often sufficient to evaluate the natural frequency of vibrations and to make sure that this frequency is remote enough from the frequency of the excitation. The following conclusions can be drawn from the performed analysis: • An expression for the response function of the maximum bending moment is obtained

for an electric lead of a heavy electronic component mounted on a PCB. This moment occurs at the clamped end of the lead and is due to the angular oscillations of this end because of the PCB vibrations. The obtained expression enables one to evaluate the bending moment occurring in the lead for the given amplitude and frequency of angular oscillations of the clamped end. • In the case of low structural damping, the response function has a strongly pronounced peak at the resonant frequency and rapidly decreases to zero with the decrease in the frequency of the excitation. As to the frequencies, exceeding the resonance ones, the response function decreases relatively slow with an increase in the excitation frequency. Therefore the most effective means to ensure high dynamic stability of the system in question is to make the frequency ratio well below the resonant values.

A.1 Appendix: Natural Vibration Frequency of a Heavy Electronic Component

31

• There exists a certain high frequency above which the response is zero, that is, no

forced vibrations are possible to occur. This frequency depends on the flexural rigidity of the lead(s), their length, the component’s mass and moment of inertia, and the position of the component’s center of gravity. In the executed numerical example, the calculated upper boundary of the response function exceeds the natural frequency of vibrations by a factor of 10.85. • The obtained results can be used in the analysis and design of external electric leads of heavy electronic components subjected to forced vibrations. If a probabilistic analysis is considered, it can be based on the obtained expression for the response function. • Although the addressed problem is a rather specific one, the way its solution has been obtained is quite typical in problems of the dynamic response of PCB systems to the vibrations applied to the PCB support contour.

A.1

APPENDIX: NATURAL VIBRATION FREQUENCY OF A HEAVY ELECTRONIC COMPONENT

A.1.1 Assumptions The following assumptions are used in this analysis: • The electronic component can be idealized as a heavy elongated rigid (nondeformable)

body attached to the lead’s outer (“free”) end. • The external electric leads can be treated, from the standpoint of structural analysis, as

• • • • •

•

•

elongated elastic prismatic cantilever beams clamped at one end and carrying a heavy elongated rigid body (electronic component) at the other (“free”) end. The lead’s length-to-diameter ratio exceeds 12, and therefore the effect of the shearing deformations of the lead need not be accounted for. The weight of the lead(s) themselves is negligibly small in comparison with the weight of the component and need not be considered. The elementary (engineering) beam theory (see, e.g., [3]) can be employed. Only the lower (fundamental) frequency of the component’s natural (“free”) vibrations is important. The lead/component structure is infinitely stiff in the axial direction, so that the lead’s displacements in this direction do not affect the component’s vibrations; therefore one can assume that the clamped ends of the leads are subjected to angular (rotational) displacements only and that the leads experience only bending deformations. The electronic component is located far enough from the short edges of the PCB, and the electric leads of this component are oriented in the direction parallel to the PCB long edges; for this reason, the leads experience vibrations only in the vertical plane parallel to the short edges of the board, that is, in the plane of the leads’ minimum flexural rigidity. Such a conservative assumption is thought to be justified for practical applications. Only one row of the leads is employed. If it is not the case, the natural frequency of the component is very high and its dynamic stability is ensured.

A.1.2 Equations of Motion The elastic curve, u(z , t), of a lead experiencing vibrations due to the rotational displacements, ψ 0 (t), of its clamped end can be sought in the form of the “equation of initial parameters” [3] as follows: M0 (t) 2 N0 (t) 3 u(z , t) = ψ0 (t)z + 0 ≤ z ≤ , (A.1) z − z , 2EI 6EI

32

Linear Response to Shocks and Vibrations where u(z, t) are the lead’s deflections, ψ 0 (t) is the PCB rotation angle at the component (lead’s) location (the angle of rotation of the clamped end of the lead), M 0 (t) and N 0 (t) are the bending moment and the lateral force, respectively, at the clamped and (“initial parameters”), EI is the total flexural rigidity of the leads, is the lead’s length, and t is time. The origin O 1 of the vertical (axial) coordinate z is at the clamped end of the lead. The rotation angle u  (z , t), the bending moment M (z , t), and the lateral force N (z , t) at an arbitrary cross-section z of the lead can be found from (1) by differentiation: N0 (t) 2 M0 (t) u  (z , t) = ψ0 (t) + z− z , (A.2) EI 2EI  (A.3) M (z , t) = EIu (z , t) = M0 (t) − N0 (t)z , N (z , t) = −EIu (z , t) = N0 (t).

(A.4)

As one can see from these equations, the bending moment changes linearly along the lead, while the lateral force remains constant. The linear and the angular displacements at the “free” end, z = , can be found, using (A.1) and (A.2), as follows: 2 3 (A.5) M0 (t) − N0 (t), 2EI 6EI 2 (A.6) u  ( , t) = ψ0 (t) + M0 (t) − N0 (t). EI 2EI Then the linear, U c (t), and the angular, ψ c (t), displacements of the component can be expressed by the formulas u( , t) = ψ0 (t) +

Uc (t) = u( , t) + u  ( , t) g 2 3 (A.7) (1 + 2 g )M0 (t) − (1 + 3 g )N0 (t) 2EI 6EI 2 c (t) = u  ( , t) = ψ0 (t) + M0 (t) − N0 (t), EI 2EI where g is the distance of the component’s center of gravity G from the “free” end, z = , of the lead and g = g / is the ratio of this distance to the lead’s length . The equations of the lateral and the rotational motions of the component can be written as M U¨ c (t) = −N ( , t) = −N0 (t), (A.8) ¨ c (t) = N ( , t) g − M ( , t) = (1 + g ) N0 (t) − M0 (t), Mρ 2  = (1 + g )ψ0 (t) +

where ρ is the radius of inertia of the component with respect to the horizontal axis. Introducing formulas (A.7) into Eq. (A.8), we obtain the following differential equations of motion: ¨ 0 (t) + (1 + 3 g ) N¨ 0 (t) − 2λ20 N0 (t) = 6EI (1 + g )ψ¨ 0 (t), −3(1 + 2 g )M  2  2 λ λ 6EI 0 0 −6M¨ 0 (t) − 2 M0 (t) + 3 N¨ 0 (t) + 2 (1 + g ) N0 (t) = ψ¨ 0 (t). ρ ρ (A.9) Here ρ = ρ/ is the ratio of the radius ρ of inertia of the component’s mass M to the lead’s length and  3EI λ0 = (A.10) M 3 is the fundamental frequency of the component’s vibrations determined for a hypothetical case when its entire mass M is concentrated at the “free” end, z = , of the lead.

A.1 Appendix: Natural Vibration Frequency of a Heavy Electronic Component

33

A.1.3 Free Vibrations In the case of “free” vibrations, one should put ψ¨ 0 (t) ≡ 0 in Eq. (A.9). In addition, having in mind undamped vibrations (or damped vibrations with very low dissipation), one can assume M0 (t) = M0 e i λt ,

N0 (t) = N0 e i λt ,

(A.11)

where M 0 and N 0 are the complex amplitudes of√the processes M 0 (t) and N 0 (t), λ is the sought fundamental vibration frequency, and i = −1 is the “imaginary unity.” Introducing (A.11) into (A.9), with ψ¨ 0 (t) ≡ 0, we obtain the following system of homogeneous algebraic equations for the complex amplitudes M 0 and N 0 :     3(1 + 2 g )λ2 M0 − (1 + 3 g )λ2 + 2λ20 N0 = 0,      2  2 λ0 λ0 (A.12) [12pt]2 3λ2 − M0 − 3λ2 − 2 (1 + g ) N0 = 0. ρ ρ The determinant of these equations must be zero in order that the amplitudes M 0 and N 0 be nonzero. This results in the following biquadratic equation for the eigenvalue (frequency) λ: λ4 − 2αλ20 λ2 + βλ40 = 0,

(A.13)

where the coefficients α and β are expressed by the formulas 2

4 2 1 + 3 g + 3 g 2 β = 1 + (1 + 3 g + 3 g ) , . β= 3 2 ρ2 3ρ 2 The lowest positive root of the biquadratic equation (A.13) is α =1+

λ = ηλ0 , where the frequency λ0 is expressed by formula (A.10) and the factor   

β  η < 1, η = α 1− 1− 2 , α

(A.14)

(A.15)

(A.16)

considers the effect of the finite distance g of the component’s center of gravity G from the lead’s “free” end, as well as the effect of the finite radius ρ of inertia of the component’s mass M . Indeed, when the ρ ratio is small, the β value, as evident from the second formula in (A.14), is large. Then, with g = 0, the first formula in (A.14) yields: α ∼ = β/2. The β/α 2 ratio is inversely proportional to the β value, and, with large β values, is small. Then formula (A.16) yields     

  

β ∼ β β ∼  = η = α 1− 1− 2 = α 1− 1− 2 = 1. α 2α 2α

A.1.4 Numerical Example Let, for instance, the component’s structure be such that g = 1 and ρ = 2. Then formulas 1 (A.14) yield α = 13 6 , β = 3 . From (A.16) we find η = 0.280. Thus, the actual frequency of “free” vibrations is significantly lower than the frequency calculated using an assumption that the entire component’s mass M is concentrated at the lead’s “free” end, z = . Let, for instance, the radius of the component’s lead be r 0 = 0.5 mm, Young’s modulus of the material be E = 16,000 kg/mm2 , the number of leads be n = 2, their length be = 5 mm, and the component’s weight be G = 50 g (so that its mass M = 5.102 × 10−6 kg × sec2 × mm−1 ).

34

Linear Response to Shocks and Vibrations In this case, the calculated flexural rigidity of the leads is EI = 1571 kg × mm2 , and formula (A.10) yields λ0 = 85,960 sec−1 . The predicted fundamental frequency of “free” vibrations of the component is λ = ηλ0 = 24,060 sec−1 = 3,829 Hz.

REFERENCES 1. Suhir, E., “Predicted Fundamental Vibration Frequency of a Heavy Electronic Component Mounted on a Printed Circuit Board,” ASME Journal of Electronic Packaging, Vol. 122, No. 1, 2000. 2. Timoshenko, S. P., and Young, D. H., Vibration Problems in Engineering, 3rd ed., VanNostrand, New York, 1955. 3. Suhir, E., Structural Analysis in Microelectronic and Fiber Optic Systems, Van-Nostrand Reinhold, New York, 1991.

CHAPTER

3

LINEAR AND NONLINEAR VIBRATIONS CAUSED BY PERIODIC IMPULSES Ephraim Suhir University of California, Santa Cruz, California University of Maryland, College Park, Maryland ERS Co., Los Altos, California

1

INTRODUCTION Dynamic response of materials and structures to shocks and vibrations has always been an important topic of the applied science and engineering (see, e.g., [1, 8]), including the field of electronic systems [2–5, 9, 10]. Electronic and photonic components and systems often experience dynamic loading, and therefore the ability to predict and, if necessary, minimize the adverse consequences of such loading is of obvious practical importance. In commercial electronics, dynamic loading can take place during handling or transportation of the equipment. In military, avionics, space, automotive, and marine electronics, dynamic loading is expected to occur even during the normal operation of the system. On the other hand, random vibrations are often applied deliberately (in addition to or even instead of thermal cycling or mechanical testing) as an effective and fast means to detect and weed out infant mortalities in the system, even though a particular system might not be intended for operation in a dynamic environment. Shock loading is part of a number of MIL specs and other qualification requirements (see, e.g., [6, 7]). In recent years, the necessity to protect portable electronics from shock loading (primarily, because of an accidental drop) triggered the development of both theoretical methods and experimental techniques for predicting and minimizing the adverse consequences of an accidental shock. This chapter contains a brief review of the recent literature on the dynamic response of micro- and opto-electronic systems to shocks and vibrations. We use, as a useful illustration of methods applied in structural dynamics, the case of vibrations caused by a train of instantaneous impulses. Vibrations caused by periodic impulses have numerous applications in engineering and physics, including the field of electronics and photonics. Various mechanical equipment (diesel engines, rolling mills, gear teeth, punch presses, pneumatic tools, such as hammers, miner’s hacks and machines or chisels, etc.); clock mechanisms; military equipment (machine guns, rapid-shooting artillery); vortex separation from unfavorably shaped civil, aircraft, space, and ocean structures (aero- and hydrodynamic flutter); ship slamming in regular seas; synchrocyclotrons and other accelerators of charged particles; biological systems (heart beating, walking, running, jumping, etc.) and certainly electronic and photonics equipment in the above systems experience vibrations due to periodic impulses (impacts). In this chapter we use the case of linear and nonlinear vibrations of a single-degree-of-freedom (SDOF) system to illustrate some basic methods of the analysis of the dynamic response of structural elements to shock excitation. Structural Dynamics of Electronic and Photonic Systems Copyright © 2011 John Wiley & Sons, Inc.

Edited by Ephraim Suhir, David S. Steinberg and T. X. Yu

35

36 2

Linear and Nonlinear Vibrations Caused by Periodic Impulses

REVIEW Goyal et al. [21] suggested a shock-protective suspension design for printed circuit boards (PCBs), including those used in portable electronics. Dynamic response of a PCB in a portable electronic product to a drop impact was addressed by Seah et al [66]. Suhir [52–55, 59–61] emphasized the nonlinear effects in the PCBs subjected to shock loads applied to the support contour of the PCB. He examined cases of periodic shocks [51, 53] and a suddenly applied constant acceleration [54]. Exact solutions to highly nonlinear problems described by von Karman’s equations (see, e.g., [52]) were obtained using elliptic functions and elliptic integrals. Later on He and Fulton [57] and He and Stallybrass [58] also addressed some nonlinear problems in the mechanical behavior of PCBs subjected to dynamic loading. Various aspects of drop impact in application to portable electronics products and chip-size packages (CSPs) were recently analyzed by numerous investigators (see, e.g., Lim and Low [67], Lim et al. [68, 69], Zhu [22, 33, 70], Zhu and Marcinkiewicz [43], Luan and Tee [27, 32, 35, 37, 41, 45, 46], and Wang [34]). The role of viscous damping in a SDOF linear system experiencing a drop impact was investigated in detail by Suhir [15]. It has been demonstrated, particularly, that although elevated damping always has a favorable effect on maximum displacements leading to lower maximum displacements, it can have a strong adverse effect on the maximum accelerations (decelerations): Elevated damping can result in accelerations (decelerations) that are significantly larger than the accelerations in a damping-free system. In many cases drop test conditions can be adequately substituted by shock tests provided that the drop test conditions are correctly predicted and the shock tester is appropriately “tuned” [20]. The incentive for substituting drop tests with shock tests is due to the fact that shock tests might be easier to conduct as well that the shock impact’s magnitude and duration can be easily prescribed and, hence, predicted, while the characteristics of the drop impact pulse have to be determined during the experiment itself. By employing nonlinear shock protection elements (materials) and multi-degree-offreedom shock protection systems, one could improve such a protection dramatically. Substantial improvement in the performance of a shock protection system can be achieved, particularly, by employing a two-degree-of-freedom system (“box-in-a-box” type). This should be done, however, with an appropriate selection of the masses and spring constants, so that resonance conditions that might lead to a “rigid impact” are avoided [55, 56]. The analysis and design concepts addressed in [62] were developed in the application to a liquid crystal display design. A detailed analysis of the shock response spectrum of a portable device with a shock protection system was carried out by Goyal et al. [64]. Many recent studies were devoted to the investigation of the effect of a short-term loading on solder materials and solder joints in portable electronic products, with an emphasis on lead-free solders and on drop test conditions [17–19, 23–49]. Ong et al. [38] addressed some general aspects of dynamic testing of materials in the application to solder interconnections. As is known, the dynamic response of a material to a short-term loading can differ considerably in its response to a statically applied load (see, e.g., [1]). Zhu [22], Sogo and Hara [23], Yu et al. [25], and Tan [29] investigated the mechanical behavior of ball grid arrays (BGAs) in CSPs. BGAs are typically the most vulnerable structure of a surface mount technology (SMT) package design. Arra [24] and Date et al. [39] investigated the reliability of lead-free solders under the action of dynamic loading caused by a drop impact. Various aspects of the drop impact and drop test conditions were addressed also by Wu [17, 18], Mishiro et al. [26], Wong et al. [28], Xie et al. [30], Yeh and Lai [40], Luan and Tee [41], Chiu et al. [42], and Syed et al. [48]. Predictive modeling and especially computer-aided evaluations (simulations) play an increasingly important role in the analysis and design of microelectronic materials and structures subjected to dynamic loading [3, 62, 63]. Some general approaches to the analysis

3

Shock-Excited Vibrations

37

of random vibrations were described in the monograph [5]. Huang, Kececioglu, and Prince [65] carried out a simplified random vibration analysis for portable electronic products. The dynamic response of some photonic structural elements was addressed in [12–14]. A reasonably good proficiency in applied math is a “prerequisite” to understand well the value of the analyses that follow.

3

SHOCK-EXCITED VIBRATIONS

3.1 Linear Response Linear response x (t) of a SDOF system to the train of periodic instantaneous impulses of the magnitude S can be found from the following equation: x¨ (t) + 2r x˙ (t) + ω02 x (t) =

n S  δ(t − kT ), M

k = 0, 1, 2 . . . ,

(1)

k =0

where δ(t) is Dirac’s delta function (see Chapter 4), M is the system’s mass, r is the damping coefficient, T is the period of the train of impulses, and t is time. Using the Duhamel integral (see Chapters 2 and 4), we obtain the following solution to Eq. (1):  t n  S e −r(t−τ ) δ(τ − kT ) sin ω∗ (t − τ ) dτ x (t) = Mω∗ 0 k =0

=

=

S Mω∗

n  t  k =0

0

e −r(t−τ ) δ(τ − kT ) sin ω∗ (t − k τ ) dτ

(2)

n S  −r(t−kT ) e sin ω∗ (t − kT ), Mω∗ k =0



where ω∗ = ω02 − r 2 is the natural frequency of the oscillator with consideration of damping. Expression (2) provides a solution in the “global” time, that is, in the time counted from the moment of the application of the first (n = 0) impulse. One can obtain the solution in the “local” time, that is, in the time counted from the moment of the application of the sequential nth impulse, by substituting in the above solution the time t with the time t + nT . This yields x (t) =

n n S  −r(t+jT ) S  −r[t+(n−k )T ] e sin ω∗ [t + (n − k )T ] = e sin ω∗ (t + jT ) Mω∗ Mω∗ (3) j =0 k =0

S −rt S −rt  e e An cos(ω∗ t + εn ), (Cn sin ω∗ t + Dn cos ω∗ t) = Mω∗ Mω∗ n

=

k =0

where Cn =

n 

e −krT cos k ω∗ T =

e rT − cos ω∗ T − e −nrT cos(n + 1)ω∗ T + e −(n+1)rT cos nω∗ T , 2(cosh rT − cos ω∗ T )

e −krT sin k ω∗ T =

sin ω∗ T − e −nrT sin(n + 1)ω∗ T + e −(n+1)rT sin nω∗ T , 2(cosh rT − cos ω∗ T )

k =0

Dn =

n  k =0

(4)

38

Linear and Nonlinear Vibrations Caused by Periodic Impulses and the amplitudes and the initial phase angles are expressed as   cos(n + 1)rT − cos(n + 1)ω∗ T 2 2 An = Cn + Dn = e −rT cosh rT − cos ω∗ T  1 + e −2(n+1)rT − 2e −(n+1)rT cos(n + 1)ω∗ T , = 1 + e −2rT − 2e −rT cos ω∗ T εn = − arctan

Cn e −rT − cos ω∗ T − e −nrT cos(n + 1)ω∗ T + e −(n+1)rT cos nω∗ T = − arctan , Dn sin ω∗ T − e −nrT sin(n + 1)ω∗ T = e −(n+1)rT cos ω∗ T

n = 0, 1, 2 . . . .

(5)

In the special case of undamped vibrations (r = 0) formulas (5) yield An = |cosec θ0 sin(n + 1)θ0 | εn = − 12 π + nθ0 ,

(6) n = 0, 1, 2 . . . . θ0 = 12 ω0 T , The obtained results show that the amplitudes of the undamped vibrations do not approach any finite limit with an increase in the number n of impulses but change periodically, within the range (0, cosec θ0 ), with the “frequency” θ0 = ω0 T /2. When θ0 = ω0 T /2 = mπ , m = 1, 2, . . . , an impulse resonance takes place. In this case the dimensionless amplitude An becomes An = n, that is, increases linearly, without any limit, with an n-fold increase after the action of the nth impulse. In another special case, when n → ∞, formulas (5) yield 1 e rT − cos ω∗ T , εn = ε∞ = − arctan An = A∞ =  . (7) sin ω∗ T 1 + e −2rT − 2e −rT cos ω∗ T As evident from these formulas, damped vibrations achieve steady-state condition in the time domain. When damping is small and has an effect only on the final steady-state character of the vibrations but not on the magnitude of the amplitude and the phase angle, these can be obtained by simply putting rT = 0 in formulas (7). This yields A∞ ∼ (8) ε∞ ∼ = 1 | cosec θ0 | , = −θ0 . 2

Hence, the amplitude of the steady-state nonresonant vibrations with low level of damping turns out to be half the maximum amplitude in the undamped case. This result was obtained first by Duffing from the conditions of periodicity: S x (0) = x (T ), x˙ (0) = x˙ (T ) + . (9) M These conditions indicate that if, because of a low but still finite (nonzero) damping, the shock-excited vibrations stabilize when time progresses and eventually become steady state, each sequential impulse does not change the displacement but changes the velocity with the step S /M . The numerator in this formula is, in effect, the additional momentum that the system gains as a result of the application of the sequential impulse and the corresponding energy compensates the loss in energy because of damping (energy dissipation). In the third special case, when θ0 = ω0 T /2 = mπ , m = 1, 2 . . . (impulse resonance in a system with damping), we obtain 1 − e −(n+1)rT π An = , εn = − . (10) 1 − e −rT 2 At the initial moment of time (n = 0), A0 = 1. When the vibrations stabilize and become steady state (n → ∞), 1 ∼ A∞ = . (11) 1 − e −rT

3

Shock-Excited Vibrations

39

Minimum amplitudes occur when θ0 =

2m + 1 π, 2

m = 1, 2 . . . .

(12)

In this case, 1 + e −(n+1)rT π , εn = − . (13) −rT 1+e 2 The minimum steady-state amplitude (n → ∞) is 1 A∞ ∼ . (14) = 1 + e −rT As evident from the obtained results, the steady-state amplitudes at resonant conditions always exceed the transient ones, while at nonresonant conditions the transient amplitudes may exceed considerably the steady state amplitudes. In the case of small damping, the ratio of the maximum transient amplitude to the steady-state amplitude obtained on the basis of formulas (6) and (8) is An,max = 2 |sin(n + 1)θ0 | . (15) A∞ Thus, the maximum transient nonresonant amplitude can be twice as large as the steady-state amplitude. This practically important conclusion should be considered when measures are taken to avoid resonance. In such a situation transient shock-excited vibrations can result in higher amplitudes, accelerations, and stresses than the steady-state ones. An =

3.2 Nonlinear Response: Steady-State Undamped Vibrations We proceed from the equation x¨ (t) + ω02 x (t) + μx 3 (t) =

n S  δ(t − kT ), M

(16)

k =0

where μ is the parameter of nonlinearity that one encounters with a nonlinear system with the cubic characteristic of the restoring force (“Duffing’s oscillator”), when analyzing the nonlinear response of PCBs to shocks and vibrations (see Chapters 6, 18, and 23). We seek the solution to Eq. (16) in the form S S x (t) = A cn(σ t + ε) = A cnu, (17) Mω0 Mω0 where cnu is an elliptic cosine, k is the modulus of the elliptic function, ε is the initial phase angle, and σ is the parameter of the nonlinear frequency (see, e.g., [71–73]). The quantity S /(Mω0 ) in the above formula is the amplitude of linear vibrations caused by a single instantaneous impulse of the magnitude S. Using the formulas (cnu) = −snudnu,

(snu) = cnudnu,

(dnu) = −k 2 snucnu

(18)

for differentiating elliptic functions, we obtain S x˙ (t) = −A σ snudnu, (19) Mω0   S x¨ (t) = −A σ 2 cnu 1 − 2k 2 sn 2 u . (20) Mω0 √ Here snu is an elliptic sine and dnu = 1 − k 2 sn 2 u is the function of delta amplitude. After substituting (17) and (20) into Eq. (16), we conclude that this equation is fulfilled if the

40

Linear and Nonlinear Vibrations Caused by Periodic Impulses parameter σ of the nonlinear frequency and the module k of the elliptic function are expressed as follows:  A μ σ = ω0 1 + μA2 , k= , (21) σ 2 where 2

S μ=μ (22) Mω02 is the dimensionless parameter of nonlinearity. Introducing Eq. (17) and (20) into the conditions of periodicity (9), we obtain the following two equations: cn(ε, k ) = cn(σ T + ε, k ), (23) −Aσsn(ε, k ) dn(ε, k ) = −Aσsn(σ T + ε, k ) dn(σ T + ε, k ) + ω0 .

(24)

From (23) we find sn(ε, k ) = ±sn(σ T + ε, k ),

dn(ε, k ) = ∓dn(σ T + ε, k ).

As it is known from the theory of elliptic functions, ⎧ snucnvdnv + snvcnudnu ⎪ ⎪ , ⎨ 1 − k 2 sn2 usn2 v sn(u + v) = ⎪ ⎪ ⎩ cnucnv − snvdnudnv . 1 − k 2 sn2 usn2 v Putting u = ε and v = σ T + ε and using formulas (25) we find that sn(σ T + 2ε, k ) = 0,

cn(σ T + 2ε, k ) = 1.

(25)

(26)

(27)

Then we obtain the following equation for the amplitude ϕ = amu of the elliptic function of interest: ϕ = am(σ T + 2ε, k ) = 2π i , i = 0, ±1, ±2, . . . . (28) 

Therefore σ T + 2ε =

2πi



0



where K (k ) =

dξ 1 − k 2 sin2 ξ

π/2

0



= 4iK (k ),

dξ 1 − k 2 sin2 ξ

(29)

(30)

is the complete elliptic integral of the first kind. From (29) we obtain the following formula for the initial phase angle: ε = 2iK (k ) − 12 σ T ,

i = 0, ±1, ±2, . . . .

(31)

In order to obtain an expression for the amplitude A of vibrations, let us substitute formulas (25) into (24). Then we obtain the following formula for the dimensionless vibration amplitudes: ω0 A= , (32) 2σ F where the notation F = F (ε, k ) = sn(ε, k ) dn(ε, k ) is used.

(33)

3

Shock-Excited Vibrations

41

Introducing the σ value from the first formula in (24) into (32) and solving the obtained biquadratic equation for the dimensionless amplitude A, we have  1 + μ/F 2 − 1 A= . (34) 2μ Since the period of an elliptic function is 4K (k ), the period of the function cn(σ t + ε, k ), that is, the period of vibrations, is (4/σ )K (k ). The corresponding frequency is π p=σ . (35) 2K (k ) In the case of linear vibrations (μ = 0), we have k = 0,

K (k ) = 12 π ,

sn(ε, 0) = sin ε = − sin

p = σ = ω0 ,

ε = i π − 12 ω0 T ,

ω0 T = sin θ0 , 2

dn(ε, 0) = 1,

ω0 T = sin θ0 , F = − sin 2 and formula (34) yields  1 + μ/F 2 − 1 1 1 A = lim = = | cosec θ0 | , μ→0 2μ 2 |F| 2

(36)

(37)

which coincides with Duffing’s formula (8). In the case of a single impulse, the initial conditions x (0) = 0,

x˙ (0) =

S M

(38)

should be applied. These yield cn(ε, k ) = 0,

−A1 σsn(ε, k ) dn(ε, k ) = ω0 ,

where A1 is the amplitude of vibrations caused by a single impulse S. Hence,  A1 σ 1 − k 2 = ω0 .

(39)

(40)

Considering formulas (21) and solving the obtained equation for the amplitude A1 , we obtain √ 1 + 2μ − 1 A1 = . μ Clearly,

√ 1 + 2μ − 1 lim A1 = lim = 1. μ→0 μ→0 μ

The function (33) in this case is 1 F= 2

   1 1 . 1+ √ 2 1 + 2μ

(41)

(42)

(43)

The dimensionless amplitude A of vibrations caused by a train of instantaneous impulses has its minimum when F = Fmax . From (33) we find √ 1 − 1 − 4k 2 F 2 sn 2 u = . (44) 2k 2

Linear and Nonlinear Vibrations Caused by Periodic Impulses Since an elliptic sine cannot exceed unity, the maximum possible value of the function F is therefore  Fmax = 1 − k 2 . (45) From (34) and (45) we conclude that the minimum value of the dimensionless amplitude A is √ 1 + μ/2 − 1 Amin = . (46) μ The maximum value of the function F can therefore be computed as  √ 1 + 1 + μ/2 . Fmax = √ 2 1 + μ/2

(47)

The calculated amplitudes of nonlinear steady-state shock-excited vibrations of a PCB subjected to periodic lateral impulsive loads applied to the PCB support contour are plotted in Fig. 1. The calculations were carried out using the relationships that could be found in Chapters 18 and 23. Both bending and reactive tensile stresses were considered in the total stress. Curves 1 and 2 correspond to a simply supported and a clamped PCB, respectively. The curve 3 was obtained for linear vibrations, in which only bending stresses were considered. It is noteworthy that while the nonlinear amplitudes are always smaller than linear ones, the total stresses might be comparable. This is due to the fact that the linear approach considers bending stresses only, although for substantially larger amplitudes, while the nonlinear approach considers both bending and in-plane (“membrane”) stresses. Figure 1 indicates also that the linear approach can be misleading when evaluating the response of the PCB to periodic shock loads for a particular frequency ratio since the nonlinear response is strongly affected by the level of the applied load. It should be emphasized that the nonlinear response is different for the nonlinear amplitudes, accelerations, frequencies, etc.

150

Dynamic stresses, s kg/mm2

42

100

2

1 3 50

0

0

0.5

1

1.5 ws ws

Figure 1

2

2.5

3

Shock-Excited Vibrations

43

3.3 Stochastic Instability Some deterministic systems reveal stochastic properties, that is, they oscillate in a quasirandom manner in spite of the nonrandom nature of the excitation. Quasi-random vibrations of deterministic systems subjected to deterministic excitation could be caused, in particular, by the stochastic instability of the phase angle. The physical nature of such instability in the case of an excitation caused by a train of instantaneous impulses can be explained as follows. The phase angle of linear vibrations occurring immediately after the action of the nth impulse is expressed by the second formula in (6) as follows: π π ω0 T π ω0 εn = − + nθ0 = − + n (48) = − + nπ , 2 2 2 2 ωs where ωs = 2π/T is the frequency of the train of impulses. Let us assume that, in an approximate analysis, this formula can be applied to nonlinear vibrations as well and that the frequency of nonlinear vibrations expressed by formula (35) can be evaluated on the basis of the first formula in (21):    p ≈ σ = ω0 1 + μA2 ≈ ω0 1 + 12 μA2 . (49) Then we have



π 1 μ 2 εn ≈ − + nπ η 1 + A , 2 2 ω02 n

(50)

where ω0 (51) ωs is the ratio of the natural linear frequency of the undamped vibrations to the frequency of the train of the impulses. Due to inevitable random disturbances in the frequency of the train of impulses, to uncertainties associated with the evaluation of the natural frequency of vibrations, and especially in the magnitude of the applied impulses, the values of η and An are never known with certainty. These values should be treated therefore as random variables, although, perhaps, with very small variances. Using expression (50), we obtain the following formula for the variance Dεn of the phase angle after the nth impulse is applied: ⎡ ⎤ 2

    1 μ 2

η2 DA2n + A2n Dη + Dη DA2n ⎦. Dεn = n 2 π 2 ⎣Dη + (52) 2 ω02 η=

Here Dη is the variance of the frequency ratio η, DA2n is the variance of the amplitude squared   A2n , η is the mean value of the frequency ratio, and A2n is the mean value of the amplitude squared. As evident from formula (52), the uncertainty in the phase angle measured by the magnitude of its variance increases in the time domain with an increase in the applied impulse number n squared. Therefore, at the moment of time when a new impulse is applied, the information about the initial phase angle for the preceding period of time is partially or completely lost. Because of that the amplitude caused by this impulse and to a great extent affected by the phase angle at the moment of the application of the impulse cannot be evaluated with sufficient certainty and should be treated as a random variable. Such a situation is especially strongly pronounced in nonlinear systems, where both variances Dη and DA2n contribute to the variance Dεn of the phase angle. A simulation procedure based on the approximate solution S −nT x (t) = An cos εn , n = 0, 1, . . . , (53) e Mω

Linear and Nonlinear Vibrations Caused by Periodic Impulses Dimensionless amplitudes An

44

5 w0T = 100

4 3 2

1500

1 0

0

20

40

60 80 Number of impulses n

100

120

Figure 2 Quasi-stochastic shock-excited vibrations. has been carried out. An example of the calculated response is shown in Fig. 2. The plot proves the quasi-stochastic nature of the oscillations. Thus, it is natural to use, right from the very beginning a probabilistic approach, to describe the behavior of systems prone to the development of stochastically unstable behaviors. How it could be done is shown, as an illustration, in the next section.

3.4 Stochastic Phase Approximation When applying a probabilistic approach, we find it more convenient to proceed from energy considerations. The velocities of oscillations prior to and after the application of a sequential impulse are related as S x˙n+ = x˙n− + . (54) M The corresponding kinetic energies are related as follows: Kn+ = Kn− + H0 + S x˙n− ,

(55)

is the energy due to a single impulse, that is, the energy absorbed where H0 = by the system after the first shock. An instantaneous impulse does not change the potential energy of the system, and therefore Eq. (55) holds for the total energy of the system as well: S 2 /(2M )

We assume

Hn+ = Hn− + H0 + S x˙n− .

(56)

x˙ (t) ≈ x˙max cos ε,

(57)



where

2H (58) M is the maximum velocity and ε is the phase angle. Then relationship (56) can be written as  Hn+ = Hn− + H0 + 2 H0 Hn− cos ε− (59) x˙max =

Assuming, for the sake of simplicity, that the energy of free damped vibrations fades away exponentially, we obtain the following formula for the total energy at the moment of time immediately following the (n + 1)th impulse:  − . (60) Hn+1 = Hn e −2rT + H0 + 2e −rT H0 Hn cos εn+1

3

Shock-Excited Vibrations

45

− Here εn+1 is the phase angle prior to the application of the (n + 1)th shock. Introducing dimensionless energy hn = Hn /H0 and using notation τ = e −rT , we obtain the following formula for the dimensionless energy h:  − hn+1 = 1 + τ 2 hn + 2τ hn cos εn+1 , n = 0, 1, . . . . (61)

If one wishes to obtain a deterministic solution for the energy evolution, it would be necessary to introduce an additional relationship for the change in the phase angle. Such a relationship could be obtained from the condition that the shock loading does not change the system’s position. This would lead to the condition xn+ = xn− . In the present case, however, when trying to find probabilistic characteristics of the oscillations, one can use, examining an extreme case, the stochastic phase approximation as an additional condition. This approximation presumes that when the standard deviation Dεn of the phase angle at the moment of an impact exceeds 2π , the phase angle can be viewed as an “absolutely random,” that is, absolutely unpredictable, variable. Assuming that this angle is a uniformly distributed random variable and treating the dimensionless energy hn+1 as a deterministic function of the random − variable εn+1 , we obtain the following equation for the probability transition function: Q(hn+1 , hn ) =

1  π

1  2 , 4τ 2 hn − hn+1, − τ 2 hn, − 1

n = 0, 1, . . . .

(62)

3.5 Smoluchowski’s Equation The probability transition function is the cornerstone of the theory of continuous Markovian processes. A random (stochastic) process is a Markovian one, or a process without an aftereffect, if the probability of any state of the system characterized by this process in the “future” (t > t0 ) depends only on its state in the “present” (t = t0 ) and is independent of how the system reached the present state, that is, independent from the “prehistory,” the “past” of the process. Markovian processes occupy a special and very important place in the theory of random processes. This is due to the fact that the behavior of many engineering and physical systems can be meaningfully modeled using the formalism of Markovian process theory. Let a transition x0 , t0 ⇒ x , t characterized by the probability f (x , t/x0 , t0 ) be realized through some intermediate state ξ , τ. The relationship   ∞      t t τ (63) f x , , t0 = f x , , τ f ξ , , t0 d ξ , x0 ξ x0 −∞ which, in effect, reflects the rule of the addition of probabilities, is known as Smoluchowski’s equation. In the mathematical literature this equation is usually referred to as the Chapman–Kolmogorov equation. The integral equation (63) expresses a more or less obvious fact that the transitional probabilities for any three sequential moments of time must be in a certain way related to each other.

3.6 Solution Based on Smoluchowski’s Equation With the probability transition function (62), the evolution of the probability density distribution function can be computed based on the Smoluchowski’s equation  hn,max fn+1 (hn+1 ) = fn (hn )Q(hn+1, hn )dhn n = 0, 1, . . . . (64) hn,min

Note that in the nondissipative case (τ = 1) the function (62) is symmetric with respect to its arguments, and the kernel of the integral equation (64) is also symmetric. This means

Linear and Nonlinear Vibrations Caused by Periodic Impulses that in such a case the random process of undamped shock-excited vibrations is reversible in the time domain. The initial probability density function defines a nonrandom “distribution” expressed by a delta function f0 (h0 ) = δ(h0 ). In this analysis we assume that the nonlinearity is weak and affects only the very fact of the development of the stochasticity but does not result in any essential change in the extreme hn values. For this reason, the limits of integration in Eq. (64) can be determined on the basis of the formula (61), so that     1 + τn+1 2 1 − τn+1 2 hn,min ≈ , hn,max ≈ n = 0, 1, . . . . (65) 1+τ 1−τ These limits define the widest range of the possible energy changes. It may occur, however, that large values of hn+1 are not possible with small values of hn,min . On the other hand, small hn+1 values might not be possible for large enough hn,max . Such hn,max values might take place, for instance, in a hypothetical case, when all the preceding impulses occurred at the moments of time, corresponding to the extreme conditions cos ε− = ±1. In actual calculations, the necessity to narrow the interval of integration will be indicated by the negative value of the expression in parentheses in (62). Such narrowed limits can be found by simply requiring that this expression is positive. This yields √ √ 2 2 hn+1 − 1 hn+1 + 1 hn,min ≈ , hn,max ≈ . (66) τ τ The governing equation (64) does not lend itself to an analytical solution, which is typical for Smoluchowski’s equation. It was solved numerically assuming the following initial

n=0 0.6 n=1 rT = 0.2

0.5 Probability density function f (h)*

46

0.4 n=2 0.3

0.2

0

2

8

∞

n

0

hnmax

1

3.308 6.195 2.120 3.043

hnmin

1

0.0328 0.725 0.410 1.492

hn

1

1.6700 2.118 2.948 3.033

0.1 n=8 0

0

5

10

15

Hn Nondimensional energy hn = H0

Figure 3

Probability density distribution functions for shock-excited vibrations.

References

47

distribution: f1 (h1 ) = Q(h1 , h0 ) = Q(h1 , 1) =

1  π



1

4τ 2 − h1 − τ 2 − 1

2

(67)

and the initial limits for the dimensionless energy are h1,min ≈ (1 − τ )2 ,

h1,max ≈ (1 + τ )2 .

(68)

Some computed probability density functions are plotted in Fig. 3 for the case rT = 0.2. In the accompanying table the magnitudes of the extreme values hn,max and hn,min are indicated as well as the mean value hn  of the dimensionless energy. As one could see from the computed data, the initial deltalike probability density function for the dimensionless energy is getting spread out when time progresses and, for damped vibrations, approaches a certain steady-state condition. One important result from the carried-out analysis is that for large enough n values the energy distribution probability density function is strongly skewed to the left: low levels of energy are much more likely than large ones, and the mean values hn  of the dimensionless vibration energy are considerably closer to hn,min than to hn,max .

REFERENCES General Publications 1. Timoshenko, S. P., Vibration Problems in Engineering, 4th ed., Wiley, New York, 1974. 2. Steinberg, D., Vibration Analysis for Electronic Equipment, 2nd ed., Wiley, New York, 1988. 3. Suhir, E., “Analytical Modeling in Structural Analysis for Electronic Packaging: Its Merits, Shortcomings and Interaction with Experimental and Numerical Techniques,” ASME Journal of Electronic Packaging, Vol. 111, No. 2, June 1989. 4. Suhir, E., “Dynamic Response of Microelectronics and Photonics Systems to Shocks and Vibrations,” INTERPack’97, Hawaii, June 15–19, 1997. 5. Suhir, E., Applied Probability for Engineers and Scientists, McGraw-Hill, New York, 1997. 6. JEDEC Standard JESD22-B104-B, Mechanical Shock, 2001. 7. JEDEC Standard JESD22-B111, Board Level Drop Test Method of Components for Handheld Electronic Products, 2003. 8. Isbell, W. M., Shock Waves: Measuring the Dynamic Response of Materials, Imperial College Press, London, 2005. 9. Suhir, E., “Dynamic Response of Micro-Electronic Systems to Shocks and Vibrations: Review and Extension,” in E. Suhir, C. P. Wong, and Y. C. Lee, Eds., Micro- and OptoElectronic Materials and Structures: Physics, Mechanics, Design, Packaging, Reliability, Springer, New York, 2007. 10. Suhir, E., “Dynamic Response of Micro-Electronic Structural Elements to Shocks and Vibrations,” keynote presentation, MicroNanoReliability Congress, Berlin-Koepenick, September 2–5, 2007.

Linear Response 11. Suhir, E., “Shock-Excited Vibrations with Application to the Slamming Response of a Flexible Ship to a Regular Wave Packet,” SNAME Journal of Ship Research, Vol. 26, No. 4, 1982.

48

Linear and Nonlinear Vibrations Caused by Periodic Impulses 12. Suhir, E., “Free Vibrations of a Fused Biconical Taper Lightwave Coupler,” International Journal of Solids and Structures, Vol. 29, No. 24, 1992. 13. Suhir, E., “Vibration Frequency of a Fused Biconical Taper (FBT) Lightwave Coupler,” IEEE/OSA Journal of Lightwave Technology, Vol. 10, No. 7, 1992. 14. Suhir, E., “Elastic Stability, Free Vibrations, and Bending of Optical Glass Fibers: The Effect of the Nonlinear Stress-Strain Relationship,” Applied Optics, Vol. 31, No. 24, 1992. 15. Suhir, E., “Dynamic Response of a One-Degree-of-Freedom Linear System to a Shock Load During Drop Tests: Effect of Viscous Damping,” IEEE CPMT Transactions, Part A, Vol. 19, No. 3, 1996. 16. Suhir, E., “Is the Maximum Acceleration an Adequate Criterion of the Dynamic Strength of a Structural Element in an Electronic Product?” IEEE CPMT Transactions, Part A, Vol. 20, No. 4, Dec. 1997. 17. Wu, J., Song, G., Yeh, C-p., and Wyatt, K., “Drop/Impact Simulation and Test Validation of Telecommunication Products,” Thermomechanical Phenomena in Electronic Systems—Proceedings of the 1998 6th Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems, ITHERM, May 27–30 1998, pp. 330–336. 18. Wu, J., “Global and Local Coupling Analysis for Small Components in Drop Simulation,” 6th International LS-DYNA Users Conference Proc., 2000. 19. Zhu, L., “Submodeling Technique for BGA Reliability Analysis of CSP Packaging Subjected to an Impact Loading”, InterPACK Conference Proc., 2001 20. Suhir, E., “Could Shock Tests Adequately Mimic Drop Test Conditions?” 52nd Electronic Components and Technology Conference (ECTC) Proc., 2002 21. Goyal, S., Buratynski, E. K., and Elko, G. W., “Shock-Protection Suspension Design for Printed Circuit Board,” Proceedings of SPIE , Vol. 4217, 2002. 22. Zhu, L., “Submodeling Technique for BGA Reliability Analysis of CSP Packaging Subjected to an Impact Loading,” ASME InterPACK Conference Proc., 2001. 23. Sogo, T., and Hara, S., “Estimation of Fall Impact Strength for BGA Solder Joints,” ICEP Conference Proc., Japan, 2001. 24. Arra, M., et al., “Performance of Lead Free Solder Joints under Dynamic Mechanical Loading,” Proc. 52nd Electronic Components and Technology Conference (ECTC), 2002. 25. Yu, Q., Kukuichi, H., Ikeda, S., Shiratori, M., Kakino, M., and Fujiwara, N., “Dynamic Behavior of Electronics Package and Impact Reliability of BGA Solder Joints,” Intersociety Conf. on Thermal Phenomena, 2002. 26. Mishiro, K., et al., “Effect of the Drop Impact on BGA/CSP Package Reliability,” Microelectronics Reliability, Vol. 42, 2002. 27. Tee, T. Y., Ng, H. S., Lim, C. T., Pek, E., and Zhong, Z. W., “Application of Drop Test Simulation in Electronic Packaging,” paper presented at the 4th ASEAN ANSYS Conf., 2002. 28. Wong, E. H., Lim, C. T., Lee, N., Seah, S. K. W., Hoe, C., and Wang, J., “Drop Impact Test—Mechanics and Physics of Failure,” 4th Electronic Packaging and Technology Conference (EPTC), Proc., Singapore, 2002. 29. Tan, L. B., “Board Level Solder Joint Failure by Static and Dynamic Loads,” Proc. 5th EPTC, Singapore, 2003. 30. Xie, D., et al., “Solder Joint Behavior of Area Array Packages in Board-Level Drop for Hand Held Devices,” 53rd Electronic Components and Technology Conference (ECTC), 2003.

References

49

31. Xu, L., et al., “Numerical Studies of the Mechanical Response of Solder Joints to Drop/Impact Load,” Proc. EPTC, 2003. 32. Tee, T. Y., Ng, H. S., Lim, C. T., Pek, E., and Zhong, Z. W., “Board Level Drop Tests and Simulation of TFBGA Packages for Telecommunication Applications,” 53rd ECTC Proc., May 2003. 33. Zhu, L., “Modeling Technique for Reliability Assessment of Portable Electronic Product Subjected to Drop Impact Loads,” 53rd ECTC, 2003, pp. 100–104. 34. Wang, Y. Q., “Modeling and Simulation of PCB Drop Test,” Proc. 5th EPTC, pp. 263–268, 2003. 35. Luan, J. E., Tee, T. Y., Pek, E., Lim, C. T., and Zhong, Z. W., “Modal Analysis and Dynamic Responses of Board Level Drop Test,” 5th EPTC Conference 4 Proc., Singapore, 2003. 36. Liping, Z., Marcinkiewicz, W., “Drop Impact Reliability Analysis of CSP Packages at Board and Product System Levels through Modeling Approaches.” paper presented at the Ninth Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems, Vol. 2, IEEE, New York, 2004. 37. Luan, J. E., and Tee, T. Y., “Analytical and Numerical Analysis of Impact Pulse Parameters on Consistency of Drop Impact Results,” paper presented at the 6th Electronics Packaging Technology Conference, Dec. 8–9, 2004, IEEE, New York. 38. Ong, K. C., et al., Dynamic Materials Testing and Modeling of Solder Interconnects, 54th ECTC, 2004. 39. Date, M., et al., “Ductile-to-Brittle Transition in Sn-Zn Solder Joints Measured by Impact Tests,” Scripta Materialia, Vol. 51, 2004. 40. Yeh, C.-L., and Lai, Y.-S., “Effect of Solder Joint Shapes on Free Drop Reliability of Chip-Scale Packages,” Proc. IMAPS Taiwan Tech. Symp., 2004. 41. Luan, J. E., and Tee, T. Y., “Effect of Impact Pulse on Dynamic Responses and Solder Joint Reliability of TFBGA Packages During Board Level Drop Test,” paper presented at the 6th EMAP Conference, Malaysia, Dec. 2004 42. Chiu, T. C., et al., “Effect of Thermal Aging on Board Level Drop Reliability for PbFree BGA Packages,” 54th Electronic Components and Technology Conference (ECTC), 2004. 43. Zhu, L., and Marcinkiewicz, W., “Drop Impact Reliability Analysis of CSP Packages at Board and Product System Levels through Modeling Approaches,” paper presented at the Intersociety Conference on Thermal and thermo-mechanical Phenomena (ITHERM) 2004, pp. 296–303. 44. Irving, S., and Liu, Y., “Free Drop Test Simulation for Portable IC Package by Implicit Transient Dynamics FEM,” 54th ECTC, 2004. 45. Tee, T. Y., Luan, J. E., Pek, E., Lim, C. T., and Zhong, Z. W., “Novel Numerical and Experimental Analysis of Dynamic Responses under Board Level Drop Test,” EuroSime Conference Proc., 2004. 46. Tee, T. Y., Luan, J. E., Pek, E., Lim, C. T., and Zhong, Z. W., “Advanced Experimental and Simulation Techniques for Analysis of Dynamic Responses During Drop Impact,” 54th ECTC Proc., 2004, pp. 1088–1094. 47. Yeh, C. L., and Lai, Y. S., “Transient Simulation of Solder Joint Fracturing Under Impact Test,” 6th EPTC, Singapore, Dec. 8–10, 2004. 48. Syed, A., et al., “A Methodology for Drop Performance Prediction and Application for Design Optimization of Chip Scale Packages,” 55th ECTC, 2005.

50

Linear and Nonlinear Vibrations Caused by Periodic Impulses 49. Marjamaki, P., Mattila, T., and Kivilahti, J., “FEA of Lead-Free Drop Test Boards,” 55th ECTC Proc., 2005. 50. Suhir, E., “Response of a Heavy Electronic Component to Harmonic Excitations Applied to Its External Electric Leads,” Elektrotechnik & Informationstechnik (Austria), Vol. 9, 2007.

Nonlinear Response 51. Suhir, E., “Linear and Nonlinear Vibrations Caused by Periodic Impulses,” paper presented at the AIAA/ASME/ASCE/AHS 26th Structures, Structural Dynamics and Materials Conference, Orlando, FL, Apr. 1985. 52. Suhir, E., “Nonlinear Dynamic Response of a Flexible Printed Circuit Board to Shock Loads Applied to Its Support Contour,” Proc. of the 41st Elect. Comp. and Techn. Conf., IEEE, Atlanta, GA, May 1991. 53. Suhir, E., “Response of a Flexible Printed Circuit Board to Periodic Shock Loads Applied to Its Support Contour,” ASME Journal of Applied Mechanics, Vol. 59, No. 2, 1992. 54. Suhir, E., “Nonlinear Dynamic Response of a Flexible Thin Plate to Constant Acceleration Applied to Its Support Contour, with Application to Printed Circuit Boards Used in Avionic Packaging,” International Journal of Solids and Structures, Vol. 29, No. 1, 1992. 55. Suhir, E., “Shock Protection with a Nonlinear Spring,” IEEE CPMT Transactions, Advanced Packaging, Part B, Vol. 18, No. 2, 1995. 56. Suhir, E., “Shock-Excited Vibrations of a Conservative Duffing Oscillator with Application to Shock Protection in Portable Electronics,” International Journal of Solids and Structures, Vol. 33, No. 24, 1996. 57. He, X., and Fulton, R., “Nonlinear Dynamic Analysis of a Printed Wiring Board,” ASME Journal of Electronic Packaging, Vol. 124, No. 2, 2002. 58. He, X., and Stallybrass, M., “Impact Response of a Printed Wiring Board,” International Journal of Solids and Structures, Vol. 39, No. 24, 2002. 59. Suhir, E., and Arruda, L., “The Coordinate Function in the Problem of the Nonlinear Dynamic Response of an Elongated Printed Circuit Board (PCB) to a Drop Impact Applied to Its Support Contour,” European Journal of Applied Physics, Vol. 48, No. 2, 2009, in preparation. 60. Suhir, E., and Arruda, L., “Could an Impact Load of Finite Duration Acting on a Duffing Oscillator Be Substituted with an Instantaneous Impulse?” Journal of Solid Mechanics and Materials Engineering, Vol. 4, No. 9, 2010. 61. Suhir, E., Vujosevic, M., and Reinikainen, T., “Nonlinear Dynamic Response of a ‘Flexible-and-Heavy’ Printed Circuit Board (PCB) to an Impact Load Applied to Its Support Contour,” Journal of Applied Physics, D, Vol. 42, No. 4, 2009.

Shock Protection of Portable Electronics 62. Suhir, E., and Burke, R., “Dynamic Response of a Rectangular Plate to a Shock Load, with Application to Portable Electronic Products,” IEEE Transactions on Components, Packaging, and Manufacturing Technology, Part B: Advanced Packaging, Vol. 17, No. 3, 1994. 63. Suhir, E., “Shock-Excited Vibrations of a Conservative Duffing Oscillator with Application to Shock Protection in Portable Electronics,” International Journal of Solids and Structures, Vol. 33, No. 24, 1996.

References

51

64. Goyal, S., Papadopoulos, J. M., and Sullivan, P. A., “Shock Protection of Portable Electronic Products: Shock Response Spectrum, Damage Boundary Approach, and Beyond,” Shock and Vibration, Vol. 4, No, 3, 1997. 65. Huang, W., Kececioglu, D. B., and Prince, J. L., “A Simplified Random Vibration Analysis on Portable Electronic Products,” IEEE Transactions on Components & Packaging Technologies, Vol. 23, No. 3, 2000. 66. Seah, S. K. W., Lim, C. T., Wong, E. H., Tan, V. B. C., and Shim, V. P. W., “Mechanical Response of PCBs in Portable Electronic Products During Drop Impact,” Proceedings 4th Electronics Packaging Technology Conference (EPTC 2002), Singapore, Dec. 10–12, 2002. 67. Lim, C. T., and Low, Y. J., “Investigating the Drop Impact of Portable Electronic Products,” 52nd ECTC Proceedings, May 28–31, 2002. 68. Lim, C. T., Teo, Y. M., and Shim, V. P. W., “Numerical Simulation of the Drop Impact Response of a Portable Electronic Product,” IEEE CPMT Transactions, Vol. 25, No. 3, Sept. 2002. 69. Lim, C. T., Ang, C. W., Tan, L. B., Seah, S. K. W., and Wong, E. H., “Drop Impact Survey of Portable Electronic Products,” 53rd ECTC, IEEE, May 2003. 70. Zhu, L., “Modeling Technique for Reliability Assessment of Portable Electronic Product Subjected to Drop Impact Loads,” 53rd ECTC, 2003, pp. 100–104.

Elliptic Functions 71. Appell, P., and Lacour, E., Principles de la Theorie des Functions Elliptiques et Applications, 2nd ed., Paris, Gauthier Villars, 1922. 72. Oberhettinger, F., and Magnus, W., Anwendung der elliptischen Funktionen in Physik und Technik (Die grundlehren der Mathematischen Wissenschafter, Vol. 55), Springer Verlag, Berlin, 1949. 73. Spanier, J., and Oldham, K. B., An Atlas of Functions, Hemisphere Washington, DC 1987.

CHAPTER

4

RANDOM VIBRATIONS OF STRUCTURAL ELEMENTS IN ELECTRONIC AND PHOTONIC SYSTEMS Ephraim Suhir University of California, Santa Cruz, California University of Maryland, College Park, Maryland ERS Co., Los Altos, California

“Chance governs all” —John Milton, Paradise Lost

1

INTRODUCTION Engineering products must have a worthwhile lifetime and operate during this time and under stated conditions successfully to match the user’s expectations. To achieve this, the engineer must understand the ways in which the useful life in service of the product can be evaluated and to incorporate this understanding in the design. His or her tasks are to analyze, design, test, fabricate, operate, and maintain the product, system, or structure at all stages of its creation and use, from the moment of conception and substantiation of the initial idea to the moment of writing off and wrapping the product, so that it does not fail during the service period [1]. When solving these problems, the engineer inevitably encounters variability in the employed materials, loads, manufacturing processes, testing techniques, and applications. The traditional engineering approaches, when dealing with these problems, are referred to as deterministic, that is, do not pay sufficient attention to the variability of the parameters and criteria used. Such approaches are acceptable and can be justified in many cases when the deviations (“fluctuations”) from the mean values are small, when the design parameters are known or can be predicted with reasonable accuracy, and when the processes and procedures that the engineer deals with are “stochastically stable,” that is, when “small causes” result in “small consequences (effects).” There are however numerous situations in which the fluctuations from the anticipated (mean) values are significant and in which the variability, change, and uncertainties play a vital role. In such situations the product will most likely fail if these uncertainties are ignored. Understanding the role and significance of the “laws of chance” and the causes and effects of variability in material properties, structural dimensions, mechanical tolerances, loading conditions, uncertainties in stresses and strengths, possible applications and environmental conditions, and a multitude of other design, manufacturing, and service parameters is critical for the creation and successful operation of a viable and reliable product or structure. In many practically important cases, the random nature and uncertainties in the design characteristics can be described and evaluated on the basis of the methods of the theory of probability (see, e.g., [2]). Probabilistic methods proceed from the fact that various uncertainties is an inevitable and essential feature of the nature of an engineering system or design and provide ways Structural Dynamics of Electronic and Photonic Systems Copyright © 2011 John Wiley & Sons, Inc.

Edited by Ephraim Suhir, David S. Steinberg and T. X. Yu

53

54

Random Vibrations of Structural Elements in Electronic and Photonic Systems of dealing with quantities whose values cannot be predicted with absolute certainty. Unlike deterministic methods, probabilistic approaches address more general and more complicated situations in which the behavior of the given characteristic or parameter cannot be determined (predicted) with certainty in each particular experiment or a situation but, for products manufactured in large quantities and for experiments which are repeated many times in identical or at least similar conditions, follow certain “statistical” (nonrandom) relationships. These manifest themselves as nonrandom trends in a large number of random events. Probabilistic predictive models reflect the actual physics of phenomena and the inevitable variability of the characteristics of an engineering system much better and in a more consistent fashion than deterministic models do. Probabilistic models enable one to establish the scope and the limitations of the deterministic theories and provide a solid and well-substantiated basis for a goal-oriented accumulation and effective use of empirical data. It would not be an exaggeration to say that all the fundamental theories and approaches of modern physics and applied science are probabilistic and contain the corresponding deterministic models as the first approximations. Realizing that the probability of failure of an engineering system is never zero, the probabilistic methods enable one to quantitatively assess the degree of uncertainty in various factors which determine the performance of the product and to design, on this basis, a product with a low and acceptable probability of failure. Probabilistic methods, underlying the modern methods of forecasting and decision making (known as probabilistic risk management), enable one to extend the accumulated experience on new products and designs, which may differ from the existing ones by type, dimensions, materials, operation conditions, and so on. We would like to emphasize that the use of the probabilistic methods and approaches in physics, engineering, and applied science is due not so much to the insufficient or unavailable information for a deterministic analyses but to the fact that the variability and uncertainty are inherent in the very nature of physical phenomena, materials properties, engineering designs, and application conditions. Having said that, we would like to point out that quite often a serious obstacle for applying probabilistic methods is indeed the difficulty in obtaining the input information necessary for a probabilistic prediction. If this is the case, the design has to be based on the worst-case scenario, that is, in effect, on a deterministic situation or on a suitable combination of probabilistic and deterministic approaches. In this chapter we provide as illustrations some problems that show how probabilistic methods are used in some problems of the dynamic response of electronic systems to shocks and vibrations. The analyses in this chapter are designed for readers that are reasonably proficient in applied mathematics.

2

ELONGATED PCB SUBJECTED TO EXTERNAL LOADING: NONLINEAR EQUATIONS Consider an elongated printed circuit board (PCB) whose long edges cannot get closer during vibrations caused by a load applied to the PCB support contour. One encounters such a situation during drop or shock testing of electronic products on the brand level. These vibrations can be described by the equation   ∂ ∂ 4w ∂ 2w ∂ 2w D 1 + 2ε + m 2 − σx0 (t)h 2 = q(x , t), (1) 4 ∂t ∂x ∂t ∂x where t is time, w (x , t) is the deflection function of the board, m is its mass per unit length, h is its thickness, q(x , t) is the external loading per unit length, ε is the coefficient of viscous damping, σx0 is the in-plane reactive normal stress acing at the PCB cross sections, D=

Eh 3 12(1 − ν 2 )

(2)

2

Elongated PCB Subjected to External Loading: Nonlinear Equations

55

is the flexural rigidity of the PCB treated as a thin-and-flexible plate, E is the Young’s modulus of the PCB material, and ν is its Poisson’s ratio. The origin of the coordinate x is at the mid–cross section of the board. For the sake of simplicity, we assume that the load q(x , t) is symmetric with respect to the origin. The first term in Eq. (1) is due to the forces caused by the elastic deformations of the board, with consideration of the role of the viscous damping (second term in parentheses). The second term in Eq. (1) is due to the inertia forces. The third term is due to the reactive in-plane forces that arise because of the inability of the PCB edges to move closer during the PCB motions in bending. The σx0 (t) is the corresponding in-plane normal stress. This stress is the same for all PCB cross sections, that is, coordinate x independent, but is time dependent: it is equal to zero when the PCB deflections are zero and reaches its maximum value when these reach their maximum values. The product σx0 (t)h is the in-plane force. The derivative ∂ 2 w /∂x 2 is the PCB curvature with respect to the x coordinate. Equation (1) can be obtained from the following von Karman equations of large deflections of a rectangular plate [3]: ∂ 2 w (x , y, t) = hL(ϕ, w ) + q(x , y, t), ∂t 2 (3) 1 1 4 ∇ ϕ(x , y, t) = − L(w , w ). E 2 Here ϕ(x , y, t) is the stress (Airy) function, which is such that the in-plane stresses σx0 (t) and 0 (t) can be found as σy0 (t), the shearing stress τxy D∇ 4 w (x , y, t) + m

σx0 (t) =

∂ 2ϕ , ∂y 2

σy0 (t) =

∂ 2ϕ , ∂x 2

0 τxy (t) = −

∂ 2ϕ ∂x ∂y

(4)

and the operators ∇ 4 w (x , y, t) and L(ϕ, w ) are expressed as ∇ 4 w (x , y, t) =

∂4 ∂4 ∂4 +2 2 2 + 4, 4 ∂x ∂x y ∂y

(5)

∂ 2w ∂ 2ϕ ∂ 2w ∂ 2ϕ ∂ 2w ∂ 2ϕ − 2 . (6) + ∂x 2 ∂y 2 ∂x ∂y ∂x ∂y ∂y 2 ∂x 2 The operator (5) is known as the biharmonic operator. In accordance with von Karman’s theory, the displacements u(x , y, t) and v(x , y, t) of the PCB points in the x and y directions, respectively, can be found, for PCBs whose elastic constants are the same for all the PCB points, by the formulas    σx0 (t) − νσy0 (t) 1 x ∂w 2 dx , u(x , y, t) = x− E 2 0 ∂x (7)    σy0 (t) − νσx0 (t) 1 y ∂w 2 v(x , y, t) = dy. y− E 2 0 ∂y If the PCB contour is nondeformable, then the following conditions should take place: L(ϕ, w ) =

u(±a, y, t) = v(x , ±b, t) = 0,

(8)

where a is half the PCB dimension in the x direction and b is half the PCB dimension in the y direction. Then formulas (7) yield         E ν b ∂w 2 1 a ∂w 2 0 dx + dy , σx (t) = 2(1 − ν 2 ) a 0 ∂x b 0 ∂y (9)         ν a ∂w 2 E 1 b ∂w 2 0 σy (t) = dx + dy . 2(1 − ν 2 ) a 0 ∂x b 0 ∂y

56

Random Vibrations of Structural Elements in Electronic and Photonic Systems In the case of an elongated PCB (b → ∞) we have   a E ∂w 2 dx , σx0 (t) = 2a(1 − ν 2 ) 0 ∂x   a ∂w 2 Eν σy0 (t) = dx = νσx0 (t), 2a(1 − ν 2 ) 0 ∂x and Eq. (3) yields      ∂ ∂ 4w ∂ 2 w a ∂w 2 ∂ 2w Eh D 1 + 2ε + m − dx = q(x , t). ∂t ∂x 4 ∂t 2 2a(1 − ν 2 ) ∂x 2 0 ∂x

(10)

(11)

We use the method of principal coordinates and Galerkin’s method to obtain the solution to Eq. (11). In accordance with the method of principal coordinates we seek this solution in the form of a series: ∞  w (x , t) = Tk (t)Xk (x ), (12) k =1

where Xk (x ) is the (known) coordinate function of the k th mode of vibrations and Tk (t) is thus far an unknown principal coordinate of this mode. The functions Xk (x ) are selected in such a way that the appropriate boundary conditions for the displacements w (x , t) are satisfied. In accordance with the procedure of Galerkin’s method, we introduce the expansion (12) into Eq. (11), multiply the obtained equation by Xj (x ), and integrate the obtained expression over half of the PCB width. The functions Xk (x ), being coordinate functions, possess the property of mutual orthogonality. This means that the following conditions are fulfilled: ⎧  a for k = j , ⎨ 0 a a Xk (x )Xj (x ) dx = (13) 2 Xk (x ) dx = for k = j . ⎩ 0 2 0 Then we obtain the following system of equations for the unknown principal coordinates Tk (t): T¨ k (t) + 2rk T˙ k (t) + ωk2 Tk (t) + Tk (t)

∞ 

δik Ti2 (t) =

i =1

where



a

Mk = m 0



a

Qk (t) =

Qk (t) , Mk

k = 1, 2, . . . ,

Xk2 (x ) dx = generalized mass of k th mode,

q(x , t)Xk (x ) dx = generalized excitation force of k th mode,

(14)

(15) (16)

0

ωk2 =

D Mk



a 0

XkIV (x )Xk (x ) dx =

D Mk



a



0

Xk (x )

2

dx

= square of frequency of undamped vibrations of k th mode, rk = ε

D Mk

 0

a

XkIV (x )Xk (x ) dx = ε

= coefficient of damping,

D Mk



a 0

(17)

  2 Xk (x ) dx (18)

3

Dynamic Response of Multi-Degree-of-Freedom Linear System to External Loading

57

 a  a   2 Eh X (x ) dx Xi (x )Xi (x ) dx k 2(1 − ν 2 )aMk 0 0  a  a   2   2 Eh = X Xi (x ) dx (x ) dx k 2 2(1 − ν )aMk 0 0

δik = −

= factor of nonlinearity (of in-pane stress), Introducing the notation



a

κk = 0



Xk (x )

2

dx ,

(19)

(20)

we obtain the following formula for the factor of nonlinearity: Eh κk κi . (21) δik = 2(1 − ν 2 )aMk As one could see from this formula, different modes of nonlinear vibrations are coupled. It should be pointed out that the behavior of multi-degree-of-freedom nonlinear systems is quite different from linear systems, that is, for systems that are characterized by amplitudes of vibrations, which are low enough, so that their behaviors could be described by liner differential equations. Some phenomena that take place in nonlinear systems do not occur in linear ones. For instance, when time progresses, the energies of different modes of vibrations are “pumped over” in the nonlinear system (1) from one vibration mode to another. This leads to a seeming “chaos” in the behavior of the system, which behaves in a “quasi-random” fashion despite the nonrandom nature of the excitation. Another important difference is that “nonlinearity destroys resonances”: the “natural” frequencies in nonlinear systems characterized by large amplitudes of vibrations are, unlike in linear systems, amplitude (loading) dependent. When, because of resonance conditions in a linear system (when the amplitudes of vibration are still low), these amplitudes increase, the resonance conditions become “destroyed,” and the vibration amplitudes decrease again. This leads to a situation known as “wandering over resonances”: Higher modes of vibrations could get generated and interfere with the lower modes. The detailed analyses of such situations are, however, beyond the scope of this chapter. In this chapter we will limit the analysis by addressing the dynamic response of multi-degreeof-freedom linear systems using as a suitable example a PCB experiencing random loading applied to its support contour. Many other examples could be found in the monograph [2].

3

DYNAMIC RESPONSE OF MULTI-DEGREE-OF-FREEDOM LINEAR SYSTEM TO EXTERNAL LOADING Omitting the last term in the left part of the Eq. (1), we have Qk (t) . (22) T¨ k (t) + 2rk T˙ k (t) + ωk2 Tk (t) = Mk Hence, in a linear system each mode can be treated in isolation of the other modes of vibrations, and the cumulative effect of the interaction of these modes can be accounted for on the basis of the expansion (12). The physical model for a single-degree-of-freedom vibrating system is schematically shown in Fig. 1. By using the notation y = y(t) = Tk (t),

m = Mk ,

c = 2rk ,

F = F (t) = Qk (t),

(23)

one can obtain Eq. (22) in the form m y¨ (t) + c y(t) ˙ + ky(t) = F (t),

(24)

58

Random Vibrations of Structural Elements in Electronic and Photonic Systems F k

c

y m

Figure 1 Single-degree-of-freedom system subjected to external excitation: m y¨ (t) + c y(t) ˙ + ky(t) = F (t), where m = mass, c = factor damping force, k = spring constant of restoring force, m y(t) ¨ = inertia force, c y(t) ˙ = damping force, ky(t) = restoring force, F (t) = excitation force, and y(t) = displacement. i.e., in the form of the equation of motion of a single-degree-of-freedom system. By comparing Eq. (22) and (24), we see that

k c rk = , ωk = . (25) 2m m

4

SOLUTION USING DUHAMEL INTEGRAL Determine the impulse response (Green function) h(t) of the system (22), that is, its response to a unit pulse excitation. The use of the impulse response function is an effective means of evaluating the response of a system to an arbitrary excitation, whether deterministic or random, steady state or transient. This function can be found for the system (22) as follows: e −rk t sin ωk∗ t, (26) hk (t) = Mk ωk∗ where ωk∗ =



ωk2 − rk2

(27)

is the frequency of damped vibrations of the k th mode. The system’s response at any moment of time t to an elementary impulse ds applied at the preceding moment of time τ can be evaluated as dTk (t) = hk (t − τ ) ds(τ ).

(28)

Treating a continuous excitation Qk (t) as a sequence of successive elementary impulses ds(τ ) = Qk (τ ) d τ , we obtain the following formula known as the Duhamel integral:  t  t 1 Qk (ξ )hk (t − τ ) d ξ = e −rk (t−ξ ) Qk (τ ) sin[ωk∗ (t − τ )] d τ. (29) Tk (t) = Mk ωk∗ 0 0 This solution is applicable starting at a moment of time sufficiently remote from the initial moment, that is, at the moment when free vibrations fade away because of damping and need not be accounted for. Therefore no initial conditions enter expression (29).

5

SOLUTION USING FOURIER INTEGRAL With few exceptions, any periodic function x(t) of period T can be expanded in a trigonometric Fourier series:

5 x (t) =

∞ 

Solution Using Fourier Integral

(ak cos ωk t + bk sin ωk t),

59 (30)

k =1

where the Fourier coefficients ak and bk are expressed as follows:   2 T /2 2 T /2 x (t) cos ωk t dt, bk = x (t) sin ωk t dt, k = 0, 1, 2, . . . , (31) ak = T −T /2 T −T /2 and 2π ωk = , k = 0, 1, 2, . . . . (32) T The function x (t) is an even one with respect to the index k , that is, does not change if the numbers k are replaced with the numbers −k . Therefore the expansion (30) can be rewritten as ∞ 1  (33) x (t) = (ak cos ωk t + bk sin ωk t ). 2 k =−∞

Using Euler’s formulas e i θ + e −i θ e i θ − e −i θ , sin θ = , 2 2i one could write the series (33) in the complex form ∞  Ck e i ωk t , x (t) = cos θ =

(34)

(35)

k =−∞

where the complex Fourier coefficients are expressed as  1 T /2 x (t)e −i ωk t dt . (36) Ck = 1/2(ak − ibk ) = T −T /2 The quantities Ak = 2Ck = ak − ibk are known as complex amplitudes, and the set of the amplitudes Ak is known as the complex-frequency spectrum. The set of modules | Ak | =  2 2 ak + bk is known as the amplitude frequency spectrum and the argument εk = arg Ak = arctan (bk /ak ) is the phase frequency spectrum of the periodic function x (t). The spectra | Ak | and εk can be represented graphically as vertical lines perpendicular to the frequency axis ωk . The Fourier series can be generalized for the case of a nonperiodic function given on an infinite interval. This can be done by treating this function as a periodic one with an infinitely large period. From Eq. (35) and (36) we obtain  ∞ 1  i ωk t T /2 x (t) = e x (t)e −i ωk t dt (37) T −T /2 k =−∞

Let us find the limit of this expression when the time T approaches infinity. Since the quantity 2π/T is, in effect, the frequency interval between two adjacent harmonics, one can use the following substitution: 2π/T → d ω, ωk → ω. This yields  ∞  ∞  ∞  ∞ 1 1 x (t) = e i ωt d ω x (t)e −i ωt dt = dω x (ξ )e −i ω(t−ξ ) d ξ . (38) 2π −∞ 2π −∞ −∞ −∞ The obtained expression is known as the Fourier integral formula, and the integral with respect to the variable ξ in the right part of this formula is the Fourier integral . Formula (38) can be written in real form as  ∞  1 ∞ x (t) = dω x (ξ ) cos ω(t − ξ ) d ξ . (39) π 0 −∞

60

Random Vibrations of Structural Elements in Electronic and Photonic Systems Formula (38) can be written also as follows:   1 ∞ 1 ∞ x (t) = Gx (ω)e i ωt d ω, Gx (ω) = x (t)e −i ωt dt. (40) 2 −∞ π −∞ These equations are known as Fourier transforms, and the function Gx (ω) is known as the complex spectrum, complex spectral density, or complex spectral characteristic of the function x (t). In the theory of Fourier integral the function Gx (ω) plays the same role as the complex amplitudes play in the theory of Fourier series. The module Gx (ω) = | Gx (ω)| of the complex spectrum is called the spectrum or the spectral density of the function x (t). The quantity εx (ω) = arg Gx (ω) is called the phase spectrum of the function x (t). Differentiating the first formula in (40) n times with respect to the argument t, we obtain  1 ∞ (n) (i ω)n Gx (ω)e i ωt d ω. (41) x (t) = 2 −∞ On the other hand, the first formula in (40), when applied directly to the function x (n) (t), yields  1 ∞ G (n) (ω)e i ωt d ω (42) x (n) (t) = 2 −∞ x Comparing formulas (41) and (42), we conclude that the following relationship should take place: Gx (n) (ω) = (i ω)n Gx (ω). (43) This formula expresses the theorem of the spectrum of a derivative. Let the behavior of a linear dynamic system be described by the following ordinary differential equation: an y (n) (t) + an−1 y (n−1) (t) + · · · + a1 y  (t) + a0 y(t) = x (t).

(44)

After applying the Fourier transform to both parts of this equation and using formula (43), we obtain Gy (ω) = Gx (ω)(i ω),

(45)

where the function 1 (46) an (i ω)n + an−1 (i ω)n−1 + · · · + a1 (i ω) + a0 is known as the complex-frequency characteristic, complex transfer coefficient, or mechanical impedance of the dynamic system. The formula  1 ∞ y(t) = Gx (ω)(i ω)e i ωt d ω (47) 2 −∞ provides the solution to Eq. (44). Formula (46), when applied to harmonic “input” x (t) and “output” y(t) processes (i ω) =

x (t) = Ai e i ωt , yields

y(t) = Ao e i ωt ,

(48)

y(t) A0 . (49) = x (t) Ai Thus, the complex-frequency characteristic of a dynamic system can be obtained as the ratio of the complex displacements or the complex amplitudes of the “output” and the “input” harmonic processes. By analogy with the static impedance (“compliance” in the case of a mechanical system), which is defined as the ratio of the static response (“displacement”) to the applied excitation (“force”), the complex-frequency characteristic can be called the dynamic impedance of the system. The modulus (absolute value) (i ω) =

5

Solution Using Fourier Integral

a(ω) = | (i ω)|

61 (50)

of the complex-frequency characteristic is known as the frequency response function of the amplitudes, or the gain factor. The argument θ(ω) = arg (i ω)

(51)

of the complex-frequency characteristic is known as the frequency response function for the phase angles, or the phase factor. As an example, examine the dynamic response of a single-degree-of-freedom system of the type given by Eq. (24) to a harmonic excitation. The motions of the system are described by the equation y¨ (t) + 2r y(t) ˙ + ω02 y(t) = x (t). (52) For n = 2 formula (46) yields 1 . (53) a2 (i ω)2 + a1 (i ω) + a0 Comparing formula (52) with formula (44), we have a2 = 1, a1 = 2r, a0 = ω02 . Thus, 1 (i ω) = 2 (54) ω0 − ω2 + 2irω and the solution to Eq. (52) is  1 ∞ Gx (ω)e i ωt y(t) = d ω. (55) 2 −∞ ω02 − ω2 + 2irω The frequency response function (transfer function) for the amplitudes is  1 a(ω) = | (i ω)| = (i ω)(−i ω) =  . (56) 2 ω02 − ω2 + (2rω)2 (i ω) =

The case ω → 0 corresponds to static conditions. In this case formula (56) yields 1 ast = 2 . ω0 The dynamic factor of the system is expressed as (Fig. 2) a(ω) 1 Z0 = =  2  2 ast 2 1 − ω2 /ω0 + 2rω/ω02

Z0

(57)

(58)

d1 d2

2 A0/w0

0

d3 d4 w0

Ω

Figure 2 Dynamic factor (resonance curves): ratios of output amplitudes to static response for different damping coefficient values. When damping is significant, no vibrations could possibly occur. Resonance condition occurs when the frequency of the harmonic excitation becomes equal to the natural frequency of the damped vibrations. When damping is zero, the resonance amplitudes are infinitely high.

62 6

Random Vibrations of Structural Elements in Electronic and Photonic Systems

COMPLEX-FREQUENCY CHARACTERISTIC AS SPECTRUM OF IMPULSE RESPONSE The unit pulse excitation can be naturally approximated by the Dirac delta function δ(t). This function was introduced to quantum mechanics by the English physicist P. A. Dirac and is often used in applied problems as a suitable means for describing highly localized effects such as instantaneous impulses, concentrated loads, or point electrical charges. The Dirac delta function possesses the following properties:  0 for t = 0, δ(t) = ∞ for t = 0, (59)  ∞  ∞ −∞

f (t − a)δ(t) dt = f (a),

−∞

δ(t) dt = 1.

The second formula in (40) leads to the following formula for the spectrum of the delta function:   1 ∞ 1 ∞ 1 Gx (ω) = x (t)e −i ωt dt = δ(t)e −i ωt dt = . (60) π −∞ π −∞ π The obtained spectrum is frequency independent. Using this spectrum as an input spectrum and applying formula (47), we obtain the following formula for the impulse response function:  ∞ 1 y(t) = h(t) = (i ω)e i ωt d ω. (61) 2π −∞ On the other hand, applying the first formula in (40) to the function h(t), we have  1 ∞ h(t) = Gh (ω)e i ωt d ω, (62) 2 −∞ where Gh (ω) is the spectrum of the impulse response function h(t). Comparing formulas (61) and (62), we conclude that the complex-frequency characteristic (i ω) of a dynamic system is related to the spectrum Gh (ω) of its impulse response function h(t) as follows: (i ω) = π Gh (ω).

(63)

Thus, instead of evaluating, experimentally or theoretically, the complex-frequency characteristic (i ω) for different excitation frequencies, one could evaluate the impulse response function for a single impulse excitation, compute the spectrum of this function on the basis of the second formula in (40), and then evaluate the frequency characteristic by formula (63).

7

DURATION OF A PROCESS AND WIDTH OF ITS SPECTRUM Equation (63) in the real domain results in the following relationship between the frequency response function for the amplitude (transfer function) a(ω) and the spectrum Sh (ω) of the impulse response of the system: a(ω) = π Sh (ω).

(64)

From (58) we have Z0 . ω02 Introducing (65) into (64), we obtain the following relationship: Z0 (ω) Sh (ω) = . π ω02 a(ω) = Z0 (ω)ast =

(65)

(66)

8 Correlation Theory and Spectral Theory of Random Processes

63

Hence, the spectrum of the impulse response function coincides, up to a constant factor, with the dynamic factor. As evident from Fig. 2, the decrease in damping, leading to an increase in the duration of the response process h(t), results in an increase in the width of the curve Z0 (ω) and, as evident from formula (66), in an increase in the width of the spectrum Sh (ω). Thus, the shorter the dynamic process, the more harmonics it generates, and the wider its spectrum is. In the extreme case of an “ideally narrow” deltalike process, the spectrum becomes, as evident from formula (60), constant, that is, ‘infinitely wide,” and is frequency independent. Conversely, for a deltalike spectrum Sx (ω) = δ(ω), the process x (t), as one could easily obtain from the first formula in (40) and formulas (59), lasts for an infinitely long time. It is impossible therefore to localize an external excitation in time and, concurrently, enhance the selectivity of this excitation to different frequencies: The more intense in time the excitation is, the larger is the number of the induced harmonics (vibration modes). This explains, in particular, the poor convergence and even divergence of the method of principal coordinates when this method is used for the evaluation of the response characteristics of an elastic system to a shock excitation. It is noteworthy that the inability to localize the external excitation in time and, at the same time, to enhance the selectivity of this excitation to a narrow frequency range is one of the manifestations of a much more general Heisenberg principle of uncertainty, which plays an important role in modern quantum physics.

8

CORRELATION THEORY AND SPECTRAL THEORY OF RANDOM PROCESSES A theory of random processes based on the use of correlation functions is known as the correlation theory of random processes [3]. Alternatively, the spectral theory of random processes can be used to analyze the dynamic response of a system to a random excitation. The relationships between the (time interval τ –dependent) correlation function Kx (τ ) of a random process X (t) and the (frequency ω–dependent) spectrum Gx (ω) of this process are established for stationary (steady-state) processes by the Wiener–Khinchin formulas   1 ∞ 1 ∞ Kx (τ ) = Gx (ω)e i ωτ d ω, Gx (ω) = Kx (τ )e −i ωτ d τ , (67) 2 −∞ π −∞ √ where i = −1 is the imaginary unity. Since Kx (τ ) is an even function, in the real domain formulas (67) are known as Fourier cosine transforms:  ∞ Kx (τ ) = Sx (ω) cos ωτ d ω, 0 (68)  ∞ 2 Kx (τ ) cos ωτ d τ. Sx (ω) = π −∞ The variance of a random process can be obtained as  ∞ Sx (ω) d ω, (69) Dx = Kx (0) = 0

From this formula we find 2dDx (ω) . (70) dω Thus, the spectrum of the random process determines the density of the distribution of variances (standard deviations squared) over the frequencies of a continuous spectrum. The variance of an elementary harmonic (mode) determines the power of this harmonic and therefore the function Sx (ω) characterizes the distribution of the power of the random process X (t) over the range of frequencies. For this reason the function Sx (ω) is often called the power spectral density or the power spectrum. Sx (ω) =

64

Random Vibrations of Structural Elements in Electronic and Photonic Systems The major incentive for the substitution of the correlation function Kx (τ ) of the random process X (t) by its power spectrum Sx (ω) is due to the fact that the use of the function Sx (ω) enables one to simplify considerably the procedure of the transformation of stationary random processes by linear dynamic systems.

9

SPECTRAL THEORY OF TRANSFORMATION OF STATIONARY RANDOM PROCESSES BY LINEAR DYNAMIC SYSTEMS Let a dynamic system (Fig. 3) whose behavior is established by its complex-frequency characteristic (i ω) be subjected to a random excitation (input) X (t) described by its power spectrum Sx (ω). The objective of the analysis in this section is to establish the relationship between the power spectrum Sx (ω) of the input random process X (t) and the spectrum Sy (ω) of the output process Y (t). Both random processes X (t) and Y (t) are assumed to be stationary with zero means. The second formula in (67), when applied to the process Y (t), yields   1 ∞ 2 ∞ Ky (τ )e −i ωτ d τ = Ky (τ )e −i ωτ d τ Sy (ω) = π −∞ π 0    2 ∞ ∞ ∞ −i ωτ = e h(τ1 )h(τ2 )Kx (τ + τ1 − τ2 ) d τ1 d τ2 d τ π 0 0 0    2 ∞ ∞ ∞ −i ω(τ +τ1 −τ2 ) i ωτ1 i ωτ2 e e e h(τ1 )h(τ2 )Kxi (τ + τ1 − τ2 ) d τ1 d τ2 d τ = π 0 0 0  ∞  ∞  2 ∞ h(τ1 )e i ωτ1 d τ1 h(τ2 )e −i ωτ2 d τ2 Kx (τ + τ1 − τ2 )e −i ω(τ +τ1 −τ2 ) d τ = π 0 0 0  ∞  ∞ h(t)e i ωt dt h(t)e −i ωt dt, = Sx (ω) 0

0

(71) where

  2 ∞ 2 ∞ Kxi (τ + τ1 − τ2 )e −i ω(τ +τ1 −τ2 ) d τ = Kx (τ )e −i ωτ d τ (72) π 0 π 0 is the spectrum of the input process X (t). In Eq. (71) we used the notation Sy (ω) instead of Gy (ω). The spectrum Gh (ω) of the impulse response function h(t) can be obtained based on the second formula in (40) as follows:   1 ∞ 1 ∞ h(t)e −i ωt dt = h(t)e −i ωt dt. (73) Gh (ω) = π −∞ π 0 Introducing this formula into the formula (63), we obtain  ∞ h(t)e −i ωt dt. (74) (i ω) = Sx (ω) =

0

Hence,



∞

(−i ω) =

h(t)e i ωt dt.

(75)

0

Formula (71) can be written as Sy (ω) = Sx (ω)(i ω)(−i ω) = Sx (ω) |(i ω) |2 = Sx (ω)a 2 (ω).

(76)

10

White Noise

65

Dynamic system whose behavior is described by its nonrandom complex-frequency characteristic (response to harmonic excitations of different frequencies) Φ (iw) = Given (see Fig. 2)

Output random process Y(t) that is characterized by its power spectrum Sy (w) = Sought

Input random process X(t) that is characterized by its power spectrum Sx (w) = Given

Figure 3 Dynamic system (that is described by its complex-frequency characteristic) is subjected to a random stationary input process X (t) and transforms it into a random stationary output process Y (t): both processes are characterized by their power spectra Sx (ω) and Sy , (ω) respectively.

Thus, the power spectrum Sy (ω) of the output process Y (t) can be obtained by multiplying the power spectrum Sx (ω) of the input process X (t) by the absolute value of the complexfrequency characteristic (transfer function) squared. The simple and physically meaningful relationship (76) underlies the spectral theory of transformation of stationary random processes by linear dynamic systems.

10

WHITE NOISE In some applied problems there is a possibility to approximate the input process as a “white noise.” A white noise is a special case of a stationary random process that is characterized by a constant, frequency-independent, power spectrum. The name “white noise” appeared by analogy with the white light, which has, in its visible portion, a uniform continuous spectrum: all the colors are equally “represented” in this spectrum. The correlation function of a white noise n(t) is extremely narrow and is expressed through the spectrum N0 of the process and the delta function δ(τ ) of the time interval τ as follows: Kn (τ ) = n(t)n(t + τ ) = π N0 δ(τ ).

(77)

Since the delta function δ(τ ) is one of the multipliers in expression (77) for the correlation function, the white noise is often referred to as a delta-correlated random process. The white noise is an “absolutely random process” since its cross sections, no matter how close they are, are uncorrelated. An actual stationary random process can be approximated as a white noise if it has a rapidly decreasing correlation function. Examine, for example, a stationary random process X (t) with a correlation function Kx (τ ) = Dx e −α| τ | , where α is a large parameter. The spectrum of this process is  2 ∞ 2 αDx Sx (ω) = Kx (τ ) cos ωτ d τ = . π 0 π α 2 + ω2

(78)

(79)

66

Random Vibrations of Structural Elements in Electronic and Photonic Systems If ω  α, that is, if the period of oscillations τ0 = 2π/ω exceeds substantially the correlation time τk = 1/α, then the spectral density can be expressed as 2 Dx (80) = N0 Sx (ω) ≈ π α and can be considered constant. In practice, it is sufficient to make sure that the power spectrum of the given stationary random process is constant or next-to-constant, within the width of the frequency response function of the dynamic system of interest. As a matter of fact, many actual dynamic systems characterized by low enough damping satisfy this requirement. These systems, when subjected to forced oscillations, behave like narrowband filters that enhance the input harmonics, whose frequencies are close to the frequencies of free vibrations of the system and suppress all the other harmonics. The response functions of such systems have narrow, strongly pronounced peaks in the vicinity of the system’s own frequencies (see Fig. 2). Within this peak even a rapidly changing spectral density function can be considered constant. The main reason for the idealization of a real random process by a white noise is due to the fact that, when a narrow-band system is subjected to the excitation of the whitenoise type, the system’s response can be satisfactorily approximated by a quasi-harmonic process with a slow changing amplitude and phase angle. Such a response can be treated as a Markovian process, and the well-developed and powerful mathematical “equipment” of the theory of Markovian processes can be effectively applied to solve many practical problems.

11

BIVARIATE CORRELATION FUNCTION AND BIVARIATE SPECTRAL DENSITY IN PCB RANDOM VIBRATIONS When addressing multi-degree-of-freedom systems subjected to random loading, there is a need to consider bivariate correlation functions and bivariate spectral densities. The bivariate correlation function for two random processes Tk (t) and Tj (t), that is, for the principal coordinates of the k th and j th modes of vibrations at two arbitrary moments t1 and t2 of time separated by the time interval τ = t2 − t1 is as follows:   ∞ ∞ 1 T 1 Tk (t1 )Tj (t2 ) dt = hk (τ1 )hj (τ2 ) KTk Tj (t1 , t2 ) = T 0 Mk Mi 0 0 

= where

1 T

1 Mk Mj



T

 Qk (t1 − τ1 )Qj (t2 − τ2 ) dt

d τ1 d τ2

(81)

0

 0

∞ ∞ 0

hk (τ1 )hj (τ2 )KQk Qi (τ + τ1 − τ2 ) d τ1 d τ2 ,

 1 T KQk Qi (τ + τ1 − τ2 ) = Qk (t1 − τ1 )Qi (t2 − τ2 ) dt = KQk Qi [(t2 − τ2 ) − (t1 − τ1 )] (82) T 0 is the bivariate correlation function for the generalized forces Qk (t) and Qi (t). In formulas (81) and (82) we have considered that the mean values of the random forces Qk (t) and Qi (t) are zero. Formula (81) enables one to obtain complete information about the system of random processes. In the case in question, this “system” is the elongated PCB subjected to random excitation.

11 Bivariate Correlation Function and Bivariate Spectral Density in PCB Random Vibrations

67

The bivariate spectral density for the principal coordinates Tk (t) and Tj (t) is STk Tj (ω) = = = = =



1 π

∞

Kx (τ )e −i ωτ d τ

−∞



2 π Mk Mj

∞

0



2 π Mk Mj

0

∞



Mk Mj

2 π



0

∞

e −i ωτ hk (τ1 )hj (τ2 )KQk Qi (τ + τ1 − τ2 ) d τ1 d τ2 d τ

e −i ω(τ +τ1 −τ2 ) e i ωτ1 e i ωτ2 hk (τ1 )hj (τ2 )KQk Qi (τ + τ1 − τ2 ) d τ1 d τ2 d τ

 hk (τ1 )e i ωτ1 d τ1

0

SQk Qj (ω)

SQk Qj (ω) =

∞

0 0 0  ∞ ∞ ∞

2 π Mk Mj

Here



∞





∞ 0

hj (τ2 )e −i ωτ2 d τ2



0 ∞

hk (t)e i ωt dt

0

∞

0

∞

KQk Qi (τ + τ1 − τ2 )e −i ω(τ +τ1 −τ2 ) d τ

hj (t)e −i ωt dt.

(83)

0

KQk Qi (τ + τ1 − τ2 )e −i ω(τ +τ1 −τ2 ) d τ =

1 π



∞

−∞

KQk Qi (τ )e −i ωτ d τ (84)

is the bivariate spectral density of the generalized forces Qk (t) and Qi (t). Since the complex-frequency characteristic of the impulse response function hk (t) is  ∞  ∞ hk (t)e −i ωt dt = hk (t)e −i ωt dt, (85) k (i ω) = −∞

0

the bivariate spectral density of the principal coordinates Tk (t) and Tj (t) can be found as STk Tj (ω) = SQk Qj (ω)

k (i ω)j (−i ω) . Mk Mj

(86)

Introducing formula (54) into this expression, we obtain STk Tj (ω) =

SQk Qj (ω) Mk Mj (ωk2

−

ω2

+ 2irk ω)(ωj2 − ω2 − 2irj ω)

.

(87)

The bivariate variances of the principal coordinates Tk (t) and Tj (t) can be found as follows:  ∞  ∞ SQk Qj (ω) d ω 1 STk Tj (ω) d ω = (88) DT k T j = 2 2 + 2ir ω)(ω2 − ω2 − 2ir ω) M M (ω − ω k j 0 0 k j j k Applying the theorem of the spectrum of a derivative expressed by the formula (43), we have ST˙ k Tj = STk T˙ j = i ωSTk Tj ,

ST˙ k T˙ j = −ω2 STk Tj ,

(89)

and therefore the bivariate variances for the processes Tk (t), Tj (t) and their velocities (time derivatives) T˙ k (t) and T˙ j (t) can be expressed as follows:  ∞ ωSQk Qj (ω) d ω i DT˙ k Tj = DTk T˙ j = , 2 2 Mk Mj 0 (ωk − ω + 2irk ω)(ωj2 − ω2 − 2irj ω)  ∞ ωSQk Qj (ω) d ω 1 DT˙ k T˙ j = . (90) Mk Mj 0 (ωk2 − ω2 + 2irk ω)(ωj2 − ω2 − 2irj ω)

68

Random Vibrations of Structural Elements in Electronic and Photonic Systems If the forces Qk (t) can be approximated by a white noise, so that KQk Qi (τ ) = π SQk Qi δ(τ )

(91)

with SQk Qi = const, then the bivariate variances of the principal coordinates Tk (t) and Tj (t) and the velocities T˙ k (t) and T˙ j (t) can be found by the formulas 4π SQk Qj rk + rj DT k T j = , 2 2 2 Mk Mj (ωk − ωj ) + 4(rk + rj )(rk ωj2 + rj ωk2 ) DT˙ k Tj = DTk T˙ j = DT˙ k T˙ j =

4π SQk Qj

rk + rj , Mk Mj (ωk2 − ωj2 )2 (rk + rj ) + (rk ωj2 + rj ωk2 )

2π SQk Qj

rk ωj2 + rj ωk2

Mk Mj (ωk2 − ωj2 )2 + 4(rk + rj )(rk ωj2 + rj ωk2 )

(92)

.

As evident from these formulas, when the damping coefficients are small (which is usually the case for lower modes of vibrations of elastic systems employed in electronic packaging engineering), one could neglect the bivariate correlations of the principal coordinates with their velocities. Such a correlation becomes appreciable and should be accounted for only for very close values of the frequencies ωk and ωj . This circumstance enables one to examine the random response of a particular principal coordinate independently of (uncoupled with) principal coordinates of other modes of vibration. For k = j we have π SQ k Q j π SQ k Q j DT k T j = , D D . (93) ˙ k Tj = DTk T˙ j = 0, ˙ k T˙ j = T T 2 2Mk2 rk 2Mk2 rk ωk

12

PROBABILITY OF EXCEEDING THE GIVEN LEVEL In practical applications of probabilistic approaches to various engineering and applied science problems, one often seeks the probability that the given random process (say, stresses, accelerations or displacements, voltage or current) exceeds a certain level. Then, by establishing the allowable probability of exceeding such a level, the reliability engineer determines the corresponding level to which his or her design should be geared. The objective of the analysis that follows is to obtain a simple and easy-to-use formula for establishing the probability of exceeding the given level by a random process. If a realization (an actual process) of a random process X (t) crosses (upward) a straight line (“level”) that is parallel to the t axis and is located at the distance x∗ from it, then the process X (t) is said to cross the level x∗ . The number of level crossing during the given time is a discrete random variable. Typically, such crossings are infrequent. If this is the case, their number can be considered to have a Poisson distribution [2]. In order to determine the number of upward crossings of the given level x∗ by the random process X (t), one should know the two-dimensional probability density function f (x , x˙ ) for the process X (t) and its time derivative X˙ (t). Indeed, if the process X (t) reaches the level x∗ , its time derivative X˙ (t) should be large enough so that this “threshold” is exceeded. The expression f (x∗ , x˙ ) dx d x˙ = f (x∗ , x˙ )x˙ d x˙ dt determines the probability that the value of the functions X (t) and X˙ (t) are found within the ˙ x˙ + d x˙ ), respectively. It is clear that upward crossings of the intervals (x∗ , x∗ + dx ) and (x, ˙ that is, for 0 < x˙ (t) < ∞. With level x∗ take place for all positive values of the derivative x, this in mind, one can write the expression for the complete probability of crossing the level x∗ during the elementary time interval dt as

12 Probability of Exceeding the Given Level  P1 = dt

∞

0

69

f (x∗ , x˙ )x˙ d x˙ .

Since the duration dt is small, one could expect that upward crossing of the level x∗ during the elementary duration of time dt does not occur more than once. With such an assumption, one can view the probability P1 as a probability of a single upward crossing of the level x∗ during the time dt, that is, within the time interval (t, t + dt). Let P0 be the probability that there will be no upward crossings of the level x∗ during the time dt. Then the mean number of upward crossings within the time dt can be found as 0xP0 + 1xP1 = P1 (no upward crossings with the probability P0 and one crossing with the probability P1 ). Thus, the mean number Nx∗ of upward crossings per unit time can be computed as  ∞ f (x∗ , x˙ )x˙ d x˙ . (94) λ = Nx∗ = 0

If the random processes X (t) and X˙ (t) are statistically independent, their joint probability density function f (x , x˙ ) can be represented as f (x , x˙ ) = f1 (x )f2 (x˙ ). In this case formula (94) can be simplified: λ = Nx∗ = f1 (x∗ )



∞

f2 (x˙ )x˙ d x˙ .

(95)

(96)

0

This formula indicates that for a stationary random process with a statistically independent time derivative the mean value of the upward crossings of the given level is proportional to the probability density function on this level. This result can be used particularly for the experimental evaluation of the probability density functions for such processes simply by counting the number of the upward crossings on different levels. If the process X (t) and its time derivative X˙ (t) are normally distributed, then the number of crossings of the level x∗ can be calculated as   1 (x∗ − x )2 exp − λ = Nx∗ = , (97) τe 2Dx where x is the mean value of the process X (t), Dx is its variance,  Dx τe = 2π Dx˙

(98)

is the effective period of the process X (t), and Dx˙ is the variance of the process X˙ (t). Indeed, if the distributions f1 (x ) and f2 (x˙ ) are normal (Gaussian) distributions with zero means, so that   1 x2 , f1 (x ) = √ exp − 2Dx 2π Dx (99)   1 x˙ 2 f2 (x ) = √ , exp − 2Dx˙ 2π Dx˙ then the probability that the process X (t) will exceed the level x∗ can be found as  ∞   ∞    t x2 x˙ 2 x2 t 2 d x˙ = √ exp − ∗ exp − exp − ∗ e −η d η. P= √ 2Dx 2Dx˙ 2Dx 2π Dx Dx˙ π 2Dx 0 0 (100)

70

Random Vibrations of Structural Elements in Electronic and Photonic Systems The Laplace function (probability integral, error function)  α 2 2 (α) = erf α = √ e −η d η π 0 is equal to 1 for α → ∞ and the formula (100) can be written as P = λt, where

(101)

(102)

 1 λ= 2π

Dx˙ −x∗2 /(2Dx ) e Dx

(103)

is the number of upward crossings of the level x∗ per unit time. If the random processes X (t) and X˙ (t) have narrow frequency spectra, then these processes could be characterized by their effective period expressed by formula (98), and formula (103) yields 1 2 (104) λ = e −x∗ /2Dx τe The only difference between this formula and formula (97) is that the nonzero mean value x of the process X (t) is taken into account in (91). Thus, the mean number of crossings by the principal coordinate (process) Tk (t) a certain level T∗ can be evaluated by formula (97) as follows:   1 T2 exp − ∗ λ = NT∗ = . (105) τe 2DTk This formula is based, of course, on the assumption that the process Tk (t) is normal, stationary, narrow banded, and characterized by a statistically independent time derivative (velocity). The mean number of zeros, that is, the number crossing the zero level Tk (t) = 0 per unit time, can be found from (105) by putting T∗ = 0: 1 N0 = . (106) τe As long as the process Tk (t) is narrow banded, the number of zeros is equal to the number of maxima. Thus, only NT∗ “attempts” (out of the total number N0 of maxima) to exceed the level T∗ are successful, and therefore the probability of the occurrence of such exceedings can be evaluated as the ratio of the “successful attempts” NT∗ to the total number N0 of maxima:     Mk2 rk ωk T∗2 T∗2 NT∗ = exp − . (107) = exp − P {Tk > T∗ } = N0 2DTk π SQ k Q j The same result could be obtained based on the consideration that the amplitudes of the random process Tk (t) follow the Rayleigh law: It has been established that the amplitudes of a normal random harmonic process are distributed in accordance with the Rayleigh law if the spectrum of this process is narrow band and, in accordance with the Rice law, if its spectrum is not narrow band (see, e.g., [3]).

13

ROLE OF HIGHER MODES (HARMONICS) Let us assess the role of higher harmonics using, as an example, the dynamic response of an elongated PCB to a shock load applied to its supports as a result of the drop impact. For the sake of simplicity, we assume that the PCB is simply supported at its contour, so that the coordinate function Xk (x ) can be accepted in the form kπx Xk (x ) = cos , k = 1, 3, 5, . . . . (108) 2a

14

Optimized Damping

71

Here a is half the PCB width, and the origin of the coordinate x is at the mid–cross section of the PCB. We seek the deflection function w (x , t) in the form (12) w (x , t) =

∞ 

Tk (t)Xk (x ) =

k =1

∞ 

Tk (t) cos

k =1

kπx . 2a

(109)

The velocities of the PCB points are as follows: w˙ (x , t) =

∞  k =1

kπx T˙ k (t) cos . 2a

(110)

At the initial moment of time, when the board touches a hard floor, all its points have the same velocity, w˙ (x , 0) =

∞  k =1

kπx T˙ k (0) cos = 2a



2gH ,

(111)

where H is the drop height and g is the acceleration due to gravity. Let us apply the Fourier  transform to this equation, that is, multiply both parts of the equation by cos k π x / (2a) and integrate over the PCB length 2a. This yields 4  T˙ k (0) = 2gH . (112) kπ The generalized mass Mk of the k th mode can be found on the basis of formula (15) as follows:  a  a kπx ma Xk2 (x ) dx = m cos2 Mk = m dx = . (113) 2a 2 0 0 The total kinetic energy of half of the PCB at the moment of touching the hard floor can be found as ∞ ∞ 1  8 mag H  1 K = Mk T˙ k2 (0) = . (114) 2 π2 k2 k =1,3,5...

k =1,3,5...

As evident from this formula, the energy of the first mode of vibrations is 8 mag H , (115) π2 while the total energy should be equal to the initial potential energy  = mag H of half of the board. The energy balance condition  = K yields K1 =

∞  k =1,3,5...

π2 1 = , 2 k 8

(116)

which is indeed the case. As evident from the obtained formulas, the first mode of vibrations contains 8/π 2 = 0.8106 = 81% of the total (strain) energy of the board. This means that an approach, when one considers the first mode of vibrations only and “assigns” the entire initial energy to this mode, is moderately conservative and, hence, acceptable in engineering calculations.

14

OPTIMIZED DAMPING An elongated simply supported PCB is subjected to random loading applied to its support contour. The loading can be approximated by a white noise. We proceed from Eq. (22) for

72

Random Vibrations of Structural Elements in Electronic and Photonic Systems the principal coordinate and consider the first mode of vibrations only, so that the equation of motion is n(t) T¨ (t) + 2r T˙ (t) + ω02 T (t) = . (117) M The frequencies of the white noise are within a finite range 0 ≤ ω ≤ ω1 . The objective of the analysis that follows is to determine the spectral functions and the variances of the principal coordinate T (t) and its time derivative T˙ (t), and determine the most advantageous value of damping that would minimize the standard deviation of the force applied to the board. In order to determine the complex-frequency characteristic of the system, we assume that the system (117) is subjected to a harmonic excitation P = n(t) = P0 e iwt

(118)

with a complex amplitude P0 and that its response is also harmonic, T (t) = T0 e iwt

(119)

with the complex amplitude T0 . Then Eq. (117) yields (−ω2 + 2i ωr + ω02 )T0 =

P0 . M

The complex-frequency characteristic of the system is T0 1 1  . = = T (i ω) = 2 P0 M (−ω2 + 2i ωr + ω02 ) ω 2r ω 2 M ω0 1 − 2 + i ω0 ω0 ω0

(120)

(121)

The spectral density of the principal coordinate T (t) can be found in accordance with formula (76) as follows:  N0 ST (ω) = N0 |T (i ω)2 = (122)  2 , 2r ω ω2 2 K 1− 2 +i ω0 ω0 ω0 where K = M ω02 is the spring constant of the PCB and N0 is the power spectrum of the white-noise process for 0 ≤ ω ≤ ω1 . Outside this region the spectrum ST (ω) is zero. The complex-frequency characteristic of the velocity T˙ (t) is T˙ (i ω) = i ωT (i ω), and therefore the spectral density of the process T˙ (t) can be determined as  S ˙ (ω) = |i ωT (i ω)2 N0 = ω2 ST (ω). T

(123)

(124)

Similarly, the spectral density of the acceleration process T¨ (t) is ST¨˙ (ω) = ω4 ST (ω).

(125)

The total force acting on the PCB is due to the combined action of the restoring and damping forces and can be calculated as    (126) F (t) = M ω02 T (t) + 2r T˙ (t) = M ω02 + 2irω T (t). The complex-frequency characteristic of the process F (t) is   F (i ω) = M ω02 + 2irω T (i ω).

(127)

14

Optimized Damping

The spectrum SF (ω) of the force F (t) transmitted to the PCB contour is  SF (ω) = |F (i ω)2 N0 = K 2 ST (ω) + 4r 2 M 2 S ˙ (ω). T

73 (128)

The variances DT and DT˙ of the response processes T (t) and T˙ (t) are as follows:  ∞  ∞   dω N0 ω1 |T (i ω)2 d ω = 2 ST (ω) d ω = N0 DT =  2     , K 0 0 0 ω2 2r 2 ω 2 1− 2 + ω0 ω0 ω0 (129)  ∞  ∞  ω1 2 2  ω dω  ˙ (i ω) d ω = N0 DT˙ = ST˙ (ω) d ω = N0  2     . T 2 K 2 0 0 0 2r 2 ω 2 ω + 1− 2 ω0 ω0 ω0 (130) The integrals (129) and (130) can be found in a closed form:   1 N0 ω0 η2 + 2η 1 − μ2 + 1 DT = ln   K2 8 1 −μ2 η2 − 2η 1 − μ2 + 1    1 η + 1 − μ2 η − 1 − μ2 + arctan + arctan , 4μ μ μ   N0 ω03 η2 + 2η 1 − μ2 + 1 1 DT˙ = ln −   K2 8 1− μ2 η2 − 2η 1 − μ2 + 1    1 η + 1 − μ2 η − 1 − μ2 + arctan + arctan , 4μ μ μ

(131)

where the following notation is used: μ=

r , ω0

η=

ω1 . ω0

(132)

For ω1 → ∞ we find π N0 ω0 π N0 ω0 π N0 ω02 π N0 ω04 = = , D = . (133) ˙ T 4 K 2μ 4 K 2r 4 K 2μ 4 K 2r Thus, although the spectrum of the white-noise loading n(t) is unlimited, the variances DT and DT˙ are not. The variance of the force F (t) transmitted to the PCB is   π 4r ω0 2 2 2 + . (134) DF = K DT + 4r M DT˙ = N0 ω0 4 r ω0 The obtained formula reflects the fact that the variance of the principal coordinate decreases with an increase in damping, while the variance in the velocity increases with an increase in damping, and therefore the variance of the total force acting on the PCB has a minimum at a certain level of damping. From (134) we find that the minimum value of the variance DF takes place for r = ω0 /2 (R = 2Mr = M ω0 ) and is equal to DT =

DF ,min = π N0 ω0 .

(135)

As evident from this formula, systems with low natural frequencies lend themselves better to vibration damping than high-frequency systems.

74

Random Vibrations of Structural Elements in Electronic and Photonic Systems

REFERENCES 1. Suhir, E., “How to Make a Device into a Product: Accelerated Life Testing It’s Role, Attributes, Challenges, Pitfalls, and Interaction with Qualification Testing,” in E. Suhir, C. P. Wong, and Y.C. Lee, Eds., Micro- and Opto-Electronic Materials and Structures: Physics, Mechanics, Design, Packaging, Reliability, Springer, New York, 2007. 2. Suhir, E., Applied Probability for Engineers and Scientists, McGraw-Hill, New York, 1997. 3. Timoshenko, S., Theory of Plates and Shells, McGraw-Hill New York, 1940.

CHAPTER

5

NATURAL FREQUENCIES AND FAILURE MECHANISMS OF ELECTRONIC AND PHOTONIC STRUCTURES SUBJECTED TO SINUSOIDAL OR RANDOM VIBRATIONS David S. Steinberg Steinbergelectronics, Inc. Westlake Village, California

1

INTRODUCTION There are many different types of mechanical devices in use today that rotate, oscillate, spin, bounce, swing, lift, drill, and dig. Some have electric motors or run on gasoline, roll on wheels, have gears, wash clothes, cut wood, forge steel, and stamp out coins. These all involve some kind of motion which often generates some type of vibration. There are good vibrations and there are bad vibrations. Good vibrations are used in conveyor belts that vibrate back and forth to sort out different sizes in gravel and fruit. Good vibrations would also be adding vibrators to the instrument panels for the first jet aircraft. Engines were so smooth on these new jets that instruments from the old propeller engines would stick and not work properly. The propellers from the old airplanes generated much more vibration so the same instruments worked very well on the old airplanes. Instruments for the new jets had to be completely redesigned to operate with minimum vibrations. Bad vibrations are vibrations that cause failures or malfunctions in equipment exposed to the vibration. Bad vibrations can often be traced back to a structural member that is not stiff enough, so it has a low natural frequency. This generates a higher dynamic displacement that increases the stress levels and reduces the effective fatigue life. Vibration failures in electronic equipment can be very difficult to trace because there are so many different factors to consider. The first item that is examined is usually the failed part, its location in the structure, how is it supported, whether it is a purchased part or a manufactured part. The size and weight of the part, the part material, plastic, metal, ceramic, vibration input level, vibration response level in area of failure, direction of vibration input, duration of vibration, type of sensors used in the failure area, type of test fixture used to hold the test specimen, any history of previous tests, any previous analyses with calculations by hand or computer. Do not overlook the possibility of sabotage by an unhappy employee. Sinusoidal vibrations are often used for testing various types of electronic equipment because this is often the actual operating environment for many types of systems. This applies to aircraft with piston engines and propellers and also to ships and submarines with screw propellers. Sine vibration tests are not as expensive as many other types of tests and they are easy to set up and run. Sinusoidal vibrations are also used because it can generate a very clear picture of how various types of structures move when their natural frequencies are excited by watching their movements with an oscilloscope. Many previous structural failures could Structural Dynamics of Electronic and Photonic Systems Copyright © 2011 John Wiley & Sons, Inc.

Edited by Ephraim Suhir, David S. Steinberg and T. X. Yu

75

76

Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures not be solved using computer analyses, along with sinusoidal vibration tests, but with no oscilloscope. When these same sinusoidal tests were repeated with the use of an oscilloscope, the failure mechanism could be observed almost instantaneously. This type of information is very helpful for finding why and how certain types of failures often occur in these electronic systems and how these failures can be avoided.

2

NATURAL FREQUENCIES FOR BEAM TYPES OF STRUCTURES Simple types of beam structures are often used to support electronic equipment. The natural frequency of the beam must be determined to obtain the displacements and forces acting on the beam structure. The stresses can then be obtained which will determine the approximate fatigue life expected in different vibration environments. The approximate geometric boundary conditions have to be known to obtain a reasonably accurate estimate of the expected natural frequency of the structure. The geometric boundary conditions are defined as the deflections and the slope. This is based upon analysis methods developed by Lord Rayleigh many years ago and published in his book, Theory of Sound . Use the displacement estimate as the deflection curve for a half sine wave (Fig. 1) with the parameters for a uniform beam structure simply supported (hinged) at both ends. Then πX Y = Y0 sin (1) L where Y = displacement at any point along length of beam, in. (cm) Y0 = maximum displacement at center of beam, in. (cm) X = length of beam taken at displacement point Y , in. (cm) L = length of beam taken at displacement point Yo , in. (cm) The geometric boundary Deflection: ⎧ ⎨ 0 0 Y = ⎩ Y0

conditions for the beam deflection and slope are satisfied below: when when when

X =0 X = L, X = 12 L

(beginning of beam), (end of beam), (maximum deflection at center).

Slope (angle of rotation of cross section): dY πX π θ= = Y0 cos , dX L L ⎧ π ⎪ Y0 when X = 0, ⎪ ⎪ ⎪ L ⎨ θ = − π Y0 when X = L, ⎪ ⎪ L ⎪ ⎪ ⎩ 0 when X = 12 L. The general bending moment equation can be used to get the strain energy relation as EI Strain energy is given as U =

EI 2

d 2Y = M. d. X 2  0

L



d 2Y dX 2

(2)

2 dX.

(3)

2

Natural Frequencies for Beam Types of Structures

77

y0

y

x L 2 Half-sine wave

Figure 1 From Eq. (1)

dY πX π = Y0 cos , dX L L

d 2Y π2 πX = − 2 Y0 sin , (4) 2 dX L L  2 2 π4 2 πX d Y = Y sin , dX 2 L4 0 L Substitute Eq. (4) into Eq. (3) to obtain the strain energy relation for the beam structure:    EI L π 4 2 2 π X Y sin U = dX 2 0 L4 0 L   EI π 4 Y02 L EI π 4 Y02 . (5) = 4 2L 2 4L3 The kinetic energy T of the vibration beam can be obtained as  w 2 L 2 2 π X T = Y sin dX 2g 0 0 L   w 2 Y02 L . (6) = 2g 2 At the resonant frequency of the beam the kinetic energy is equal to the strain energy so Eq. (6) is equal to Eq. (5): EI π 4 Y02 w2 Y02 L = . 4L3 4g Then

π 4 EIg wL4 and since  = 2π fN , the natural frequency of a uniform beam with simply supported (hinged) ends becomes π 2 EIg π EIg fN = or f =: , (7) N 2π wL4 2 WL3 where E = beam modulus of elasticity, lb/in.2 I = moment of inertia, in.4 g = acceleration of gravity, in./sec2 w = weight of uniform beam per length, lb/in. 2 =

78

Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures W = total weight of uniform beam, lb L = length of beam between end supports, in.

3

SAMPLE PROBLEM: CALCULATING THE NATURAL FREQUENCY OF A UNIFORM BEAM Find the expected natural frequency of the simply supported beam (hinged at both ends) shown in Fig. 2 if it is made of aluminum and if it is made of steel. The beam dimensions are 12 in. long, 2 in. wide, and 0.50 in. thick. Vibration is in the vertical direction. Solution: For the aluminum beam: E = 10.5 × 106 lb/in.2 , modulus of elasticity h = 0.50 in., height or thickness b = 2.0 in., width L = 12 in., length g = 386 in./sec2 , acceleration of gravity ρ = 0.10 lb/in.3 , density of aluminum I = bh 3 /12 = (2.0)(0.50)3 /12 = 0.0208 in.4 , moment of inertia W = Lbh, ρ = (12)(2.0)(0.50)(0.10) = 1.20 lb, total beam weight w = W /L = 1.20/12 = 0.10 lb/in., weight of beam per length Substitute into Eq. (7) for the aluminum beam natural frequency: π fN = 2

 10.5 × 106 (0.0208) (386) (1.2) (12)3

= 316 Hz.

(8)

Solution: For the steel beam: E = 29 × 106 lb/in.2 , modulus of elasticity ρ = 0.283 lb/in.3 , density of steel W = (12)(2.0)(0.50)(0.283) = 3.39 lb, total beam weight w = W /L = 3.39/12 = 0.282 lb/in., weight per length Substitute into Eq. (7) for the steel beam natural frequency:

 29 × 106 (0.0208) (386) π fN = = 313 Hz. 2 (3.39) (12)3

(9)

The steel modulus of elasticity is almost three times greater than the aluminum modulus. A casual observer might believe that the steel natural frequency would be much higher than the aluminum natural frequency. Comparing Eq. (8) and Eq. (9) shows they are about the

0.50

2

12

Figure 2

4 Simple Method for Finding the Natural Frequency of Complex Structures

79

same. A closer examination shows that the steel beam is almost three times heavier than the aluminum beam. The ratio of the modulus of elasticity to the density of the materials must be considered to get a true picture of the natural frequency. This ratio is shown for several different materials to demonstrate the effect on the frequency: Magnesium:

E 6.5 × 106 = = 100 × 106 in., ρ 0.065

Aluminum:

E 10.5 × 106 = = 105 × 106 in., ρ 0.10

Steel

E 29.0 × 106 = = 102 × 106 in., ρ 0.283

Beryllium

E 42.0 × 106 = = 619 × 106 in. ρ 0.068

(10)

The materials selected above show that it may be difficult to increase the natural frequency of a given structure by simply selecting a material with a high modulus of elasticity without considering the material density. Major structural changes may have to be made to increase the natural frequency, which can be very expensive, and may add too much weight in a weight-sensitive application. Lightweight stiffening ribs can be added to some designs to increase the stiffness with a small increase in the weight and cost.

4

SIMPLE METHOD FOR FINDING THE NATURAL FREQUENCY OF COMPLEX STRUCTURES It is often easy to find the approximate natural frequency of a complex structure from the strain energy and the work done on the structure during vibration. For a linear system where the maximum deflection Y is directly proportional to the maximum force P, the area under the curve in Fig. 3 represents the work done on the structure. The strain energy U then becomes (11) U = 12 PY. The spring rate K can be defined in terms of the maximum force and deflection: K = P/Y

so

P = KY .

(12)

Substitute Eq. (12) into Eq. (11) and solve for the maximum strain energy: U = 12 KY 2 .

(13)

The maximum kinetic energy T in a vibrating system is defined in physics with the mass m and the velocity V as (14) T = 12 mV 2 .

p

y

Figure 3

80

Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures The tangential instantaneous velocity V can then be combined with the rotational velocity  as shown below: V = Y . (15) Substitute Eq. (15) into Eq. (14) to obtain the kinetic energy of the system: T = 12 mY 2 2 .

(16)

When the damping is zero, the kinetic energy T is equal to the strain energy U so Eq. (16) is equal to Eq. (13). This can be used to find the simplified natural frequency equation as shown below: 1 1 2 2 2 where  = 2π fN . 2 mY  = 2 KY

1 K K = and fN = . (17) m 2π m The above equations can be modified so they can be used to obtain the approximate natural frequency of a structure based upon its static displacement δ due to its own weight W and a structural spring rate K as shown below: W . (18) K = δ The mass m can be expressed in terms of the weight and the acceleration of gravity g: W . (19) m= g Substitute Eqs. (18) and (19) into Eq. (17) to obtain some another simple equations for obtaining the natural frequency of a structure based upon its static displacement: 1 g fN = . (20) 2π δ Then

5

SAMPLE PROBLEM Find the natural frequency of lateral vibrations for a cantilever aluminum beam with length 8.0 in., width 3.0 in., and thickness 0.40 in. with end mass of 5 lb (Fig. 4). Solution: The weight of the aluminum beam is small compared to the weight of the end mass. The weight of the aluminum beam can be ignored with little error. The cantilevered end mass can be evaluated on a massless beam that has all the elastic properties. The static displacement δ of the end mass can be obtained from reference books as shown below: δ= where

WL3 , 3EI

(21)

W = 5 lb, end weight L = 8 in., length of cantilevered beam

8

0.40 5 lb d

Figure 4

3

6 Sample Problem

81

E = 10.5 × 106 lb/in.2 , aluminum modulus of elasticity I = bh3 /12 = (3)(0.40)3 /12 = 0.016 in.4 , moment of inertia aluminum g = 386 in./sec2 , acceleration of gravity Then δ=

(5)(8)3 = 0.0051 in. static displacement. 3(10.5 × 106 )(0.016)

Substitute Eq. (22) into Eq. (20) where acceleration of gravity g is 386 in./sec2 : 1 386 fN = = 43.8 Hz. 2π 0.0051 The natural frequency can also be obtained from the spring rate K using Eq. (21):  3 10.5 × 106 (0.016) W 3EI = 984 lb/in. K = = 3 = δ L (8)3 The mass m is defined as W 5 m= = = 0.013 lb sec2 /in. g 386 Substitute into Eq. (17) to obtain the natural frequency another way: 1 k 984 1 fN = = = 43.8 Hz. 2π m 2π 0.013 Equations (23) and (25) match very well.

6

(22)

(23)

(24)

(25)

SAMPLE PROBLEM Find the natural frequency, displacement, force, stress, and fatigue life of a simply supported (hinged) 9.0-in.-long, 2.0-in.-wide, 0.30-in.-thick 6061 T6 aluminum beam with a 2.0-lb transformer mounted at its center; the beam is required to withstand a 2-hr sine resonant dwell with a 6G peak input in the vertical direction (Fig. 5). Is the design acceptable? Solution Aluminum density: 0.10 lb/in.3 Beam: length L = 9.0 in., width b = 2 in., thickness h = 0.3 in. Weight for beam only: (9)(2)(0.3)(0.10) = 0.54 lb Transformer weight: 2.0 lb Weight of beam and transformer: W = 0.54 + 2 = 2.54 lb Aluminum modulus of elasticity: E = 10.3 × 106 lb/in.2 Input acceleration level: G = 6.0, dimensionless gravity units Beam moment of inertia: I =

2 (0.3)3 bh 3 = = 0.0045 in.4 12 12 0.30

2 lb

2

9

Figure 5

82

Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures Beam static displacement: δ=

WL3 2.54 (9.0)3  = = 0.00083 in. 48EI 48 10.3 × 106 (0.0045)

Natural frequency: fN =

1 2π



g 1 = δ 2π



386 = 109 Hz. 0.00083

(26)

Dynamic force acting on aluminum beam: P = WGQ,

(27)

where Q is the Estimated transmissibility of beam at its resonant frequency. (For more vibration analysis on electronic equipment, see [1], p. 369.) The best results are for frequencies over 100 Hz. Note: The constant for beams is 1 and the constant for plates is 1 2 because beams typically have less damping than plates, so that beams have a higher Q:

 

109 0.76 fN 0.76 = (1) = 15.6 (dimensionless) (28) Q = (1) G 0.6 (6)0.6 So

P = (2.54)(6.0)(15.6) = 238 lb total dynamic force.

(29)

The bending moment at the beam center is given as P(L) (238) (9) = = 535 lb in. 4 4 and the beam center bending stress as M =

(30)

MC 535 (0.3/2) (31) = = 17,830 lb/in.2 I 0.0045 The approximate fatigue life expected for the aluminum beam with 6G peak sine input is based upon the definition of damage as defined by Crandall [2], as shown below (See Fig. 6 for more data):  b S2 N1 S1b = N2 S2b so N1 = N2 , (32) S1 Sb =

where

N1 = number of cycles expected for beam to fail at position 1 S1 = calculated beam stress level 17,830 lb/in.2 expected at position 1

2

b = 6.4

45,000

1

17,830 psi 7500

s K=2 Stress, psi 103

108 N Cycle

Figure 6

8 Sample Problem

83

N2 = number of cycles (1000) for beam to fail at 45,000 lb/in.2 at position 2 S2 = ultimate aluminum beam strength of 45,000 lb/in.2 at position 2 b = 6.4 expected slope of fatigue curve when stress concentration K factor of 2 is included in calculating fatigue life Thus

 N1 = (1000)

45,000 17,830

6.4 = 3.74 × 105 cycles to fail.

(33)

The expected aluminum beam fatigue life is L= and

3.74 × 105 cycles to fail (109 cycles/sec) (3600 sec/hr) Time to fail = 0.95 hr.

(34)

The aluminum beam must be able to survive a 2-hr dwell at its resonant frequency of 109 Hz for a period of 2 hr. The proposed beam design is expected to have a fatigue life of only about 1 hr. Therefore the proposed design is not acceptable. It may be possible to add some damping material to the beam or add snubbers to reduce the transmissibility factor Q at the resonant frequency. This would reduce the dynamic force acting on the beam, which would decrease the stress level and increase the fatigue life.

7

EFFECTS OF SINE VIBRATIONS ON FORCES, STRESSES, AND FATIGUE LIFE OF LEAD WIRES AND SOLDER JOINTS ON ELECTRONIC COMPONENTS SOLDERED TO PCBs Electronic component parts are manufactured in a wide variety of sizes and shapes that are often combined with many different plastics and ceramics for through-hole mounting and surface mounting on printed circuit boards (PCBs) and other applications. Many of these devices will be used for commercial, industrial, and military applications. This can include an extremely large number of industries such as automobiles, telephones, radios, airplanes, and space applications, just to name a few. In all cases the buyer, customer, or end user wants to have a reliable, low-cost product that is easy to use and will not harm the user. Sometimes they get it and sometimes they do not. Electronic products that look the same can often be very different in their performance and reliability. Manufacturing and assembly processes for the same product are often performed at different facilities and in different countries. This makes it very difficult to ensure the reliability and quality of the product. It often seems to be a matter of good luck when the product is good and bad luck when the product is bad. Satisfied customers are often willing to pay slightly more for a product they know from past experience will be a good reliable product. This chapter examines the effects of sinusoidal vibrations on the reliability and fatigue failure mechanisms associated with the electrical lead wires and solder joints on electronic component parts.

8

SAMPLE PROBLEM A small rectangular package 0.50 in. long, 0.30 in. wide, 0.20 in. thick (high) that weights 0.0030 lb with a Kovar wire 0.015 in. diameter at each of the four corners (a total of four wires) is through-hole soldered to a PCB leaving an air gap of 0.10 in. between the PCB and the package, as shown in Fig. 7. The PCB must pass a qualification test with a sine vibration

84

Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures Solder Wire

0.30

0.50 0.20

0.10 Gap

Wire

0.30

Base

Solder

Figure 7 resonant dwell of 10 min with a 2.0 G peak input parallel to the plane of the PCB; determine the forces, stresses, and expected fatigue life of the electrical lead wires and solder joints. Solution: The package is expected to bounce back and forth on its electrical lead wires when it is exposed to the sine vibration. This will force the four lead wires to bend back and forth, producing alternating bending stresses in the electrical lead wires and overturning forces and moments in the through-hole solder joints. If the alternating stresses in the lead wires and solder joints are high enough, failures can occur in the wires and solder joints. The maximum forces and stresses will occur at the natural (resonant) frequency of the system. The natural frequency of the package bouncing on its wires can be obtained from Eq. (20): 1 g fN = , 2π δ where g = 386 in./sec2 , acceleration of gravity δ = in., static displacement of package due to its own weight, L = 0.5 in., length of transformer w = 0.30 in., transformer width t = 0.20 in., package thickness (height) d = 0.10 lb/in.3 , package density W = (L)(w )(t)(d ) = (0.5)(0.3)(0.2)(0.10) = 0.003 lb weight The static displacement of the package acting on the four lead wires will force them to bend through a distance δ: WL3W δ= , (35) 12EI where W = 0.0030 lb, package weight LW = 0.30 in., length of each of four wires dW = 0.015 in., wire diameter (π ) (dW )4 (π ) (0.015)4 IW = = = 2.48 × 10−9 in.4 , one wire moment of inertia 64 64 = (4)(2.48 × 10−9 ) = 9.92 × 10−9 in.4 , four-wire moment of inertia E = 20 × 106 lb/in.2 , Kovar modulus of elasticity Then δ=

(0.0030) (0.30)3   = 3.4 × 10−5 in. package displacement. (36) (12) 20 × 106 2.48 × 10−9 (4 wires)

85

9 Sample Problem

Substitute into Eq. (20) to find the approximate natural frequency of the package, using the acceleration of gravity, g = 386 in./sec2 . 1 g 386 1 = 536 Hz. (37) = fN = 2π δ 2π 3.4 × 10−5 The dynamic force P acting on the four-package lead wires can be obtained from the 2G peak sine vibration input, the estimated transmissibility Q, and the weight W as follows: P = WGQ where

acting on four wires,

(38)

W = 0.003 lb, package weight G = 2.0 √ acceleration input √ peak sine Q = 3 fN = 3 536 = 69.4 estimated transmissibility Q value and the dynamic force acting on the four lead wires is given as P = (0.003)(2.0)(69,4) = 0.416 lb.

(39)

The dynamic bending moment M acting on the four electrical lead wires each with a fixed end on the top of the package and a fixed end at the PCB solder joint for a wire length of 0.30 in. will be as follows:   PLW (0.416) (0.30) = = 0.0624 lb in., on 4 wires. (40) M = 2 2 A stress concentration factor K = 2 was used for the wires which had a sharp bend where they were soldered to the top of the package. The bending stress in each wire was obtained from the standard bending stress equation SB as KMCW SB = , (41) IW where K = 2, stress concentration at sharp bend in each wire CW = 0.015/2 = 0.0075 in., radius of package wire [Eq. (35)] IW = 2.48 × 10−9 × 4 = 9.94 × 10−9 in.4 , four-wire moment of inertia [Eq. (35)] Thus, for four wires   (2) (0.0624) (0.0075) SB = = 9.4 × 104 lb/in.2 bending stress. 9.94 × 10−9

9

(42)

SAMPLE PROBLEM Determine the approximate fatigue life of the four Kovar electrical lead wires with a sharp 90◦ bend soldered at the top of the package. The approximate number of bending cycles required to produce a failure at the sharp bend in the wires can be obtained from Eq. (32) and Fig. 7. Solution

N1 = N2

where

S2 S1

b [Eq. (32)]

N1 = number of cycles required to produce failure at sharp wire bend S1 = 9.4 × 104 lb/in.2 , bending stress calculated in wire N2 = 1000, number of cycles required for wire failure at 84,000 lb/in.2

86

Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures S2 = 84,000 lb/in.2 , ultimate strength of Kovar wire b = 6.4, expected slope of fatigue curve for stress concentration factor K = 2, stress concentration factor at sharp bend in wire at top of package Thus

N1 = (1000)

84,000 9.4 × 104

6.4 = 5.12 × 105 cycles to fail.

(43)

The approximate time it will take for the wire to fail for a resonant dwell with a 2G peak sine vibration input can be obtained from the 536 Hz package natural frequency shown in Eq. (37) and the number of cycles to fail shown above:  5.12 × 105 cycles to fail = 0.26 hr (16 min). (44) Time to fail = (536 cycles/sec) (3600 sec/hr) At first glance it would appear the fatigue life of the package wires for the 10-min resonant dwell requirement would be acceptable. The calculated fatigue life appears to be about 16 min, which provides a small safety factor of about 1.6 times the testing requirements. This would be the typical conclusion of a design and analysis engineer with little or no vibration testing experience. However, an examination of the calculations shows that the dynamic forces acting on the four package wires were assumed to be divided equally among the four wires. Vibration test data shows this condition is almost impossible. Test data with a strobe flashing light shows that these systems will bend and twist with six degrees of freedom. No two systems ever respond in exactly the same way. This means that the forces on the wires are constantly changing with time. The standard method of analysis for these types of systems is to assume a distribution of 60 and 40% for the forces on the wires to compensate for unknown force increases in local areas. This then results in general local force increase of 60/40, which increases the maximum expected force by a factor of 1.5 times in some wires, from Eq. (42). The new stress becomes 31,690 × 1.5, or about 47,530 lb/in.2 rounding off the number. The 1.5 times increase in the dynamic force will act on Eqs. (39) and (40) and (42). This changes Eq. (43) as shown below:

 84,000 6.4 N1 = 1000 = 38,269 cycles to fail, (45) 47,530 38,269 cycles to fail 0.02 hr (1.2 min). (46) Time to fail = (536 cycle/sec) (3600 sec/hr) The more realistic fatigue life analysis of 0.02 hr to fail shows there is a high probability of a failure in the electrical lead wires on the package during the 10-min dwell. Therefore, the package mounting on the PCB may have to be changed, or the 2G peak sine vibration dwell duration may be reduced to a lower level. It may also be possible to use snubbers or dampers under the package. This can reduce the package displacements, which will reduce the lead wire stresses and increase the fatigue life.

10

SAMPLE PROBLEM What follows shows a method for obtaining the fatigue life of solder joints where electrical lead wires are soldered to copper in through-hole circuit boards, similar to the one shown in Fig. 7. This sample problem will use a different type of package that is similar in appearance to Fig. 7. A different set of parameters was assumed. These were used to demonstrate a method of analysis for calculating the approximate fatigue life of various structural units.

10

Sample Problem

87

0.018 PCB hole dia.

Wire dia.

0.030 Solder

Figure 8

Solder Shear tear out

Figure 9 Solution: Holes for solder joints in PCBs must have a diameter slightly larger than the diameter of a round electrical lead wire. This will allow capillary action to pull the liquid solder up into the spaces between the wire and the hole. The dimensions of the wires and the holes in the PCB are shown in Fig. 8. During vibration, the electrical lead wires at the through-hole solder joints in the PCB will be forced to bend back and forth, producing shear tearout forces and stresses in the solder joints shown in Fig. 9. These are called shear tearout forces and stresses. The analysis shown here can be used for eutectic solders (63/37 tin/lead) and lead-free solders. The basic shear tearout equation used for these analyses is shown below: M (shear tear out stresses). (47) hA The bending moment M at the PCB through-hole solder joint was assumed to start by considering the electrical lead wires to be clamped (fixed) at both ends, where they are soldered at the top of the package and where they are soldered at the through holes in the PCB. This was evaluated as a beam fixed at both ends and forced to move laterally. Using a system similar to Fig. 7 the sum of the moments M at the top of the wire and at the bottom of these solder joints must be equal to the moment produced by the force P acting at each end of the wire over the length L of the wires. See [3], page 166, Fig (10.2) and Eq. (10.42), where M = Ph/2, to be used in the equations below, where M = 0.062 lb in., bending moment at PCB solder joint (assumed to start) h = 0.065 in., height of solder joint to be equal to the PCB thickness DS = 0.030 in., outer diameter of solder joints in PCB through holes DW = 0.018 in., outer diameter of package wires (Fig. 8). STO =

An examination of many vibration and thermal cycling failures in through-hole solder joints on electrical lead wires has shown that the solder joints appear to fail most often at locations between the outer diameter of the through-hole solder joint and the outer diameter of the lead

88

Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures wires. Therefore an average diameter is used based on the diameter of the through hole in the PCB and the outer diameter of the electrical lead wire: 0.030 + 0.018 Average diameter DAV = = 0.024 in. (48) 2 The average solder joint area is given as π AAV = (DAV )2 4 So π AAV = (0.024)2 = 4.5 × 10−4 in.2 4 From Eq. (47) the shear tearout stress per wire is 0.062  STO = (0.065) 4.5 × 10−4 (4 wires) and

Solder STO = 530 lb/in.2

(49)

The solder joint approximate fatigue life (see [1], p. 44, Fig. 3.2) is  b S2 N1 = N2 , S1 where N1 = number of vibration stress cycles to produce solder joint failure S1 = 530 lb/in.2 , calculated shear tearout stress [Eq. (49)] N2 = 1000, number of stress cycles for solder joint failure at 6500 lb/in.2 S2 = 6500 lb/in.2 , stress level expected for solder failures at 1000 cycles b = 4, slope of solder fatigue curve for rapid vibration stress cycles Note that solder has very little creep during rapid vibration cycles (Fig. 10). Thus   6500 4 = 2.26 × 107 cycles to fail, N1 = (1000) 530

(50)

(51)

2.26 × 107 cycles to fail = 11.7 hr. (52) (536 cycles/sec)(3600 sec/hr) The above time to fail assumes the four package solder joints had exactly the same stress levels during vibration. Extensive testing on these types of systems shows this condition is Time of fail =

6500

Sold

er vib

ratio

2900 b=4 The

rma

Stress lb/in.2 102

n 650

Sol

der

l cy

cle

b = 2.5

200 29

103

84 N Cycle to fail

Figure 10

107

11

Octave Rule for Reducing Severe Sine Vibration Dynamic Coupling

89

almost impossible to achieve. The package will have six degrees of freedom so the dynamic forces on the wires and solder joints will be constantly changing during a sine vibration dwell condition so the solder will have very little creep effects. When the industry standard analysis distribution of 60–40% for the dynamic forces for this type of system is used, the maximum force acting on the typical wire will be increased as shown below: 60 Maximum dynamic force = (53) = 1.5 × 530 lb/in.2 = 795 lb/in.2 40 Substitute into Eq. (50) using the 795 solder joint shear tearout stress value:   6500 4 = 4.47 × 106 cylces to fail. (54) N1 = (1000) 795 The approximate time for a solder joint to fail will be 4.47 × 106 cycles to fail = 2.3 hr. (55) (536 cycles/sec) (3600 sec/hr) Comparing Eq. (52) with Eq. (55) shows the solder joint vibration fatigue life of 2.3 hr to fail for the variable stress condition is far more critical than the solder joint uniform shear tearout stress condition fatigue life of 11.7 hr. Time to fail =

11

OCTAVE RULE FOR REDUCING SEVERE SINE VIBRATION DYNAMIC COUPLING WITH A CHASSIS HOUSING AND REDUCING FAILURES IN INTERNAL PCBs Electronic components are being used in a wide range of applications in many different industries throughout the world. Different methods are used for packaging these devices depending upon many factors associated with the size, weight, cost, health, safety, and operating environments. One of the most common type of packaging involves the use of PCBs, where the electronic components can be attached using a wide range of materials. These include solders, screws, straps, epoxy, crimping, and tying, just to name a few. Operating environments can include vibration, shock, thermal, space, humidity, sand, dust, liquids, as well as combinations of these and others. PCBs are often enclosed within different types of chassis and outer housings for added protection from severe external environments. The PCBs may be plug-in types in sockets or bolted or clamped in place. This type of packaging can be the source of many problems for systems that are required to operate with a high reliability in severe sine vibration environments. Problems here involve structural conditions where the outer housing may have a natural frequency that is very close to the natural frequencies of some PCBs contained within the housing. If the sine vibration input acceleration G level excites the natural frequency of the outer housing, and if the outer housing natural frequency is close to the natural frequency of several internal PCBs, severe dynamic coupling can occur between the housing and the PCBs. This condition can amplify the acceleration G forces acting on the sensitive PCBs that may produce rapid failures in these PCBs. The force increase depends upon the damping in the system which determines the transmissibility Q. Several different ideas or methods can be used in this application to reduce the forces and stresses produced in the PCBs that will increase the fatigue life of the sensitive structural members. The first method shown here uses the octave rule, which can be difficult to implement This requires the structural members of the outer housing to have a natural frequency that is separated from the internal PCB natural frequency by an octave. This is a factor of at least 2 to 1 in either direction above or below the coupling frequency.

90

Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures PCB

(Housing chassis)

Support

Vibration input

Figure 11 When the PCBs are mounted in a housing and the housing assembly is exposed to a sinusoidal vibration input, the housing receives the energy first so the housing becomes the first degree of freedom. When the internal PCBs are attached to the housing, the response of the housing will be the input to the PCBs, so the PCBs become the second degree of freedom. The vibration response of the housing includes the input acceleration G level that will be multiplied by the transmissibility Q, which depends upon the damping of the housing. For example, if the sine vibration input value to the housing is 3G peak and the transmissibility Q of the housing is 5, the total force on the housing will be 3 × 5, or 15G. This now becomes the input to the PCBs. The internal PCBs will also have their own natural frequency and a transmissibility Q that also depends upon the PCB damping. When the natural frequencies of several PCBs are very close to the natural frequency of the housing, severe coupling can occur that can produce high acceleration G levels and rapid failures in the PCBs. For example, if the transmissibility Q of the sensitive PCBs is 4, the real response of the sensitive PCB will be 4 ×15, or 60G, which could cause problems. A drawing that shows how the first degree of freedom for the housing interacts with the second degree of freedom for the PCBs is shown in Fig. 11. See Section 18 for other methods, such as snubbers, for improving the PCB fatigue life when the octave rule cannot be followed.

12

SINUSOIDAL VIBRATIONS A series of sinusoidal vibration tests were combined with extensive finite element model analyses for typical types of electronic component parts that are used extensively by the electronics industry. These tests show that component attachment points, such as electrical lead wires and solder joints, were able to achieve a fatigue life of about 10 million stress cycles when the peak single-amplitude displacement Z (zero to peak) of the PCB was limited to the value shown in Eq. (56). Note that the 0.00022B constant included a safety factor of 1.3. The purpose of this safety factor was to provide enough fatigue life in the electronic system to pass a typical qualification test at least five times. Extensive qualification testing experience with electronic equipment has shown that unexpected problems always seem to appear that require the tests to be repeated over and over again. If there is not enough fatigue life available for repeated tests, a new test system must be provided to complete the qualification test program. This can be very expensive and time consuming. This can also make upper management executives very unhappy.

13

Large Dynamic Displacements in PCBs

91

Designing electronic systems for reliable operation in severe environments normally requires the use of safety factors. Safety factors will almost always result in a small increase in the size, weight, and cost of the system. It is not a good idea to mention this to any upper management people, because they always desire smaller, lighter weight, and lower costs for these systems for improved profits. They also want increased reliability for these products. Most upper management people have a business background with very little engineering experience. They are there to make a lot of money for the company. They want the design engineer to design rugged, reliable, lightweight, low-cost electronic systems for improved profits. They do not want to hear that there are large variations in the quality and manufacturing tolerances in the materials that must be purchased from outside suppliers. They do not want to hear that even their own factories find it difficult to machine high-strength materials with close tolerances at a low cost. They do not want to hear that the engineers must introduce safety factors in their designs to compensate for the large variations in the manufacturing processes, materials, and quality of the work to ensure a high reliability for operation in severe environments. An examination of Eq. (33) shows that the fatigue life equation for lead wires with a stress concentration factor of 2 has an exponent of 6.4. The safety factor of 1.3 taken to the 6.4 power then shows the solder joints and the lead wire fatigue life can be increased by a factor of 5.4 times. This is a very important requirement to ensure there will be enough fatigue life in the structure when certain tests must be repeated several times. The exception to this rule involves designing electronic equipment for space programs that will orbit around the earth. It takes 1000 lb of fuel to put 1 lb into orbit. The severe launching environment in this case cannot be repeated five times, so high safety should not be used because the added weight will require more fuel for the launching.

13

LARGE DYNAMIC DISPLACEMENTS IN PCBs Large dynamic displacements in PCBs at the natural frequency rapidly increase the solder joint and lead wire forces and stresses that reduce their fatigue life. The following equation is used to ensure the reliability of the electronics by limiting the maximum allowable singleamplitude vibration displacement Z for a sine vibration environment: Z = where B L h C R

= = = = =

0.00022B √ ChR L

(56)

length of PCB edge parallel to rectangular component edge, in. length of rectangular component, in. thickness or height of PCBs, in. constant for different types of electronic components, dimensionless approximate relative location of components on PCB (see Fig. 12 for value of location constant)

A second equation (56a) using English units can be combined with Eq. (56) to add to the reliability extension of the general fatigue life. A third equation that was approximate and based upon extensive vibration testing Eq. (57), shown below, was added to include the damage generated by the transmissibility Q factor at the natural frequency ( fN ) of the PCB. This relation is fairly accurate for a sine vibration input level of 5G peak. The accuracy is somewhat reduced at higher input G levels and at lower input G levels: Single-amplitude displacement:

(9.8) (G) , f2  Q = fN .

ZSA =

(56a) (57)

92

Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures The minimum desired natural frequency for the PCB to obtain a fatigue life of about 10 million stress cycles can be obtained by combining Eqs. (56), (56a), and (57). The final equation shown below, Eq. (58), will provide a fatigue life for a typical PCB with electronic components, electrical lead wires, and solder joints based upon a sine vibration environment:  √ 2/3 9.8.GIN ChR L fD = (minimum desired PCB frequency). (58) 0.00022B

14

SAMPLE PROBLEM: VIBRATION FATIGUE LIFE OF LARGE BALL GRID ARRAY A large ball grid array (BGA) 1.25 in. square is mounted at the center of a plug-in type of multilayer epoxy fiberglass PCB that measures 8 × 7 × 0.20 in. thick. Find the minimum desired PCB natural frequency required to provide a fatigue life of about 4 hr for a resonant dwell condition when subjected to a 5 G peak sinusoidal vibration input. Solution: The input information is as follows: GIN = 5 G, peak input sine vibration, resonant dwell C = 1.75, component type for BGA h = 0.20 in., height or thickness of PCB R = 1.0, characteristic for component mounted at center of PCB B = length of PCB edge parallel to square component edge = 7, for square component, use shorter of PCB length or width L = 1.25 in., length of square BGA component side Then

 fD =

2/3 √ (9.8)(5)(1.75)(0.20)(1.0)( 1.25) = 534 Hz. (0.00022)(7)

(59)

The approximate fatigue life expected for the BGA solder joints is  10 × 106 (cycles to fail) = 5.2 hr to fail. (60) Life = (534 cycles/sec) (3600 sec/hr) The desired fatigue life of the BGA is 4 hr. Equation (60) shows the expected fatigue life of the BGA is only about 5.2 hr. This is acceptable. If the fatigue life must be increased, this can be accomplished by mounting the BGA away from the center of the PCB where the dynamic displacements are maximum. See Fig. 12. The relative curvature of the PCB can be related to the displacment of the PCB and the basic equations relating these functions (see [1], p. 178): πX π, Y Z = Z0 sin sin . a b When the component is at the center of the PCB, π π Z = Z0 sin sin = Zo = 1. 2 2 When the component is at X = a/2 and Y = b/4, π π Z = Z0 sin sin = 0.707. 2 4 When the component is at X = a/4 and Y = b/4, π π Z = Z0 sin sin = 0.50. 4 4

15 Using Sine Sweeps through a Resonance to Evaluate Electronic Equipment 93 Component Space large

y

Component

Space small

Sine curve Base

= 1.0 = 0.707 = 0.50

x

Figure 12 Change the mounting position of the BGA to one of the quarter-mounting points near one of the four side-corner edges away from the center of the PCB. The location constant for these four areas is only 0.5 instead of 1.0 because the relative dynamic displacements of the PCB near the corner edges is much lower than at the center of the PCB. The reduced dynamic displacements also reduce the solder joint stresses, which increases the fatigue life of the BGA as shown below:  ⎤2/3 √ ⎡ 1.25 (9.8) (5) (1.75) (0.20) (0.5) ⎦ = 336 Hz. fD = ⎣ (61) (0.00022) (7) The expected fatigue life of the BGA for the new mounting position will now be increased as shown below:  10 × 106 (cycles to fail) = 8.26 hr to fail. (62) Life = (336) (cycles/sec) (3600 sec/hr) Since this fatigue life is well above the 5.2 hr shown in Eq. (60), this condition is more acceptable for the required fatigue life.

15

USING SINE SWEEPS THROUGH A RESONANCE TO EVALUATE ELECTRONIC EQUIPMENT Sinusoidal vibration tests are often used to evaluate the response characteristics of new electronic system designs, or the effects of design changes on the fatigue life of a PCB, or the effects of mounting a large heavy transformer on a sheet metal cover, or to check the accuracy of a finite element computer model analysis. These types of tests are used extensively as described in military specifications such as MIL-E- 5400, MIL-T-5422, and MIL-STD-810. A logarithmic sweep is typically recommended from 20 to 2000 Hz, then back to 20 Hz. The sweep rate is specified in octaves per minute. Octave means to double. Therefore the sweep rate from 20 to 40 Hz will be the same time as the sweep rate from 40 to 80 Hz will be the same time as from 1000 to 2000 Hz. It is very important to make plots of these sweeps up and down using the same frequency and sweep rate for both directions. The sweep-up plot can then be compared with the sweepdown plot. It is desirable to have very similar looking plots which will then show the system

Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures

Transmissibility Q

94

Q

p1

p2 Q 2

Δf =

fn Q

Figure 13 is linear. If the plots are substantially different and they show the resonant peaks have sharply different heights and at different frequencies, it is a sign of a nonlinear system. Most nonlinear systems are more difficult to analyze and evaluate accurately. The failure rates are often higher because they are difficult to evaluate accurately. Most of the damage will occur at the peak response points during the sweep through the resonant frequency, where the transmissibility Q factors are the highest. A convenient reference point is the half power points. This is a reference used extensively by the electrical engineers. These are the points where the power that can be absorbed by the damping is proportional to the square of the amplitude at a given frequency. For a lightly damped system, where the transmissibility Q is greater than about 10, the curve in the region of the resonance is approximately symmetrical. The half-power points are often taken to define the bandwidth, as shown in Fig. 13. The time T it takes to sweep across the half-power points can be obtained from the equation   1 + (1/2Q) (loge ) 1 − (1/2Q) T = , (63) R(loge 2) where T = time, min R = sweep rate, octaves/min Q = transmissibility at resonance, dimensionless

16

SAMPLE PROBLEM: FATIGUE DAMAGE ACCUMULATED BY A SINE SWEEP THROUGH THE HALF-POWER POINTS Sinusoidal vibration test data on an electronic assembly showed a fatigue life of about 20 min during a 400-Hz resonant dwell condition with a 5G peak input. A reliability demonstration test was proposed to sweep back and forth through the half-power points using a sweep rate of 1.0 octave/min. How many fatigue cycles are expected to be accumulated during one sweep through the half-power points? How many single sweeps through the half-power points will be required to produce a failure? Solution: Equation (63) can be used to find the time it will take to sweep through the halfpower points if the transmissibility Q can be obtained. Extensive test data have shown that a

18

When Octave Rule Cannot Be Followed, Other Options Are Available

95

good approximation for the Q factor can be taken as the square root of the natural frequency when a 5G peak sine vibration input is used. Then   1 + 1/40 log e √ 1 − 1/40 Q = 400 = 20, T = = 0.072 min. (64) 1.0 loge 2 The approximate number of stress fatigue cycles N that will be accumulated during one sweep through the resonant frequency of 400 Hz can now be obtained: N = (400 cycles/sec)(0.072 min)(60 sec/min) = 1728 cycles/sweep.

(65)

The approximate number of single sweeps NS that will be made during a period of 0.072 min through the half-power points to produce a failure can now be determined. 20 min to fail NS = = 277. (66) 0.072 min/sweep

17

PROPERTIES OF VARIOUS TYPES OF ENCLOSURES AND PCBs Support structures are often fabricated using flat panels with various shapes such as rectangular and circular for enclosures, chassis, and housings that will provide protection for sensitive electronic equipment in the enclosure. This sensitive electronics is usually mounted on flat PCBs that are attached to the inside walls of the chassis. Many different types of PCBs are manufactured by the electronics industry. The most common types of PCBs are fabricated of epoxy fiberglass. Electronic components are usually soldered to the PCBs, sometimes on one side and sometimes on both sides. Electronic components can be surface mounted or throughhole mounted on the PCBs. These PCBs can be attached to the inside walls of the enclosure in many different ways. Plug-in types of PCBs using bottom-edge connectors and side guides are popular because the PCBs are easy to insert and remove or to replace defective PCBs. Side wedge clamps are popular because they provide a good conduction heat flow path from the PCBs to the inside walls of the chassis to outside ambients where the heat can be removed by natural convection and radiation. The side wedge clamps can also add support the PCBs that can increase the PCB natural frequency and improve the vibration and shock fatigue life. Side wedge clamps can be attached to the sides of the PCB or they can be attached to the inside walls of the chassis. If the chassis is made as an aluminum casting, the side edge guides can be part of the casting. If the chassis is machined out of an aluminum block, the side edge guides can be part of the machining operation. Side edge guides are often formed from beryllium copper and epoxy bonded to the inside walls of the chassis.

18

WHEN OCTAVE RULE CANNOT BE FOLLOWED, OTHER OPTIONS SUCH AS SNUBBERS OR ULTRASMALL PING-PONG BALLS OR ISOLATION SYSTEMS ARE AVAILABLE When the octave rule cannot be followed, it is still possible to have a safe design by adding snubbers to the PCBs to reduce the PCB dynamic displacements. High PCB displacements can produce high forces and stresses in the electrical lead wires, solder joints, and electronic component parts. High stresses reduce the fatigue life of the lead wires and solder joints, which is not desirable. Effective snubbers can be made from small epoxy fiberglass dowel rods about 0.25 in. in diameter that can be epoxy bonded to the center

96

Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures of each PCB on both sides of each PCB. A small clearance of about 0.005–0.010 in. should be allowed between the adjacent PCB snubbers.. This allows the snubbers to strike each other when the PCBs are forced to bend back and forth when they are exposed to vibration and shock. This action reduces the dynamic displacements of the PCBs so the stresses are reduced and the fatigue life is increased. Care must be used in attaching the snubbers to the PCBs to make sure there is room for the snubbers. They must not hit the electronic components on adjacent PCBs when a PCB is inserted or removed from the chassis. When the PCB snubbers strike each other during vibration and shock, the striking action may generate forces as high as 50G. These forces can act on the sensitive electronic parts in the general area of the snubbers. Some people believe these forces can damage the parts very close to the snubbers. Extensive test data show no components are ever damaged. The reason for this can be found in the MIL-STD 883 document. This specifies that all military semiconductor electronic component parts must be able to withstand an acceleration force of 30,000G when tested in a centrifuge to ensure the component structural integrity. So a force of 50G will have little effect on electronic components. Note: Extreme care must be used when electronic components are tested at high acceleration G levels in a rotating centrifuge. Components that have unsupported internal die bond wires must be installed in the centrifuge so the force acting on the die bond wires tends to pull the wires out of their sockets. When the electronic components are inserted in the centrifuge backward, high compression forces will act on the die bond wires. This can cause the wires to buckle and collapse and possibly cause internal short circuits during electrical operation in shock and vibration environments. Very small, ultra lightweight, hollow plastic spheres similar to ping-pong balls are available that can be dropped into the open top of an electronic enclosure before the top cover is fastened to the enclosure. These small lightweight spheres can fill in around the PCBs and their electronic components to sharply reduce their dynamic displacements and stresses for an improved fatigue life. The enclosure should be vibrated at a low frequency when the spheres are being added. This will allow them to fill in around the various PCBs and other electronic components in the housing to sharply reduce the displacements and stresses in the structural members. If repairs or replacements have to be made, the small spheres can be easily removed by turning the housing over and tapping it until all of the spheres have been removed. The spheres can be replaced in the housing after the repairs have been made.

19

ISOLATION SYSTEMS FOR IMPROVED FATIGUE LIFE IN VIBRATION AND SHOCK Another method that can be used to reduce the stresses and improve the system fatigue life is to isolate the system. Isolators can be purchased from companies such as Lord Corp. They will help you select a set of isolators that will protect you from vibration and shock. If the electronic system is enclosed in a chassis about the size of a shoe box, with a weight of about 20 or 30 lb, a set of four isolators can be used, mounting one isolator at each of the four corners at the bottom of the chassis. The isolators will perform better if they can be mounted on the sides of the chassis near the center of gravity (CG) of the chassis. This type of mounting takes up more room, but a CG mount reduces the rotational sway amplitude experienced by the chassis when it is exposed to vibration at its natural frequency or to shock. The CG mount reduces the dynamic forces and stresses acting on the electronic system, which increases the fatigue life. A fundamental natural frequency of about 30–35 Hz for the isolation system usually works well for vibration and shock protection. These devices will reduce the vibration and shock levels to improve the fatigue life during operations in harsh environments.

20

20

Determining Natural Frequencies of Uniform Flat Plates and PCBs

97

DETERMINING NATURAL FREQUENCIES OF UNIFORM FLAT PLATES AND PCBs Printed circuit boards are fabricated in a wide range of shapes as rectangular, circular, oval, triangular, hexagonal, and many other shapes to fit into unusual shapes such as airplane wings and tails, under the hood of an automobile, or in the handle of an electric knife, a tooth brush, a wrist watch, or a missile. The most popular shape is the simple rectangle. A wide variety of different materials are also used to fabricate the various PCBs. The most popular PCB material is epoxy fiberglass. Thicknesses often will vary from about 0.006 to 0.125 in. In order to control warping or to increase the natural frequency, the PCB thickness may be increased to 0.20 in. Thin copper circuits are often used in several layers to provide electrical interconnects. Additional copper layers and thin aluminum panels may also be used to improve the thermal conductivity of the PCBs or to act as heat sinks to carry away the heat and reduce high temperatures. Adding the copper and aluminum layers also increases the stiffness of the PCB, which will increase the PCB natural frequency. PCB sizes can vary from about 2 in. for instrument panels to about 36 in. for large cabinets. High-power electronic systems that include transformers and power supplies that can generate large amounts of heat are often mounted on PCBs. These high-power devices often require forced convection with fans to carry away the excessive heat. Flat rectangular plates with different edge conditions and different loading conditions can use the strain energy and the kinetic energy properties of the plate, with the use of the general plate equations, to determine the natural frequencies of the plate. One method of evaluation that is very convenient was shown by Lord Rayleigh [4]. In this method of analysis, a deflection curve is assumed that satisfies the geometric boundary conditions of the deflection and the slope for a specific plate. When this has been completed, the assumed deflection curve is used to obtain the strain energy and the kinetic energy of the specific plate. The Rayleigh method of analysis produces natural frequencies that are slightly higher than the correct natural frequencies for a given set of conditions for the plate geometry. Start by assuming a flat rectangular plate with a uniformly distributed load and with four simply supported (same as hinged) edges. This plate will be vibrated in a direction that is perpendicular to the plane of the plate. The resulting deflection curve for the simply supported plate can be represented by a double-trigonometric series: Z =

∞ 

∞ 

Am,n sin

m=1,3,5 n=1,3,5

mπ X nπ Y sin . a b

Most of the damage will typically occur at the fundamental resonant mode where the displacements and the stresses are the maximum. Therefore, the above equation can be simplified as Z = Z0 sin

πX πY sin . a b

(67)

The geometric deflection boundary condition can be checked to make sure it meets the assumed deflection curve. This requires the edges of the plate to have a zero deflection and the center of the plate to have the maximum deflection using the above equation: At At At At At

X X X X X

= 0 and Y = 0 and Y = a and Y = a and Y = a/2 and

= 0, Z = 0. = b, Z = 0. = 0, Z = 0. = 0, Z = 0. Y = b/2, Z = Z0 .

98

Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures The boundary conditions for deflections are satisfied since there are no deflections at the four edges and the maximum deflection is at the center of the plate. The slope of the plate must be checked for the boundary conditions at various points. All four edges must have a finite slope and the center of the plate must have a zero slope. Examine the slope (θX ) along the X axis first. An equation can be obtained using the partial derivative of Z with respect to X from Eq. (67): ∂Z π πX πY θX = = Z0 cos sin ∂X a a b Then At X = 0 and Y = b/2, θX = Z0 (π/a). At X = a/2 and Y = b/2, θX = 0. At X = a and Y = b/2, θX = −Z0 (π/a). Boundary conditions for the slope are satisfied for the edges and the center along the X axis. The same conditions are also satisfied for the slope along the Y axis. The total strain energy for the vibration plate () will be as shown in the following strain energy equation:  2 2  2  2   2 2     2  2 D a b ∂ Z ∂ Z ∂ Z ∂ Z ∂ Z = dX dY, + + 2μ + 2 (1 − μ) 2 0 0 ∂X 2 ∂Y 2 ∂X 2 ∂Y 2 ∂X ∂Y (68) where D E h μ

= = = =

Eh 3 /12(1 − μ2 ) plate stiffness factor modulus of elasticity, lb/in.2 plate thickness, in. Poisson’s ratio, dimensionless

The total kinetic energy T of the vibrating plate is shown by the equation   ρ2 a b 2 T = z dX dY , 2 0 0 where a = length of plate, in. b = width of plate, in. h = plate thickness, in. δ = material density, lb/in.3 g = acceleration of gravity, 386 in./sec2 w = total weight of plate, lb ρ = w/abg = δh/g (mass per unit area)  = circular frequency, rad/sec Perform the operations on Eq. (67) as defined by Eq. (68) and outlined below: ∂Z π πX πY = Z0 cos sin , ∂X a a b πX π2 πY ∂ 2Z = −Z sin sin , 0 2 2 ∂X a a b  2 2 π4 πX πY ∂ Z = Z02 4 sin2 sin2 , 2 ∂X a a b

(69)

21

Sample Problem

99

∂Z π πX πY = Z0 sin cos , Y b a b πX π2 πY ∂ 2Z = −Z0 2 sin sin , 2 ∂Y b a b  2 2 4 πX πX ∂ Z 2π = Z sin2 sin2 , 0 ∂Y 2 b4 a b  2  2  4 ∂ Z ∂ Z πX πY 2 π = Z sin2 sin2 , 0 ∂X 2 ∂Y 2 a 2b2 a b  2 2 π4 πX πY ∂ Z = Z02 2 2 cos2 cos2 . ∂X ∂Y a b a b The eight equations shown above must be integrated using the following relations:  a πX a sin2 dX = , a 2 0  b π Y b sin2 dY = . b 2 0 It can be shown that the strain energy () of the vibrating plate will be given as    D abZ02 π 4 1 2μ 2 (1 − μ) 1 . = + + + 2 2 4 a4 b4 a 2b2 a 2b2 So π 4 DZ02 ab = 8



2 1 1 + 2 2+ 4 a4 a b b

(70)

 .

(71)

The kinetic energy of the vibration plate can be obtained with the use of the deflection equation (67) and the strain energy equation (69). This will result in the kinetic energy of the vibrating plate,   ρ2 2 ab Z0 . (72) T = 2 4 The kinetic energy of the vibrating plate must equal the strain energy of the plate at the resonant condition if there is no energy lost. So Eq. (71) must equal Eq. (72):     π 4 DZ02 ab 1 ρ2 Z02 ab 1 2 + 2 = 8 a2 b 8 These equations can be combined and simplified, which will yield the basic natural frequency (fN ) of a flat uniform plate simply supported (hinged) on all four sides:

  π D 1 1 . (73) + fN = 2 ρ a2 b2

21

SAMPLE PROBLEM Determine the natural frequency of a flat rectangular plate fiberglass PCB that measures 8.0 in. long, 5.0 in. wide, 0.08 in. thick, with a uniformly distributed load and a total weight of 0.90 lb, The plate is simply supported (hinged) on all four edges; the properties of the board are shown below.

100 Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures Solution: a b h w g E μ

= = = = = = =

8.0 in., PCB length 5.0 in., PCB width 0.08 in., PCB height or thickness 0.90 lbs, PCB weight 386 in./sec2 , acceleration of gravity 2.0 × 106 lb/in.2 , PCB modulus of elasticity 0.12 Poisson’s ratio, dimensionless

and

ρ=

where Mass = and D=

mass , area

weight w = gravity g

Eh 3 2 × 106 (0.08)3 =  = 86.7 lb in.  12 1 − μ2 12 1 − (0.12)2 

w 0.90 = = 5.83 × 10−5 lb sec2 /in.3 gab (386)(8.0)(5.0) Substitute into Eq. (73) to solve for the expected natural frequency of the PCB:   86.7 1 π 1 = 106 Hz. + fN = 2 5.83 × 10−5 82 52 ρ=

22

(74)

NATURAL FREQUENCY EQUATIONS The natural frequency equations for several different types of flat rectangular plates with uniformly distributed loads and three different types of perimeter edge guides were derived and tested on a vibration shaker. The various edge guides evaluated used combinations of simple supports (or hinged) as well as fixed (or clamped) and free edges:     π D 1/2 1 1 fN = + 11 ρ a2 b2  1/2   π D 1 1 = + 2 2 ρ 4.a 2 b  1/2   π D 1 1 = + 2 ρ a2 b2 1/2

 3.2 1 π D 1 + 2 2+ 4 = 5.42 ρ a 4 a b b 

 π D 0.75 2 12 1/2 = + + 3 ρ a4 a 2b2 b4

 1/2 π D 3 2 3 = + + 1.5 ρ a 4 a 2b2 b4

 1/2 π 8 3 D 16 = + + . 3.46 ρ a 4 a 2b2 b4

23

Effects of Loose Edge Guides on Plug-in Types of Rectangular Circuit Boards fN = = = = = = = = = = = =

23

101

  D 2.08 1/2 ρ a 2b2   0.56 D 1/2 a2 ρ  1/2 3.55 D a2 ρ  1/2 0.78π D a2 ρ  1/2 π D 2a 2 ρ

 1/2 π 1 1 D 4 + 2 2+ 1.74 ρ a 4 2a b 64b 4

 1/2 π D 0.127 0.20 + 2 2 4 2 ρ a a b 

 π D 1 0.608 0.126 1/2 + + 2 ρ a4 a 2b2 b4 

 π D 2.45 2.90 5.13 1/2 + + 2 ρ a4 a 2b2 b4

  π D 0.127 0.707 2.44 1/2 + + 2 ρ a4 a 2b2 b4

  π D 2.45 2.68 2.45 1/2 + 2 2+ 4 2 ρ a4 a b b 1/2

 π D 2.45 2.32 1 + + . 2 ρ a4 a 2b2 b4 π 2

EFFECTS OF LOOSE EDGE GUIDES ON PLUG-IN TYPES OF RECTANGULAR CIRCUIT BOARDS Printed circuit boards are often installed in enclosures and housings to protect them from damage during operation in severe environments. Rectangular types of PCBs are often used because they are easy to insert and remove for changes and repairs. Plug-in PCBs that will be exposed to any type of vibration must be carefully supported with guides that firmly grip the sides of the PCBs. This is needed to control the position and maintain the alignment accuracy for the blind mating electrical connectors at the bottom edges of the PCBs. This is also needed to prevent the PCBs from flopping back and forth freely when they are exposed to vibration. If the PCBs are allowed to flop back and forth during vibration, this will increase the PCB dynamic displacement. This will increase their period of vibration and tend to decrease the PCB natural frequency. Vibration test data have shown that this condition often brings the natural frequency of the PCBs closer to the housing natural frequency. This increases the dynamic coupling between the housing and the PCBs, which is very dangerous. It can increase the acceleration G level acting on the PCBs, resulting in more rapid PCB failures.

102 Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures Loose edge guides that do not grip the PCBs firmly should be avoided in order to improve the reliability of the electronic equipment. Many different types of side edge guides are available in the form of spring clips and adjustable wedge clamps. These devices often have another benefit. They can improve the heat flow path from the PCB to the side walls of the enclosure that will improve the transfer of heat to the outside environment. This can reduce the hot-spot temperatures and improve the reliability of the electronic component parts. Sometimes slots or grooves are cast or machined into the side walls of the housing to act like edge guides. These slots and grooves are usually made extra wide to allow for manufacturing and assembly tolerances to reduce costs. Sometimes arguments are made that extra wide PCB slots and grooves will allow the PCBs to float in these wide slots which can act like an isolation system and reduce the forces on the PCBs. Vibration test data show that this is not true. The acceleration forces are much greater. These types of slots and grooves should not be used if the equipment will be exposed to any vibration or shock conditions. It will result in a loose edge guide with all of the problems and failures as outlined above.

24

ADDITIONAL DETAILS Additional details for determining the maximum desired displacements, minimum desired natural frequencies, and improved reliability for different types of PCBs with different types of electronic components that are mounted at different locations on the PCBs are given below. One of the most important factors involving the fatigue life and reliability of PCBs and their electronic components operating in a sine vibration environment is the maximum desired dynamic displacement and the minimum desired natural frequency of the PCB. When plug-in types of PCBs are mounted within a chassis or housing, the vibration force path will typically act on the exterior chassis first, because the chassis is attached to the vibration machine. Since the PCBs are attached to the chassis, the response of the chassis becomes the input to the various PCBs inside the chassis. A typical force path would then represent the chassis as the first degree of freedom since it receives the vibration energy first. The PCB would act as the second degree of freedom since it receives the energy from the chassis. A typical system is shown in Fig. 11. High dynamic forces can be produced in the PCBs if the natural frequency of the chassis is close to the natural frequency of the PCBs. It was pointed out earlier that the dynamic transmissibilities between coupling structures will not add; they will multiply. It was also pointed out earlier that if these frequencies are not separated by at least one octave (a ratio of 2 to 1), then snubbers (or something similar) should be added to the PCBs to prevent excessive dynamic coupling between the chassis and the PCBs that lead to high stresses and rapid failures. Computer-generated finite element models (FEMs) are being used extensively to find more accurate details associated with dynamic forces, stresses, and fatigue life for a large variety of new designs and applications. Many people assume any information coming from a computer FEM must be accurate. This is generally true if the modeling is done by an experienced engineer with extensive testing experience. Problems are often generated by FEM analysis engineers who are very familiar with modeling but do not have much “hands-on” vibration testing experience. There have been many field failures that can be traced back to a lack of experience related to boundary conditions that must be entered into the FEM analysis. A classic example involves finding the expected natural frequency of a rectangular, flat epoxy fiberglass plate that measures 10 in. square and 0.15 in. thick. The perimeter of the plate is bolted on all four sides to a stiff perimeter test fixture that is bolted to a vibration machine. The epoxy fiberglass plate is fastened to the fixture using large screws (No. 8-32 screws 0.164 in. in diameter) spaced 1 in. apart.

24

Additional Details

103

Note: The following sample problem is for demonstration purposes only. This type of close screw spacing as well as large screw sizes would never be used in a real application. It would take up too much space on the PCB, which would reduce the number of electronic components that could be placed on each PCB and might require additional PCBs to support the extra components. The large screw size and close spacing are used only to generate a mental picture for an engineer with FEM computer experience but with very little vibration testing experience when asked to analyze a typical PCB. Question: When the FEM is being generated for the above plate design, select which of the following geometric boundary conditions best describes the geometry at the edges of the plate needed to find the plate natural frequency: (a) Fixed (clamped) (b) Supported (hinged) (c) Free Most engineers with limited vibration experience will choose the Fixed-edge condition. When a fixed-edge condition is used in the FEM model, the answer will result in a much higher plate stiffness and a much higher natural frequency. Test data show the true edge condition will be very close to the hinged condition. When a microscope and a strobe light are used to view the vibrating plate edge, it is obvious that the plate edges are rotating and sliding back and forth under the screw heads This type of extensive testing and FEM analysis has been used extensively for many years to determine the maximum desired dynamic single-amplitude displacement Z and the natural frequency required for the PCB to ensure a fatigue life of about 10 million stress cycles in a sine vibration environment as shown below: 0.00022B Z = (75) √ = in. (maximum desired PCB displacement), Chr L where B = length of PCB edge parallel to a rectangular component on the PCB, in. L = length of the electronic component, in. h = height or thickness of PCB, in. C = constant Values of C for different electronic components are as follow: 0.75 for axial leaded components, resistors, capacitors, and fine-pitch semiconductors under 0.75 in. long 0.75 for a leadless ceramic chip carrier with added perimeter J wires 1.0 for axial leaded components, resistors, capacitors, and fine-pitch semiconductors over 1.0 in. 1.0 for a standard dual inline package (DIP) over 1.0 in. 1.0 for a PGA with lead wires around the perimeter extending from the bottom surface of the PGA 1.0 for a leaded ceramic chip carrier where the lead length is about the same as for a standard DIP 1.26 for a DIP over 1.5 in. with side brazed lead wires 1.26 for a pin grid array (PGA) with two parallel rows of wires extending from the bottom surface of the PGA 1.15 for a square ball grid array (BGA) under 1.0 in. 1.25 for a square ball grid array (BGA) over 1.25 in. 2.25 for a leadless ceramic chip carrier (LCCC)

104 Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures The peak single-amplitude displacement is usually at the center of the PCB. For a singledegree-of-freedom system the following equation can be added to Eq. (75) to obtain the desired frequency: Z =

9.8GIN Q 9.8G = . f2 fN2

Another important equation for estimating the transmissibility Q at resonance can also be added to Eq. (75) (see [1], p.173, Eq. 8.10):  Q = fN . The two equations shown above can be combined with Eq. (75) to obtain the desired PCB natural frequency that will provide a fatigue life of about 10 million stress cycles that includes the component electrical lead wires and solder joints:  √ 2/3 9.8GIN Chr L . (75a) Minimum desired PCB frequency: fD = 0.00022B

25

SAMPLE PROBLEM: FINDING MINIMUM DESIRED PCB NATURAL FREQUENCY A 40-pin dual inline package (DIP) with standard electrical lead wires is to be mounted at the center of a 6.0 × 8.0 × 0.10-in.-thick plug-in PCB. The DIP will be mounted parallel to the 8-in. edge. The system must operate in a sine vibration qualification test where the input level will be 6G peak. Determine the minimum desired PCB natural frequency. Solution: Care must be used here because the PCB might be mounted in an enclosed chassis or the PCB might be mounted on a holding fixture. In either case the natural frequency of the PCB must be about one octave (octave means to double) away from the holding device, chassis, or fixture. If the resonant frequencies are close to each other, they can multiply and damage the PCB. Or the PCB can use snubbers on the front and back faces to prevent any severe coupling. Then: B = 8.0 in., length of PCB parallel to component h = 0.10 in., PCB height or thickness L = 2.0 in., length of 40-pin DIP C = 1.0, constant for standard DIP r = 1.0, relative position factor for component at center of PCB G = 6.0, input vibration acceleration level Substitute into Eq. (75A) for the minimum desired PCB natural frequency:  √ 0.666 9.8.(6.0)(1.0)(0.10)(1.0) 2 = 280 Hz. fD = 0.00022(8.0) The DIP expected fatigue life can be estimated from the fatigue life of 10 million cycles expected life: Life =

10 × 106 cycles to fail = 9.9 hrs to fail. (280 cycles/sec)(3600 sec/hr)

27

26

Vibration Characteristics of Box Structures and Frame Structures

105

HOW BENDING CURVATURE OF PCB CAN AFFECT PCB FATIGUE LIFE The relative curvature of the PCB at different locations can be used to find the PCB dynamic displacements, forces, stresses, and fatigue life of the electronic component at these various locations when the PCB is vibrated at its natural frequency. This is shown in Fig. 12. When the PCB edges are simply supported (or hinged), the single-amplitude displacement of the PCB at any location can be obtained from the following: πX πY Z = Z0 sin sin . (76) a b The forces and stresses in the electrical lead wires and solder joints for the electronic components mounted on the PCBs can be related to the bending curvature of the PCBs as they are excited at their natural frequency. When the electronic component is located at the center of the PCB, X is at a/2 and Y is at b/2. The maximum displacement will be at the center of a uniform PCB and the relative position vibration factor R for the component will be 1.0: π π Z = Z0 sin sin = Z0 so R = 1.0. (77) 2 2 When the electronic component is located off the PCB center, at the position where X is at a/2 and Y is at b/4 the PCB relative position vibration factor R for the off center will be 0.707: π π Z = Z0 sin sin = 0.707Z0 so R = 0.707. (78) 2 4 When the electronic component is located off the PCB center, at the position where X is at a/4 and Y is at b/4 the PCB relative position vibration factor R for the off the center will be 0.50: π π Z = Z0 sin sin = 0.50Z0 so R = 0.50. (79) 4 4 The displacement of a square PCB with a uniformly distributed load supported on all four sides will be at the maximum at the center of the PCB when it is vibrated at its natural frequency. The relative curvature of the PCB can be very high, as well as the bending stress in this area. The maximum PCB displacement will produce maximum stresses in the electronic components and the maximum relative position factor R. This will produce the lowest fatigue life in the electronics mounted at the center of the PCB. The fatigue life of the electronics in the vibration environment can be increased by moving the electronic component away from the center, closer to one of the sides of the PCB. The displacement and the relative curvature at the PCB sides will be much lower at the new location. This will also reduce the relative position factor R and the bending stress in the PCB at the new location. It will also increase the vibration fatigue life of the electronics. This method for increasing the vibration fatigue life for electronic systems was shown in the sample problem for a BGA where it was moved from the center of the PCB to one of the side edges. It is shown in the analysis section for Eqs. (58)–(62).

27

VIBRATION CHARACTERISTICS OF BOX STRUCTURES AND FRAME STRUCTURES A large manufacturing and testing company was having a problem with the failures of small glass diodes during sine vibration tests using a 2G peak input level. The diodes had a rating of 25G peak. The manufacturing and testing company complained to the diode company about the poor quality of the diodes. The diode company insisted that its 25G rating was accurate and strongly recommended the use of a consultant who was very familiar with vibration testing to solve the problem.

106 Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures Z Input Response X

Figure 14 When the vibration consultant arrived, a meeting was held at the testing facility so everyone could view the test specimens, the test holding fixture, the vibration shaker, and the accelerometer instrument locations. The testing fixture was an open welded frame made up of 12 aluminum bars each 1 in. square and 5 in. long all welded at the ends of each bar to form an open box frame. Only one accelerometer was used, mounted at the top of one of the four corners where the three aluminum bars were welded together (see Fig. 14). A small aluminum plate was mounted across the top two parallel aluminum bars to hold the diode test specimens. The test consultant asked why only one accelerometer was being used to monitor the vibration in the vertical Z direction and not in the X and Y lateral directions. This arrangement would only get acceleration data for the vertical Z direction and no data for the two lateral X and Y directions. Their answer was that they are only vibrating the test specimens in the vertical Z direction and there is no response in the two lateral X and Y directions. The consultant asked if they instrumented the lateral X and Y directions to prove there was no significant vibration response in the X and Y directions. Again their answer was that when you vibrate in the vertical direction the system can only respond in the vertical direction. Therefore, it is not necessary to monitor any other direction. The consultant said it does not matter which vibration direction is used because there will always be some responses in all three X , Y, and Z axes. The consultant also said, when one vibrates a flexible structure in any direction, the response will be in the direction that gives the minimum strain energy. This means it will respond in the direction that gives the lowest natural frequency, whether you like it or not. Their answer was, “You are deliberately stalling for more time so you can charge us more money for your consulting services. We hired you to solve our problem and all you do is waste our time. I want action to solve our problem.” The consultant said the only way to solve the problem is to install two more accelerometers to monitor the X and Y responses for vibrations in the vertical Z axis. Then there will be substantial motion along the Y and Z axes when the input vibration is only in the Z direction. The two additional accelerometers were fastened to the frame to monitor the X and Y directions, and a 2G peak sine vibration test was run in the vertical Z direction. The response results showed that the 2G peak sine vibration input produced a lateral Y response of about 100G peak. The company engineers, looked at the test data and thought the holding vibration fixture was broken. An examination showed it was not broken. The data were accurate. They were finally convinced. They asked one question, “How do we fix this problem?” The consultant recommended the welded frame fixture be scrapped. The natural frequency was too low, which produced very large vibration displacements and a very high acceleration G response. Recommendations were made to use a block of aluminum about 4 in.3 for the vibration fixture. This will produce a very high natural frequency, with small dynamic displacements, very low acceleration G levels, and low stresses. The new test fixture solved the problem.

29

28

Effects of Random Vibration on Various Types of Electronics and Their Structures

107

SAMPLE PROBLEM Find the natural frequency of an epoxy fiberglass plug-in PCB that has a uniformly distributed load of 1.5 lb, wedge clamps on both vertical sides, a width of 5 in., a height of 7 in., 0.10 in. thick, and a free top edge with no supports. Vibration test data show the side wedges clamps are very stiff so the vertical sides will act like they are fixed. The plug-in connector at the bottom of the PCB is flexible so it will act like a hinge. The top edge of the PCB is not supported so it will act like a free edge. Solution: The equation for finding the natural frequency of this plug-in PCB is

 1/2 π 1 1 D 4 fN = + + , 1.74 ρ a 4 2a 2 b 2 64b 4 where a = 5.0 in., PCB width b = 7.0 in., PCB height (or length) W = 1.5 lb, PCB weight g = 386 in./sec2 , acceleration of gravity h = 0.10 in., PCB thickness E = 2 × 106 lb/in.2 , PCB modulus of elasticity μ = 0.12, Poisson’s ratio (dimensionless) and

 2 × 106 (0.10)3 Eh 3  =  = 169.1 lb in.  D= 12 1 − μ2 (12) 1 − (0.12)2 ρ=

Then π fN = 1.74

29

(80)

mass W 1.5 = = = 1.11 × 10−4 lb sec2 /in.3 . area gab (386)(5)(7) 

169.1 1.11 × 10−4



4 1 1  +  2  2 + 4 5 2 5 7 64 74

1/2 = 183 Hz.

(81)

EFFECTS OF RANDOM VIBRATION ON VARIOUS TYPES OF ELECTRONICS AND THEIR STRUCTURES Random vibration is more closely associated with the dynamics of electronic equipment in everyday life than sinusoidal vibration. Some people have suggested that sinusoidal vibration should not even be considered for dynamic conditions, since random vibration has proven to be the true environment in which electronic systems must operate. But sinusoidal vibration has a very important property that random vibration does not have: the ability to use strobe lights to examine the complex motion of systems being vibrated at their natural frequency to get a better understanding of how and why so many failures often occur in electronic systems. The purpose of this chapter is to provide engineers and designers with the information they need to get a better understanding of the fundamental nature of random vibration so they can design and produce cost-effective, lightweight, reliable electronic systems. A few basic failure modes can occur in random vibration that must be controlled to produce a reliable electronic system. These modes may interfere and distort the electronics but not damage the electronic system. These are high acceleration levels, high stress levels, large dynamic displacements, and electrical signals out of tolerance. For example, relays can chatter when high acceleration G levels are produced. This can distort the electrical signals without

108 Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures producing any failures in the electrical connections. High acceleration G levels may generate large dynamic displacements in the PCBs that can cause impacting between adjacent PCBs that are spaced too close together. Again, this may not cause any failures in any structural members, but it may distort the electrical signals and contaminate the accuracy of the signals. High stress levels can crack solder joints and electrical lead wires on electrical components and cause the electronic system to fail. The most obvious property of random vibration is that it is nonperiodic so that the knowledge of its past history is only adequate to predict the probability of occurrences of various accelerations and displacement magnitudes. However, it is not sufficient to predict the precise magnitude at a specific instant, as it is with sine vibration.

DIFFERENCES BETWEEN RANDOM VIBRATION AND SINUSOIDAL VIBRATION

Acceleration or displacement

A typical random vibration plot of displacement versus time is shown in Fig. 15. This figure is made up of many overlapping sinusoidal frequencies which are superimposed in Fig. 16. Random vibration is very different from sine vibration because at any given time a bandwidth, say from 10 to 1000 Hz, every frequency in random vibration from 10 to 1000 Hz is present at the same time instantaneously and simultaneously. This means that all of the fundamental resonant frequencies between 10 and 1000 Hz and all the higher harmonics between 10 and 1000 Hz will be excited at the same time by the random vibration. When the same electronic system is vibrated across the same bandwidth from 10 to 1000 Hz using a sinusoidal sweep, each resonance in the given frequency band will be excited individually. Since the electronic systems respond very differently with random vibration than with sinusoidal vibration, the fatigue failures due to random vibration will be very different than the failures generated by sinusoidal vibration.

Time

Figure 15

Displacement

30

Time

Figure 16

30 Differences between Random Vibration and Sinusoidal Vibration 1

Vibration 2

109

Vibration machine

Oil film

Figure 17 Resonance mass 1

Resonance mass 2

Isolation area Q

f1

f2

f1

f2

Figure 18 The different properties between random vibration and sinusoidal vibration can be demonstrated with the use of Fig. 17. This figure shows a vibration machine with an oil film slider plate that supports two different cantilever beams, with two different end masses, that have different natural frequencies. Mass 1 is the larger mass and it is supported by a thinner bar. Mass 2 is the smaller mass and it is supported by a thicker bar of the same height. Therefore mass 1 will have a lower natural frequency and mass 2 will have the higher natural frequency, as shown in Fig. 18. During a sinusoidal sweep, mass 1 with the lower natural frequency will develop its resonant peak first and show a large bending displacement, while mass 2 with the higher natural frequency will be quiet. Therefore mass 1 will not be able to strike mass 2 while mass 2 is quiet, so there will be no damage to the system. As the sweep frequency is increased to a higher level, it will reach the natural frequency of mass 2 so mass 2 will show a large bending displacement. However, the lower natural frequency of mass 1 will force mass 1 into its isolation area where the bending displacement is very small, while the displacement of mass 2 will be high at its natural frequency. Therefore mass 1 with the small displacement will still not be able to strike mass 2 with the large displacement so there will be no system damage. This sinusoidal sweep can be made back and forth from low frequency to high frequency, but mass 1 will never be able to strike mass 2 using a sinusoidal vibration sweep. When that same system is tested with a broad-band random vibration over the same frequency range, the lower frequency mass 1 and the higher frequency mass 2 will be excited at their natural frequencies at the same time. Both masses will be forced to move through large displacements at the same time, so they can now strike each other and cause damage. This shows that random vibration environments can produce various types of structural failures that cannot be produced in sinusoidal vibration environments.

110 Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures 15 G2/Hz G2/Hz

20

1500

Figure 19

31

LOG–LOG INPUT AND RESPONSE CURVES USED FOR RANDOM VIBRATION Many different types and shapes of random vibration input curves are used for tests on a wide variety of hardware to ensure the reliability of the hardware. One of the most typical curves is the white-noise flat top curve shown in Fig. 19. The curves may be plotted on log–log charts. The vertical axis lines are often expressed in terms of the power spectral density (PSD) in units of acceleration squared per hertz, or G 2 per hertz, and the space between the vertical axis lines is the frequency bandwidth in hertz. Random vibration acceleration G levels are typically expressed in terms of the root-mean-square (RMS) and they are related to the area under the curve. The input RMS acceleration level is obtained by integrating under the input random vibration curve and the output (or response) RMS acceleration level is obtained by integrating under the output random vibration curve. The square root of the area under these input and response curves then determines the RMS acceleration level for these conditions with the units shown below:  √ G2 Area = (82) X Hz = G 2 = G RMS. Hz

32

SAMPLE PROBLEM Calculate the expected input G RMS acceleration level. Determine the expected input acceleration level for the white-noise (flat top) random vibration PSD 0.15G 2 -Hz−1 curve shown. Solution: G RMS =

33



PSD X f =

 (0.15) (1500 − 20) = 14.9 RMS input acceleration.

(83)

RANDOM VIBRATION PSD IN TERMS OF MEAN-SQUARED ACCELERATION DENSITY The PSD is also known as the mean-squared acceleration density, which is measured in terms of gravity units (G) and so is dimensionless. This is done by dividing the acceleration (a) by the acceleration of gravity (g): a acceleration G= = dimensionless units. g gravity Therefore, an acceleration level of 10G dimensionless units means that the acceleration has a magnitude that is 10 times greater than the acceleration of gravity. The most common way for evaluating random vibration in the United States is in terms of the PSD, which is measured in G 2 per hertz. However, in other countries random vibration is often expressed in terms of velocity spectral density and displacement spectral density.

34 Simplified Method for Calculating Areas

111

Random vibration can also have positive slopes and negative slopes plotted on log–log chart paper with straight lines. A positive slope will have a straight line going up to the right. A negative slope will have a straight line going down to the right.

34

SIMPLIFIED METHOD FOR CALCULATING AREAS UNDER DIFFERENT-SHAPED RANDOM VIBRATION INPUT PSD CURVES Random vibration input power spectral density curves can come in a wide variety of sizes and shapes. The area under these curves must be calculated in order to obtain the input RMS acceleration levels acting on sensitive structural members that can malfunction and fail. The area A under the more complex random vibration input PSD curves shown in Fig. 20 can be obtained using integral calculus:  2 A= Y dX = area in G 2 (dimensionless gravity units2 ). 1

Vertical lines must be used to break up the areas on the complex input PSD curves. These vertical lines must be drawn starting from the top at each break point down to the bottom. Here Y represents the vertical axis and X the horizontal axis for the breakup lines that define the individual small sections. Horizontal lines must not be used. The equations below are convenient for calculating the areas of the individual smaller areas between the vertical break points. The sum of all the smaller areas will be used to find the total area under the input PSD curve. In the equations subscript 1 refers to the left side and subscript 2 to the right side of the area being analyzed. These equations are for positive and negative slopes when the slope is not −3.   S /3  3P2 f1 A= (84) f1 f2 − 3+S f2 or 3P1 A= 3+S

   f2 S /3 f2 − f1 . f1

(84a)

When the sloped section of the PSD curve being analyzed is −3, then a new set of equations shown should be used to find the area under the curves: f1 (85) A = −f2 P2 loge f2 or f2 (85a) A = f1 P1 loge . f1

G 2/Hz PSD

Frequency

Figure 20

112 Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures When the area under the flat top input PSD curve is being calculated, the slope at the top will be zero so the area can be calculated using the following equation: A = P(f2 − f1 ).

35

(86)

SAMPLE PROBLEM Find the input RMS acceleration G level from the area under the input PSD acceleration curve with the sloped sections shown in Fig. 21. Solution: Vertical lines can be used to break up the area under the sloped sections into three separate sections to find their areas using Eqs. (84) and (86). For area 1, use Eq. (84), where f1 = 10 Hz f2 = 100 Hz P2 = 0.20G 2 Hz−1 Slope at area 1, S = 3 db/octave Substitute into Eq. (84) to obtain area 1:     3 (0.20) 10 3/3 A1 = 10 = 99G 2 . 100 − 3+3 100

(87)

Substitute into Eq. (86) to obtain area 2: A2 = (0.20) (500 − 100) = 80G 2 .

(88)

Substitute into Eq. (84a) to obtain area 3:    3 (0.20) 2000 −6/3 A3 = (2000) − (500) = 75G 2 . 3−6 500

(89)

The RMS input acceleration can be obtained from the total area under the curve: AT = A1 + A2 + A3 = 99 + 80 + 75 = 254G 2 ,  GRMS 254G 2 = 15.93G (RMS input acceleration).

36

(90)

LOCATING BREAK POINTS ON PSD RANDOM VIBRATION CURVE Input random vibration PSD curves are often specified in terms of the break points at different frequencies, and the slope of the decibels curve line. Only one G 2 -per-hertz reference point

G 2/Hz PSD

0.20 G 2/Hz −6 dB

3 dB 1

2

3

PSD

1 2 Area 1

Frequency

Figure 21

Area 2

Area 3

38

Gaussian and Rayleigh Probability Distribution Functions

113

is often defined. The P (PSD) values at these break points can be found using the following relation:  dB/3 f1 . (91) P1 = P2 f2

37

SAMPLE PROBLEM Find the PSD values at the break points for the sloped lines on the left the right sides of the PSD curve shown in Fig. 21a. Solution: Equation (91) can be used to find the break points on the left side of Fig. 21a by area A1 and the right side of Fig. 21a by area A3. For area A1, on the left side of the figure break point use Eq. (91), where Frequency f1 f2 P2 Slope S

= = = =

10 Hz, left side of area 1 100 Hz, right side of area 1 0.20 G 2 /Hz, PSD right side of area 1 3 dB/octave, left side of area 1

Substitute into Eq. (91) and solve for P1 . The PSD value at the left-side break point is given as   10 3/3 = 0.02G 2 Hz−1 . (92) P1 = 0.20 100 Substitute into Eq. (91) and solve for the P2 PSD value at the right-side break point, area 3, where Frequency f1 f2 P1 Slope S

= = = =

500 Hz, left side of area 3 2000 Hz, right side of area 3 0.20 G 2 Hz2 PSD left side of area 3 −6 dB/octave, right side of area 3

Then P2 =

38

P1 (f1 /f2 )

−6/3

=

0.20 (500/2000)

−6/3

=

0.20 (2000/500)2

= 0.0125G 2 Hz−1 .

(93)

GAUSSIAN AND RAYLEIGH PROBABILITY DISTRIBUTION FUNCTIONS FOR ESTIMATING FATIGUE LIFE OF DIFFERENT TYPES OF ELECTRONIC EQUIPMENT EXPOSED TO RANDOM VIBRATION The Gaussian distribution curve is shown in Fig. 22. This is often called the normal distribution curve and is widely used in statistics and mathematics dealing with the analysis of numerical data. The total area under the curve is unity. The curve goes from negative infinity to positive infinity. The area under the curve between any two points then represents the probability that the acceleration will be between these two points. The instantaneous accelerations will be between +1σ and −1σ 68.3% of the time. The 1σ value represents the RMS. It will be between +2σ and −2σ 95.4% of the time. It will be between +3σ and −3σ 99.73% of the time. The 1σ value is the RMS value. The maximum acceleration levels for random vibration are considered to be the 3σ levels 99.73% of the time because it is specified in MIL-STD 810 for electronic equipment. This specifies that the maximum acceleration level is to be limited

114 Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures y 4 Probability density

3 2 68.3% 1 0 −4

+4

0 a/s

x

Figure 22

to 3 times the RMS level. This is very close to 100% of the time. Higher acceleration levels of 4σ and 5σ can occur in the real world. However, these are usually ignored because most of the vibration shakers for random vibration are equipped with 3σ clippers. These limit the input acceleration levels to 3 times greater than the RMS input levels. It is interesting to note that aircraft structural design and analysis engineers use the Rayleigh distribution function as their guide for limiting their maximum allowable input acceleration levels. The Rayleigh distribution is based upon the peak accelerations that often have input acceleration levels of 5G peak or higher for their analyses. The Rayleigh distribution is shown in Fig. 23. Aircraft structural engineers state that peak stresses cause the failures so all analyses should use the Rayleigh peak distribution functions. However, the electronics engineers are guided by MIL-STD 810. Another interesting factor is the differences between aircraft structural analysis and electronics structural analysis requirements. Aircraft structural stress managers will always hold a meeting for new aircraft structural stress analysis engineers to inform them of their company’s stress analysis requirements. The requirements are usually the same for all aircraft structural analysis engineers. When new structural analysis engineers finish their design and stress analysis on a new aircraft structural member, a mockup will be made of that part. That part will then be tested to determine if it passes or fails the required load forces expected for the particular aircraft and environment. If the structural member passes all the tests, the engineers will be fired. This may sound strange to the design and analysis engineers in the electronics area who want their designs and analyses to pass the stress tests. Here is the logic. In an aircraft structural analysis it is

Probabiity density

0.6 0.5 0.4 0.3 0.2 0.1 0

0

1

2

3

4

a/s Rayleigh distribution

Figure 23

39

Characteristics of Single-Degree-of-Freedom System and Fatigue Life in Random Vibration

115

important to have strong and lightweight structures because they are very profitable with high sales volume. When a new part passes the stress tests, it means the part is too heavy. There is no quick way to know where material can be added or subtracted to make it lighter and stronger. However, if the new part fails the strength tests, they now know where the weakest structural sections are located. These can be reinforced with a little added structure in the right places to provide a lightweight strong member.

CHARACTERISTICS OF SINGLE-DEGREE-OF-FREEDOM SYSTEM AND FATIGUE LIFE IN RANDOM VIBRATION Single-degree-of-freedom systems very seldom occur in the real world. However, they are often used to obtain the approximate natural frequency of a structural member operating in a random vibration environment. In theory, a single-degree-of-freedom system can only vibrate at one frequency, the natural frequency. In a broad-band random vibration condition all the various frequencies are present at the same time. Therefore, the system will still vibrate at its natural frequency, but it will have a varying amplitude, as shown in Fig. 24. The approximate G RMS response for the single-degree-of-freedom system can be obtained from the following equation shown below (see [1], p. 207): π GOUT = (RMS), (94) (P)(fN )(Q) 2 where P = input PSD in G 2 per hertz at natural frequency fN = natural frequency, Hz Q = transmissibility at natural frequency

Random input

C

k Mass

Displacement

The equation above is most accurate when the random vibration PSD input is flat in the area of the natural frequency, as shown in Fig. 25. The amount of error is small when the slope

Time

Single-degree-of-freedom random vibration

Figure 24 Random vibration curve

Transmissibility Q

39

Frequency Hz

Figure 25

116 Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures of the random vibration input curve is 3 dB/octave in the area of the natural frequency. There is a larger error when the slope of the random vibration input curve is 6 dB/octave or higher.

40

SAMPLE PROBLEM Find the approximate random vibration fatigue life of a cantilevered 6061 T6 aluminum beam measures 6 in. long by 0.78 in. of wide by 0.40 in. thick with a transformer mounted at the free end, as shown in Fig. 26. The total weight of the assembly is 0.50 lb and it is restricted to vibrate only in the vertical direction with no lateral or rotation motion. The system will be exposed to broad-band white-noise random vibration with an input PSD of 0.45G 2 Hz−1 from 20 to 2000 Hz for a period of 1 hr. Find the expected dynamic stress in the beam along with the approximate fatigue life. Two different analysis methods are shown. Solution: The three-band method and Miner’s cumulative fatigue damage ratio are used here. The static displacement of the system due to its total weight can be used to calculate the approximate natural frequency of the cantilever beam with the end mass as shown below. The restricted motion in the vertical direction will make the beam act like a single-degreeof-freedom system. The static displacement Y can be obtained using the equation. Y = where W L E b h I

= = = = = =

WL3 , 3EI

(95)

0.50 lb, total weight of system 6.0 in., length of aluminum beam 10.5 × 106 lb/in.2 , aluminum modulus of elasticity 0.78 in., width of aluminum beam 0.40 in., height of aluminum beam bh 3 /12 = (0.78)(0.40)3 /12 = 0.0041 in.4 , beam moment of inertia

Then Y =

(0.50)(6.0)3 = 0.00083 in. (static displacement). (3)(10.5 × 106 )(0.0041)

The natural frequency of the aluminum beam ( fN ) based upon static displacement and acceleration of gravity g = 386 in./sec2 is given as g 386 1 1 fN = = = 108 Hz. (96) 2π Y 2π 0.00083

0.50 lb

0.78

A

A A

A

0.40 6.0

Figure 26

41

41

Response of Cantilever Beam to Random Vibration

117

RESPONSE OF CANTILEVER BEAM TO RANDOM VIBRATION The approximate response of the cantilever beam to the broad-band random vibration can be obtained from Eq. (94) using the following information: P = 0.45G 2 Hz−1 , input PSD fN = 108 Hz, beam natural frequency √  Q = 4 fN = 4 108 = 41, dimensionless beam transmissibility Note that Q is based upon the damping expected in the beam assembly during vibration at the natural frequency. This is a difficult number to calculate because there are many factors that contribute to this property. Tests have shown that very similar assemblies can have large differences in their Q values. The tests have also shown that higher frequencies will have higher Q values. The theory of damping says that a structure with zero damping will have a transmissibility Q factor of infinity. This is impossible for a real structure. Tests have also shown that simple structures similar to a simple beam will have low damping so it will result in a higher Q factor. The average Q value given above was based upon extensive testing data accumulated over a period of about 20 years. Substitute into Eq. (94) to find the approximate RMS acceleration response of the beam: π Acceleration response = (0.45)(108)(41) = 56.0G RMS. (97) 2 The dynamic 1σ bending stress is expected to act on the beam 68.3% of the time for a Gaussian distribution. The bending stress is obtained from the bending moment M acting on the beam: M = W (weight) × G (RMS) × L (length) where W G L M

= = = =

0.50 lb, weight 56.0 RMS 6 in., beam length (0.50)(56.0)(6) = 168 lb in., bending moment in beam

The maximum 1σ bending stress SB in the cantilever beam will be at the support. This value can be obtained from the linear formula MC (1σ ) 1σ : SB = I where M = 168 lb in., bending moment in beam C = 0.40/2 = 0.20 in., half thickness of aluminum beam I = 0.0041 in.4 , beam moment of inertia Then (168)(0.20) (98) = 8195 lb/in.2 RMS, bending stress. 0.0041 The approximate number of stress cycles required to produce a fatigue failure in the aluminum beam can be obtained from the three-band method of analysis. This is based upon the 1σ , 2σ , and 3σ acceleration G levels in the Gaussian curve with the use of Fig. 27 and the equations shown below:  b  b N1 S2 S2 or = or N1 = N2 , (99) N1 S1b = N2 S2b N2 S1 S1 1σ :

SB =

118 Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures K=2 2

Stress S lb/in.2

45,000

b = 6.4

8195 lb/in.2 1

103

108 N1 Cycles to fail

Figure 27 where N1 N2 S1 S2 b

= = = = =

number of cycles to produce beam failure, see Fig. 27 1000 cycles for beam failure at 45,000 lb/in.2 (reference point) 8195 lb/in.2 (1σ RMS calculated bending stress, Eq. 5.98) 45,000 lb/in.2 stress for beam failure (reference point) 6.4 slope of fatigue line on log–log plot with stress concentration K = 2

Solve for the number of cycles N needed to produce a fatigue failure in the beam for the 1σ , 2σ , and 3σ conditions:   45,000 6.4 = 5.4 × 107 cycles to fail, RMS 1σ : N1 = (1000) 8195   45,000 6.4 (100) RMS 2σ : N2 = (1000) = 6.4 × 105 cycles to fail, (2)(8195)   45,000 6.4 RMS 3σ : N3 = (1000) = 4.8 × 104 cycles to fail. (3)(8195) Next find the actual number of stress cycles (n) accumulated during the 1-hr, random vibration test along the vertical axis for the 1σ , 2σ , and 3σ acceleration levels based upon the time exposure for the three-band method and Gaussian distribution:

42

1σ :

n1 = (108 cycles/sec)(3600 sec/hr)(1 hr)(0.683) = 2.65 × 105 cycles,

2σ :

n2 = (108 cycles/sec)(3600 sec/hr)(1 hr)(0.271) = 5.50 × 104 cycles,

3σ :

n3 = (108 cycles/sec)(3600 sec/hr)(1 hr)(0.043) = 1.70 × 10 cycles.

(101)

4

MINER’S CUMULATIVE FATIGUE DAMAGE RATIO CAN BE USED TO ESTIMATE FATIGUE LIFE Miner’s fatigue damage ratio examines the effects of alternating stresses on the formation of microscopic cracks in the load-carrying members of structures exposed to alternating stresses. These can be due to vibration, shock, thermal cycling, or acoustic noise. Small initial cracks often grow over time until their large size results in a complete structural failure. A fatigue cycle ratio is then developed in terms of the actual number of cycles (n) for a specific environment divided by the number of cycles required for a failure (N ) for a specific condition, shown as n/N . This ratio gives the percentage of life that has been used up causing the

44

Quick Method for Finding Approximate Fatigue Life for Structures Exposed

119

structure to fail. The cumulative damage ratio RD then takes the form shown below. Structural failure should occur when the ratio equals 1: n1 n2 n3 n4 RD = + + + + · · · = 1.0. (102) N1 N2 N3 N4 Substitute Eqs. (100) and (101) into Eq. (102) to obtain Miner’s cumulative damage ratio: 2.65 × 105 5.50 × 104 1.70 × 104 + + 7 5 5.4 × 10 6.4 × 10 4.8 × 104 = 0.0049 + 0.086 + 0.354 = 0.445.

RD =

The above equation shows that about 44.5% of the life is used up by the 1-hr vibration test. It also shows that the 1σ RMS level does very little damage even though it acts 68.3% of the time. Most of the damage is produced by the 3σ value, which acts 4.33% of the time. The structural analysis of the beam shows that all of the beam life will be used up and the beam should fail when Miner’s damage ratio reaches a value of about 1.0/0.445: 1.0 Approximate fatigue life = = 2.2 hr to fail. (103) 0.445 Real-life random vibration test data on similar types of structures shows there can be a very large variation in the fatigue life. This is due to large variations in the material properties and manufacturing tolerances of mass-produced electronic assemblies that look exactly alike. Poor vibration test fixtures, poor accelerometer calibration, old vibration shakers with loose bearings, and out-of-calibration 3σ clippers that can go up to 4σ can also cause problems. These can all lead to failures that cannot be calculated by hand or on a computer. The only way to find the true fatigue life is to run vibration tests. Every real system will have a different natural frequency and also a different fatigue life. Test data show there can be substantial differences in these systems that look the same.

43

A BIT OF HISTORY REGARDING RANDOM VIBRATION METHODS OF ANALYSIS AND TESTING Years ago, when the Atlas missile was being developed for the U.S. government, there was a strong desire to use the new random vibration method for the test program to replace the old sine vibration testing method. There were no large computers available to do any finite element modeling for vibration and stress analysis. The slide rule and a few small hand-held calculators were the instruments used for analysis. Random vibration tests were being run so random vibration analysis methods were required. The government requested the assistance of mathematics professors from several universities to provide guidelines and analysis methods that could be used by engineers. Each professor had a different idea on how to work with this new method of analysis. Many meetings were held over a period of many months but they were never able to agree on methods of analysis. Even today there are still arguments regarding Gaussian distribution and Rayleigh distribution for use in the analysis and design of electrical and mechanical systems typically used for commercial, industrial, and military applications.

44

QUICK METHOD FOR FINDING APPROXIMATE FATIGUE LIFE FOR STRUCTURES EXPOSED TO RANDOM VIBRATION USING THREE-BAND TECHNIQUE The three-band method of analysis for random vibration was developed over a period of many years using vibration test data from many different programs. These were associated with commercial, industrial, and military applications at many different companies. The work involved

120 Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures different types of electronics for aircraft, helicopters, missiles, spaceships, submarines, satellites, automobiles, and communication, just to name a few. Extensive test and analysis data were taken, modified, and evaluated. This went on for many years, always searching for a simple method of analysis with reasonable accuracy. Finally such a simple method of analysis was found. It is based on the time exposure for a Gaussian distribution based upon the total damage D, which is defined as NS b , where N is the number of cycles required to produce a failure and S is the stress level for failure accumulated for the 1σ , 2σ , and 3σ values:  Damage D = NS b = N1 S1b + N2 S2b + N3 S3b . Substitute the Gaussian probability distribution function for the three bands with the time spent at each level using the b exponent of 6.4: D=

(0.683)(1.0)6.4 + (0.271)(2.0)6.4 + (0.0433)(3.0)6.4 = 72.55 1σ 2σ 3σ

(104)

What is required now is to find one random σ acceleration level that will generate the same total damage as the three-band technique for the full Gaussian (G) distribution shown above, so that it acts 100% of the time: (G)6.4 = 72.55. So

G = (72.55)1/6.4 = 1.95σ acceleration G level.

(105)

This is slightly less than the 2σ level that was assumed to act about 27.1% of the time based upon the three-band technique for the quick analysis method for finding the approximate fatigue life to fail and the approximate time to fail. The previous sample problem can now be evaluated using the simplified method of analysis to compare the accuracy of both methods [1]. The approximate natural frequency of the cantilever beam can be obtained from the static displacement Y of the beam due to its own weight using the equation (W )(L)3 Y = [Eq. (95)] 3(E )(I ) where W = 0.5.0 lb, weight L = 6.0 in., length of beam E = 10.5 × 106 lb/in.2 , beam modulus of elasticity b = 0.78 in., width of aluminum beam h = 0.40 in., height of beam I = bh 3 /12 = (0.78)(0.40)3 /12 = 0.0041 in.4 , beam moment of inertia g = 386 in./sec2 , acceleration of gravity 

 (0.50)(6.0)3 Y= = 0.00083 in. static displacement (3)(10.5 × 106 )(0.0041)     1 g 386 1 = = 108 Hz [Eq. (96)] Beam frequency fN = 2π Y 2π 0.00083

Then

The approximate number of stress cycles (N ) that will be required to produce fatigue failure in the aluminum beam can be obtained using Eq. (100) with a small change using the 1.95σ level:  6.4 45,000 1.95σ : N = 1000 = 7.54 × 105 cycles to fail. (106) (1.95)(8195)

45

Special Applications for Designing PCBs to Operate in Random Vibration

121

The approximate fatigue life of the beam can be obtained from the 108 Hz natural frequency shown above: 7.54 × 105 cycles to fail = 1.94 h to fail. (107) Fatigue life = (108 cycles/sec) (3600 sec/h) The fatigue life for solution 1 was shown in Eq. (103) as 2.2 hr to fail and can be compared with the short-cut solution 2 in Eq. (107) as 1.94 hr to fail. Vibration test data on these types of structures have shown that the fatigue life difference calculated above is very common.

45

SPECIAL APPLICATIONS FOR DESIGNING PCBs TO OPERATE IN RANDOM VIBRATION Designing electronic equipment for operation in random vibration environments must take into consideration that the various frequencies in the random bandwidth are always changing. This means that the random vibration acceleration forces are also changing. These changes in the random vibration forces will also change the random vibration stresses and therefore the fatigue life in the PCBs, the electronic components mounted on the PCBs, their electrical lead wires, and their solder joints. Extensive random vibration testing experience has shown that the fatigue life of the PCBs and their components can be increased by decreasing the dynamic displacement of the PCB. This is very similar to the work that was done for the sine vibration fatigue life that was shown in Eq. (75) and can be used for random vibration as shown below for the maximum allowable single-amplitude PCB dynamic displacement. For a fatigue life of about 20 million random vibration cycles Z is given as 0.00022B maximum desired PCB displacement, Z = √ Chr L where the parameters defined in Eq. (75). The minimum desired PCB natural frequency for a 20-million-cycle random vibration fatigue life can be obtained when the analysis is based upon the properties of a single-degreeof-freedom system. This can then be used instead of the number of positive zero crossings normally associated with random vibration. Three additional equations must be combined with the equation above so it can be used for random vibration. Two of the additional equations that were also used for the sine vibration in Eq. (75) can now be used for the random vibration. The first equation is 9.8GRMS ZRMS = . fN2 The second equation is shown Q=



fN

approximate value.

The third equation is used to obtain the random vibration minimum desired natural frequency: π GOUT = [Eq. (94)]. (P)(fN )(Q) 2 These three equations can be combined with Eq. (75) to obtain the minimum desired PCB natural frequency ( fD ) that will provide a fatigue life of about 20 million cycles: √  0.8 29.4Chr (π/2)PL fD = (minimum desired PCB frequency). 0.00022B Notice that the equation for displacement ZRMS has a first number of 9.8G for the RMS value. In order to ensure a good fatigue life, a safety factor of 3 was included here. This

122 Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures was in the form of the 3σ value for the maximum random vibration level used. Then the 3 × 9.8 GRMS = 29.4 peak value was used in the above equation.

46

SAMPLE PROBLEM Find the desired PCB natural frequency. A 40-pin DIP component with side-brazed lead wires is to be soldered to a typical 7.0 × 9.0 × 0.090-in.-thick plug in a PCB. The DIP will be mounted parallel to the 7.0-in. edge. The input PSD in the area of the PCB natural frequency is expected to be flat at 0.070 G 2 /Hz. Find the minimum desired PCB natural frequency and the approximate fatigue life for the DIP wires and solder joints for a uniform load with the component mounted at the center of the PCB. With these conditions the fatigue life of the component should be about 20 million stress cycles. Solution: The input data are as follows: B = 7.0 in., length of PCB parallel to DIP h = 0.090 in., PCB thickness L = 2.0 in., length of 40-pin DIP C = 1.26 in., constant for side-brazed DIP P = 0.070G 2 /Hz−1 input PSD to PCB r = 1.0 for component at center of PCB Substitute into the desired minimum-frequency equation for the PCB natural frequency: √

0.8 (29.4)(1.26)(0.090)(1.0) (π/2)(0.070)(2.0) fD = = 254 Hz. (0.00022)(7.0) The approximate fatigue life is given as Life =

47

20 × 106 cycles to fail = 21.87 hr to fail. (254 cycles/sec) (3600 sec/hr)

EFFECTS OF SHOCK ON VARIOUS TYPES OF ELECTRONIC SYSTEMS AND STRUCTURES

47.1 Introduction Anytime a structure is exposed to a rapid change of energy, it often results in a large increase in the internal forces, stresses, displacements, and accelerations within the body of the structure. The time duration of the energy transfer is often related to the natural frequency or the natural period of the structure. This rapid change of energy is often called shock. It may be due to impact or collision with another body in motion or a rapid acceleration from a rocket-firing system or simply dropping an electronic assembly on a hard floor. Shock can also excite the natural frequencies of internal structural members in complex systems that can also cause electronic failures. The most common failures in electronic systems are cracked solder joints, loose and broken pins on plug-in electrical connectors, chattering relays, potentiometers that slip, loose bolts, breaking cables and harnesses, large displacements in circuit boards that impact against adjacent boards, and cracking electronic components mounted on the boards. Operating systems that are exposed to high shocks can experience a temporary short circuit that disappears when the shock energy dissipates. These types of failures can occur in crystal oscillators, capacitors, and hybrids and are very difficult to trace and therefore difficult to correct. Shock failures are very seldom associated with fatigue failures unless a few million or more high stress cycles are involved, as in a rifle on a tank. Stress concentration factors in

47

Effects of Shock on Various Types of Electronic Systems and Structures

123

structural members are also associated with a million or more fatigue stress cycles so they are not normally associated with shock-induced failures. Various types of isolation systems are available to help improve the operating life of the electronic structures that are required to operate with a high reliability in severe shock and vibration environments. Care must be exercised when the same isolation system will be used to protect an electronic assembly for both vibration and shock. A good vibration isolation system is bad for shock and a good shock isolation system is bad for vibration. The proper isolation system must be one that will protect both the electronics for the vibration environments and the electronics for the shock environments. Many different methods of analysis are being used to evaluate the characteristics of different types of shock. Finite element computer models are popular for obtaining the approximate forces, stresses, and fatigue life of different types of structures exposed to different types of shock. Hand analysis calculations are also extensively used to design and evaluate different types of shock. However, results of computer analyses and hand calculations will usually be poor if the engineers performing the analyses do not have extensive vibration and shocktesting experience on real electronic hardware. The dynamics of electronic structures for vibration and shock can be very complex. This type of detailed information cannot really be obtained from a book. It requires several years of hands-on testing experience to understand the structural dynamics and the failure mechanisms. Sophisticated electronic systems are often simulated by using lumped simple masses, springs, and dampers to demonstrate the dynamic response properties of structures that are exposed to vibration and shock A simplified single-degree-of-freedom system with one mass and a two-degree-of-freedom system with two masses are shown in Fig. 28. One mass

C

K

(a) PCB C2

K2

Cnassis

C1

K1

Two masses (b)

Figure 28

124 Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures 47.2 Properties of Typical Shock Environments There are many different types of shocks, shock motion, and shock effects. Methods for describing some types of shock can often be confusing. The most common types of shock are pulse shocks, velocity shocks, and shock response spectrum. Pulse shocks that are associated with accelerations or displacement are usually shown as geometric shapes, such as a square wave, a half-sine wave, and triangular waves such as vertical rise and symmetrical and vertical decay. The mathematics of pulse shocks is very simple and convenient so they are used quite often. However, they do not represent the real world so they are not very accurate. Test data show that simple pulse shocks as shown in Fig. 29 are often quite effective in revealing weak areas in many different types of structures. Velocity shocks are used to examine structural systems that will experience a sudden velocity change when they are dropped on a hard floor or are involved in an earthquake. They may be involved in delivery vehicle accidents, hitting a tree in heavy rain, snow, or ice, or poor packaging, causing bouncing in vehicles driving on bumpy roads or even a very hard airplane landing or an airplane crash. Various types of drop shock tests are used to evaluate electronic equipment. The U.S. Navy has a lightweight shock machine with a 400-lb hammer for testing electronic assemblies with a weight of less than 250 lb, shown in Fig. 30. This machine produces a high-velocity impact to test the structural integrity of the electronic assembly. This machine is used to simulate the effects of high explosions on ships and submarine equipment. Shock response spectra use a different approach to evaluate shock by looking at how the structure responds to the shock instead of at the shock itself. The shock response spectrum is a plot of the peak acceleration response of an infinite number of single-degree-of-freedom systems to a complex transient waveform. The individual single-degree-of-freedom masses are usually assumed to have a transmissibility Q value of 10 when they are excited at their natural frequency using a sinusoidal vibration input. This method of analysis is more closely related to the real world, but the method of analysis is far more complex compared to a simple shock pulse. A typical shock response system showing the development of the frequency and acceleration G-level spectrum is shown in Fig. 31.

47.3 Evaluating Most Common Shock Pulse—Half-Sine Shock Pulse The half-sine shock pulse is the easiest pulse to generate so it is used extensively in tests for almost every type of commercial, industrial, and military product application. It can be analyzed and evaluated very quickly when the proper instruments and calibrations are used. Although the half-sine shock pulse is often used to evaluate the performance and reliability of a structure, it does not mean that the half-sine shock pulse accurately represents the dynamic acceleration forces expected in the real operating environment. These types of tests should always be run by people who are familiar with the product being tested and with the test methods being used.

Square

Half sine

Various types of shock pulses Versed Vertical Symmetrical sine rise triangle

Figure 29

Vertical decay

Effects of Shock on Various Types of Electronic Systems and Structures

Hammer Anvil plate Positioning springs

Hammer

Positioning springs

Positioning springs

Figure 30 Shock response acceleration

47

G2

G3

G4 G5

G1

f1

G6

f2

G1

G2

f3

f4

f5

f6

G3

G4

G5

G6

Acceleration response Spring–mass system

f1

f2

Transient shock input

Figure 31

f3

f4

f5

f6

125

Displacement velocity acceleration ft ft/sec G

126 Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures 100 Time, sec −0.020

Full rebound 20.5 Time Zero rebound

−20.5 −41.0

Full rebound

−0.13

Zero rebound

−0.41

Figure 32 The drop shock test is probably the most common shock test used here. This is used over a broad range from a zero-rebound condition to a full-rebound condition. The area under the acceleration curve represents the velocity change. When the acceleration and the time are the same, the velocity change will be the same. The velocity curves for the zero rebound and full bound will look different because there is a change of direction in the full-rebound curve. The displacements produced by the zero rebound will not look like the displacements produced by the full rebound even when the drop heights are the same. The relation between the acceleration, velocity, and displacement is shown in Fig. 32 for a 100G shock pulse with a 0.020-sec half-sine shock pulse for a condition where the rebound is zero. Sensors are secured inside the box to record the data.

47.4 Sample Problem: Half-Sine Shock Pulse Drop Test Shock tests are required for an electronic box assembly using a 100G half-sine shock pulse with a time base of 0.020 sec. Find the expected velocity changes, dynamic displacements, and drop heights for a zero-rebound test condition. Two different methods are examined for the proposed test. The first test method uses an air pressure guided drop shock machine that supports the electronic box and smashes into a lead pellet to generate the half-sine shock pulse. The second test method uses a balanced electronic box for a free-fall drop height onto a rigid wooden base to generate the half-sine shock pulse.

47.5 First Test Method: Drop Shock Analysis Zero Rebound The area Z under the half sine acceleration shock curve represents the velocity change V . This area can be calculated using the integration method shown below, where Y is for the vertical axis, X is for the horizontal axis, t represents time, and A0 is the maximum acceleration peak for the 100 G: π.x Y = A0 sin for the shock pulse, (108) t where  X =t Y dx . (109) Area Z = X =0

47

Effects of Shock on Various Types of Electronic Systems and Structures

This will result in the following velocity change V : (2)(AO )(t) Area = V = , π where AO = maximum acceleration peak for 100 G = (100)(32.2) = 3220 ft/sec2 t = 0.020 sec time duration of pulse

127

(110)

Then

(2) (3220) (0.020) = 41.0 ft/sec. (111) π The area under the velocity curve will represent the displacement of the lead pellet when the drop shock machine smashes it down. Integrating the area under the velocity curve will result in the displacement of the lead pellet, shown in two different ways: V =

Area = Z =

Vt Ao t 2 = = displacement. π 2

(112)

Then

(3220)(0.020)2 = 0.41 ft. π The displacement can also be obtained from the velocity change: (41ft/ sec) (0.020 sec) Z = = 0.41 ft 2 Z =

47.6 Second Test Method: Free-Fall Drop Height Zero Rebound Start with the standard free-fall drop equation for the velocity change ( V ) seen in most high school physics classes, where g is gravity and h is the drop height:  V = 2gh. (113) Equation (110) also represents the velocity change V so this equation is equal to the above equation:  2AO t 2gh = . π These two equations can be combined to produce three separate equations that will each show the calculated drop height h required to obtain the properties of the half-sine shock pulse: 2A2O t 2 V 2 2t 2 G 2 g = . = 2 gπ 2g π2 Use the first expression to find the drop height h: 2  (2) 3220 ft/sec2 (0.020)2  = 26.1 ft. h= 32.2 ft/sec2 π 2 h=

(114)

(115)

Now use the second expression: h=

(41 ft/sec)2  = 26.1 ft. (2) 32.2 ft/sec2

(116)

Then use the third expression: h=

 (2) (0.020)2 (100G)2 32.2 ft/sec2 = 26.1 ft. π2

(117)

128 Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures The above solutions for calculating the drop height appear to be simple. However, there is a big problem here. In order to produce the required 100G half-sine shock pulse with the time base of 0.020 sec on the electronic test box, the drop height must be 26.1 ft. In order to produce a good half-sine shock pulse, the test box must have a flat perimeter impact base. During the drop the box must stay flat and not rotate or oscillate while it is falling. The bottom of the test box must impact the hard wood base so there is a minimum bounce and minimum rotation at impact in order to avoid multiple shock pulses that can affect the acceleration and velocity data. In the free-fall drop test it is very difficult to obtain good test data on any size box or plate for such a high drop. The rotation and oscillation during the drop is very hard to control. Many drops usually have to be made before acceptable data are obtained. Sometimes string or wire guides can be set up through three or four holes in the test box to guide the box along the drop height to reduce oscillation during the drop. This can also cause problems if the wire guides or string guides rub against the test box while it is falling 26.1 ft. Excessive rubbing of the box on the drop guides may generate friction that may reduce the velocity of the test box that can affect the accuracy of the test data. The free-fall drop height test of 26.1 ft is not recommended if the tests will be performed by people without extensive previous testing experience.

47.7 How Some Simple Systems Respond to Common Shock Pulses When a common shock pulse excites a simple single-degree-of-freedom system, the system can amplify the magnitude of the pulse or it can attenuate the magnitude of the pulse. The way that the simple system responds to any particular shock pulse will depend upon the ratio R defined as natural frequency of structure R= . (118) natural frequency of shock.pulse The response of a simple system to a half-sine shock pulse that includes damping is shown in Fig. 33. Notice that the damping has very little effect on the shock amplification. When the damping is zero, the maximum shock amplification is only about 1.76. Also notice that for a lightly damped system, where the ratio R is less than about 0.5, the shock amplification is about two times the frequency ratio, or 2R. The responses of some other simple single-degree-of-freedom systems to other types of shock pulses are often used to verify their reliability. Figure 34 shows the response of a simple zero-damped system to a square-wave shock. Figure 35 shows the response of a simple zero-damped system excited by a vertical-decay sawtooth shock pulse.

47.8 How PCBs Respond to Shock Pulses Printed circuit boards can come in a wide variety of shapes and sizes to fit a large number of applications that can vary from rectangular to circular to triangular to any shape or combinations of shapes you can think of. They fit in automobiles, airplane wings, missiles, spacecraft, washing machines, toasters, cell phones, computers, garage door openers, calculators, and a million other applications. Shock pulses can be produced in many different ways that can vary from a hammer impact, a rebound drop on a rubber pad, sliding down an inclined plane, using compressed air to generate a high impact at a high velocity, or even simply throwing a small package against a brick wall. Shocks can also be generated by poor manufacturing and assembly processes to careless operation of transportation and shipping containers and of course to accidents.

Effects of Shock on Various Types of Electronic Systems and Structures 1.8 1.5

RC = 0 RC = 0.01 RC = 0.10 RC = 0.30

1.4 1.2 1.0

RC = 0.50 RC = 1.0

0.8 0.6

Response to half sine shock pulse

0.4 0.2 0 .5 1

2

3 4 5 6 7 8 Frequency rate R = fstruct /fpulse

9

10

Figure 33

Shock pulse Square wave

2

RC = 0

1

Input

Isulation Area 0

0

1 2 3 Frequency ratio R = fstruct /fpulse

4

Figure 34

2 A = Gout/Gin

0

Isolation area

Amplification A Gout/Gin

Amplification A = Gout/Gin

47

Sawtooth shock

1

RC = 0 Isolation area

0

0

1 2 3 Sawtooth shock pulse

Figure 35

4

129

130 Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures When PCBs first experience a shock pulse they usually respond by bending in the same direction of the shock pulse. When the pulse energy diminishes, the PCBs will then resonate at their own individual natural frequency. This is usually the fundamental or the lowest natural frequency. Good clearances must be provided between adjacent PCBs to prevent them from striking each other when they respond to various shock pulses. These clearances must include the sizes and manufacturing tolerances of the components on the PCBs as well as the tolerances on the locations of adjacent PCBs. This is to make sure the PCBs do not strike each other and damage the electronic components, the solder joints, or the electrical lead wires on the PCBs. When high shock or vibration forces are expected in severe environments, then snubbers should be considered for mounting them on both sides of each PCB, near the center of each PCB. The snubbers should be aligned and spaced so they strike each other instead of the components mounted on the PCBs to reduce the PCB displacements and increase the PCB fatigue life. See the section on PCB snubbers for more details on snubber design, materials, attachment, location, assembly, and tolerances. Experience has shown that extensive damage can be generated in PCBs when they are allowed to impact each other during operation in high shock and vibration conditions. This can result in cracked electronic components, solder joints, electrical lead wires, and extensive damage to large and heavy components such as transformers and capacitors. Figure 36 shows how large and heavy electronic components can produce large PCB displacements that also produce high stresses and displacements in the electrical lead wires and solder joints that can produce rapid electrical failures.

47.9 Finding Desired PCB Natural Frequency for Shock Environments Small electronic components mounted on PCBs very seldom cause problems when they are exposed to various types of shock environments. The heavy components cause most of the problems because they generate higher acceleration forces and stresses with larger dynamic PCB displacements. Extensive shock test data have shown that the displacement of the PCB has a strong influence on the fatigue life of the electronics. This is the same method of evaluation that was used for sine vibration and random vibration to establish first the desired PCB displacements. This information was then used to find the desired PCB natural frequency. This same method can be used to evaluate the shock environment where only a few thousand shock stress reversals are normally expected. Then fatigue is no longer a major factor and stress concentration effects are sharply rduced. Nonferrous alloys such as aluminum, with a fatigue factor that is about one-third of the ultimate tensile strength, and stress concentration

Wires bending component L

Z

B PCB bending

Figure 36

47

Effects of Shock on Various Types of Electronic Systems and Structures

131

factors of about 2 for holes and notches can now be ignored. Equation (75) can now be modified for the shock environment by a factor of 3 × 2, or 6. Applying this correction factor to the shock condition results in the adjusted PCB desired displacement 0.00132B Z = (desired PCB displacement). (119) √ Chr L The dynamic shock displacement expected for the shock environment can be obtained by using a shock amplification factor A to replace the normal vibration transmissibility Q factor, as shown below. The A factor typical value varies from about 0.5 to 1.5: 9.8GIN A Z = . (120) fN2 These two equations can be combined to obtain the minimum desired PCB natural frequency for the shock environment that will avoid component failures due to excessive displacements in the PCBs:  √ 0.5 9.8GIN AChr L fD = (minimum desired PCB resonantfrequency). (121) 0.00132B

47.10 Sample Problem: Finding Response of PCB to Half-Sine Shock Pulse Qualification tests must be run on the plug-in type PCB shown in Fig. 37 that consists of 19 half-sine shock pulses using an input acceleration level of 90G, with a time base of 0.008 sec. The most critical components on the PCB are the 40-pin DIPs that have side-brazed electrical lead wires and are mounted near the center of the PCB. The plug-in PCB will be mounted on a test fixture that acts like a simple support (similar to a hinge) only around the perimeter of the PCB so the center of the PCB is free to flex when the shock is imposed. Determine if the proposed 0.90-lb PCB will be able to provide satisfactory operation in the shock environment. Solution: The expected natural frequency fN of the PCB for a rectangular plate with a uniform distributed load and simply supported on four sides can be obtained using the equation

 π D 1 1 fN = + , (122) 2 ρ a2 b2

0.093 Acceleration

PCB b = 8.0

a = 6.0

DIP.

L

Half-sine shock pulse 90 G Time, sec 0.008 sec

2.0

Figure 37

132 Natural Frequencies and Failure Mechanisms of Electronic and Photonic Structures where a b W h g E μ

= = = = = = =

6.0 in., PCB width 8.0 in., PCB length 0.9 lb, PCB total weight 0.093 in., PCB height or thickness 386 in./sec2 , acceleration of gravity 3.5 × 106 lb/in.2 , PCB modulus of elasticity with three copper planes 0.20, Poisson’s ratio, dimensionless, PCB with three copper planes

The flexural stiffness D of the PCB with three copper planes can be obtained as follows: D=

Eh 3 (3.5 × 106 )(0.093)3  = 245 lb. in. stiffness,  = 12(1 − μ2 ) 12 1 − (0.20)2

(123)

W 0.90 = 4.8 × 10−5 lb sec2 /in.3 mass/area. (124) = gab (386) (6) (8) The expected natural frequency can now be obtained from the above equations:   π 1 245 1 Expected fN = + = 153 Hz. (125) 2 4.8 × 10−5 62 82 The minimum desired PCB natural frequency fD can be obtained from Eq. (121) using the following data: ρ=

Gin fN fP R A C B r h L

= = = = = = = = = =

90G peak, input acceleration 153 Hz, sec Eq. (125) 1/(2)(0.008) = 62.5 Hz, shock pulse frequency 153/62.5 = 2.4, ratio that determines shock amplification A 1.6, shock amplification from Fig. 33 1.26, constant for DIP with side-brazed lead wires 8.0 in., length of PCB parallel to component 1.0, factor for components mounted at center of PCB 0.093 in., PCB thickness 2.0 in., length of 40-pin DIP

Substitute into Eq. (12) to obtain the minimum desired PCB natural frequency fD : √  ⎤0.5 ⎡ 2 (9.8) (90) (1.6) (1.26) (0.093) (1.0) ⎦ = 148 Hz. (126) Minimum desired fD = ⎣ (0.00132) (8) Since the expected PCB natural frequency of 153 Hz is greater than the minimum desired PCB natural frequency of 148 Hz, the design is acceptable for the desired fatigue life. Shock and vibration tests often have to be repeated many times before the test data will be accepted by the customer. Many problems often occur during qualification test programs that require rerunning different sections of the test over and over due to improper calibration of the electronic equipment. Accelerometers may break loose and fall off or be mounted in the wrong areas. Holding fixtures may not be attached properly or they may have the same natural frequency as the test specimen, which can cause early failures. The wrong channels may be monitored so the test has to be rerun. Other test programs may be run in the same area at the same time and cause transient electrical spikes that change the power settings on the test equipment. The electronic equipment must therefore have built-in safety factors that provide enough fatigue life that will allow the qualification tests to be rerun at least 10 or

References

133

more times without failing. These extra built-in safety factors will increase the size, weight, and cost of the electronic equipment slightly, but they must be designed into the system to ensure the reliability of the system.

REFERENCES 1. Steinberg, D. S., Vibration Analysis for Electronic Equipment, 3rd ed., Wiley, 2000. 2. Crandall, S., Random Vibration, Wiley. 3. Steinberg, D. S., Preventing Thermal Cycling and Vibration Failures in Electronic Equipment, Wiley, 2001. 4. Lord Raleigh, The Theory of Sound , Dover, 1945.

CHAPTER

6

DROP/IMPACT OF TYPICAL PORTABLE ELECTRONIC DEVICES: EXPERIMENTATION AND MODELING T. X. Yu and C. Y. Zhou Hong Kong University of Science and Technology Hong Kong, People’s Republic of China

1

INTRODUCTION Portable electronics, such as, for example, cellular phones, PDAs (personal digital assistants), digital cameras, and notebook computers, have a substantial market share worldwide. It should be noted, however, that portable devices are often exposed to relatively harsh operating environments because of the high likelihood of dropping these devices and also of making impact with other objects. As a result, drop/impact-induced failures are reported to be one of the most dominant causes of damage for portable devices. Therefore, particular concern regarding drop/impact reliability must be taken into account for the relevant product design. In engineering practice, on the other hand, due to the lack of design guidelines, shock proofing of products is obtained through a design–failure–redesign process. That means that only after the prototype is produced are the reliability tests able to be carried out and physical drawbacks detected. Then the design needs to be modified and weak parts need to be changed. Subsequently, new specimens are produced and tested until all parts pass the environmental tests. Obviously, this cycle causes a tremendous waste for product development time and cost considering the complexity of modern electronic products and the supply chain. In order to improve the drop reliability design, instead of making designs on a trial-and-error basis, one must pay attention to the shock reliability at the very early stage of design. Therefore, there is a fundamental need to understand how electronic components or portable electronic devices respond to shock and impact loading and which parameters may govern their shock performance. Several researchers were engaged in the investigation of product-level impact tests of portable electronic devices. Seah et al. [1] and Lim et al. [2, 3] conducted comprehensive tests to investigate the drop impact response of several models of PDAs and mobile phones. Impact force, strains, and acceleration induced at the printed circuit board (PCB) were measured and compared in different orientations. Their results offered the basic information about the strains and impact forces during drop impact. However, since these studies were aimed at testing some specified products, little analysis was conducted to interpret the testing data. And conclusive results on this subject are still inadequate. In general, three basic problems still remain untouched: 1. For a given product, how severe will the dynamic loading be in a typical drop/impact, and how will the product as a whole respond to the drop/impact? Structural Dynamics of Electronic and Photonic Systems Copyright © 2011 John Wiley & Sons, Inc.

Edited by Ephraim Suhir, David S. Steinberg and T. X. Yu

135

136 Drop/Impact of Typical Portable Electronic Devices: Experimentation and Modeling 2. How will the drop/impact load be transmitted to internal electronic components? 3. How do typical components respond to the drop/impact? The objective of the present chapter is to address the fundamental issues related to drop/impact from the viewpoint of mechanical sciences. Systematic study has been carried out both experimentally and analytically to understand the basic mechanical problems described above. First of all, a dynamic test rig was designed, enabling the performance of repeatable impact tests. A simplified system which contained an outer case and a PCB with an attached packaged chip was adopted as a specimen. By employing a Hopkinson bar in the dynamic test rig, the actual impact force pulses were measured. Tests were conducted to correlate the magnitude and duration of the impact pulse with the input parameters such as impact velocity and impact orientation. Dynamic strains at several locations of the PCB were recorded simultaneously to explore the relationship between the dynamic strains and the impact pulse. Particular attention was paid to observing how the key parameters in the dynamic responses of the devices (e.g., the maximum strain and impact force) varied with the input parameters (e.g., the impact velocity and drop orientation). Based on the experimental observations and measurements, an analytical model was proposed for generic impact scenarios of portable electronic devices. By combining rigid-body impact mechanics and the Hertz contact theory in the model, the dependence of the impact magnitude and duration on the system parameters such as the mass and dimension of the specimen as well as the velocity and orientation of the impact were formulated. The impact transmitted to individual components inside the device can be predicted thereafter according to their position and configuration. The comparison between the experiments and the analytical results obtained from the model showed good agreement.

2

EXPERIMENTAL SET-UP The drop test rig, as shown in Fig. 1, consists of a pendulum and a polyvinyl chloride (PVC) bar. An 180◦ air gripper MHY2 produced by SMC Corp. was assembled at the free end of a 1.5-m pendulum arm. This air gripper can be rotated and then fastened so as to hold the specimen in any desired orientation. During tests, the gripper together with a specimen was swung from a specified height. A pair of laser beams and laser detectors together with an electronic control system was designed to open the gripper. The laser beams were placed at an appropriate position so that the gripper released the specimen just before it hit the end of the horizontal PVC bar. Consequently, the specimen freely impinged onto the horizontal bar in a desired orientation while the gripper swung away. To ensure coherent test conditions, red ink was spread on the bar end to mark the contact point for each test. Moreover, a high-speed digital camera was used to monitor the impact process. Due to wave propagation and reflection, most commercial load cells placed in test machines are unable to provide the precise force pulse generated by a drop/impact. In our dynamic test rig, a PVC bar of diameter 26 mm was employed to accurately record the actual impact pulse. The impact pulse duration in our tests was normally less than 1.2 msec while the measured wave speed in the PVC bar was 1.730 m/sec. To ensure the impact pulse was not interfered with by the reflection from the far end of the bar, the length of the bar was chosen as 2.0 m. In order to eliminate the bending effect of the bar, four strain gauges were attached around the circular profile of the bar to record the strain waves passing by. To simplify the structure of typical electronic devices, the specimen used in our tests, as shown in Fig. 2, was composed of a PCB and an outer case. The outer case made of PVC had the external dimensions of 110 × 60 × 20 mm and a wall thickness of 5 mm. A 40 × 90 × 1-mm single-layer PCB was fastened in the case by four screws at its four corners. A 64-ball 8 × 8-mm BGA package with pad size 0.4 mm and pitch 0.8 mm was attached to the

2 Experimental Set-Up

137

Dynamic strain meter

Pendulum Block Air gripper

Oscilloscope

Specimen Strain gauges

PVC bar

Velocity sensor Laser sensors High-speed camera

Figure 1

Schematic illustration of the drop tester.

Figure 2 Photographs of the test specimen. center of the PCB. The mass of the whole system was 173.5 g, of which the outer case was 167.8 g, the PCB was 5.6 g, and the attached package was 0.1 g. As reported in the literature, failure of solder joints is mainly caused by bending of PCBs, so four strain gauges were mounted on the PCB inside our specimen to measure the strains along the PCB’s longer symmetric axis, as indicated in Fig. 3. One stain gauge (Ch2) was mounted in a position close to the chip, and another one (Ch3) was mounted on the opposite surface at the same location. The other two strain gauges (Ch1 and Ch4) were mounted near the screws to detect the wave transferred from the outer case. At the same time, in order to

138 Drop/Impact of Typical Portable Electronic Devices: Experimentation and Modeling

Ch6 Ch5 Impact direction

Ch4

Ch3 Impact direction

Ch2

Ch1

Impact direction

Figure 3 Schematic diagram showing strain gauge locations. measure the impact pulse applied to the outer case, two more strain gauges (Ch5 and Ch6) were mounted on the case along its longitudinal direction.

3

REPEATABILITY OF IMPACT TESTS Through random drops of the specimen by hand, it was found that there was hardly any chance in which the specimen may contact with the ground with one of its edges or one of its faces. Accordingly, in our experiments the impact between the specimen and the bar end was so appropriately controlled that it was capable of mimicking the point contact impact in real drops. The photographs taken by a high-speed camera, as shown in Fig. 4, confirmed that there was no change in the orientation after the specimen was released from the air gripper. To ensure the consistent test results, three to five drops were repeatedly performed for each specified test condition. The results demonstrated that the measurements of both impact force and strains on the PCB were highly repeatable, as shown in Fig. 5. From the strain

Grip

Grip

Grip

Specimen

Figure 4

Bar

Bar

Grip

Bar

Specimen

Specimen

Specimen Bar

Grip

Grip

Specimen

Bar

Specimen Bar

Impact process shown in photographs taken by high-speed camera.

3 Repeatability of Impact Tests 139 1.5

test 1

test 2

test 3

1.0

Impact Force (KN)

0.5 0.0 −0.5 −1.0 −1.5 −2.0 −2.5 −3.0 −2

0

4

2

6

8

Time (ms) (a) 0.4 Ch2 (PCB, strain at the middow) Test 1 Test 2 Test 3 0.3

Strain (10−3)

0.2 0.1 0.0 −0.1 −0.2 −0.3 −5

0

5

10

Time (ms) (b)

Figure 5

Repeatability of impact pulse and strains on PCB.

15

140 Drop/Impact of Typical Portable Electronic Devices: Experimentation and Modeling signals obtained on the bar, it is evident that the incipient wave and the reflected wave were successfully separated. The reflective tensile wave reached the measurement point on the bar (mounted at a location 70 mm away from the impact end) after 2 msec, while the impact had finished within a brief duration of less than 1.2 ms.

4

EXPERIMENTAL RESULTS AND DISCUSSION Two groups of tests were performed in our experimental study. The first one aimed to investigate the effect of impact orientation. The second aimed to correlate the dynamic responses of the specimens with the drop height.

4.1 Impacts in Different Orientations In the following, the length, width, and thickness of the specimen are denoted by a, b, and c, respectively. As shown in Fig. 6, two angles, β and γ , are chosen to specify the drop/impact orientation, where β denotes the angle between the projection of the bar axis on the main face (a × b) and the side of length a, while γ is the angle between the bar axis and the side of length c. Because the thickness of the device was much smaller than its length and width, it was reasonable to choose γ as the major variable when examining the effect of orientation. It is known from rigid-body dynamics that when the axis of the bar (anvil) passes through the center of the specimen, the impact pulse would be the most intense among all the impact conditions. Therefore, while varying angle γ in the tests, angle β was fixed at 26o , which was the angle between the diagonal of the main face and line a. Moreover, in all drop/impact tests, the drop velocity remained constant at 3.58 m/sec (i.e., from a drop height of 0.65 m). Figures 7a –c depict the dynamic strain response with different angle γ . Figure 8 plots the peak value of the strain in relation to angle γ . From the test results it is interesting to observe that: (a) Ch1 had the highest strain on the PCB, but its location was the farthest from the impact point.

Specimen

b c

Bar

a

β γ (a)

β

γ

(b)

Figure 6

(c)

Impact orientation parameters.

4 Experimental Results and Discussion

141

CH1 0° 1.2

15° 30° 45°

0.8 Strain (10−3)

60° 75° 0.4

90°

0.0

−0.4

−0.8

−2

0

2

4

6

8

Time (ms) (a) 1.4

CH3 0° 15° 30° 45° 60° 75° 90°

1.2 1.0

Strain (10−3)

0.8 0.6 0.4 0.2 0.0 −0.2 −0.4 −0.6 −0.8

−2

0

2

4

6

8

Time (ms) (b)

Figure 7

Strain response to drop/impact.

(b) Under various angle of γ , the phase and the peak value of the strains on the PCB and strains on the outer case displayed no significant variation. (c) The impact force increased with the increase of angle γ . Figure 8c shows the variation of the peak bending strain, (Ch2 − Ch3)/2, with the impact angle. It can be seen that the minimum bending strain appeared when the impact angle was

142 Drop/Impact of Typical Portable Electronic Devices: Experimentation and Modeling 0.8

CH6 0° 15° 30° 45° 60° 75° 90°

0.6 0.4

Strain (10−3)

0.2 0.0 −0.2 −0.4 −0.6 −0.8 −2

0

2

4

6

8

Time (ms) (c)

Figure 7

(Continued )

90o while the impact force almost passed through the PCB plane. The maximum bending strain happened when the impact angle was about 75o . A vibration spectrum analysis, as summarized in Table 1, indicates that the two lowest frequencies of the outer case are 1099 and 1587 Hz, while the two lowest frequencies of the PCB were 488 and 854 Hz. For the strain gauges Ch1 and Ch4 located near the screws, the response contained both the fundamental frequencies of the PCB and those of the outer case. For Ch2 and Ch3, the response can be resolved into two parts: a flexural wave with frequencies 488 and 854 Hz as represented by the difference of the two strains, Ch2 − Ch3, and a small component of in-plane strain wave of 1656 Hz as represented by the sum of the two strains, Ch2 + Ch3. According to the spectrum analysis, wave propagation in the outer case had very little effect on the dynamic performance of the central part of the PCB mounted inside the case.

4.2 Impacts under Different Velocities The second group of tests was aimed to investigate the effect of drop height or impact velocity on the dynamic response of the specimens. While the impact orientation was confined to β = 26◦ and γ = 60◦ , the drop height was gradually increased from 0 to 1.5 m. The major concern was the peak values of the strains, believed to be essential in the previous studies for failure of the solder balls. Figures 9 and 10 plot the peak values of the strains on the PCB, the peak strain on the outer case, as well as the peak impact force as functions of impact velocity. As illustrated by Fig. 9a, an interesting phenomenon is that the peak strains on the PCB may approach a steady state after the impact velocity reaches a certain level. This may be caused by the slippage of the bolt joint interface, where stress concentrated. When the impact was so severe that the

143

4 Experimental Results and Discussion

Peak Load (kN)

0.6

0.4

β = 26° V = 3.58 m/s

0.2

0.0 0

15

30 45 60 Impact Angle, γ (degree)

75

90

(a) 1.4

Ch1 Ch2 Ch3 Ch4

Peak Strain (10−3)

1.2

1.0

0.8

0.6

0.4

0.2 0

15

30 45 60 Impact Angle, γ (degree) (b)

Figure 8

Peak strains in relation to angle γ .

75

90

144 Drop/Impact of Typical Portable Electronic Devices: Experimentation and Modeling 0.50

Experimental data Polynomial fit

Peak Bending Strain (X 10−3)

0.48

0.46

0.44

0.42

0.40

0.38 0

15

30

45 Angle, γ

60

75

90

(c)

Figure 8

Table 1

(Continued )

Spectrum Analysis of Impact Responses (Hz)

Ch1 Ch4 Ch2 + Ch3 Ch2 − Ch3 Ch5 Ch6

1

2

3

4

488 488 1656 488 1099 1587

854 854

1099 1587

1587

854 1587 3296

3296

strain level at the interface exceeded the yield strain, slippage may occur and the acceleration transferred to the PCB was then alleviated. Further research would be carried out to explore the detailed facts behind this phenomenon. This observation is highly valuable. Once it is understood which factors affect the level of the steady state, effective methods may be able to be proposed to protect the PCB. According to Fig. 9b, the peak strain of the outer case increased proportionally to the impact velocity to the power 1.39. From the results shown in Fig. 10 and the subsequent analysis, it can also be concluded that the peak load was proportional to the impact velocity to the power 1.14. This implies that the magnitude of the impact force is mainly governed by the momentum of impact (proportional to the impact velocity) rather than the impact energy (proportional to the drop height).

4 Experimental Results and Discussion

145

Figure 11 displays the impact duration varying with the impact velocity. It is evident that the higher the impact velocity, the shorter the impact duration. Figure 12 demonstrates the relationship of the impact impulse (i.e., the integration of the impact force within the impact duration) with the peak impact force. The results display an approximate linear relationship that will be verified by the analysis presented in Section 5.

1.6

CH1 Ch2 CH3 CH4

1.4

Peak Strain (10−3)

1.2 1.0 0.8 0.6 0.4 0.2 0.0 0

1

2

3

4

5

6

Velocity (m/s) (a) 0.5

Ch6 Longitudinal strain near the impact point at the outer case Power fit

Peak Strain (X 10−3)

0.4

0.3

0.2

y = 0.0349x1.386

0.1

0.0 0

1

2

3 Velocity (m/s)

4

5

6

(b)

Figure 9 (a) Peak strains on PCB in relation to impact velocity. (b) Peak strains on outer case in relation to impact velocity.

146 Drop/Impact of Typical Portable Electronic Devices: Experimentation and Modeling Experimental results Power fit

1.0

Peak Force (kN)

0.8

0.6

0.4 y = 0.119x1.14

0.2

0.0 0

1

2

3 Velocity (m/s)

4

5

6

(a) Experimental results Linear fit

0.0

ln(Peak Force)

−0.5 −1.0 −1.5 −2.0 y = −2.15 + 1.14x −2.5 −3.0 −1.0

−0.5

0.0

0.5 ln(Velocity)

1.0

1.5

(b)

Figure 10 Peak impact load in relation to impact velocity.

2.0

5 Dynamic Model 1.15

147

Impact duration vs impact velocity Curve fitting of y = ax−0.2

1.10

Impact Duration (ms)

1.05 1.00 0.95 0.90 0.85 0.80 0.75 0

1

2

3

4

5

6

Impact Velocity (m/s)

Figure 11 Impact duration vs. impact velocity.

5

DYNAMIC MODEL In the following analysis based on the theory of impact dynamics, a simple model is proposed to predict the impact response of every package hierarchy, as shown in Fig. 13. First, in the context of rigid-body dynamics, the overall dynamic response of the system can be determined provided the impact pulse is known. Then, by combining rigid-body dynamics with the Hertz contact theory, the impact pulse can be formulated in terms of the material properties and the dimension of the impact area as well as the impact velocity and orientation. The impact imparted to individual components inside the device can therefore be estimated according to their positions. Finally, the dynamic response of the components can be calculated based on the acceleration pulses imposed at the components’ boundaries as obtained from the above analyses.

5.1 Rigid-Body Dynamics and Impact Impulse In a real drop of a portable electronic device, most likely one of its corners will first make contact/impact with the ground, as first contact on an edge or on a whole face rarely occurs. Therefore, the contact area is very small compared with the dimension of the device. As illustrated by Stronge [4], the rigid-body impact theory can thus be applied to predict the impact event. Based on rigid-body dynamics, the current problem can be treated by considering a rigid body under a specified impact pulse at one corner. To simplify the analytical model, the system is reduced to a one-dimensional uniform rod, although in future work two- or threedimensional rigid-body dynamics would be considered to incorporate the effect of the width and thickness, if they are not small compared with the length of the product. Denote the overall mass by m, the length by l , the initial impact velocity by V0 , and the drop/impact orientation by γ . The contact/impact period is set to be from t = 0 to t = T . If

148 Drop/Impact of Typical Portable Electronic Devices: Experimentation and Modeling Experimental results Power fit

0.5

Impact Impulse (Ns)

0.4

0.3 y = 0.495x1.035 0.2

0.1

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Peak Force (kN) (a) −0.5

Experimental results Linear fit

ln (Impact Impulse)

−1.0

−1.5

−2.0

y = −0.703x1.035

−2.5

−3.0 −3.0

−2.5

−2.0

−1.5

−1.0

−0.5

ln (Peak Force) (b)

Figure 12 Impact impulse vs. peak impact force.

0.0

5 Dynamic Model

149

ac αl/2

B

C

αl/2

γ

A

Figure 13 Sketch of a beam–rod system. an impact force pulse F (t) is specified, the acceleration of the mass center of the system is given by F (t) ac (t) = . (1) m In addition to the translational motion, the impact force also produces a rotation of the rod. The angular acceleration is F (t)l cos γ , (2) α(t) = 2Jc where α is the angular acceleration and Jc is the moment of inertia calculated by 1 2 (3) Jc = ml . 12 Therefore, the vertical acceleration at any point x (0 ≤ x ≤ l ) of the rod can be deduced by     l F (t) 6 cos2 γ (x − l /2) a⊥x (t) = ac (t) − α(t) x − cos γ = 1− . (4) 2 m l It is worth mentioning that the magnitude of the acceleration pulse at different positions of the device is totally different though the shape and duration are the same. For example, when γ = 0, the acceleration at the impact end is four times that at the center, while the acceleration at the other end is two times that of the center with inverse direction. Using this notation, the angular velocity of this rotation is found as  t P(t)l cos γ ω (t) = α(τ ) d τ = , (5) 2Jc 0 where P(t) is the impulse of the impact force pulse given by  t P(t) = F (τ ) d τ . (6) 0

Furthermore, from Eqs. (4) and (5) the vertical velocity of contact point A is  P(t)  (7) 1 + 3 cos2 γ V⊥A (t) = −V0 + m where V0 is the impact velocity. Let ξ denotes the coefficient of restitution; then the rebound velocity is V⊥A (T ) = ξ V0 .

(8)

Thus, a relationship between the impulse and the impact angle can be deduced from Eqs. (7) and (8), (1 + ξ ) mV0 P(T ) = . (9) 3 cos2 γ + 1

150 Drop/Impact of Typical Portable Electronic Devices: Experimentation and Modeling 0.7

0.7

Impact Force Impact Impulse

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.0

0.0

−0.1

−0.1

−0.2 −1.0

−0.5

0.0

0.5

1.0

Impact Impulse (kg m/s)

Impact Force (KN)

V = 3.58 m/s γ = 90°

−0.2 2.0

1.5

Time (ms)

Figure 14 Calculation of experimental impulse.

0.4

Experimental results Theoretical fitting results

Impulse (kg m/s)

0.3

0.2

0.1

0.0 0

15

Figure 15

30

45 60 Impact angle (degree)

75

Comparison of experimental impulse with theory.

90

5 Dynamic Model

151

Figure 14 depicts a typical impulse based on the experimentally measured force history, and Fig. 15 compares the experimental impulse with the theoretical prediction, Eq. (9). The results show good agreement, although the rotation about the second principal axis of inertia is neglected.

5.2 Impact Force In the derivations of the above section, the impulse of the impact force remains undetermined. However, combining the Hertz contact theory with the rigid-body dynamics theory, the duration and magnitude of the impact force can be explicitly determined. In the theory of solid-body impact, it is assumed that the deformations of the solid bodies generated by the impact are concentrated in small contact regions, and the force–deformation relationship can be determined by the well-known Hertz contact theory. Therefore, the behavior of the small material volumes close to the contact area can be simplified as a nonlinear spring, and then the system can be modeled as two rigid bodies connected by a deformable contact spring. Suppose the impact is elastic. From Hertz contact theory [5], the contact force between a sphere of radius R and a massive body with a plain contact surface is given by F = R 1/2 K δ 3/2 ,

(10)

where δ denotes the compression of the contact spring when the two bodies are in contact and K is given by −1  1 − ν22 4 1 − ν12 K = + (11) 3 E1 E2 in which Ei and νi denote the Young’s modulus and Poisson’s ratio, respectively, of the body i (i = 1, 2). Subscript 1 refers to the sphere and subscript 2 refers to the massive body with the plain surface. Considering a solid rod with a spherical head of radius R, the vertical displacement at the spherical head of the rod can be regarded as the compression of the contact spring. By Eq. (4), it is found that   1 + 3 cos2 γ d 2δ = −F (t) . (12) dt 2 m When the impact angle is γ , define the equivalent mass as m m∗ = . (13) 1 + 3 cos2 γ Then, the governing equation of the rod–plane contact problem can be found by combining Eqs. (10)–(13) as d 2δ 1 = − ∗ R 1/2 K δ 3/2 . dt 2 m

(14)

Using the initial conditions dδ (15) = V0 , dt and integrating Eq. (14) twice, the deformation process and the impact force can be obtained consequently by using Eq. (10). Although the exact solution of Eq. (14) is quite complicated, through some approximations and derivations, Hunter [6] obtained some simple results as follows:   π V0 t 2.94δ0 δ (t) ≈ δ0 sin , (16) , 0
152 Drop/Impact of Typical Portable Electronic Devices: Experimentation and Modeling where δ0 is the maximum compression determined by  2/5 5m ∗ V02 . δ0 = 4R 1/2 K

(17)

The impact duration is found as 2.94δ0 = T = V0



5m ∗ 4R 1/2 K

2/5

−1/5

V0

while the impact force pulse is found as     π V0 t π V0 2 sin F (t) = −m ∗ δ0 2.94δ0 2.94 δ0  2 π V0 = −m ∗ δ (t) 2.94δ0   π V0 t ∗3/5 6/5 −1/5 2/5 = −3.07m V0 R K sin , 2.94 δ0

(18)

0
2.94δ0 . V0

(19)

These equations provide reasonable predictions of the impact force and the acceleration imposed on the electronic device. Figure 10 demonstrates the power-fitting results of the peak load with respect to velocity. The resulted power value from experiments is 1.14, which is quite close to the value of 1.2 based on this theory.

5.3 Response of Internal Component Because the mass of the PCB was much smaller than that of the outer case, it can be assumed that the PCB had little effect on the overall performance of the outer case during the impact. Then the dynamic response of the PCB can be determined by the motion of its supporting ends and the accelerations applied there. According to the above analysis, the acceleration of the mass center of the PCB–case system can be obtained by calculation or simple tests. However, because of the angular velocity, acceleration at the supporting ends of the PCB is notably different from that at its center. Taking the rod system discussed in the above section as an example, when angular acceleration is taken into account, it can be seen that the total acceleration in the region close to the impact point is much higher than the average translational acceleration of the system, as depicted in Fig. 16. Therefore, particular concern should be shown when selecting the test method for the shock test at the component level. Most of the conventional constraint conditions used in the shock test at the component level can only reproduce the dynamic response under the uniform-distributed acceleration pulse, so that the asymmetric response mode generated by angular acceleration displayed in a real situation could not be simulated.

x=0

α(x, t)

x=l x

w

Figure 16 Sketch of beam under both translational and angular accelerations.

5 Dynamic Model

153

In order to compare the response under the test conditions to that in a real drop, a simplified system with a clamped–clamped beam attached to a rod is employed to illustrate the analytical model. Here the beam represents the PCB and the rod of higher stiffness represents the outer case. Both the rod and the beam have length l , and the impact angle is denoted by γ as used before. This simple model is adopted to examine the correlation of the impact response of the system with the input impact parameters. Suppose a certain translational acceleration Gc (x , t) and a given angular acceleration α(x , t) are applied to the base where the beam’s ends are supported, and the strain and relative displacement along the beam are to be determined. Two approaches can be employed to tackle the problem. One is to directly solve the dynamic response of the beam under the given boundary accelerations. The other is to deduce the dynamic response of the beam under a noninertia coordinate system which is fixed to the base. The second approach is adopted in the following analysis. Lateral vibration of an elastic beam is governed by the equation ∂ 4w ∂ 2w (x , t) + ρA 2 = f (x , t), (20) 4 ∂x ∂t where w is the lateral displacement of the beam, I is the second momentum of the crosssectional area about the neutral axis, A is the cross-sectional area, ρ and f (x , t) are the density of the beam and the distributed force per unit length, respectively. The boundary conditions for Eq. (20) are ∂w w (0, t) = 0, (0, t) = 0, ∂x (21) ∂w w (l , t) = 0, (l , t) = 0. ∂x Before impact the beam is almost stationary with respect to the base rod; so the initial conditions are ∂w (x , t = 0) = 0. (22) w (x , t = 0) = 0, ∂t The downward direction is taken as the positive direction for the lateral displacement w , and the distributed lateral inertia force produced by the translational acceleration is expressed as f1 (x , t) = ρAa⊥c (t) = ρA cos γ ac (t) . (23) EI

Neglecting the axial force and Coriolis force, the distributed force caused by the angular acceleration can be expressed as   l α(t). (24) f2 (x , t) = −ρA x − 2 The natural mode of free vibration for this clamped–clamped beam is x x

x x

sinh κn − sin κn + , n = 1, 2 . . . , cosh κn − cos κn Wn (x ) = sinh κn − sin κn l l cos κn − cosh κn l l (25) where κn denotes the nth root of cos κn cosh κn − 1 = 0

(26)

(κ1 = 4.730041, κ2 = 7.853205, κ3 = 10.995608, κ4 = 14.137165 . . . ). Accordingly, the natural frequency is  EI 2 n = 1, 2, . . . ωn = κn , (27) ρAl 4

154 Drop/Impact of Typical Portable Electronic Devices: Experimentation and Modeling The deflection of the beam is assumed to be ∞ Wn (x )qn (t). w (x , t ) =

(28)

n=1

Hence, the dynamic response of the beam can be determined using the mode superimposition theorem. Thus, for the nth mode d 2 qn (t) 1 + ωn2 qn (t) = Qn (t), dt 2 ρACn where the constant Cn is given by



(29)

l

Cn = 0

Wn2 (x ) dx

(30)

and Qn (t) is the generalized force which can be calculated by  l Qn (t) = f (x , t)Wn (x ) dx 0



= ρAa⊥c (t)

l

Wn (x ) dx − ρAα(t)

0

 l 0

 l Wn (x ) dx . x− 2

Using the Duhamel integral [7], the solution can be expressed as  t 1 qn (t) = An cos ωn t + Bn sin ωn t + Qn (τ ) sin ωn (t − τ ) d τ ρACn ωn 0

(31)

(32)

From the initial conditions, it is determined that An = Bn = 0, qn (t) =

1 ρACn ωn



(33a) t

Qn (τ ) sin ωn (t − τ ) d τ .

(33b)

0

According to the derivation in the previous section, the impact force has the form of a half sine; therefore the acceleration also has the same shape, that is,

πt Gm sin , 0 ≤ t ≤ T , a⊥c (t) = T 0 T ≤ t, (34)

πt αm sin , 0 ≤ t ≤ T , α(t) = T 0 T ≤ t. When n is odd, the natural mode is symmetric with respect to the middle section; otherwise, the mode is antisymmetric. It follows that ⎧  l ⎪ ⎪ T W2k −1 (x ) dx π G ⎪ m ⎪ − (ω2k −1 T /π) sin (π t/T ) + sin ω2k −1 t ⎪ 0 ⎪ ⎪ , 0 ≤ t ≤ T, ⎪ 2 2 ⎪ C2k −1 ω2k −1 π 2 − ω2k ⎨ −1 T q2k −1 (t) =  l ⎪ ⎪ ⎪ ⎪ ⎪ π G T W (x ) dx ⎪ ⎪ m 0 2k −1 sin ω2k −1 (t − T ) + sin ω2k −1 t ⎪ ⎪ , T ≤ t, ⎩ 2 2 C2k −1 ω2k −1 π 2 − ω2k −1 T (35)

5 Dynamic Model ⎧  l ⎪ ⎪ ⎪ T π α (l /2 − x ) W2k (x ) dx m ⎪ ⎪ − (ω2k T /π) sin (π t/T ) + sin ω2k t ⎪ 0 ⎪ , ⎪ 2 ⎨ C2k ω2k π 2 − ω2k T2 q2k (t) =  l ⎪ ⎪ ⎪ ⎪ T π α (l /2 − x ) W2k (x ) dx ⎪ m ⎪ sin ω2k (t − T ) + sin ω2k t ⎪ 0 ⎪ , ⎩ 2 C2k ω2k π 2 − ω2k T2

155

0 ≤ t ≤ T,

T ≤ t. (36)

Consequently, the bending moment is found as ∂ 2w . (37) ∂x 2 Note that the bending strain on the surface of the PCB has the following relationship with bending moment: M (h/2) h ∂ 2w . (38) ε=± =± EI 2 ∂x 2 To simplify the deduction, the following nondimensional parameters are defined: w Gm ε t ω1 t w∗ = , , ε∗ = 2 2 , t∗ = = Gm∗ = , (39) h h /l T1 2π hω12 M = EI

where T 1 denotes the fundamental period of the beam and h denotes the thickness of the beam. By choosing a shock duration as T = 0.1T1 and Gm∗ = 1, the nondimensional displacement and strain are calculated. Figures 17 and 18 compare the calculated results with and without consideration of the angular acceleration. Under the translational acceleration alone, the maximum displacement occurs at the center of the beam, while under both translational and angular accelerations,

Dimensionless displacement at x*=0.50 under uniform acceleration Dimensionless displacement at x*=0.68 under both translational acceleration and angular acceleration

Dimensionless Displacement, w *

0.518 0.399

0

0

1

2

3

4

Time (t/T0)

Figure 17 Comparison of maximum displacement under translational acceleration and under both translational and angular accelerations.

156 Drop/Impact of Typical Portable Electronic Devices: Experimentation and Modeling Dimensionless strain at x* = 0 under uniform acceleration Dimensionless strain at x* = 0 under both translational acceleration and angular acceleration 500

Dimensionless Strain, ε*

400 197.7

300 200 100 0 −100 −200 −300 −400

370.0

−500 0

1

2 Time, t/T0

3

4

Figure 18 Comparison of maximum bending moment under translational acceleration and under both translational and angular accelerations.

the maximum displacement occurs at a cross section which is 0.7 beam length away from the impact end. The results indicate that, when taking the rotation into consideration, the maximum displacement will increase by 30%, and the maximum strain, which is proportional to the maximum moment, will be 87% higher. However, these increases may be different for impact pulses with different durations. Because of the symmetry of the system, the second-order mode generates zero-bending moment in the midsection of the PCB. Therefore, the response of the midsection of the PCB can be assumed to be governed mainly by the fundamental natural mode while the effect of higher order modes is negligible. Thus, Eqs. (34)–(36) can be combined into a simple expression for the strains on the top and bottom surfaces of the midsection of the PCB as ⎧  l ⎪ ⎪   W1 (x ) dx hG ⎪ m ⎪ T −ω1 T /π sin π t/T + sin ω1 t  l ⎪ 0 ⎪ W1 , 0  t  T, ⎪ ⎪   ⎨ T1 2 C1 ω12 1 − (ω1 T /π)2 l ε =±  l ⎪ 2 ⎪ ⎪ ⎪   hG W1 (x ) dx ⎪ m ⎪ ⎪ T sin ω1 (t − T ) + sin ω1 t  l 0 ⎪ ⎩ W , T  t. 1 C1 ω1 T1 2 1 − (ω1 T /π )2 (40) Taking our specimen as an example, when the impact angle is γ = 0 and impact velocity is 3.58 m/sec, the peak force recorded in the test is 0.203 kN and impact duration is 1.08 msec. As the mass of the sample is 173.5 g, the translational acceleration is then found to be Gm = 1170 m/sec2 . The dimension of the PCB is 40 × 90 × 1 mm. From Table 1, the fundamental natural frequency of the PCB is 488 Hz. By substituting all these parameters into Eq. (40), the longitudinal strains at the midsection of the PCB can be calculated, which were recorded by Ch2 and Ch3 in the experiments. The calculated results are depicted in

157

6 Conclusions Calculation of bending strain

0.4 0.3

Bending Strain (X10−3)

0.2 0.1 0.0 −0.1 −0.2 −0.3 −0.4 0

1

2

3

4

5

6

7

8

Time (ms)

Figure 19

Calculated longitudinal strain at center of PCB.

Fig. 19. The peak strain found from this model is 0.28 × 10−3 , while the peak bending strain calculated from the measurements at Ch2 and Ch3 in experiments is 0.42 × 10−3 . While the two values are in the same order, the difference between them may be attributed to the nonuniformity of the PCB caused by the dummy chip attached at the center.

6

CONCLUSIONS Fundamental understanding is essential for the design of robust portable electronic products. A drop test system was designed to perform repeatable drop tests with arbitrarily specified orientation. In this test system, the drop impact was generated by a pendulum while the actual impact force pulse was measured by strain gauges attached to a Hopkinson bar. Although the impact force increased dramatically when the impact angle γ changed from 0◦ to 90◦ , no significant change was observed for the strains measured on the PCB and the outer case. Spectrum analysis of the signals showed that near the screw points the response of the PCB may interfere with the outer case. However, in its central region, the dynamic responses were governed by the PCB itself. Both flexural waves and in-plane compressive waves were detected near the chip. Based on the low-velocity impact theory, the impact peak load is found to be proportional to the impact velocity to the power 1.2. After the impact velocity reaches a certain level, the peak strains on the PCB no longer increase linearly with the impact velocity but approach constant levels. Further analysis demonstrates that the contact/impact force produces not only a translational acceleration but also an angular acceleration of the case–PCB system, which may lead to higher stresses in the PCB. Hence, if this rotational effect is not incorporated in shock tests at the component level, then those shock tests cannot truly simulate the real drop/impact situations.

158 Drop/Impact of Typical Portable Electronic Devices: Experimentation and Modeling A simplified beam–rod model is proposed to provide a guideline to the drop/impact protection design of typical portable electronic devices. Designers may first calculate the impact force based on the Hertz contact theory and find the acceleration level (in g) for the base (i.e., the case or the rod); then estimate the acceleration level on the PCB by the rigid-body impact theory as well as the boundary conditions. Thus, the shock reliability level can be determined for each component before a prototype is produced.

Acknowledgment The results reported in this chapter are part of a research project CERG grant 6189/03E. Financial support from the Hong Kong Research Grant Council is gratefully acknowledged.

REFERENCES 1. Seah, S. K. W., Lim, C. T., Wong, E. H., Tan, V. B. C., and Shim, V. P. W., “Mechanical Response of PCBs in Portable Electronic Products during Drop impact,” Proc. IEEE 4th Electron. Packaging Technol., Singapore, 10–12 December 2002. pp. 120–125. 2. Lim, C. T., and Low, Y. J., “Investigating the Drop Impact of Portable Electronic Products,” Proc. IEEE 52nd Electron. Components Technol., San Diego, CA, 28–31 May 2002. pp. 1270–1274. 3. Lim, C. T., Ang, C. W., Tan, L. B., Seah, S. K. W., and Wong, E. H., “Drop Impact Survey of Portable Electronic Products,” Proc. IEEE 52nd Electron. Components Technol., New Orleans, LA, 27–30 May 2003. pp. 113–120. 4. Stronge, W. J., Impact Mechanics, Cambridge, UK: Cambridge University Press, 2000. 5. Love, A. H. E., A Treatise on the Mathematical Theory of Elasticity, New York: Dover, 1944, pp. 193–203. 6. Hunter, S. C., “Energy Absorbed by Elastic Waves during Impact,” J. Mechanics and Physics of Solids, Vol. 5, pp. 162–171, 1957. 7. Clough, R. W., and Penzien, J., Dynamics of Structures, New York: Mc-Graw Hill: 1993.

CHAPTER

7

SHOCK TEST METHODS AND TEST STANDARDS FOR PORTABLE ELECTRONIC DEVICES C. Y. Zhou, T. X. Yu, and S. W. Ricky Lee Hong Kong University of Science and Technology Hong Kong, People’s Republic of China

1

Ephraim Suhir University of California Santa Cruz, California University of Maryland College Park, Maryland ERS Co. Los Altos, California

INTRODUCTION: NECESSITY OF DROP/IMPACT TESTING OF PORTABLE ELECTRONIC PRODUCTS Due to the advancement of new technologies, electronic components have become smaller, thinner, and lighter. As a result, a large variety of electronic devices have become portable and can now be carried around. However, along with the obvious benefits of miniaturization and portability, there are hazards due to the likelihood of dropping these devices. According to 1998 research conducted by the IDC (International Data Corporation), 64% of notebook damage is caused by drops, 38% is due to spillages, and 28% to crushing, with liquid crystal displays (LCDs) and hard disk drives (HDDs) reported to be the two components most often cited as suffering damage. Portable devices are often expensive and may contain impactsensitive electronic components, such as the flat-panel displays and packages with tiny and vulnerable solder joints. Uncontrolled shocks can result in unacceptable cost and time losses for repairing or replacement of the equipment, loss of operation downtime, and possibility of losing valuable data or even business opportunities. To be able to meet rigorous customer expectations and demands, there is intensive competition among manufacturers to develop new approaches and new methods for assuring product shock reliability. According to a study, “The Global Markets for Rugged Mobile Computers, Volume 1: Rugged Mobile Computer Products,” of the research firm Venture Development Corp. (VDC), Natick, Massachusetts, the world rugged mobile computer market is expected to grow 10.2% annually for many years to come. On the other hand, manufacturers have taken pains recently to develop rugged mobile models to meet the specific needs for field work, military patrols, law enforcement, telecommunications, and health care operations. Among the rugged and semirugged mobile computers are the Itronix GoBook II, the Panasonic Toughbook CF-29, the Rugged Notebooks Rough Rider Junior, the Xplore Technologies iX104-TPC, Twinhead N15RN, Rugged Notebooks Hawk, and HP nr3600. Due to the awareness of costly damage to portable electronic devices caused by mechanical shock, shock reliability has become a key product attribute. At the same time, with the incessantly increasing product complexity and performance requirements, engineers constantly face the challenge to design Structural Dynamics of Electronic and Photonic Systems Copyright © 2011 John Wiley & Sons, Inc.

Edited by Ephraim Suhir, David S. Steinberg and T. X. Yu

159

160 Shock Test Methods and Test Standards for Portable Electronic Devices more reliable products as packages become smaller and smaller. To sum up, the following reasons make research into drop impact reliability of great importance for portable devices: • Portable electronics often contain many shock-sensitive components such as LCDs,

ceramic capacitors, and fine solder interconnections. • With technology diminishing in size, less energy-absorbing materials can be added

• •

•

•

2

into the packages, so that higher impact force is sustained directly by the electronic components. Increasing components integrated into products raises the risk of interknocking of the components or subsystems when subjected to mechanical shock. To design environmentally friendly products, portable electronic devices usually adopt new packaging materials, such as polymeric based adhesive interconnections, which are more susceptible to drop impact. To allow for increase in functions, more and more opto-electronic modules are being integrated into portable electronics, while opto-electronics are intrinsically sensitive to misalignment and vulnerable to drop impact. Special rugged portable electronics are needed to meet the rigorous needs for severing in harsh mechanical environments, such as for field work, military patrols, law enforcement, telecommunications, and health care applications.

TEST CRITERIA AND SPECIFICATIONS To evaluate the dynamic performance of electronic products, various experimental techniques and test methods have been proposed. The existing test methods can be broadly classified into tests on the component level and on the product level. For tests on the component level, many standards and protocols can be followed, for example, military standard MIL-STD-810 [1, 2], American National Standards Institute/American Society for Testing and Materials (ANSI/ASTM) D3332 [3], and International Electrotechnical Commission (IEC) 68-2-471999 [4]. In recent years, the JEDEC (Joint Electron Device Engineering Council) has made extensive efforts to set up test protocols [5–7], especially for hand-held electronic products. Among all the protocols and standards, the military standard MIL-STD-810 [1-2] and IEC 68-2-47-1999 [4] have been most widely adopted by the industry. MIL-STD-810 method 516.5 deals with shock testing, including such test topics as functional shock, material to be packaged, fragility, and transit drop. IEC 68-2-47-1999 recommends test methods for shock, bump, and free-fall tests. In recent years, extensive attention has been attracted to several standardized test protocols, especially for hand-held electronic products issued by JEDEC [5–7]. However, due to the weak correlation between the performance of the components in test and in field, it is still a challenge for a designer to select the appropriate test parameters. No matter how different the particular test procedures are, all the existing test procedures and methods can be divided into two major categories: (1) drop tests (or free-fall test) and (2) shock tests.

2.1 Drop Tests (Free Fall Tests) The objective of the drop test is to assess the structural and functional integrity of the entire product for safety purposes for some or all the handling, transportation, and service environments/conditions. A traditional test method is to drop the products from a certain height (e.g., 122 cm recommended by MIL-STD-810) with a quick-release hook or by using a drop tester (Fig. 1). Usually, the drop surface (e.g., plywood), rude drop orientation (surfaces, edges, and corners), and number of drops (e.g., two times for IEC 68-2-47) are specified by these standards. In general, there is no instrumentation calibration for the test and the measurement

3 Shock Test Machine

161

Figure 1 Drop test machine (from the website of Yoshida Seiki Co. Ltd., Japan, http://www. yoshida-seiki.co.jp/). Table 1 A Typical Example of Product Technical Specifications Gravity-drop test standard Drop height Drop surface Number of drops

MIL-STD-810F 516.5 120 cm Plywood 1

information is minimized. Analysis of the results consists of visual and operational comparisons of the “before-test” and “after-test” conditions. Table 1 shows typical product technical specifications for a drop test.

2.2 Shock Tests: Shock Table Tests A shock test is carried out to determine a component’s ruggedness or fragility level so that packaging could be adequately designed to protect the component or so that the component could be redesigned to meet transportation and/or handling requirements. This test is used to determine the critical shock conditions at which there is reasonable chance of structural and/or functional system degradation.

3

SHOCK TEST MACHINE To perform a shock test, a specially designed and appropriately tuned shock test apparatus is employed. A shock-producing apparatus should be capable of delivering a certain kind of shock input as determined by the standard methods, for example, an external half-sine acceleration pulse with the maximum value of 1500 g acting for 1 ms. A simple shock test machine is the drop table recommended by ANSI/ASTM D3332 [3] (Fig. 2). The machine

162 Shock Test Methods and Test Standards for Portable Electronic Devices

Figure 2 Shock test machine (from the website of Yoshida Seiki Co. Ltd., Japan, http://www. yoshida-seiki.co.jp/). consists of a flat horizontal surface (carriage) of sufficient strength and rigidity. The test surface is guided to fall vertically without rotation or translation in other directions. A certain method (e.g., a cushion under the surface) is to be equipped to produce shock pulses as specified by the existing standards. MIL-STD-810F 516.5 also introduces other types of shock apparatus, such as hydraulic, compressed gas, electrodynamic shaker, electrohydraulic shaker, rail car methods, or other activating types capable of producing a test specimen’s response over the time, amplitude, and frequency ranges specified, as sketched in Fig. 3. When conducting shock tests, the product (component) is mounted firmly on the support structure of the test equipment. Then a shock impulse of the given shape, magnitude, and duration is applied to the supporter and then imparted to the test sample. Noting that the dynamic response of the component or product is strongly affected by its orientation, support position, and support methods, multidirectional tests are recommended by some standards (see, e.g., ASTM D3332). For JEDEC standards, even the material and the size of screws to be used are specified. The fragility of the component (product) is determined by the critical acceleration and critical velocity. This definition is based on the damage boundary theory (Fig. 4) suggested by Newton [8] in 1968. Under this concept the fragility of a product is measured by two factors: critical velocity and critical acceleration. The critical velocity represents the velocity change (which can be related to drop height) below which no damage could possibly occur, regardless of the magnitude of the peak pulse acceleration. The critical (“threshold”) acceleration represents the acceleration above which the product will be most likely damaged.

Figure 3

Classical uniaxial shock pulses: half sine, trapezoidal, and terminal peak sawtooth.

Critical velocity line

Acceleration, g

4 Comparison of Different Test Methods

163

Damage region

Critical acceleration line

No-damage region

Velocity, m/s

Figure 4

Typical damage boundary curves.

A typical product technical specification for a shock test is a trapezoidal shock of magnitude 50 g (ASTM D3332).

4

COMPARISON OF DIFFERENT TEST METHODS The drop test is the most popular testing method for the evaluation of the dynamic response of a device to an impact load. The obvious advantage of this method is that it can directly represent the field impact scenario. However, test repeatability cannot be guaranteed in drop tests because it is difficult to control the orientation of the object at the moment of impact and to instrument it. Moreover, drop tests are not applicable to test the shock resistance of individual components. Shock table tests are free of such problems. A shock impulse with a given shape, magnitude, and duration (e.g., an external acceleration with maximum value of 1500 g, acting for 0.5 msec) is applied to the shock table and then imparted to the sample product or component fastened onto the table. However, the shock table test is just a more or less satisfactory simulation of the real-life drop impact process. To make the shock table tests adequately mimic the real drop impact, it is critical to make correlations between the shock table test parameter and field conditions. Studies have been conducted to assess various dynamic testing methods. Goyal and Buratynski [9] summarized the advantages and disadvantages of constrained drop testing (termed “shock test” in this chapter) and free drop testing (termed “drop test” in this chapter), which are cited in Table 2. Goyal and Buratynski [9] proposed that the object being tested should be suspended on a guided drop table in the precise desired impact orientation. The drop table is dropped from a specific height to the ground. The test object is released from the suspension right before the drop table hits the floor so that it can move unrestrained thereafter. Ong et al. [10] initiated experiments comparing the mechanical response of a PCB subjected to product-level and board-level drop tests. The test results showed that the mechanical response of the PCB in the product-level test differed greatly from that of board-level testing because of the deformation of the casing and intercontacts between internal components in free drop tests. Suhir [11] has preliminarily examined the feasibility of adopting shock tests to mimic the drop test conditions. By employing a simple, yet physically meaningful, analytical (“mathematical”) predictive model, Suhir had determined that, in order to adequately mimic drop test conditions, the shock test loading, at least in a situation when a linear approach is applicable, should be as close as possible to an instantaneous impulse, and the duration of the shock load should

164 Shock Test Methods and Test Standards for Portable Electronic Devices Table 2

Comparison of Different Test Methods

Test Method

Advantages

Drawbacks

Constrained drop test (shock test)

Fairly controlled shock pulse shape, amplitude, and duration applied to object Predetermined orientation of object Repeatable and instrumentable Best replication of real drop impact

Not able to mimic multiple impacts resulting in real drop impact Difficult to subject impact on objects’ vertices and edges

Free drop test

Not able to precisely determine impact orientation Not very suitable for individual components Poor repeatability

be established based on the lowest (fundamental) natural frequency of vibration of the most vulnerable structural element of the device. He showed that, for practical purposes, it is sufficient to consider the fundamental mode of vibration only. After the appropriate duration of the shock impulse is established, the time dependence and the maximum value of the imposed acceleration in shock tests should be determined in order to best mimic the drop test conditions, with consideration of the most likely drop height. In the analysis that follows we describe an experimental study aimed at the evaluation of the feasibility of using an appropriate shock table test to mimic the shock environment for components and systems adopted in portable electronics. First, real drop impact tests were conducted to assess the dynamic characteristics of typical portable electronic systems. Subsequently, a series of shock table tests with different constraint conditions were designed to mimic the real-life impact state. By comparing the typical results from shock table tests and those from drop tests, the correlation of the shock table test parameters and the drop test conditions is analyzed. The results reveal that the conventional, fully constrained shock table tests cannot mimic the actual drop impact conditions, while an appropriate shock table test method should allow the sample to rotate freely. Theoretical analysis is carried out to explain the mechanics of the impact scenarios, revealing that, due to the Hertz contact spring effect and the rotational acceleration during the impact, the acceleration at the centroid of the sample is significantly different from that of the table. The acceleration estimated by traditional force divided by mass may underestimate the real acceleration of components inside the products. We believe that the results of this study will be valuable to the engineering practice, since the portable electronics manufacturers could reduce the testing time of their products by using fewer testing rigs.

4.1 Drop Tests Drop tests were first conducted to observe the real dynamic response of a typical portable device under drop impact. A drop test system was designed to perform drop tests under various specifications. The test system and instrumentations are described in Chapter 6, Section 2. Impact force pulse and strains of the insider components can be instantaneously measured. To facilitate the tests, the specimens used in our tests represent a simplified structure of actual and typical electronic devices. This is also given in Chapter 6 (see Fig. 2 in that chapter). We assume that the failure of the solder joints is caused mainly by the bending of the PCB,

4 Comparison of Different Test Methods

165

and therefore four strain gauges were mounted on the PCB to measure the strains along its longer symmetric axis, as indicated in Fig. 3, Chapter 6. The locations of other strain gauges are shown in Fig. 3, Chapter 6, as well. To ensure the consistency of the test data, three to five drops were performed for every specified test condition. The results demonstrated that the measurements of both impact force and strains on the PCB were highly repeatable (consistent).

4.2 Shock Table Tests The shock test system, as shown in Fig. 5, is installed on a Dynatup 8250 drop test machine. An accelerometer is fixed at the shock table to record the pulse. Soft materials are put on the anvil to shape the impact pulse. In the following description, the length, width, and thickness of the specimen are denoted by l , b, and c, respectively. As shown in Fig. 6, Chapter 6, which indicates the drop/impact direction, two angles, β and γ , are chosen to identify the impact orientation, where β denotes the angle between the projection of the bar axis on the main face (l × b) and the side of length l and γ is the angle between the bar axis and the side of length c. When carrying out the tests, we have taken as a typical orientation γ = 60◦ , and β = 26◦ , which is the angle between the diagonal of the main face and line a. Moreover, in all drop/impact tests, the drop velocity remained constant, 2.0 m/sec (drop height 0.2 m). Three groups of shock table tests were conducted and compared with impact test results. For shock test condition 1, the specimen was fixed using a stiff fixture to the shock table, as shown in Fig. 6. This was done to ensure that the predefined shock pulse would be endured by the overall specimen. This test method is a modification of Ong’s [9] experiments. Ong’s shock table test considered a similar shock pulse but did not account for the influence of the outer case. For the shock test condition 2, the specimen was fixed by a pair of air grippers connected to the shock table, as shown in Fig. 7. The specimen was placed at a predetermined orientation, the same as that in the impact test. During the tests, the specimen, together with the gripper and shock table, was released from a certain height. Just before the table hit the anvil at the bottom, the gripper was opened and the specimen was free to hit with the shock table. This was done to implement the test method suggested by Goyal [9].

Dynamic strain meter Accelerometer Laser sensor

Oscilloscope Pulse shaper

Oscilloscope

Figure 5 Schematic illustration of shock tester.

166 Shock Test Methods and Test Standards for Portable Electronic Devices

Figure 6

Fixation of shock table test condition 1.

Figure 7

Fixation of shock table test condition 2.

For shock test condition 3, the specimen was glued at several corners. The glue was used only in such small amounts so as just to sustain the orientation of specimen, as shown in Fig. 8. Therefore, during the tests, the glue would lose effect and the specimen was free to rotate. This design was to ensure that the shock pulse sustained at the connecting corner is similar to the impact pulse during impact tests. And the specimen had only slight constraint on free movements as in a real drop impact.

4 Comparison of Different Test Methods

Figure 8

167

Fixation of shock table test condition 3.

4.3 Comparison of Drop Test and Shock Test Results From the drop test, the impact pulse was measured by the strain gauge (CH7) on the PVC bar, and the average peak acceleration was 1.31 kg and average impact duration 1.0 msec. Figure 9 plots the acceleration pulse at the centroid of the sample resulting from the impact force divided by the total mass.

0.25

Acceleration for drop tests

Acceleration (1000g)

0.20

0.15

0.10

0.05

0.00

−0.05 −1.0

−0.5

0.0

0.5

1.0

1.5

Time (ms)

Figure 9 Average impact acceleration for drop tests.

2.0

168 Shock Test Methods and Test Standards for Portable Electronic Devices 0.35

Acceleration for shock table test

0.30

Acceleration (1000g)

0.25 0.20 0.15 0.10 0.05 0.00 −0.05 −1.0

−0.5

0.0

0.5

1.0

1.5

2.0

Time (ms)

Figure 10 Impact acceleration for shock table tests.

To make a comparison, we adapted a pulse shaper to tune the impact duration of the shock table to be similar to those in the drop tests, as shown in Fig. 10. It should be noted that the acceleration we obtained from the drop test was the force divided by the overall mass of the sample. Therefore this acceleration was the acceleration of the centroid of the sample. However, the acceleration of the shock test was that of the shock table, which is different from real acceleration of the sample. If we assume that the bottom of the sample does not detach from the shock table during the impact, it is reasonable to guess that the shock duration was the same. During the tests, the response of group 1 was found to be totally different from the drop test results. Figures 11a–d show the comparison of the response on a PCB for a drop test and shock test under conditions 2 and 3. The results indicate that the response of the shock test under condition 2 is quite different from that of the drop test. From the obtained curve, however, it is clear that the shock table test under condition 3 is very similar to that of the drop result despite the approximately 20% discrepancy in the peak value. Therefore, the shock table test can possibly mimic the situation of real drop tests if a proper test condition is chosen. However, it should be emphasized that impact duration may not be chosen as a parameter to mimic the real drop test. This will be discussed in detail in the next section. Further investigation reveals that in test condition 2 the gap between the specimen and the shock table was the cause of the discrepancy. For example, if a gap of only 0.5 mm exists between the bottom of the specimen and the table, a lag in time t = 0.3 mm/(2 m/ sec) = 0.15 msec would occur. That means, after the shock table has made contact with the pulse shaper and decelerated for 0.15 msec, the specimen hit the table. To observe the gap, we placed a PVDF piezoelectric film between the specimen and the shock table. Figure 12 shows the shock table acceleration and the contact force record by the PVDF. It is evident that the contact force pulse had 0.15-msec lag behind the acceleration pulse. But the gap was not easy to eliminate since there always exited some small perturbation during the dropping process.

4 Comparison of Different Test Methods 1.0

v = 2.0 m/s Drop test Shock test condition 2 Shock test conditon 3

0.8

Ch1 ministrain

0.6 0.4 0.2 0.0 −0.2 −0.4 −0.6 −2

0

2

4

6

Time (ms) (a) v = 2.0 m/s Drop test Shock test condition 2 Shock test conditon 3

0.5 0.4 0.3

Ch2 ministrain

0.2 0.1 0.0 −0.1 −0.2 −0.3 −0.4 −0.5 −0.6

−2

0

2

4

6

Time (ms) (b)

Figure 11 Comparison of PCB response for shock table tests and drop tests.

169

170 Shock Test Methods and Test Standards for Portable Electronic Devices 0.8

v = 2.0 m/s Drop test Shock test condition 2 Shock test conditon 3

0.7 0.6 0.5

Ch3 ministrain

0.4 0.3 0.2 0.1 0.0 −0.1 −0.2 −0.3 −0.4 −0.5 −0.6

−2

0

2

4

6

8

Time (ms) (c) v = 2.0 m/s Drop test Shock test condition 2 Shock test conditon 3

0.4

Ch4 ministrain

0.2

0.0

−0.2

−0.4 −2

0

2

4 Time (ms) (d)

Figure 11 (Continued )

6

8

5 Discussion

171

0.8 Acceleration of shock table Contact force between specimen and table

0.7

Acceleration (1000g)

0.6 0.5

0.15ms

0.4 0.3 0.2 0.1 0.0 −0.1 −0.2 −0.3 −1.0

−0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Time (ms)

Figure 12

5

Acceleration of shock table and contact force.

DISCUSSION The process of shock transmission can be illustrated as in Fig. 13. According to the theoretical model presented in Chapter 6, the internal dynamic response of the PCB was determined mainly by the acceleration of its connecting points to the outer case. That acceleration was composed by both translational acceleration vertical to the contact face and the relative rotational acceleration of the sample to the contact point. Therefore, it is easy to understand why the response of condition 1, in which the sample was fully constrained, was totally different from real drop tests. The contact conditions of the sample can be regarded as a Hertz contact spring, so the input force from the table to the sample is mainly determined by the shock acceleration and the Hertz contact spring. Although the bottom of the sample has the same acceleration as the shock table, the real acceleration of the sample’s centroid during impact is different, as there is relative acceleration of the sample’s centroid with respect to the bottom of the sample itself. That relative acceleration contains both the relative vertical acceleration and the rotational acceleration.

Hertz contact spring

Figure 13

Illustration of drop impact of typical device.

172 Shock Test Methods and Test Standards for Portable Electronic Devices Duration 0.24 ms Duration 0.38 ms Duration 0.48 ms Duration 1.8 ms

0.8

Ch1 ministrain

0.6

0.4

0.2

0.0 −0.2 −0.4 0.000

Figure 14

0.002 Time (s)

0.004

0.006

Comparison of PCB response under different shock duration.

From the theory of the mass–spring model proposed by Newton [11], when the shock pulse is shorter than one-sixth of the natural period of the sample, the response of the sample is similar to that under the condition in which the sample is dropped onto a hard surface. Here, the natural period of the sample can be considered as the drop impact duration, which is determined by the Hertz contact conditions and the mass of the sample. Therefore, when the shock duration is shorter than one-sixth of the basic period of the sample, the shock test may mimic the harsh impact conditions of a real drop test. Figure 14 shows the PCB response to different shock durations. It is seen from the results that the shock duration may affect the first peak of the response, and the shorter the duration, the higher the strains on the PCB. It also shows that the peak strain would approach certain values when the duration decreases to a very short one. This result verifies the conclusion made by Suhir [11]: When the duration of the shock pulse is sufficiently short, the shock table test may mimic adequately the severe shock conditions of drop impact.

6

CONCLUSIONS Shock table tests with different constraint conditions were designed to mimic real drop tests. The results reveal that the conventional fully constrained shock table test cannot mimic the real-life drop impact condition. The shock table test method suggested by Goyal [9] would be feasible provided the possible gap between the sample and the table is eliminated. The test results with different shock durations have verified the previous estimation made by Suhir [11] on the basis of analytical (“mathematical”) modeling of the drop test and the shock test situations. Therefore, to fully mimic the real drop test conditions, an appropriate shock table test method should allow the sample to rotate freely and the shock duration should be much shorter than the contact period in the real drop.

References

173

Acknowledgment The results reported in this chapter are part of a research project CERG grant 6189/03E. Financial support from the Hong Kong Research Grant Council is gratefully acknowledged.

REFERENCES 1. MIL-STD-810F, “Military Standard: Test Method Standard for Environmental Engineering Consideration and Laboratory Tests,” U.S. Department of Defense, Washington, DC, 2000. 2. MIL-STD-883F, “Military Standard: Test Methods and Procedures for Microelectronics,” U.S. Office of Naval Publications, Washington, DC, 2004. 3. ANSI/ASTM D3332-93, “Standard Test Methods for Mechanical-Shock Fragility of Products, Using Shock Machines,” American Society for Testing and Materials, Philadelplua, PA 1996. 4. IEC 68-2-27, “International Standard: Basic Environmental Testing Procedures,” International Electrotechnical Commission, Geneva, Switzerland, 1987. 5. JESD22-B104-B, “Mechanical Shock Test Method,” JEDEC Solid State Technology Association, Arlington, VA, 2001. 6. JESD22-B110, “Subassembly Mechanical Shock Test Method,” JEDEC Solid State Technology Association, Arlington, VA, 2001. 7. JESD22-B111, “Board Level Drop Test Method of Components for Handheld Electronic Products,” JEDEC Solid State Technology Association, Arlington, VA, July 2003. 8. Newton, R. E., Fragility Assessment Theory and Test Procedure, Monterey Research Laboratory, Monterey, CA, 1968. 9. Goyal, S., and Buratynski, E. K., “Methods for Realistic Drop-Testing,” International Journal of Microcircuits and Electronics Packaging (IJMEP), Vol. 23, No. 1, pp. 45–52, 2000. 10. Ong, Y. C., Shim, V. P. W., Chai, T. C., and Lim, C. T., “Comparison of Mechanical Response of PCBs Subjected to Product-Level and Board-Level Drop Impact Tests,” Proc. IEEE 5th Electron. Packaging Technol., Singapore, 10–12 December 2003, pp. 223–227. 11. Suhir, E., “Could Shock Tests Adequately Mimic Drop Test Conditions?” Proc. IEEE 52nd Electron. Components Technol., San Diego, CA, May 28–31, 2002, pp. 563–573. 12. Newton, R. E., Theory of Shock Isolation, Shock & Vibration Handbook , McGraw-Hill Book Company, New York, 2002, Chapter 31.

CHAPTER

8

DYNAMIC RESPONSE OF SOLDER JOINT INTERCONNECTIONS TO VIBRATION AND SHOCK David S. Steinberg Steinbergelectronics, Inc. Westlake Village, California

1

SOLDERS MUST BE CAREFULLY ATTACHED TO STRUCTURAL MEMBERS TO AVOID FAILURES Electronic components are often used with a large variety of different types and sizes of solder joint for through-hole mounting and surface-mounting components. Printed circuit boards (PCBs) are used extensively for many different commercial and military applications that require light weight, low cost, and reliable hardware. Some types of equipment must be capable of operation in severe sine vibration, random vibration, and shock conditions. Some systems must require easy and rapid repairs in the field. In all cases the consumers always want a low-cost, reliable, easy-to-use piece of hardware. Luck appears to play a part in the quality of a product, which may be due to poor quality control in manufacturing and assembly areas. Most electrical failures appear to be mechanical in nature. Many of these failures tend to occur in component lead wires and solder joints. Extensive military tests over a period of several years have shown that about 80% of the electromechanical failures are due to some type of thermal conduction and about 20% of the failures are due to some type of vibration and shock.

2

VIBRATION PROBLEMS WITH ELECTRONIC COMPONENTS MOUNTED ON PCBs Electronic components are usually mounted on plug-in-type PCBs with the use of solder to secure the components. Many different types of solders are available, such as dip solder, wave solder, vapor phase, oven, and hand, just to name a few. Most solders are eutectic type, which is a mix of 63% tin and 37% lead with a melting temperature of about 183◦ C. A wave solder machine for through-hole mounts uses a solder temperature of about 230◦ C. This provides good wicking action up through the plated through holes for the lead wires. Semiconductor electronic components are typically rated below a temperature of 100◦ C. Some types of components are only rated at 85◦ C ambient. Caution must be used here when they are soldered to PCBs because the solder heat can be conducted up the lead wires directly into sensitive silicon semiconductor chips, which can cause them to malfunction. This problem can often be cured by placing the component above the PCB with the use of longer wires. The longer lead wires have a higher thermal resistance, so the hot-spot temperatures are reduced. Care must be used here to avoid making the lead wires too long, which can cause a decrease in the natural frequency of the wires. This could increase the wire-bending displacement during vibration. This may increase the bending stresses in the wires, which can reduce the fatigue life in the wires. Structural Dynamics of Electronic and Photonic Systems Copyright © 2011 John Wiley & Sons, Inc.

Edited by Ephraim Suhir, David S. Steinberg and T. X. Yu

175

176 Dynamic Response of Solder Joint Interconnections to Vibration and Shock 3

PROBLEMS WITH LEAD-FREE SOLDERS Lead is a soft commercial material that is used in many industrial applications. The soft lead is used in many manufacturing processes. This includes paint, because it is very easy to apply. Lead has caused many deaths, especially in young children who tend to eat the soft lead that peels off the walls in old homes. Lead is a poison that is slow acting and makes children, sick before parents notice the change. This condition is very common in poor families. The federal government is trying to reduce the use of lead in many commercial programs, including military programs. This has turned out to be more difficult than expected. in military programs. The problem with lead-free solders is the growth of tin whiskers. These are very small and very thin metallic hairlike growths that emerge from the surface of the tin (Sn). On lead-free tin surfaces, tin whisker can grow quite long. The growth length is sufficient to make contact with other electronic circuits, causing short circuits and component failures. Some scientists believe that whisker growth in ordinary bright tin finishes is due to various impurities that combine with the tin during the purifying process in the plating solutions. This produces a matte finish that covers the bright original finish. This matte finish will reduce whisker growth and increase the operating life of the solder a bit, but the extended life will not ensure the long-term reliability requirements. The U.S. government has issued the Restriction of Hazardous Substances (ROHS) in July 2006 to reduce the use of lead in electronic products because of environmental concerns. This was to cover component lead plating as well as solder and solder pastes. The original thoughts were that it would not be difficult to change from 63% tin and 37% lead (called 63/37). As a result of the ROHS program, component manufacturers began to supply parts with lead-free plating made from ordinary tin so the old 63/37 solder coated products were no longer available. Replacing tin–lead with pure tin has turned out to be a problem. The lead-free assembly is worse for the environment, according to the U.S. Environment Protection Agency (EPA). The lead-free assembly is less reliable than the lead-based assembly. This is why exceptions were made that allow military programs and high-reliablility applications to continue using the 63/37 solders. Many people still believe that the lead-free solder is safe to use because the whiskers only grow in a vacuum environment. This was shown many years ago to be false. Some people believe that tin whiskers only grow in temperature cycling conditions. This was also shown to be false. Conformal coatings are used extensively with no problem so they should be very safe for lead-free applications. Again this was proven to be false. No inexpensive coatings have been found, but many companies are still running tests on various materials searching for a solution. A mixture of tin, silver, and copper has been used with some success for short periods, but it always fails during long-term use. Military and high-reliability programs still prefer to use the 63/37 solders.

4

METHODS FOR PREDICTING FATIGUE LIFE OF SOLDER JOINTS Printed circuit boards are used extensively to hold electronic components on a wide variety of equipment that must operate in severe environments with high reliability. In many cases the equipment must also meet the required limits for size, weight, and cost. Many of these systems will be expected to operate in areas that will be exposed to rapidly changing temperatures, which will produce rapid changes in the stresses associated with electrical lead wires, solder joints, cables, and harnesseses. Differences in the thermal coefficient of expansion (TCE) between the PCBs and the components can produce deflection during the thermal cycling

5

How Poor Manufacturing and Poor Assembly Methods Might Affect Reliability

177

conditions. These deflection differences may result in high stresses in the component body, the lead wires, the solder joints, and possible cables and harnesses. These factors depend on the geometry of the different types of leaded or leadless components that can be surface mounted or through-hole mounted on the PCB, the strain relief in the cables and harnesses, as well as the thermal cycling conditions. Solder can cause a lot of problems during thermal cycling conditions. This is due to its elastoplastic and creep properties at temperatures near 100◦ C. Displace a cantilevered bar of solder rapidly through some amplitude A at 100◦ C and assume it generates an initial stress of about 1000 lb/in.2 Hold that amplitude A for a while. The solder will start to creep and relieve the stress. In a period of about 2 hr the solder stress will be close to zero. Now bring the bar of solder rapidly back to its original starting position through the same amplitude A. The solder stress will now be 2000 lb/in.2 The creepand stress-relieving properties of solder in this case resulted in the doubling of the solder stress. When solder is displaced and held in the displaced condition for long periods at high temperatures, the creep effect can relax the stresses down to a near-zero condition. When the solder is then displaced in the opposite direction through the same amplitude, the solder stress can be doubled. Higher stress levels will shorten the effective fatigue life of the solder. Solder has a short-term ultimate strength of about 6500 lb/in.2 at room temperature. Extensive testing and analysis experience with solder creep have shown that it is good practice to keep solder joint stress down to a value of about 400 lb/in.2 to ensure a long failure-free operating life in thermal cycling conditions. The standard method for evaluating the fatigue life of a structure is to examine the singleamplitude stress compared to the number of cycles required to produce a failure. In a typical sine wave the single amplitude refers to the displacement from zero to the peak. This is the same as half of the double amplitude, which is half of the peak-to-peak amplitude. The same philosophy is applied to thermal cycling. The single-amplitude stress will then be related to the single-amplitude temperature cycle. This is half of the double amplitude or half of the peak-to-peak temperature cycle. These analysis methods are based on linear structures. But solder is not a linear material; it can creep especially at high temperatures above about 100◦ C. Therefore, the methods shown above are intended to be more accurate for evaluating the approximate fatigue life of solder when some creep is included. There are many arguments related to the analysis methods for evaluating solder joint strains and stresses. Should the peak-to-peak temperature range be used in the analysis of a thermal cycling event or should half the peak-to-peak temperature cycle range be used for the analysis? In a rapid temperature cycle event, where the solder does not have a chance to creep, the solder should be treated like a linear material. In this case half of the peak-to-peak temperature cycle range should be used. In a condition where there is extensive creep in a thermal cycling event, the peak-to-peak temperature range should be used for the analysis. When there is no information available as to the use of the electronic equipment, or when a conservative design is desired, the peak-to-peak temperature cycle range should be considered, When a simple conservative analysis is desired, a safety factor should be considered. by using the double amplitude or the peak-to-peak temperature cycling range. This is equivalent to a safety factor of 2 on a rapid temperature cycle, where little or no creep will be involved.

5

HOW POOR MANUFACTURING AND POOR ASSEMBLY METHODS MIGHT AFFECT RELIABILITY An examination of most electronic failures shows they are often mechanical in nature. Some typical hard failures often include cracked solder joints, broken wires, cracked circuit traces, cracked plated through holes, broken connector pins, broken screws, cracked components, and solder creep, just to name a few. Condensation and high humidity without proper protection

178 Dynamic Response of Solder Joint Interconnections to Vibration and Shock have also been known to cause failures. Radio frequency interference (RFI), electromagnetic interference (EMI), and chattering relays are usually included as soft failures because they will often work after the disturbance is removed. Some of these failures are due to poor design practices and some are due to poor manufacturing. It is often difficult to determine which is the major fault. Manufacturing costs are often a major factor in the success or failure of a project and often a major factor in the success of a company. One big cost driver is tolerance control. Loose tolerances are desirable since they can reduce costs, but they can also reduce the fatigue life of the product. This usually ends up increasing the design cost since the design function is a one-time cost, but the manufacturing is a recurring cost.

6

TYPICAL TOLERANCES IN ELECTRONIC COMPONENTS AND EFFECTS ON VIBRATION AND THERMAL CYCLING AND FATIGUE LIFE Dimensional tolerances must be controlled very carefully to ensure the reliability of sensitive electronic equipment. The most critical tolerance variations are often the greatest in the electronic components themselves. An examination of a typical dual inline package (DIP) shows there are large variations in the dimensions of the component body and the electrical lead wires. Large variations in these areas mean there will be large variations in the fatigue life of these components during operation in severe shock, vibration, or thermal cycling conditions. Variations in the dimensions of just the lead wires can easily change the fatigue life of many components by a factor of about 100 to 1. Many different companies fabricate components with the same general form, fit, and function. Every component manufacturer has different sets of tools and dies which are used to produce a particular component. It becomes extremely expensive for different component manufacturers to meet periodically to try to establish close dimensional tolerances on a wide variety of electronic components. Pity the purchasing manager in a large company who must purchase a wide variety of components at a reasonable cost. Her or she may get a request from the mechanical design engineering department to purchase a particular component, with a particular geometry, from a particular company. The purchasing department says NO, because they do not want to use a single source. What happens if the single source has a fire, or if that company goes on a strike, or there is a flood and that plant closes for 6 months? The purchasing department can go broke waiting to get delivery for the particular component. The mechanical design group must make the component design more fault tolerant, so the component purchase can be made from one of the available companies. A design that is more fault tolerant is a design that has a higher safety factor that can compensate for the higher stresses that may be produced by the operating environment. Some large companies with several different divisions often take advantage of companies that require large purchases of special hardware by offering reduced prices to get the contract. Another area where large dimensional tolerances can cause reliability problems is in the PCBs themselves. The most critical tolerance is usually the thickness, which has a normal value of about 0.062 in., with a tolerance of about ±0.0070 inc. for multi layers. Extreme care has to be used when the PCBs are being fabricated, to make sure that the proper materials are being used. There have been many cases where very thin structural members were used, well below the minimum allowed thickness. Many PCBs failed during the vibration and shock environments. If the PCB normal thickness is reduced by about 40%, these tolerances can reduce the natural frequency by about 40%. This can sharply increase PCB displacements, which will sharply increase dynamic stress levels, which sharply reduces the fatigue life of the PCBs. This is a large variation that can lead to rapid PCB failures if the electronic equipment is exposed to any severe or extended vibration or shock

7

7

Problems Associated with Thermal Cycling on Solder Joints and Lead Wires

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PROBLEMS ASSOCIATED WITH THERMAL CYCLING ON SOLDER JOINTS AND LEAD WIRES Differences in the TCE of various materials used in electronic assemblies can result in very high stresses and strains in critical mechanical structural elements during thermal cycling events. Relative displacement differences of only 0.00050 in. between the PCB and components soldered to the PCB have been known to crack solder joints and lead wires. Most electrical lead wires will only fail when they are loaded in tension in thermal cycling events. Wires loaded in direct tension will fail when the ultimate tensile stress is exceeded. Lead wires loaded in bending during thermal cycling very seldom break. Even when the bending stresses in the wires exceed the ultimate strength and the wires experience plastic bending, testing experience shows the wires do not break. There are very seldom enough thermal cycles accumulated in these events to cause a fatigue failure unless there is a sharp and deep cut in the wire at a high stress point. This can be demonstrated by bending a paper clip back and forth through a large displacement several times. The paper clip will experience permanent deformation since the bending is in the plastic range, but the paper clip does not break, because there are not enough stress cycles accumulated to produce a fatigue failure. It may take many dozen stress cycles to produce a fatigue failure with large displacements. It may take tens of thousands and perhaps millions of stress cycles to produce a fatigue failure with very small displacements. Most electronic systems will never develop that many thermal stress cycles in their lifetime. A computer that is turned on twice a day every day for 15 years will accumulate about 11,000 thermal fatigue cycles. A television set that is turned on 10 times a day every day for 15 years will accumulate about 55,000 thermal fatigue cycles. An automobile that is started 10 times a day everyday for 20 years will accumulate 73,000 thermal stress cycles. A satellite in orbit around the earth experiences a thermal cycle about every 90 min. In 20 years it can accumulate about 117,000 thermal cycles. Electronic systems for aircraft are often designed to provide an operating life of about 10,000 hr. In this period, a PCB with a natural frequency of 100 Hz will accumulate about 3.6 billion vibration fatigue cycles. The very large number of stress cycles associated with vibration means that stress risers in the form of small holes and sharp notches will be more sensitive in vibration than they will be in thermal cycling. Transformers can carry high electrical current so their electrical lead wires are usually made of copper, with a large diameter to reduce the electrical resistance. When a transformer is mounted on a PCB, differences in the TCE between the body of the transformer and the PCB will force the wires to bend. Thermal cycling test data over a temperature range from –55 to + 95◦ C shows that failures will occur in about 12 cycles. Most engineers (who are not familiar with thermal cycling failures in electronic equipment) believe that failures will occur in lead wires or solder joints and are surprised to hear these are not the failure points in the system. They are more surprised when they find the failures are in the copper circuit traces on the PCB that are used as solder pads for making electrical connections to the PCB. The bending action of a very stiff wire can procuce an overturning moment that will lift the solder pad off of the PCB. The solder pad is held down on the PCB only by an adhesive which is not strong enough to resist the overturning moment action on the stiff lead wire. One way to reduce the expansion forces generated in thermal cycle events is to reduce the stiffness in the wire. The stiffness of the wire can be reduced by looping the wire to make it longer, or the wire can be coined to make it flat to reduce the moment of inertia, or the wire pad can be reinforced to solve the problem. Copper-plated through holes are often added to the solder pads for reinforcement to prevent them from lifting off of the PCB during thermal cycling events. Test data show the fatigue life of the assembly with plated through holes will now be inceased to about 150 thermal cycles. There are only two places where the failures can now occur, in the lead wire or the

180 Dynamic Response of Solder Joint Interconnections to Vibration and Shock solder joint. Engineers not familiar with thermal cycling tests are surprisd to find that bending failures almost never occur in the lead wires. When the calculated bending stresses in the wires are well above their ultimate tensile strength, they bend in the plastic bending range. They very seldom break as long as there are no sharp and deep cuts in the wires to act as a severe stress concentration.

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VIBRATION AND THERMAL CYCLING ENVIRONMENTS ON LEAD WIRES AND SOLDER JOINTS Vibration environments can often involve millions of stress cycles because natural frequencies in electronics can range from 50 Hz to well over 1000 Hz. Stress risers and stress concentrations can be very severe when millions of cycles are accumulated. Electrial lead wires loaded in tension or bending will fail very often in vibration events because a large number of stress cycles can be accumulated in a short period of time. Stress concentrations in the solder joints do not appear to have much effect on the fatigue life. The relatively soft solder tends to strain relieve the high local stresses associated with stress concentrations by plastically deforming in these areas. Testing has shown that thermal cycling events typically produce many more solder joint failures than lead wire failures for surface-mounted and through-hole mounted PCB componnents. Vibration events tend to produce many more electrical lead wire failures than solder joint failures in surface-mounted and through-hole-mounted PCBs. When a solder joint failure shows up during vibration, most of the time the crack initiation was due to thermal cycling. Since thermal cycling is slow, the crack propagation is also very slow. However, vibration can often develop several hundred stress cycles in a second so the crack propagation is very rapid in vibration. The solder joint failure then appears to be due to vibration. An engineer with very little experience will immediately begin to investigate vibration-related stresses in a failed solder joint. An experienced engineer will first look for thermal-cycling-related stresses in the failed solder joint.

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COMPARING RELIABILITY PERSPECTIVE MEAN TIME BETWEEN FAILURES (MTBF) WITH FAILURE-FREE OPERATING PERIOD (FFOP) The term reliability often means different things to different people. In electronics it refers to the failure, and sometimes to the failure rate, of some type of electronic equipment. The dictionary defines reliable as dependable and trustworthy. A reliable electronic system would then be a dependable and trustworthy system. A system with these properties would be expcted to provide trouble-free service for a long time. It is probably safe to say that a long time in a military program would be many years of trouble-free service. Can an electronic system for a military program be defined as reliable when there are many failures in the system in a relatively short period of time? The answer has to be NO. Many people’s lives are at risk in any military attack. Yet this is the way that almost every military and most commercial organizations have set up their training programs when they use the MTBF (Mean Time Between Failures) schedules. The MTBF method typically uses the MIL-HDBK-217 computer program to obtain the failure rates of different equipment. This makes the calculations for the MTBF very simple, like working from a cook book. Very little original thinking goes into the analysis so there is no need to have a technical degree to perform the calculations. Engineers with a college degree who have worked with the MTBF method of analysis for several years often find they can no longer remember how to perform a detailed technical analysis they learned in college. They can only work with the MIL-HDBK-217 programs. The problem is that the MIL-HDBK-217 program is not very accurate. It cannot be used to obtain a detailed

9 Comparing Reliability Perspective MTBF with FFOP 181 Table 1

Environmental Failures Outlined in MIL-A-87244

A. Fatigue failures Due To vibration, shock, and thermal cycling Electrical lead wires Solder joints Water vapor Plated through holes Component case Electrical drift B. High operating temperatures C. Aging effects Corrosion Water vapor Embrittlement Electrical drift

description of a complex structure to find the natural frequency in critical areas, and it cannot find the expected dynamic displacements or the stresses or the expected fatigue life. The MTBF is not an accurate measure of when or where a failure may occur. There appears to be something wrong with the probabilistic MTBF approach for military programs where the lives of many soldiers and civilians may depend on the reliability of their electronic equipment. There should be no failures during the operating life of the system. When a button is pushed in an airplane to fire a missile at an enemy and there is a malfunction, the battle can be lost along with many lives. A more reliable method, FFOP (failure-free operating period), is also more accurate and is being used with the new Lockheed F-22 airplane. This was developed at Wright Patterson Air Force Base, Dayton, Ohio, by John Halpin, Amar Bungalia, and Dave Steinberg. This is also the goal of a new Avionics Integrity Program (AVIP) aimed at dramatically improving the reliability of the electronic equipment through better understanding of the various failure mechanisms. The Air Force has outlined its avionics integrity program specifications in MIL-A-87244. This document describes three environments that are known to produce many electronic failures, as shown in Table 1. The AVIP function uses Miner’s cumulative damage ratio to add the damage from many different environments, including thermal, thermal cycling, vibration, shock, and acoustic noise. A scatter factor (or safety factor) of 2 was used for the fatigue life evaluation. The electronic systems were evaluated for an FFOP of 20,000 hr since the operating life of the airplane was expected to be 10,000 hr. The fatigue life evaluation of the component lead wires and solder joints included the effects of manufacturing tolerances associated with the dimensional variations in the component lead wires and circuit boards for the environments outlined above. Preliminary reports for the reliability of the F-22 electronic system have shown outstanding results. The AVIP fatigue life evaluation requires special training in the art and science of analysis methods using computers and hand calculations to obtain stresses and strains in the various load-carrying elements of the electronic system. The steady-state and transient properties of the materials used in the electronic equipment must be well documented. This requires more work than the MTBF method so the AVIP method will be slightly more expensive. Most people believe the small increase in cost is worth the big increase in reliability.

CHAPTER

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TEST EQUIPMENT, TEST METHODS, TEST FIXTURES, AND TEST SENSORS FOR EVALUATING ELECTRONIC EQUIPMENT David S. Steinberg Steinbergelectronics, Inc. Westlake Village, California

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COMPONENT FAILURES PRODUCED BY THERMAL CYCLING AND VIBRATION CYCLING Any time a structure experiences stress reversal, part of its life is used up. The two most common types of stress cycles that cause the most damage in electronic equipment are thermal/temperature cycling and vibration cycling. These two types of cycling can occur separately or they can be combined. Alternating stress cycles can be generated with a zero mean stress or it can also be superimposed on a steady stress. The stresses may be thermal or vibration and sometimes shock. In electronic and photonic systems various stresses are often generated together, but they can be generated separately as well. Most of the time the steady-state stress is thermal and the alternating stress is vibration. Predicting the fatigue life of electronic equipment exposed to any combination of an alternating stress superimposed on a steady stress is very difficult, even with the use of a computer program. The fatigue life analysis made with any computer evaluation, no matter how sophisticated, should not be trusted unless there are test data from well-instrumented prototype test models to prove the accuracy of the data. Thermal cycling and vibration fatigue damage test data on printed circuit board (PCB) through-hole pin grid arrays (PGAs) showed no damage when the vibration tests were performed at 95 and 25◦ C. However, when the tests were performed at −55◦ C, many of the PGA wires fractured. The tests were always run with new assemblies to avoid any problems with possible fatigue damage. Some of the conclusions from these tests were shown as follows. Solder creep at elevated temperatures, and even at room temperature, allows the PGA wires to relax. This reduces the magnitude of the bending moments and bending stresses on the PGA wires. This increases the fatigue life of the wires. Solder creep at low temperatures is sharply reduced, so high thermal stresses are locked in the PGA lead wires. When the vibration is imposed at low temperatures, the PGA wires experience an alternating vibration stress superimposed on the sustained thermal stress. This increases the magnitude of the maximum stress acting on the PGA wires, which reduces the fatigue life of the wires. Slow thermal cycling events allow the solder to creep and to relax stresses in a manner that results in higher solder joint stresses and strains than rapid thermal cycle events. Slow thermal cycling therefore will result in more solder joint failures than rapid thermal cycling over the same temperature range. The slower solder is cycled the weaker it gets because of the creep and stress relaxation properties of solder. Solder creep at higher temperatures presents a problem in trying to establish a laboratory thermal cycling test program that produces the same type of solder joint failures that are Structural Dynamics of Electronic and Photonic Systems Copyright © 2011 John Wiley & Sons, Inc.

Edited by Ephraim Suhir, David S. Steinberg and T. X. Yu

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184 Test Equipment, Test Methods, Test Fixtures, and Test Sensors experienced in the actual operating environment. Time is money. To reduce costs, accelerated laboratory life tests are usually run for short periods to try to generate the same amount of damage that is generated for a much longer period of actual operating environment. This is called accelerated life testing. The normal practice is to increase the temperature-cycling range along with a decreased temperature-cycling time. A decreased temperature-cycling time means a more rapid thermal cycle. The problem here is that the slower solder is cycled the weaker it gets, so a more rapid thermal cycle will be a much longer fatigue life. Therefore the data gathered in the accelerated thermal cycling fatigue life test must be examined very carefully to try to get better correlation with the fatigue failures produced in the actual operating conditions. Solder tends to become more plastic with more creep as the temperature increases. At very low temperatures below 0◦ C, solder is quite rigid with very little creep and stress relaxation. The approximate time it takes 63/37 tin lead solder to creep and relax at temperatures of 125◦ C from stress levels of 3000 psi to about 1000 psi is a period of about 2 min. Vibration with a very rapid cycle does not appear to have any creep effects on the solder. If electrical lead wires are made of copper, Kovar, or nickel, their mechanical properties are not affected by the normal high electronic temperature exposure. There appears to be little creep in the lead wires at the same elevated temperatures. Very few thermal cycling or vibration-related failures occur in the PCBs themselves. Epoxy fiberglass and polyimide glass PCBs have expansions that are very similar to the copper traces in the x –y plane, so the thermal cycling stress levels in the copper circuit traces are typically very low. Vibration can cause large dynamic displacements in the PCBs when their natural frequencies are excited, but this is very seldom a problem. Tests have shown that almost all of the electrical lead wires on the electronic component parts will fail long before there are any vibration fatigue failures in the circuit board itself. Copper in the PCB-plated through holes can fail in thermal cycling events when there is not enough copper to carry the thermal expansion forces produced in the z axis normal to the plane of the PCB. Vibration does not seem to have any effect on plated through holes. Extra care must be used to strengthen the plated through holes when materials such as copper–invar–copper are used to restrain thermal expansions in the x –y plane, when leadless ceramic chip carriers are surface mounted on the PCBs. The copper–invar–copper reduces the thermal expansions in the x –y plane, which substantially reduces the shear stresses developed in the solder joints on the leadless chip devices. But when the x –y expansions are reduced, Poisson’s ratio comes into effect and the z-axis expansions are increased. This increases the stresses in the copper-plated through holes. The copper can be made slightly thicker, which will make it stronger, to prevent plated through-hole failures.

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HOW MANUFACTURING METHODS AND MATERIAL PROPERTIES CAN PRODUCE FAILURES IN ELECTRONIC EQUIPMENT An examination of most electronic equipment failures shows they are really mechanical in nature. Typical hard failures include cracked solder joints, broken wires, cracked circuit traces, cracked plated though holes, broken connector pins, broken screws, cracked components, cracked hermetic seals, electrical short circuits, and cracked silicon chips, just to mention a few. Electromagnetic pulse (EMP), molecular migration, and solder creep may be added to the list. High humidity and condensation may be borderline. Radio frequency interference (RFI) and electromagnetic interference (EMI), noise, and relay chattering are considered to be soft failures since the system will still work after the disturbance is removed. Some of these failures are due to poor design practices and some due to poor manufacturing. It is often quite difficult to determine which is the major report. Manufacturing and material costs are often a

3 Viscoelastic Damping Materials Cause Problems If They Are Not Used Carefully

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major factor in the success or failure of a project and often a major factor in the success or failure of the company itself. Tight tolerance controls may improve the quality of a company project, but it may come at the expense of high costs and poor sales. These factors must be examined and controlled if the company is to survive. Damping is often used to reduce the forces and stresses in electronic equipment due to sine vibration, random vibration, and shock. Damping is usually defined as the conversion of kinetic energy into heat. Material damping considers the energy lost due to internal friction or hysterics in the molecular structure of the material and structural damping relates to the energy lost due to vibration in rubbing, scraping, slapping, and impacting at various interfaces and joints. The total system damping is the sum of the material damping and the structural damping. All real systems will produce some damping when they are vibrated. When a system with light damping is disturbed, it will continue to oscillate for a long time after the disturbing force has been removed. A system with heavy damping may oscillate once or twice after the disturbing force has been removed. The damping in all systems will always bring the system back to the state of rest sometime after the disturbing force has been removed. Damping removes some of the energy within the system so there is less energy available to distort and damage the system. This means there is less energy available to deform the structure. A reduction of the kinetic energy reduces the force and stress in PCBs, which increases the fatigue life, so damping increases PCB reliability. Increasing the damping in the PCB increases the amount of energy that is lost, which decreases the transmissibility and further increases the fatigue life and reliability of the PCB. Internal energy is lost, or converted into heat, anytime a structure is deformed. Applying a tensile load to a bar does not instantly deform the bar. There is a small time lag as the bar elongates to its new stabilized length. The entire process is reversed when the applied load is removed. However, the bar never returns to its exact original length, since some of the strain energy has been converted into heat so less energy is available to restore the original bar length. Applying an equal compressive load on the bar reverses the direction of the forces that were described above. The positive and negative strains can be plotted on a graph to show the typical hysteresis loop formed by materials that are deformed. The area in the enclosed hysteresis loop becomes a measure of the energy lost with each stress cycle. A larger area enclosed in the loop shows there is more damping. It was previously shown that more damping reduces PCB transmissibility, which increases the fatigue life and the reliability.

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VISCOELASTIC DAMPING MATERIALS CAUSE PROBLEMS IF THEY ARE NOT USED CAREFULLY Electronic equipment is used extensively in many different areas and in many different types of equipment that can vary from cell phones and computers to tooth brushes and airplanes to automobiles and space programs, manufacturing, health care, food preparation, and many others. Many of these applications will be exposed to vibration, shock, heat, acids, and liquids, to name a few environments. The industries are always looking for ways to improve reliability; reduce size, weight, and cost; and improve testing methods. Many of these applications make use of viscoelastic damping materials with constrained layers to increase the damping for improved performance. When the damping is increased, PCB reliability and life are also increased. More damping in a PCB is desirable if it does not increase the size, weight, and cost of viscoelastic materials that are very similar to rubber materials. They are very flexible and can deform, stretch, and compress over large displacements without failing. These materials can also dissipate a large amount of heat when they are deformed. Even more energy can be dissipated when these viscoelastic materials are combined with other materials such as asbestos, and then carbon black, for greater improvement in reliability.

186 Test Equipment, Test Methods, Test Fixtures, and Test Sensors Viscoelastic damping materials have been used to improve the vibration and shock performance of so many electronic systems for so many programs that engineers often tend to think this material will be safe to use under almost any condition. This has resulted in many failures where engineers were careless in their applications. One example is the case of a large electronic company with a good reputation that received a large contract for a military electronic system. This system had to provide a lightweight structure that could withstand a severe shock and vibration environment over a long period of time. Similar types of electronic systems always used a vibration and shock isolation mount. This new application did not have enough sway space for an isolation system so it was decided to use a hard-mounted system. A special program was set up to use an internal viscoelastic damping material cover around each PCB, to isolate it, and to protect it from external vibration and shock forces. Several full-scale models were made up and tested in the laboratory. The results were very good. There were no failures recorded so the design was released to the production group. When the production equipment was completed, normal vibration and shock tests were run on several hard-mounted electronic boxes. Some tests were run at a low temperature with good results. But when the tests were run at high temperatures, many electronic components failed and broke loose. The tests were a disaster. A new structural system had to be designed because the engineers did not understand the properties of the viscoelastic materials and their damping properties. Viscoelastic materials have good damping at lower temperatures; the materials are more rigid so they provide good damping. At high temperatures the material is soft so the damping is sharply reduced and the internal components no longer have any good support. This results in more rapid failures. The engineers did not understand the unusual properties of the viscoelastic material. The prototype electronic systems were all run at room temperature. The original engineers did not research the properties of the viscoelastic materials. They only depended on comments from other engineers that viscoelastic materials worked very well and were safe to use for their equipment in their environments. The prototype models that were tested by the original engineers were tested at room temperature. No one thought of doing any research on the material properties at other temperatures. They did not know that the viscoelastic materials they were using and testing became very soft at high temperatures. The entire system had to be completely redesigned using special stiffening ribs on the PCBs to maintain a high natural frequency. This was to reduce the forces and stresses in order to improve the required fatigue life. A lot of money was lost on this program, and the product delivery time was substantially delayed because the mechanical engineering group was too lazy to do a little research.

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DIFFERENT TYPES OF TEST EQUIPMENT ARE OFTEN REQUIRED Different types of holding fixtures may be required to hold and support different types of mechanical and electrical equipment to evaluate their ability to withstand high forces and stresses. Vibration frequencies for high-speed vehicles such as missiles and airplanes generally extend to 1000 Hz and often up to 2000 Hz. Many specifications for these vehicles can go to 3000 Hz. In order to produce harmonic motion over such a broad frequency range, special adapters are often required to ensure the accuracy of the proposed tests. Electrodynamic shaking equipment is generally used for these tests. These shakers are often called vibration machines. They are very much like loudspeakers that have a moving coil. Instead of connecting to a speaker cone, the moving coil connects to an armature, which joins the shaker head that simulates the desired harmonic motion. The armature has a driving coil that is cylindrical in shape and the shaker head is usually an extension of this shape. The shaker head itself must be very rigid in order to control the displacement amplitudes at high frequencies. Electrodynamic shakers that are capable of producing frequencies up to

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Vibration Test Fixtures and Adapters

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2000 Hz normally have their shaker head resonance above 3000 Hz. In order to provide a pure translatable motion for the shaker head, various types of flexures or springs, metal, and rubber are often used between the shaker head and the support frame. Even with these devices many rotational modes often develop in the shaker head during severe resonant conditions. Vibration machines are generally rated in terms of the peak force in pounds based on sinusoidal wave motion. These machines are available with force ratings ranging from 25 to 2500 lb. The choice of the vibration machine depends on the maximum weight of the system to be tested and the maximum acceleration force required by the test. Most vibration tests on electronic units require acceleration inputs along each of the three mutually perpendicular axes of the system. In order to provide this type of adaptability, the vibration machine usually has the shaker head mounted on a structure that permits the head to be rotated and locked in different positions.

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VIBRATION TEST FIXTURES AND ADAPTERS The shaker head on vibration machines usually has some form of a hole pattern that permits the installation of machine screws. These holes can often be used to mount small electronic components for vibration testing. Large electronic boxes require some sort of mechanical adapter that will permit the shaker head to transfer vibration motion to the electronic box. The adapter is now the vibration test fixture, which is an extension of the armature in the form of a very rigid structure that can transfer the required force at the required frequency. An optimum fixture would have its lowest natural frequency about 50% higher than the highest required forcing frequency in order to avoid fixture resonances during the test. For example, if the vibration test requirements had a range of 5–500 Hz, the vibration test fixture should have its fundamental resonant mode at about 750 Hz when the mass of the test specimen is included in the loading on the test fixture. Since most boxes for electronic systems are not very heavy, a vibration fixture resonance of 750 Hz is not too difficult to obtain. However, if the vibration test must go to 2000 Hz, the desirable fixture resonance would be about 3000 Hz. This may be very difficult to obtain unless the test specimen happens to be quite light, probably less than about 20 lb. If the test specimen is relatively heavy, perhaps greater than 50 lb, a test fixture with a resonance of 3000 Hz may be so massive that the force needed to vibrate the test specimen may exceed the force rating on the available vibration machine. Under these conditions a compromise has to be made. Either reduces the required force input to the test system or reduces the weight of the fixture and tries to live with the resulting resonances that may develop in the future. Very often severe fixture resonances can be reduced by introducing a highly damped figure structure. This may be in the form of laminated structures where energy is dissipated at several interfaces. Laminated wooden fixtures have been used successfully. Highly damped castings such as zirconium magnesium are often used for structures that require high damping and stiffness with light weight. If there is any doubt as to why it is desirable to keep the natural frequency of a fixture at least 50% higher than the highest forcing frequency, remember that a resonance can magnify acceleration forces. If an improperly designed vibration fixture is used to support a sensitive electronic component during a 5G sinusoidal vibration test, it is possible for this component to receive 100G if the fixture has a transmissibility of 20 at its resonance. If the fixture were not properly monitored with accelerometers, a casual observer could conclude that this component failed at 5G. Therefore, before any vibration fixtures are designed, it is necessary to know the basic fundamentals of vibration. This requires a familiarity with the natural frequency formulas for simple systems such as beams, plates, and multiple spring–mass systems. Without this knowledge even the best designer will be groping in the dark. The end result may be an inadequate fixture that must be redesigned, modified, or changed in some way to make it work. This could be very time consuming. Rigid

188 Test Equipment, Test Methods, Test Fixtures, and Test Sensors mathematical solutions are not necessary to solve for the natural frequencies of various types of structures. If the natural frequency of a particular structure cannot be found in a reference book, it may be possible to derive the necessary equation using approximate methods such as the Rayleigh or work and strain energy method.

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BASIC VIBRATION FIXTURE DESIGN CONSIDERATIONS Sharp changes in the cross section of any vibration test fixture should be avoided. These sharp changes often result in a reduction of the effective spring rate without a proportional reduction of the mass. This will result in a lower natural frequency for the fixture. The design of the test fixture should be kept simple since it can help to reduce the costs. It can also be used to compare with standard handbook equations that can provide good estimates for the performance of various types of test fixtures. The stiffness-to-weight ratio of any proposed vibration fixture must always be considered to make maximum use of the fixture mass. Understand the basic frequency equations to know what factors affect the resonant frequency. The materials used to fabricate the vibration fixture are an important part of the design function. Is it better to make fixtures out of castings, solid plates, or welded plates than bolted assemblies? At high frequencies, bolted assemblies tend to slide and to separate so that the calibrated stiffness does not really exist. If a large increase in stiffness is not required, bolted assemblies provide a substantial amount of damping during resonant conditions that effectively reduce the transmissibility. If bolted assemblies are also cemented with epoxy cements, a very rigid fixture can be made. Vibration test data on bolted assemblies indicate that the typical efficiency of a bolted joint is about 25%. This will vary depending upon the relative stiffness of the structure as well as the size, spacing, and number of bolts. With many bolts this factor may go as high as 50%, but very few designers use enough large, properly spaced bolts to reach that efficiency. Bolts are generally threaded into tapped holes to a depth of about two diameters, so there appears to be many threads holding the bolt. However, a close examination of the engaged threads reveals that for a normal class 2 fit, only a few threads are actually holding the bolt. If these threads happen to be near the tip of the bolt, the effective length of the bolt can extend well into the tapped hole. When bolts are loaded in an axial direction and the spring rate of the bolt must be determined, it is suggested that the effective length of the bolt should be considered as extending at least one diameter into the tapped hole. All bolts should be installed with a predetermined torque value, which depends upon the bolt material and the function. The torque should be checked periodically on the bolts that can influence the vibration characteristics of the system, because bolts that are inserted and removed often tend to loosen more easily during vibration.

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OIL FILM SLIDER TABLE VIBRATION FIXTURES Most vibration test laboratories make use of oil film slider tables when vibration tests must be run in a horizontal plane. An oil film slider, as the name implies, has a large flat plate that slides on a film of oil. The plate usually rests on a very rigid foundation of concrete, steel, or granite, where one edge of the plate is bolted to the vibration shaker head. There is a test specimen that consists of the vibration fixture and a test package that is normally bolted to the slider plate. The test specimen can usually be rotated 90◦ without changing the slider plate to permit vibration along the horizontal axis. Oil film slider tables are capable of very pure translation motion with very little crosstalk (rotational modes in either the horizontal or vertical planes). This is due to the viscous nature of the oil film acting on a very large surface area. Overturning moments in the vertical plane and rotational moments in the horizontal plane can be effectively damped out in most cases. If an attempt is made to vibrate a very

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Summary for Good Vibration Fixture Design

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tall test specimen with a high center of gravity, the surface area of the plate may have to be increased and a thicker slider plate may have to be used in order to prevent lifting and slapping of the slider plate. Counterweights may also have to be added to prevent the lifting and slapping. Before slider plates were used, horizontal vibration tests were run using flexure tables and suspension systems. These systems are very difficult to control during resonant conditions and their use is not recommended if an oil film slider can be used instead.

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VIBRATION FIXTURE COUNTERWEIGHTS Oil film slider plates can eliminate many difficulties related to overturning moments developed during vibration in the horizontal plane. However when tall masses with a high CG must be vibrated, counterweights may have to be used to lower the CG. Counterweights are normally made of a dense material such as steel or lead to keep the overall size down. Counterweights may be good for the static balance on the vibration system, but they may not be good for the dynamic balance due to the lack of dynamic similarity to the test specimen. Consider a tall nose cone that must be vibrated in a direction perpendicular to the axis of the cone. A severe resonance in the nose cone could shift the CG so that the counterweight would not be able to compensate for the overturning moment. Under these conditions, dynamic similarity could be obtained if two nose cones were vibrated back to back simultaneously. Any dynamic change in one nose cone would be duplicated in the opposite nose cone so that severe overturning moments can be reduced.

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SUMMARY FOR GOOD VIBRATION FIXTURE DESIGN Vibration test fixtures have the capability to provide adequate performance if the following few simple rules can be followed: 1. Understand the test specimen and the test specification. 2. Analyze preliminary designs and try to keep the lowest natural frequency about 50% higher than the highest forcing frequency. 3. Avoid sharp changes in the cross section. 4. Consider the stiffness-to-mass ratio for optimum design. 5. Keep fixtures as small as possible. 6. Avoid bolted fixture assemblies, except where they may be required for stiffness and damping. 7. Keep fixture designs simple. 8. Design symmetrical fixtures. 9. Design for dynamic similarity. 10. Consider the effective length of the bolt thread engagement when calculating the effective bolt spring rate. 11. Torque all bolts. 12. Use fine threads instead of coarse threads on bolts whenever possible. 13. Balance fixtures with the test specimen. 14. Positioning the test sensor servocontrol accelerometer. The G force transmitted from the shaker head to the vibration test specimen can be controlled directly from the shaker head or from an accelerometer sensor placed near the test specimen. When the G force is controlled from the shaker head, a constant G force can be maintained on the shaker head over the entire frequency band. If a resonance should develop

190 Test Equipment, Test Methods, Test Fixtures, and Test Sensors in the slider plate or the vibration fixture, the G forces will build up at resonance so that the input to the test specimen will be far greater than the output to the shaker head. At frequencies above the resonance the opposite situation occurs, and the input to the test specimen will be far less than the output to the shaker head. When the G force is controlled from an accelerometer placed near the test specimen, the vibration machine can vary the output of the shaker head to hold a constant G force on the control accelerometer over the entire frequency band. If the control accelerometer is placed on a resonant structure, the output of the shaker head will be reduced when the structure passes through its resonance. At frequencies above the resonance, the output of the shaker head will be increased to hold a constant G force on the control accelerometer. Standard piezoelectric accelerometers, the same types that are generally used to monitor and record acceleration data, are normally used as servocontrol accelerometers. These small devices are usually equipped with screw studs to fasten them to the test equipment. Many testing laboratories use adhesives such as dental cement, quick-drying Eastman 910 cement, and even double-back tape to mount accelerometers. These methods all work quite well on accelerometers that are used to monitor data, but they should never be used to fasten a control accelerometer to a vibrating system. The servocontrol accelerometer actually controls the G force output of the shaker head. If the servocontrol accelerometer should fall off during the vibration test, the shaker head will lose the feedback provided by this accelerometer. This results in a rapid buildup of the acceleration force until the limit displacement switch of the shaker head cuts off the power. By this time the damage has already been done. Very high G forces can be developed very rapidly. Once an expensive electronic system has been subjected to accelerated loads that far exceed its specifications, there is no good way to determine the actual and potential damage. In most cases where this has happened, and it has happened at many places, the electronic equipment has to be rebuilt because the customer will not accept an item that may fail shortly after it is placed in service. Since the position of the servocontrol accelerometer can determine the actual G force received by the tested specimen, the location selected for this accelerometer can be quite important to the success of a vibration test. If the top of the resonant fixture has a transmissibility of 10, then it will see 20G at its resonant peak when the control accelerometer is held at 2G. This means the electronic box will probably see an average of about 11G at the fixture resonance point instead of the required input of 2G. If the control accelerometer is mounted at the top of the resonant fixture, the bottom of the fixture will only see 0.2G at the resonant peak. This means the electronic box will probably see an average of about 1.1G at the fixture resonant point instead of the required input of 2G. The best solution to this problem is to design resonance-free fixtures. Since this is not always possible, a compromise is the only possibility, so the control accelerometer will probably be placed halfway up the fixture.

10

EFFECTS OF SHOCK ON ELECTRONIC EQUIPMENT Shock is often defined as a rapid transfer of energy to a mechanical system, which results in a significant increase in the stress, velocity, acceleration, or displacement in the system. The time in which the energy transfer takes place is usually related to the resonant frequency or the natural period of the system. Shock will often excite many of the natural frequencies in a complex structure, which can produce four basic types of failures in electronic systems. These failures are due to: 1. High stresses, which can cause fractures or permanent deformations in the structure 2. High acceleration levels, which can cause relays to chatter, potentiometers to slip, and bolts to loosen

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Specifying the Shock Motion and Environment

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3. High displacements, which can cause impact between adjacent circuit boards, cracking components and solder joints, breaking cables and harnesses, and fracturing castings 4. Electrical malfunctions that occur during the shock but disappear when the shock energy dissipates This last effect can occur in crystal oscillators, capacitors, and hybrids. A large thin hybrid cover can displace and cause a temporary short circuit with the internal die bond wires. The hybrid typically appears normal after the shock exposure, which makes it difficult to find and correct the failure. Fatigue is usually not important in shock unless a million or more stress cycles are involved. When less than a few thousand stress cycles are involved, fatigue stress concentrations are ignored because they do not have a great influence on how or when the structure will fail. Stress concentration factors must be included in any shock evaluation where fatigue effects are expected to accumulate. The impact sensitivity, notch sensitivity, and brittleness of a structure are important, especially when high-strength steels and castings are used to carry high shock loads. When the ductility of any structural load-carrying material is less than about 3%, problems can develop unless the dynamic stresses are carefully evaluated with respect to the anticipated environment. However, in most structural elements the ductility is normally greater than about 5%, so the limiting factor in the design is usually based on the yield strength of the material. Isolation systems are often used to protect sensitive equipment in severe shock environments. Care must be exercised to allow sufficient sway space around the equipment to prevent impact against other surrounding structures. Shock analysis techniques can become quite complex unless some simplifying approximations are made. For this reason simple electronic systems are often simulated using simple masses, springs, and dampers to estimate the dynamic characteristics of the system.

11

SPECIFYING THE SHOCK MOTION AND ENVIRONMENT Many different methods have been used to specify shock motion or its effects. The three most popular are (1) pulse shock, (2) velocity shock, and (3) shock response spectrum. Pulse shock deals with accelerations or displacements in the form of well-known shapes such as square waves, half-sine waves, and various types of triangular waves (vertical rise, vertical decay, and symmetrical). Pulse shocks are easy to work with because the mathematics is simple and convenient. But the pulse shocks do not represent the real world. The true shock environment is seldom a simple pulse. Nevertheless, simple pulses are often effective in revealing weak areas in many different types of structures. Velocity shock is concerned with systems that experience a sudden velocity change, such as a falling package whose velocity abruptly goes to zero when the package strikes the ground. This is a common test that is called the drop shock. Sometimes an inclined plane is used, where a package gains velocity as it slides down the plane and hits a rigid wall. Another type of velocity shock makes use of a heavy hammer that slams into a fixture that supports a test specimen. The hammer imparts a sudden velocity to the fixture and the test specimen. This type of velocity shock test is used extensively by the navy to simulate the effects of explosions on ships and submarines. The shock response spectrum deals with the way in which a structure responds to the shock motion, rather than trying to describe the shock motion itself. The spectrum is a plot of the peak acceleration response of an infinite number of single-degree-of-freedom systems to a complex transient waveform. The individual single-degree-of-freedom masses are usually specified as having a transmissibility Q of 10 when excited at their resonant frequency with a sinusoidal vibration input. This method of analysis is more representative of the real world, but the mathematics is far more complex than the mathematics of the simple shock pulse.

192 Test Equipment, Test Methods, Test Fixtures, and Test Sensors 12

PULSE SHOCKS IN ELECTRONIC EQUIPMENT Various types of shock pulses are often used to excite electronic assemblies to simulate transportation environments, bench-handling conditions, and pyrotechnic events used to separate multiple stages on missiles and spacecraft. The manner in which the various electronic components respond to these shocks will determine if the components will survive the environment. It is often convenient to represent various structural elements in the electronics as simple masses and springs so the approximate response characteristics can be evaluated quickly and cheaply. This type of analysis yields fast results, but the accuracy is reduced, since it is not possible to accurately represent a complex structure and a few masses and springs. A real electronic system will typically have many major resonant frequencies. The purpose of the simplified analysis is to try to simulate the first major resonances where most of the damage normally occurs. When approximations are adequate, a lot of time and money can be saved with a small decrease in the accuracy. Plug-in PCBs can quickly be evaluated by approximating them as a single degree of freedom using a single mass, spring, and damper. This produces good results, since test data on many different types of PCBs show that most of the damage is produced by the fundamental resonant frequency of the PCB. Dynamic displacements, stresses, and accelerations are usually maximum under these conditions. Therefore, it is necessary to understand how single-degree-of-freedom systems respond to various types of shock pulses in order to determine if these systems will be damaged by shock pulse environments.

13

HOW PCBs RESPOND TO SHOCK PULSES When PCBs are excited by shock pulses, they will respond by bending initially in the same direction as the pulse. When the pulse diminishes, the PCBs will resonate at their own resonant frequencies, where the lowest resonant frequency is usually the most prominent. Extreme care must be used in the determination of clearances for PCBs that must operate in severe shock environments. Sufficient clearances must be provided to account for tolerance accumulations in the thickness of the PCBs, the component size, component lead wire protrusions on the back side of the PCBs, location tolerances, and possible displacement amplitudes of adjacent PCBs moving in opposite directions at the same time. It is important to keep the dynamic displacement low, so the dynamic stresses will be low and the chances for impact between adjacent PCBs will also be low. Experience has shown that high shock acceleration levels can result in cracked solder joints and fractured lead wires on large or heavy electronic components such as transformers and capacitors. These must be mounted carefully to avoid failures in the support structure and mounting hardware. The greatest damage will occur when the large and heavy components are mounted at the center of a plug-in PCB where the curvature changes and the acceleration levels are the greatest.

14

CASE HISTORIES OF FAILURES AND FAILURE ANALYSES Failures in electronic systems are often very difficult to analyze because they can be caused by many different factors. These include poor design, unusually severe environments, poor manufacturing, carelessness, accidents, improper testing, improper handling, and even sabotage. When a failure occurs, an investigation is usually made to find the cause of the failure so that corrective actions can be taken to prevent more failures. People involved with the failed hardware are usually questioned to try to pinpoint the reason or reasons for the failure. People become very defensive when they are questioned because they do not want to be accused of causing the failure. Engineering groups and manufacturing groups often end up accusing each other for causing the failures. The testing group may also get involved in the argument.

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Failures in Small Cantilever Shafts of Spinning Gyro

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Any association with a product failure can cause hard feelings and friction between working groups that can interfere with workers’ morale and productivity. It can result in dismissals, transfers, or quitting. Many will deliberately lie or mislead the investigators to protect themselves and their friends. Some group leaders and managers who think they are very close to the hardware will make what they consider to be statements of fact that are not really true. The investigators usually have to sort through lies, half truths, and opinions to try to find the real cause of the failures. They do not always succeed. Some failures take years to solve. Some failures are never solved. The failure histories, analyses, evaluations, and problems that were encountered during the various investigations described here are all true. The cases were taken from personal experiences at a number of different industries involved with the design and manufacturing of commercial, industrial, and military electronic systems.

15

FAILURES IN SMALL CANTILEVER SHAFTS OF SPINNING GYRO A large number of failures were occurring in the shafts of spinning gyros in the guidance system of a major missile program during low-level vibration acceptance tests. The failures were occurring at a step in the small diameter of the shaft. A structural dynamic load stress analysis and a fatigue life analysis of the small shaft were made that included a high stress concentration factor. This showed that the stress levels were not high enough to produce the testing failures. There had to be another reason for the failures in the shaft. The investigation turned to the manufacturing process for the possible source of the failures. A theory was proposed where the cantilever shafts might be machined using a lathe with a single cutter. This cutting action could generate a bending moment in the small shaft. The rotating action of the lathe could generate many stress reversals that might initiate small cracks at the step in the shaft. The low-level vibration could then cause the small cracks to propagate the fracture in the shaft. The machining process crack initiation and propagation failure for the shaft were presented to the supervisor of the manufacturing facility. He vehemently rejected the idea that his methods and manufacturing processes could in any way affect the fatigue life of the shaft. He took great pride and great pains that he knew as much, and probably a lot more, than most engineers on the causes and effects of improper machining practices on the fatigue life of manufactured parts. The shop supervisor explained in great detail how the shafts were being machined using three cutters spaced 120◦ apart so there would not be any bending generated in the shafts during the machining process. He went on to say that the failures were probably caused by poor engineering design, so the failures should not be blamed on his poor manufacturing processes. The investigation turned away from the manufacturing process to other possible sources for the failures, such as overstressing during vibration. The dynamic analysis was approached in a couple of different ways, but the results were always that the shaft should not fail in the low-level vibration. Nothing else was uncovered that could cause the shaft failures. The investigator decided to return to the area of the manufacturing to see how the shafts were being machined. He waited until the supervisor left the area. Then putting on his safety glasses he walked over to the area where several shafts were being machined. What the investigator saw at one of the machining stations did not really surprise him. One of the machinists was using a single cutter to machine the shaft. The shop supervisor walked in and saw the investigator he had chased out the previous day and asked him to leave the area because he was disturbing the machinist. The investigator did not move. Instead he suggested that the supervisor take a look at the machining operation in progress. One quick look and the supervisor’s face turned white. He could see that a single cutter was being used for the machining operation. The supervisor demanded to know why a single cutter was being used for the machining operation when there were clear instructions to use three cutters for that operation. The machinist replied that using three cutters took too much time and he could not

194 Test Equipment, Test Methods, Test Fixtures, and Test Sensors make his daily quota. However, when he used a single cutter he could easily make his daily quota and the end result was just as good as with three cutters. The circumstances described here are true and were presented to demonstrate that people will typically tell you what they believe to be true, even though they may not know all of the facts. The shop supervisor gave specific instructions to the machinists to use three cutters on that particular shaft. He did not think it was important to inform the machinists of the critical nature of the part. He assumed the machinists were all following his instructions so there was no need to check back to see if his instructions were being followed. If the investigator really believed the supervisor at his first meeting and he did not go back to check the operation himself, there is a good chance the shaft failures would never have been solved. The investigator in this case was lucky because the machine shop was on a double shift. If the machinist using the single cutter was on the second shift, there is a good chance the problem would never have been solved. The lesson learned here is that a good investigator must always try to see as much as possible before any questions are asked.

CHAPTER

10

CORRELATION BETWEEN PACKAGE-LEVEL HIGH-SPEED SOLDER BALL SHEAR/PULL AND BOARD-LEVEL MECHANICAL DROP TESTS WITH BRITTLE FRACTURE FAILURE MODE, STRENGTH, AND ENERGY Fubin Song and S. W. Ricky Lee Hong Kong University of Science and Technology Hong Kong, People’s Republic of China

Keith Newman Oracle Corporation Santa Clara, California

Bob Sykes and Stephen Clark Dage Precision Industries Aylesbury, Bucks, United Kingdom

1

INTRODUCTION Reliability is the ability of a product to be consistently good in performance and so elicit trust of both the manufacturer and the customer. Reliability is usually defined as the ability of a product or a system to survive and to perform a required function, without failure or breakdown, for a specific envisaged period of time under stated operation and maintenance conditions [1–3]. Reliability considerations play an importante role in design, material selection, and manufacturing decisions. Reliability is conceived during product design, implemented during its manufacturing, qualified by testing, screened during mass production, and, if necessary, maintained during the product’s use. There are different levels of reliability in electronic devices: package, board, and system level. Package-level reliability concerns the failures found internal to the package, like die cracking, wire bond failures, delamination, and package cracking. Board-level reliability concerns the failures at the subassembly level. For example, interconnect failures at the intermetallic or solder joints, trace failures, delamination, dielectric cracking, plated through hole (PTH) cracking, and via cracking. Solder joint reliability depends upon component quality, carrier quality, and assembly process control. These include package-side solder volume, solder ball planarity, flatness of the substrate and carrier, lead dissolution into the solder volume, and attachment reflow time. System-level reliability concerns the failures of the outer casing or interconnection like connector defects, card warpage, and case cracking [4, 5]. Structural Dynamics of Electronic and Photonic Systems Copyright © 2011 John Wiley & Sons, Inc.

Edited by Ephraim Suhir, David S. Steinberg and T. X. Yu

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196 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests Solder joints were initially intended to be simple electrical interconnections between mechanically interlocked components in electronic packages. As technology advanced, mechanically interlocked components are being replaced by surface mount packaging where the solder joints are both the electrical and mechanical interconnection. Additionally, a solder joint with a smaller and smaller feature size is being incorporated in electronics packages. With the small size of these electrical-mechanical interconnections using solder balls, the intrinsic reliability, especially the long-term reliability, of solder balls becomes a critical issue for area array packages like ball grid array (BGA), chip scale package (CSP), and flip chip (FC) using solder balls (bumps). To evaluate the reliability of BGA, CSP, and FC packages, the strength of solder balls (bumps) is one piece of the prerequisite data. Currently the most popular method to evaluate the strength of the solder ball attachment is the ball shear test [6–13], which is adopted from the Joint Electron Device Engineering Council (JEDEC) standard JESD22-B117A [6]. Currently, a new method called the ball pull test has emerged as an attractive alternative to the traditional ball shear test as an interconnect monitor [11, 12]. The ball pull test is a relatively new development, and there is a little information on the method. The JEDEC standard about the ball pull test, JESD22-B115A, was published in August 2010. Many researchers and engineers used the conventional ball shear test (shearing speed is less than 0.8 mm/sec) to evaluate the reliability of solder joints in the past decade [7–13]. Huang et al. (2001 and 2002) first studied the ball shear test in terms of the effects of shear speed and shear height [8, 9]. He employed both experimental and finite element analysis techniques to investigate the effects on the ball shear test of the conventional SnPb solder with a NSMD (non–solder mask defined) type substrate. The optimization of the shear test method for all kinds of BGA packages was restrictively discussed. Huang et al. (2002) also reported on the progressive failure mechanism of solder balls during the ball shear test with experimental investigation. Kim et al. (2003) and Kim and Jung (2004) reported on the shear speed effects in the shear test of In 48% Sn, which has a low melting temperature, and SnAg and SnAgCu lead-free solder balls, which have high melting temperatures, using similar experimental and simulation techniques with Huang et al. (2001). However, the research objective of these studies was focused on the ball shear test. The comparison of ball shear and pull tests was not included. As the ball pull test method is relatively new and the equipment is not widely available, so far there are very few publications in the literature reporting the experimental results of solder ball pull tests. From this standpoint, it is necessary to investigate the use and effects of the ball shear and ball pull tests on monitoring packaging reliability. Solder joint reliability concerns are increasing exponentially with the continuous push for device miniaturization and the expanded use in portable electronic products. The reliability of lead-free solder joints under mechanical shock loading is a major concern. Brittle fracture at the interfaces between solder balls and bond pads are considered unacceptable. In principle, this kind of solder joint reliability should be characterized by board-level drop testing (BLDT). However, such testing has some major drawbacks. First, each drop test will consume several packages and hundreds of solder joints and the cost of samples are rather high. Second, the crack in the solder joint may close after the impact, and as a result, the failure is undetectable unless there is a high-speed real-time data acquisition system for monitoring during drop testing. Third, analysis of the data is very time consuming and therefore expensive. Furthermore, due to the nature of dynamic impact testing, the scatter of experimental data is usually rather wide. Therefore, it is imperative to find alternative methods for evaluating solder joint integrity under mechanical shock loading. In order to predict solder joint reliability under drop conditions, it is important to increase

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Experimental Investigation of High-Speed Ball Shear and Pull Tests

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the testing speed of package-level test methods, such as high-speed solder ball shear and pull [15–17]. Traditional ball shear and pull tests are not considered suitable for evaluation of joint reliability under drop loading, since the applied test speeds, usually lower than 5 mm/sec, are well below the impact velocity applied to the solder joint in a drop test [8–13, 18, 19]. Recently, high-speed shear and pull test equipment with testing speeds up to several meters per second (maximum. speed of 4 m/sec for shear and 1.3 m/sec for pull) has become available. Thus, it is necessary to investigate the characterization of high-speed ball shear and pull tests, including the failure mode and brittle fracture mechanism during these high-speed tests.

2

EXPERIMENTAL INVESTIGATION OF HIGH-SPEED BALL SHEAR AND PULL TESTS A comprehensive evaluation was conducted to investigate the effects of solder ball alloy, package substrate surface finish, high-temperature storage, and package construction on highspeed solder ball shear/pull tests. Due to space constraints, this chapter only describes the results for a single 316 plastic ball grid array (PBGA) (27 mm × 27 mm) construction, using Sn–4.0% Ag–0.5% Cu (SAC405) and Sn–37% Pb solder balls, but fabricated with different surface finish options: electroless nickel immersion gold (ENIG) and organic solderability preservative (OSP). Experimental investigation of the effects of test speed on high-speed solder ball shear and pull tests for the BGA package was performed in this chapter first. The effects of shear height on the solder ball shear test were also discussed. An advanced, state-of-the-art machine, the DAGE 4000HS, was used to perform all of the tests. This high-speed testing machine was equipped with the most updated control and analysis software and a new generation of force transducers, which are now able to evaluate the fracture energy of solder balls in both the ball shear and ball pull tests. Second, the brittle fracture surfaces of SnAgCu and SnPb solder balls have been systematically analyzed using scanning electron microscopy/energy-dispersive X-ray (SEM/EDX) characterization of brittle fracture surfaces following high-speed ball shear/pull testing. Detailed images were recorded for the fractured surfaces of both solder balls and their corresponding package pads. In order to compare brittle fracture failures in both high-speed ball shear and pull tests, the morphologies and elemental distributions of the brittle fracture surfaces from these two methods were also compared. Finally, the samples were divided into groups which were subjected to thermal aging at 125◦ C (0–500 hr) in order to accelerate the formation of intermetallic compound (IMC). Then, the effects of IMC growth on the high-speed ball shear and pull tests were investigated. The ball shear tests ranged from 10 to 3000 mm/sec and the ball pull tests ranged from 5 to 500 mm/sec.

2.1 Experimental Procedures The ball shear test speeds ranged from 10 to 3000 mm/sec, and the ball pull test speeds ranged from 5 to 500 mm/sec. New test equipment (DAGE 4000HS) was employed to perform both types of tests. The test machine (see Fig. 1) used the most recent updates for the control and analysis software as well as the latest generation of high-speed transducers to capture both the peak force and total fracture energy.

198 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests

(a) DAGE 4000HS Bond tester

(b) Ball shear system

(c) Ball pull system

Figure 1 DAGE 4000HS bond tester system.

A large matrix of test packages was evaluated employing various combinations of solder alloys, surface finishes, substrate materials, solder ball sizes, and package dimensions. In this chapter only a single 316 PBGA (27 mm × 27 mm) construction was discussed using ENIG and OSP substrate surface finishes, but with two BGA solder alloy options: Sn–4.0% Ag–0.5% Cu (SAC405) and Sn–37% Pb (SnPb). The packages were fabricated using standard 0.76 mm (0.030 in.) diameter solder spheres attached to a BT laminate substrate with soldermask-defined pad openings of 0.635 mm diameter. A summary of the various sample and test parameters is given in Table 1.

Table 1

Description of Mechanical Tests for Solder Balls (High-Speed)

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10–3000 mm/sec 50 μm — Sn–4.0% Ag–0.5% Cu Sn–37% Pb ENIG/OSP As reflowed (two times reflow)

5–500 mm/sec — 2.2 bars Sn–4.0% Ag–0.5% Cu Sn–37% Pb ENIG/OSP As reflowed (two times reflow)

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Experimental Investigation of High-Speed Ball Shear and Pull Tests

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2.2 Failure Modes of High-Speed Ball Shear/Pull Tests In this investigation, the solder ball shear and pull test samples were evaluated both quantitatively (force and fracture energy) and qualitatively (failure mode). The area percentage of solder remaining on the pad is used as a semiquantitative criterion for distinguishing different failure modes. The failure modes were identified in five categories. In addition to the ductile and brittle modes, there are two mixed failure modes, that is, quasi-ductile (or <50% area with exposed pad) and quasi-brittle (or >50% area without solder) modes. The pad lift failure was also observed after the high-speed ball shear/pull tests. Representative photographs of the five failure modes (ductile, quasi-ductile, quasi-brittle, brittle, and pad lift) are shown in Fig. 2.

2.3 Effects of Testing Speed on High-Speed Ball Shear/Pull Tests Figure 3 provides a comprehensive graphical summary of the solder ball shear and pull testing failure mode observations at all test speeds, excluding the non-solder-related failure mode (pad lift). Despite the wide range of sample constructions and materials, the failure mode trends are quite similar. Although the failure mode responses are remarkably similar between the shear and pull testing, the transition from ductile to brittle occurred at lower speeds in solder ball pull testing. At low shear and pull test speeds, the ductile (bulk solder) failure mode predominates. Given that the typical failure mode in BLDTs is interfacial failure, correlations between BLDT and low-speed solder ball shear/pull tests are necessarily problematic. As shear and pull test speeds increased, however, a parallel increase in brittle failure mode percentage was observed. This is essential for the correlation between high-speed ball shear/pull and drop tests. The transition between ductile and brittle fractures occurred at higher shear and pull test speeds for the BGA sample with SnPb solder, compared to the SAC solder samples. This higher transition speed and lower brittle failure rate for the SnPb solder sample suggests that SnPb solder may prove more resistant to BLDT fracture than SAC solder; this relationship has also been proven by some previous studies [20, 21]. The failure mode trends between the solder ball shear test and pull test were remarkably similar across all test parameters and package configurations. The most noticeable difference is that the transition speed between ductile and brittle falure modes is lower for solder ball pull testing. A less obvious difference is a general reduction in mixed ductile/brittle failure modes for the solder ball pull test samples. As noted in a previous study [18], secondary interactions between the fractured solder ball surface and the surrounding solder mask in solder ball shear testing tend to result in solder residue at the solder mask edge and on the fractured pad surface. The pull test avoids these secondary interactions with the surrounding solder mask and appears to reduce the level of mixed ductile/brittle failure modes. Other parameters such as Sn grain orientation and IMC uniformity may also contribute to mixed-mode failures and cannot be avoided. Figures 4 and 5 graphically summarize the measured peak force and total fracture energy of the test samples for solder ball shear and pull testing, respectively. The graphs provide separate force and energy values for each observed failure mode. The error bars on the graphs identify the minimum and maximum values observed for each failure mode. These figures illustrate that the measured peak force increases relatively steadily for all failure modes with increased test speed. Moreover, this trend is consistent for both the shear and pull tests. In most cases, however, the variation in peak force between ductile solder joint failures and brittle failures falls within the min/max range and therefore the peak force does not correlate strongly with failure mode.

200 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests

(a) Ductile mode (100% area with solder left)

(b) Quasi-ductile mode (<50% area with exposed pad)

(c) Quasi-brittle mode (>50% area without solder)

(d) Brittle mode (almost no solder left)

(e) Pad lift

Figure 2 High-speed ball shear and pull failure modes.

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Experimental Investigation of High-Speed Ball Shear and Pull Tests

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Failure mode distribution in ball shear and pull tests of various samples with different loading

By contrast, solder joint fracture energy appears to correlate well with failure mode for both the shear and pull tests. Indeed, mixed failure modes show intermediate fracture energy values between extreme high values (ductile) and low values (brittle). Interestingly, the progressive increase observed for peak force with increased test speed does not occur in solder joint fracture energy. Instead, fracture energy values appear to peak at an intermediate shear/pull speed.

202 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests Ductile

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Ball shear strength and energy for different failure modes as function of test speed.

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Experimental Investigation of High-Speed Ball Shear and Pull Tests 25

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20 15 Pull Energy (mJ)

2

Pull Force (kgf/mm )

20

10

5

15 10 5

0 5

50

100 250 Pull Speed (mm/s)

0

500

5

50

100 250 Pull Speed (mm/s)

500

Pull Energy

Pull Force (b) SAC405+ENIG (PBGA, 0.76 mm solder ball) 25 20

15

Pull Energy (mJ)

Pull Force (kgf/mm2)

20

10

5

15 10 5

0

0 5

50

100 250 Pull Speed (mm/s)

500

5

50

Pull Force

100 250 Pull Speed (mm/s)

500

Pull Energy

(c) SAC405+OSP (PBGA, 0.76 mm solder ball)

Figure 5 Ball pull strength and energy for different failure modes as function of test speed. Figure 6 illustrates how the brittle solder ball fracture often occurs prior to the shear tool traveling the entire distance of the solder ball or pad diameter, in contrast to ductile failure. Although the peak force values of the brittle and ductile solder failures shown in Fig. 6 differ by approximately 20%, the integrated force–distance area (fracture energy) is many times greater for the ductile failure. Besides being a more sensitive quantitative measure of fracture strength, fracture energy also correlates much better to the failure mode type. A similar comparison of ductile and brittle failures is shown in Fig. 7 for solder ball pull test

204 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests 3500

500 mm/s Ductile Failure Mode 500 mm/s Brittle Failure Mode

Shear Force (g)

3000

Solder Ball Diameter (760 µm)

2500 2000 1500 1000 500 0

Figure 6

0

200

400 600 Displacement (µm)

800

Typical shear force–displacement curve of ductile and brittle failure modes.

4500

50 mm/s Ductile Failure Mode 50 mm/s Brittle Failure Mode

4000

Pull Force (g)

3500 3000 2500 2000 1500 1000 500 0 0

200 400 Displacement (µm)

600

Figure 7 Typical pull force–displacement curve of ductile and brittle failure modes. samples. The earlier solder joint rupture of a brittle fracture, relative to a ductile failure, is once again apparent. Again, fracture energy correlates better than peak force with the failure modes. Figures 8 and 9 show the data distribution of ball shear force and energy on an SAC405+ENIG sample at 500 mm/sec, respectively. These two plots present data of all failure modes together. Compared to the shear force distribution presented in Fig. 8, it is clear that there are two peaks shown in the shear energy distribution of Fig. 9. Figures 10 and 11 plot the data distribution in another mode, showing the data by grouping according to failure modes. The ductile and quasi-ductile failure modes belong in one group; quasi-brittle and brittle modes are in another group. It can be found that the degree of correlation between fracture energy and failure mode is obviously much better than using the traditional measure of force. The introduction of fracture energy as an alternative measure of solder joint strength (beyond the conventional force metric) at high shear/pull speeds represents a major shift in the test equipment industry.

2

Experimental Investigation of High-Speed Ball Shear and Pull Tests

205

10

8

Count

6

4

2

0 1500

Figure 8

2000

2500 3000 Shear Force (g)

3500

4000

Data distribution of ball shear strength (total, SAC405+ENIG, 500 mm/sec).

10

8

Count

6

4

2

0 0

2

4

6

8 10 12 14 Shear Energy (mJ)

16

18

20

Figure 9 Data distribution of ball shear energy (total, SAC405+ENIG, 500 mm/sec).

2.4 Comparison of Brittle Failure of Lead-Free and Lead–Tin Solder Balls Failure analysis of fracture surfaces of solder balls following the ball shear and pull tests usually consists of visual evaluation using an optical microscope to identify the failure mode and possible fracture mechanisms. Although it is common to see micrographs of solder ball fracture surfaces illustrating failure modes from shear and pull tests, the actual mechanism of brittle fracture failure is obscure, especially for high-speed testing. Previous studies typically did not discuss the detailed mechanisms of brittle interfacial fracture between solder balls and bond pads.

206 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests 10

8 Ductile and Quasi-Ductile

Count

6 Quasi-Brittle and Brittle 4

2

0 1500

2000

2500 3000 Shear Force (g)

3500

4000

Figure 10 Data distribution of ball shear strength (separated, SAC405+ENIG, 500 mm/sec).

10 8

Count

Ductile and Quasi-Ductile 6 Quasi-Brittle and Brittle

4 2 0

0

2

4

6

8 10 12 14 Shear Energy (mJ)

16

18

20

Figure 11 Data distribution of ball shear energy (separated, SAC405+ENIG, 500 mm/sec). High-speed ball shear/pull tests are relatively new, and therefore there are few publications reporting detailed failure analysis, especially for the brittle fracture interface between the solder ball and bond pad. In this study, the brittle fracture surfaces of SnAgCu and SnPb solder balls have been systematically analyzed using SEM/EDX characterization of brittle fracture surfaces following high-speed ball shear/pull testing. As shown in Figs. 3a and b, for both shear and pull testing of SAC405 and SnPb solder balls on ENIG pads, the same general trends can be observed: The percentage of brittle fracture mode failures increases with increased test speed, and the SAC405 solder alloy shows a markedly increased rate of brittle fracture, initiating at lower test speeds, relative to eutectic SnPb solder. In order to assess the composition of the solder ball shear brittle fracture surfaces, a series of detailed SEM and EDX images were captured for both the solder ball and package substrate fracture interfaces (see Figs. 12–15 and Tables 2 and 3). The failure interface for all samples occurred between the Ni layer and IMC.

2

Experimental Investigation of High-Speed Ball Shear and Pull Tests

207

(a) SEM images

(b) Sn element distribution

(c) Ni element distribution

(d) Cu element distribution

Figure 12 SAC405 EDX mapping analysis after high-speed ball shear (1000 mm/sec, quasibrittle).

208 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests

(a) SEM images

(b) Sn element distribution

(c) Ni element distribution

(d) Pb element distribution

Figure 13 SnPb EDX mapping analysis after high-speed ball shear (3000 mm/sec, brittle).

2

Experimental Investigation of High-Speed Ball Shear and Pull Tests

209

(a) SEM images

(b) Close-up view of location indicated by rectangle in (a)

(c) EDX analysis of the image area in (b)

Figure 14 Brittle fracture surface and EDX analysis after high-speed ball shear (SAC405, 3000 mm/sec). Table 2 EDX Analysis of Brittle Fracture Surface After High-Speed Ball Shear as Shown in Fig. 14 (SAC405, 3000 mm/sec) (wt %) Element Ni K Cu K Sn L

Solder Ball

Corresponding Pad

6.03 9.67 84.30

19.70 2.84 77.46

Table 3 EDX Analysis of Brittle Fracture Surface After High-Speed Ball Shear as Shown in Fig. 15 (SnPb, 3000 mm/sec) (wt %) Element Ni K Sn L Pb M

Point 1

Point 2

Point 3

Point 4

5.02 84.03 10.95

— 56.47 43.53

16.75 83.25 —

— 72.03 27.97

210 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests

(a) SEM images

(b) Close-up view of location indicated by rectangle in (a)

(c) EDX as shown at point 1 in (b)

(d) EDX as shown at point 2 in (b)

(e) EDX as shown at point 3 in (b)

(f) EDX as shown at point 4 in (b)

Figure 15 Brittle fracture surface and EDX analysis after high-speed ball shear (SnPb, 3000 mm/sec).

2

Experimental Investigation of High-Speed Ball Shear and Pull Tests

211

Compared to the brittle fracture surfaces of SAC405 solder joints, solder residues were found by EDX analysis as shown in Fig. 15 and point 4 of Table 3 on the pad of SnPb specimens after high-speed ball shear testing. SEM and EDX analysis for the solder ball pull brittle fracture surfaces is shown in Figs. 16–19 and Tables 4 and 5. The elemental mapping and SEM imaging of the fracture surfaces are essentially identical between the pull and shear test samples. Brittle failures

(a) SEM images

(b) Sn element distribution

(c) Ni element distribution

(b) Cu element distribution

Figure 16 SAC405 EDX mapping analysis after high-speed ball pull (500 mm/sec, quasibrittle).

212 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests

(a) SEM images

(b) Sn element distribution

(c) Ni element distribution

(d) Pb element distribution

Figure 17 SnPb EDX mapping analysis after high-speed ball pull (500 mm/sec, quasibrittle).

2

Experimental Investigation of High-Speed Ball Shear and Pull Tests

213

(a) SEM images

(b) Close-up view of location indicated by rectangle in (a)

(c) EDX analysis of the image area in (b)

Figure 18 Brittle fracture surface and EDX analysis after high-speed ball pull (SAC405, 500 mm/sec). Table 4 EDX Analysis of Brittle Fracture Surface after High-Speed Ball Pull as shown in Fig. 18 (SAC405, 500 mm/sec) (wt %) Element Ni K Cu K Sn L

Solder Ball

Corresponding Pad

11.57 13.43 75.00

31.26 3.54 65.20

Table 5 EDX Analysis of Brittle Fracture Surface after High-Speed Ball Pull as shown in Fig. 19 (SnPb, 500 mm/sec) (wt %) Element Ni K Sn L Pb M

Point 1

Point 2

Point 3

Point 4

16.57 72.14 11.29

— 93.14 6.86

23.31 76.69 —

— 25.30 74.70

214 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests

(a) SEM images

(b) Close-up view of location indicated by rectangle in (a)

(c) EDX as shown at point 1 in (b)

(d) EDX as shown at point 2 in (b)

(e) EDX as shown at point 3 in (b)

(f) EDX as shown at point 4 in (b)

Figure 19 Brittle fracture surface and EDX analysis after high-speed ball pull (SnPb, 500 mm/sec).

3

Effects of IMC Growth on High-Speed Ball Shear and Pull Tests after Thermal Aging

215

occurred between the Ni layer and IMC, both on the SAC405 and SnPb specimens. From a comparison of Figs. 15b and 19b, it can be seen that more solder residues are left on the pad after the ball shear tests than after the ball pull tests. In addition, more Ni element was found on the pad after the ball pull tests compared with the ball shear tests, regardless of solder composition (see Tables 2 and 4, SAC405, and point 3 of Tables 3 and 5, SnPb). The high-speed pull test revealed the brittle fracture interface of the solder joints more clearly than the ball shear tests, especially for SnPb solders.

3

EFFECTS OF IMC GROWTH ON HIGH-SPEED BALL SHEAR AND PULL TESTS AFTER THERMAL AGING Due to space constraints and the focus on the lead-free solder, this section describes only the results for a single 316 PBGA (27 mm × 27 mm) construction using Sn–4.0% Ag–0.5% Cu (SAC405) solder balls but fabricated with different surface finish options: ENIG and OSP. The samples were divided into groups which were subjected to thermal aging at 125◦ C (0–500 hr) in order to accelerate the formation of IMC at the package substrate/solder-joint interface. The ball shear tests ranged from 10 to 3000 mm/sec and the ball pull tests ranged from 5 to 500 mm/sec. An advanced machine, the DAGE 4000HS, was used to perform all of the tests. This high-speed testing machine was equipped with the most updated control and analysis software and a new generation of force transducers, which are now able to evaluate the fracture energy of solder balls in both ball shear and ball pull tests.

3.1 IMC Growth after Thermal Aging Figures 20a, c, and e show the interfacial intermetallic compound on the ENIG substrates aged at 125◦ C for 0, 300, and 500 hr, respectively, which exhibits Ni–Cu–Sn intermetallic compounds detached from the substrate. Copper and Sn from SAC solder play a very important role in the IMC growth at the interface of ENIG pad finish while Ni layer provides a good barrier to inhibit detrimental growth of Cu-Sn intermetallics as mentioned above. The SEM micrographs of OSP showing the IMC growth at the substrate/solder ball interface are given in Figs. 20b, d , and f . The microstructures of IMC formed by SAC solder and OSP pad finish are very different from those of solder and ENIG substrate. EDX analysis revealed that the scallop-shaped intermetallic phase seen in these micrographs is Cu6 Sn5 . A thin layer of Cu3 Sn was observed adjacent to the Cu substrate in the specimens subjected to 300 hr of thermal aging. Given the highly nonuniform topography of the IMC layer, the average thickness was determined by dividing the cross-sectional area of the IMC by its base length. Figure 21 plots the measured IMC thickness against time, yielding the expected linear fit when plotted against the square root of the aging time. The experimental data follow classical diffusion theory (Fick’s law), which specifies a linear relationship between the thickness of the IMC layer and the square root of time. Figure 21 also shows that the IMC growth rates in solders on OSP surface finish are higher than those on ENIG. This demonstrates that the Ni layer in ENIG serves as a good barrier to inhibit growth of Cu–Sn IMC [18].

3.2 Failure Mode Percentage of High-Speed Ball Shear and Pull Tests after Thermal Aging Graphical summaries of the solder ball shear and pull failure mode results are shown in Figs. 22 and 23, respectively. Representative photographs of the failure modes (ductile, quasiductile, quasi-brittle, brittle, and pad lift) are shown in Fig. 2. Similar to the results of previous

216 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests

(a) ENIG, 0 hr

(b) OSP, 0 hr

(c) ENIG, 300 hr

(d) OSP, 300 hr

Ni–Cu–Sn

Cu6Sn5 Cu3Sn

(e) ENIG, 500 hr

(f ) OSP, 500 hr

Figure 20 IMC growth and morphology changes subject to thermal aging at 125◦ C (ENIG and OSP). studies, a clear transition occurred between the ductile mode typical of low shear/pull speeds and a brittle mode at higher test speeds. Although the failure mode responses are remarkably similar between the shear and pull testing, the transition from ductile to brittle occurred at lower speeds in solder ball pull testing. Figures 22 and 23 also demonstrate that the samples with an OSP finish showed an increased incidence of brittle failure with increased themal exposure. By contrast, the samples with ENIG plating yielded only a slight increase in brittle fracture frequency even at the highest test speeds. The different brittle fracture response between ENIG and OSP may not be surprising given the comparative IMC growth rates shown in Fig. 21. Figures 24 and 25 plot the failure mode data in another format, showing the percentage of brittle failure mode for each test speed against aging time, excluding the non-solder-related failure mode (pad lift). This transformation allows the rate of brittle fracture incidence to be expressed as a function of time and aging temperature to be described by simple linear curve fitting. The curve fitting follows the relationship y = ax + b, where y is the brittle failure percentage of the ball shear and pull tests after aging, x equals aging time, a represents the change in brittle mode percentage with aging, and b is the brittle failure percentage for the zero time sample. It should be noted that the different slopes are a measure of the sensitivity

3

Effects of IMC Growth on High-Speed Ball Shear and Pull Tests after Thermal Aging 7 SAC405+OSP SAC405+ENIG

Average Thickness (µm)

6 5

y = 0.164(t1/2) + 2.19

4 3 y = 0.043(t1/2) + 1.12 2 1 0 0

5

15

10

20

25

(Aging Time)1/2 (hr1/2)

Figure 21 Correlation between IMC thickness and aging time. Ductile

Quasi-Ductile

Quasi-Brittle 100% Failure Mode Percentage

Failure Mode Percentage

100% 80% 60% 40% 20%

80% 60% 40% 20% 0%

0% 0

100

300

0

500

Aging Time (hr)

(a) 10 mm/s (ENIG)

300

500

(b) 10 mm/s (OSP) 100%

80%

80%

Failure Mode Percentage

Failure Mode Percentage

100

Aging Time (hr)

100%

60% 40% 20% 0%

60% 40% 20% 0%

0

100 300 Aging Time (hr)

(c) 100 mm/s (ENIG)

Figure 22

Brittle

500

0

100 300 Aging Time (hr)

500

(d) 100 mm/s (OSP)

Failure mode distribution in ball shear of specimens with ENIG and OSP pad finishes.

217

218 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests 100% Failure Mode Percentage

Failure Mode Percentage

100% 80% 60% 40% 20%

80% 60% 40% 20% 0%

0% 0

100

300

500

0

Aging Time (hr)

(e) 500 mm/s (ENIG)

500

100% Failure Mode Percentage

Failure Mode Percentage

300

(f) 500 mm/s (OSP)

100% 80% 60% 40% 20% 0%

80% 60% 40% 20% 0%

0

100 300 Aging Time (hr)

500

0

(g) 1000 mm/s (ENIG)

100 300 Aging Time (hr)

500

(h) 1000 mm/s (OSP) 100% Failure Mode Percentage

100% Failure Mode Percentage

100 Aging Time (hr)

80% 60% 40% 20% 0% 0

300 100 Aging Time (hr)

80% 60% 40% 20% 0%

500

(i ) 3000 mm/s (ENIG)

0

300 100 Aging Time (hr)

500

(j ) 3000 mm/s (OSP)

Figure 22

(Continued )

of the test system for detecting changes in brittle fracture with time or test speed. Observe that the line with the highest slope in each graph represents the optimal test speed for each sample to detect changes in brittle fracture rate with time. Figure 26 transforms the linear curve fitting of Figs. 24 and 25 into simplified brittle fracture rate response curves versus solder ball shear and pull test speeds. These graphs provide a technique to quantitatively define the optimal test speed (transition point) at which a solder joint fracture is most prone to shift from ductile to brittle failure mode. Additionally, Fig. 26 can be used to define a recommended test speed for evaluating brittle solder joint fractures. It can be found that the optimal speed for shear test is 500 mm/sec, both on ENIG and OSP substrates. However, for the ball pull test, the different optimal testing speeds were presented on different substrates. For SAC+ENIG, the optimal pulling speed is 50 mm/sec; and 5 mm/sec is the best one for the SAC+OSP system.

3

Effects of IMC Growth on High-Speed Ball Shear and Pull Tests after Thermal Aging Quasi-Ductile

Quasi-Brittle 100%

80%

80%

Failure Mode Percentage

Failure Mode Percentage

Ductile 100%

60%

40%

20%

0%

60%

40%

20%

0% 0

100

300

0

500

Aging Time (hr)

300

500

(b) 5 mm/s (OSP)

100%

100%

80%

80%

Failure Mode Percentage

Failure Mode Percentage

100

Aging Time (hr)

(a) 5 mm/s (ENIG)

60%

40%

20%

60%

40%

20%

0%

0% 0

100

300

500

0

Aging Time (hr)

100

300

500

Aging Time (hr)

(c) 50 mm/s (ENIG)

(d ) 50 mm/s (OSP)

100%

100%

80%

80%

Failure Mode Percentage

Failure Mode Percentage

Brittle

60%

40%

20%

0%

60%

40%

20%

0% 0

100

300

Aging Time (hr)

(e) 100 mm/s (ENIG)

500

0

100

300

500

Aging Time (hr)

(f ) 100 mm/s (OSP)

Figure 23 Failure mode distribution in ball pull of specimens with ENIG and OSP pad finishes.

219

100%

100%

80%

80%

Failure Mode Percentage

Failure Mode Percentage

220 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests

60%

40%

20%

60%

40%

20%

0%

0% 0

300

100

500

0

(h) 250 mm/s (OSP)

80%

80%

Failure Mode Percentage

100%

60%

40%

20%

60%

40%

20%

0% 300

0%

500

0

100

Aging Time (hr)

300

500

Aging Time (hr)

(i) 500 mm/s (ENIG)

(j) 500 mm/s (OSP)

Figure 23

10 mm/s 100%

(Continued )

100 mm/s

500 mm/s

y = 0.005x + 0.7961 80% y = 0.0011x + 0.3532 60%

40%

20%

1000 mm/s

y = 0.0002x + 0.0296 y=0

0%

3000 mm/s

y=1

100%

y=1 Shear Brittle Failure Percentage(%)

Shear Brittle Failure Percentage(%)

Failure Mode Percentage

(g) 250 mm/s (ENIG)

100

500

Aging Time (hr)

100%

0

300

100

Aging Time (hr)

y = 0.003x + 0.8953 80% y = 0.0009x + 0.6224

60%

y = 0.0014x + 0.885 40% y = 0.0004x − 0.0203

20%

0% 0

100

200

300

400

Aging Time (hr)

(a) Ball shear (ENIG)

500

600

0

100

200

300

400

500

600

Aging Time (hr)

(b) Ball shear (OSP)

Figure 24 Brittle failure percentage of ball shear tests at various testing speeds and aging time.

Effects of IMC Growth on High-Speed Ball Shear and Pull Tests after Thermal Aging 5 mm/s

100 mm/s

y = 0.0004x + 0.8463 80% y = 0.0007x + 0.5217 60%

40% y = 0.0003x + 0.139 20%

500 mm/s

y=1 y = 0.0003x + 0.8888

y = 0.001x + 0.5239

60%

y = 0.0015x + 0.0282

40%

20%

0% 0

100

200

300

400

500

600

0

100

200

300

400

Aging Time (hr)

Aging Time (hr)

(a) Ball pull (ENIG)

(b) Ball pull (OSP)

500

600

Brittle failure percentage of ball pull test at various testing speeds and aging

0.035

0.035 0.03 a (Ball Shear Test)

0.03 a (Ball Shear Test)

221

80%

0%

Figure 25 time.

250 mm/s

100% Pull Brittle Failure Percentage(%)

Pull Brittle Failure Percentage(%)

50 mm/s

y=1

100%

0.025 0.02 0.015 0.01

0.025 0.02 0.015 0.01 0.005

0.005

0

0

0 0

500 1000 1500 2000 2500 3000 3500

500 1000 1500 2000 2500 3000 3500 Ball Shear Speed (mm/s)

(a) Ball shear (ENIG)

(b) Ball shear (OSP)

0.035

0.035 a (Ball Shear Test)

0.03 a (Ball Shear Test)

3

0.025 0.02 0.015 0.01

0.03 0.025 0.02 0.015 0.01 0.005

0.005 50 mm/s 0

100

200 300 400 500 Ball Pull Speed (mm/s)

(c) Ball pull (ENIG)

5 mm/s

0

0 600

0

100

200 300 400 500 Ball Pull Speed (mm/s)

600

(d ) Ball pull (OSP)

Figure 26 Sensitivity of transition level to brittle mode in ball shear and pull tests at different testing speeds.

222 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests 3.3 Effects of Thermal Aging on Ball Shear/Pull Force and Energy Although all of the previous figures relate to qualitative observations of failure mode, electronics companies would clearly prefer a quantitative assessment method which is independent of subjective operator classification and is highly repeatable. The introduction of fracture energy as an alternative measure of solder joint strength (beyond the conventional force metric) at high shear/pull speeds represents a major shift in the test equipment industry. Figures 27 and 28 provide a graphical summary of the fracture force and energy data (shear/pull) of SAC405 + ENIG, while Figs. 29 and 30 display the same parameters for the samples with OSP pad finish. The degree of correlation between fracture energy and failure mode is obviously much higher than using the traditional measure of force, confirming early observations using high-speed shear testing. For example, the difference of force data at various failure modes is not significant. In contrast, it can be found that the fracture energies of various failure modes are obviously different, on the specimens with both ENIG and OSP pad finishes as shown in Figs. 28 and 30 for ball pull tests. Specifically, compare histograms (e) and (f ) of Fig. 28; forces (e) are virtually uniform for all the aging times, even though

Ductile

Quasi-Ductile

Quasi-Brittle

Brittle

25

20

Shear Energy (mJ)

2

Shear Force (kgf/mm )

20 15

10

15

10

5 5

0

0 0

100 300 Aging Time (hr)

0

500

(a) Shear force (10 mm/s)

100 300 Aging Time (hr)

500

(b) Shear energy (10 mm/s) 25

20

Shear Energy (mJ)

2

Shear Force (kgf/mm )

20 15

10

15

10

5 5

0

0 0

100 300 Aging Time (hr)

(c) Shear force (100 mm/s)

500

0

100 300 Aging Time (hr)

500

(d) Shear energy (100 mm/s)

Figure 27 Ball shear strength and energy of different failure modes as function of aging time and test speed (SAC405 + ENIG).

3

Effects of IMC Growth on High-Speed Ball Shear and Pull Tests after Thermal Aging 25

20

15 Shear Energy (mJ)

2

Shear Force (kgf/mm )

20

10

15

10

5 5

0

0 0

100 300 Aging Time (hr)

500

0

100

300

500

Aging Time (hr)

(e) Shear force (500 mm/s)

(f) Shear energy (500 mm/s) 25

20

Shear Energy (mJ)

Shear Force (kgf/mm2)

20 15

10

5

15

10

5

0

0 0

100 300 Aging Time (hr)

0

500

(g) Shear force (1000 mm/s)

500

(h) Shear energy (1000 mm/s)

20

25

20

15 Shear Energy (mJ)

Shear Force (kgf/mm2)

100 300 Aging Time (hr)

10

15

10

5 5

0

0 0

100 300 Aging Time (hr)

0

500

(i) Shear force (3000 mm/s)

Figure 27

100 300 Aging Time (hr)

(j) Shear energy (3000 mm/s)

(Continued )

500

223

224 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests Ductile

Quasi-Ductile

Quasi-Brittle

Brittle

25

20

15 Pull Energy (kgf/mJ)

Pull Force (kgf/mm2)

20

10

5

15

10

5

0

0 0

100

300

500

0

(a) Pull force (5 mm/s)

300

500

(b) Pull energy (5 mm/s)

20

25

20

15

Pull Energy (kgf/mJ)

Pull Force (kgf/mm2)

100

Aging Time (hour)

Aging Time (hr)

10

15

10

5 5

0

0 0

100

300

500

0

100

300

500

Aging Time (hour)

Aging Time (hr)

(c) Pull force (50 mm/s)

(d) Pull energy (50 mm/s) 25

20

Pull Energy (kgf/mJ)

Pull Force (kgf/mm2)

20 15

10

15

10

5 5

0

0 0

100

300

500

0

100

300

Aging Time (hr)

Aging Time (hour)

(e) Pull force (100 mm/s)

(f) Pull energy (100 mm/s)

500

Figure 28 Ball pull strength and energy of different failure modes as function of aging time and test speed (SAC405 + ENIG).

3

Effects of IMC Growth on High-Speed Ball Shear and Pull Tests after Thermal Aging 20

225

25

Pull Energy (kgf/mJ)

Pull Force (kgf/mm2)

20 15

10

5

15

10

5

0

0 0

100

300

0

500

100

Aging Time (hr)

300

500

Aging Time (hour)

(g) Pull force (250 mm/s)

(h) Pull energy (250 mm/s) 25

20

Pull Energy (kgf/mJ)

Pull Force (kgf/mm2)

20 15

10

5

0

15

10

5

0 0

100

300

0

500

100

300

Aging Time (hr)

Aging Time (hour)

(i ) Pull force (500 mm/s)

(j ) Pull energy (500 mm/s)

Figure 28 Ductile

500

(Continued ) Quasi-Brittle

Quasi-Ductile

20

Brittle

25

Shear Energy (mJ)

Shear Force (kgf/mm2)

20 15

10

15

10

5 5

0

0 0

100

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Figure 29 Ball shear strength and energy of different failure modes as function of aging time and test speed (SAC405 + OSP).

226 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests 20

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Figure 29

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(h) Shear energy (1000 mm/s)

(Continued )

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3

Effects of IMC Growth on High-Speed Ball Shear and Pull Tests after Thermal Aging 25

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Figure 29 Ductile

500

(Continued )

Quasi-Ductile

Quasi-Brittle

Brittle

25

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Figure 30 Ball pull strength and energy of different failure modes as function of aging time and test speed (SAC405 + OSP).

228 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests 25

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Figure 30

(Continued )

500

4 Correlation of High-Speed Ball Shear/Pull Tests and Board-Level Drop Test 229 failure modes have shifted from ductile to brittle, but there is a distinct shift from highenergy values for ductile failures to lower energies for brittle mode, as shown in (f ). A similar argument applies to histograms (c) and (d ) in Fig. 30. Corresponding examples can be seen for the shear tests in Figs. 27 and 29.

4

CORRELATION OF HIGH-SPEED BALL SHEAR/PULL TESTS AND BOARD-LEVEL DROP TEST Portable electronic devices such as personal digital assistants (PDAs), cellular phones, and potable audio/video devices are becoming more and more popular in our daily lives. Due to the nature of portable devices, the chance of electronic packages being subjected to mechanical impact such as free falling is increased. Dynamic loading often plays a critical role in the functional performance and mechanical reliability of micro-electronic components and devices. In principle, the impact loading condition may be simulated by various mechanical tests. Although mechanical shock tests may be used for the qualification or life assessment of micro-electronic and opto-electronic products, shock conditions such as the duration of the shock impulse, the time dependence, and the maximum value of imposed acceleration have to be carefully determined in order to adequately mimic the real impact conditions. This task is rather difficult in general. Therefore, instead of performing mechanical shock tests, actual drop tests should be conducted whenever it is possible. The reliability of lead-free solder joints under mechanical drop loading is a major concern. Brittle fracture at the interfaces between solder balls and package substrate bond pads are considered unacceptable. In principle, this kind of solder joint reliability should be characterized by BLDT. However, such testing has some major drawbacks. First, each drop test will consume several packages and hundreds of solder joints, incurring considerable expense. Second, the cracks in the solder joint may close after the impact, resulting in undetectable failure unless there is a high-speed real-time data acquisition system available for in situ monitoring. Third, analysis of the data is very time consuming, adding significant expense. Therefore, it is imperative to find alternative methods for evaluating solder joint integrity under mechanical drop loading. The present study was performed to compare high-speed solder ball shear and pull tests with BLDT. Emphasis has been placed on the correlation of failure mode and energy absorption between the two methods. The objective was to investigate the feasibility of using high-speed solder ball shear and pull tests as an alternative method of evaluating solder joint integrity under dynamic loading. During the course of this study, a comprehensive testing program was conducted which included BGA package constructions employing various combinations of solder alloys, surface finishes, substrate material, solder ball size, and package dimensions. During each drop test, electrical resistance, circuit board strain, and fixture acceleration were recorded. Detailed analyses were performed to identify the failed solder joints and corresponding failure modes. The failure modes and loading speeds of solder ball shear and pull tests were cross-referenced with the mechanical drop tests for comparison. From the test results, various correlations between failure mode and loading speed have been observed. Also, the energy absorption value recorded during the solder ball shear and pull tests is considered an effective index to interpret the solder joint failure mode. This chapter will document more thoroughly the analytical relationships observed between BLDT, high-speed shear/pull, and solder joint fracture energy.

230 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests 4.1 Experimental Procedures Two types of substrate pad finishes were investigated in this study: OSP and ENIG. The chemical composition of the lead-free solder alloy used in this work is SAC405. The objective of this chapter is to investigate the correlation of the board-level drop test and solder ball shear/pull tests. The 316 PBGA samples used standard 0.76 mm (0.030 in.) diameter spheres. The package substrates were composed of BT laminate, with a thickness of 0.36 mm. The solder bond pads were solder mask defined with an opening of 0.635 mm in diameter. The solder balls were attached to the substrates in a hot-air convective reflow oven. The Pb-free soldering profile has a 150 ± 2◦ C preheat, with a peak temperature of 260◦ C. Afterward, thermal aging to accelerate IMC growth was conducted at 125◦ C in an oven for a number of different durations (100, 300, and 500 hr). The ball shear and pull tests were performed with different speeds on the lead-free solder balls after thermal aging using a DAGE 4000HS bond tester. Each set of ball shear or pull test data consisted of a series of 20 individual measurements. The high-speed ball shear and pull tests were already discussed in Section 3. In this section the focus is on the correlation of high-speed ball shear/pull tests and the drop test.

4.2 Test Vehicles of Drop Test Similar BGA samples were assembled on test boards and dropped using a dual-rail guided device. Since the packages used in the present study were relatively large, the specifications of the test boards (142 × 142 mm, eight-layer Cu, 2.35 mm thick, as shown in Fig. 31) were different from those given in JESD22-B111 [22]. Three strain gauges were attached to the back side of the test printed circuit board (PCB) for monitoring during the drop test, as shown in Fig. 31b. Some board-level test samples were also subjected to thermal aging, as above. All samples were equipped with daisy chains and subjected to real-time data acquisition monitoring. During the drop test, it is common practice to verify the status of the solder joints by reading the resistance of the solder joints. Daisy chain is a circuit that connects the solder joints of the electronic package together. The monitoring of daisy chain resistance can be done either before or after testing. The completed daisy chain circuits of 27 × 27 mm PBGA is shown in Fig. 32.

4.3 Drop Test Setup The schematic diagram of the equipment setup is illustrated in Fig. 33. The test facility consists of three major parts: the drop tower, the daisy chain monitoring system, and the strain-measuring system, as shown in Fig. 33. The test board was mounted onto the steel fixture where it is fixed to the drop table, as shown in Fig. 34c. The drop orientation is horizontal with packages in a face-down position. During the test, the drop table is raised and dropped from the desired height along the two guiding rods of the drop tester, and in situ event detection with a threshold resistance of 1000  was used to monitor the discontinuity of the daisy chain. An accelerometer was used to characterize the drop test apparatus. In this work, the test acceleration has a peak value of 500g with duration of 1.0 msec as shown in Fig. 35 following the JEDEC standard B110A [23]. The resistance of the daisy-chained solder joints was monitored during the drop test. The drop lifetime was monitored by the change of dynamic resistance for the daisy-chained solder joints in real time during the drop test. For each drop, the solder joint resistance for the PBGA packages and PCB bending strain were captured. A typical curve response is shown in

4 Correlation of High-Speed Ball Shear/Pull Tests and Board-Level Drop Test 231

(a) Top side

(b) Bottom side with strain gauge

Figure 31

Test board for drop test.

Fig. 36. Test results for the as-reflowed board after the first failed drop is used for illustration. Three channels are used for dynamic strain measurement at the package center (also the PCB center) and corner. Three channels measured the dynamic response of the package threedaisy-chain loops. It was found that an initial failure in the package has occurred as shown in Fig. 36. A resistance drop was registered after impact. The following fluctuation in drops was constituted by the upward/downward flexing of the PCB leading to the opening/closing mode of the crack in the solder joints. This observation describes the intermittent solder joint failure.

4.4 Effects of Thermal Aging on Drop Test As highlighted in the introduction, the primary objective of this research was to investigate the feasibility of using the high-speed solder ball shear and pull tests as alternative methods to board-level drop testing for evaluating solder joint integrity under dynamic loading. In order

232 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests EMC Loop (Loop 2)

Outer Loop (Loop 1)

Die Loop (Loop 3)

Figure 32

Completed daisy chain of 27 × 27 mm PBGA for the drop test.

Daisy Chain Monitoring System

Video Monitoring System

Drop Tower

Strain Monitoring System

Figure 33 Schematic diagram of drop test setup.

4 Correlation of High-Speed Ball Shear/Pull Tests and Board-Level Drop Test 233

(b) Fixture on gliding rods

(a) Drop tower

(c) Fixture of drop test

Figure 34 Drop test setup.

600

400 300 200

500 g

Acceleration (g)

500

100 0

−100 −0.0005

1.0 ms 0

0.0005 Time (s)

0.001

0.0015

Figure 35 Testing condition of the drop test. to provide a basis for comparison to the shear and pull testing of the components, a series of drop tests were conducted using board assemblies of identical package lots evaluated by the DAGE 4000HS. Moreover, these assemblies were aged at 125◦ C (0–500 hr). Following preliminary studies, a JEDEC standard JESD22-B110A service condition A—500G, 1.0-msec half-sine pulse—was selected for all drop testing in this evaluation. Choosing this drop condition was necessarily a compromise; too severe a drop and relative assessment of various constructions and aging exposures would be difficult, and too gentle a drop could introduce potentially significant solder joint cyclical fatigue effects.

234 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests 1500 Gauge 1 (Board Center) Gauge 2 (Package Corner) Gauge 3 (Package Corner) Resistance of Loop 1 Resistance of Loop 2 Resistance of Loop 3

1000

0 −500

−1000

1000

−1500

500

−2000

0 0.06 0.08 0.1 0.12 0.14 Time (s) (a) Dynamic strain of PCB and resistance change at failure 0

0.02

Resistance (Ω)

Microstrain

500

0.04

1500 Gauge 1 (Board Center) Gauge 2 (Package Corner) Gauge 3 (Package Corner) Resistance of Loop 1

1000

0 −500

−1000

1000

−1500

500

−2000 0

0.01

0.02 0.03 0.04 Time (s) (b) Close-up view as shown in (a )

0 0.05

Resistance (Ω)

Microstrain

500

Figure 36 Typical dynamic responses for the drop test.

BLDT test boards were fabricated with both NSMD and solder-mask-defined (SMD) pad geometries. In both cases, the solder-wetted pad diameter was 0.684 mm (90% solder ball diameter). Although NSMD is more typical of the actual production circuit boards, SMD has the advantage for this correlation study that the BLDT fracture locations are more likely to occur at the package side; this is significant because solder ball shear/pull testing can only evaluate the package side fractures as the component is not attached to a PCB [24]. This chapter only reports the results for the SMD board configuration. A summary of the drop testing results is shown in Fig. 37, identifying the mean value (eight assemblies per data point) of the drops to failure for the test board assemblies. It can be found from Fig. 37 that the drop test lifetime decreased significantly after thermal aging, both on the OSP and ENIG packages. Repeating observations recorded in earlier work, the drop fracture strength of devices with an OSP package substrate surface finish showed a more rapid degradation with thermal aging than those with an ENIG finish [25–28]. For the as-reflowed specimens, as shown in Fig. 37, the lifetime of SAC+OSP (110 drops) was

4 Correlation of High-Speed Ball Shear/Pull Tests and Board-Level Drop Test 235 120 27X27

SAC+ENIG

27X27

SAC+OSP

Drop Number

100 80 60 40 20 0

Figure 37

0

100

200

300 400 Aging Time (hr)

500

600

Board-level drop lifetime with thermal aging of SAC405 on different pad finishes.

better than SAC+ENIG (93 drops). While for the specimens after thermal aging, SAC+ENIG showed better performance than SAC+OSP. The drop impact lifetime siginificantly decreased from 110 to 66 for SAC+OSP only after 100 hr of aging, and after 500 hr, the specimen only endured 34 drops before failure occurred. The drop lifetimes for SAC+ENIG after 100 and 500 hr of aging were 83 and 55, which are obviously better than those of SAC+OSP. Drop impact failure was characterized from both the plane view and the cross section. For plane-view observation, ink penetration (dye and pry) was used to mark the failure site. The PBGA unit was then peeled off from the board, and the marked failure site displayed the dye color under an optical microscope as shown in Fig. 38 (SAC+ENIG without aging is used as an example). Some specimens were cleaned by acetone to remove the dye solution for SEM analysis. After inspection of the dye and pry process, it can be found that the dominant failure mode of the drop test is brittle failure at the package side, on both SAC+ENIG and SAC+OSP samples. This is also expected by using a special design on the PCB pad as mentioned before. In this section, the cross sections of failed samples were observed under high-resolution SEM when SAC+ENIG and SAC+OSP were subjected to thermal aging and the drop test. Figure 39 shows the enlarged drop test failure interface for the SAC+ENIG specimens after 0 and 500 hr of thermal aging and a drop test. It can be found that the failure on ENIG substrate occurred between the IMC and Ni layers, regardless of aging time. Figure 40 shows the drop impact failure interface for the SAC+OSP specimens after 0 and 500 hr of thermal aging. Although both types of failure occurred in the IMC, the location of cracks was different after 0 and 500 hr of aging. For the as-reflowed specimen shown in Figs. 40a and b, the crack path is between Cu6 Sn5 IMC and Cu layers. After 500 hr of aging, as shown in Figs. 40c and d , a dominant crack can be seen between the Cu6 Sn5 and Cu3 Sn IMC phases in the specimen. The crack is more flat compared to the one in the as-reflowed sample.

4.5 Comparison of Failure Mode of High-Speed Ball Shear/Pull Tests and Drop Test Brittle fractures have been observed in the previous evaluations of high-speed solder ball shear and pull testing, which appeared similar to the brittle fracture mode observed in BLDT

236 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests

PCB Side

Figure 38

Package Side

Dye-and-pry analysis after the drop test.

(a) 0 hour of aging

(b) Close-up view of location indicated by rectangle in (a)

(c) 500 hours of aging

(d) Close-up view of location indicated by rectangle in (c)

Figure 39 IMC interfacial fracture during drop test with and without 500 hr of aging (SAC405 + ENIG).

4 Correlation of High-Speed Ball Shear/Pull Tests and Board-Level Drop Test 237

(a) 0 hour of aging

(b) Close-up view of location indicated by rectangle in (a)

(c) 500 hours of aging

(d ) Close-up view of location indicated by rectangle in (c )

Figure 40 IMC interfacial fracture during drop test with and without 500 hr of aging (SAC405 + OSP). assemblies but microstructure evidence has not yet been provided. This has been due in part to the difficulty of such studies, in terms of retrieval of both retrieval of individual sheared or pulled balls and matching them to their corresponding pad and the subsequent cross-sectional work. However, the correlation between the high-speed ball shear/pull tests and drop test cannot be established if their failure modes or locations are different. This is also the reason why the shear/pull test speed needs to be increased to find the brittle failure. In this study, the effort has resulted in the images shown in Figs. 41–48. From the failure analysis of drop test specimens, brittle failure on the ENIG was induced between the IMC and the Ni layers, on the samples both without and with 500 hr of thermal aging, as shown in Figs. 42a and 44a. Figures 42b,c and 44b,c show the failure location of high-speed ball shear/pull test at the optimal testing speed after 0 and 500 hr of aging. The failure location after the ball shear/pull tests was also between IMC and Ni layers, which was quite similar to that of the drop test. For the OSP specimens without aging (with two times reflow), brittle failure was found between Cu6 Sn5 IMC and Cu layers as shown in Fig. 46a. Brittle failure of the OSP specimens after 500 hr of thermal aging occurred between the Cu6 Sn5 and Cu3 Sn IMC phases, as shown in Fig. 48a. Again, the same failure locations were also found on the related OSP samples after the high-speed ball shear and pull tests, as shown in Figs. 46b,c and 48b,c. From the comparison of fracture surfaces (Figs. 41, 43, 45, and 47) and cross-section (Figs. 42, 44, 46, and 48, the related sheared/pulled solder balls and pads are shown at the top right corner of each SEM image), it is clear that brittle fracture interfaces from drop testing show a striking similarity with those from high-speed ball shear and pull tests. These figures provide clear evidence of a close resemblance between the brittle fracture modes of BLDT and high-speed solder ball shear and pull. It can be said there should be a correlation between high-speed solder ball shear/pull tests and drop test from the microstructure analysis.

238 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests

(a) Brittle fracture surface of a solder joint

(b) Brittle fracture surface of the matching pad of (a)

(c) Brittle fracture surface of a sheared ball (500 mm/s)

(d ) Brittle fracture surface of the matching pad of (c )

(e) Brittle fracture surface of a pulled ball (50 mm/s)

(f) Brittle fracture surface of the matching pad of (e)

Figure 41 Top-view comparison of brittle fracture surface during drop test, HS ball shear/pull tests (ENIG, 0 hr of aging).

4 Correlation of High-Speed Ball Shear/Pull Tests and Board-Level Drop Test 239

(a) IMC failure during the drop test

(b) IMC fracture failure during the HS ball shear (500mm/s)

(c) IMC fracture failure during the HS ball pull (50 mm/s)

Figure 42 Cross-sectional comparison of failure fracture during drop test, HS ball shear/pull tests (ENIG, 0 hr of aging).

240 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests

(a) Brittle fracture surface of a solder joint

(b) Brittle fracture surface of the matching pad of (a)

(c) Brittle fracture surface of a sheared ball (500 mm/s)

(d) Brittle fracture surface of the matching pad of (c)

(e) Brittle fracture surface of a pulled ball(50 mm/s)

(f) Brittle fracture surface of the matching pad of (e)

Figure 43 Top-view comparison of brittle fracture surface during drop test, HS ball shear/pull tests (ENIG, 500 hr of aging).

4 Correlation of High-Speed Ball Shear/Pull Tests and Board-Level Drop Test 241

(a) IMC failure during the drop test

(b) IMC fracture failure during the HS ball shear (500 mm/s)

(c) IMC fracture failure during the HS ball pull (50 mm/s)

Figure 44 Cross-sectional comparison of failure fracture during drop test, HS ball shear/pull tests (ENIG, 500 hr of aging).

242 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests

(a) Brittle fracture surface of a solder joint

(b) Brittle fracture surface of the matching pad of (a)

(c) Brittle fracture surface of a sheared ball (500 mm/s)

(d) Brittle fracture surface of the matching pad of (c)

(e) Brittle fracture surface of a pulled ball (50 mm/s)

(f ) Brittle fracture surface of the matching pad of (e)

Figure 45 Top-view comparison of brittle fracture surface during drop test, HS ball shear/pull tests (OSP, 0 hr of aging).

4 Correlation of High-Speed Ball Shear/Pull Tests and Board-Level Drop Test 243

(a) IMC failure during the drop test

(b) IMC fracture failure during the HS ball shear (500 mm/s)

(c) IMC fracture failure during the HS ball pull (50 mm/s)

Figure 46 Cross-sectional comparison of failure fracture during drop test, HS ball shear/pull tests (OSP, 0 hr of aging).

244 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests

(a) Brittle fracture surface of a solder joint

(b) Brittle fracture surface of the matching pad of (a)

(c) Brittle fracture surface of a sheared ball (500 mm/s)

(d) Brittle fracture surface of the matching pad of (c)

(e) Brittle fracture surface of a pulled ball (50 mm/s)

(f) Brittle fracture surface of the matching pad of (e)

Figure 47 Top-view comparison of brittle fracture surface during drop test, HS ball shear/pull tests (OSP, 500 hr of aging).

4 Correlation of High-Speed Ball Shear/Pull Tests and Board-Level Drop Test 245

(a) IMC failure during drop test

(b) IMC fracture failure during HS ball shear (500 mm/s)

(c) IMC fracture failure during HS ball pull (50 mm/s)

Figure 48 Cross-sectional comparison of failure fracture during drop test, HS ball shear/pull tests (OSP, 500 hr of aging).

246 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests 4.6 Correlation of High-Speed Ball Shear/Pull Tests and Drop Test In this section, some mathematical correlations relating the solder ball shear/pull and drop test results are presented. An innovative approach is graphically summarized in Figs. 49–54. Figures 49 and 50 relate the brittle fracture percentages from shear and pull solder ball testing to the drops to failure for the specific packages and drop test conditions used in this study. These two plots are achieved by plotting the drops-to-failure number for each time point against the equivalent data from shear or pull testing, followed by power law curve fitting. It should be noted that the curves in Figs. 49 and 50 correspond to the optimal shear or pull test speed. These curves can be employed to estimate the drops-to-failure number from the brittle fracture percentage obtained in either the ball shear or pull test with the optimal test

120

Drop Number

100 80 60 y = 50.778x −0.4659 40 20 0 0%

20% 40% 60% 80% Shear Brittle Failure Percentage (%)

100%

(a) ENIG (500 mm/s) 120 100 Drop Number

y = 34.492x −0.4659 80 60 40 20 0 0%

20%

60% 80% 40% Shear Brittle Failure Percentage (%)

100%

(b) OSP (500 mm/s)

Figure 49 Correlation of drops to failure and brittle failure percentage of ball shear test at optimal test speed.

4 Correlation of High-Speed Ball Shear/Pull Tests and Board-Level Drop Test 247 120

Drop Number

100 80 60 y = 52.324x −0.8212 40 20 0 0%

20% 40% 60% 80% Pull Brittle Failure Percentage (%)

100%

(a) ENIG (50 mm/s) 120 100 Drop Number

y = 31.009x −0.8212 80 60

2 1

40 20 0 0%

20%

40%

60%

80%

100%

Pull Brittle Failure Percentage (%) (b) OSP (5 mm/s)

Figure 50 Correlation of drops to failure and brittle failure percentage of ball pull test at optimal test speed.

speed. For demonstration purpose, the 5-mm/sec ball pull test curve in Fig. 50b is used as an example. If 60% of brittle fracture is observed during this kind of test, then a vertical line can be drawn at 60% on the horizontal axis and pointed upward as shown at point 1 of Fig. 50b. When the vertical line meets the designated curve, it can turn left and find its corresponding vertical coordinate (or just plug 60% into the fitting function), which is 39. About 60 drops to failure was estimated if 20% brittle failure was found in the ball pull test, as shown at point 2 in Fig. 50b. These values are the estimated drops-to-failure number for the same type of specimens subject to a mechanical drop test with the conditions used in the present study. In other words, the curves in Figs. 49 and 50 may provide us with a certain “prediction” capability to estimate the drop test results from the high-speed ball shear or pull test data.

248 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests 120 500 mm/s

Drop Number

100

Expon. (500mm/s)

80 60 y = 1.0144e 4.5087x 40 20 0 120%

100% 80% 60% 40% 20% Normalized Ball Shear Force (%)

0%

(a) ENIG 120 500 mm/s

Drop Number

100

Expon. (500 mm/s)

80 60 y = 1.011e 4.3796x 40 20 0 120%

100%

80%

60%

40%

20%

0%

Normalized Ball Shear Force (%)

(b) OSP

Figure 51 Correlation of drops to failure and normalized ball shear force with optimal test speed. Figures 51 and 52 show the exponential relationship between normalized solder ball shear force/energy and drops to failure for the brittle failure mode. Figures 53 and 54 show similar graphical summaries for the solder ball pull force/energy. The shear and pull data are from the optimal speed for each time point. The optimal shear speeds of SAC+ENIG and SAC+OSP are the same: 500 mm/sec. For the ball pull test, the optimal pulling speed of SAC+ENIG is 50 mm/sec, and the one of SAC+OSP is just 5 mm/sec. These graphs demonstrate that a moderate shift in the brittle fracture rate, fracture energy, or force for high-speed solder ball shear/pull testing can have a significant impact on the predicted drop testing lifetime. Moreover, it was found that the myriad configurations of devices using SAC solder ball tend toward the same exponential curve fit. This suggests that the relative relationship between drops to failure and shear/pull strength or energy is closely associated with the solder ball

5 Conclusions

249

120 500 mm/s Expon. (500 mm/s)

Drop Number

100 80 60

y = 1.0203e4.3196x 40 20 0 120%

100% 80% 60% 40% 20% Normalized Ball Shear Energy (%) (a) ENIG

0%

120 500 mm/s Expon. (500 mm/s)

Drop Number

100 80 60

y = 1.323e4.4896x 40 20 0 120%

100%

80%

60%

40%

20%

0%

Normalized Ball Shear Energy (%)

(b) OSP

Figure 52 Correlation of drops to failure and normalized ball shear energy with optimal test speed. alloy and not the specific package construction, such as the package size, package mass, and solder ball diameter. Then, the manufacturers might establish a solder ball shear/pull failure mode, fracture energy, or force acceptance criteria for a particular product based upon a similar analysis.

5

CONCLUSIONS The effects of test conditions on high-speed ball shear and pull tests were investigated in detail. The failure mechanisms of the solder ball shear/pull tests on the PBGA package with 0.76 mm lead-free SAC and lead–tin solder balls were discussed. Different substrate pad

250 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests 120 50 mm/s 100

Drop Number

Expon. (50 mm/s) 80 60 y = 1.0016e 4.3429x 40 20 0 120%

100% 80% 60% 40% 20% Normalized Ball Pull Force (%)

0%

(a) ENIG 120 5 mm/s

Drop Number

100

Expon. (5 mm/s)

80 60 y = 0.9552e4.4808x 40 20 0 120%

100% 80% 60% 40% 20% Normalized Ball Pull Force (%)

0%

(b) OSP

Figure 53 speed.

Correlation of drops to failure and normalized ball pull force with optimal test

finishes, ENIG and OSP, were used for the evaluation. The present study is performed to compare the high-speed ball shear/ball pull tests with the board-level mechanical drop test. The emphasis is placed on the correlation of the failure mode, force, and energy absorption between these two testing methods. The feasibility of using the high-speed ball shear/ball pull tests as an alternative method to evaluate solder joint integrity under dynamic loading was investigated. The detailed conclusions are summarized as follows: • The high-speed ball shear/pull tests were investigated in detail. Two kinds of solder

alloys (SAC405 and SnPb) with ENIG/OSP package substrate pad finishes were studied. The effects of testing conditions on the high-speed ball shear and pull tests were studied. As the shear and pull test speeds increased, a continual increase in the brittle failure mode percentage was observed, regardless of solder composition.

5 Conclusions

251

120 50 mm/s

Drop Number

100

Expon. (50 mm/s)

80 60 y = 1.0181e4.457x 40 20 0 120%

100% 80% 60% 40% 20% Normalized Ball pull Energy (%)

0%

(a) ENIG 120 5 mm/s

100

Drop Number

Expon. (5 mm/s) 80 60 y = 0.9963e4.5094x 40 20 0 120%

100% 80% 60% 40% 20% Normalized Ball pull Energy (%)

0%

(b) OSP

Figure 54 speed.

Correlation of drops to failure and normalized ball pull energy with optimal test

• The transition between ductile and brittle fracture occurred at higher shear and pull test

speeds for the BGA sample with SnPb solder, compared with the SAC solder samples. SAC solder showed more susceptibility to brittle fracture than SnPb solder alloy, in both the ball shear and pull tests. Solder joint fracture energy appears to correlate better to the failure mode for both the shear and pull tests. Fracture energy data suggest that a lower shear tool height reduces secondary interactions and extended deformation. • The high-speed pull test revealed the brittle interface with a greater sensitivity than in the ball shear test, especially for SnPb solders. High-magnification SEM/EDX analysis of the fracture surfaces indicated that solder residue on the package pad is more prevalent for SnPb solder compared to SnAgCu solder and for the high-speed shear test compared with the pull test. • The solder ball shear and ball pull tests produced a high incidence of brittle fracture with increasing test speed, independent of pad finish or aging time. OSP pads generated

252 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests more brittle solder joint failures in the high-speed ball shear/pull tests after thermal aging. Compared to the specimens with ENIG pad finish, the ball shear/pull strength and fracture energy of specimens with OSP decreased more rapidly with aging time. • Through detailed failure analysis and data comparison, correlation between the drop test and high-speed ball shear/pull tests is provided in this study. Thermal aging showed a significant effect on board-level drop reliability for both ENIG and OSP package substrate pad finishes. SAC + ENIG exhibits a longer drop impact life after thermal aging than SAC + OSP. • A comparison between brittle fracture interfaces from the drop test and those from high-speed shear ball shear and pull tests was performed. It was found that there is a striking similarity between the high-speed ball shear/pull tests and drop test from the top view and cross section of the fracture surfaces and failure locations. Thus, based on these microstructure observations, the high-speed ball shear and pull tests can be used as an indicator of board-level drop test performance. • Power law relationships can be obtained at different shear/pull testing speeds using the combined results of the drops to failure and brittle failure percentages of the ball shear/pull tests with thermal aging time. Drop test lifetimes can be estimated according to the brittle failure percentages obtained from the high-speed ball shear and pull tests. The exponential relationship between normalized solder ball shear force/energy and drops to failure for the brittle failure mode was also established in this study.

REFERENCES 1. Lemieux, P., “The Implication and Realities of Furnace Processing Using Lead-Free solder Systems,” Proc. 6th SMT/SMD Workshop, Shanghai, China, September 2001. 2. Suhir, E., “Accelerated Life Testing (ALT) in Microelectronics and Photonics: Its Role, Attributes, Challenges, Pitfalls, and Interaction with Qualification Tests,” Journal of Electronic Packaging, Vol. 124, No. 3, pp. 281–293. 3. Syed, A., and Pang, J. H. L., “Solder Joint Reliability: Materials, Modeling and Testing,” Proc. International Symposium on Advances in Packaging (APACK 2001), Singapore, December 2001. 4. Syed, A., and Doty, M., “Are We Over Designing for Solder Joint Reliability? Field vs. Accelerated Conditions, Realistic vs. Specified Requirements,” Proc. 49th Electronic Components & Technology Conference, San Diego, CA, May 1999, pp. 111–117. 5. Lau, J. H., and Lee, S. W. R., Chip Scale Package, McGraw-Hill, New York, 1999. 6. JESD22-B117A, “Solder Ball Shear,” JEDEC Solid State Technology Association, October 2006. 7. Coyle, R. J., Solan, P. P., and Serafina, A. J., “The Influence of Room Temperature Aging on Ball Shear Strength and Microstructure of Area Array Solder Balls,” Proc. 50th Electronic Components & Technology Conference, Las Vegas, NV, May 2000, pp. 160–169. 8. Huang, X., Lee, S. W. R., and Yan, C. C., “Characterization and Analysis on the Solder Ball Shear Testing Conditions,” Proc. 51th Electronic Components & Technology Conference, Lake Buena Vista, FL, May, 2001, pp. 1065–1071. 9. Lee, S. W. R., Yan, C. C., and Karim, Z., “Assessment on the Effect of Electroless Nickel Plating on the Reliability of Solder Ball Attachment to the Bond pads of PBGA Substrate,” Proc. 50th Electronic Components & Technology Conference, Las Vegas, NV, May 2000, pp. 868–873.

References

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10. Jang, S. Y., and Paik, K. W., “Comparison of Electroplated Eutectic Sn/Bi and Pb/Sn Solder Bumps on various UBM Systems,” Proc. 50th Electronic Components & Technology Conference, Las Vegas, NV, May 2000, pp. 64–68. 11. Sane, S., Tandon, S., and Chandran, B., “On Measurement of Effective Silicon Backend Strength using Bump Pull/Shear Techniques,” Proc. ASME InterPACK 2005, San Francisco, CA, July 2005. 12. Raiser, G. F., and Amir, D., “Solder Joint Reliability Improvement Using the Cold Ball Pull Metrology,” Proc. ASME InterPACK 2005, San Francisco, CA, July 2005. 13. Coyle, R. J., and Serafina, A. J., “Ball Shear Versus Ball Pull Test Methods for Evaluating interfacial Failures in Area Array Packages,” Proc. International Electronics Manufacturing Technology (IEMT) Symposium, San Jose, CA, July 2002, pp. 200–205. 14. JESD-B115A, “Solder Ball Pull,” JEDEC Solid State Technology Association, August 2010. 15. Newman, K., “BGA Brittle Fracture—Alternative Solder Joint Integrity Test Methods,” Proc. 55th Electronic Components & Technology Conference, Lake Buena Vista, FL, June 2005, pp. 1194–1200. 16. Chia, J. Y. H., et al., “The Mechanics of the Solder Ball Shear Test and the Effect of Shear Rate,” Materials Science and Engineering A, Vol. 417, 2006, pp. 259–274. 17. Chai, T. C., Wong, E. H., et al, “Board Level Drop Test Reliability of IC Packages,” Proc. 55th Electronic Components & Technology Conference, Lake Buena Vista, FL, June 2005, pp. 630–636. 18. Song, F. B., and Lee, S. W. R., “Investigation of IMC Thickness Effect on the Lead-Free Solder Ball Attachment Strength-Comparison between Ball Shear Test and Cold Bump Pull Test Results,” Proc. 56th Electronic Components & Technology Conference, San Diego, CA, June 2006, pp. 1196–1203. 19. Song, F. B., Lee, S. W. R., Newman, K., Sykes, B., and Clark, S., “High Speed Solder Ball Shear and Pull Tests vs. Board Level Mechanical Drop Tests: Correlation of Failure Mode and Loading Speed,” Proc. 57th Electronic Components & Technology Conference, Reno, NV, June 2007, pp. 1504–1513. 20. Brizer, C., et al., “Drop Test Reliability Improvement of Lead-Free Fine Pitch BGA Using Different Solder Ball Composition,” Proc. 55th Electronic Components & Technology Conference, Lake Buena Vista, FL, June 2005, pp. 1194–1200. 21. Yeh, C. L., and Lai, Y. S., “Insights into Correlation between Board-Level Drop Reliability and Package-Level Ball Impact Test,” Proc. 56th Electronic Components & Technology Conference, San Diego, CA, June 2006, pp. 455–461. 22. JESD22-B111, “Board Level Drop Test Method of Components for Handheld Electronic Products,” JEDEC Solid State Technology Association, July 2003. 23. JESD22-B110A, “Subassembly Mechanical Shock,” JEDEC Solid State Technology Association, Nov. 2004. 24. Song, F. B., Lee, S. W. R., Newman, K., Sykes, B., and Clark, S., “Brittle Failure Mechanism of SnAgCu and SnPb Solder Balls during High Speed Ball Shear and Cold Ball Pull Test,” Proc. 57th Electronic Components & Technology Conference, Reno, NV, June 2007, pp. 364–372. 25. Liu, Y. L., “High Temperature Aging Effects on Lead Free CSPs—Drop Test Reliability,” Proc. IPC/ JEDEC Conference 2006, Santa Clara, CA, August 2006. 26. Chiu, T. C., et al., “Effect of Thermal Aging on Board Level Drop Reliability for PbFree BGA Packages,” Proc. 54th Electronic Components & Technology Conference, Las Vegas, NV, June 2004, pp. 1256–1262.

254 Correlation between Package-Level Ball Shear/Pull and Board-Level Drop Tests 27. Xu, L., and Pang, J. H. L., “Effect of Imtermetallic and Kirkendall Voids Growth on Board Level Drop Reliability for SnAgCu Lead-Free BGA Solder Joint,” Proc. 56th Electronic Components & Technology Conference, San Diego, CA, June 2006, pp. 275–282. 28. Wong, E. H., et al, “Drop Impact: Fundamentals and Impact Characterization of Solder Joints,” Proc. 55th Electronic Components & Technology Conference, Lake Buena Vista, FL, June 2005, pp. 1202–1209.

CHAPTER

11

DYNAMIC MECHANICAL PROPERTIES AND MICROSTRUCTURAL STUDIES OF LEAD-FREE SOLDERS IN ELECTRONIC PACKAGING V. B. C. Tan, K. C. Ong, and C. T. Lim National University of Singapore Singapore

1

J. E. Field University of Cambridge Cambridge, United Kingdom

INTRODUCTION For more than 50 years, tin–lead (Sn–Pb) solder has been used almost exclusively throughout the world in the electronics industry to attach electronic components onto printed circuit boards (PCBs). However, there are concerns about the hazardous effects of lead on the environment. Once the electronic devices are discarded, the fear is that the lead will find its way into the garbage and landfill and may contaminate groundwater and the local environment. Japanese companies such as NEC, Hitachi, and Sony have been marketing some lead-free products since 2000 [1]. Hitachi, Sony, Fujitsu, and Matsushita have been lead free since 2002. In June 2000, the European Union (EU) introduced legislation to minimize lead usage and thus promote the use of lead-free solders [1, 2]. Finally, in the United States, the National Electronics Manufacturing Initiative (NEMI) has held lead-free initiative meetings since 1999. At the same time, the advancement of the portable electronics industry in the past 20 years has been astounding. Processing power that once required a whole room to house can now fit onto the palm of your hand. Greater portability also means that electronic devices are more prone to experiencing severe physical shock than before; for example, when electronic devices are dropped or struck. The increasing global demand for miniaturization, cost-effectiveness, and multifunctionality of electronic devices has encouraged the development of surface mount technology (SMT) to replace the less space-efficient through-hole technology (THT). With chip scale packaging (CSP) and ball grid array (BGA) technologies developing rapidly, the size and pitch of interconnects have also shrunk. As a result, solder interconnects play an ever more significant role in providing physical support. Zhu [3] found that impact-induced BGA solder interconnect cracking is the most dominant cause of failure in portable phone drop and tumble verification testing. Going “small” and “green” are the primary motivations for research into the dynamic mechanical properties and microstructure of lead-free solders in electronic packaging. Many organizations and institutions from Europe (IDEALS, ITRI, NPL), the United States (NEMI,

Structural Dynamics of Electronic and Photonic Systems Copyright © 2011 John Wiley & Sons, Inc.

Edited by Ephraim Suhir, David S. Steinberg and T. X. Yu

255

256 Dynamic Mechanical Properties and Microstructural Studies NIST), and Japan (JEITA)∗ have been doing similar research and have initiated studies to look for the best lead-free replacement for eutectic Sn–37Pb solder. Several solder compositions were short-listed by these institutions and organizations. With reference to their findings, two lead-free solders (binary Sn–3.5Ag and ternary Sn–3.8Ag–0.7Cu) were selected for the purposes of this research. Eutectic Sn–37Pb solder was also studied to serve as a benchmark for comparison. The Sn–Ag–Cu (tin–silver–copper) close eutectic ternary solder is the most promising and popular choice among many institutions [1, 4, 5]. The large-volume telecommunication industry has targeted this alloy [1]. The Sn–3.8Ag–0.7Cu solder was identified by the European IDEALS consortium as the best lead-free alloy for reflow due to its baseline advantages of reduced melting temperature (as compared to Sn–3.5Ag) and additional strengthening phase. It is also reported to have reliability equivalent to, if not better than, that of Sn–Pb and Sn–Pb–Ag solders [4]. Tin–Silver (Sn–3.5Ag) solder is another lead-free solder that is believed to have high potential [4] along with others such as Sn–Cu and Sn–Ag–Bi [5]. The Sn–3.5Ag solder is said to have good fatigue resistance and overall good joint strength [6]. With one of the longest histories of use as a lead-free alloy, it also has good mechanical properties and better solderability than Sn–Cu. Ford (Visteon Automotive Systems) has reported using Sn–3.5Ag solder successfully in production (module assembly) by wave soldering since 1989 [4]. This is due to its higher melting temperature (221◦ C) as compared to the tin–lead solder (183◦ C). The Sn–Ag solder has been used for many years in certain electronic applications [5] and thermal fatigue testing of the alloy has often shown it to be more reliable than Sn–Pb solder. There has been a great deal of research on different aspects of solder interconnect properties in the past decade. The emphasis has been on areas such as product-level tests [7, 8], boardlevel tests, simulation involving drop tests [9–11], bending tests [12, 13], thermomechanical effects [11, 14, 15], low-strain-rate tensile properties [15–17], creep and stress relaxation [18–21], vibration [22, 23], and microstructure [19, 20, 24]. There has been rising interest and emphasis on board-level and product-level drop tests due to increased awareness and major concerns surrounding possible failure caused by drop impact of portable electronic devices. The ultimate aim is to be able to predict the behavior and response of electronic devices when they are subjected to such loads and, in doing so, to improve their reliability. Research concerning solder deformation with varying strain rates is not new. However, experiments have always been conducted at relatively low strain rates. There have been several reports on the range of strain rates that solder interconnects experience during drop experiments: 1 × 10−5 to 1 × 10−3 sec−1 by Wei et al. [15], 2.66 × 10−5 to 1.33 × 10−2 sec−1 by Grivas et al. [16], and 1 × 10−5 to 0.1 sec−1 by Nose et al. [17]. During drop impact scenarios, solder joints experience deformation at high strain rates; consequently, the high-strain-rate response of solder material is needed. Geng [12] concluded that solder joint failure is dependent on strain rate, and at high strain rates, solder joints fail at lower board deflections. The report also agrees that traditional quasi-static bending experiments are not sufficient to quantify solder joint failure under shock loading. A better understanding of the dynamic response of solder materials is crucial, but to date high-strain-rate studies using Hopkinson bars are relatively sparse [25–37], and only Siviour et al. [30] and Williamson et al. [37] have researched lead-free solders. In their research, Williamson et al. [37] found that the high-strain-rate shear strengths of some lead-free BGA solder joints could not be predicted from extrapolation of low-strain-rate ∗

IDEALS, Improved Design Life and Environmentally Aware Manufacturing of Electronics Assemblies by Lead-Free Soldering; ITRI, International Tin Research Institute; NPL, National Physical Laboratory (UK); NEMI, National Electronics Manufacturing Initiative; NIST, National Institute of Standards and Technology; JEITA, Japan Electronics and Information Technology Industries Association.

2

Microstructure of Solder Joints

257

data. The reason was attributed to a change in failure mechanism (failure from within the bulk solder to at the solder–PCB interface). A novel tensile experiment, which tested the spall strength of the solder–PCB interface, confirmed this observation. These results further underline the need for experimental data at dynamic rates of strain.

2

MICROSTRUCTURE OF SOLDER JOINTS The microstructure of a material describes the constitution of that material down to the atomic level. It is important in the research of a material response because it provides a link between the mechanical behavior and physical structure of the material. Not many researchers of the microstructures of solder materials specifically state and show micrographs of solder grains and their grain boundaries. Instead, most focus on the size of different phases (e.g., tin-rich and lead-rich phases in Sn–Pb solder) in the solder rather than grain sizes. It has also been mentioned [38] that some published research on solder microstructures considers the diameters of the Sn or Pb phases to represent the grain size. However, the phase diameters are not equivalent to the diameters of the grain. By definition, a grain refers to an element of a material which has a single crystallographic domain. In a eutectic structure, many individual phase regions may, in fact, constitute a single eutectic grain. Description of a eutectic microstructure is not straightforward, especially in a solder (unlike single-phase materials) where individual grains are not readily apparent. By comparison, phase diagrams of Sn–Ag and Sn–Ag–Cu appear to be much more complex than that of Sn–Pb solder. As a result, it is a greater challenge to understand the microstructural behavior of Sn–Ag and Sn–Ag–Cu as compared to that of the simpler Sn–Pb solder. Unlike Sn–Pb solder, which has relatively clear definition of Sn-rich and Pb-rich areas, the Sn–Ag and Sn–Ag–Cu solders have complex intermetallic phases such as Ag3 Sn and Cu5 Sn6 . Wiese et al. [19] attributed the high level of creep resistance found in Sn–3.5Ag and Sn–4Ag–0.5Cu (as compared to Sn–37Pb solder) to small precipitates of these intermetallics, finely dispersed in the β-Sn matrix. In Sn–Pb solder, Sn and Pb solidify into a simple eutectic system of limited miscibility. This leads to a solid solution strengthened by Sn and Pb mixed crystals that have very similar deformation resistances. In contrast, the bimaterial systems Sn–Ag and Sn–Cu solidify in a complex system and in doing so form various intermediate phases. The two most significant intermetallics are Ag3 Sn and Cu6 Sn5 . Since the deformation resistances of Ag3 Sn and Cu6 Sn5 are much higher than that of the β-Sn matrix, the Ag3 Sn and Cu6 Sn5 phases act as hard particles within an inherently soft β-Sn matrix. These particles can slow down or even arrest mobile dislocations [20]. The ambient temperature shear strength of joints made from Sn–Ag–Cu solders is suggested [39, 40] to be weakened by Sn dendrites within the joint microstructure, especially by the coarse Sn dendrites in solute-poor Sn–Ag–Cu. Anderson [40] suggests that optical microscopy produces better micrographs as compared to scanning electron microscopy (SEM) in terms of revealing β-Sn dendritic structures. In Sn–Ag–Cu solder, Chen et al. [41] noted that binary and ternary eutectics are dispersed at the boundaries of these tin dendrites, including some large Ag3 Sn and Cu6 Sn5 intermetallic compounds. It is suggested that Cu6 Sn5 was found in the middle of the β-Sn dendrites, possibly behaving as a heterogeneous nucleation site. In his review of recently published papers on Sn–Ag–Cu lead-free solder materials by six different authors, Syed [42] noted great variation in the reported Young’s modulus of solder ranging from 10 to 50 GPa. This shows that there is no general agreement on the properties of lead-free solders. Solder, being used at high homologous temperatures, is subjected to creep most of the time. The three basic mechanisms that contribute to creep in metals are grain boundary sliding, dislocation slip and climb, and diffusional flow. It has been reported by Mavoori et al. [21] that grain boundary sliding and dislocation glide and climb are most active in solder. Wiese

258 Dynamic Mechanical Properties and Microstructural Studies and Meusel [19] reported that at room temperature Sn–37Pb and Sn–3.5Ag solders show nearly the same absolute creep rate at stresses beyond 15 MPa, whereas Sn–Ag–Cu solder only reaches that level of creep above 40 MPa. That Sn–Ag–Cu solder shows significantly higher creep resistance is suggested to be due to the presence of η-Cu6 Sn5 precipitates. Here, the different microstructures of bulk solder specimens obtained from casting at different cooling rates (SC, slow cooled; MC, moderately cooled; and QC, quenched cooled) were compared for each of the three materials studied: Sn–37Pb, Sn–3.5Ag, and Sn–3.8Ag–0.7Cu. The microstructures of commercially available 0.76-mm-diameter solder balls before and after reflow are also studied. A comparison between the solder specimens and virgin solder balls of the respective solder compositions was made to determine which cooling rate produced microstructures most similar to that of the solder balls before and after reflow.

3

SPECIMEN PREPARATION To reveal bulk solder and solder ball microstructure, the specimens were ground and polished before being etched. Specimens of solder were mounted in a co-cast resin and progressively ground using 320-, 600-, then 1200-grade silicon carbide abrasive papers until the surfaces of the solder specimens were flat and smooth, after which they were polished with 5 μm alumina solution (a mixture of water and Al2 O3 powder) to remove most of the scratches followed by 1 and 0.3 μm alumina to give a smooth and specular finish. Specimens were etched using the following solutions obtained from the literature [43]: SnPb SnAg SnAgCu

Diluted nitric acid (4%) (for several minutes) 2% HCl, 5% HNO3 , 93% isopropanol (for several seconds) 2% Nital (2% HNO3 , 98% isopropanol) (for several seconds)

Optical and scanning electron microscopy was used to study and acquire images of the specimen microstructures. Optical microscopy was used to perform visual inspection. Scanning electron microscopy was employed when higher magnification images were needed.

4

MICROSTRUCTURE OF Sn–37Pb SOLDER BALL SPECIMENS Micrographs obtained from SEM of virgin solder balls (Fig. 1) show “island”-shaped lead phases suspended in a tin solution. SEM micrographs of solder balls after reflow (Fig. 2) also show similar island-shaped lead phases; however, the sizes of these islands are smaller than those of the virgin solder balls. Virgin solder balls are formed by releasing droplets of molten solder into a cold medium, forcing the molten droplets to cool rapidly and form characteristic spherical shapes. Reflow uses a controlled temperature environment to slow down cooling rates and prevent electronic components from experiencing thermal shock. Solder balls which have gone through reflow were expected to possess lead phases of smaller sizes. However, a comparison between Figs. 1 and 2 seems to suggest that solder balls after reflow are cooled faster than virgin solder balls since the latter possess lead phases of larger size. A possible explanation is that the virgin solder could have been produced and left untouched for a long period. Since the solder balls are stored at room temperature, which is more than half of the absolute melting temperature of solder, significant aging could have occurred over this period of time, thus causing the virgin solder balls to have more coarse

5

(a)

Microstructure of Sn–3.5Ag Solder Specimens

(b)

259

(c)

Figure 1 Scanning electron micrographs of Sn–37Pb virgin solder balls.

(a)

Figure 2

(b)

(c)

Scanning electron micrographs of Sn–37Pb solder balls after reflow.

grains than solder balls after reflow. In the case of virgin and reflowed Sn–37Pb solder balls, MC bulk solder possesses the most similar microstructure.

5

MICROSTRUCTURE OF Sn–3.5Ag SOLDER SPECIMENS Although Sn–Pb solders have been studied extensively for the past few decades, knowledge of Sn–Ag is still quite limited. Besides having a much higher eutectic temperature of 221◦ C, Sn–Ag solder is also very different from Sn–Pb in terms of its phase fractions and the solubility behavior of the phases. Lead (Pb) accounts for more than 30% volume fraction in Sn–Pb solder whereas silver (Ag)–formed intermetallics (Ag3 Sn) comprise less than 4% of its total volume [44] in Sn–Ag solder. Also, Pb-rich phases in Sn–Pb solder are relatively ductile when compared to Ag3 Sn intermetallics, which are stronger and more brittle [45, 46]. SEM micrographs as given in Fig. 3 show the microstructure of virgin Sn–3.5Ag solder balls. They reveal a pool of Sn matrix dotted with spherical holes which used to be filled by specks of Ag3 Sn intermetallics, similar to QC bulk as-cast solder. The Sn–3.5Ag solder balls after reflow shown in Fig. 4 exhibit similar microstructures to those of the virgin solder balls. The holes that used to be filled with Ag3 Sn intermetallics here seem slightly larger than that of virgin solder balls. This matches well with the expectation that at a lower cooling rate during the reflow process the Ag3 Sn intermetallics are given slightly more time to nucleate, thus forming larger and longer Ag3 Sn phases as compared to virgin solder balls. It is apparent that the effect of aging on Sn–Ag and Sn–Ag–Cu solder balls is less significant when compared to Sn–Pb (see next section). This can be explained by the higher melting temperature of Sn–Ag solder, resulting in less significant aging effects. However, despite the slight differences, the microstructures both before and after reflow are best represented by the bulk as-cast solder formed following quench cooling.

260 Dynamic Mechanical Properties and Microstructural Studies

(b)

(a)

Figure 3

Scanning electron micrographs of virgin Sn–3.5Ag solder balls.

(a)

Figure 4

6

(c)

(b)

(c)

Scanning electron micrographs of Sn–3.5Ag solder balls after reflow.

MICROSTRUCTURE OF Sn–3.8Ag–0.7Cu SOLDER SPECIMENS Having a composition very similar to Sn–Ag solder, Sn–Ag–Cu solder has a slightly lower melting temperature of 217o C. This makes the Sn–Ag–Cu solder (as compared to Sn–Ag solder) more desirable for most of the electronic industry in their efforts to minimize cost and thermal shock damage to their electrical components during the reflow process. The addition of copper does more than just lower the melting point of the solder; adding another material to the binary alloy makes the phase diagram much more complex. Several researchers have focused their attention on the eutectic point of the ternary alloy [45, 48, 49]. The incorporation of copper also introduces a new intermetallic phase into the solder microstructure: Cu6 Sn5 [39, 47, 49, 50]. It is noticed from the difference in microstructure of Sn–Ag and Sn–Ag–Cu solders that Ag3 Sn intermetallics are more prominent and larger in Sn–Ag–Cu than in Sn–Ag solder. This was also noted by Kang et al. [50]. It is observed that in the presence of Cu larger and more numerous Ag3 Sn intermetallic phases are detected in the Sn–Ag solder. However, there is no report of any large difference when the content of Cu is changed from 0.35Cu to 0.7Cu wt % [50]. Only small amounts of copper may be needed to promote large Ag3 Sn plate formation. Thus, 0.35% Cu may be more than sufficient, since no significant differences were observed when 0.7% Cu was used [50]. Therefore there is no apparent relationship between the concentration of Cu and the formation of Ag3 Sn, but the presence of Cu does result in the formation of more and larger Ag3 Sn plates. Larger Ag3 Sn may result in a local reduction of Ag in some regions of the Sn–Ag–Cu solder. For example, it is suspected to reduce the Ag content dissolved in the β-tin dendrites, which in turn is believed to reduce its hardness [50]. Figures 5 and 6 show the microstructure of the solder balls before and after reflow, respectively. SEM micrographs in Fig. 5 show tiny spheres of Ag3 Sn that have not been etched out of the Sn matrix. The virgin solder balls exhibit similar microstructures to those

6 Microstructure of Sn–3.8Ag–0.7Cu Solder Specimens

(a)

Figure 5

(b)

261

(c)

Scanning electron micrographs of virgin Sn–3.8Ag–0.7Cu solder balls.

(a)

(b)

(c)

Figure 6 Scanning electron micrographs of Sn–3.8Ag–0.7Cu solder balls after reflow.

found in quench-cooled bulk Sn–Ag–Cu bulk solder (not shown). This again suggests that the virgin solder balls were formed by rapid cooling. However, the cooling rate must have been much higher than for the quenched specimens as the Ag3 Sn intermetallics seen in Fig. 5c appear significantly smaller. Solder balls after reflow (Fig. 6) exhibit a microstructure resembling that of Sn–Ag–Cu bulk solder samples formed from moderate cooling. Dendrite-like features similar to the βSn dendrites are observed. This signifies that it has experienced a more rapid cooling rate as compared to those Sn–3.8Ag–0.7Cu bulk solder specimens formed via moderate cooling [50]. Experiments have confirmed that for Sn–37Pb solder high cooling rates produce spherical Pb phases. When the cooling rate falls, the Pb phases will cluster and eventually form laminar layers with the Sn matrix. The Sn3 Ag intermetallics in Sn–3.5Ag and Sn–3.8Ag–0.7Cu solder are evenly dispersed as spheres when quench cooled. As the cooling rate decreases, the Sn3 Ag intermetallic phases become needle- or platelike. The presence of Cu and lower cooling rates encourages the growth of thicker Ag3 Sn intermetallics. The microstructures of virgin and reflowed solder balls were also studied and compared with the bulk solder microstructure. Table 1 states the method of cooling bulk solder which results in microstructures that most closely resemble the two types of solder balls.

Table 1

Microstructure of Bulk Solder Most Similar to Solder Balls before/after Reflow

Closest Matching Microstructure

Sn–37Pb

Sn–3.5Ag

Sn–3.8Ag–0.7Cu

Virgin solder balls Solder balls after reflow

Moderate cooling Moderate cooling

Quench cooling Quench cooling

Quench cooling Moderate cooling

262 Dynamic Mechanical Properties and Microstructural Studies 7

QUASI-STATIC MATERIAL PROPERTIES OF SOLDER SPECIMENS Bulk solder specimens prepared by casting were compressed at a strain rate of 8.3 × 10−4 sec−1 to study their response under quasi-static loading. A Shimadzu AG-25TB testing machine was used to perform the quasi-static compression tests. Specimens were approximately 21 mm in length and 7 mm in diameter. The specimens were loaded to 3% strain to keep within the limits of the strain gauges. Although the ASTM standard [51] states that an aspect ratio of 1.5–2 is sufficient, we found that an aspect ratio of 3, together with strain gauges, would yield more consistent results. Using an aspect ratio of more than 3 would cause buckling.

8

QUASI-STATICALLY COMPRESSED SOLDER SPECIMENS Emphasis is placed on the first 3% strain of the specimen since the strain gauges are most accurate in this region. The graphs shown represent the averages of three to five experiments performed on each specimen type. The histograms in Figures 7–9 display the mean Young’s modulus, yield stress, and tangential modulus of each type of specimen obtained from measuring the gradient of the stress–strain curves. Error bars reflect the maximum and minimum values from each pool of three to five specimens tested.

YOUNG’S MODULUS OF SOLDER SPECIMENS From the summary of Young’s modulus in Fig. 7, the lead-free solders (Sn–3.5Ag and Sn–3.8Ag–0.7Cu) appear to have a higher Young’s modulus than that of leaded solder. This is expected due to the low hardness of Pb (which is comparable to Sn in Sn–37Pb solder) as

50 45 40 Young’s Modulus, GPa

9

35 30 25 20 15 10 5 0

SC

Figure 7

MC Sn - 37Pb

QC

SC

MC Sn - 3.5Ag

QC

SC MC QC Sn - 3.5 Ag - 0.7Cu

Young’s modulus of bulk solders of three different compositions.

9 Young’s Modulus of Solder Specimens

263

40 35

Yield Stress, MPa

30 25 20 15 10 5 0

SC

MC

QC

SC

Sn - 37Pb

Figure 8

MC

QC

Sn - 3.5Ag

SC

MC

QC

Sn - 3.8 Ag - 0.7Cu

Yield stresses of bulk solder (0.2% strain offset).

Tangent Modulus (1% – 3% strain), MPa

250

200

150

100

50

0

SC

MC Sn - 37Pb

QC

SC

MC Sn - 3.5Ag

QC

SC MC QC Sn - 3.8 Ag - 0.7Cu

Figure 9 Tangent modulus of bulk solder between 1 and 3% plastic strain.

compared to the Ag and Cu components in lead-free solder, which possess stronger atomic bonds resulting in greater resistance during elastic deformation. The Sn–37Pb bulk solder shows an increase in Young’s modulus when cast at higher cooling rates. The large fluctuation in results obtained from moderately cooled specimens is due to the multiple-step cooling procedure. However, by comparing all three cooling rates

264 Dynamic Mechanical Properties and Microstructural Studies 50 45 40

Stress, MPa

35 30 25

SC MC QC

20 15 10 5 0 0

0.005

0.01

0.015 Strain

0.02

0.025

0.03

Figure 10 Stress–strain curves of bulk Sn–37Pb solder under quasi-static loading. (SC, MC, and QC), it is clear that a faster cooling rate yields a significantly higher Young’s modulus. Bulk Sn–3.5Ag solder appears to have a Young’s modulus which is inversely related to cooling rate. Figure 7 shows that higher cooling rates result in smaller grain sizes, and the Ag3 Sn intermetallics might have caused the elastic modulus to decrease. The elastic modulus of bulk Sn–3.8Ag–0.7Cu solder increases as cooling rate increases. This is shown in Fig. 10, with QC specimen having a Young’s modulus approximately 10 MPa higher than SC specimens.

10

YIELD OF SOLDER SPECIMENS The elastic–plastic transition of leaded Sn–37Pb solder is not sensitive to microstructure or cooling rate. This is observed in Fig. 10 where all three specimens of different microstructure have the same distinct transition. However, this is not the case in lead-free solder (Figs. 11 and 12). It is observed that the elastic–plastic transition of the lead-free solder becomes more gradual with larger microstructures (lower cooling rate). For lead-free solder, only QC specimens have a distinct transition. The MC and SC specimens undergo elastic–plastic transition in a more gradual manner. The elastic–plastic transition is the change of deformation mechanism from the stretching of interatomic bonds (elastic deformation) [52] to dislocation movement (plastic deformation) [53]. Plastic deformation in polycrystalline metals occurs by the glide of dislocations, and hence the critical shear stress at the onset of plastic deformation is the stress required to move dislocations [53]. However, in the case of the lead-free solders shown in Figs. 11 and 12, significant nonlinear microplasticity occurs in the preyield region due to limited dislocation motion. This means that for the two lead-free solders the occurrence of the two deformation mechanisms significantly overlap. The Sn–Pb solder and the quench-cooled lead-free solder, by comparison, have more distinction between the occurrences of the two deformation mechanisms. The two lead-free solders cast at high cooling rates must have achieved a microstructural state whereby the distinct transition to dislocation movement is permitted. Since the yield stress is defined as the point of initial departure from linearity of the stress–strain curve, the exact location of the yield point of lead-free solder can be ambiguous.

10

265

Yield of Solder Specimens

50 45 40

Stress, MPa

35 30 25 SC MC QC

20 15 10 5 0 0

0.005

0.01

0.015 Strain

0.02

0.025

0.03

Figure 11 Stress–strain curves of bulk Sn–3.5Ag solder under quasi-static loading. 50 45 40

Stress, MPa

35 30 25 SC MC QC

20 15 10 5 0 0

0.005

0.01

0.015 Strain

0.02

0.025

0.03

Figure 12 Stress–strain curves of bulk Sn–3.8Ag–0.7Cu solder under quasi-static loading. Here, the 0.2% strain offset method was used for all three materials in order to standardize yield stress identification. Using the 0.2% strain offset method shows that the lead-free solders have greater fluctuation in yield stress (Fig. 8) as compared to leaded solder, which shows only small differences in yield stress when cast at different cooling rates. The yield stress of bulk Sn–37Pb solder appears to be relatively independent of cooling rate. As seen in Fig. 8, the yield stresses of the three different Sn–37Pb solders, cooled at different rates, possess relatively similar values (of approximately 34 MPa), with limited variation. For Sn–3.5Ag solder, the yield stresses of the QC specimens are distinctly higher than the specimens cast at lower cooling (SC) rates. Figure 11 shows that, although the transition between elastic and plastic deformation is not distinct, the onset of significant

266 Dynamic Mechanical Properties and Microstructural Studies plastic deformation (dislocation movement) of QC specimens occurs at higher strains and at much higher stresses than those of the other Sn–3.5Ag solder specimens. This leads to much higher yield stresses. The yield stresses of the Sn–3.8Ag–0.7Cu solder specimens in Fig. 8 do not display any consistent trend between cooling rate and yield stress. Although QC specimens possess slightly higher yield stresses as compared to SC specimens, the yield stresses of the MC specimens show a sudden dip. Hence, it is not conclusive as to whether higher yield stresses result from higher cooling rate or not.

11

TANGENTIAL MODULUS OF SOLDER SPECIMENS

40

45

35

40

Stress at 3% Strain, MPa

Stress at 1% Strain, MPa

The summary of tangent modulus of plastic deformation, or plastic modulus (taken between 1 and 3% strain), of the three solders is shown in Fig. 9. A significant variation in strain hardening occurs in all three solders, making it difficult to observe any trends. However, the average plastic modulus of Sn–Ag–Cu solder is larger than the rest. The plastic modulus of the leaded Sn–37Pb solder is noticed to have a slightly higher variation (of about 120 MPa) compared to that of the lead-free solders (about 70 MPa). The Sn–Ag–Cu solder has a slightly higher average tangential modulus of plastic deformation than the other two materials. As dislocation movement is the mechanism for plastic deformation, work hardening during plastic deformation is caused by an increase in the glide resistance of these dislocations. This reduction in dislocation mobility could be due to interactions with other dislocations, particles within the solder, and/or grain boundaries in the polycrystalline material. A significant decrease in the plastic modulus of bulk Sn–37Pb solder is observed with a faster cooling rate, as seen in Fig. 9. The plastic modulus of slow-cooled specimens drops from 162 to 36 MPa when quench cooled. This implies that strain hardening of Sn–37Pb becomes less significant with smaller grains and smaller spherical Pb-rich phases. Charts of flow stress at 1 and 3% strain (Figs. 13(a and b) show that flow stress of bulk Sn–37Pb solder is also higher when slow cooled. This direct correlation between grain size and flow stress seems to imply a reverse Hall–Petch effect, but this is unlikely to be the case. Dislocation theory states that grain boundaries act as obstructions to dislocation movement. With smaller grain size and the presence of phased particles (in this case, Pb-rich phases), the number of grain boundaries increases, leading to more obstruction to dislocation motion; that is, flow stress increases due to strain hardening [52, 53] and thus the Hall–Petch relation. However, the result of QC

30 25 20 15 10 5 0

35 30 25 20 15 10 5 0

SC

MC (a)

QC

SC

MC (b)

QC

Figure 13 Quasi-static flow stresses of Sn–37Pb solder at (a) 1% strain and (b) 3% strain.

11

Tangential Modulus of Solder Specimens

267

40

45

35

40

Stress at 3% Strain, MPa

Stress at 1% Strain, MPa

specimens (more refined microstructure) seems to be reversed for the case of quasi-statically compressed Sn–37Pb bulk solder. The prominent creep effect seen in Sn–Pb solder may be the answer to this unexpected behavior. When compressed at such a low rate, significant grain boundary sliding occurs. With smaller grains, more grain boundary sliding may occur. In the case of Sn–37Pb solder, the weakening effect of grain boundary sliding may outweigh the strengthening effect of strain hardening caused by grain boundaries obstructing dislocation movement. Thus, instead of Sn–37Pb solder experiencing stronger strain hardening effect, smaller grain size results in weaker strain hardening. Results of Sn–3.5Ag specimens in Fig. 9 show that the tangent modulus of QC specimens is lower than that of slower cooled specimens. Findings by Wiese and Meusel [19] show that at stress levels greater than 15 MPa, creep rates of Sn–37Pb and Sn–3.5Ag solder (at room temperature) are very similar. This suggests that bulk Sn–3.5Ag solder also has a similar grain boundary sliding effect akin to Sn–37Pb solder, that is, with grain boundary sliding being dominant over strain hardening caused by obstruction to dislocation movement in specimens with smaller grain sizes. However, Sn–3.5Ag exhibits a less significant decrease in tangential modulus with decreasing grain size (compared to Sn–37Pb). This could be due to stronger precipitation strengthening by harder Ag3 Sn intermetallics [20] as compared to Pbrich phases. However, since QC specimens have higher yield stresses, Sn–3.5Ag solder cast at higher cooling rates continues to show higher flow stress up to 3% strain (Figs. 14a and b). Results of quench-cooled and slow-cooled Sn–3.8Ag–0.7Cu solder specimens show a significant increase in strain hardening with higher cooling rate as the specimens begin to deform plastically (Fig. 9). At the same time, the MC specimens once again show values different from specimens cast at the two extreme cooling rates. With reference to the two previous materials and findings from Wiese et al. [19], there is a good possibility that the results from MC are merely due to variations in casting conditions. The Sn–Ag–Cu solder shows significant creep only at much higher levels of stress (40 MPa) as compared to Sn–Pb and Sn–Ag solder (15 MPa) [19]. This implies that below 40 MPa, plastic deformation should be mainly dominated by obstruction to dislocation movement rather than grain boundary sliding. Thus, the effect of work hardening in Sn–Ag–Cu solder increases as the cooling rate rises. Figures 15a and b show increases in flow stress with higher cooling rates at 1 and 3% strain. Although MC specimens show lower yield stresses than SC specimens, they seem to possess higher work hardening rates, thus achieving higher flow stresses as compared to SC specimens (overtaking the flow stresses of SC specimens at 1.3% strain), but by 2.5% strain,

30 25 20 15 10 5 0

35 30 25 20 15 10 5 0

SC

Figure 14

MC (a)

QC

SC

MC

QC

(b)

Quasi-static flow stresses of Sn–3.5Ag solder at (a) 1% strain and (b) 3% strain.

268 Dynamic Mechanical Properties and Microstructural Studies 40

Stress at 3% Strain, MPa

Stress at 1% Strain, MPa

45 35 30 25 20 15 10 5 0

SC

MC

QC

50 45 40 35 30 25 20 15 10 5 0

SC

(a)

MC

QC

(b)

Figure 15 Quasi-static flow stresses of Sn–3.8Ag–0.7Cu solder at (a) 1% strain and (b) 3% strain. the work-hardening rate of MC specimens appears to have reduced and become similar to the others (Fig. 12).

12

DYNAMIC MATERIAL PROPERTIES OF SOLDER SPECIMENS A compressive split Hopkinson pressure bar (SHPB) [54] was used in this project to determine the dynamic response of solder specimens. A typical SHPB comprises three coaxial cylindrical bars of identical cross section and material—a striker bar, an input bar, and an output bar. A small disc of specimen material (of thickness two orders of magnitude smaller than the length of the bars) is sandwiched between the input and output bars. The striker bar is propelled and impacted against the incident bar to create a stress pulse in the input bar. This incident pulse travels down the input bar toward the specimen and is partially reflected back up the input bar and partially transmitted across the specimen into the output bar. The magnitudes of the reflected and transmitted pulses are dependent on the properties of the specimen. Strain gauges mounted at midlength of the input and output bars measure the incident, reflected, and transmitted pulses which can then be used to infer the stress–strain response of the specimen material. The idea of using two Hopkinson bars to measure dynamic properties of materials in compression was developed by Taylor [55], Volterra [56], and Kolsky [57]. An extensive list of references pertaining to the study of the SHPB can also be found in a review by Field et al. [58]. Wang et al. [25] and Siviour et al. [35] obtained strain rates reaching up to a maximum of 3000 sec−1 from SHPB experiments on solder material. However, numerical simulation by Ong [59] shows that certain parts of the solder balls will experience higher strain rates (close to 10,000 sec−1 ) when the solder balls are compressed at a deformation rate of approximately 5 m/sec. Different striker bar velocities ranging from 5 to 15 m/sec were used with the different specimen lengths to attain strain rates ranging from 102 to 104 sec−1 . The specimens prepared for the Hopkinson bar tests had an aspect ratio of 1. The specimen lengths ranged from 2 to 9 mm. Approximately 30 specimens were tested per material per cooling rate.

13

MECHANICAL PROPERTIES OF Sn–37Pb, Sn–3.5Ag, AND Sn–3.8Ag–0.7Cu SOLDER BALLS Virgin Sn–37Pb, Sn–3.5Ag, Sn–3.8Ag–0.7Cu solder balls of 0.76 mm in diameter were obtained from manufacturers and tested without additional treatment. The solder balls were compressed at quasi-static and dynamic rates and their responses recorded and compared.

13

Mechanical Properties of Sn–37Pb, Sn–3.5Ag, and Sn–3.8Ag–0.7Cu Solder Balls

269

The more sensitive Instron Micro-Force tester and load cell were used to perform the quasi-static compression test on the solder balls as compared to bulk specimen tests. The loading rate was set to be 0.038 mm/min. As for the dynamic experiments, a miniaturized SHPB set-up with bars of 5 mm diameter was used. Since the solder balls were much smaller than the diameter of the Hopkinson bar (0.76 vs. 5 mm), the transmitted stress wave was very small in magnitude. Thus, semiconductor strain gauges (with gauge factors of 120) were used for their greater sensitivity to the transmitted strain signal. Figure 16 shows typical loading curves of 0.76-mm virgin solder balls at two different compression rates—a slow rate of 3.6 × 10−5 m/sec and a high rate of 12.5 m/sec for each solder composition. Similar data on lead-free solder balls were reported by Siviour et al. at 8 m/sec [35]. Since they are spheres and not cylindrical specimens with uniform crosssectional area), force–displacement curves are shown instead of stress–strain curves. The curves clearly illustrate how the solder balls respond to a difference in compression rate; a greater degree of force was required to deform the solder balls at the high strain rate as compared to the low strain rate. A summary of experiments done at different strain rates on the three types of virgin solder balls is shown in Fig. 17. The result clearly shows the strain rate dependence of the force required to deform a solder ball to half of its original diameter (0.38 mm). This confirms the response found in bulk SHPB experiments; dynamically loaded bulk solder specimens require distinctly higher stresses to deform them than do quasi-statically loaded specimens. These results are in agreement with those from [35] on bulk solder properties, which again showed an increase in Sn–Pb solder strength with strain rate and also with reduction in temperature. The same was true of Sn–Ag, albeit to a lesser extent; see Figs. 18 and 19.

300 SnPb at 3.6 × 10−5 m/s

250

SnAg at 3.6 × 10−5 m/s SnAgCu at 3.6 × 10−5 m/s SnPb at 12.5 m/s

200 Force, N

SnAg at 12.5 m/s SnAgCu at 12.5 m/s 150

100

50

0 0

0.00005 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004 0.00045 0.0005 0.38 mm Displacement, m

Figure 16 Force vs. displacement graph of virgin solder balls undergoing low (3.6710 × strain rates (12.5 m/sec).

−5

m/sec) and high

270 Dynamic Mechanical Properties and Microstructural Studies 160

Force at 0.38 mm deformation, N

140 120 100 80 60 SnPb SnAg SnAgCu

40 20

0.0001

0 0.01 0.1 1 Compression Rate, m/s

0.001

10

100

Figure 17 Plot of force required for 0.38 mm deformation of solder balls at different compression rates.

140

200

2720 ± 80 s−1 100

940 ± 35 s−1 450 ± 5 s

50

−1

True stress, MPa

True stress, MPa

120 150

100 80 60

−40°C 20°C 60°C

40 20

0

0

0.05 0.1 0.15 0.2 0.25 0.3

0

0

0.02

0.04

True strain

True strain

(a)

(b)

0.06

Figure 18 (a) Plot of room temperature Sn–Pb compressive stress–strain behavior at three different dynamic strain rates; (b) at three different temperatures and strain rate of 950 ± 50 sec−1 . (Reprinted with permission from [35].) Their research showed Sn–Pb to be much more sensitive to strain rate than the lead-free solders. At low temperatures there was evidence that the tin-rich solders had experienced a phase change (from β- to α-tin below 13.4◦ C). The plateau region above 10% strain was observed for all solders studied. Similar to Fig. 17, the shear strengths of BGA solder joints have been found to have dependency on loading rate; see Fig. 20.

200

200

150

150

100

3600 ± 40 s−1 1500 s−1 955 ± 5 s−1 570 ± 40 s−1

50

0

0

True stress, MPa

True stress, MPa

Mechanical Properties of Sn–37Pb, Sn–3.5Ag, and Sn–3.8Ag–0.7Cu Solder Balls

100 −40 ± 1°C 23 ± 1°C 60 ± 1°C

50

0

0.05 0.1 0.15 0.2 0.25 0.3 True strain (a)

271

0

0.02

0.04 0.06 True strain (b)

0.08

Figure 19 (a) Plot of room temperature Sn–Ag compressive stress–strain behavior at three different dynamic strain rates; (b) at three different temperatures and strain rate of 980 ± 40 sec−1 . (Reprinted with permission from [35].)

100

100 SnAg SnPb

Maximum Shear Force, N

Maximum Shear Force, N

13

10

1 0.001

1000 0.1 10 Blade Speed, mm/min (a)

105

Castin SnAgCu

10

1 0.001

1000 0.1 10 Blade Speed, mm/min

105

(b)

Figure 20 Shear force required for a blade to remove BGA solder joints over a range of loading rates (quasi-static to dynamic) for (a) binary alloys and (b) higher order alloys. (Reprinted with permission from [37].)

Figure 20 illustrates the previous point, that the high-rate data cannot be reliably extrapolated from low-rate data. In the above, the reduced strength of the higher order alloys at the greatest loading rate was found to derive from a change in failure mode: from bulk solder failure to interfacial failure at the solder–PCB interface. The comparison between the microstructures of solder balls before and after reflow, with the microstructure of bulk solder cast using different cooling rates was shown in Table 1. The microstructures obtained from bulk solders which are most similar to the microstructures of the solder balls before and after reflow are listed there.

272 Dynamic Mechanical Properties and Microstructural Studies 14

CONCLUSIONS Three distinctly different microstructures were obtained by cooling the commonly used tin–lead solder, Sn–37Pb, and two lead-free solder materials, Sn–3.5Ag and Sn–3.8Ag–0.7Cu, at three different cooling rates. It is found that for fast cooling rates Pb phases in Sn–37Pb solder specimens tend to form spheres. At low cooling rates, Pb phases tend to cluster into laminar layers. For Sn–3.5Ag and Sn–3.8Ag–0.7Cu solder specimens, needle/plate-shaped Ag3 Sn intermetallics increase in length at slower cooling rates. Slower cooling rates and the presence of Cu appear to encourage the growth of Ag3 Sn intermetallic phases. The closest bulk solder microstructural matches to virgin and reflowed solder balls are given in Table 1. Quasi-static compression experiments reveal that: • Sn–3.5Ag and Sn–3.8Ag–0.7Cu lead-free solder specimens have a higher Young’s

modulus than Sn–37Pb solder specimens. • The yield stresses of Sn–3.5Ag and Sn–3.8Ag–0.7Cu lead-free solder specimens are

significantly more dependent on microstructure than Sn–37Pb solder specimens. • The tangent modulus of Sn–37Pb solder specimens between 1 and 3% strain is sig-

nificantly more dependent on microstructure than Sn–3.5Ag and Sn–3.8Ag–0.7Cu lead-free solder specimens. • The flow stresses of lead-free Sn–Ag and Sn–Ag–Cu solder specimens increase with cooling rate, whereas the flow stress of Sn–Pb solder specimens appears to decrease with cooling rate. In the dynamic SHPB experiments: • Distinct differences were observed between quasi-static and dynamic properties of

solder; all required measureably greater forces to cause deformation at high strain rates as compared to low strain rates. • Generally, dynamic loading of all solder specimens showed higher rates of strain hardening and higher maximum flow stresses with faster cooling rates. • There are also important differences with changes of temperature. Lower temperatures are analogous to higher rates and vice versa. The results clearly show a need for experimental data to be taken at high rates of strain, which are relevent to the loading environment portable electronic devices experience when subject to being dropped or struck.

ACKNOWLEDGMENTS The research on this topic at the National University of Singapore (NUS), the University of Cambridge, UK, and the Institute of Microelectronics (IME), Singapore, was part of a collaborative project. We thank E. H. Wong (IME) for overseeing the program and also the support of Micron Technology Foundation. We thank Dr. S. M. Walley (Cambridge) for preparing a database of references and Dr. D. M. Williamson and Dr. C. R. Siviour for their many inputs to the research at Cambridge. DMW and SMW are also thanked for their valuable comments on the manuscript.

References

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34. Gomatam, R. R., and Sancaktar, E., “The Interrelationships between Electronically Conductive Adhesive Formulations, Substrate and Filler Surface Properties, and Joint Performance. 1: The Effects of Adhesive Thickness,” Journal of Adhesion Science and Technology, Vol. 18, pp. 1225–1243, 2004. 35. Siviour, C. R., Walley, S. M., Proud, W. G., and Field, J. E., “Mechanical Properties of SnPb and Lead-Free Solders at High Rates of Strain,” Journal of Physics D: Applied Physics, Vol. 38, pp. 4131–4139, 2005. 36. Zhi, J. Z., Zheng, J., Zhao, Q. L., and Yao, Z. J., “Dynamic Strength and Constitutive Equation of Sn-Ag Pb-Free Solders,” in T. D. Wen, Ed., Proc. 6th Int. Symp. on Test and Measurement, International Academic Publishers, Beijing, (2005), pp. 4263–4266. 37. Williamson, D. M., Field, J. E., Palmer, S. J. P., and Siviour, C. R., “Rate Dependent Strengths of Some Solder Joints,” Journal of Physics D: Applied Physics, Vol. 40, pp. 4691–4700, 2007. 38. Frear, D. R., et al., The Mechanics of Solder Alloy Interconnect, Van Nostrand Reinhold, New York, 1994. 39. Xiao, Q., Bailey, H. J., and Armstrong, W. D., “Aging Effects on Microstructure and Tensile Property of Sn3.9Ag0.6Cu Solder Alloy,” Journal of Electronic Packaging, Vol. 126, pp. 208–212, June 2004. 40. Anderson, L. E., Cook, B. A., Harringa, J., and Terpstra, R. L., “Microstructure Modifications and Properties of Sn-Ag-Cu Solder Joints Induced by Alloying,” Journal of Electronic Materials, Vol. 31, No. 11, pp. 1166–1174, 2002. 41. Chen, Z. G., Shi, Y. W., Xia, Z. D., and Yan, Y. F., “Study on the Microstructure of a Novel Lead-Free Solder Alloy SnAgCu-Re and Its Soldered Joints,” Journal of Electronic Materials, Vol. 31, No. 10, pp. 1122–1128, 2002. 42. Syed, A., “Accumulated Creep Strain and Energy Density Based Thermal Fatigue Life Prediction Models for SnAgCu Solder Joints,” Proceedings, 54th Electronic Components and Technology Conference, June 1–4, 2004, Las Vegas, Nevada, pp. 737–746. 43. Petzow, G., Metallographic Etching, 2nd ed., The Materials Information Society, Portland, OR, 1999. 44. Choi, S., Subramanian, K. N., Lucas, J. P., and Bieler, T. R., “Thermomechanical Fatigue Behavior of Sn-Ag Solder Joints,” Journal of Electronic Materials, Vol. 29, No. 10, pp. 1249–1257, 2000. 45. Kim, K. S., Huh, S. H., and Suganuma, K., “Effects of Cooling Speed on Microstructure and Tensile Properties of Sn-Ag-Cu Alloys,” Materials Science and Engineering A, Vol. A333, pp. 106–114, 2002. 46. Maveety, J. G., Liu, P., Vijayen, J., Hua, F., and Sanchez, E. A., “Effect of Cooling Rate on Microstructure and Shear Strenght of Pure Sn, Sn–0.7Cu, Sn–3.5Ag, and Sn–37Pb Solders,” Journal of Electronic Materials, Vol. 33, No. 11, pp. 1355–1362, 2004. 47. Chada, S., Fournelle, R. A., Laub, W., and Shangguan, D., “Copper Substrate Dissolution in Eutectic Sn-Ag Solder and Its Effect on Microstructure,” Journal of Electronic Materials, Vol. 29, No. 10, pp. 1214–1221, 2000. 48. Liu, C. M., Ho, C. E., Chen, W. T., and Kao, C. R., “Reflow Soldering and Isothermal Solid-State Aging of Sn-Ag Eutectic Solder on Au/Ni Surface Finish,” Journal of Electronic Materials, Vol. 30, No. 9, pp. 1152–1156, 2001. 49. Moon, K.-W., Boettinger, W. J., Kattner, U. R., Biancaniello, F. S., and Handweker, C. A., “Experimental and Thermodynamic Assessment of Sn-Ag-Cu solder Alloys,” Journal of Electronic Materials, Vol. 29, No. 10, pp. 1122–1136, 2000.

276 Dynamic Mechanical Properties and Microstructural Studies 50. Kang, S. K., Choi, W. K., Shih, D. Y., Henderson, D. W., Gosselin, T., Sarkhel, A., Goldsmith, C., and Puttlitz, K. J., “Ag3 Sn Plate Formation in the Solidification of NearTernary Eutectic Sn-Ag-Cu,” JOM , June 2003, pp. 61–65. 51. ASTM E9, “Standard Test Methods of Compression Testing of Metallic Materials at Room Temperature,” American Society for Testing and Materials, Philadelphia, PA, reapproved 2000. 52. Callister, W. D., Jr., Materials Science and Engineering : An Introduction, 4th ed., Wiley, New York, 1997. 53. Hull, D., and Bacon, D. J., Introduction to Dislocations, 4th ed., Butterworth Heinemann Woburn, MA, 2001. 54. Hopkinson, B., “A Method of Measuring the Pressure Produced in the Detonation of High Explosives or by the Impact of Bullets,” Philosophical Transactions of the Royal Society A, Vol. 213, pp. 437–456, 1914. 55. Taylor, G. I., “The Testing of Materials at High Rates of Loading,” Journal of the Institution of Civil Engineers, Vol. 26, pp. 486–519, 1946. 56. Volterra, E., “Alcuni Risultati di Prove Dinamichi sui Materiali,” Riv Nuovo Cimento, Vol. 4, pp. 1–28, 1948. 57. Kolsky, H., “An Investigation of the Mechanical Properties of Materials at Very High Rates of Loading,” Proceedings of the Physical Society B , Vol. 62, pp. 676–700, 1949. 58. Field, J. E., Walley, S. M., Proud, W. G., Goldrein, H. T., and Siviour, C. R., “Review of Experimental Techniques for High Rate Deformation and Shock Studies,” International Journal of Impact Engineering, Vol. 30, pp. 725–775, 2004. 59. Ong, K. C., Tan, V. B. C., Lim, C. T., Wong, E. H., and Zhang, X. W., “Dynamic Materials Testing and Modeling of Solder Interconnects,” Proceedings, 54th Electronic Components and Technology Conference, June 1–4, 2004, Las Vegas, Nevada, pp. 1075–1079.

CHAPTER

12

FATIGUE DAMAGE EVALUATION FOR MICROELECTRONIC COMPONENTS SUBJECTED TO VIBRATION T. E. Wong Raytheon Company El Segundo, California

1

INTRODUCTION The demand for surface mount packages and assemblies has become the mainstream in today’s electronic packaging industry. These packages include the leaded and leadless components, such as gull-wing lead quad flat pack (QFP), ball grid array (BGA), chip scale package (CSP), direct chip attach (DCA), and leadless chip capacitor/resistor/carrier, and are directly soldered onto the substrates without the need for through holes in the printed wiring board (PWB). This reduces the substrate size, complexity, and cost and facilitates automation of the substrate assemblies. A gull-wing lead soldered onto the PWB, shown in Fig. 1, is one of the techniques used for surface mount packages. In general, the lead is more vulnerable than the solder joint when the package is subjected to vibration. For leadless packages, Fig. 2 shows a BGA attached onto the PWB with the solder balls. A lack of compliant interconnect between the BGA package and the PWB results in a higher failure risk in the BGA solder joints in vibration environments. One of the possible ways to minimize the lead and the solder ball vibration fatigue damages is to underfill the packages. The effects of the underfill materials on the vibration durability/reliability of the lead and the BGA solder joint are then required to be evaluated. Vibration fatigue life predictive models of the packages with and without underfills have been developed for this evaluation, and it is the objective of the present study to address these models. In the life prediction model development, test vehicles (TVs), on which various sizes of BGAs and leaded components with daisy-chained interconnects are soldered, are first designed, fabricated, and subjected to random vibration tests. A destructive physical analysis (DPA) is then followed to verify the failure locations and the failure modes of the interconnects. Next, three-dimensional finite element models (FEMs) combined with linear dynamic and static analyses have been developed to calculate the effective strains of the interconnects. Finally, vibration fatigue life prediction models have been established and the TV test results combined with the derived effective strains of the interconnects are used to calibrate the proposed life prediction models. In the calculation process, several FORTRAN programs, in conjunction with the outputs obtained from MSC/NASTRANTM static and frequency response analyses, were developed and then used to perform the required computations. Some understanding of lead and solder joint vibration fatigue and life-predicting capabilities can be obtained from Barker et al. (1990, 1991), Basaran and Chandaroy (1999), Jih et al. (1993, 1998), Lau and Pao (1997), Lee and Ham (1999), Liguore and Followell (1995), Structural Dynamics of Electronic and Photonic Systems Copyright © 2011 John Wiley & Sons, Inc.

Edited by Ephraim Suhir, David S. Steinberg and T. X. Yu

277

278 Fatigue Damage Evaluation for Microelectronic Components Subjected to Vibration Toe tp Toe

Foot Length

Horizontal Load Length (Thigh)

Vertical Load Length (Shin)

Ankle Radius

Toe Angle °

Air Gao

Knee Radius

(a) QFP Configuration

Y Z

X

(b) Solder Joint Assembly

Figure 1

Gull-wing lead solder joint configuration.

BGA Package Solder Joint PWB

Figure 2

BGA solder joint assembly.

Pitarresi and Akanda (1993), Steinberg (1988), Wong et al. (1997a,b, 1999a,b, 2000a–c, 2001–2005), and Yang et al. (1998). A theoretical background of the random vibration can be found in Elishakoff (1983), Suhir (1997), Lin (1973), Meirovitch (1967), Shinozuka and Jan (1972), and Wirsching et al. (1995).

2

TEST VEHICLE DESIGN Each of two TVs, shown in Fig. 3 and 4 as TV1 and TV2, respectively, has been designed and constructed. These TVs consist of one single-sided circuit card assembly (CCA) and

2 352-pin TBGA

304-pin CBGA

600-pin SBGA

313-pin PBGA

Test Vehicle Design

279

Header

44-lead TSOP 56-lead SSOP

30-lead ceramic cap

28-pin CLCC 240-lead PQFP

20-lead PLCC (J) 46-pin CSP 96-pin DCA (perimeter) FC

RF48-pin CBGA

Connector

Figure 3

Wedge Lock Location

Commercial off-the-shelf test vehicle 1 (TV1) circuit card assembly.

352-pin TBGA 1.27 mm 600-pin SBGA 1.27 mm

313-pin PBGA 1.27 mm 256-pin PBGA 1.00 mm 280-pin flexBGA A0.8 mm

96-pin flip chip 0.46 mm 2-pinv chip capacitor

580-pin PBGA 1.00 mm

360-pin ceramic package w/SCI 1.27 mm 28-pin LCCC 1.27 mm

Figure 4

28-pin ceramic flat pack 1.27 mm

Commercial off-the-shelf test vehicle 2 (TV2) circuit card assembly.

one aluminum heat sink. The CCA is bonded on one side of the aluminum heat sink with an extremely compliant adhesive whose elastic modulus is only 2.76 MPa. These TVs, configured in this manner, are easier to rework with this compliant adhesive. Various types of commercial off-the-shelf (COTS) surface-mounted electronic components are soldered onto the CCAs.

280 Fatigue Damage Evaluation for Microelectronic Components Subjected to Vibration In the TV1 design, three design/manufacturing process parameters (shown in Table 1), each with either two or three levels of variation, are three underfilled types (including no underfilled material), two coating types (including no coating), and two PWB materials, which result in 12 different types of TV1 configurations (shown in Table 2). Each type of configuration includes two test modules (see the last column in Table 2). A total of 24 singlesided CCAs have been built for vibration tests. The elastic moduli of the underfills in Table 1 are 8.96 GPa for non-reworkable 1 and 0.07 GPa for reworkable 1, and the conformal coating is parylene. Table 3 is a BGA/CSP/DCA parts list spreadsheet of TV1 in which the reference designation for each device is also included, for example, U1 corresponds to 304-pin CBGA. Part configurations of BGA/CSP/DCA are also summarized in this table. The rest of the packages Table 1

TV1 Design/Manufacturing Process Parameters and Their Variation Levels

Parameter A B C

Description

Level 1

Level 2

Level 3

PWB material Conformal coating Underfill material

Polyimide Parylene None

Thermount None Non-reworkable 1

— — Reworkable 1

Table 2 Type

Types of TV1 Design Configures Configuration PWB Coating Underfill TV1-1 Polyimide Parylene None TV1-2 Non-reworkable TV1-3 Reworkable 1 TV1-4 None None TV1-5 Non-reworkable TV1-6 Reworkable 1 TV1-7 Thermount Parylene None TV1-8 Non-reworkable TV1-9 Reworkable 1 TV1-10 None None TV1-11 Non-reworkable TV1-12 Reworkable 1

a Tested

Table 3

1

1

1

1

Module No. 20a, 21 26, 27 23, 24 29a, 30a 35, 36a 32, 33a 2a, 3 8, 9 5, 6 11, 12 17, 18 14, 15

up to 72 hr, all other modules only tested up to 2 hr.

TV1 BGA/CSP/DCA Parts List Spreadsheet

Component

Reference Designation

Pitch (mm)

Ball Diametr (mm)

Body Size (mm)

Die Size Squared (mm)

CBGA-304 SBGA-600 PBGA-313 TBGA-352 CSP-46 DCA-96

U1 U3 U4 U5 U11 U12

1.27 1.27 1.27 1.27 0.75 0.46

0.89 0.75 0.76 0.75 0.40 0.20

21 × 25 45 × 45 35 × 35 35 × 35 5.76 × 7.87 12.7

N/A N/A 7.73 8.5 5.36 × 7.46 N/A

2

Test Vehicle Design

281

mounted on the TV1 are the components of gull-wing lead, J-lead, and leadless chip carrier. The location of each part mounted on the TV1 is shown in Fig. 3. The BGA/CSP/DCA solder joint failures most likely occurred at the solder joints at the package corners during vibration tests. To determine the solder joint failure locations, the solder joints at each package corners are combined as one group. Figure 5 illustrates the daisy chain pattern of the 313-pin PBGA with a total of 10 groups. In TV1, the 600-pin SBGA will be excluded from the evaluation since its PWB solder pad is offset due to a design error. Table 4 lists the information of the 313-pin PBGA, 304-pin CBGA, 352-pin TBGA, and 600-pin SBGA, whose solder balls’ pitch is 1.27 mm, and their solder joint assemblies. For the 313-pin PBGA, whose microsection of solder joint assembly is shown in Fig. 6, the solder balls are made of 63Sn/37Pb solders. Figure 7 shows

1

1

3 2

5 4

7 6

9 11 13 15 17 19 21 23 25 8 10 12 14 16 18 20 22 24

2

A B C D E

5

G J

F

6

H K

L

10

M

N R

P T

7

U V W Y AA AB AC AD AE

4 Figure 5 Table 4

8

3

9

A 313-pin PBGA daisy chain pattern.

TV1 BGA Geometry and Solder Joint Assembly

Package

Array

PWB Pad Diameter Design Measure

Microsection Pad Diameter Solder PWB Package Joint Height

304-pin CBGA 600-pin SBGA 313-pin PBGA 352-pin TBGA

Full matrix (19 × 16) Perimeter (5 rows/side) Full matrix (staggered) Perimeter (4 rows/side)

0.6604

0.7112

0.7112

0.6198

0.6731

0.7620

0.7112

0.7188

0.5791

Offset

0.9398

0.8890

0.8890

0.5080

0.5334

0.7620

0.7112

0.7201

0.5799

0.5588

Note: Unit: mm.

282 Fatigue Damage Evaluation for Microelectronic Components Subjected to Vibration PBGA Substrate

PWB

Figure 6 Microsection of 313-pin PBGA solder joint in TV1. CBGA Substrate

90Pb/10Sn Solder Stub

PWB

Figure 7

Microsection of 304-pin CBGA solder joint in TV1.

the microsection of the 304-pin CBGA solder joint assembly. The solder joint consists of 90Pb/10Sn solder stub (in dark) with eutectic solder (in white) on the bottom of the stub. The use of a finer pitch (≤1 mm) BGA in electronic package design becomes increasingly more popular due to the significant reduction in substrate interconnect area. The smaller the pitch, the smaller the BGA solder ball. This condition can further reduce the already low compliance interconnect between the BGA component and the substrate and could result in a higher failure risk in the BGA solder joint in vibration environments. To further study this risk, TV2, on which various sizes of BGA daisy-chained packages (with 0.8-, 1.0-, and 1.27-mm pitches) are soldered, is then designed and fabricated. TV2 includes three design/manufacturing process parameters, each with either two or three levels of variation, which are summarized in Table 5. Three underfilled types (including no underfilled material), three solder pad sizes, and two rework part types (including no rework part) were tested. Note Table 5 Parameter D E F a

TV2 Design/Manufacturing Process Parameters and Variation Levels Description

Level 1

Level 2

Level 3

Underfilled material Solder pad size Rework parts (U3, 5, 7)a

No Nominal No

Reworkable 2 Larger Yes

Non-reworkable 2 Smaller —

The reference designation for each part is listed in Table 3.

2

Test Vehicle Design

Table 6

Types of TV2 Design Configures

Type

Module No. D

TV2-1 TV2-2 TV2-3 TV2-4 TV2-5 TV2-6 TV2-7 TV2-8 TV2-9 TV2-10 TV2-11 TV2-12 TV2-13 TV2-14

N016, N023 L012, L015 N010, N013 S014, S015 S018, S019 L017, L018 L009 L021 N021 S010 S008 N007 S001, S012 N015, N019

1 3 3 3 1 1 1 3 3 3 1 1 2 2

283

Parameter E F 1 2 1 3 3 2 2 2 1 3 3 1 3 1

1 1 1 1 1 1 2 2 2 2 2 2 1 1

that the three different solder pad sizes are nominal, 10% larger and 10% smaller than the nominal size. Only three components were selected to be reworked. They were first removed from the PWB; then the PWB surface was cleaned at the component mounting locations and new components were resoldered onto the PWB without applying solder paste on the solder pads. Table 6 lists 14 different types of TV2 configurations and a total of 22 single-sided CCAs were built for carrying out vibration tests. The elastic moduli of the underfills are shown in Table 5 and are 8.5 GPa for non-reworkable 2 and 1.12 GPa for reworkable 2. No conformal coating is used on TV2. The selection of BGA and CSP devices is predicated on incorporation of all popular array styles surveyed in the high-performance electronics industry. Several different types of BGAs/CSPs are chosen: (1) molded encapsulent (the wirebonded top-of-die type); (2) wirebonded bottom of die, thermally enhanced, “super” BGA utilizing a nickel-plated copper cover type; (3) full array and staggered bump interposer pattern types; (4) fine-pitch PBGA and FlexBGA; and (5) full-array symmetrically patterned ceramic column grid array (CCGA) with solder column interposer (SCI). A small input/output (I/O) perimeter bumped flip-chip component was selected. In addition, leaded flat packs, leadless ceramic chip capacitors (LCCCs), and tantalum and ceramic capacitors were included. Table 7 is a parts list spreadsheet in which the reference designation for each device is also included, for example, U3 corresponding to a 600-pin SBGA. Part configurations are also summarized in this table. The location of each part mounted on the TV2 is shown in Fig. 4. The shape of the SCI is similar to the solder mask, on which there are many through holes. Each CCGA solder column is inserted through the corresponding hole and then attached to the hole with solder. The representative cross sections of the solder ball or column with or without the interposer (at the top or the middle position) are shown in Fig. 8. In TV2, all solder materials are 63Sn/37Pb solder except the 360-pin CCGA with SCI, whose solder columns are made of 90Pb/10Sn solder. High-T g FR-4 PWB is selected. The FR-4 epoxy resin selected is considered to be a preferred mainstream high-temperature PWB material. The conventional plated-through-hole board technology was used in the board design for BGA/CSP-style components. The use of solder-mask-defined (SMD) pads is abandoned.

284 Fatigue Damage Evaluation for Microelectronic Components Subjected to Vibration Table 7

TV2 Parts List Spreadsheet

Component TBGA-352 LCCC-28 SBGA-600 PBGA-313 CCGA-360 with SCI flexBGA-280 PBGA-256 PBGA-580 Flip Chip-96 Ceramic flat pack-28 Tantulam chip cap. a 1.62 mm

Reference Designation

Pitch (mm)

Ball Diameter (mm)

Body Size (mm)

Die Size (mm)

U1 U2 U3 U4 U5

1.27 1.27 1.27 1.27 1.27

0.75 N/A 0.75 0.76 0.89a

35 × 35 12 × 12 45 × 45 35 × 35 25 × 25

N/A N/A N/A 7.73 N/A

U6 U7 U8 U9 U10/ U11

0.80 1.00 1.00 0.46 1.27

0.48 0.50 0.63 0.20 N/A

16 × 16 17 × 17 35 × 35 13 × 13 9.2 × 19

9.76 5.13 20.36 N/A N/A

C1-C4

N/A

N/A

3.8 × 7.2

N/A

height.

Substract

Substract

Substract

PWB

PWB

PWB

(a) Solder Ball without Interposer

(b) Solder Ball with Interposer at Top

Substract

Substract

(c) Solder Ball with Interposer at Middle

Interposer Note: (1) 90Pb/10Sn solder ball or column (in dark).

PWB (d) Solder Column without Interposer

PWB (e) Solder Column with Interposer at top

(2) 63Sn/37Pb solder (in white) on both top & bottom of the solder ball or column

Figure 8 Cross section of solder ball/column with/without interposer at top/middle position.

3 Experiment

285

Figure 9 Bare PWB design of TV2. The current literature suggests that this design practice leads to inconsistent wetting of liquid solders and the resultant joint volumes. The non-solder-mask-defined (NSMD) pads were then used in the PWB. Figure 9 is a bare PWB design which illustrates the solder ball pattern for each BGA/CSP-type device. The design of the daisy chain nets in the PWB permits continuity of select solder joints. The location of singular nets allows precise resistance fluctuations at predicted maximumstress sites. Emphasis is placed on the design of test nets at maximum distances from neutral points (DNP) on the substrates of the packages. Other nets are designed to permit further testing after openings have occurred in parallel traces. All daisy chain nets are terminated in a standard pinned connector, precisely identifying continuity to individual and “ganged” nets.

3

EXPERIMENT The setup for vibration testing is shown in Fig. 10. The vibration test fixture is bolted directly to the vibration shaker head and the TV is installed in the test fixture using wedge locks along the two side edges. Vibration testing was performed in the direction normal to the plane of the test vehicle only. One control accelerometer was mounted on the fixture, adjacent to the test vehicle, which served as the measure of the vibration input to the test vehicle. Nine accelerometers, evenly spaced, were mounted on the surface of the test vehicle using capton tape to measure the dynamic response characteristics of the test vehicle (resonant frequencies and transmissibilities) for the vibration input test profiles shown in Fig. 11. The test vehicle is connected to an Anatec event detector and computer to monitor electrical opens during vibration via the bottom connector and harness shown in Fig. 10. A resistance threshold of 1000  and spike duration of 0.2 μsec for 15 consecutive evidences constituted a failure. This higher resistance threshold is selected to overcome the noise effects during the vibration. Each test vehicle is sequentially installed on the fixture and subjected to vibration tests consisting of a pre- and post-1g sinusoidal sweep from 10 to 2000 Hz at a rate of 1 octave/min

286 Fatigue Damage Evaluation for Microelectronic Components Subjected to Vibration Electrical Harness

Bottom Connector M2

M1

Control Accel M3

M4 M6

M5

M7

M8

M9

040-4005180

(b) TV2

(a) TV1

Figure 10

Vibration test set-up.

A

B

Amplitude, g 2/Hz

C

Frequency, Level

Freq. (Hz)/Amplitude (g 2/Hz)

Total

Duration,

A

B

C

Grms

hr

2000/.01

1

50/.04

1000/.04

7.71

1

2

50/.08

1000/.08 2000/.020 10.90

1

3

50/.16

1000/.16 2000/.040 15.41

2 or 72

Figure 11 Random vibration inputs and test duration.

and random vibration at three different levels and durations or until failure occurred (shown in Fig. 11). The pre- and postsinusoidal sweeps are used as a signature to determine whether any structural anomalies/failures have occurred. In TV1, a total of 12 types of TVs are tested with the vibration inputs normal to the PWB. Figure 12 shows a typical sinusoidal vibration response measured at the center of the test vehicle during a 1g sinusoidal sweep. This figure indicates that the first: 3mode natural frequency of the TV1 is about 400 Hz with transmissibility equal to 38.6. In addition, the accelerometers on the test vehicles are monitored in real time during random vibration testing to determine the approximate time of failure. Vibration time and level were also recorded at the time when electrical failures occurred, required for determining component durability. To further investigate underfill effects on the BGA solder joint fatigue life of TV1, an additional 72 h of vibration is applied to modules 33 (polyimide PWB, no coating, reworkable 1 underfill) and 36 (polyimide PWB, no coating, non-reworkable 1 underfill).

3 Experiment Sweep Number: 1.00 Sweep Rate 1: 1.000 oct/min Compression: 90% 100 Hz g (0-pk)

5

Remaining Time: 000:00:01 Test Range: 10.000, 2000.000 Hz Points Per Sweep: 450

Elapsed Time: 000:07:38 Filter Type: Proportional Fundamental: 40.000 % BB RMS: 1.02 cyc

1 401.8

2 1598.

38.637

13.646

287

1 + 2 +

10

1

0.1

0.01 10

1000

100 Log

2000

Frequency (Hz) M5-VERTICAL

PROJECT: SEM-E MODULE ASSY. MODULE#29 PRE-SINE SURVEY Sine Data Review Name: SEM_E_MODULE_ASSY.070

Figure 12

Typical transmissibility plot at center of TV1 during a 1g sinusoidal sweep.

Tables 8a and 8b summarize the failure locations (or four corners in the package) and duration of the solder joints. This duration corresponds to the level 3 vibration input. The test results shown in Tables 8a and 8b indicate that: 1. CBGA are more susceptible to failure than the PBGAs under the same conditions. 2. The use of underfill materials (including non-reworkable 1 and reworkable) improve the life expectancy of the PBGA by more than a factor of 10. 3. The use of underfill material of non-reworkable 1 could also improve the life expectancy of the CBGA by more than a factor of 10 (by comparing type 5 with type 4). 4. Only a slight increase in the lifetime of the CBGA using the underfilled material of reworkable 1 (by comparing type 6 with types 1 and 4). 5. The use of conformal coating and underfill prolongs the BGA solder joint vibration fatigue life. 6. The use of polyimide or thermount PWB has an insignificant effect on BGA solder joint fatigue life.

288 Fatigue Damage Evaluation for Microelectronic Components Subjected to Vibration Table 8a

TV1 BGA Failure Locations and Duration (Types 1–6)

BGA

Locationa 1

304-pin CBGA

1 2 3

313-pin

4 1

107 107 107 1831 — —

2 3 4 1 2 3 4

— — — — — — —

PBGA

352-pin TBGA

Type, min.@failure 2 3 4 5 * * 120

* * *

* *

* *

* * * * * * *

* * * * * * *

50 — 94 149 149 56 65 — — — 109 39 — 149 144

6

— — —

190 190 122

— —

190 —

— — — — — — —

— — — — — — —

Note: Asterisks mean tested up to 120 min and no failure observed. left, 2 = upper right, 3 = lower left, 4 = lower right.

a 1 = upper

1

2

4

3

Failure locations at BGA

Table 8b BGA

TV1 BGA Failure Locations and Duration (Types 7–12) Location 7

304-pin CBGA 313-pin PBGA 352-pin TBGA

1 2 3 4 1 2 3 4 1 2 3 4

489 2862 1548 153 — — — — — — — —

Type, min.@failure 8 9 10 11

12

* * * * * * * * * * * *

* * * * * * 56 * * * * *

* * * * * * * * * * * *

* * * * 64 * * * * * * *

* * * * * * * * * * * *

Note: Asterisk mean tested up to 120 min and no failure observed.

3 Experiment

289

A destructive physical testing was conducted to further verify the failure locations and crack paths of the solder joints. The results include test measurements (by Anatec Event Detector) and also indicate that the solder cracks are at the interfaces of the solder/package for 313-pin PBGA and the solder/PWB for 304-pin CBGA, respectively. In TV2, a total of 14 types of TVs (see Table 6) were subjected to normal vibration inputs. Prior to actual testing, each module was subjected to a presurvey 1g sine sweep, 10–2000 Hz, and 1 octave/min. Figure 13 shows a typical sinusoidal vibration response at the center of the tested module. This figure shows that the first mode frequency is 573 Hz with transmissibility equal to 37. Random vibration response data were recorded every 15 min as well as after each level test. Figure 14 shows a representative vibration response measurement which indicates that the first-mode frequency and the transmissibility are compatible with the measurements obtained from the sine sweep test. A similar measurement technique as TV1 was implemented to record the time the electrical failures occurred. The failure map of the TV2 random vibration test results are shown in Fig. 15. Table 9 shows that the 352-pin TBGA (U1) and 600-pin SBGA (U3) are more susceptible to failure than PBGAs under the same conditions and that the use of underfill material (including reworkable 2 and non-reworkable 1) appears to improve the life expectancy of all the components. The stiffer packages of TBGA and SBGA, which have copper heat spreaders, may account for higher BGA solder joint stress/strain during random vibration tests. A finite element analysis (FEA) was conducted to verify this hypothesis and will be presented later. No solder joint failure was observed in finer pitch BGAs, whose pitches were smaller than or

50 H(t) Magnitude Ch 5 /Ch 10 10

Sweep Number: 1.00 Sweep Rate 1: 1.000 oct/mit Compression: 90% 1 Hz 572.5 Ratio 36.728

Elapsed Time: 000:07:38 Filter Type: Proportional Fundamental: 40.000 %, BB RMS: 1.02 cyc 2 1070 4.6774

Remaining Time: 000:00:01 Test Range: 10.000, 2000.000 Hz Points Per Sweep: 450 1 +

2 +

1 Log Ratio 0.1

0.01

0.001 10 Frequency (Hz)

PROJECT: COTS TV2 CCA TEST MSN: L015 PRE-SINE SURVEY .5G Sine Test Name: COTS_TV2_CCA_TEST.019

Figure 13

1000

100 Log

5/10

Typical transmissibility plot at center of TV2 during a 1g sinusoidal sweep.

2000

290 Fatigue Damage Evaluation for Microelectronic Components Subjected to Vibration Test Level: 0.000 dB Test Time: 001:00:00 200 100 H(f) Magnitude Ch 5/Ch 10 Auxiliary

Hz dB

1 575 38.982 39.625

Reference RMS: 10.782 Clipping: Off

Test Range: 50.000, 2000.000 Hz Resolution: 5,000 Hz

2 1090 6.620 15.403

10 Transmissibility

Log

1

DDF 120 0.1

RMS:

Vibration Response

64.576 g 0.01

0.001 50

100 Log

14.22.45 26-Apr-2001

1000

2000

Frequency (Hz)

PROJECT: COTS TV2 CCA TEST MSN: L015 Level II Test Name: COTS-TV2_LEVEL_II.007

5/10 MS–Vertical

042-4005180

Figure 14 Representative random vibration response at center of TV2.

MSN S008, TBGA U1-2, Level 3, 111 Minutes MSN L018, Super BGA U3-3, Level 3, 89 Minutes

MSN N016, TBGA U1-2, Level 3, 157 Minutes*

MSN S001^, Leaded U10/11-1, Level 3, 73 Minutes

MSN N016, Super BGA U3-4, Level 3, 378 Minutes*

MSN N015^, Leaded U10/11-1, Level 3, 58 Minutes

* Extended Test Time

^ No Visible Solder Joint Failure 046-4005180

Figure 15 Failure map of TV2 random vibration test.

4 Failure Analysis Table 9

291

Comparison of Failures in TV2 Random Vibration Test

Underfilled Material

PWB Rework Solder Parts Pad Size (U3,5,7)

None

Nominal No Yes +10% No Yes –10% No Yes Reworkable 2 Nominal No Yes +10% No Yes –10% No Yes Non-Reworkable 2 Nominal No Yes +10% No Yes –10% No Yes

TV2 CCA No.

Min.@failure U1:352-pin TBGA 3

4

1

2

3

4

U10/ U11

* 157 * * * * * * * * * * * * * * 111 * * * * — — — — — — — — — * * * — — — * * * * * * * * * * * * * * * * * *

* * * * * * * — — — * — * * * * * *

* * * * * * * — — — * — * * * * * *

* * * * * * * — — — * — * * * * * *

* * 89 * * * * — — — * — * * * * * *

378 * * * * * * — — — * — * * * * * *

* * * * * * 58 — — — 73 — * * * * * *

1 N016a, N023 N007 L017, L018 L019 S018, S019 S008 N015, N019 N/A N/A N/A S001, S012 N/A N010, N013 N021 L012, L015 L021 S014, S015 S010

2

U3:600-pin SBGA

Note: ‘157’ means solder ball failed under level 3 random vibration (15.4 Grms) at 157 minutes. Asterisks mean testing only performed to 120 min, no failure observed. a Module subjected to extended testing (up to 8 hr). 1

2

4

3

Failure locations at BGA

equal to 1.0 mm. Due to insufficient data on solder joint failures, the mean cycle to failure [or N (50%)] cannot be derived.

4

FAILURE ANALYSIS Visual inspection of the solder joints under the BGAs is limited due to the small size of the bumps and the recession of the row of bumps inward from the edge of the device. Only the outermost rows of joints can be visually inspected. Therefore, a DPA is required to identify the failure locations and determine the failure mechanisms of the solder joints. Typically, the row(s) of solder bumps that show electrical opens are selected for sectioning. The devices are first cut out from the boards and then encapsulated in clear epoxy. In general, the sections are taken first through the outermost row of pins of the nets that are indicated as having electrical opens by measurements from the Anatech event detector. If no physical evidence of the opens is found in the first level of sectioning, then additional sectional levels are taken, either to deeper rows of pins within the same net or through rows of pins on the opposite side of the component. Evidence of opens is taken to be cracking that completely traverses the bump. For most of the devices, it is found that within the group of solder joints that exhibited opens there is at least one solder joint that shows cracking completely through the joint.

292 Fatigue Damage Evaluation for Microelectronic Components Subjected to Vibration In some cases, virtually every solder joint within the row shows either partial or complete cracking. For TV1, a DPA is conducted to verify the failure locations and crack paths of the solder joints based on the measured results using the Anatec event detector. All solder joint failures detected during the vibration tests were confirmed by DPA results. In general, the cracks in the BGA/CSP solder joints occur at the component side and are initiated from the interface between the SMD and the solder ball, arising from the stress concentration there. This failure mode is illustrated in Fig. 16 and 17, which are the microsections of the failed solder joints for the 313-pin PBGA and 352-pin TBGA in TV1. These figures show that (1) a significant amount of solder voids due to the manufacturing process defect; (2) the solder is precluded from wetting to the sidewall of the PWB solder pad, which would result in possible weaken stress points at the feather edges of the solder joints; and (3) the size of PWB solder pad for 313-pin PBGA is significantly larger than that of the component solder pad (see Fig. 16). The “unbalanced” solder pad sizes for 313-pin PBGA would result in a stiffer interconnection, which could reduce solder joint fatigue life, and the weak link or crack propagation would be

u4 33-05 A1 C-2 (50%) 100X

C-2

250X

Figure 16 Solder joint cracks of 313-pin PBGA in TV1.

Figure 17 Microsection of failed solder joint (upper left corner of 352-pin TBGA in module 29 of TV1).

4 Failure Analysis

293

at the smaller solder pad, that is, at the package side. All three defects have been corrected in the TV2 design. Excessive solder voids were also observed in the TV1 304-pin CBGA solder joint (see Fig. 18), whose crack surface is at the PWB side of the solder joint assembly and is propagated through the eutectic solder. In Fig. 18, the dark color is a 90Pb/10Sn solder stub and the light (or white) area is eutectic solder. This figure shows insufficient solder flow up along the solder stub, which could be induced by insufficient solder paste and/or smaller size of the PWB solder pad. Table 4 provides information of BGA solder joint assemblies. This table shows that the difference in the PWB solder pad diameters between design and measurement could be up to 0.0508 mm (2 mils). This observation will be incorporated into the BGA solder joint reliability evaluation. For TV2, based on measurement taken during vibration tests, two modules with electrical opens in the test components were selected for a DPA to further isolate the failure locations and determine the failure mechanisms of the solder joints. The modules selected were L018 (U3 600-pin SBGA) and S008 (U1-352 pin TBGA). The DPA reveals solder cracks, which is consistent with the test observations, that is, electrical opens. Figure 19 shows a microsection of one of the corner solder joints in the 600-pin SBGA (U3 in module L017). Solder cracks are observed along the PWB solder pad and at the package side. A small number of tiny voids are found in the solder joint. In general, the severity of the solder crack gradually decreases from the package corner toward the inside of the package (along the package outer row) and

90Pb/10Sn Solder Stub

Figure 18 Microsection of failed solder joint (lower right corner of 304-pin CBGA in module 29 of TV1).

Figure 19 Microsection of failed solder joint in 600-pin SBGA for module L018 in TV2.

294 Fatigue Damage Evaluation for Microelectronic Components Subjected to Vibration Package Corner

Inside of Package

Note: solder cracks at the left end of the PWB solder pad

Figure 20 Severity of solder crack decreased from package corner toward inside of package 600-pin SBGA in module L018 of TV2. is shown in Fig. 20 for the U3 600-pin SBGA, module L018. This observation indicates that the solder crack is initiated at the corner solder joint. For the 352-pin TBGA (U1, module S008), the solder crack is along the package side. No cracks were observed along the PWB solder pad. A small number of voids are also found in the solder joint. Note that no visible solder joint failure was observed in flat packs (U10/U11). With hand probes, an electrical open is found between pins 21 and 22 in both failed modules (U11/N015 and U10/S001). A DPA is conducted to further exclude the solder joint failure mode.

5

METHODOLOGY DEVELOPMENT FOR DETERMINATION OF VIBRATION FATIGUE DAMAGE In current analysis tools, the outputs of random vibration analysis are limited to spectral densities and root-mean-square (rms) values of the strain components with no phase information, which is required to be able to correlate the strains and thus calculate the effective strain using a type of von Mises relationship. This section presents a methodology to determine the effective strains of the interconnects of the components, such as leads and solder joints, resulting from exposure of modules to random vibration environments. It addresses the problem associated with the analysis of almost microscopic elements attached to large physical structures, for example, BGA solder balls and/or QFP leads attached onto PWB. Note that the effective strain is traditionally utilized in component strength durability analyses to predict fatigue damage, as opposed to any single component of strains. The proposed methodology (flowchart shown in Fig. 21) includes two analysis paths. First, it is necessary to create a FEM of the complete module, including packages, with microcomponents such as solder balls modeled as beams everywhere except at the corners of the packages. Some of the microcomponents are grouped together for simplicity (e.g., five solder balls may be represented by one beam). Two MSC/NASTRANTM analysis bulk data decks are then created, one with the beams present and one without. Second, generic FEMs of the microcomponents are then created for each type of component. An in-house MSC/NASTRANTM “DMAP” is then utilized to generate reduced stiffness and mass matrices for these components. The analysis bulk data deck which includes beam elements is then utilized along with an in-house FORTRAN program to modify the generic stiffness and mass matrices so as to make their connectivity equivalent to that of the corresponding beam elements. Another in-house FORTRAN program is employed to renumber the FEMs of the microcomponents so that they may be incorporated into the macromodel bulk data deck of the module (without beam elements) at the corner locations of the packages. The modified stiffness and mass matrices for all other microcomponents are also incorporated into the analysis deck through direct matrix input techniques.

5 Methodology Development for Determination of Vibration Fatigue Damage

295

Create micromodel of a single solder joint for the BGA currently under investigation. Then run customized version of MSC/Nastran to extract the stiffness matrix connecting the package to the PWB through the solder joint.

Create global model of module with beam elements representing all solder joints on all BGA components.

Run customized version of MSC/Nastran to insert the micromodel into the four corner solder joint locations of the BGA under analysis and to represent all other solder joints as multiples of the stiffness matrix extracted from the micromodel. The resulting model is known as the “combined model.”

Run a customized version of MSC/Nastran using the combined model and the specified output frequencies to determine the complex-valued frequency response functions of strain in every element of the solder joint fracture planes in the four embedded solder joint micromodels.

Run MSC/Nastran to extract the natural frequencies of the combined model up to 2000 Hz, and use these frequencies, along the PSD, to set the output frequencies for a frequency response analysis.

Use in-house program to assemble complex-valued frequency response functions of effective strain to average them by volume over the potential fracture planes and then integrate them with the PSDs over the frequency range to determine RMS strain levels and the expected number of strain cycles per second.

Figure 21 Flowchart for evaluating BGA solder joint (or interconnect) vibration-induced effective strain.

After verification of the model integrity by means of static analyses, equilibrium checks, and so on, a final static run was conducted during which volumes of the solid elements of the microcomponents were calculated and printed. A modal analysis was then conducted to obtain requisite frequencies and to generate a database for subsequent restarts. The complex frequency response of the six principal components of strain (represented as ε1 , ε2 , ε3 , τ4 , τ5 , and τ6 ) to a sinusoidal one-gravity input acceleration is then obtained for each solid element in a critical layer of the microcomponent using the MSC/NASTRANTM analysis code. These strain transfer functions are then used to develop the effective strain transfer functions by employing the von Mises relationship     9 2 2 2 2 + 1.5 τ 2 + τ 2 + τ 2 , (1) 4 5 6 2 εeff = (ε1 − ε2 ) + (ε2 − ε3 ) + (ε3 − ε1 ) where we substitute the principal strain transfer functions for the strain components to derive the following: 9 2 2 |H (εeff )|

= |H (ε1 ) − H (ε2 )|2 + |H (ε2 ) − H (ε3 )|2 + |H (ε3 ) − H (ε1 )|2   + 1.5 |H (τ4 )|2 + |H (τ5 )|2 + |H (τ6 )|2 .

(2)

The effective strains resulting from exposure of the module to random vibration are then obtained using another in-house FORTRAN program which calculates the power spectrum density (PSD) of the response by the usual formula: S0 (ω) = H (εeff )2 Si (ω),

(3)

296 Fatigue Damage Evaluation for Microelectronic Components Subjected to Vibration where Si (ω) is the input PSD of the source excitation and derives a corresponding curve which weights the response by the frequency squared:  ω 2 So (ω). (4) Fo (ω) = 2π The next step is to integrate the power spectral density curve of the responses as well as the weighted corresponding curve:

PSε = So (ω) dω, (5)

FSε = Fo (ω) dω. (6) The rms value of the effective strain for each element was then obtained as the absolute value of the complex square root of the integral,   εeff,e = (PS)0.5  , (7) and the measure of the average frequency (“number of positive crossings”) as    FSε 0.5  .  f+x =  (8) PSε  It is well known that the derived strains are heavily dependent on the finite element mesh size. The smaller the element size, the higher the resulting strains. This finite element mesh size dependency is primarily due to the stress/strain singularity at the edge of a bimaterial (Kuo, 1990: Kuo et al., 1997; Yin, 1992, 1993). Therefore, to minimize the mesh dependence problem, a volume-weighted average method (Kuo et al., 1997) is used to compute an average effective strain over all the elements in the critical layer: εeff,e Ve , (9) ε= Ve where Ve is the element volume. Note the method described herein can be used in the multi-degree-of-freedom system. The method described above has been programmed using FORTRAN as a series of MSC/NASTRANTM run and FORTRAN executions. Note that no singularity occurs if analytical modeling based on an approximate engineering approach is used (Suhir, 1986).

6

FINITE ELEMENT ANALYSIS A three-dimensional (3D) modeling technique is utilized to estimate the stress/strain of the interconnects, such as BGA solder joints. The FEMs are constructed with a MSC/PATRANTM code. Linear dynamic and static analyses are performed with a MSC/NASTRANTM finite element code. In the construction of 3D FEMs, many microelectronic parts, such as solder joints and connecting leads, are not only very small compared to the surrounding parts but also usually large in quantity. As a result, it is difficult if not impractical to include their representations or FEMs with sufficient degree of complexity to permit accurate determination of the stress or strain state resulting from exposure of an overall model of the complete package (macro model) to random vibration environments. It is more practical to include them in the macromodel of the complete package by simplistic representations, such as beam elements, which are sufficient to account for their effect on package response and to provide output information as to displacements of the connection points which can then be utilized as input source displacements to the micromodel (or submodel) if necessary. A 3D macro/micro FEM of the electronic module of TV2, shown in Fig. 22, is then constructed to simulate major structural elements and determine the module dynamic responses

6 Finite Element Analysis

297

Macromodel

Micromodel

Z Y X

Figure 22 Table 10

A 3D macro/micro-FEM of TV2.

Material Properties

Material

Tensile Modules, GPa

Shear Modulus, GPa

Poisson’s Ratio

Aluminum FR-4 PWB

68.9 16.9 (X, Y ) 7.44 (Z )

25.9 3.03 (XY ) 2.41 (YZ, ZX )

Copper 63Sn/37Pb solder 90Pb/10Sn solder Solder mask Polyimide tape Alumina Silicon Mold compound Heat sink adhesive Die adhesive Non-reworkable underfill 2 Reworkable underfill 2

117 30.2 9.78 4.89 14.5 303 131 12.5 0.00276 3.45 8.5 1.12

44.0 10.8 3.49 1.72 6.27 125 50.4 4.82 0.00095 1.31 3.04 0.4

0.33 0.129 (XY ) 0.417 (YZ ) 0.183 (ZX ) 0.33 0.4 0.4 0.4 0.16 0.21 0.3 0.3 0.45 0.35 0.4 0.4

when subjected to an excitation normal to the TV2. Five types of BGA packages soldered onto the PWB are shown in this figure. Table 10 lists the material properties of aluminum, polyimide/glass, copper, 63Sn/37Pb and 90Pb/10Sn solders, dry film solder mask, polyimide tape, alumina, silicon, mold compound, heat sink and die adhesives, and underfills of reworkable 2 and non-reworkable 2 (ASM, 1979; IFI/Plenum, 1977; Rohde and Swearengen, 1980; Darveaux and Mawer, 1995; Lau and Pao, 1997). In the macro-/micromodel, the considered structural elements include PWB, heat sink adhesive, heat sink, five BGAs, header, two wedge locks, and two connectors. The adhesive is modeled using CBAR elements which in some cases are offset from the grid points. The CCGA package is simulated by plate elements. Each PBGA package is modeled with a composite plate which includes the layers of heat spreader, polyimide tape, copper ring, die adhesive, die, mold compound, copper pad, and solder mask. In Fig. 22, the X and Y axes

298 Fatigue Damage Evaluation for Microelectronic Components Subjected to Vibration define the in-plane of the package and the PWB (or module). The Z axis is perpendicular to the PWB surface. For the boundary conditions, three edges of the module (two opposite sides of wedge locks and one side of connectors) are constrained. Note that the model is simply supported (Z axis) at all grid points along the connector edge of the PWB. At the wedge lock edges, grid points near the 45◦ wedges are clamped while all other grids along the wedge locks are simply supported. In-plane loads are also reacted at the clamped grid points. The weights of the various parts (except BGAs) are included in the model by specifying either a material density or a nonstructural weight per unit area in the model to achieve the same weight. A random vibration analysis is applied in the macro-/micromodel. This model is then calibrated by matching the analysis results with the vibration test measurements obtained in the section of the Experiment, that is natural frequencies and their corresponding amplitudes. Note that this modeling calibration is achieved by adjusting the model boundary conditions, that is, the numbers of clamped grid points in the wedge lock edges and the damping for each natural frequency. Linear static and dynamic finite element analyses with MSC/NASTRANTM computer code combined with the developed method described in Fig. 21 and the previous section with the macro-/micromodel shown in Fig. 22 are conducted to calculate the volume-weighted average effective strains of the solder joints (four outermost corners) at the interfaces of package/solder and solder/PWB. Table 11 summarizes the calculated effective strains for the 352-pin TBGA and 600-pin SBGA when subjected to level 3 random vibration input. In the calculation process, several in-house developed FORTRAN programs, in conjunction with the outputs obtained from MSC/NASTRANTM static and frequency response analyses, are used to perform the required computations.

7

VIBRATION FATIGUE DAMAGE MODEL DEVELOPMENT AND VALIDATION To estimate solder joint fatigue life, an empirically derived formula (with some modifications) of universal slopes based on high-cycle fatigue test data (Manson, 1965) is used and is described by the equation

 3.5Su −0.12 Ai .12 ε , (10) = N ε= 2 2E AD where N = expected cycles to failure, ε = strain amplitude, ε = total strain range, S u = ultimate tensile strength = 37.9 MPa for 63Sn/37Pb (or eutectic) solder, E = modulus of elasticity = 30.2 GPa for 63Sn/37Pb (or eutectic) solder, Ai = solder crack surface in mm2 , and AD = characteristic area in mm2 and will be determined from the test results. Note that the values of Ai correspond to the interfaces of the solder ball/package and the solder ball/PWB for each BGA. The solder vibration fatigue curve is then modified as

 Ai c . (11) ε = N −c AD Equation (11) relates the amount of solder strain amplitude ε developed during one vibration cycle to the number of cycles needed to induce solder failure, N . Two unknown parameters, c and AD , were determined by correlating the derived solder effective strains to the TV2 test data, which is summarized in Table 9. To evaluate the random vibration fatigue failure, a three-band technique (Steinberg, 1988) is used. The basis for this technique is the Gaussian distribution. The 1σ , 2σ , and 3σ strains occur 68.31% of the time, 27.1% of the time, and 4.33% of the time, respectively. The

7 Vibration Fatigue Damage Model Development and Validation Table 11 Input

299

Solder Effective Strains of Critical Corner Solder Joints under Level 3 Vibration

(a) 352-Pin TBGA without Underfill Solder Ball Location

Positive Crossing, N0 (Hz)

Upper left Upper right Lower right Lower left

577 595 629 577

Solder Strain, 10−4 Solder/Package Solder/PWB 1.31 1.40 1.51 2.56

0.57 0.63 0.68 1.11

(b) 352-Pin TBGA with Reworkable 2 Underfill Solder Ball Location

Positive Crossing, N0 (Hz)

Solder Strain at Solder/Package, 10−4

Upper left Upper right Lower right Lower left

613 595 628 768

0.42 1.00 1.18 0.69

(c) 600-Pin SBGA without Underfill Solder Ball Location

Positive Crossing, N0 (Hz)

Upper left Upper right Lower right Lower left

586 669 688 589

Solder Strain, 10−4 Solder/Package Solder/PWB 3.66 1.30 1.17 3.51

3.26 1.15 1.03 3.11

(d) 600-Pin SBGA with Reworkable 2 Underfill Solder Ball Location

Positive Crossing, N0 (Hz)

Solder Strain at Solder/Package, 10−4

Upper left Upper right Lower right Lower left

582 673 702 585

3.07 0.98 0.84 2.91

vibration fatigue lives of the solder with the 1σ , 2σ , and 3σ strains can be obtained as

 Ai c k ε = Nk−c , (12) AD where k = 1, 2, 3. The corresponding number of fatigue cycles of random vibration is obtained by multiplying the times by the maximum of the number of positive zero crossings (N0+ ), obtained from Eq. (8). Thus, for a total of T hours of random vibration, the number of applied

300 Fatigue Damage Evaluation for Microelectronic Components Subjected to Vibration cycles is calculated from the equations n1 = N0+ T (3600 sec/hr)(0.6831), n2 = N0+ T (3600 sec/hr)(0.271), n3 =

(13)

N0+ T (3600 sec/hr)(0.0433).

The cumulative damage index (CDI) using Miner’s cumulative damage law, assuming a linear summation of damage, can be obtained as n1 n2 n3 CDI = + + . (14) N1 N2 N3 “Failure” is predicted when the CDI has a value greater than or equal to a critical value, usually chosen as 0.5. The determination of c and AD is described as follows. To solve the m equations, 3  nk =1 for j = 1, . . . , m (m > 2), (15) Nk ,j k =1

 Ai for k = 1, 2, 3 and j = 1, . . . , m. Nk ,j = (k εj ) AD This problem is converted to the following minimization problem:  2 m 3   nk . F= 1− Nk ,j

where

−1/c

j =1

(16)

(17)

k =1

It can be accomplished by setting ∂F ∂F = 0. (18) =0 and ∂c ∂AD Since three defects are found in TV1, described in Section 4, only TV2 test results are used to calibrate the life prediction model. In addition, for the conservative purpose and the insufficient data on solder joint failures to generate the fatigue life at 50% failure rate (through the Weibull plot), the first failures of the solder joints for each TV2 BGA (e.g., 1.85 hr for 352-pin TBGA and 1.48 hr for 600-pin SBGA) are used to calibrate the life prediction model. Using the analysis and test results of TV2, c and AD can be calculated to be 0.5128 and 0.2555 mm2 , respectively. These two values will be used in Eq. (11) to predict BGA solder joint vibration fatigue life. Using the experimentally validated fatigue life prediction model of Eq. (11), the CDIs of the 352-pin TBGA and 600-pin SBGA with and without reworkable underfilled material are calculated when the TV2 subjected to the random vibration input, and the CDIs along with predicted and tested fatigue lives are summarized in the last three columns of Table 12. This table indicates that (1) the predicted fatigue lives are reasonably consistent with the tested observations, (2) the use of the reworkable underfilled material can improve BGA fatigue life, (3) the fatigue life of the 600-pin SBGA is less than that of the 352-pin TBGA, and (4) the first failure location of the 352-pin TBGA is predicted at the lower right corner solder ball, which is inconsistent with test observation (this inconsistency could be due to the varied configurations in the solder joints induced by the manufacturing process, package warpage, etc.). In addition, the analysis results indicate that the solder fatigue lives of the 313-pin PBGA (1.27 mm pitch) and 580-pin PBGA (1.0 mm pitch) without underfill are calculated to be larger than 16 hr and are one order of magnitude higher than those of the 352-pin TBGA and 600-pin SBGA. The calculated results are consistent with test observations, which are that no solder joint failure occurred in the 313-pin and 580-pin PBGAs for the module of N016

7 Vibration Fatigue Damage Model Development and Validation Table 12

301

Predicted and Tested Vibration Fatigue Life Comparison

Package

Solder Ball Location

CDI

Predicted Fatigue Life, hr

Tested Fatigue Life, hr

TBGA 352 without underfill

1 2 3 4

0.167 0.203 0.524 0.142

TBGA 352 with reworkable underfill

1 2 3 4

⎫ 0.086 ⎬ ⎪ 0.126 0.055 ⎪ ⎭ 0.016

7.94

>2 (no failure observed)

SBGA 600 without underfill

1 2 3 4

⎫ 0.122 ⎪ ⎬ 0.101 0.743 ⎪ ⎭ 0.803

1.25

1.48

SBGA 600 with reworkable underfill

1 2 3 4

0.070 0.055 0.513 0.567

1.74

>2 (no failure observed)

⎫ ⎪ ⎬ ⎪ ⎭

1.91

⎫ ⎪ ⎬ ⎪ ⎭

1.85

Note: under level 3 random vibration (15.4 Grms) 1

2

4

3

Failure locations at BGA

tested up to 8 hr duration. These analyses results further verify the previous hypothesis in Section 3, which is the stiffer packages of TBGA and SBGA (having copper heat spreaders) to induce higher BGA solder joint stress/strain during the random vibration. The effect of the underfilled stiffness impacts on the BGA fatigue life is evaluated and the predicted result for the 600-pin SBGA is shown in Fig. 23. This figure indicates that (1) the solder joint vibration fatigue life increases as the elastic modulus of underfilled material increases and (2) the fatigue life significantly increases when the elastic modulus of the underfilled material is larger than 7 GPa. Therefore, the experimentally validated fatigue life prediction model [Eq. (11)] can be used to select the underfill material for improving BGA solder joint vibration fatigue life. In addition, using the experimentally validated life prediction model, the vibration fatigue life of the 313-pin PBGA (without underfill) in TV1 is estimated as 49 min, which is consistent with the test data of the fatigue life of 56–65 min shown in Tables 8a and 8b. Another example for the product design is to use the above techniques to select BGA mounting locations that have a better chance of surviving the vibration environment. The macro/micro FEM, shown in Fig. 24, is first constructed. Linear static and dynamic finite element analyses combined with the developed methodology and the experimentally validated fatigue life prediction model are then followed to determine the solder joint CDIs for each corresponding BGA location, as summarized in Table 13. This table shows that location 1 has a better chance of surviving the vibration environment. For the electronic packages (e.g., QFP), with gull-wing leads soldered onto the PWB, shown in Fig. 1, the leads are generally most vulnerable instead of the solder joints when

302 Fatigue Damage Evaluation for Microelectronic Components Subjected to Vibration 14

Fatigue Life, hr

12 10 8 6 4 2 0 0.01

0.1

1

10

100

Elastic Modulus, GPa

Figure 23 material.

600-pin SBGA solder joint vibration fatigue life vs. elastic modulus of underfilled

BGA (location #1) BGA (location #2)

Micro-FEM Package mid-plane

Polymide Tape Copper Polymide Tape

BGA Solder Mask (location #3) Copper Pad

z y x

Copper Pad

Solder Ball Polymide/Glass

s: Simply supported c: Clamped

PWB mid-plane z

Copper

y x

Figure 24 Example of selecting BGA mounting locations. Table 13 Location

Solder Joint CDI vs. BGA

BGA Location

CDI

1 2 3

0.03 0.09 0.84

the packages are subjected to vibration environments. A 3D FEM of an electronic module, shown in Fig. 25, is first constructed to simulate the major structural elements. A random vibration analysis using MSC/NASTRANTM computer code is then conducted to determine the strains/stresses of the gull-wing leads when the module is subjected to the vibration environments. These strains, combined with an empirically derived formula [with some modifications and shown in Eq. (19)] of universal slopes based on high-cycle fatigue test data (Manson, 1965), are used to determine the lead vibration fatigue life N : ε=

3.5Su −.12 ε , = N 2 2E

(19)

7 Vibration Fatigue Damage Model Development and Validation

303

Packages

Lead

Figure 25 A 3D FEM of electronic module with gull-wing-lead packages. where N = expected cycles to failure, ε = strain amplitude, ε = total strain range, Su = ultimate tensile strength = 517 MPa for Kovar lead, and E = modulus of elasticity = 138 GPa for Kovar lead. The lead vibration fatigue life prediction model shown in Eq. (19) is generally conservative in comparison with other validated life prediction models [see Eq. (4) in Wong and Fenger, 2004]. The analysis technique for the QFP is used for design optimization in a gull-wing-lead package (Wong et al., 1997a,b). Four critical design parameters and their variation levels are first selected and summarized in Table 14. The elastic modulus of the bonding material between the package and the PWB is 0.031 GPa. A Taguchi design-of-experiment (DOE) technique (Phadke, 1989; Ross, 1988) is adopted for establishing the analysis matrix with nine study cases (or L9 orthogonal array) to evaluate the criticality of these four parameters. Note that the possible combination of these four parameters with three levels of variation is 34 , or 81. However, with the assumption of noninteraction effects between these parameters, only nine cases are required. With this approach, the number of the required FEA runs is reduced by more than 85% as compared to the full factorial DOE approach. Figure 26 shows the parameter effects on the lead fatigue life. Critical parameters can be identified by simply comparing the differences among three levels for each parameter. The larger the difference is, the more critical that parameter is. Consequently, it can conclude that the lead thickness is the most critical parameter affecting the lead fatigue life, followed by the lead thigh length and adhesive bonding thickness. However, the change in the lead fatigue life becomes negligible when the lead thigh length increases beyond a threshold value, that is, 1.778 mm. Note that the evaluation on the criticality of the above four parameters was conducted only for comparison since the lead vibration fatigue life prediction model has not been validated by tests.

Table 14

Design Parameters and Variation Levels in Gull-Wing-Lead Package

Parameter

Description

Level 1

Level 2

Level 3

A B C D

Lead thickness, mm Lead width, mm Package-to-PWB bond thickness, mm Lead thigh length, mm

0.1016 0.1016 0.254 1.524

0.1270 0.2540 0.381 1.778

0.1524 0.4064 0.508 2.032

304 Fatigue Damage Evaluation for Microelectronic Components Subjected to Vibration 155

Lead Fatigue 128 Life (dB) 101 A1

A2

A3

Lead Thickness

B1

B2 Lead Width

B3

C1

C2

C3

Package-to-PWB Bond Thickness

D1

D2

D3

Lead Thigh Length

Figure 26 Parameter effects on lead fatigue life optimization.

8

CONCLUSIONS Linear dynamic and static analyses, combined with the TV test results, were conducted to support the development of a vibration fatigue damage model for the BGA solder joint assembly (with/without underfill material). TVs, on which various types of surface mount daisy-chained devices such as BGAs, CSP, flip chip, lead flat pack, and leadless capacitor are soldered, are designed, constructed, and tested under the pre– and post–sine sweep and random vibration environments, continuously monitoring solder joint integrity. Several design/manufacturing process parameters, including PWB materials, conformal coating, BGA underfilled materials, solder pad sizes on PWB, and BGA rework, with each having either two or three levels of variation, are used to construct 26 different types of TV configurations (12 types for TV1 and 14 types for TV2). Based on solder joint integrity measurements during the vibration tests, a DPA was conducted to further isolate the failure locations and determine the failure mechanisms of the solder joints. The vibration tests and DPA results show that 1. CBGA, 352-pin TBGA, and 600-pin SBGA mounted to the same modules will be more susceptible to failure than the PBGA under the same conditions. 2. Significant “unbalanced” solder pad sizes (at package and PWB) could reduce solder joint fatigue life. 3. The use of conformal coating and underfill would prolong the BGA solder joint vibration fatigue life. 4. The use of polyimide or thermount PWB would result in an insignificant effect on BGA solder joint fatigue life. 5. The PBGA solder joint fatigue life could be increased by one order of magnitude (or more) using underfilled material in the package versus using no underfilled material. 6. The use of non-reworkable 1 underfill could improve the life expectancy of the CBGA by more than a factor of 10 and only a slight increase in the life of the CBGA with using the reworkable 1 underfill. 7. Only a limited number of electrical openings are observed in TV2, which indicates that the TV2 is robust enough to survive the random vibration inputs (one possible reason is that the TV2 is very stiff, whose first mode of natural frequency is about 573 Hz therefore, the curvature change of the TV2 is minimal, which resulted in smaller relative motion between the package and the PWB, and less solder joint stresses).

Bibliography

305

8. BGA solder joint failure is initiated at the corner solder joint of the package and then gradually toward the inside of the package. 9. DPA results confirm test observations for solder joint failure locations. Next, a 3D modeling technique is used to estimate the strains of the BGA solder joints resulting from the exposure of electronic modules to the random vibration environments. This technique is implemented by utilization of several in-house-developed FORTRAN computer programs, which, in conjunction with the outputs obtained from MSC/NASTRANTM static and frequency response analyses, perform the required computations. The FORTRAN computer codes allow users to obtain the average effective strains of the BGA solder joints. A vibration fatigue life model has been established. This model, combined with a threeband technique and the derived solder effective strains, is then used to predict the BGA solder joint survivability/durability. This prediction was compared to test results to calibrate the proposed BGA solder joint vibration fatigue damage model. The analysis results obtained from the calibrated model show that (1) the predicted fatigue lives are reasonably consistent with the tested observations and (2) the use of the underfilled material can improve BGA fatigue life. Since this updated fatigue damage model has been qualitatively validated by test, this model could be used and is recommended to serve as an effective tool to determine the solder joint integrity of BGAs (with/without underfilled materials) during vibration and also to optimize the electronic packaging interconnect design. In addition, the relationship between the BGA solder joint fatigue life and the elastic modulus of underfilled material is evaluated. The analysis results show that BGA fatigue life exponentially increases as the elastic modulus of underfilled material increases to a certain threshold value, and this relationship can be used to select the underfilled material to improve BGA solder joint vibration fatigue life.

Acknowledgments The author would like to thank H. S. Fenger, Dr. F. W. Palmieri, Dr. I. C. Chen, Dr. R.-C. Yu, Dr. L. A. Kachatorian, Dr. E. Jih, H. M. Cohen, K. T. Teshiba, M. J. Miranda, E. L. Craig, M. D. Walley, H. L. Zauss, J. M. Ickes, K. A. Kirchner, D. R. Nelson, B. E. Oliver, D. W. Chu, T. Y. Jue, A. H. Matsunaga, S. H. Lee, B. A. Reed, D. T. Winslow, and Professor J. M. Pitarresi for their valuable contributions and technical discussions. Supports from my wife, Chen-Ping, and my two sons, Ning and Hsiang, have also been invaluable to the completion of this chapter.

BIBLIOGRAPHY ASM International (ASM), Electronic Materials Handbook , Vol. 1, Packaging, ASM, Materials Park, OH, 1979. Barker, D., Vodzak, J., Dasgupta, A. and Pecht, M. G. “Combined Vibrational and Thermal Solder Joint Fatigue—A Generalized Strain Versus Life Approach,” ASME Transactions: Journal of Electronic Packaging, Vol. 112, pp. 129–134, June 1990. Barker, D., Dasgupta, A., and Pecht, M. G., “PWB Solder Joint Life Calculations Under Thermal and Vibrational Loading,” 1991 Proceedings, Annual RELIABILITY and MAINTAINABILITY Symposium, pp. 451–459. Basaran, C., and Chandaroy, R., “Thermomechanical Analysis of Solder Joints under Thermal and Vibration Loading,” EEP-Vol. 26-1, in D. Agonafer et al. (Eds.), Advances in Electronic Packaging, The 1999 ASME International Mechanical Engineering Congress & Exposition, Nashville, TN, Nov. 14–19, 1999, pp. 419–426. Darveaux, R., and Mawer, A., “Thermal and Power Cycling Limits of Plastic Ball Grid Array Assemblies,” 1995 SMI Proceedings, San Jose, CA, Aug. 1995, pp. 1–12.

306 Fatigue Damage Evaluation for Microelectronic Components Subjected to Vibration Elishakoff, I., Probabilistic Methods in the Theory of Structures, Wiley New York, 1983. IFI/Plenum, Thermal Expansion, Nonmetallic Solids, Thermophysical Properties of Matter, Vol. 13, New York, 1977. Jih, E., Brown, G. M., Blair, H. D., Pan, T. Y., and Oh, K., “Vibration Fatigue Life Assessment of Surface Mounted PLCC through the Application of the Finite Element Modeling and Computer-Aided Holometry,” paper presented at the ASME Winter Annual Meeting, New Orleans, LA, 93-WA/EEP-18, 1993. Jih, E., and Jung, W., “Vibration Fatigue of Surface Mount Solder Joints,” paper presented at the 1998 InterSociety Conference on Thermal Phenomena, Seattle, WA, May 1998, pp. 246–250. Kuo, A. Y., “Thermal Stresses at the Edge of a Bi-Metallic Thermostate,” ASME Journal of Applied Mechanics, Vol. 57, pp. 585–589, 1990. Kuo, A. Y., Yin, W. L., Newport, D., and Chiang, M. Y. M., “Generalized Stress Intensity Factor Concept for Fatigue and Fracture Evaluations of IC Package Solder Joints,” EEPVol. 19-2, in E. Suhir et al. (Eds.), Advances in Electronic Packaging, The 1997 ASME International Mechanical Engineering Congress & Exposition, Dallas, TX, Nov. 16–21 Nov. 1997, pp. 1451–1460. Lau, J. H., and Pao, Y. H., Solder Joint Reliability of BGA, CSP, Flip Chip, and Fine Pitch SMT Assemblies, McGraw-Hill, New York, 1997. Lee, S. B., and Ham, S. J., “Fatigue Life Assessment of Bump Type Solder Joint under Vibration Environment,” EEP-Vol. 26-1, in D. Agonafer et al. (Eds.), Advances in Electronic Packaging, The 1999 ASME International Mechanical Engineering Congress & Exposition, Nashville, TN, Nov. 14–19, 1999, pp. 699–704. Liguore, S. S., and Followell, D., “Vibration Fatigue of Surface Mount Technology (SMT) Solder Joints,” 1995 Proceedings, Annual RELIABILITY and MAINTAINABILITY Symposium, pp. 18–26. Lin, Y. K., Probabilistic Theory of Structural Dynamics, McGraw-Hill, New York, 1973. Manson, S. S., “Fatigue: A Complex Subject—Some Simple Approximations,” Experimental Mechanics, Vol. 5, No. 7, 1965, p. 193. Meirovitch, L., Analytical Methods in Vibrations, Macmillan Company, Collier-Macmillan Limited, London, 1967. Phadke, M. S., Quality Engineering Using Robust Design, Prentice Hall, Englewood Cliffs, NJ, 1989. Pitarresi, J. M., and Akanda, A., “Random Vibration Response of a Surface Mount Lead/Solder Joint,” ASME EEP-Vol. 4-1, in Advances in Electronic Packaging, 1993, pp. 207–215. Rohde, R. W., and Swearengen, J. C., “Deformation Modeling Applied to Stress Relaxation of Four Solder Alloys,” Transaction of ASME: Journal of Engineering Material Technology, Vol. 102, pp. 207–214, 1980. Ross, P. J., Taguchi Techniques for Quality Engineering, McGraw-Hill, New York, 1988. Shinozuka, M., and Jan, C.-M., “Digital Simulation of Random Process and Its Applications,” Journal of Sound and Vibration, Vol. 25, pp. 111–128, 1972. Steinberg, D. S., Vibration Analysis for Electronic Equipment, Wiley 2nd ed., New York, 1988. Suhir, E., “Stresses in Bi-metal Thermostats,” ASME Journal of Applied Mechanics, Vol. 53, No. 3, 1987. Suhir, E., Applied Probability for Engineers and Scientists, McGraw–Hill, New York 1997. Wirsching, P. H., Paez T. L., and Ortiz, K., Random Vibration—Theory and Practice, Wiley, New York, 1995.

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Wong, T. E., Lee, S. H., Cohen, H. M., Winslow, D. T., and Reed, B. A., “Electronic Interconnection Fatigue Life Sensitivity Evaluation,” NEPCON West ’97 Proceedings, Feb. 1997a, pp. 53–59. Wong, T. E., Cohen H. M., and Kachatorian, L. A., “Robust Design for Electronic Interconnection Fatigue Life,” paper presented at the 43rd Annual Technical Meeting and Exposition, Institute of Environmental Science, Los Angeles, CA, May 4–8 1997b, pp. 142–147. Wong, T. E., Reed, B. A., Cohen H. M., and Chu, D. W., “Development of BGA Solder joint Vibration Fatigue Life Prediction Model,” paper presented at the 49th Electronic Components & Technology Conference, San Diego, CA, June 1–4 1999a, pp. 149–154. Wong, T. E., Palmieri, F. W., Reed, B. A., and Cohen, H. M., “Vibration Fatigue Damage Assessment on Multi-Degree-of-Freedom Macro/Micro System,” paper presented at the 1999 ASME Winter Annual Meeting, 11th Symposium on Mechanics of Surface Mount Assemblies, Nashville, TN, Nov. 14–19 1999b. Wong, T. E., Palmieri, F. W., Reed, B. A., and Cohen, H. M., “BGA Solder Joint Vibration Fatigue Damage,” paper presented at the APEX Conference, Long Beach, CA, Mar. 14–16, 2000a, pp. P-AP9/1–5. Wong, T. E., Palmieri, F. W., Reed, B. A., Fenger, H. S., Cohen, H. M., and Teshiba, K. T., “Durability/Reliability of BGA Solder Joints under Vibration Environment,” paper presented at the 50th Electronic Components & Technology Conference, Las Vegas, NV, May 21–24, 2000b. Wong, T. E., Palmieri, F. W., and Kachatorian, L. A., “Experimentally Validated Vibration Fatigue Life Prediction Model for Ball Grid Array Solder Joint,” in G. J. Kowalski et al (Eds.), Packaging of Electronic and Photonic Devices, EEP-Vol. 28, Nov. 2000c, pp. 113–119. Wong, T. E., Miranda, M., Cohen, H. M., Fenger, H. S., Teshiba, K. T., Chen, I. C., and Zauss, H. L., “BGA Vibration Fatigue Test,” Proceedings of IPACK’01, IPACK200115502, Kauai, HI, July, 8–13, 2001. Wong, T. E., Palmieri, F. W., and Fenger, H. S., “Under-filled BGA Solder Joint Vibration Fatigue Damage,” ITherm 2002, IEEE, San Diego, CA, 5/29-6/1/2002, pp. 961–966. Wong, T. E, and Fenger., H. S., “Vibration Fatigue Test of Surface Mount Electronic Components,” InterPack ’03, Maui, HI, July 6–11, 2003. Wong, T. E, and Fenger, H. S., “Vibration and Thermo-Mechanical Durability Assessments in Advanced Electronic Package Interconnects,” paper presented at the 54th Electronic Components and Technology Conference, Las Vegas, NV, June 1–4, 2004. Wong, T. E., and Yu, R.-C., “Vibration Fatigue Evaluation on Solder Joints of Under-filled BGA,” paper presented at the 2005 APEX and IPC Printed Circuits Expo Conference, sponsored by IPC SMEMA, Feb. 22–24, 2005, Anaheim, CA. Yang, Q. J., Wang, Z. P., Lim, G. H., and Pang, H. L. J., “Characterization of PBGA Assemblies’s Dynamic Properties and Vibration Fatigue Failures,” paper presented at the ASME Winter Annual Meeting, Anaheim, CA, 98-WA/EEP-16, 1998. Yin, W. L., “Refined Variational Solutions of the Interfacial Thermal Stresses in a Laminated Beam,” ASME Journal of Electronic Packaging, Vol. 114, pp. 125–140, 1992. Yin, W. L., “The Effects of Inclined Free Edges on the Thermal Stresses in a Layered Beam,” ASME Journal of Electronic Packaging, Vol. 115, pp. 208–213, 1993.

CHAPTER

13

VIBRATION CONSIDERATIONS FOR SENSITIVE RESEARCH AND PRODUCTION FACILITIES E. E. Ungar Acentech, Incorporated Cambridge, Massachusetts

H. Amick Colin Gordon & Associates Brisbane, California

J. A. Zapfe Acentech, Incorporated Cambridge, Massachusetts

1

INTRODUCTION Most items of instrumentation and equipment used in research and production of microelectronic and opto-electronic systems are sensitive to vibration, as well as to other disturbances, with excessive vibrations leading to such effects as blurring of images and misalignments of features or components. Such items need to be provided with environments in which the vibrations they experience are within acceptable limits. This chapter is intended to serve as an aid to owners and users who need to guide the specification, procurement, and mobilization of sensitive facilities. It is also intended to inform architects and engineers concerning the specification of vibration criteria for equipment and facilities, the design of facilities with relatively benign vibration environments, and the means for protecting sensitive equipment from vibrations.

2

PLANNING AND DESIGN GUIDANCE

2.1 Sources of Vibration The designer of a facility that is to house sensitive equipment needs to take account of the various sources of vibrations that may affect the equipment. These sources are delineated below, followed by discussion of approaches to reducing their effects. The sources of vibrations that generally are of concern in a sensitive facility fall into three categories: (1) external sources, (2) in-facility activities and process equipment, and (3) building services. Sources external to a facility typically comprise ambient microseismic activity; truck, automobile, and rail traffic (including that in tunnels and on bridges and viaducts) in the area; construction Structural Dynamics of Electronic and Photonic Systems Copyright © 2011 John Wiley & Sons, Inc.

Edited by Ephraim Suhir, David S. Steinberg and T. X. Yu

309

310 Vibration Considerations for Sensitive Research and Production Facilities activities; and machinery (such as generators and cooling towers) operating outdoors or in nearby buildings. Generally significant in-facility sources of vibration include walking personnel; in-facility traffic of carts, dollies, and such vehicles as forklifts; facility users’ equipment (e.g., vacuum pumps) that serves research or production tools; and some such tools themselves. Key production tools that can be significant sources include such items as scanners, implanters, probers, robots, and materials-handling systems. The building services category includes mechanical and electrical equipment, notably HVAC systems (air-conditioning and distribution fans, chillers, cooling towers, furnaces) and transformers, plumbing, distribution systems for liquids and gases, elevators, mechanically actuated doors, and loading docks.

2.2 Dealing with External Sources Microseismic activity—ambient vibrations that usually are of rather small magnitude—are present everywhere. The ultimate causes of these vibrations generally are not known but may involve geological tremors, waves in bodies of water, and wind blowing through trees and past buildings. A facility designer can do nothing about these sources and thus would be well advised to select a site where measurements show microseismic activity to be relatively small. Vibrations associated with street traffic result from vehicles bouncing on their suspension systems and from tires interacting with the road surface. Significant vibrations may result from aggressive tire treads acting on relatively smooth road surfaces, but more severe vibrations tend to result as the tires traverse irregularities in the road surfaces, such as cracks, bumps, debris, and potholes. Considerable vibrations may also be generated as a vehicle crosses joints in a bridge and as its motion along a bridge span deflects the span and causes it to vibrate. Vehicle-induced vibrations generally increase as the vehicle speed increases and tend to be greater for heavier vehicles. The vibrations that a vehicle generates locally propagate through the soil and typically decrease with increasing distance. Thus, a sensitive facility should be located as far as possible from highways, bridges, and busy streets. Maintenance of the surfaces of nearby streets to reduce potholes and bumps typically helps to reduce the vibrations produced by traffic on these streets. Limiting the speed of traffic on these streets also is helpful in general. Clearly, speed bumps and curbs that vehicles could impact should not be employed in the vicinity of a sensitive facility. The vibrations produced by rail vehicles result primarily from interaction of asperities of the wheels and rails. The vibrations induced in the rails propagate through the soil (along the ground or, for rail lines in tunnels, through the ground) and, like the vibrations due to street traffic, decrease with increasing distance. Thus, sensitive facilities should be located as far away from rail lines as possible, particularly from areas where the rails include such wheel-impact-causing devices as switches and crossovers. Although the designer of a facility in general has no control over the conditions of the rails and the vehicles that traverse them, the developer in some instances may be able to convince the rail system entity to improve the smoothness of the wheels and rails by better maintenance or even to undertake such modifications as replacing jointed by continuously welded rail and/or installing resilient rail supports or similar means for reducing the ground or tunnel vibrations produced by rail systems. Construction activities in the neighborhood of a sensitive facility tend to be sources of considerable disturbance. In some cases the developer of the facility may be able to influence these activities and to have construction equipment and methods employed that produce comparatively little vibration, for example, having vibratory pile drivers used instead of impact pile drivers. In instances where nearby construction activity is beyond the control of the users of a sensitive facility, however, it may be necessary to stop sensitive activities while disturbing construction work is in progress. Power plants and factories that employ heavy machinery tend to produce severe environmental vibrations. Although isolation of machinery is technically feasible, retrofitting

2 Planning and Design Guidance

311

machinery in an operating plant of a factory with vibration isolation systems generally involves interruption of operations and usually is not practical. On the other hand, it often is feasible to retrofit emergency generators and similar items that do not operate continuously with vibration isolation. Ideally, vibration-sensitive facility should not be located in the vicinity of power plants or heavy factories. It should be noted that vegetation, berms, sheet piles, or trenches of practical dimensions do not attenuate significantly any low-frequency vibrations that propagate along the ground. These low-frequency vibrations typically are most troublesome for sensitive equipment, not only because this equipment tends to be more sensitive to such vibrations, but also because equipment isolation systems do not provide much protection at these frequencies. Higher frequency vibrations, which may be attenuated somewhat by the aforementioned means, tend to be of lesser concern.

2.3 Taking In-Facility Sources into Account In-facility sources generally produce less severe vibrations on ground-supported slabs than on above-grade floors. It thus is advisable to place sensitive equipment on grade-supported slabs, rather than on higher floors. In all cases, sensitive equipment should be located as far as possible from both internal and external sources of disturbances. If sensitive equipment is to be located on supported floors, it should be placed near cores, columns, and/or heavy girders, where floor structures tend to vibrate less than in areas near the middles of the structural bays. (A structural bay is defined as the area enclosed by adjacent column lines.) Since the impacts from footfalls of walking personnel constitute a significant source of vibrations on supported floors—particularly in areas such as busy corridors, where numbers of people may walk at a rapid pace—corridors should not be located in the same bays as sensitive equipment and should be kept far from equipment areas. In order to limit the vibrations generated by cart traffic, the floor areas traversed by such traffic should be kept smooth, that is, free of bumps and joint gaps against which cart wheels would impact. Carts with soft pneumatic wheels of relatively large diameter are preferable.

2.4 Consideration of Building Services As implied by the foregoing discussions, the separation between areas that house sensitive equipment and the locations of potentially disturbing machinery should be kept as large as possible. Building services machinery should be installed on efficient vibration isolation systems and the floors that support such machinery must not be so flexible that they negate the isolation effectiveness of the isolation systems appreciably. It should be noted that not only the mechanical equipment items themselves need to be provided with vibration isolation, but also the piping, ducts, and conduits that are connected to this equipment. This implies that the facility layout needs to allow for the installation of resilient pipe hangers, duct supports, and so on. On the whole, one needs to have a vibration isolation strategy that is properly conceived, designed, specified, and implemented. In some cases one may be able to choose alternative machinery that is a less significant source of vibrations than other machinery that performs the same function. For example, one may opt for a traction elevator whose machinery is on a building’s roof rather than for a hydraulic elevator whose machinery is on a ground floor that supports sensitive installations. Similarly, one might consider using rotating pumps or compressors in place of their reciprocating counterparts. It also is useful, where possible, to employ machines that rotate about vertical axes instead of equivalent machines that rotate about horizontal axes. Since floors are most flexible and vibrate most readily in the vertical direction, machines that rotate about vertical axes (and that generate vibrations predominantly in the horizontal direction) tend to induce less vibrations in the floors on which they are supported than similar machines that rotate about horizontal axes (and generate vibrations in the vertical direction).

312 Vibration Considerations for Sensitive Research and Production Facilities 2.5 More About Protecting Sensitive Equipment As implied by some of the foregoing discussions, sensitive equipment may be protected at least to some extent by placing as much distance between this equipment and the significant vibration sources. But it is not only distance that is important—the more circuitous a path that vibrations have to traverse from a source to the equipment, the less vibration reaches the equipment. Thus, for example, a vibrating machine located three floors above a sensitive installation subjects equipment in that installation to lesser vibration than does the same machine if it is located on the same floor as the equipment at the same separation distance; the vibration propagation path along the upper floor slab, down columns, and along the lower floor slab is more circuitous than a propagation path that extends only along the lower floor slab. It also is useful to interrupt propagation paths, for example, by introducing a gap (or resilient joint) between the floor structures that support sensitive equipment and neighboring areas. Such gaps in grade-supported slabs typically attenuate only vibrations at relatively high frequencies, which generally are of limited concern. However, joints in the slabs of supported floors—for example, between a bay that houses sensitive equipment and neighboring bays that support busy corridors or mechanical equipment—can provide significant protection even at the lower frequencies that are of primary importance. Items of sensitive equipment may also be mounted on vibration isolation systems. In the simplest cases, such a system consists of arrangements of pads (or other configurations) of rubber or of other resilient materials, steel coil springs, and/or commercially available combinations of these. In some cases it is advantageous to mount the sensitive equipment on an “inertia base”—typically a massive concrete block—that itself is supported on an arrangement of resilient elements. Such an inertia block provides improved vibration isolation and also permits one to mount the equipment on a relatively rigid support. Additional information on vibration isolation systems appears in Appendix A. Massive inertia bases, of course, can best be accommodated in basements or at other grade-supported locations. For particularly sensitive equipment it often is useful to provide a concrete “bathtub” in which an inertia base is mounted on springs so that its top is at the level of the adjacent floor slab. It is particularly advantageous to support inertia bases in such installations on air springs (essentially air-filled cylinders with pistons at their tops), which can be obtained with automatic devices for keeping the inertia base level under different load distributions. Caveat —Vibration isolation systems protect their payloads from floor vibration, but not from other disturbances, such as those associated with acoustic noise or with air movements. Thus, providing sensitive equipment with low-vibration environments may require the use of acoustical enclosures, baffles, and the like, in addition to effective vibration isolation systems.

3

VIBRATION CRITERIA

3.1 Equipment-Specific Criteria The suppliers of commercial instrumentation and equipment that are sensitive to vibrations (and other disturbances) generally specify limits on the vibrations to which the items they supply may be subjected and prescribe how these limits are to be verified. These limits in general apply to the floors or bases on which the equipment is to be supported. A complete specification for an item of sensitive equipment should prescribe the following: 1. At what locations on or near the equipment the vibrations are to be measured and in which directions. 2. For steady vibrations (i.e., in the absence of transient disturbances), what the limits are as a function of frequency; the frequency bands in which these limits apply;

3

Vibration Criteria

313

how the vibration spectra (frequency distributions) are to be measured, including the measurement durations, averaging, and spectral windowing. 3. How transient (short-duration) disturbances are to be distinguished from steady disturbances (e.g., in terms of their durations, repetition rates, frequencies of occurrence). 4. For transient vibrations, how the corresponding vibration signals should be captured and analyzed and what the corresponding limiting amplitudes are. Vibration criteria for noncommercial equipment are rarely at hand and often need to be deduced from criteria for similar commercially available items. In some instances the vibration sensitivity of a given experimental installation may be determined analytically; for example, the sensitivity of an optical array mounted on a table may be calculated from an analysis of the dynamics of the table and the mounted item and from the effect of the dynamic deflection on critical optical alignments. One may also be able experimentally to determine limits on the floor vibrations that are acceptable for a given item of equipment by subjecting the floor on which such an item is supported to slowly increasing vibrations and noting at what point unacceptable conditions result. For some classes of vibration-sensitive equipment—most notably photolithography scanners—the criteria include limits on the dynamic resistance of the supporting structures in addition to limits on the support vibration in order to ensure that the equipment itself will not induce excessive vibrations in the supporting structures.

3.2 Facility Vibration Criteria A facility—be it an entire building, a floor of a building, or a portion of a floor—obviously should be designed to accommodate the equipment that is to be housed in that facility. Thus, the vibration criterion for a given facility should be determined from the criteria for the relevant equipment. However, in many practical situations the specific equipment items that are to be located in a facility are not known, particularly in the facility’s early design stages. In such situations one may need to rely on generic criteria that pertain to certain broad categories of equipment and facility usage. Generic criteria that have been used widely in recent years recommend that the velocities associated with steady vibrations of floors that support sensitive installations not exceed the values indicated by the curves in Fig. 1 in any one-third-octave frequency band between 1 and 80 Hz. Correspondingly, the timewise maximum (peak) velocity associated with a transient vibration (determined from a velocity√signal bandpass filtered in one-third-octave bands between 1 and 80 Hz) should not exceed 2 ≈ 1.4 times the indicated values. Table 1 shows the criteria recommended for various equipment types and facility usages. The velocity values given in the table correspond to those portions of the criteria that are represented by horizontal lines in Fig. 1. It should be noted that these generic criteria initially were developed in the 1980s on the basis of equipment data available at that time. The early criteria have been updated to some extent more recently, largely on the basis of qualitative observations, by extending the most stringent criteria (VC-C, -D, and –E) down to 1 Hz, as shown in the figure. These criteria intend to take account of the fact that the instruments to which these curves apply may have low-frequency resonances, perhaps associated with internal vibration isolation. Note that any of the generic criteria may be either too stringent or too lenient for some specific items of equipment. The NIST-A criterion, shown in Fig. 2 together with the VC-B through VC-E curves, originally was developed for metrology work at the National Institute for Science and Technology but has gained popularity in the nanotechnology community. It is identical to the constant velocity curve of VC-E at frequencies above 20 Hz but corresponds to the constant

314 Vibration Considerations for Sensitive Research and Production Facilities 10000

Workshop (WS)

rms Velocity, μm/s

1000

Office (OF) Non-Sensitive Labs (TP)

100

VC-A (50 μm/s) VC-B (25 μm/s) VC-C (12.5 μm/s)

10

VC-D (6 μm/s) VC-E (3 μm/s)

1 1

10

100

One-Third Octave Band Frequency, Hz

Figure 1 Table 1

Generic vibration criteria.

Generic Vibration Criteria

Facility Use or Equipment

Ordinary workshops Offices Nonsensitive laboratories, bench microscopes up to 100X magnification, laboratory robots Ordinary laboratories, bench microscopes up to 400X magnification, precision balances, optical comparators Bench microscopes up to 400X magnification, optical equipment on isolation tables, epitaxy, confocal microscopes Magnetic resonance imagers, photolithography with line widths of 1 μm or greater, electron microscopes up to 30,000X magnification Nanofabrication, photolithography with line widths of 0.5 μm Electron microscopes, precision metrology

Criterion Designation

Velocity μ-m/sec μ-in/sec

WSa OF TPb

800 400 100

32,000 16,000 4,000

VC-A

50

2,000

VC-B

25

1,000

VC-C

12

500

VC-D

6

250

VC-E

3

125

a From Guide for the Evaluation of Human Exposure to Whole-Body Vibration, American National Standard ANSI S3.29-1983, and Guide for the Evaluation of Human Exposure to Whole-Body Vibration, ISO Standard 2631, 1978. b This criterion corresponds to the whole-body threshold of perception of the most sensitive humans. The average threshold of perception is greater than this by a factor of 2.

4 Design Considerations

315

100

rms Velocity, µm/s

VC-B (25 µm/s) VC-C (12.5 µm/s)

10

VC-D (6 µm/s) VC-E (3 µm/s) 1

NIST-A 0.1 1

10

100

One-Third Octave Band Frequency, Hz

Figure 2 Comparison of stringent generic criteria. rms displacement amplitude of 0.025 μm = 1 μin. between 1 and 20 Hz. This criterion is intended to accommodate some of the ultra-high-precision metrology, probe, and lithography equipment used in nanotechnology but is very difficult to meet at some sites with significant low-frequency environmental vibrations. Since the vibration environment in a facility usually varies to some extent with time as well as from location to location, the specification for a facility should delineate how acceptability of the facility is to be judged. That is, the specification should state where vibration measurements are to be made, for how long, and at what intervals. It should also indicate what data analysis is to be used and what will be considered acceptable. Since one typically cannot determine from a limited set of measurements that a given vibration magnitude will never be exceeded, one may, for example, specify that a certain number of vibration samples are to be taken and that the mean plus one standard deviation of the values measured in each frequency band is not to exceed the stated limit.

4

DESIGN CONSIDERATIONS

4.1 Criteria Assignment It is rarely cost-effective to design an entire facility to meet a single stringent criterion. It generally is appropriate to assign different criteria to different spaces, depending on their usage. Typical semiconductor facilities, for example, may have two classes of clean-room space, perhaps VC-D or VC-E for photolithography and VC-A for nonlithography, as well as nonclean space that may include VC-E areas for metrology, VC-A for support space, and space without a criterion for mechanical rooms and offices.

4.2 Facility Configurations Several types of configurations are widely found in technology buildings. As noted previously, an on-grade slab is the simplest way to achieve satisfaction of VC-D or -E or NIST-A

316 Vibration Considerations for Sensitive Research and Production Facilities criteria. Placement of the most sensitive equipment on on-grade slabs is common practice for nanotechnology and research facilities but is rare in current-generation commercial semiconductor facilities. In the latter a “subfab” area, which may consist of two levels, is generally required below the vibration-sensitive fabrication area for through-the-floor air flow and/or support equipment. However, in some facilities with subfabs, the most vibration-sensitive space (generally a metrology laboratory) is placed on grade in the subfab, generally away from the production area and mechanical equipment. The floor structure at the production level of a modern semiconductor production facility with a subfab generally consists of either a “waffle slab” (a concrete slab with ribs in two directions on its underside) or a “grillage” (concrete ribs in two directions without a slab). In both cases, raised access floors (often called “computer floors”) are used to carry people and nonsensitive tools above the floor structure, whereas the vibration-sensitive tools are supported on the structural concrete floor via specially designed pedestals. The space between the raised access floors and the structural floor typically is used to route piping, cables, and air flows. A “stacked fab” configuration is often used where space on the building site is limited or where there is a need for some processes to be directly above others. In a stacked fab, there are generally two or three fabrication modules arranged vertically, with each module containing a subfab level, a clean-room level, and an interstitial level. Thus, a three-level stacked fab actually contains nine levels. In a stacked fab, the most sensitive processes (other than those placed on grade) typically are located in the lowest clean room, the floor of which is supported directly from the ground by closely spaced columns.

5

TESTING AND EVALUATION

5.1 Vibration Surveys One may consider five stages in the construction and operation of a facility at a given location: (1) site without the facility building in place, (2) completed structure without mechanical systems, (3) completed building with operational mechanical systems but without operational process equipment, (4) fully operational facility, and (5) “mature” facility, after it has been in operation for some time and multiple modifications may have been made to tool configurations and the mechanical plant. In general, vibrations get more severe as a facility progresses through these states. The site vibrations limit how quiet a facility can be, and vibration energy tends to be added in each successive stage. An on-grade or below-grade floor may exhibit less severe vibrations than its empty “greenfield” site, particularly in the absence of operating mechanical systems. The reverse generally is true for above-grade structures. In either case, the operation of even well-isolated mechanical systems generally contributes vibrations that may be significant. Surveys tend to be carried out over the life of a facility for a variety of purposes, including (1) assessment of the site, (2) determination of vibration conditions at various stages of construction, (3) evaluation of a proposed tool location, (4) diagnosis of problematic vibrations, and (5) documentation of the state of the facility. The purpose of a preconstruction site survey is to aid in the site selection or acceptance process. Since the amount of vibration attenuation that can be achieved at or inside a building is limited, it may be necessary to verify that vibrations due to external sources (rail and automobile traffic, industrial facilities, etc.) will permit functioning of the planned facility and processes. Differences may exist between a “greenfield” site at which there are no existing facilities and a “developed” site with one or more existing facilities (e.g., a “campus”); future development of a site should be kept in mind when a site is being selected. Site assessment is discussed further in the next section.

5 Testing and Evaluation

317

At various stages of construction there may be carried out evaluations of specific structural parameters related to the structural dynamics, such as the stiffness and resonance frequencies of floors, performance of structural isolation breaks, and attenuation or amplification between various points (such as between mechanical equipment and sensitive areas). Such evaluations are useful because they make it possible to remedy any deficiencies that may be found relatively easily, before the facility is fully completed and equipment is installed. Further details concerning these structural evaluations are presented in Section 5.3. A “final evaluation” is generally carried out near the time the owner takes possession to demonstrate that the facility performs as intended. The timing of this evaluation may be important if it is necessary to make a distinction between vibrations that are the responsibility of the designer or builder and those due to equipment provided by the facility’s owner. The final evaluation should document the “as-built” state, defined by the International Organization for Standardization (ISO) standard 14644 as where “all construction is complete, all services connected and functioning but with no production equipment, materials or personnel present.” In this state, tools have not yet been installed, but equipment associated with the facility itself (e.g., fans, pumps, chillers) is running normally and there is flow in the ducting and piping. Tool site evaluations generally are rather limited in scope and typically are carried out to evaluate whether the planned locations of particular tools meet the tools’ specific installation requirements. These evaluations usually are done on a tool-by-tool basis and may involve determination of the dynamic stiffness of the tool’s support structure, in addition to measurement of the vibrations at a tool’s site. Diagnostic investigations generally are performed to identify the sources of vibrations that exceed a particular tool’s tolerances or that cause any other problems. Surveys to document the current state of a facility may be carried out to satisfy an owner’s requirement to update his or her records periodically regarding the environmental status of a production area as a facility ages. These surveys may address vibration and acoustics, as well as particulates.

5.2 Site Assessment Assessment of the vibrations at a site that is being considered for construction of a sensitive facility should include measurement of the contributions of any obvious vibration sources near the site, such as roads or rail lines. One needs to take into account that traffic may vary with the time of day and perhaps also with the weather and with the seasons. Vibrations of the ground generally are most severe in the vertical direction; thus, measurements in the vertical direction usually suffice for a site assessment. If the planned building is to rest on soil-supported spread footings, then measuring the vibrations at or near the soil surface may be expected to provide relevant data. However, if the building is to be supported on piles or caissons that communicate well with deeper soil layers, or if the vibration-sensitive equipment is to be housed in a basement, then it is advisable to measure vibrations in holes that extend to the approximate depth of the piles, caissons, or basement. Use of such measurements is of particular value if the piles or caissons are to rest on layers of rock or soil that do not extend to the surface. It is important that the vibration sensors be well connected to the soil whose vibrations they are to measure. Good connection may be obtained, for example, by attaching a sensor to a concrete curb or to a rock outcropping that is imbedded in the ground, by connecting the sensor to a stake that has been driven into the soil, or by burying the sensor in a shallow hole. In planning site evaluation measurements and in evaluating their results one also should note that weather and groundwater conditions may affect the transmission of vibrations. Since water transmits vibrations well, wet or frozen soil and/or a high water table typically imply relatively good vibration transmission.

318 Vibration Considerations for Sensitive Research and Production Facilities The vibrations in a building that is placed at a given site generally will differ from the vibrations observed at the site before the building is in place. As a first, very rough, approximation the building may be considered as a mass and the soil that supports it may be considered  as a resilient element, with the combination exhibiting a natural frequency f = (1/2π ) k /m (Hz), where k denotes the stiffness of the resilient element and m the mass of the building. At frequencies well below this natural frequency one may expect the vibrations of the building not to differ significantly from the vibrations of the ground, at frequencies well above this natural frequency one may expect the vibrations of the building to be of lesser magnitude than those of the ground, and at frequencies in the vicinity of this natural frequency the vibrations of the building may exceed those of the ground. The effective stiffness k is difficult to estimate for an entire building, particularly if it is supported on piles or caissons. The same is true for the effective mass, since the entire mass of a building may not participate in vibrations induced by a given motion of the soil or foundation. However, for a soil-supported slab one may take m as the slab’s mass per unit area and k as the supporting soil’s effective stiffness per unit area, called “modulus of subgrade reaction” by structural engineers: values of k are available in the literature.

5.3 Structural Evaluations It generally is prudent to determine the vibrations of a facility in its “as-built” state, before its sensitive equipment is installed, so that one can determine whether the facility meets the equipment’s criteria and so that the facility can be modified, if necessary, before installation of the equipment. These measurements ideally should be made in the presence of all significant sources, such as external traffic, the building’s mechanical equipment, personnel walking near the sensitive areas, and carts traveling along corridors. However, one needs to consider that the presence of the sensitive equipment may change the facility’s vibrations. The magnitude of this change may be computed if the dynamics of the equipment and of the structure that supports it are known well enough. These dynamics rarely are known in much detail, unfortunately, so that one usually needs to rely on rather rough estimates based on the equipment’s mass and on the quasi-static stiffness of the supporting structure. Verification of the facility’s meeting its criteria once the equipment is installed—when the facility is in its “operational” state—is rather straightforward, on the other hand. One merely needs to measure the vibrations in the manner prescribed in the facility specifications.

APPENDIX: VIBRATION ISOLATION OF SENSITIVE EQUIPMENT A.1

Mass–Spring–Damper Model The purpose of a vibration isolation system is to protect a sensitive payload—an item of sensitive equipment—from vibration of its supporting structure. This supporting structure in most cases is a floor of the facility that houses the equipment. In its very simplest form, isolation is provided by a soft spring that supports the payload from the floor. The most important principles of vibration isolation systems can be understood in terms of the simple model that is shown schematically in Fig. A1. The rigid mass M, which is assumed to be able to move only in the vertical direction x , represents the payload, and a spring of stiffness K in parallel with an energy dissipation element characterized by the viscous damping coefficient C represents the isolator.

Appendix: Vibration Isolation of Sensitive Equipment

319

M (payload) x

K

C

u

Figure A.1

Mass–spring–damper model of payload and isolation system.

For steady sinusoidal motion of the floor U = u sin ωt with amplitude u at radian frequency∗ ω, the payload motion may be expressed as X = x sin ωt with amplitude x . These amplitudes may represent displacement, velocity, or acceleration, with both representing the same type of parameter. The ratio of the payload’s amplitude to that of the floor motion is known as the transmissibility T and obeys x 1 + 2ζ (ω/ωn ) T = = , (A.1) u 1 − (ωn /ω)2 + 2ζ (ω/ωn ) where ωn represents the radian natural frequency of the system and ζ is called the damping ratio. These are given by  K 1 C ωn = . (A.2) , ζ = √ M 2 KM The natural frequency is the frequency at which the system oscillates if it is deflected from its rest position and released, much like a guitar string oscillates after being plucked. The damping ratio relates the energy dissipation capability of the isolator to the energy dissipation magnitude that is just large enough so that the mass, after being deflected, will drift back toward its equilibrium position without oscillating. Figure A.2 shows how the transmissibility of the system of Fig. A.1 varies with the ratio of the floor’s vibration frequency to the system’s natural frequency. One may differentiate between three regions, depending on the value of the frequency ratio ω/ωn compared to unity—or on the value of the floor vibration frequency ω compared to the natural frequency ωn . At floor vibration frequencies that are well below the natural frequency of the system, the transmissibility is practically equal to 1; the payload motion is very nearly the same as the floor motion and thus the arrangement provides no protection to the payload. At floor vibration frequencies near the natural frequency, the payload motion is greater than the floor motion; the system actually amplifies the floor vibration. The maximum amplification—that is, the greatest value of T —is called the “amplification at resonance” and is often designated by the symbol Q. It depends only on the energy dissipation (damping) inherent in the isolator and obeys  1 1 + 4ζ 2 Q= ≈ , (A.3) 4ζ 2 2ζ ∗ The radian frequency ω (in radians/second) is equal to 2π f, where f denotes the cyclic frequency in Hz (cycles/second).

320 Vibration Considerations for Sensitive Research and Production Facilities 10

Transmissibility

Q

1

Isolation Region 0.1

0.01 0.1

1 Frequency Ratio, ω/ωn

10

Figure A.2 Transmissibility of mass–spring–damper system. where the approximate expression applies for small damping, which usually is the case in practice. The regime of principal interest is that where the floor’s driving frequency is at least 1.4 times the system’s natural frequency. In this “isolation region” the transmissibility decreases (that is, the isolation increases) in proportion to the square of the frequency ratio. In this region a 9-fold increase in the frequency ratio, for example, results in an 81-fold decrease in transmitted vibration. Note that such an increase in the frequency ratio may result from a 9-fold increase in the driving frequency or from a 9-fold decrease in the natural frequency, where the latter may be obtained by replacing the spring with one that has one-third of the stiffness of the original one. In general, better isolation results from use of more flexible, “softer,” springs. Unfortunately, there are limits on the flexibility of practical springs or similar elastic supports because softer supports may deflect excessively under the static loads they need to support. For a mathematically linear spring whose deflection is proportional to the applied load, the spring’s static deflection due to the gravity load imposed by the mass it carries is given by Mg , (A.4) δ= K where g denotes the acceleration of gravity. In view of Eq. (A.2), this static deflection may be expressed in terms of the natural frequency of the system as g g . (A.5) δ= 2 = ωn (2π fn )2 Thus, for example, a system with a 1 Hz natural frequency would require a linear spring to deflect about 25 cm (10 in.) due to the mass it supports. Such large deflections generally are impractical. Very soft conventional springs also have the drawback that they permit the payload to vibrate a long time after a brief disturbance; a system that is inadvertently bumped may take many vibration cycles, and many seconds, to return to rest. For these reasons linear springs are rarely used in situations where extremely effective isolation is needed.

Appendix: Vibration Isolation of Sensitive Equipment

A.2

321

Isolation System Types A.2.1 Springs, Pads, and Pendulum Systems Systems consisting of conventional springs and/or pads of resilient materials are routinely used in many applications. Many types are commercially available, and well-qualified vendors typically provide application and design guidance. Such systems work best if the mass and mass distribution of the payload are relatively constant; otherwise, particularly when soft springs are employed, small changes in the payload’s center of gravity can result in “listing” of the payload to one side or the other, requiring repositioning or adjustment of the springs. Bungee cords often are useful for suspending relatively small fixed-mass payloads, in that they act as very soft springs that can provide considerable isolation. A bungee cord suspension also can provide excellent isolation of horizontal vibration due to the system’s pendulum action. A pendulum arrangement acts like a spring in relation to horizontal vibrations, and a pendulum of reasonable length generally can act like a very flexible spring and thus can provide very good isolation of such vibrations. The natural frequency of a pendulum-supported payload is given by  g , (A.6) ωn = L where L represents the length of the pendulum. Note that this natural frequency depends only on the length of the pendulum and not on the mass of the payload. For example, a 1-m pendulum system has a horizontal natural frequency of about 0.5 Hz. A.2.2 Air–Spring Systems An air–spring system or “pneumatic isolator” in essence consists of a piston that is located at the end of a cylinder and that rests on a cushion of air, where the air is compressed by the weight of the piston and the payload it supports. A typical commercial air–spring system employs a resilient diaphragm to provide a seal between the piston and the chamber in which it moves. Many types of such air–spring systems make use of two air chambers connected by a small orifice, as shown schematically in Fig. A.3, so that air moving between the chambers as the piston oscillates provides some energy dissipation (damping). Most commercial air–spring systems can be equipped with automatic leveling arrangements that compensate for any changes in the weight or load distribution of the payload. Air–spring systems that are not provided with automatic leveling generally lose little air, so that it usually is necessary to add air only periodically to make up for leakage. However, air–spring systems that include leveling systems use air in the leveling process and typically need to be connected to a source of compressed air. An air–spring system with an automatic leveling system has the advantage that the system’s natural frequency is virtually independent of the payload. Since the stiffness of the Diaphragm

Piston

Upper chamber Orifice

Lower chamber

Figure A.3

Basic air–spring configuation.

322 Vibration Considerations for Sensitive Research and Production Facilities air in the chamber is proportional to its absolute pressure and since this pressure is very nearly proportional to the weight of the payload (if, as usual, the pressure in the chamber is much greater than atmospheric pressure), then the ratio of the stiffness to the payload mass is essentially constant, and consequently the natural frequency of the system is essentially constant. Typical commercial air–spring isolators provide natural frequencies for vertical motions between about 1 and 2.5 Hz and Q (amplification at resonance) values in the 3–10 range. Most commercial air–spring isolators are combined with devices, such as pendulum arrangements, that provide efficient horizontal isolation in addition to the vertical isolation due to the air–spring system itself. Air–spring systems are generally sized according to their load capacity. Once the payload weight is known, the appropriate size and number of air–spring mounts can be determined. Although a stable system requires at least three mounts, four-mount systems are used most commonly, with multimount systems used to support large payloads. A.2.3 Negative-Stiffness Systems So-called negative-stiffness systems make use of preloaded structural configurations that are on the edge of mechanical instability (buckling) and that act like very flexible elements for payloads within a limited weight range. In some such systems changes in the payload weight can be accommodated by adjustments of the preloading mechanisms. A.2.4 Active Isolation Systems Active isolation systems in essence are feedback devices that employ sensors, actuators, and a controller and that are configured to minimize the motion of the payload. They are most useful for isolation of low-frequency vibrations and generally are used in conjunction with conventional passive systems that provide good isolation at the higher frequencies. Commercial active isolation systems can provide effective vibration isolation at frequencies below 1 Hz without much amplification at resonance.

A.3

Common Isolation System Configurations A.3.1 Benchtop Platforms Benchtop platforms are intended for small and light equipment, such as bench microscopes or analytical balances, which would typically be located on a laboratory bench. Figure A.4 is a photograph of a typical benchtop isolation platform. A.3.2 Isolated Workstations Isolated workstations, as illustrated in Fig. A.5, are stand-alone floor-mounted systems that resemble a desk or small table. Such workstations often are equipped with arm rests that limit the equipment operator’s contact with the isolated platform.

Figure A.4

Benchtop platform. (Photo courtesy of Technical Manufacturing Company.)

Appendix: Vibration Isolation of Sensitive Equipment

Figure A.5

323

Isolated workstation. (Photo courtesy of Kinetic Systems Incor.)

A.3.3 Optical Tables Optical tables employ platforms of great flexural stiffness intended to maintain the alignment of optical elements mounted on them. The platforms usually are supported atop legs that contain vibration isolation arrangements, often of the air–spring type; see Fig. A.6. A.3.4 Isolation Cradles Isolation cradles are spring-supported platforms intended for use where it is desired to minimize the payload’s height above the floor. Figure A.7 illustrates an isolation cradle that is supported by four air–spring legs.

* Table and legs sold separately

Figure A.6

Isolated optical table. (Photo courtesy of Newport Corp.)

324 Vibration Considerations for Sensitive Research and Production Facilities

Figure A.7

Isolation cradle. (Photo courtesy of Kinetic Systems Inc.)

A.3.5 Inertia Blocks Many payloads contain vibration-sensitive elements that are provided with their own vibration isolation systems, internal to the payload. If it is desired to provide greater vibration protection for these elements, one may consider mounting the entire payload on a supplementary isolation system. In such cases, however, care must be taken to ensure that the supplementary isolation system does not interact adversely with the internal isolation systems. Avoidance of such interaction generally can be accomplished most readily by mounting the payload on massive inertia block. An inertia block is a large, stiff mass (usually of reinforced concrete), which as a rule of thumb should be about 10 times as great as the mass of the payload. Figure A.8 shows a schematic cross section of an inertia block system using a massive “keel” to lower the center of gravity of the assembly and thus to increase its stability. Ideally, the center of gravity

Figure A.8

Inertia Block

Isolator

Isolator

Payload

Schematic sketch of system with inertia block.

Appendix: Vibration Isolation of Sensitive Equipment

325

should be in the same horizontal plane as the tops of the isolators in order to minimize coupling between vertical and horizontal motions. Although Fig. A.8 shows the top of the inertia block above the floor, one can devise a configuration in which the inertia block’s top is even with the floor by providing a suitable pit. The design often includes walk-on floors that are supported from the unisolated areas, so that personnel can interact with the payload without compromising the isolation arrangement. In Fig. A.8 there is shown a small gap between the bottom of the inertia block and the structural floor. This gap must not be too small; otherwise the stiffness of the air entrapped in the gap may exceed the stiffness of the air–spring system, negating the isolation they provide. It is common practice to cast concrete inertia blocks in place before the isolators are installed, with a bond breaker between the underside of the block and the floor, and then to use the air–spring system to lift the block off the floor to the desired height. Figure A.9 shows a

Figure A.9

Large inertia block installation.

326 Vibration Considerations for Sensitive Research and Production Facilities large inertia block installation for a nuclear magnetic resonance (NMR) imaging instrument from above and below.

BIBLIOGRAPHY Amick, H., “On Generic Vibration Criteria for Advanced Technology Facilities; with a Tutorial on Vibration Data Presentation,” Journal of the Institute of Environmental Science, Vol. 40, No. 5, pp. 35–44, 1997. Amick, H., and Bayat, A., “Meeting The Vibration Challenges of Next-Generation Photolithography Tools,” Sound & Vibration, Vol. 36, No. 3, pp. 22–24, March 2002. Amick, H., and Gendreau, M., “On the Appropriate Timing for Facility Vibration Surveys,” Semiconductor Fabtech, No. 25, Mar. 2005, Cleanroom Section. Amick, H., and Stead, M., “Vibration Sensitivity of a Laboratory Bench Microscope,” Sound and Vibration, Vol. 41, No. 2, pp. 10–17, Feb. 2007. Gendreau, M., “Specification of the Effects of Acoustic Noise on Optical Tools,” Noise and Vibration Worldwide, Vol. 32, No. 4, pp. 17–22, Apr. 2001. Gordon, C. G., and Ungar, E. E., “Vibration as a Parameter in the Design of Microelectronics Facilities,” Transactions, Inter-Noise ’83 , pp. 483–486, 1983. Guide for the Evaluation of Human Exposure to Whole-Body Vibration, American National Standard ANSI S3.29-1983. Guide for the Evaluation of Human Exposure to Whole-Body Vibration, ISO Standard 2631, 1978. Murray, T. M., Allen, D. E., and Ungar, E. E., Floor Vibrations due to Human Activity, Steel Design Guide 11, American Institute of Steel Construction, Chicago, IL, 1997 (and errata). Ungar, E. E., Sturz, D. H., and Amick, C. H., “Vibration Control Design of High Technology Facilities,” Sound and Vibration Vol. 24, pp. 20–27, July 1990. Ungar, E. E., “Vibration Criteria for Sensitive Equipment,” Transactions, Inter-Noise ’92 , pp. 737–742, 1992. Ungar, E. E., and Gordon, C. G., “Vibration Criteria for Microelectronics Manufacturing Equipment,” Transactions, Inter-Noise ’83 , pp. 487–490, 1983.

CHAPTER

14

APPLICATIONS OF FINITE ELEMENT ANALYSIS: ATTRIBUTES AND CHALLENGES Metin Ozen Ozen Engineering, Inc. Sunnyvale, California

1

HISTORY OF FINITE ELEMENT ANALYSIS Finite element analysis (FEA) techniques have been successfully used in the engineering world since the 1960s. The wide range of applications mainly started in the aerospace industry. Today, FEA is being used also in automotive, biomedical, civil engineering, electronics, semiconductor, sporting equipment and goods, toys, and so on. In addition, what initially started out as an FEA of one part is now being performed at the system and assembly level. Today’s engineers can analyze system-level performance with the advancement in software as well as hardware. A FEA structural analysis that took one week 25 years ago can be accomplished in less than a day in the engineering environment today.

2

LINEAR AND NONLINEAR FEA ANALYSES A static structural FEA requires solution of the equation [K ]{u} = {F }, where [K ] is the stiffness matrix, {u} is the displacement matrix, and {F } is the matrix of the externally applied load. Today’s FEA programs automatically assemble the [K ] matrix from the mesh and material property inputs. The right part, {F }, of the above equation is also assembled from the applied loads. The above equation, solved for the displacements {u}, can be written as {u} = [K ]−1 {F }. The sought matrix, {u}, is known as “primary unknowns” since the displacements are calculated at the nodes as a direct solution of the above equation. As one can see from the above equations, the stresses and strains are not calculated as a direct manipulation of the matrices. Since both stresses and strains are derivates of the displacements, they are calculated as a secondary operation and stored in the results file (“secondary unknowns”). In a linear elastic analysis, the equation [K ]{u} = {F } is solved once since the material properties, boundary conditions, geometries, and so on, are all assumed to be constant and not a function of any other property (like time or temperature). Structural Dynamics of Electronic and Photonic Systems Copyright © 2011 John Wiley & Sons, Inc.

Edited by Ephraim Suhir, David S. Steinberg and T. X. Yu

327

328 Applications of Finite Element Analysis Therefore, for a linear elastic analysis, the equation can be solved only once, and the results can be stored in a results file. However, in a nonlinear analyses, the above equation needs to be solved multiple times, since the stiffness matrix [K ] may be changing as a result of load. In other words, as the load increases, the stiffness matrix could change due to the changes in one of the conditions discussed below.

2.1 Geometric Nonlinearity Geometric nonlinearity inherently exists in micro- and opto-electronic systems due to the fact that most of these systems have one or more components that are at least one order of magnitude larger in one direction. When a structure is much longer in one direction than the other(s), it deflects in a nonlinear manner since the orientation of the each element coordinate system changes with respect to the global coordinate system. This results in the violation of the small-deflection theory and thus, the stiffness matrix [K ] needs to be reformulated as the load is increased. Therefore, iterations and possibly substeps are required to reach a solution.

2.2 Material Nonlinearity Especially under shock loading, a material’s yield limit may be exceeded. In this case, a nonlinear stress–strain curve must be specified to calculate the plastic zones generated. When plasticity occurs in a structure, the stresses and strains are redistributed, which may require multiple load steps for accurate calculation of these plastic zones. The material stress–strain curve may be entered in one of the following forms: • Bilinear: This is the simplest form of specifying a nonlinear stress–strain curve. The

modulus of elasticity E , the yield stress SY, and the tangent modulus ET are the only three parameters required to specify the material’s behavior. • Multilinear: For materials exhibiting highly nonlinear behavior, the stress–strain curve may be specified as a combination of multiple linear lines where each point must be entered as a pair of stress and strain readings in the table. • For either bilinear or mutilinear, there are more options to specify the material’s response: (a) Isotropic hardening: If the yield curve grows with increasing plasticity, then isotropic hardening must be specified. (b) Kinematic hardening: If the yield curve shifts with plasticity, then kinematic hardening must be specified. (c) Combined kinematic and isotropic hardening can also be specified if the material’s yield curve shifts and grows at the same time.

2.3 Contact Nonlinearity In an assembly, when multiple components are in contact with each other, the contact condition must be specified with special contact elements. These contact elements behave in a linear manner when all contacts are “bonded.” When contacts between the components are specified as “frictionless” and/or “frictional,” then multiple load steps are required to calculate the final contact conditions resulting in a nonlinear simulation. In the case of “frictionless” contacts, the FEA program has to calculate the regions where the components are sliding with respect

3 Commercial FEA Codes

329

to each other and where they are open (no contact). In the other case of “frictional” contacts, the regions of “stick,” “slip,” and “open” regions must be calculated. In either case, the calculation of the “stick,” “slip,” and “open” regions may require extensive load steps and iterations since these regions may change drastically depending on the magnitude of shock and transient loads. The nonlinearities listed above add more complexity when the full set of dynamics equations are considered: [M ]{u  } + [C ]{u  } + [K ]{u} = {F (t)}, where [M ] is the mass matrix, {u  } is the nodal acceleration vector, [C ] is the damping matrix, {u  } is the nodal velocity vector, [K ] is the stiffness matrix, {u} is the nodal displacement vector, and {F (t)} is the load vector which is a function of time. The FEA codes that use the above equation to determine the dynamic response of a structure subject to any general time-dependent loads are called “implicit” codes. When solving problems in the time transient domain, implicit code limitations are subject to the following conditions: • For linear problems, the time step size can be large. • For nonlinear problems, the time step size may become very small depending on the

degree of nonlinearity. When the physics of the simulation require the solution of events happening in very short time periods (like crashes subject to very large deformation), then “explicit” codes may be more efficient. The implicit FEA codes require matrix inversion at every single time step and evaluate the displacements at time t + t :  a  {ut+t } = [K ]−1 Ft+t . However, in the explicit formulation, the displacements are evaluated at time t + t according to the following formula (no matrix inversion):   {ut+t } = {ut } + vt+t/2 t t+t/2 . In the explicit formulation, first the accelerations are calculated and then the velocities. Once the nodal velocities are known, the displacements are calculated from the above equation.

3

COMMERCIAL FEA CODES There are many commercial FEA codes in the market than can handle shock and vibration loads. However, the following FEA companies dominate the industry: • ANSYS: ANSYS, Inc. is becoming the leader in the computer-aided engineering

industry (http://www.ansys.com/). The following ANSYS products are widely used in the simulation of micro- and opto-electronic systems: ANSYS (Implicit), Autodyn (Explicit), and ANSYS/LS-DYNA (Explicit) • MSC/Nastran: MSC/Nastran has been widely used in the aerospace industry. Nastran is an implicit code (http://www.mscsoftware.com/). • Simulia (formerly Abaqus): Abaqus/Standard (implicit) and Abaqus/Explicit are the primary codes from the former Abaqus company which has been sold to Dassault Systems (http://www.simulia.com/products/abaqus_fea.html).

330 Applications of Finite Element Analysis 4

CHALLENGES IN FEA MODELING AND SIMULATION This section will discuss the five basic steps in FEA simulation.

4.1 Geometry The geometry can be either created in the preprocessor of the FEA package or it can be imported from an external solid modeler. The geometry can be 2D or 3D. Usually, the geometry creation is the first and the biggest hurdle in an FEA procedure. If the geometry is created in an external solid modeler, the geometry transfer into the FEA package may result in loss of some geometry. This is usually true if the geometry transfer is done through IGES and/or STEP format. Lately, the ACIS (*.sat) or Parasolid (*.x_t) formats have been successful in geometry transfers. Sometimes, the FEA vendors have direct links into the solid modeler packages from which the geometry is brought into the FEA package in its native format. This procedure mostly results in “clean” geometry transfer. If the geometry is created inside the FEA package, the issues with respect to geometry transfers are eliminated. However, this may result in the duplication of engineering effort if the geometry already is created in a solid modeling package.

4.2 Mesh Once the geometry is created, meshing is the next step. Top commercial FEA packages usually have good meshers that will enable local and global controls for the meshing process. The user should have control over what shape elements can be specified for the mesh. For micro- and opto-electronic systems, hexahedral (hex) meshes are preferred. This is due to the fact that the geometries in these systems are predominantly long in one direction. When meshing these geometries, elements with relatively high aspect ratios will be generated. Tetrahedral (tet) elements should not have aspect ratios more than 1 : 10 since high aspect ratio tet elements tend to cause artificial stiffness. Hex elements, on the other hand, will not cause artificial stiffness issues for aspect ratios larger than 1 : 10. This does not mean that hex elements will work for any aspect ratio. The users of the FEA packages should try simple geometries with a hex mesh and increase aspect ratios of the hex mesh to test out the aspect ratio limits of the hex elements. Once the mesh is created, then the geometry is not needed anymore. The geometry does not participate and is not needed in the solution phase. The geometry may still be needed in the next step (material property and boundary condition specifications).

4.3 Material Properties and Boundary Condition Specifications Material properties must be specified since they will be used in the creation of stiffness, damping, and mass matrices. For the stiffness matrix, the modulus of elasticity and Poisson ratio of the material are needed. For the damping matrix, the damping coefficient is needed. The material density is needed for the mass matrix. In the case of an assembly, these material properties must be specified for each component in the assembly. As for the boundary conditions, they can be specified either on the solid model or on the finite element model. For parametric studies, it is easier to specify boundary conditions on the solid model since, as the parameters are changed and the new geometry is remeshed, there is no dependency on the new node-numbering scheme which will have to change with each new mesh. There are variety of boundary conditions in the dynamic response of micro- and optoelectronic systems. The boundary conditions can be time-dependent displacements, forces, pressures, and so on.

5

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331

4.4 Solution The solution phase may seem trivial to the user. However, there is a lot of technology implemented in the background when the user hits the solve button. The current trend in computers is the implementation of dual and quad cores and clusters. This means that the solvers should take advantage of parallel and distributed processing. With the increasing demand to include as much detail in the FEA models as possible, the mesh size, that is, number of nodes in the FEA model, keeps going up. It is not unusual now to solve FEA problems that have multiple millions of degrees of freedom (DOF = number of nodes times the degrees of freedom per node). Under these conditions, the solvers should be readily available to solve the problems in distributed or parallel form. The solution phase may take the least amount of effort as long as the problem is well set up in the FEA package. If the user has not set up the mesh and/or the boundary conditions properly, the solution phase will demand a lot of user interaction; the solution may not converge or may not even run. This will result in the user going back and rechecking the model.

4.5 Postprocessing The postprocessing phase is the last step in an FEA project. When the analysis is run properly, the results file is written. The extraction of sometimes gigabytes of information and preparing a report is the postprocessing phase. Although the postprocessing may seem easy, the user actually has to spend much time here to make sure that the solution obtained is actually the correct solution. The obtained results must be carefully reviewed for accuracy. The user will have to extract the relevant information from the results file and prepare a report which is a very small subset of the information that the FEA solution provides.

5

TYPES OF SIMULATION

5.1 Modal Modal analysis determines the natural frequencies of a system. In addition, the user can also ask for mode shapes for each natural frequency. Modal analyses are linear; nonlinearities such as contact, material, and geometric are ignored. The user must determine if the contacts are closed or open and designate them as such before the modal analysis is performed. Modal analysis must be performed for most dynamic analyses. By performing a modal analysis, the following benefits are obtained: • Allows the design to avoid resonant vibrations or to vibrate at a specified frequency. • Gives engineers an idea of how the design will respond to different types of dynamic

loads. • The assumptions in performing a modal analysis are:

(a) The structure is linear. (b) There is no damping. (c) There are no time-varying boundary conditions. Therefore, the linear general equation of motion is [M ] {u} ¨ + [C ] {u} ˙ + [K ] {u} = {F } . Given the assumptions that there is no damping and no time-varying boundary conditions, 0

0

  [M ] {u} ˙ + [K ] {u} = {F } , ¨ + [C ] {u} [M ] {u} ¨ + [K ] {u} = {0} .

332 Applications of Finite Element Analysis

Figure 1 Natural frequency of the box at 109 Hz and the corresponding mode shape. Assuming harmonic (periodic) motion for the solution, {u} = {φ}i sin (ωi t + θi ) , {u} ˙ = ωi {φ}i cos (ωi t + θi ) , {u} ¨ = −ωi2 {φ}i sin (ωi t + θi ) Substituting and rearranging the terms, [M ] {u} ¨ + [K ] {u} = {0} , −ωi2

[M ] {φ}i sin (ωi t + θi ) + [K ] {φ}i sin (ωi t + θi ) = {0} ,  2 −ωi [M ] + [K ] {φ}i = {0} ,

lead to the following equation being equal to zero:  det [K ] − ωi2 [M ] = {0} . This is an eigenvalue solution which can be solved for the eigenvalues (ωi ) and corresponding eigenvectors (φi ). Here the ωi ’s are the natural frequencies and the φi ’s are the mode shapes after normalization. In Fig. 1, we can see the natural frequency of the box at 109 Hz (ωi ) and its corresponding mode shape (φi ) plotted as a vector sum of displacements.

5.2 Transient (Time Domain) Simulation When the loads are specified as a function of time, a transient solution in the time domain is required. The transient solution can be linear or nonlinear. Contact, material, and geometric nonlinearities may all be activated during a time transient solution.

5

Figure 2

Types of Simulation

333

Micro-electro-mechanical system (MEMS).

In a transient solution, the general equation of motion is solved: [M ] {x¨ } + [C ] {x˙ } + [K (x )] {x } = {F (t)} . It can be observed that the loads and boundary conditions can be functions of time and space. In addition, inertial and damping effects are included. Also, since the full equation of motion is being solved, all nonlinearities can be included (geometric, contact, and material) just like the MEMS example shown in Fig. 2.

5.3 Harmonic (Frequency Domain) Simulation The FEA models can be analyzed for solutions not only in the time domain but also in the frequency domain. The solution can be obtained as magnitude and angle or as real and imaginary components. In a harmonic analysis, the loading is cyclic and linear. A harmonic analysis determines the steady-state response of structure to sinusoidal (harmonic) loads of known frequency. The inputs can be pressures, forces, and displacements of known amplitude and frequency (in phase or out of phase with each other). The outputs from the model are harmonic displacements and corresponding stresses and strains. A harmonic analysis is usually performed to (1) make sure that a given design can withstand sinusoidal loads at different frequencies and (2) examine the response of the design to the given harmonic excitation. The assumptions of a harmonic excitation are the following: • The structure has constant or frequency-dependent mass, damping, and stiffness effects. • All boundary conditions vary in a sinusoidal manner at a known frequency

334 Applications of Finite Element Analysis

Figure 3

Deformations for the given frequency and phase angle.

In the general equation of motion, the displacement and external forces are substituted with harmonic frequency expressions:     { u} = umax e i ψ e i t { F } = Fmax e i ψ e i t = {Fmax (cos ψ + i sin ψ)} e i t = {umax (cos ψ + i sin ψ)} e i t = ({ F1 } + i { F2 } ) e i t , = ({u1} + i { u2 } ) e i t . Rearranging the general equation of motion yields [M ]{u} ¨ + [C ]{u} ˙ + [K ]{u} = {F }, −2 [M ]({u1 } + i {u2 })e i t + i [C ]({u1 } + i {u2 })e i t + [K ]({u1 } + i {u2 })e i t = ({F1 } + i {F2 })e i t , (− [M ] + i [C ] + [K ])({u1 } + i {u2 }) = ({F1 } + i {F2 }). 2

As can be seen from the last equation above, the displacement solution is complex if and only if: 1. Damping exists. 2. Applied loads are complex. The results can be plotted as displacements, stress, strain contours at a specified frequency, and phase angle, as in Fig. 3 Or same entities can be plotted at specific points as a function of frequency, as in Fig. 4.

5.4 Response Spectrum Simulations When a system is subject to any type of excitation that can be specified as displacement or velocity of acceleration or force as a function of frequency, then the system’s response due to

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Types of Simulation

335

Frequency Response 7.96e-4

Amplitude (m)

7.e-4 6.e-4 5.e-4 4.e-4 3.e-4 2.e-4 1.24e-4 504 550

600

650

700

750

800

850

900

Frequency (Hz)

Figure 4

Amplitude-frequency relationship (response function).

this excitation can be calculated in terms of displacements and stresses. The response spectrum can be either “single point” or “multipoint” meaning the whole system can be excited by one spectrum at all support points or by different spectra at different support points. These types of analyses are deterministic. Response spectrum analysis is a fast alternative to solving a full transient solution. The time domain boundary conditions (function of time) are converted to frequency domain (function of frequency) and used as input excitation. The most important assumption of performing a response spectrum is that the stiffness and mass matrices are linear (constant). The output can be plotted in terms of displacements, stresses, and strains as one generalized (combined) solution: ⎛ ⎞1/2 N N {R} = ⎝ εij {R}i {R}j ⎠ , i =1 j =1

where R is the total modal response and Ri Rj is the product of modes i and j . The modal correlation coefficients εij are dependent on the method chosen for evaluating the correlation coefficient: For completely correlated modes i and j , εij = 1. For partially correlated modes i and j , 0 < εij < 1. For uncorrelated correlated modes i and j , εij = 0.

5.5 Power Spectral Density The power spectral density (PSD) analysis is also called “random vibration” analysis; it is a special case of “spectrum” analysis as described in Section 5.4. It is a probabilistic analysis; that is, the displacement and stress results are 1σ results. PSD analysis is performed for cases when transient analysis is not an option since the time history is not deterministic; that

336 Applications of Finite Element Analysis

Figure 5

Example of aircraft electronics.

is, the input is not repeatable. When this is the case, the sample time histories are converted to PSD by using a statistical representation of load time history. A PSD analysis computes the probability distribution of different results, such as displacement or stress, due to some random excitation. An internal combination is done to compute the combined effect from each mode and their interactions (similar to response spectrum simulations). The PSD type can be displacement, velocity, force, pressure, or acceleration. The PSD tables are specified as a frequency table. This methodology is commonly used for airborne electronics, airframe parts, optical equipment, and so on (Fig. 5). The natural frequencies and mode shapes from a modal analysis are calculated first and then single or multiple PSD excitations are applied to the ground node. The 1σ results can be contoured like any other analysis, such as the contour plot shown in figure 5. In addition, the response PSD at one point in one direction can also be evaluated.

6

CONCLUSIONS The dynamic response of micro- and opto-electronic systems to shock and vibrations can be and is often analyzed using the finite element method. These simulations employ advanced features of the FEA programs, including nonlinearities and almost the whole spectrum of the program’s capabilities. These simulations provide valuable information to the engineers and help them in the design of micro- and opto-electronic systems.

CHAPTER

15

SHOCK SIMULATION OF DROP TEST OF HARD DISK DRIVES D. W. Shu, B. J. Shi, and J. Luo Nanyang Technological University Singapore

1

INTRODUCTION

1.1 Background The Hard-disk drive (HDD) is one of the most important and most interesting components within a computer. It has a long and interesting history dating back to the early 1950s. Perhaps one reason they are so fascinating is how well engineers have improved them as far as reliability, capacity, speed, power usage, and so on. In 1956, IBM invented the first computer disk storage system. This system could store 5 MB (megabytes) and had fifty 24-in. -diameter disks. In 1961, IBM invented the first disk drive with air-bearing heads and in 1963 introduced the removable disk pack drive. In 1973, IBM shipped the model 3340 Winchester sealed HDD, the predecessor of all current HDDs. In 1980, Seagate Technology introduced the first HDDs for microcomputers. It was a fullheight 5- 41 -in. drive with a stepper motor and held 5 MB. In 1997, Seagate introduced the first 7200-RPM drive for desktop. In 2000, IBM tripled the capacity of the world’s smallest drive. This drive holds 1 GB on a disk, which is the size of an American quarter. In 2004, Seagate announced the industry’s first 2.5-in. enterprise disk drive and launched 12 new products—an unprecedented array of HDDs to target a wide range of applications like MP3 players, consumer electronics, notebook computers, PCs, servers, and corporate data centers. Figure 1 shows a photograph of a modern small computer system interface (SCSI) hard disk, with major components annotated. The platters are mounted by cutting a hole in the center and stacking them onto a spindle. The platters rotate at high speed, driven by a special spindle motor connected to the spindle. Special electromagnetic read/write devices called heads are mounted onto sliders and used to either record information onto the disk or to read information from it. The sliders are mounted onto suspension and arms, all of which are mechanically connected into a single assembly and positioned over the surface of the disk by a device called an actuator. A logic board controls the activity of the other components and communicates with the rest of the PC. In terms of their capacity, storage, reliability, and other characteristics, HDDs have probably improved more than any other PC component. The areal density of hard-disk platters continues to increase at an amazing rate, even exceeding some of the optimistic predictions of a few years ago. The latest achievement in areal density is 100 Gb/in.2 (gibabits per inch square) in production and 160 Gb/in.2 at the laboratory level. Great efforts are being made to push technology toward 1000 Gb/in.2 or 1 Tb/in.2 (terabits per inch square). Researchers are pushing technology innovation and development in all aspects Structural Dynamics of Electronic and Photonic Systems Copyright © 2011 John Wiley & Sons, Inc.

Edited by Ephraim Suhir, David S. Steinberg and T. X. Yu

337

338 Shock Simulation of Drop Test of Hard Disk Drives Cover Mounting Holes (Cover not shown) Base Casting Spindle Slider (and Head) Suspension Actuator Arm Actuator Axis

Case Mounting Holes

Actuator

Platters Ribbon Cable (attaches heads to logic board)

SCSI Interface Connector Jumper Pins Jumper

Figure 1

Power Connector

Tape Seal

Photograph of a modern SCSI hard disk (www.pcguide.com).

as so to keep an annual areal density increase rate at 35–45%. The trend of HDD development is to have more and more high areal density as the speed, size of the HDD, and its components become smaller and smaller. The improved performance of the mechanical parts has been of great significance in this progress. Among these, the shock resistance of the more and more elaborate parts is one of the most important guideline in the design of HDDs.

1.2 Shock Robustness of HDDs The shock performance of drives under operation and nonoperation status is becoming an increasingly important issue across all form factors, and HDD manufacturers have been steadily increasing the threshold of shock level that the drivers can withstand. Today’s high-performance drives, with their lower flying heights, more sensitive medias, and read/write heads as well as more delicate suspensions must be able to withstand the rigors of manufacturing—hence the imposition of relatively high shock requirements destined for HDD design. Current technological improvements require more robust shock testing of HDD. This type of testing is most relevant to applications in portable computers and other mobile products with an HDD. Desktop computers also need to withstand shock due to shipping and mishandling. Dropping, striking, or bouncing a drive against a hard surface can damage it internally with no external evidence of damage. A drive that is subjected to this type of shock may fail on initial use. Or the damage could simply cause the reliability of the drive to degrade over time. When an HDD is subjected to a high level of acceleration shock, the head suspension system lifts off the disk and lands on it in a very short time, and the impact during this slap often leads to failure of the magnetic head and disk.

1

Introduction

339

Several authors have examined the problem of shock on the head–disk interface both experimentally and numerically. Kumar et al. (1994) studied the mechanics at the head–disk interface caused by an input shock. Experimental results and a dynamic impact model were used to analyze the problem. A six–degree–of–freedom (DOF) model, which included disk bending as well as the bending mode of the suspension, was developed. Their paper addressed the head-disk interface (HDI) damage in a drive designed for contact start–stop (CSS) when exposed to shock under nonoperational conditions. The HDI failure occurs when the slider overcomes the preload and separates from the disk surface upon impact. Although this type of failure can be largely avoided through the use of a load–unload design (Sagar, 1993), it may still be desirable to retain a CSS design to other considerations such as space and reliability. Kumar et al.’s results indicated that disk vibration becomes significant for narrow-pulse-width shocks. In such cases, the energy transfer from the disk to the slider can be large enough to lift the slider even when it is on the top surface of the disk. Their analysis underscored the importance of all individual components toward a single failure mechanism and highlighted the need to consider the entire drive as a single unit. They measured the slider motion using the capacitance between the slider and the disk. However, the relationship between the slider and the slider motion was not addressed. Harrison and Mundt (2000) studied the flying height response to mechanical shock during operation of an HDD. The suspension and slider were simplified as a lumped multiple-degreeof-freedom spring–mass system. They modeled the air bearing by the usual Reynolds equation and solved it numerically. They measured the disk’s response at the point under the slider and the arm’s response at the end of the arm. This measured data were used as input into the airbearing simulation code, and they obtained the dynamic flying height. However, it is inconvenient to apply their method because it requires the measurement data for each case of interest. The previously recognized industry shock specifications (half-sine pulse acceleration profiles of 100g to 300g amplitudes and 3–11 msec in duration) have proven to be insufficient for the identification of the mechanisms that cause damage to the components of small portable drives. Currently new standards are being developed to adequately test and evaluate the roughness of portable drives and their components. These new standards include higher shock amplitudes of 500g to 1000g and shorter durations of 0.5–2.5 msec. Allen and Bogy (1996) designed and constructed a uniaxial shock testing apparatus to produce high-amplitude, shortduration shock levels. The effects of shock on the HDD substrates, suspensions, arms, and sliders were studied by these apparatus. In addition to the experimental apparatus, a finite element model of the experiment was created to emulate the apparatus closely and provide numerical correlation for the observations made during the experiment. Because the amplitude and frequency of the vibration of the disk are governed by the material properties, they found that the potential for disk damage is affected by these properties since the relative velocity of the slider and disk, which govern the force and severity of the impact, are directly related to the disk and suspension vibrations being either in or out of phase at the time of contact. More energy was imparted to the slider when the disk velocity was increasing at the time of contact. Higher likelihood of damage to the disk is the result of higher impact velocity and energy. In their finite element model, they only considered the effects of the disk and the suspension arm on the magnetic head for the case of a linear shock where the impact surface is parallel to the disk. Ishimaru (1996) also developed an experimental set-up to study an HDI subjected to half-sine shock acceleration during nonoperation. using a dummy drive. He measured the degradation of recorded signals to investigate the permanent damage to the disk surface due to the collision between head and disk. He found that the slap movement of the head actuator assembly (HAA) is the dominant phenomenon governing the shock-proof performance of an HDI. Similar works also have been conducted by Sohn et al. (2000) and found that there are two important factors for the actuator to endure high shock level. One is shock

340 Shock Simulation of Drop Test of Hard Disk Drives transmissibility and the other is the beating between the arm and the suspension, and the first bending natural frequency of the arm was found to be the most important factor for the low shock transmissibility. The dynamic performance of three different suspensions during shock was studied by Jen et al. (1997) through both finite element simulations and experiments to determine the frequency response function for in-plane and out-of-plane vibrations of different suspensions during shock. Lee et al. (2001) built a theoretical model for predicting the shock behavior of the suspension in order to get a shock-improved design. Their model consisted of a single-DOF (SDOF) mass–spring system and a continuous system model with a single disk and spindle. The calculated results obtained from their model were compared with the results of finite element simulations and experimental data and found that the head lift-off entirely depended on the dynamic behavior of the suspension at nonoperation; the suspension geometry design was found to be very important to improve the shock performance of the HDI, especially the length of suspension. On the other hand, the dynamic response of the head was fully controlled by the shock response of the disk when the head was put on the middle or outer diameter of the disk. Eight parameters of the suspension shape were also investigated by Takahashi et al. (2002) through a series of experiments to optimize the design of the suspension for high shock resistance. The thickness of the suspension was found to be the most sensitive factor. Shu et al. (2006, 2007) performed a drop test simulation of an HDD. A pseudoresonance phenomenon was observed and investigated by finite element simulation and a simplified SDOF model. They found that, for both finite element simulation and theoretical analysis by the SDOF model, when subjected to half-sine acceleration shock, the peak relative displacement occurred at a critical frequency ratio of the characteristic frequency of the acceleration shock and the natural frequency of the dynamic system. Shi et al. (2004–2007) investigated the pulse shape effect on the shock responses of an actuator arm subjected to three types of different acceleration shocks. Their numerical results showed that for the three acceleration pulses the peak displacements have opposite behaviors for 0.1- and 1-msec pulse widths. The pseudoresonance phenomena occurred for the maximum relative displacement, but at different pulse widths for these different acceleration shocks. They found that the relative magnitude of the peak displacement of the actuator arm is mainly determined by the power magnitude of the acceleration pulse at the resonant frequency. Many researchers have tended to reduce the investigation to the component level while also noting that the drive dynamics are complex and the problem cannot be fully understood without evaluating the entire system [Allen and Bogy (1996); Kumar et al. (1994); Lee et al. (2001)]. Edwards (1999) attempted to evaluate the HDD system as a whole; his finite element model in ANSYS includes models of the HDD enclosure base and cover, the HAA, the disk pack/spindle motor assembly, and the voice coil motor assembly (VCM). He studied the shock response of an HDD dropped from a height onto a surface with a specific contact stiffness. In each simulation, the HDD model was dropped from a 25.4-mm height, but the contact stiffness of the impact surface was varied. His simulations demonstrate the effect the surface stiffness characteristics have on both the magnitude and the pulse width of the impact shock received by the HDD and the dramatic changes in the response of the internal components of the HDD to these different shocks. A finite element model of an HDD was developed by Jayson et al. (2003) to investigate the response of the HDD to a shock impulse. Two types of shock were of interest in their model, a linear shock and a rotary shock. The linear model corresponds to an HDD being dropped flat onto an impact surface. The rotary model was constrained to rotate about an axis and simulated the HDD standing on one edge that was allowed to drop and impacted the opposite edge. Comparison of the simulation results for the two models was used to develop a correlation between the linear and rotary shock tests. Murphy et al. (2006) investigated the shock performance of two form factor HDDs, 3.5 and 2.5 in., respectively. The displacement of the actuator arm, the suspension, and the disk

2 Finite Element Modeling

341

due to linear shock loads was studied experimentally for both nonoperating and operating states of the HDDs. A finite element model was developed to simulate the shock response. Their numerical results compared well with their experimental results. However, they did not investigate the pulse width effect and pulse shape effect on the shock responses. Due to the relatively complicated components in the HDD, a full finite element analysis is time consuming, not only in building the model, but also in performing the system-level analysis. Gao et al. (2005) adopted a flexible multibody dynamics formulation for nonoperational analysis. They pointed out that this method is significantly faster compared to a full finite element approach. Harmoko et al. (2007) proposed a more efficient method for predicting the shock tolerance of the HDD using state space formulation to model the structural components of the HDD and quasi-static concept to model the nonlinearity of the air bearing. They also proposed a procedure to conduct parametric study with this method and investigated the effect of overmold and voice coil stiffness on the shock tolerance. As nontraditional applications of HDDs emerge, their mechanical robustness during the operating state is of greater concern. In recent years, there has been an increasing application of small form factor (1 in. and smaller) HDDs in portable consumer appliances. Bhargava and Bogy (2007) proposed a procedure for simulating the operational shock response of a disk–suspension–slider air bearing system. The modeling of the structural components was done in ANSYS. The air-bearing modeling was done using the CML dynamic air bearing simulator. The two modules are coupled and each is iterated to convergence at every time step. They simulated shocks using a half-sine acceleration pulse with a pulse width of 0.5 msec and varying amplitude from –800g to 600g. However, this method was inefficient and computationally expensive due to the exchange of data between the two modules at each time step.

2

FINITE ELEMENT MODELING The finite element method (FEM) has been successfully used to study the dynamic responses of assemblies in HDDs. The finite element model of the actuator arm of an HAA (without suspension) of Seagate hard disk type C2, which is shown in Fig. 2, was created and gradually improved using a commercial finite element software package (HYPERMESH) and analyzed with LS-DYNA3D software for vertical drop test simulation. The overall dimensions of the actuator arm are about 65 × 35 × 1.4 mm in length, width, and thickness, respectively. The material of the arm is aluminum, and stainless steel is used for the pivot. The VCM consists

B

A

Figure 2 Finite element model of an actuator arm.

342 Shock Simulation of Drop Test of Hard Disk Drives of epoxy, copper, and bobbin materials. The copper (C2 CU-CLAD+10%) is an anisotropic material and was simplified as an isotropic material. The pivot bearings were simulated by using spring elements with stiffnesses being defined in the radial and axial directions. For a given distance between the two rows of balls, the torsional stiffness is directly proportional to the radial stiffness. The eight-node hexahedron solid elements (SOLID164) are used for the elements in the finite element model except that the one-dimensional spring elements are used for the ball bearing; 22 springs represent the radial stiffness (along the radials of the shaft) and 22 springs for the axial stiffness (along the axis of the shaft). The finite element model includes 15,041 nodes and 10,002 elements. This model is divided into 15 different parts and created one by one for ease to define the material and connection between the different parts and then combined in the whole model. The nodes in the interfaces between different parts were used together by the different parts. All these parts are connected directly except the part tail to the plate; they are defined as a pair contact with the CONTACT_AUTOMATIC_SURFACE_TO_SURFACE function of the LS-DYNA. It means that the tail can move freely and contact the plate during the drop test. Thus, it is a nonlinear model with contact defined and small deformation involved. A linear bulk viscosity coefficient of 0.06 is applied in the finite element model through the hourglass control in the explicit algorithm of the LS-DYNA software. The properties of the materials used in this model are listed in Table 1. This model consists of several parts with different materials. All the components were assumed to be connected directly with the same element nodes on common interfaces between them. The anisotropic material was simplified as an isotropic material in order to avoid too complex simulation of an anisotropic material. The boundary conditions (BCs) of the HAA finite element model are as follows. As the HAA connects to the base through the shaft and after it contacts the ground, the shaft is fixed and cannot move; the inner surface of the pivot is restrained by the shaft and cannot move along any direction. The whole model has an initial velocity v = 5.0 mm/msec at time t = 0 to simulate the HDD dropping about 1250 mm to the ground. In order to verify the finite element model of the HAA in the drop test using the LS-DYNA package, a typical cantilever beam model (similar as the HAA model) was used to verify the convergence and accuracy of the finite element model. The beam model was meshed with three different layers of solid elements in the vertical direction, that is, two, three and four-layers of solid elements. Theoretical results and numerical results were compared. From the comparison with the theoretical analyses and the FEM results with different layers of the model, it can be found that the three- and four-layer model can obtain the results with enough accuracy and good convergence. As for the convergence in the time domain, the minimum time step size for explicit time integration depends on the minimum element length and the sonic speed. Table 1

Properties of Materials

Material Aluminum Stainless steel Copper Epoxy Bobbin material

Young’s Modulus (E) [GPa] 73.1 200 88 2.5 3.5

Poisson Ratio (ν)

Density (ρ) (kg/m3 )

0.33 0.28 0.29 0.29 0.29

2.7 × 103 7.6 × 103 8.1 × 103 1.6 × 103 1.65 × 103

3

3

Drop Test Simulation and a Pseudoresonance Phenomenon

343

DROP TEST SIMULATION AND A PSEUDORESONANCE PHENOMENON

3.1 Drop Test Simulation In order to study the effects of the pulse width and amplitude of a single half-sine acceleration pulse loading applied on the inner surface of the pivot shaft on the relative displacement between the tip of the actuator arm (A in Fig. 2) and the pivot (B in Fig. 2) of the actuator arm (connected to the base), several half-sine acceleration pulse loadings with the same amplitude of 600g (g being the gravitational acceleration) and different pulse widths/durations from 0.04 to 4 msec were inputted into the numerical model. The acceleration level of 600g is close to the design shock limit of HDDs. Figure 3 shows three cases of the input acceleration pulse loadings in which the abscissa represents the pulse widths and the y coordinate represents the amplitudes of these loadings. Figure 2 shows the time histories of the relative displacement for the three different pulse widths of a single half-sine acceleration pulse loading with 600g amplitude, corresponding to Fig. 3. In Fig. 4, the abscissa is the time of response and the y coordinate is the shock response of the relative displacement. From Fig. 4, it can be found that the maximum relative displacements occur at different times for different pulse widths with the same amplitude. For different pulse widths, the maximum relative displacement occurs at the first oscillation of vibration for some cases and/or at the second or third oscillation for some other cases depending on the pulse width. To investigate the pulse amplitude effect on the shock responses, the amplitude of single half-sine pulse loadings were increased from 600g to 900g. Similar numerical simulations were performed on the actuator arm subjected to another group of half-sine acceleration pulse loadings with 900g amplitude and different pulse widths from 0.1 to 4 msec, Three cases are shown in Fig. 5. The coordinates are the same as those shown in Fig. 3. Figure 6 shows the relative displacement historical data for different pulse durations with 900g amplitude corresponding to Fig. 5. The behavior of relative displacement responses, as shown in Fig. 6, of the actuator arm subjected to this group of shock loadings is similar to that in Fig. 4. By using the finite element analysis for the actuator arm, the relative displacements were obtained for different pulse widths and different pulse amplitudes of single half-sine

Acceleration pulse (mm/ms2)

7 6 5

0.5 ms

600g

1 ms

600g

2 ms

4 3 2 1 0 0.0

Figure 3

600g

0.5

1.0 1.5 2.0 Pulse duration (ms)

2.5

3.0

Half-sine pulses with 600g and different pulse widths/durations.

344 Shock Simulation of Drop Test of Hard Disk Drives Relative displacement (mm)

0.7 600g 600g 600g

0.5

2 ms 1 ms 0.5 ms

0.3 0.1 −0.1 −0.3 −0.5

1

0

2

3

4

5

6

Time (ms)

Figure 4

Relative displacement with 600g and different widths/durations.

Acceleration pulse (mm/ms2)

10 9 8 7 6

900g

0.5 ms

900g

1 ms

900g

2 ms

5 4 3 2 1 0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Pulse duration (ms)

Figure 5 Half-sine pulses with 900g and different widths/durations. acceleration pulse loadings. It is found that, for the same amplitude of the loadings, the maximum relative displacements may be quite different for different pulse widths. As expected, for different amplitudes, a similar behavior but different amplitude of relative displacement responses occurs.

3.2 Pseudoresonance Phenomenon The variation of the maximum relative displacements (y max ) of the actuator arm subjected to half-sine acceleration pulses with amplitude of 600g and different pulse widths from 0.1 to 4 msec is shown in Fig. 7. From Fig. 7, it can be found that the maximum relative displacement increases sharply for pulse widths less than 0.5 msec, reaches the peak value at the pulse width of about 0.6 msec, decreases quickly to 0.2 mm at the pulse width of 1.0 msec, then decreases slowly and approaches a constant value after about 2 msec pulse width. The dependence of the maximum relative displacements (y max ) to the pulse amplitudes of 600g and 900g for different pulse widths from 0.1 to 4 msec is compared in Fig. 8. It is noted that, from Fig. 8, for the different amplitudes of a single half-sine acceleration shock, the maximum relative

3

Drop Test Simulation and a Pseudoresonance Phenomenon

345

Relative displacement (mm)

0.8 900g 900g 900g

0.6 0.4

2 ms 1 ms 0.5 ms

0.2 0 −0.2 −0.4 −0.6 −0.8

0

1

2

3

5

4

6

Time (ms)

Figure 6 Relative displacement with 900g and different pulse widths/durations.

0.5

ymax (mm)

0.4 600g

0.3 0.2 0.1 0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Pulse duration (ms)

Figure 7 Variation of the maximum relative displacement (y max ) versus pulse width/duration of 600g amplitude. displacements have similar trends versus pulse width. For both amplitudes of 600g and 900g, the maximum relative displacements reach their peak values of y max at the same pulse width of about 0.6 msec except that their numerical values increase with the amplitude. Figure 9 shows the maximum relative displacement subjected to single half-sine acceleration shocks of an amplitude of 600g, which is a transformation of Fig. 7, in which the abscissa is transformed from pulse duration/width into frequency ratio (β = ω/ωn ), which is the frequency ratio of the characteristic frequency of a half-sine acceleration pulse, defined as ω = π/T , where T is the pulse duration/width of the pulse loading to the first natural frequency of the system, ωn . From Fig. 7, it can be found that, for the frequency ratio β of less than a critical value of about 0.6 (β = ω/ωn ≈ 0.6), the maximum relative displacement increases sharply as the frequency ratio increases and reaches the peak value at the critical frequency ratio and then decreases gradually after the critical point. Figure 10 shows the dimensionless maximum relative displacements y max = ymax /(A0 /ωn2 ) subjected to the same input loadings, which is a transformation of Fig. 7 or 9, in which the abscissa is transformed from the pulse widths into the frequency ratio, and the y coordinate is transformed from a physical dimension into a dimensionless one, showing the relationship of the dimensionless maximum relative displacement versus the frequency ratio.

346 Shock Simulation of Drop Test of Hard Disk Drives

ymax (mm)

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

900g 600g

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Pulse duration (ms)

Figure 8 Variation of the maximum relative displacement (y max ) versus pulse width/duration of 600g and 900g amplitudes.

0.50 0.45 0.40 600g

ymax (mm)

0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00

0

0.5

1

1.5

2

2.5

3

3.5

4

β

Figure 9

Maximum relative displacement (y max ) versus frequency ratio β of 600g amplitude.

Figure 11 shows the maximum relative displacement of the actuator arm subjected to a series of single half-sine acceleration pulses with two amplitudes of 600g and 900g, which is a transformation of Fig. 8, in which the abscissa is transformed from the pulse durations into the frequency ratio. Figure 12 shows the dimensionless maximum relative displacements subjected to the same input loadings, which is a transformation of Fig. 11, in which the y coordinate is transformed from a physical dimension into a dimensionless one. It can be found that they are almost coincident with each other for the dimensionless maximum relative displacements subjected to the two sets of acceleration shocks with the two amplitudes of 600g and 900g. From Figs. 7–12, it can be found that, for both acceleration pulse amplitudes of 600g and 900g, the maximum relative displacements of the actuator arm reach the peak value at the same frequency ratio of about 0.6. This resembles a resonance phenomenon in vibration

3

Drop Test Simulation and a Pseudoresonance Phenomenon

347

7 6

ymax

5

600g

4 3 2 1 0

0

0.5

1

1.5

2

2.5

3

3.5

4

β

Figure 10 Dimensionless maximum relative displacement (y max ) versus the frequency ratio β of 600g amplitude.

0.8 0.7

900g

ymax (mm)

0.6

600g

0.5 0.4 0.3 0.2 0.1 0

0

0.5

1

1.5

2

2.5

3

3.5

4

β

Figure 11 Maximum relative displacement (y max ) versus the frequency ratio β of two amplitudes 600g and 900g.

and is termed pseudoresonance for convenience in this chapter. It can be concluded that the dimensionless peak relative displacement of the actuator arm only depends on the frequency ratio β and not the amplitude of the acceleration pulse. This may be useful in some cases. For example, if we obtain the responses of a single half-sine acceleration pulse with a given pulse amplitude and pulse width, we can then predict the responses of another acceleration pulse with the same pulse width but at different amplitude. This could be useful in those cases where measurements are not possible. This suggests that a scaling law exists between the maximum relative displacement and the amplitude of the acceleration pulse loading. The pseudoresonance phenomenon was further investigated and explained using a simplified SDOF model (Shu et al., 2006).

348 Shock Simulation of Drop Test of Hard Disk Drives 7 6 600g 5 ymax

900g 4 3 2 1 0 0.0

0.5

1

1.5

2

2.5

3

3.5

4

β

Figure 12 Dimensionless maximum relative displacement (y max ) versus the frequency ratio β of two amplitudes 600g and 900g.

4

PULSE SHAPE EFFECTS AND POWER SPECTRUM ANALYSIS

4.1 Pulse Shape Effect Although the half-sine acceleration pulse is accepted as the industry standard, the recorded acceleration pulse usually does not have the same shape as that of the half-sine one. Therefore, it is essential to study the influence of the pulse shape changes on the dynamic response of the drive. The finite element model of the actuator arm, as shown in Fig. 2, was analyzed to study the effects of the pulse shape during the dropping process. Here, three types of simple acceleration shocks different in shape (namely, the half-sine, triangular, and dual-quadratic waveforms) with an amplitude of 600g and with different pulse widths from 0.1 to 1 msec (1 msec case is shown in Fig. 13) were applied onto the finite element model. Note that the dual-quadratic waveform consists of a rising portion in the form of βt 2 , where t is time and β a constant, and a falling portion as a mirror image of the foregoing. The response history data of the relative displacement between the tip of the actuator arm (point A in Fig. 2) and the pivot (point B in Fig. 2) of the HAA for 0.1- and 1-msec pulse widths are shown in Figs. 14 and 15, respectively. From Fig. 14, it is observed that, for the 0.1-msec pulse width, the half-sine acceleration shock produces the largest peak displacement, the triangular shock gives the second largest, and the dual-quadratic shock gives the smallest peak displacement. Letting M (·) represent the magnitude of a quantity under consideration (peak relative displacement in this case) for a specific pulse shape, the results from the above comparison for the 0.1-msec shock duration can be written as M (half sine) > M (triangular) > M (dual quadratic).

(1)

However, an abnormal phenomenon is observed for the 1-msec shock duration from Fig. 15, that is, the relationship/sequence of the peak displacements for the 1-msec duration is shown as M (dual quadratic) > M (triangular) > M (half sine).

(2)

4

Pulse Shape Effects and Power Spectrum Analysis

349

Acceleration (mm/ms2)

7 6 5

Half-Sine

4

Triangular

3

Dual-Quadratic

2 1 0

0

1

2

3

5

4

Time (ms)

Figure 13 Three acceleration shocks with 1-msec pulse width applied on the inner surface of the pivot.

Relative Displacement (mm)

0.15 Half-Sine Triangular Dual-Quadratic

0.10 0.05 0.00 −0.05 −0.10 −0.15

0

1

2

3

4

5

Time (ms)

Figure 14 Shock responses of input acceleration shocks with 0.1-msec pulse width applied on the inner surface of the pivot.

Let M (·) represent the root-mean-square (RMS) for the acceleration pulses of 1- and/or 0.1-msec pulse widths. From Fig. 13, it can also be noticed that, in the time domain, the sequence of the RMS for the three different acceleration pulses from large to small has the same relationship as shown in relation (1) for both pulse widths. Meanwhile, from Fig. 13, let M (·) represent the area under the pulse curve for acceleration pulse widths of 1 and/or 0.1 msec. The sequence of the areas for the three different acceleration pulses from large to small also has the same relationship, as shown in relation (1) for both pulse widths. The above opposite phenomena in the responses for the three different acceleration pulses with different pulse widths are summarized in Fig. 16, in which relations (1) and (2) are shown as those corresponding to pulse widths of 0.1 and 1.0 msec, respectively. The abnormal phenomena observed above, as given by relations (1) and (2), will be explained with a power spectrum analysis in the next section.

350 Shock Simulation of Drop Test of Hard Disk Drives

Relative Displacement (mm)

0.4 Half-Sine Triangular Dual-Quadratic

0.3 0.2 0.1 0.0 0.1 −0.2 −0.3 −0.4

0

1

2

3

4

5

Time (ms)

Peak Relative Displacement (mm)

Figure 15 Shock responses of input acceleration shocks with 1-msec pulse width applied on the inner surface of the pivot. 0.40 0.35 0.30 0.25 0.20

Dual-Quadratic

0.15

Triangular

0.10

Half-Sine

0.05 0.00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 Pulse Width of Acceleration (ms)

Figure 16 Peak response of relative displacement versus pulse widths. Figure 17 shows the variation of the maximum relative displacements (y max ) of the actuator arm subjected to these three types of acceleration shocks with different pulse widths of 0.1, 0.2, 0.4, 0.6, 0.8, and 1.0 msec. It can be noted that the maximum relative displacement experiences a peak value with an increase of pulse width, that is, a pseudoresonance phenomenon occurs for each case of the three pulse shapes. For the acceleration shock in a triangular shape, the pseudoresonance takes place approximately at the same pulse width (∼0.6 msec) as that for the acceleration shock in a half-sine shape. However, for the acceleration shock in a dualquadratic shape, it occurs at a different pulse width of about 0.8 msec. There is a critical pulse width for each type of acceleration shock at which the pseudoresonance phenomenon appears.

4.2 Power Spectrum Analysis The response power spectra, as shown in Figs. 18 and 19, were obtained by performing a fast Fourier transform (FFT) on the relative displacement responses (Figs. 14 and 15) for

Max Relative Displacement (mm)

4

Pulse Shape Effects and Power Spectrum Analysis

351

0.50 0.45 0.40 0.35 0.30 0.25 0.20 Half-Sine Triangular Dual-Quadratic

0.15 0.10 0.05 0.00 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Pulse Width (ms)

Figure 17 Maximum relative displacement (y max ) versus pulse width.

× 10−4

Response Power (mm2)

6 5

Half-Sine

4

Triangular Dual-Quadratic

3 2 1 0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Frequency (kHz)

Figure 18 Shock response power spectra for input acceleration shocks with 0.1-msec pulse width applied on the inner surface of the pivot. 0.1- and 1-msec shock durations, respectively. In Figs. 18 and 19, the abscissa represents the frequency in kilohertz while the y coordinate is the response power in millimeters squared. From Figs. 18 and 19, it is found that, at the natural frequency of 1.3 kHz of the actuator arm, the sequence of the response power for the three different acceleration pulses of 0.1-msec pulse width from large to small has the same relationship as shown in relation (1) if the symbol M (·) is used to the represent the magnitude of the response power. However, the case of the 1-msec pulse width is the reverse, exhibiting an opposite sequence as shown in relation (2). These observations of the power spectra are consistent with the relationship of the relative displacement responses in the time domain, as shown in Figs. 14 and 15. Next, the power spectra of the acceleration pulses with three different shapes for the 0.1and 1-msec pulse widths are shown in Figs. 20 and 21, respectively, in which the abscissa represents the frequency in kilohertz while the y coordinate is the acceleration power in mm2 /msec4 . From Fig. 20, it is found that, for the 0.1-msec shock duration, at the natural frequency of 1.3 kHz the power magnitude of the half-sine acceleration waveform is the largest

352 Shock Simulation of Drop Test of Hard Disk Drives × 10−4 30

Response Power (mm2)

27 24 Half-Sine

21

Triangular

18

Dual-Quadratic

15 12 9 6 3 0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Frequency (kHz)

Figure 19 Shock response power spectra for input acceleration shocks with 1-msec pulse width applied on the inner surface of the pivot. followed by those of triangular and dual-quadratic waveforms; that is, the power magnitudes have the same relationship as that shown in relation (1) in this case. Consequently, the corresponding peak displacement for the half-sine waveform is the largest followed by the triangular and dual-quadratic waveforms, as shown in Fig. 14; that is, the relationship of the peak displacements is consistent with that shown in relation (1). From Fig. 21, for the 1-msec shock duration, still at the natural frequency of 1.3 kHz, the reverse phenomenon occurs. The power magnitude of the dual-quadratic waveform is the largest, followed by those of the triangular and half-sine waveforms, that is, the power magnitudes have the same relationship as that shown in relation (2). Therefore, the corresponding peak displacement for the dual-quadratic waveform is the largest, followed by the triangular and half-sine waveforms, as shown in Fig. 15, that is, the relationship of the peak displacements is consistent with that shown in relation (2). In Figs. 20 and 21, it is also noticed that there is a crossover point or small interval for the power spectrum curves of the three pulse shapes. When the natural frequency falls in the region to the left of the crossover point, the peak displacement of the half-sine acceleration waveform is the largest followed by those of the triangular and dual-quadratic waveforms. However, when the natural frequency falls in the region to the right of this crossover point (but to the left of any possible second crossing point), the peak displacement of the dual quadratic is the largest followed by those of the triangular and half-sine waveforms. Near the crossover point, the three pulse shapes give similar magnitudes of the peak displacement responses and the power spectra, as shown in Figs. 16 and 22, corresponding to the pulse width of 0.8 msec. It can also be found that the largest displacement response and response power occur when the pulse width is 0.8 msec for the same pulse shape which is close to the resonant frequency of the system. The above conclusions can be summarized as shown in Figs. 16 and 22, in which relations (1) and (2) are shown as those corresponding to the pulse widths of 0.1 and 1.0 msec, respectively. Figure 23 shows the variations of acceleration power at the natural frequency of 1.3 kHz for these three types of acceleration shocks with pulse widths of 0.1, 0.2, 0.4, 0.6, 0.8, and

4

Pulse Shape Effects and Power Spectrum Analysis

353

× 10−4 Acceleration Power (mm2/ms4)

16 14 12 Half-Sine

10

Triangular

8

Dual-Quadratic

6 4 2 0

0

5

10

20 15 Frequency (kHz)

25

30

Figure 20 Acceleration power spectra for input acceleration shocks with 0.1-msec pulse width applied on the inner surface of the pivot.

Acceleration Power (mm2/ms4)

0.16 0.14 0.12

Half-Sine

0.10

Triangular

0.08

Dual-Quadratic

0.06 0.04 0.02 0.00

0

1

2

3

4

5

Frequency (kHz)

Figure 21 Acceleration power spectra for input acceleration shocks with 1-msec pulse width applied on the inner surface of the pivot. 1 msec. Comparing Figs. 17 and 23, it can be found that the relative relation of the acceleration powers is consistent with that of the corresponding maximum relative displacements, as shown in Fig. 17. It can also be noted from Fig. 23 that, for both the half-sine and triangular acceleration shocks, the maximum acceleration powers for a resonant frequency of 1.3 kHz are reached at the same pulse width of about 0.6 msec. However, for the dual-quadratic acceleration shock, it is reached at about the 0.8-msec pulse width. This observation of the acceleration powers gives a reasonable explanation for the pseudoresonance phenomena observed in Fig. 17. In summary, we conclude that the relative magnitude of the peak displacement of the actuator arm is mainly determined by the power magnitude of the acceleration pulse at the resonant frequency.

Power of Acceleration at Resonant Frequency (mm2/ms4)

354 Shock Simulation of Drop Test of Hard Disk Drives 0.010 0.009 0.008 0.007 0.006 Dual-Quadratic 0.005 Triangular 0.004 Half-Sine 0.003 0.002 0.001 0.000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 Pulse Width of Acceleration (ms)

Figure 22 Power of acceleration at resonant frequency versus pulse width.

0.017

Acceleration Power at Resonant Frequency (mm2/ms4)

0.015 0.013 0.011 0.009 0.006 0.004 Half-Sine Triangular Dual-Quadratic

0.002 0.000 −0.002 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Pulse Width (ms)

Figure 23 Power of acceleration at resonant frequency versus pulse width.

4.3 Theorem From Figs. 20 and 21, it has been noted that a crossover point (or small interval) for the three different power spectrum curves occurs at a frequency value (location) of 1/T , where T is the pulse width (pulse duration). When T equals 0.1 and 1 msec, the crossover points occur at about 10 and 1 kHz, respectively. A simple mathematical theory was developed to explain this phenomenon. Theorem: If a given waveform h(t) can be expressed as a sum of an arbitrary reference waveform q(t) with a pulse width of T and two identical and adjacent waveforms g(t) and g(t - T /2) with the pulse width of each equal to T /2, then the power spectrum curve of the given waveform will coincide with the power spectrum curve of the reference waveform at the frequency values given by fn = (2n − 1) /T , n = 1, 2, 3 . . . .

References

355

Considering the triangular pulse as the reference waveform, for n = 1, the frequency of the first coincident point is f1 = 1/T , just as that of the observed crossover point. It can be shown that the dual-quadratic waveform satisfies the conditions of the theorem exactly, with many coincident points being observed at the predicted frequency values. The conditions of the theorem are an approximation to the situation of the half-sine waveform. Nevertheless, the numerical results shown in Figs. 20 and 21 demonstrate that the theorem predicts the frequency of its first crossover (coincident) point with reasonable accuracy.

5

CONCLUSIONS To investigate the shock dynamic characteristics of HDDs, the relative displacement of the actuator arm of an HDD subjected to a single half-sine acceleration pulse during the drop test was simulated by using finite element analysis. The pulse shape effect of acceleration pulses on shock responses was investigated and explained in terms of power spectrum analysis. Based on the results obtained in this study, the following conclusions are made: 1. The dimensionless peak relative displacement of the actuator arm, that is, y max , occurs at a critical frequency ratio (i.e., β = ω/ωn ≈ 0.6), which is the frequency ratio of the characteristic frequency of the half-sine acceleration pulse to the first natural frequency of the system. In other words, a pseudoresonance phenomenon occurs at the critical frequency ratio. 2. In the shock responses of the actuator, an abnormal phenomenon was observed where a stronger single half-sine acceleration pulse results in a lower relative displacement compared with those of other two pulse shapes. This has been explained in terms of a power spectrum analysis. We conclude that the relative magnitude of the maximum relative displacement is mainly determined by the power magnitude of the acceleration pulse at the resonant frequency. 3. A simple theorem was developed to illustrate the existence and location of a crossover point (or small interval) observed in the acceleration power spectrum curves. When the resonant frequency resides in the region to the left of the crossover point, the half-sine pulse gives the largest peak displacement followed by the triangular and dual-quadratic pulses; the reverse is true when the resonant frequency resides in the right of the crossover point but to the left of any higher frequency crossover points.

REFERENCES Allen, A. M., and Bogy, D.B., “Effect of Shock on the Head-Disk Interface,” IEEE Transactions on Magnetics, Vol. 32, No. 5, pp. 3717–3719, 1996. Bhargava, P., and Bogy, D. B., “Numerical Simulation of Operational-Shock in Small Form Factor Hard Disk Drives”, Journal of Tribology, Vol. 129, pp. 153–160, 2007. Edwards, J. R., “Finite Element Analysis of the Shock Response and Head Slap Behavior of a Hard Disk Drive,” IEEE Transactions on Magnetics, Vol. 32, No. 2, pp. 863–867, 1999. Gao, F., Yap, F. F., and Yan, Y., “Modeling of Hard disk Drives for Vibration Analysis Using a Flexible Multibody Dynamics Formulation,” IEEE Transactions on Magnetics, Vol. 41, pp. 744–749, 2005. Harmoko, H., Yap, F. F., Vahdati, N., Gan, S., Liu, M., and Shi, B. J., “A More Efficient Approach for Investigation of Effect of Various HDD Components on the Shock Tolerance,” Microsystem Technologies, Vol. 13, pp. 1331–1338, 2007.

356 Shock Simulation of Drop Test of Hard Disk Drives Harrison, J. C., and Mundt, M. D., “Flying Height Response to Mechanical Shock during Operation of a Magnetic Hard Disk Drive,” Journal of Tribology, Vol. 122, pp. 260–263, 2000. Ishimaru, N., “Experimental Studies of a Head/Disk Interface Subjected to Impulsive Excitation During Nonoperation,” Journal of Tribology, Vol. 118, No. 4, pp. 807–812, 1996. Jayson, E. M., Murphy, J., Smith, P. W., and Talke, F. E., “Head Slap Simulation for Linear and Rotary Shock Impulses,” Tribology International , Vol. 36, Nos. 4–6, pp. 311–316, 2003. Jen, C. W., Talke, F. E., Ohwe, T., and Gordon, A., “On Suspensions Dynamics for PicoSliders,” IEEE Transactions on Magnetics, Vol. 33, No. 5, pp. 3172–3174, 1997. Kumar, S., Khanna, V., and Sri-Jayantha, M. “A Study of the Head Disk Interface Shock Failure Mechanism,” IEEE Transactions on Magnetics, Vol. 30, No. 6, pp. 4155–4157, 1994. Lee, S. J., Hong, S. K., and Lee., J. M., “A Study of Shock-Resistance Design of Suspension Subjected to Impulsive Excitation,” IEEE Transactions on Magnetics, Vol. 37, No. 2, pp. 826–830, 2001. Murphy, A. N., Feliss, B., Gillis, D., and Talke, F. E., “Experimental and Numerical Investigation of Shock Response in 3.5 and 2.5 in. from Factor Hard Disk Drives,” Microsystems Technology, Vol. 12, pp. 1109–1116, 2006. Sagar, A., “Mobile Environment Demands Tougher Small Form Factor Drivers,” Computer Technology Review , Spring, 1993, pp. 35–39. Shi, B. J., Wang, S., Shu, D. W., Luo, J., Meng, H., Ng, Q. Y., and Zambri, R., “Power Spectrum Analysis of the Excitation Pulse Effects in Drop Test Simulation of Hard Disk Drives” paper presented at the Asia-Pacific Magnetic Recording Conference 2004, Seoul, Korea, Aug.16-19, 2004 Shi, B. J., Shu, D. W., Wang, S., Luo, J., Ng, Q. Y., Lau, J. H. T., and Zambri, R., “Shock response and power spectrum analysis of a head actuator assembly,”. Transactions on Engineering Science, Vol. 49, pp. 470–479, 2005. Shi, B. J., Wang, S., Shu, D. W., Luo, J., Meng, H., Ng, Q. Y., and Zambri, R., “Excitation Pulse Shape Effects in Drop Test Simulation of the Actuator Arm of a Hard Disk Drive,” Microsystem Technologies, Vol. 12, No. 4, pp. 299–305, 2006. Shi, B. J., Shu, D. W., Wang, S., et al. “Drop Test Simulation and Power Spectrum Analysis of a Head Actuator Assembly in a Hard Disk Drive,” International Journal of Impact Engineering, Vol. 34, No. 1, pp. 120–133, 2007. Shu, D. W., Shi, B. J., Meng, H., Yap, F. F., Jiang, D. Z., Ng, Q. Y., Zambri, R., Lau, J. H. T., and Cheng, C. S., “The Pulse Width Effect of Single Half-Sine Acceleration Pulse on the Peak Response of an Actuator Arm of Hard Disk drive,” Material Science and Engineering. A, Vol. 423, Nos. 1/2, pp. 199–203, 2006. Shu, D. W., Shi, B. J., Meng, H., Yap, F. F., Jiang, D. Z., Ng, Q. Y., Zambri, R., Lau, J. H. T., and Cheng, C. S., “Shock Analysis of a Head Actuator Assembly Subjected to HalfSine Acceleration Pulses,”. International Journal of Impact Engineering, Vol. 34, No. 2, pp. 253–263, 2007. Sohn, J. S., Choa, S. H., Hong, M. P., Lee, H. S., and Jang, D. H., “Experimental Analysis of HDD Actuator for the Improvement of Shock Reliability,” Paper Presented at the Asia-Pacific Magnetic Recording Conference. Digests of APMRC2000 on Mechanical and Manufacturing Aspects of HDD (Cat. No.00EX395), p. MP12/1-2, 2000. Takahashi, H., Shindo, H., Saegusa, S., Nakamura, S., and Matsuda, Y., “Adopting Taguchi Method for Designing High Shock Resistant Head Suspension Assembly for Mobile Hard Disk Drives,” paper presented at the 2001 IEEE International Magnetics Conference, RAI Congress Centre, Amsterdam, the Netherlands, DP 11, Apr. 28–May 2, 2002.

CHAPTER

16

SHOCK PROTECTION OF PORTABLE ELECTRONIC DEVICES USING A ‘‘CUSHION’’ OF AN ARRAY OF WIRES (AOW) Ephraim Suhir University of California, Santa Cruz, California University of Maryland, College Park, Maryland ERS Co., Los Altos, California

1

INTRODUCTION The ability to predict and, if necessary, minimize the adverse consequences of dynamic loading in electronic, opto-electronic, and photonic assemblies, components, devices, and systems is of obvious practical importance (see, e.g., [1, 2]). In commercial electronics and photonics, dynamic loading typically takes place during handling or transportation of the equipment. In military, avionic, space, automotive, marine electronics, and photonics engineering, dynamic loading is expected to occur even during normal operation of the system. Shock loading is part of military specifications and other qualification requirements (see, e.g., [3, 4]). Random and/or harmonic (sinusoidal) vibrations are often applied deliberately in addition to, or even instead of, thermal cycling or mechanical testing as an effective means to detect and weed out infant mortalities, not necessarily caused by shocks and vibrations. In recent years, the necessity to protect portable electronics from impact loading (e.g., because of an accidental drop) triggered the development of numerous theoretical models and experimental techniques for the prediction of the response of the system to, and minimizing the adverse consequences of, accidental shocks. Various aspects of drop impact in the application to portable electronic products were analyzed, for instance, by Lim and Low [5], Lim et al. [6], Zhu and Marcinkiewicz [7], and Luan and Tee [8]. It is the maximum acceleration (deceleration) that is usually viewed as an adequate criterion of the dynamic strength of an assembly or a device. It has been shown, however [9], that in some cases such an approach might be misleading, because a structural element that experiences high accelerations or decelerations might not necessarily experience high dynamic stresses, and although high accelerations are usually responsible for the functional (electrical, optical) performance of the device, it is the dynamic stresses that cause material and structural damage, thereby affecting functional reliability as well. Periodic (repetitive) impact loading could occur in micro- and opto-electronic devices during operation of some micro- and opto-electronic equipment, particularly those used in some military systems, such as rapidly shooting artillery or machine-guns [10, 11]. The repetitive nature of loading (i.e., the interaction of the vibrations caused by sequential impacts) should be considered if the damping in the system is low and/or the impacts are significant and therefore the vibrations due to the given impact do not fade away completely by the time of the application of the next impact. In nonlinear systems, even in single-degree-offreedom ones, stochastically unstable (“chaotic”) vibrations could be generated despite the Structural Dynamics of Electronic and Photonic Systems Copyright © 2011 John Wiley & Sons, Inc.

Edited by Ephraim Suhir, David S. Steinberg and T. X. Yu

357

358 Shock Protection of Portable Electronic Devices Using a “Cushion” of an Array of Wires nonrandom characteristics of the system and the deterministic external excitation caused by periodic repetitive impacts [10, 12]. Suddenly applied constant acceleration can also cause appreciable dynamic loading on a device mounted on a thin (flexible) printed circuit board (PCB). Such accelerations could occur, for instance, during take-off of a jet fighter or a space vehicle and in equipment installed in marine torpedoes and could cause significant dynamic loading on the micro- and opto-electronic equipment mounted on thin PCBs. It has been shown [13] particularly that significant induced accelerations can be due to the elevated in-plane stresses arising in a thin PCB supported by nondeformable support contour. An exact solution to the highly nonlinear equation of motion has been obtained using elliptic functions and integrals. Based on this solution, it has been found that surface-mounted devices placed in the midportion of the PCB could be subjected to an acceleration that is threefold higher than the acceleration of the PCB support contour. This solution provides guidance for the safe arrangement of surfacemounted devices on a thin PCB whose support contour is subjected to suddenly applied constant acceleration. Shock tests are often used in addition to, or even instead of, drop tests for analyzing the dynamic robustness of electronic and photonic equipment. Drop tests are usually more complicated than the shock tests, especially when there is a need to measure dynamic accelerations and/or the induced stresses. Indeed, the only independent parameter that could be “assigned” in drop tests is the drop height, while in shock tests one can “prescribe,” in an independent fashion, the duration, shape, and magnitude of the applied shock. In addition, the test equipment used for drop tests is more complicated than shock testers. Drop test conditions can be adequately substituted, however, by shock tests, provided that these conditions are correctly predicted and the shock tester is appropriately “tuned” in terms of the duration of loading and maximum acceleration of the shock impulse [14]. Certainly, thorough modeling of the dynamic response should be carried out in order to adequately “tune” the shock tester. Many recent studies were devoted to investigation of the effect of short-term loading on the reliability of solder joints in portable products, with an emphasis on lead-free solders and on drop test conditions [15, 31]. Ong et al. [15] addressed some general aspects of dynamic testing of materials in application to solder interconnections. Zhu [16], Sogo and Hara [17], Yi et al. [18], and Tan [19] examined the mechanical behavior of ball grid arrays (BGAs) in chip size packages (CSPs). Arra et al. [20] and Date et al. [21] investigated the behavior and reliability of lead-free solders under dynamic loading. Drop impact and drop test conditions were addressed also by Wu [22], Mishiro et al. [23], Wong et al. [24], Xie et al. [25], Yeh and Lai [26], Luan and Tee [27], Chiu et al. [28], Syed et al. [29], and Wu et al. [30]. As has been indicated, random vibrations play an important role in the reliability evaluations of micro- and opto-electronic equipment [1, 2, 12]. Huang, Kececioglu, and Prince [31] recently carried out a useful simplified analysis of the dynamic response of portable electronic products to random vibrations. It should be pointed out that one of the major challenges in employing a random vibration technique is the selection of the width and intensity of the input power spectrum: This spectrum should contain all the (low) frequencies of interest and the intensity of the spectrum should be, on the one hand, strong enough to produce meaningful results but, on the other hand, weak enough not to cause permanent damage to the materials and structure under evaluation. Dynamic loading does not lend itself easily to meaningful testing, and it is often impossible to separate the influence of different factors affecting the dynamic response of a system to shock loading. Therefore predictive modeling, both analytical and computer aided (simulations), plays an important role in the analysis and design of micro-electronic and photonic materials and structures subjected to dynamic loading [30–48]. Shock protection of portable devices is of particular importance and has therefore occupied an especially important place in current research efforts. Goyal et al. [49] and

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Introduction

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Seah et al. [50] addressed dynamic stresses arising in PCBs, including those used in portable electronic devices, and discussed ways to protect these devices from shocks. Goyal et al. [51] evaluated the shock response spectrum of a shock protection system in a portable electronic product. Substantial improvement in shock protection can often be achieved by employing a multi-degree-of-freedom system with or without appreciable damping. This has been demonstrated particularly using an example of a two-degree-of-freedom system (“box-in-a-box” type). It has been shown [52] how to select the masses and spring constants of the structural elements in the system so as to avoid resonance conditions. These might lead to a highly undesirable “rigid impact” for the vulnerable element to be protected. Viscous damping can have an appreciable effect on the induced accelerations in a system experiencing a drop impact. It has been shown [53] that, although elevated damping always leads to lower maximum displacements (lower “breaking distances”), excessive damping can result in accelerations (decelerations) that are significantly larger than the accelerations in a damping-free single-degree-of-freedom system. It has been found also that there is a certain limit to what could be achieved by optimizing viscous damping in a single-degree-of-freedom system as far as minimizing the “breaking distance” is concerned. Using examples of nonlinear springs with rigid or soft cubic characteristics of the restoring force, it has been shown [54–56] that nonlinear springs, particularly springs with soft characteristics of nonlinearity, offer certain advantages as effective shock protective elements. Wire-type structural elements [56] behave indeed, even when subjected to very small deflections, as highly nonlinear springs with soft characteristics of the restoring force. In the analysis that follows we consider an array of cantilever wires that are fabricated on a substrate that is attached to a heavy mass to be protected from a drop impact. An equivalent loading situation is shown schematically in Fig. 1. We evaluate the dynamic response of the mass to a drop impact. We address small deflections of initially curved wires embedded into a low-modulus elastic media and treat these wires as beams lying in a continuous elastic foundation, which is provided by this medium. We use the ratio of the initial velocity squared to the product of the maximum displacement and the maximum acceleration (deceleration) during the impact process as a suitable criterion (figure of merit) of the quality of the dynamic system, that is, its ability to minimize the maximum displacement (the “breaking distance”) and/or the maximum acceleration/deceleration [54–56].

Mass, M, under protection V h0 H Substrate

Wire array embedded into a low– modulus medium

Rigid floor

Figure 1 Wire array embedded into a low-modulus material and acting as a nonlinear cushion for a heavy mass subjected to a drop impact.

360 Shock Protection of Portable Electronic Devices Using a “Cushion” of an Array of Wires 2

ANALYSIS Examine a long cantilever beam lying on a continuous elastic foundation and subjected to compression. It is presumed that the condition (see Appendix A)  4 K l ≥ 3.5 (1) 4EI is fulfilled for this beam. This means that the beam is long and flexible enough so that an array of such beams could provide a satisfactory cushion. Since the conditions at the clamped end are not critical in this case, we place the origin of the coordinate, x , at the free end of the beam. Then the expression for the deflection function can be obtained as   sin γ2 x −γ1 x cos γ2 x −  , (2) w(x ) = w0 e η2 − 1 where w0 is the deflection at the free end. The notation is the same as in Appendix A. The axial displacement can be found, for small enough displacements, as

3/2  η 1 ∞   2 , δ= w (x ) dx = δ0 2 0 η−1

(3)

where 1 2 (4) γw 8 0 is the initial (stress-free) axial displacement due to the initial (also stress-free) deflection of the beam at its free end. The displacement δ changes from δ√0 to 2.83 δ0 when the compressive force T changes from zero to its critical value T = Te = KEI . The further increase in the compressive force leads to the rapid increase in the axial displacements and is not advisable. From (3), considering the first formula in (A.3), we obtain the following relationship between the axial compressive force T and the axial displacement δ:  2/3 δ0 T = 2Te 1 − . (5) δ δ0 =

Let a lumped mass M mounted at the free end of a cantilever beam be dropped from the height H on a rigid floor (Fig. 1). The equation of motion can be written, considering relationship (5), as  2/3 δ0 ¨ M δ(t) + 2Te 1 − = 0. (6) δ Damping is not considered in this analysis. Since ¨ = δ˙ δ(t) the first integral of Eq. (6) is δ˙2 = 2gH − 4

Te M



d δ˙ , dδ



δ + 2δ0 − 3 3 δ02 δ .

(7)

The maximum axial displacement δ = δmax can be found, assuming δ˙ = 0, from the equation  δmax δmax −33 − 2q = 0, δ0 δ0

(8)

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Numerical Example

361

where the following notation is used: q= Introducing a new variable ξ as

MgH − 1. 4Te δ0

(9)



δmax 1 3 = ξ+ , δ0 ξ

(10)

ξ 6 − 2qξ 3 + 1 = 0.

(11)

we obtain Eq. (A.3) in the form ξ3

and its solution is as has follows: This is a quadratic equation with respect to

 3 ξ = q + q 2 − 1.

(12)

After the maximum displacement δmax is determined, the maximum acceleration (deceleration) could be found from Eq. (6): 

2Te δ0 2/3 δ¨max = − , (13) 1− M δmax and the “quality” [54–56] of the nonlinear dynamic system in question could be found as 2gH Q =− . (14) δmax δ¨max The numerator in this formula is the initial velocity squared, and the numerator is the product of the maximum “breaking distance” and the maximum deceleration. The smaller this product is, the greater is the “quality” of the shock protection.

3

NUMERICAL EXAMPLE Let a weight P = 0.1 kg be dropped from height H = 1.5 m (its potential energy is MgH = 0.15 kg · m = 150 kg · mm) and an array l0 = 100 μm = 0.1 mm long cantilever nanowires of diameter d = 200 nm = 0.0002 mm is used as an appropriate protective “cushion” (Fig. 1). Young’s modulus of the material is E = 20, 000 kg/mm2 . The maximum initial deflection of the stress-free nanowires is f0 = 0.01 mm, that is, significantly larger than the wire diameter. The objective of the calculation that follows is to establish the required number of the nanowires to provide adequate shock protection for the mass M . The flexural rigidity of a single nanowire is πd4 3.14159 × 0.00024 = 20,000 × = 1.5708 × 10−12 kg mm2 . 64 64 Its critical force for a single nanowire is EI =

π 2 EI 3.141592 × 1.5708 × 10−12 = = 3.8758 × 10−10 kg. 4 × 0.12 4l02 Let us select the maximum force that the nanowire will experience as, say, 75% of the critical force, that is, Tmax = 2.9068 × 10−10 kg. This corresponds to the “quality” 1 = 1.6. Q= 1 − 0.75/2 The δ0 value for the nanowires is

 π 2 f 2 3.14159 2 0.012 0 δ0 = = = 0.00061685 mm. 4 l0 4 0.1 Te =

362 Shock Protection of Portable Electronic Devices Using a “Cushion” of an Array of Wires Then formula (30) suggests that a single nanowire is able to effectively provide protection to an external energy of MgH =

T e δ0 (Te /Tmax − 1)2

=

3.8758 × 10−10 × 0.00061685 (1/0.75 − 1)2

= 2.1517 × 10−12 kg mm.

Since, however, the potential energy that should be “fought against” is as high as 150 kg·mm, the required number of the nanowires that should be grown (employed, manufactured) is 150 ∼ = 7 × 1013 . 2.1517 × 10−12 The maximum acceleration (deceleration) is δ¨max Tmax 2.9068 × 10−10 × 7 × 1013 = = = 203,476. g P 0.1 The maximum displacement (“breaking distance”) is     1 1 − 1 = 0.00061685 −1 δmax = δ0 (1 − Tmax /Te )2 (1 − 0.75)2 = 0.009253 mm = 9.253 μm, that is, about 9% of the wire span (height).

APPENDIX A: COMPRESSED CANTILEVER BEAM OF FINITE LENGTH LYING ON AN ELASTIC FOUNDATION We mimic the effect of the low-modulus medium in which the nanowires are embedded as an elastic foundation for the wires in the array. Let a cantilever beam with a flexural rigidity EI and of finite length l be lying on a continuous elastic foundation with a spring constant K and be subjected to a compressive force T applied to its free end. The equilibrium condition of the beam can be written as EIwIV (x ) + Tw  (x ) + Kw(x ) = 0,

(A.1)

where w(x) is the deflection function. The origin √ of the coordinate x of the beam is at its clamped end. For low enough T values (T < 2 KEI ), the solution to Eq. (A.1) is    η+1 w(x ) = C1 cosh γ1 x sin γ2 x − (A.2) sinh γ1 x cos γ2 x + C2 sinh γ1 x sin γ2 x . η−1 where the following notation is used: √ 2 KEI k η= η ∓ 1, , γ1,2 = T 2

 k=

T . EI

(A.3)

Solution (A.2) satisfies the boundary conditions w(0) = 0, w (0) = 0 at the origin, where the displacement and the angle of rotation must be zero. The constants C1 and C2 of integration can be found form the conditions w (l ) = 0,

EIw (l ) + Tw (l ) = 0

at the free end. The first condition indicates that the curvature (bending moment) at the free end must be zero, and the second condition is, in effect, the condition of equilibrium of the

Appendix B: Unembedded Cantilever Wire (Beam) Subjected to Axial Compression

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forces acting at the free end of the beam. These conditions result in the following equations for the constants C1 and C2 of integration:       η+1 tanh γ1 l cos γ2 l C1 − tanh γ1 l sin γ2 l − η2 − 1 cos γ2 l C2 = 0, η sin γ2 l + η−1       η+1 2 tanh γ1 l cos γ2 l C2 = 0. −2 tanh γ1 l sin γ2 l + η − 1 cos γ2 l C1 + (η − 1) sin γ2 l − η−1 (A.4) For the constants C1 and C2 to be nonzero, the determinant of Eq. (A.4) must be zero. Hence,

η+1 2 2 2 tanh γ1 l cos γ2 l η(η − 1) sin γ2 l − η−1 (A.5)     −2 tanh2 γ1 l sin2 γ2 l − η2 − 1 cos2 γ2 l = 0 For long enough (large l values) and highly flexible (low EI values) beams lying on highly compliant elastic foundations (low K values), the hyperbolic tangents in Eqs. (A.4) and (A.5) can be put equal to 1. Equation (A.5) has in this case the solution η = 2. Note that this solution applies when the condition γ l > 3.5 is fulfilled, that is, when the hyperbolic tangent and cotangent √ are practically equal to 1. In the condition γ l > 3.5, the parameter γ is expressed as γ = 4 K /(4EI ) and is the parameter of a beam lying on a continuous elastic foundation and experiencing bending deformations. Note that when the compressive force T [in the second formula in (A.3)] tends to zero, the parameters γ1 and γ2 tend to γ .

APPENDIX B: UNEMBEDDED CANTILEVER WIRE (BEAM) SUBJECTED TO AXIAL COMPRESSION It is desirable (and practically inevitable) that a wire (beam) has an initial curvature so that it performs as a spring even at very small axial displacements (loading). Let a cantilever wire be characterized by an initial curvature πx w0 (x ) = f0 sin , (B.1) 2l0 where f0 is the maximum initial deflection of the wire (at its free end) and l0 is the wire’s span, that is, the distance measured along the x axis from the wire’s tip to its clamped end. The origin of the coordinate, x, is at the wire’s free end. The curvature (B.1) satisfies the boundary conditions w0 (0) = 0,

w0 (l0 ) = 0,

w0 (l0 ) = f0 .

If the axial force T is applied to the free end of the wire (beam), the condition  EIw 0 (0) + Tw0 (0) = 0

should also be fulfilled. This condition results in the following well-known formula for the critical (Euler) force: Te =

π 2 EI 4l02

(B.2)

for a cantilever beam. We assume that the induced axial displacement δ = l0 − lt

(B.3)

364 Shock Protection of Portable Electronic Devices Using a “Cushion” of an Array of Wires is significantly smaller than the initial, l0 , and final, lt , wire spans. Such a (linear) approach is considered accurate enough if the angle of rotation of the wirecross section at its free end does not exceed 20◦ ; otherwise the well-known Euler’s “elastica” solution would have to be applied. In formula (B.2), EI is the flexural rigidity of the wire, E is Young’s modulus of the wire material, and I = π d 4 /64 is the moment of inertia of the cross section for the case of a wire with a circular cross section of diameter d . When the wire experiences small axial displacements, the induced deflections can be sought in a form similar to (B.1): πx , (B.4) w1 (x ) = f1 sin 2lt where f1 is the maximum induced deflection (at the free end of the wire). Since the axial displacement is small compared to the wire spans prior to and after the application of the compressive force, the final span, lt , in the denominator of the argument in the formula (B.4) can be assumed equal to the initial span l0 : πx . (B.5) w1 (x ) = f1 sin 2l0 The relationship between the initial, f0 , and the force-induced, f1 , deflections can be established from the equilibrium equation EIw1 (x ) + T [w1 (x ) + w0 (x )] = 0.

(B.6)

Introducing (B.1) and (B.5) into this equation, we obtain the following formula for the maximum induced deflection, f1 : f0 f1 = , (B.7) Te /T − 1 where Te is the critical force given by formula (B.2). The total maximum deflection is therefore f0 (B.8) 1 − T /Te and the total deflection function, which is due to both the initial and the induced curvatures, is πx . (B.9) wt (x ) = ft sin 2lt The length st of the compressed and deflected wire can be evaluated, for small enough curvatures, as follows:  lt   lt y 1 1 lt  st = 1 + [wt (x )]2 dx ∼ [1 + (wt (x ))]2 dx = lt + [wt (x )] = 2 2 0 0 0

  π 2 f 2  π 2 f 2 πx 1 π ft 2 l t t ∼ t cos2 dx = lt + . (B.10) = lt + = lt + 2 2lt 2lt 4 lt 4 l0 0 Similarly, the length s0 of the stress-free wire can be found as ft = f0 + ft =

s0 = l0 + δ0 , where

(B.11)

 π 2 f 2 0 (B.12) 4 l0 is the difference between the total length of the wire and its span. Assuming that the lengths (B.10) and (B.11) are the same, we conclude that the axial displacement δ due to the applied compressive force T is expressed as follows:   ft 2 T 1 − T /(2Te ) − 1 = 2δ0 . (B.13) δ = δ0 Te (1 − T /Te )2 f02 δ0 =

Appendix B: Unembedded Cantilever Wire (Beam) Subjected to Axial Compression The axial spring constant can be found as the ratio

T δ KT = = K0 χ , δ δ0 where Te 2EI K0 = = 2 2δ0 f0 l0

365

(B.14)

(B.15)

is the initial spring constant (prior to the application of the compressive force) and the factor



δ 1 δ0 χ 1− √ =2 (B.16) δ0 δ 1 + δ/δ0 considers the effect of the axial displacement. As evident from this formula, the axial spring “constant” of the wire rapidly decreases with an increase in the induced displacement: The initial spring constant decreases by a factor of 3 when the induced axial displacement is three times larger than the δ0 value. Let a single-degree-of-freedom system with mass M which is protected by a spring element of the type considered in the previous sections be dropped from the height H on a nondeformable (hard) floor. If the mass M is supported by a cantilever wire or an array of such wires (AOW), then the equation of motion of the system can be written as

T 1 ¨ + e 1− √ δ(t) = g, (B.17) M 1 + δ(t)/δ0 ˙ δ/dδ) ˙ where g is the acceleration due to gravity. Since δ¨ = δ(d and    δ δ dδ √ = 2δ0 1+ −1 . (B.18) δ0 1 + δ/δ0 0 Eq. (B.17) has the following first integral:     T δ e = 2g(δ + H ) ∼ δ˙2 + 2 δ − 2δ0 −1 + 1 + = 2 gH . M δ0

(B.19)

The maximum displacement can be found from (B.19) by putting δ˙ = 0. This yields ⎤ ⎡  2 MgH δmax = δ0 ⎣ 1 + − 1⎦ . (B.20) T e δ0 Solving Eq. (B.13) for the T /Te ratio and using formula (B.20), we obtain the following formula for the ratio of the maximum force to the critical force: 1 Tmax =  , (B.21) Te T e δ0 1+ MgH  where Tmax = T δ=δmax is the value of the compressive force that corresponds to the maximum axial displacement. Introducing formula (B.20) for the maximum axial displacement into Eq. (B.17), we obtain the following formula for the maximum acceleration (deceleration): Te 1 Tmax ∼ Tmax δ¨max = g − . (B.22)  =g− =− M M M T e δ0 1+ MgH Since both the maximum displacement and the maximum acceleration (deceleration) should be made as small (low) as possible, the ability of a dynamic system to provide the most

366 Shock Protection of Portable Electronic Devices Using a “Cushion” of an Array of Wires effective protection to the vulnerable mass M could be assessed on the basis of the following dimensionless parameter (quality):  MgH 1+ 1 2gH T δ =2 . (B.23) Q =−  e 0 = ¨ Tmax 1 MgH δmax δmax 1− 2+ 2 Te T e δ0 In a linear system with the √ natural frequency ω0 the√maximum displacement and the maximum acceleration are δmax = 2gH /ω0 and δ¨max = ω0 2gH , respectively, so that the quality of a linear system is always 1 and is independent of the system’s characteristics. As to the quality of an ideal/perfect protection system, it is as high as 2. Indeed, such a system should be characterized by a constant acceleration (deceleration) that is “turned on” at the moment of the beginning of breaking and is “turned off” at the moment when the velocity of the mass to be protected becomes equal to zero. Then the corresponding “breaking distance” can be found as δmax = −

v20 2δ¨max

(B.24)

where v0 is the initial velocity. From this equation, using notation (B.23), we find Q = 2. Thus, the quality of a nonlinear system can be, as evident from formula (B.23), considerably higher than 1. For the maximum force value approaching the critical force, the system’s quality can be almost twice as high as the quality of a linear system, that is, not be very far remote from the ideal situation.

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43. 44. 45.

46.

Systems—Proceedings of the Intersociety Conference, Conference: Proceedings of the 1998 6th Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems, ITHERM, May 27–30 1998, pp. 330–336. Huang, W., Kececioglu, D. B., and Prince, J. L., “A Simplified Random Vibration Analysis on Portable Electronic Products,” IEEE Transactions on Components & Packaging Technologies, Vol. 23, No. 3, 2000. Lim, C. T., Teo, Y. M., and Shim, V. P. W., “Numerical Simulation of the Drop Impact Response of a Portable Electronic Product,” IEEE CPMT Transactions, Vol. 25, No. 3, Sept. 2002. Tee, T. Y., Ng, H. S., Lim, C. T., Pek, E., and Zhong, Z. W., “Application of Drop Test Simulation in Electronic Packaging,” paper presented at the 4th ASEAN ANSYS Conf., 2002. Xu, L., et al., “Numerical Studies of the Mechanical Response of Solder Joints to Drop/Impact Load,” Proc. EPTC, Singapore, 2003. Tee, T. Y., Ng, H. S., Lim, C. T., Pek, E., and Zhong, Z. W., “Board Level Drop Tests and Simulation of TFBGA Packages for Telecommunication Applications,” 53rd ECTC Proc., New Orleans, LA, May 2003. Zhu, L., “Modeling Technique for Reliability Assessment of Portable Electronic Product Subjected to Drop Impact Loads,” 53rd ECTC, New Orleans, LA, 2003, pp. 100–104. Wang, Y .Q., “Modeling and Simulation of PCB Drop Test,” Proc. 5th EPTC, Singapore, 2003, pp. 263–268. Luan, J. E., Tee, T .Y., Pek, E., Lim, C. T., and Zhong, Z. W., “Modal Analysis and Dynamic Responses of Board Level Drop Test,” Proc. 5th EPTC, Singapore, 2003. Zhu, L., and Marcinkiewicz, W., “Drop Impact Reliability Analysis of CSP Packages at Board and Product System Levels through Modeling Approaches,” paper presented at the Intersociety Conference on Thermal and Thermo-Mechanical Phenomena, 2004, pp. 296–303. Irving, S., and Liu, Y., “Free Drop Test Simulation for Portable IC Package by Implicit Transient Dynamics FEM,” Proceedings—Electronic Components and Technology Conference, Vol. 1, Proceedings—54th Electronic Components and Technology Conference (ECTC), Las Vegas, NV, 2004. Tee, T. Y., Luan, J. E., Pek, E., Lim, C. T., and Zhong, Z. W., “Novel Numerical and Experimental Analysis of Dynamic Responses under Board Level Drop Test,” EuroSime Conference Proc., 2004. Tee, T. Y., Luan, J. E., Pek, E., Lim, C. T., and Zhong, Z. W., “Advanced Experimental and Simulation Techniques for Analysis of Dynamic Responses During Drop Impact,” 54th ECTC Proc., Las Vegas, NV, 2004, pp. 1088–1094. Yeh, C. L., and Lai, Y. S., “Transient Simulation of Solder Joint Fracturing Under Impact Test,” 6th EPTC, Singapore, Dec. 8–10, 2004. Marjamaki, P., Mattila, T., and Kivilahti, J., “FEA of Lead-Free Drop Test Boards,” 55th ECTC Proc., Orlando, FL, 2005. Suhir, E., “Predicted Fundamental Vibration Frequency of a Heavy Electronic Component Mounted on a Printed Circuit Board,” ASME Journal of Electronic Packaging, Vol. 122, No. 1, 2000. Suhir, E., “Free Vibrations of a Fused Biconical Taper Lightwave Coupler,” International Journal of Solids and Structures, Vol. 29, No. 24, 1992.

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47. Suhir, E., “Vibration Frequency of a Fused Biconical Taper (FBT) Lightwave Coupler,” IEEE/OSA Journal of Lightwave Technology, Vol. 10, No. 7, 1992. 48. Suhir, E., “Elastic Stability, Free Vibrations, and Bending of Optical Glass Fibers: The Effect of the Nonlinear Stress-Strain Relationship,” Applied Optics, Vol. 31, No. 24, 1992. 49. Goyal, S., Buratynski, E. K., and Elko, G. W., “Shock-Protection Suspension Design for Printed Circuit Board,” Proceedings of SPIE, Vol. 4217, 2002. 50. Seah, S. K. W., Lim, C. T., Wong, E. H., Tan, V. B. C., and Shim, V. P. W., “Mechanical Response of PCBs in Portable Electronic Products during Drop Impact,” Proceedings 4th Electronics Packaging Technology Conference (EPTC 2002), Singapore, Dec. 10–12, 2002. 51. Goyal, S., Papadopoulos, J. M., and Sullivan, P. A., “Shock Protection of Portable Electronic Products: Shock Response Spectrum, Damage Boundary Approach, and Beyond,” Shock and Vibration, Vol. 4, No. 3, 1997. 52. Suhir, E., and Burke, R., “Dynamic Response of a Rectangular Plate to a Shock Load, with Application to Portable Electronic Products,” Components, Packaging, and Manufacturing Technology, Part B: Advanced Packaging, IEEE Transactions on, Vol. 17, No. 3, 1994. 53. Suhir, E., “Dynamic Response of a One-Degree-of-Freedom Linear System to a Shock Load During Drop Tests: Effect of Viscous Damping,” IEEE CPMT Transactions, Part A, Vol. 19, No. 3, 1996. 54. Suhir, E., “Shock Protection with a Nonlinear Spring,” IEEE CPMT Transactions, Advanced Packaging, Part B, Vol. 18, No. 2, 1995. 55. Suhir, E., “Shock-Excited Vibrations of a Conservative Duffing Oscillator with Application to Shock Protection in Portable Electronics,” International Journal of Solids and Structures, Vol. 33, No. 24, 1996. 56. Suhir, E., “Dynamic Response of Micro-Electronic Systems to Shocks and Vibrations,” in E. Suhir, C. P. Wong, and Y. C. Lee (Eds.), Micro- and Opto-Electronic Materials and Structures: Physics, Mechanics, Design, Reliability, Packaging, Vol. 1, Materials Physics/Materials Mechanics, 2006. 57. Timoshenko, S. P., and Gere, J. M., Theory of Elastic Stability, 2nd ed., McGraw-Hill, New York, 1988. 58. Suhir, E., Structural Analysis in Microelectronics and Fiber Optics, Van-Nostrand, 1991.

CHAPTER

17

BOARD-LEVEL RELIABILITY OF LEAD-FREE SOLDER UNDER MECHANICAL SHOCK AND VIBRATION LOADS Toni T. Matilla, Pekka Marjamaki, and Jorma Kivilahti Helsinki University of Technology Helsinki, Finland

1

INTRODUCTION Reliability of portable electronic devices is becoming an important factor of success in the highly competitive global markets due to the fact that decreased solder interconnection volumes, increased current densities, increased power consumption, and increased heat dissipation of novel high-density electronics make electrical interconnections more vulnerable to failures. Furthermore, these high-density and high-power applications experience diverse operation environments when they are simultaneously exposed to several different loads. The vast majority of electronic applications experience thermal-mechanical loads during normal operation but especially portable consumer electronic products are exposed, in addition, to mechanical shocks and vibrations. Since portable electronic products encounter such diverse environments in ordinary daily use, their reliability should be studied with tests that simulate real-use strains and stresses as realistically as possible. Drop impact studies carried out with commercial portable electronic products have shown that impact forces generated by the strong deceleration pulse are transmitted through the product casing to the component boards and high-frequency vibration is initiated [1–11]. The high-frequency bending of the component board induces strains at the solder interconnections at very high strain rates. The magnitudes of forces and decelerations experienced by products are, however, highly dependent on the orientation at which they hit the ground, design of their casing, as well as their weight and dimensions [7, 9–11]. Even though the shock impacts caused by dropping create highly complex bending motion of the component boards, certain generalizations can be made on the impact forces, strains, and accelerations based on the detailed investigations published by Lim et al., who studied the drop impact responses of various portable electronic devices [6–11]. Both vertical and horizontal impact orientations result in much larger impact forces and decelerations transmitted to the component boards inside the product casings as compared to other orientations. This is due to the fact that when a device falls on one of its corners, not all the kinetic energy is converted and the device tends to continue its motion (and rotate as the center of mass will change the direction of motion) and further collisions follow. It is to be noted, however, that vertical drops cause the highest impact force but horizontal ones induce the highest bending strains [10, 11]. This is one of the most important findings of the product-level tests and also the reason why many of the board-level drop tests are performed horizontally. Another important observation is that although the bending and deformation of the component board at the moment of impact Structural Dynamics of Electronic and Photonic Systems Copyright © 2011 John Wiley & Sons, Inc.

Edited by Ephraim Suhir, David S. Steinberg and T. X. Yu

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372 Board-Level Reliability of Lead-Free Solder under Mechanical Shock and Vibration Loads can be highly localized, the “free” vibration of the component board thereafter (their modes and frequencies) is independent on the location of the impact and the drop orientation of the device [11]. This is because the natural modes of vibration depend on the geometry of the board as well as its weight distribution (i.e., component layout) and support. The natural frequencies, on the other hand, depend on the mode, stiffness (thickness and elastic modulus) of the structure, and total mass. The results from product-level tests have been used to develop board-level drop tests such as the widely employed Joint Electron Device Engineering Council (JEDEC) standard JESD22-B111, “Board-Level Drop Test Standard for Handheld Electronic Products” [12–15]. The reliability of electronic assemblies under shock loading conditions has been studied extensively with the JEDEC-compliant standardized drop test, which aims to evaluate and compare drop performance electronic components at the board level without considering the effects of product covers or other adjacent structures in an accelerated manner. The method is widely employed mainly because at the time of its introduction: in 2003 it was the first attempt to create a common procedure for the portable electronics industry, where different companies had so far employed their own self-developed test methods and the results of which were difficult to compare. However, there are many inconveniences related to the method of drop testing. First of all, testing can be very time consuming and laborious, especially if the components are relatively small, when the number of drops required to determine their lifetime may increase to hundreds. In such cases faulty signals due to failures of the measurement cables become common and their repair further increases testing time. In addition, the possibilities to measure the component board behavior during testing are limited to strain gauges and more accurate contact-free measurements of the board, such as optical measurements of the bending amplitude with a laser, are very difficult due to the motion of component boards during testing. In order to avoid many of the difficulties inherent to the “impact-based methods” of drop testing, an alternative testing method based on continuous harmonic vibration of the test board has been introduced. It has been shown that highly similar loading conditions can be produced by applying harmonic loading to assemblies at their natural (resonance) frequencies and, furthermore, the failure modes in the vibration tests have been found to be the same as those in the drop tests [16–20]. Perhaps the most important benefit of the vibration test method is that the time required for testing one component board is decreased from dozens of minutes (sometimes even hours), taking place during drop testing, the range of a few seconds. Other benefits include (1) the contact-free measurements enabled by the smaller distance that component boards travel during testing, (2) the bending mode(s) can be determined by adjusting (sweeping) the loading frequency, and (3) vibration testing can be carried out in a chamber with temperature and atmosphere control. The adoption of high-density chip scale packaging technologies and larger scales of integration have enabled the introduction of numerous new functions into portable electronic equipment (see Fig. 1). New functions and higher performance of the devices have increased their power consumption and, more importantly, increased internally generated heat dissipation from high-power-density packages. Thus, it is likely that the temperatures inside products are well above their ambient values when they experience the mechanical shock caused by the accidental drops. Furthermore, portable products are seldom dropped soon after they are assembled. Most likely their components have been exposed to elevated temperatures and experienced thermomechanical loadings to some extent. It is well known that different loading conditions will evoke different failure mechanisms that produce dissimilar failure modes. The reliability of electronics is to a large extent determined by the ability of electrical interconnections to withstand the various loadings during products’ operational lifetime.Thermomechanical reliability is controlled by the microstructures of solder interconnections formed during soldering and their subsequent evolution during use [21–26]. The solder interconnections fail under thermal cycling by cracking through the bulk

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solder because the as-soldered microstructure, which is composed of only a few large eutectic colonies, undergoes local recrystallization that produces networks of grain boundaries along which the intergranular cracks damage solder interconnections. Mechanical shock impacts, on the other hand, are known to produce entirely different kinds of failure modes: cracks in the newly soldered interconnections do not propagate through the bulk of the solder but propagate mainly along the brittle intermetallic compound layers formed between solder and contact metallizations [27–31]. If we consider the entire life span of portable products, the accidental shock loads caused by dropping take place between the thermal cycles caused by internal heat production. Dense continuous networks of grain boundaries may have been produced in the solder interconnections by recrystallization and the above-mentioned two failure modes are mixed when the products are dropped [31, 32]. Thus, even though the thermomechanical loads are not always considered as the primary cause of failures in portable electronic products, the changes of microstructures initiated by the cyclic thermomechanical loads make the reliability of these devices dependent on their loading histories. Hence, by understanding the fundamental relations between real service loading conditions of electronic equipment and the mechanisms behind the observed failure modes—caused during accelerated reliability tests as well as during use of the products—the potential reliability risks in present and future designs can be identified and solved. In this chapter we discuss many details on the loading conditions evoked by drops, the failure modes produced with different material combinations, and the physics of failure of solder interconnections under fast deformation rates. The focus is on board-level reliability evaluation. A method based on continuous harmonic vibration that is able to produce highly similar loading condition as compared to the JEDEC board-level drop test is presented and examined. Furthermore, because temperatures during normal operation are well above half of the melting temperatures of commonly used solders, plastic properties are significantly dependent on temperature and strain rate. The analysis of the failure mechanisms will cover also the effect of temperature on the reliability of component boards. The effect of temperature can be rather complex since the mechanical properties of printed wiring boards, solders, and component packages can have very different sensitivities to the change of temperature. Thus, for example, the increase of testing temperature can have very different effects on the reliability performance depending on the materials and structures of the component boards. This will be exemplified with two case examples.

374 Board-Level Reliability of Lead-Free Solder under Mechanical Shock and Vibration Loads 2

METHODS OF SHOCK IMPACT TESTING Many different high-strain-rate methods ranging from the classical split Hopkinson pressure bar [34–37] and miniature Charpy testers [38–40] for single solder balls to sophisticated drop testers for populated component boards have been proposed for studying the impact reliability electrical components and assemblies (see Fig. 2). The single joint testers are typically component-level tests where individual solder bumps are either sheared [38–44] or pulled [45–49] from the packages by using striker probes or mechanical jaws and the force required to detach the bump is recorded. The advocates of the component-level tests often mention shorter testing times, lower cost, and smaller dispersion of the measured data as some of the most important benefits of the approach as compared to the board-level drop tests. However, the approach has an undisputable drawback: In functional products the impact loads are transmitted to the interconnections via the vibration motion of the printed wiring board (PWB) and, therefore, the stress states of the solder interconnections during the component-level tests differ greatly from those experienced in soldered structures [49]. Furthermore, electrical failures of the solder interconnections are not necessarily near the package side interfaces of the interconnections. Few methods have been developed that are still considered as component-level tests but utilize components soldered to a small-sized PWB substrate and the component package is shared from the PWB [50, 51]. The benefit of this approach is that the structures of the test samples as well as the microstructures of the solder interconnections closely resemble those in functional products. In addition, they can provide reasonably good values of the mechanical properties of soldered interconnections that can be utilized in the finite element calculations, for instance. However, despite the recognized benefits of the component shear tests, they tell little about the reliability of the component boards under shock loading since the loading condition of the interconnections, as we shall discuss later in this chapter, is highly complex and governed by the normal strains rather than the shear strains. Due to the highly complex response of the component board to the shock loads, it seems inevitable to include a relatively large PWB in the test structure and load the interconnections by its response to the shock impulse. A variety of techniques that apply the principle of the historical American Society for Testing and Materials (ASTM) “Falling Weight Test” standards [52, 53] have been introduced for electronic component boards [54, 55]. However, the impact load of the weight produced highly localized deformation of the component board at the vicinity of the impact location. Furthermore, as discussed above, the modes and frequencies of vibration after the shock impact are dependent on the structure of the component board: Bending modes are determined by the geometry and weight distribution whereas the frequencies are dependent on the mode, stiffness, and total mass. Therefore, in order to produce similar loading of the interconnections on the test board as takes place in functional products, the test boards should have approximately the same size and weight. Furthermore, in order to avoid the direct impact of the falling weight on the components, the impact load should be subjected to the device under test from locations far enough from the region or components of interest. This is why supporting the component boards from the four corners or clamping along the edges and applying the shock impact through them is much more preferred. This way the inertia of the component board at the shock impact makes the board bend between the supports and the modes and frequencies of vibration determined by the component board. This is an approach adopted by a few research groups [27, 56–59] as well as the widely employed JESD22-B111 board-level drop test standard for portable applications [12]. When the component boards are loaded with a shock impulse, several natural bending modes are excited simultaneously and, as a consequence of their simultaneous vibration, the bending of test assemblies can be highly complex and the mechanical analysis by the

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Figure 2 Some high-strain-rate methods developed for shock impact testing of surface mount devices and assemblies: (a) split Hopkinson pressure bar [34–37]; (b) miniature Charpy test [38–40]; (c) single bump shear or pull test [38–49]; (d ) die/package shear [51]; (e) die/package double shear [52].

376 Board-Level Reliability of Lead-Free Solder under Mechanical Shock and Vibration Loads finite element method (FEM) can become challenging. The finite element analysis is needed, first, to evaluate the stress and strain states of the interconnections during loading since their measurement is nearly impossible due to the sizes the interconnections and, second, to establish a correlation between the lifetime under accelerated tests and the lifetime under operation conditions. Therefore, the possibilities to mimic the drop loading condition with the (monotonic or cyclic) four-point bending test have been studied [60–64]. The advantage of the four-point bending setup is that the board behavior would be much more simplified. In such tests the resistance of a daisy-chained component and board strains are measured continuously and the failure strain is recorded as the resistance measurement increases above a predefined threshold values. The IPC (The Association Connecting Electronics Engineers) and JEDEC have published a joint standard for monotonic bend characterization on board-level interconnects [65]. There are many benefits to the test; for example, the tests are very fast and easy to carry out, the loading condition is very well defined, and strain rates up to 7–8 sec−1 have been reported [61]. However, especially in the monotonic form, the loading condition of the solder interconnections is almost pure share and therefore differs greatly, as we shall discuss later on, from that experienced by those under free vibration of the component board.

2.1 Bending of Component Board Under JESD22-B111 Drop Test Several studies carried out with commercial portable electronic products have shown that impact forces generated when products are dropped onto the ground are transmitted through the product casing to the component boards and make the boards bend and vibrate excessively. This bending motion can cause various failure modes in the device, most common of which are cracking of the circuit board, cracking of copper traces on the PWBs, cracking of solder interconnections, and component cracks. However, when complete products are drop tested, many different factors affect the bending of the component boards inside the casings and, thereby, can cause large dispersion in the number of drops needed to cause an electrical failure. For example, rigidity of product covers, method of attachment of the component boards to the body and casings, layout of components on the PWB, rigidity of the component boards, proximity of other rigid structures (battery, display, mechanical reinforcement, etc.), and the orientation of the product at the moment of impact have strong effects on the vibration (modes and frequencies, bending amplitudes, damping of vibration, etc.) of the component boards. Therefore, in order to reduce the number of such factors, board-level drop test standards have been developed. JEDEC was one of the first organizations to come out with a standard for portable hand-held devices [12]. It is currently very widely used and therefore we will be discussing it in more detail in the following. In order to fully assess the test results, one needs to understand the behavior of the component boards under the shock loads. Therefore we will look into the details of how the JESD22-B111-compliant component boards experience the shock loads during testing. The JEDEC-compliant drop tester is composed of a mechanism to drop the board repeatedly in a specified orientation and a high-speed data acquisition system to record deceleration, strains on the component boards, and number of drops to failure. The condition B of the JESD22-B111 standard defines the deceleration pulse as having a shape of half sine with a 0.5-mse width and maximum at 1500G (see also [66, 67]). The shape of the deceleration pulse not only is a function of the drop height but also depends on the characteristics of the strike surface: Drop height affects mainly the maximum deceleration and strike surface pulse width. The standardized layout of the test board along with its dimensions is presented in Fig. 3. The test board is attached horizontally on a fixture, where the components are facing downward during the test (see Fig. 4). The fixture is mounted on a sledge that is dropped down on a rigid surface from a height that is needed to produce the required maximum deceleration in a controlled manner with the help of guiding rails. The drop height depends

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on the structure of the tester and varies from model to another, but it is most importantly affected by the friction of the bearings between the guide rails and the drop table, but the height is typically in the range of 60–80 cm. Placing the component boards horizontally causes maximum bending amplitude of all possible orientations. The test can be considered as an accelerated test because the test structure lacks the support provided by product casings and other adjacent (rigid) structures such as display and battery. At the moment of impact the component board bends downward forced by the inertia and oscillates with rapidly decreasing bending amplitude. Figure 5 shows an example of the measured longitudinal strain on the JESD22-B111 component board in the vicinity of the centermost component (refer to Fig. 3 for gauge location). Since the component board is allowed to bend freely, there are numerous different modes in which it can bend. The natural mode of the component board describes the shape in which the board bends, and the natural frequency describes how fast the bending takes place. The frequency of the shock pulse (0.5 msec half sine) is 1 kHz and, hence, only the natural modes that vibrate at about this frequency or lower contribute to the total bending of the board. In principle, when a structure is loaded, all the natural modes take part in the vibration. Figure 6 shows all the natural modes (measured and computed) of the JESD22-B111 component board and Table 1 lists their measured vibration frequencies. The effect of the pulse shape on the response of a printed wiring board is studied by Suhir [68], who concluded that the shape of the pulse does not have a significant effect on the amplitude of a mode if the duration of the pulse is shorter than 12% of the period of the natural mode. Component boards that are in accordance with the JEDEC standard typically have their first natural frequency close to 200 Hz. Thus, not only the energy and duration of the pulse but also its shape can affect on the vibration. The modal analysis of the component board was carried out by attaching the test board to a fixture similar to that in the drop test vehicle by using four screws, and an instrumental hammer (Br¨uel & Kjær, type 8203) was utilized to given the test board a short and light impact. The initiated vibration was measured optically with a vibrometer (Polytec OVD-01, OFV302/3000). The estimates of the natural modes were obtained by repeating the measurement at 10 different locations (see the white rings in Fig. 6). The measured waveforms were analyzed by executing a Fourier transformation (Hewlett-Packard 3566A/67A FFT analyzer), and the natural frequencies were determined from the peak values of the frequency response graphs. See [69] for more experimental details. It should be noted that the bending modes illustrated in Fig. 6 intend only to characterize the shape of the different bending modes and they are not drawn to scale with respect to each other.

378 Board-Level Reliability of Lead-Free Solder under Mechanical Shock and Vibration Loads

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Figure 4 Example of the JESD22-B111-compliant drop tester: (a) structure of the tester; (b) test board is attached from the four corners on support pins and components are facing downward. The bending amplitudes are significantly decreased with increasing vibration frequency. The shapes of the natural modes depend on the support structure of the component board, whereas the natural frequencies depend on the stiffness and mass of the component board. Since each of the natural modes vibrates simultaneously at their characteristic frequency, the total bending of the component board is their sum. The contribution of the natural modes with highest frequencies to the total bending of the board is usually not significant because their amplitude is relatively small and their vibration is attenuated relatively fast. Only the lowest frequencies, in the case of the JESD22-B111 board the lowest three, can be considered significant.

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Figure 5 Measured deceleration and longitudinal strain on the component board at the center of the component board. Owing to the simultaneous action of many different natural modes and the fast attenuation of the vibration amplitude, the strain distribution on the test board changes very rapidly and the location of the highest stress changes constantly. Figure 7 shows the longitudinal strain measured in the middle of the board layout on the opposite side of the board from the components. Figure 7a shows how the strain develops after the drop impact. The macroscopic oscillation is due to the natural mode with the lowest frequency. Oscillations at higher frequencies are embedded in the larger strains. The strains due to natural modes with higher frequencies are shown in Fig. 7b, which presents only the 0–5-msec time interval from Fig. 7a. Figure 8 shows the calculated stress distribution on the PWB 1.2 msec after the drop impact. Due to the simultaneous action of the many vibration modes, the distribution as well as the location of the highest stress changes very rapidly.

380 Board-Level Reliability of Lead-Free Solder under Mechanical Shock and Vibration Loads

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2.2 Loading of Solder Interconnections During Drop Testing If one compares the strains and stresses produced by thermomechanical loads with those produced by the (high-frequency) bending of the PWB during the drop tests, there are number of obvious similarities (see Fig. 9). It is well known that as a consequence of changes in temperature, the solder interconnections experience strains and stresses, as illustrated in Fig. 9b, because component packages and the PWBs have the unequal coefficients of thermal expansion (CTE). In drop tests, on the other hand, strains and stresses are also concentrated in the solder interconnections, but they are caused by bending of the PWB instead of the CTE mismatch. The common operation temperatures of electronic devices are relatively speaking high for all the tin-rich solder alloys commonly used in the electronics assembly processes. This

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relative temperature is called homologous temperature and is defined as the ratio of the prevailing temperature to the melting point of a material expressed in Kelvin. At high homologous temperatures, above about 0.45, the nonconservative motion of dislocation becomes possible because the diffusion of vacancies (and interstitial atoms) becomes faster. Therefore, at high homologous temperatures plastic deformation becomes time dependent and materials can be deformed plastically even at stress levels below their macroscopic yield stress. This phenomenon is often called creep, and it can be defined as time-dependent plastic deformation of materials. Readers interested in more thorough discussion of the subject are referred to the literature (e.g., [70]). Standardized thermal cycling tests commonly place the extreme temperatures to −45◦ C and +125◦ C. For the eutectic Sn3.4Ag0.8Cu alloy this temperature interval equals 0.5–0.8 in terms of homologous temperature, which means that creep processes constantly contribute to plastic deformation during thermal cycling. (However, creep processes are much slower at low temperatures and therefore stress relaxation is also less extensive at lower temperatures.) At high homologous temperatures solders deform relatively fast and the CTE mismatch is primarily accommodated by deformation of the solder interconnections. The generated loads are mainly shear loads (normal strain ε ≈ 0). When the components and board expand by different amounts (the CTE of most PWBs is many times that of components), the amount of shear strain (γ ) can be approximated according to the simple analytical formula   L γ = T αT ,comp − αT ,PWB , hjoint where T is the change of temperature, αT ,comp and αT ,PWB are the CTEs of the component body and the PWB, respectively, L is the half width of the component (or the distance from the neutral point), and hjoint is the height of the interconnection. Cracking of lead-free solder interconnections under the thermal cycling conditions occurs through the bulk of the solder interconnections. This failure mechanism involves considerable cyclical plastic deformation of the solder interconnections that is typically followed by a transformation of microstructure by recrystallization and the growth of cracks between the high angle boundaries between the newly formed grains. Figure 10 shows an example of a failed BGA solder interconnection under thermal cycling conditions. See [21, 22, 29] for more details and discussion.

382 Board-Level Reliability of Lead-Free Solder under Mechanical Shock and Vibration Loads S, Mises [MPa] Multiple section points − + 9.3 − + 8.5 − + 7.7 − + 6.9 − + 6.2 − + 5.4 − + 4.6 − + 3.8 − + 3.1 − + 2.3 − + 1.5 − + 0.7 − + 0.0

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Figure 9c illustrates how the component board responds when it is being placed under shock loading (JESD22-B111). Bending causes displacement between the board and the component package and the solder interconnections are loaded in a similar manner as in static bending: The component resists the bending of boards and the interconnections between the package and the board carry the induced strains and stresses. The inertia forces of packages in drop testing are typically negligible due to their relatively light weight (about 0.3 g vs.

2

Methods of Shock Impact Testing

383

2L Si chip hjoint Printing Wiring Board

hPWB

Thermal Cycling: Increase of Temperature Si chip

Drop Testing Si chip

Figure 9 (a) Component board at rest, (b) displacements caused by thermal cycling, and (c) displacements caused by bending of the PWB (drop testing).

about 3 mm2 of contact surface to solder interconnections) and, therefore, the inertia of the components does not have a significant effect on the stresses at the interconnection level. The inertia of the component package should not be confused with the inertia of the entire component board, which is responsible for the vibrating motion after the drop impact. Even though bending of the component board causes strains and stresses in the solder interconnections between the component packages and the PWB in a very similar manner as in thermal cycling tests, the stress states differ considerably between the tests in the following sense: Under vibratory bending the loading of the solder interconnections on the package perimeter experiences higher tensile than shear stresses. As discussed in more detail in the Appendix, strains and stresses in the cornermost solder interconnections are mostly tensile: γ 3 hPWB = , ε 2 L where γ is the shear strain, ε is the normal strain, hPWB is the height of the printed wiring board, and L is the half width of the component (or the distance from the neutral point). On the other hand, when components and board expand by different amounts during thermal cycling (the CTE of most PWBs is many times that of components), the stresses formed in the interconnections are dominantly share. In addition, the time to achieve the maximum strain is commonly several, in some cases even tens of, minutes. Thus, there are three major differences between thermal cycling and drop testing: (1) the duration of the load, (2) the magnitude of load, and (3) temperature. Drop tests are carried out at room temperature, while during thermal cycling the temperature often raises above 100◦ C. High temperatures together with a long loading time make the microstructural evolution possible during thermal cycling, but the duration of drop testing is much too short to cause any such observable change of microstructures.

384 Board-Level Reliability of Lead-Free Solder under Mechanical Shock and Vibration Loads

(a)

(b)

Figure 10 Common failure mode of thermally cycled BGA solder interconnection (brightlight image on left and cross-polarized light image on right).

2.3 Failure Modes and Mechanisms Under Fast Deformation Rates The commonly observed failure modes of drop-tested component boards differ greatly from those observed in the same structures after thermal cycling. Besides temperature the most important difference between drop tests and thermal cycling tests is the deformation rate. The drop tests are typically carried out at room temperature, which is relatively high as compared to the melting point of the commonly used solder alloys (0.6 in terms of homologous

2

Methods of Shock Impact Testing

385

90 Sn3.4AgO.8Cu [72]

Ultimate Tensile Strength, MPa

80 70

Sn [73] Sn1.SBi[74]

60 50 40 30 20 10 0 1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

Strain Rate, s−1

Figure 11 Effect of strain rate on strength of Sn and two Sn–based solders [69, 71–73]. temperature). At such high homologous temperatures the plastic behavior of the solder is strongly strain rate dependent. In general, strength increases with strain rate. This increase can be attributed to the average number of dislocations and their limited velocities at high strain rates. Readers interested in a more comprehensive treatment of the subject are again referred to the literature (e.g., [74, 75]). Figure 11 illustrates the increase of the ultimate tensile strength of different materials as the strain rate increases from that used in thermal cyclic tests (typically in the range of 10−7 –10−5 sec−1 ) to that used in drop tests (typically in the range of 1–10 sec−1 ). As can be seen, the flow stress of solder is about two to three times as high when the strain rate is increased from that occurring in thermal cycling to that occurring in drop tests. Both the ultimate tensile strength and the yield strength increase with strain rate, but the yield stress is typically more strain rate sensitive. As a consequence of the increased strength (by strain rate hardening) the magnitudes and distributions of the stresses in the solder interconnections under drop test conditions become different as compared to those under thermal cycling conditions. Our finite element calculations shown in Fig. 12 illustrate how the strain rate not only increases the stresses in solder interconnections but also becomes more concentrated on the component side of the interconnections [28, 69]. Due to the much higher stress levels in the solder interconnections during the drop tests, the intermetallic compound layers will experience significantly higher stresses than those in thermal cycling. The tensile strength of the solder increases above the fracture strength of the IMC, and this ultimately makes the fractures propagate inside the IMC layers, instead of the bulk solder. On the other hand, in thermal cycling, where the strain rates are relatively low, the cyclic thermomechanical loading of the interconnections generates plastic deformation, which ultimately leads to propagation of fatigue cracks through the solder interconnections. Figure 10 shows an image taken with polarized light from the same interconnection as the bright-light image in Fig. 10. The reflection of the polarized light is dependent on the grain orientation of the surface and therefore the different colors in the images represent different orientations.

386 Board-Level Reliability of Lead-Free Solder under Mechanical Shock and Vibration Loads

(a)

(b)

(c)

(d)

(e)

(f)

Figure 12 Strain rate hardening of solder forces cracks propagating in brittle intermetallic layers instead of bulk solder.

3

VIBRATION TEST AS METHOD TO TEST TO REPLACE DROP TEST The JESD22-B111 drop test standard is very commonly employed for studying the reliability of components under shock loading conditions. However, there are several drawbacks related to the method of drop testing. Some of the most important ones are (1) it can be laborious and time consuming, (2) faulty signals due to failures of measurement cables are common, and (3) the possibilities to measure the component board behavior during testing are limited to strain

3

Vibration Test as Method to Test to Replace Drop Test 387

gauges. In order to avoid the drawbacks inherent in drop testing, an alternative testing method based on continuous harmonic vibration of the test board has been introduced earlier [69, 76]. The initial hypothesis that started the development work was the following: If the bending of the component board is the primary source for the interconnection strains and stresses in the drop tests, then matching the bending amplitude of the PWB and the frequency of the board vibration to those taking place during the drop tests should result in closely similar strains and stresses in the interconnections during the vibration tests. Furthermore, since is it most likely that the solder interconnections fail during the first few large-amplitude vibrations following the drop impact (see Fig. 7), the bending amplitude of the PWB during vibration testing should be closely similar. Thus, by knowing what was discussed in Section 2.3 and adjusting the frequency and magnitude of the excitation force, one should in principle be able to initiate the same failure mechanism and produce the same failure modes with the vibration tester as takes place with the drop tester. In the following, the loading conditions produced with the two testers are compared with the help of finite element calculations as well as experimental measurements. Results of the board-level submodel of the finite element calculations are verified by measuring strains on the component boards during the tests with three directional strain gauges. The bending of the boards during the tests is examined in order to evaluate how strong excitation is required in the vibration test to produce similar bending as during the drop tests. In the present study replacement of the JESD22-B111 drop test with the vibration test has been investigated by implementing the same experimental design with both testers and using the number of load cycles to failure and the produced failure modes as evaluation criteria.

3.1 Principal Similarities in Loading Conditions Figure 13 illustrates the principle of applying the harmonic vibration of the component boards to produce similar loading of the components as in the JESD22-B111 drop test. The diagram shows the measured longitudinal strain on the PWB close to the corner of the component (see Fig. 3 for gauge location), where the load force of the vibration tester has been adjusted to produce strains of closely similar magnitude as take place (on the same location) during the first few vibrations during the drop test (1500G, 0.5-msec pulse width). In the drop test, after Longitudinal Strains on the PWB 0.3

Vibration Test Drop Test

0.2

Strain, %

0.1 0.0 −0.1 −0.2 −0.3 0.0

0.1

0.2

0.3

0.4

0.5

Time, s

Figure 13 Measured longitudinal strains on PWB during the tests (see Fig. 1 for gauge location).

388 Board-Level Reliability of Lead-Free Solder under Mechanical Shock and Vibration Loads the moment of impact the component board bends first downward as forced by the inertia and then oscillates at the characteristic frequencies and attenuating bending amplitude. Since the first large-amplitude vibration frequency most significantly contributes to the strains and stresses in the solder interconnections, vibrating the component boards harmonically at closely similar frequency and equivalent levels of maximum strains on the PWB should, in principle, produce a highly similar strain state and stress levels at the solder interconnections and, thereby, the same failure modes are expected. The statement being made here is based on what was discussed in Section 2.3: By producing closely similar maximum strains and applying the strain at closely similar rates, the levels of maximum stress in the solder interconnections are closely similar. However, there is a fundamental difference that makes the strain rate and ultimately the strains and stresses at the interconnections during the vibration test a little bit smaller than those at the drop tests: In the drop tests several natural bending modes act simultaneously while in the vibration tests only one mode is initiated. When a sinuous loading force is applied to the component board in the vibration test, bending of the board is forced and only the mode whose natural frequency is closest to the frequency of the loading force takes part in the vibration. Figure 14a compares the strain produced during the drop tests to that on the PWB during vibration testing when the component board is loaded by using the 217 Hz frequency of the loading force (217 Hz is the frequency of the lowest natural vibration mode of the component board under study). As can be observed from the diagram, the strain history of the vibration test is much more simplified due to the fact that a single mode of vibration is initiated. Figure 14b shows the calculated strain rates from the strain measurements in the diagram above. The calculation assumes linear strain on the PWB over the period of 0.2 msec. The graph shows that, due to the more complex strain history of the drop test, the maximum strain rate during the drop test is clearly above 300% sec−1 whereas that during the vibration test remains below 300% sec−1 . Furthermore, the strain rate during the drop tests changes much more as a consequence of the simultaneous action of many different bending modes. (The calculated strain rate emphasizes the fact that the measured board strain during the vibration test is not pure sinuous even though the loading force is of the pure sinuous form.) The resonance frequencies of the component boards to be vibration tested can be determined by various methods. Perhaps the simplest way to determine it is to continuously measure the displacement of the PWB with the help of a laser while sweeping the frequency of the loading force from zero to 1 kHz, for instance. The same result can be achieved by measuring strains or acceleration of the PWB, but, being a contact-free measurement, the laser gives the most accurate results. Another way would be to apply a mild shock (with a lightweight instrumental hammer, for example) or a drop on the component board while measuring the strain and performing the Fourier transformation for the measured strain. The frequency of the lowest vibration mode should be chosen for the frequency of the loading force. The amplitude of the loading force is then adjusted to produce equivalent maximum levels of strain as take place in the drop test.

3.2 Configuration Vibration Test System The vibration test system used in this work consists of a computer with a signal generator, a signal amplifier, a shaker, and a high-speed data acquisition system to record the number of load cycles to failure and to stop the test when a failure is recorded (see Fig. 15). The failure was recorded when the resistance of the daisy chain network exceeded 1.0 k. The shaker consists of a set of permanent magnets and a coil through which the amplified excitation waveform is conducted. The magnets are attached to the body of the shaker, while the mounting table and the coil are suspended by the sheet springs on top of the barrel surrounding the magnets. The cyclic force generated between the magnet and the coil is linearly dependent

3

Vibration Test as Method to Test to Replace Drop Test 389 Longitudinal Strains on the PWB

0, 3

Vibration Test Drop Test

0, 2

Strain, %

0, 1 0, 0 −0, 1 −0, 2 −0, 3

0

2

4

6

8

10

Time, ms Strain Rates on the PWB 500

Vibration Test

400

Drop Test

Strain Rate, %s

300 200 100 0 −100 −200 −300 −400 −500

0

2

4

6

8

10

Time, ms

Figure 14 (a) Comparison of longitudinal strains in the vibration test and the drop test. (b) Strain rates calculated from the measured strains (linear strain assumed over 0.2-msec time interval). on the excitation voltage. The component boards are fastened to a fixture (see Fig. 16a) that was composed of a rigid lightweight “honeycomb sandwich” plate which is composed of two glass fiber sheets attached on both sides of the honeycomb cells. The component boards were mounted on the four support pins at the corners of the sandwich plate with screws as defined in the JESD22-B111 drop test standard. The fixture was attached from the center of the honeycomb plate to the modal exciter. Figure 16b shows an image from a test where the loading force is adjusted to produce equal strain on the PWB during the vibration testing as takes place during the drop test under condition B (1500G, 0.5-msec pulse width). The 4.7-mm bending amplitude of the component was measured by the laser.

3.3 Behavior of Test Board Under Vibration Testing The following measurements were carried out with assemblies having one component at the center of the board layout and a small loading force. Therefore, the strain magnitudes and

390 Board-Level Reliability of Lead-Free Solder under Mechanical Shock and Vibration Loads Number of load cycles to failure Component board

Sheet springs

High-speed data acquisition (to detect failure) Cyclic force

Stop the test

Mounting table

Signal generator Magnets

Sinuous wave

Amplifier

Coil

Amplified excitation current to the coil

Figure 15 Operating principle of the vibration tester.

displacement amplitudes are irrelevant in the following. It should be noted that even though the loading signal (i.e., the loading force) is increased the vibration frequency (resonant) remains approximately constant. Figure 17 shows the measured velocities of the component board of the frequency sweep over the range of 100–1600 Hz. The sweep is carried out to find out the frequency of the natural mode(s) initiated within the range. The measurements show the first significant increase of bending amplitude at the frequency of 224 Hz. As compared to the measurements shown in Table 1, which are made with a different component board (same board layout, different component), one can observe that the frequency of the lowest vibration modes is closely similar, but the difference increases with frequency. The bending behavior of component boards at vibration testing was compared to the behavior at drop testing with the help of the finite element method [69]. Stresses in interconnections of the centermost component were calculated under the vibration tests and the drop test and the results are presented in Fig. 18. Even though the bending amplitudes are nearly equal in the two tests, there are differences in the shape and peak values of the stress history curve. They are mainly caused by the fact that under the drop tests there are several different vibration modes acting simultaneously while under the vibration testing there is only one mode. This also makes the maximum stress under drop tests somewhat larger than that under vibration tests. Despite this difference, very similar loading of interconnections can be achieved when assemblies are subjected to harmonic loading with a vibration tester at their natural resonance frequencies as compared to the loading in drop testing. From a practical point of view vibration testing has many other advantages over the drop test. For instance, because of the smaller distance that component boards travel during the test, the contact-free measurements are easier to carry out. Furthermore, the bending mode can be specified by setting the loading frequency. Vibration testing can also be carried out in an environmental chamber with temperature and atmosphere control, for example.

4

Reliability under Combined Loading Conditions

391

(a)

(b)

Figure 16 (a) Component board attached to the tester. (b) Bending of the component board during vibration testing (scale in centimeters, bending amplitude about ±5 mm). The bending amplitude of the board is matched to that of the same board under the JEDEC drop test at 1500G, 0.5 msec.

4

RELIABILITY UNDER COMBINED LOADING CONDITIONS Portable electronic devices are exposed to varying operational environments where mechanical shocks can be a critical threat to their operation. On the other hand, an increased number of new high-performance functions integrated into novel portable electronic devices has increased internally generated heat dissipation from high-power-density ICs (see Fig. 19). Therefore, novel high-performance electronics such as electrical notebooks and hand-held computers are exposed to temperature fluctuations caused by internally generated heat. Thus, it is very likely that the temperatures inside products are above the ambient temperature when they experience

392 Board-Level Reliability of Lead-Free Solder under Mechanical Shock and Vibration Loads 224 Hz 409 Hz 0.1

616 Hz

561 Hz

1291 Hz

810 Hz

Velocity, dB

975 Hz 891 Hz 0.01

0.001 100

500

1000 Frequency, Hz

1500

Figure 17 Vibration velocities at different frequencies reveal the frequencies of the natural bending modes. Table 1

Measured Natural Frequencies of Mending Modes Mode

Natural Frequencies

1st

2nd

3rd

4th

5th

6th

7th

15 components 226.8 Hz 413.0 Hz 484.8 Hz 578.4 Hz 768.8 Hz 882.9 Hz 926.4 Hz No components 235.5 Hz 387.6 Hz 475.7 Hz 586.6 Hz 728.7 Hz 783.6 Hz 876.4 Hz

0.4 s33, vibrated s33, dropted

Tensile stress, GPa

0.3 0.2 0.1 0 −0.1 −0.2 −0.3

0

5

10

15 20 Time, ms

25

30

35

Figure 18 Calculated vertical tensile stress in the corner interconnection of the centermost component during the vibration test and the drop test.

4

Figure 19

Reliability under Combined Loading Conditions

393

Thermal image of a portable 3G terminal device during data transfer.

the mechanical shock caused by accidental drops. Furthermore, if one considers the entire life span of portable electronic products, the accidental shock loads take place randomly over the lifetime of a product or, in other words, between the thermal-mechanical loads caused by their normal operation. Due to the complex nature of the loading conditions during ordinary use of electronic devices, it is important that their reliability is also studied by employing test procedures that replicate the real operational loadings as realistically as possible. Conventionally, reliability has been studied by loading electronic assemblies with a single type of stress, typically either purely mechanical or purely thermomechanical. This, however, can form an incomplete understanding of the failure mechanisms in real service environments, where the various loadings act simultaneously. Therefore, in this section we discuss the reliability of component boards under sequentially combined loadings but also discuss the effects of temperature on the drop reliability of portable electronic devices.

4.1 Consecutively Combined Thermomechanical and Mechanical Loads The increase of internal heat production during operation produces not only thermomechanical fatigue but also changes in the microstructures of solder interconnections. Changes of microstructures can, on the other hand, change mechanical properties and crack paths of solder interconnections. It is well known that the microstructures produced during soldering are not stable and evolve continuously, and many of the microstructural changes take place faster at higher temperatures. Therefore, for example, drop testing of as-soldered component boards covers a very narrow time frame of the entire operation lifetime of electronic products. Thus, it is not unreasonable to assume that usually components of electronic products have undergone thermomechanical loadings and their microstructures may have changed from the

394 Board-Level Reliability of Lead-Free Solder under Mechanical Shock and Vibration Loads as-soldered state before they are accidentally dropped. However, currently employed standard drop reliability tests are carried out with as-soldered microstructures and, thus, the effects of such changes on the properties of solder interconnections are neglected. In the following, the reliability of chip-scale packaged area array components will be discussed under consecutively combined thermal(mechanical) and mechanical shock loads. To simulate the temperature caused by use of products before drop impact, the component boards were thermally cycled (−45◦ C/+125◦ C, 15 min dwell time, 750 cycles) or isothermally annealed (at 125◦ C for 500 hr) before the standard JESD22-B111 drop test. The component used in the study was a 12 × 12-mm chip-scale packaged ball grid array with 144 Sn0.2Ag0.4Cu bumps (500 μm bump diameter and 800 μm pitch). The underbump metallization of the component was electrochemical copper. The JESD22-B111-compliant PWBs were coated with either Ni(P)|Au [Ni: 2 μm, immersion Au: ∼0.02 μm; 9 wt% P in Ni] or organic soldering preservative (OSP) (0.2–0.5 μm). The component boards were reflow soldered with Multicore’s Sn3.8Ag0.7Cu (wt%) solder paste. The nominal interconnection composition is approximately Sn0.5Ag0.5Cu when the solder paste and the component bump alloy have been mixed at the reflow (perfect mixing assumed). For more details, refer to [32]. The as-soldered thermally cycled and isothermally annealed component boards showed different failure modes and lifetimes in failure analyses carried out after the drop tests. Figure 20 shows the average lifetimes of each combination of the studied variables with standard deviations. In the as-soldered state component boards with the Cu|OSP-coated PWB were more reliable than those with the Ni(P)|Au coating (at 0.2 % risk level, Wilcoxon ranksum test). The failure analyses revealed that cracking of the reaction layers on either side of the interconnections was the primary failure mode in the as-soldered state (see Fig. 21). The Ni(P)|Au-metallized boards failed at the PWB side exhibiting cracking of the brittle reaction layers between the Ni(P) metallization and the bulk solder. The reasons for the inferior reliability performance of the Ni(P)|Au PWB coating has been discussed in more detail elsewhere [27, 31, 78]. The solder interconnections on the Cu|OSP-coated PWBs failed at the component side, where cracks propagated through the Cu6 Sn5 reaction layer. The primary reason for this failure mode is that under fast deformation rates the flow stress of the solder is increased and stresses concentrate at the corner regions of the interconnections where the fracture strength of the intermetallic reaction zone is exceeded [27, 28]. Average Drops to Failure 20 18 Number of Drops

16 14 12 10 8 6 4 2

8

0

14

4

14

16

4

After Reflow Thermally Isothermally After Reflow Thermally Isothermally Cycled Annealed Cycled Annealed Ni(P) Au on PWB Pads

Figure 20

Cu OSP on PWB Pads

Average number of drops to failure with standard deviations.

4

Reliability under Combined Loading Conditions

395

Cracking of the P-rich reaction layer underneat the (Cu, Ni)6Sn5 layer

SnAgCu solder (Cu, Ni)6Sn5

Ni(P) TKK

SEI

15.0kV X2,500 10µm WD 15.2mm

(a)

Cracking of the Cu6Sn5 intermetalic layer

Cu

Cu6Sn5 SnAgCu solder TKK

COMPO 15.0kV X1,000 10µm WD 15.0mm

(b)

Figure 21 Failure modes of the as-soldered interconnections after drop testing: (a) Ni(P)Au, brittle intermetallic fracture through the reaction layers between the Ni(P) and the bulk solder; (b) Cu|OSP, fracture through the Cu6 Sn5 reaction layer.

396 Board-Level Reliability of Lead-Free Solder under Mechanical Shock and Vibration Loads However, when the thermal cycling preceded the drop test, the failure mode changed to intergranular cracking through the bulk solder, and the failure mode was the same regardless of the board-side metallization (see Fig. 22). The cracks propagated either entirely through the bulk solder or first through the bulk and later moved to the intermetallic layers. The change in the crack propagation can be explained in terms of the formation of continuous networks of grain boundaries by recrystallization [21, 22, 29]. The microstructures of the solder interconnections that were formed during reflow soldering consist of relatively few (typically one to five) colonies of uniformly oriented Sn cells (separated by low-angle boundaries). But due to the recrystallization during thermal cycling, numerous new (high-angle) boundaries are generated near the corner region of the interconnections, where the structure is the most heavily deformed plastically. It seems that the local recrystallization of solder interconnections enhances cracks to nucleate in and propagate through the recrystallized solder

Figure 22 Failure modes of the interconnections with thermal cycling pretreatment before drop testing: cracks propagate intergranularly through the bulk of the solder (instead of the intermetallic layers), where the recrystallization has created favorable paths for cracks to propagate.

4

Reliability under Combined Loading Conditions

397

interconnections. The formation of a continuous network of grain boundaries enables intergranular cracking of interconnections during the subsequent drop tests. The average number of drops to failure in Fig. 20 shows an increase in the average number of drops to failure when the thermal cycling treatment is carried out before the drop tests as compared to the average number of drops to failure of the component boards in the as-soldered state. The difference, however, was not statistically significant (19.3% risk level) due to the relatively large dispersion of the drops to failure of the thermally cycled assemblies. The change of fracture mode from brittle intermetallic cracking to intergranular cracking can explain both the apparent increase in the average drops to failure as well as the large scatter in the lifetimes. The assemblies isothermally annealed before the drop tests were significantly weaker than the others. The failure mode was again different from what had previously been observed in this study, and the mechanism was the same regardless of the board-side coating. The cracks nucleated in the bulk solder as before, but directly after the nucleation they had entered the Cu3 Sn layer (see Fig. 23). The large numbers of voids being formed in the Cu3 Sn layer during the isothermal annealing constitute almost continuous paths along which cracks can propagate through the entire interconnections. The voids locating at the vicinity of the Cu|Cu3 Sn interface indicate that they could be Kirkendall voids, as suggested by some authors [79–81], but the fact that in this study the voids were observed only on the component side of the interconnections, and not on the PWB side, suggests that the formation mechanisms

Silicon

Epoxy coating

200 µm 3.0 mm

6 5 4 3.0 mm 3 2

500 µm

1 F

E

D

C

B

A

300 µm 500 µm

Figure 23 Failure modes of the interconnections with isothermal annealing pretreatment before drop testing: Cracks propagate inside the Cu3 Sn intermetallic layer assisted by the voids formed during the annealing treatment.

398 Board-Level Reliability of Lead-Free Solder under Mechanical Shock and Vibration Loads may be more complicated than that and, for example, the amount of impurities may play a role in their formation.

4.2 Drop Reliability of Component Boards at Different Temperatures It is well known that mechanical properties of materials are dependent on temperature. Therefore, it is important to pay attention also to the temperatures of the components during the drop impact. In this section the drop reliability of component boards at different temperatures will be discussed in light of two structurally different component boards: (1) the wafer-level chip scale package (WL-CSP) on a conventional FR4 PWB and (2) the chip-scale packaged (CSP) ball grid array on the “1+6+1” buildup multilayer structure defined by the JESD22B111 standard (see Figure 24). The results of the two case studies are discussed in more details individually in [17, 82, 83]. 4.2.1 Materials and Methods The WL-CSP component was a 3 × 3 mm-package with 36 Sn4.0Ag0.5Cu bumps (6 × 6 fullarea array), pitch of 500 μm, and bump size of 300 μm (see Fig. 19a). The underbump metallization of the chip was Al|Ni(V)|Cu. The component was equipped with an on-chip heater element, an on-chip aluminum resistor for temperature measurements, and a daisy chain network for time-to-failure measurements. The PWB was a 1-mm-thick double-sided FR4 board with no resin-coated copper (RCC) layers. The dimensions and component layout of the board were otherwise based on the JESD22-B111 drop test standard (see Fig. 25). The OSP surface finish was used on the copper-soldering pads. The reliability tests were carried out at four different temperatures: room temperature, 75◦ C, 100◦ C, and 125◦ C. Ten component boards were tested at each temperature below 100◦ C and four at 125◦ C. The CSP–ball grid array (BGA) component was a 12 × 12-mm package with 168 Sn3.0Ag0.5Cu bumps (14 × 14 peripheral array, pitch of 800 μm, and bump size of 500 μm. There was no underbump metallization between the bumps and the copper pads on the Silicon

Epoxy coating 0.3 mm

200 µm

14

1.2 mm

1 12.0 mm 0.8 mm

3.0 mm

0.4 mm

6

4 3.0 mm 3

10.4 mm

5

2

500 µm

1 300 µm

F

E

D

C

B

A

500 µm

Figure 24 Schematic of the components used in the studies of drop reliability at different temperatures: (a) the WL-CSP; (b) the CSP-BGA.

4

Reliability under Combined Loading Conditions

399

Figure 25 Layout of the WL-CSP drop test board (dimensions according to JESD22-B111). The single-component configuration was used in the tests.

redistribution layer of the component. The silicon chip was equipped with two heater elements, a PN junction for temperature measurements and a daisy chain network for time-to-failure measurements. The PWB was designed according to the JESD22-B111 drop test standard (RCC layers included). An integrated heater was designed underneath the centermost component to provide additional heating (see Fig. 26). The drop tests were carried out at room temperature (23◦ C), 70◦ C, and 110◦ C and 10 component boards per temperature were drop tested. The one-component configuration defined by the JESD22-B111 standard was used; that is, one component per test board was soldered onto the centermost component location of the PWB. The one-component configuration was chosen in order to produce a similar temperature distribution on the test boards as that around the hottest components in functional products.

Figure 26 Layout of the CSP–BGA component board and description of the integrated heater element in the PWB.

400 Board-Level Reliability of Lead-Free Solder under Mechanical Shock and Vibration Loads Number of Drops

1000 800 600 400 200 0

112 RT

458

568

711

75 °C

100 °C

125 °C

Test Temperature

Number of Drops

Figure 27 WL-CSP: average number of drops to failure (n=10) with standard deviations at the four different temperatures (only four boards were testes at 125◦ C).

70 60 50 40 30 20 10 0

54

33

10 110 °C

70 °C RT Test Temperature

Figure 28 CSP-BGA: average number of drops to failure (n=10) with standard deviations at the three different temperatures.

4.2.2 Results The numerical lifetime analysis of the drop tests as well as the physical failure analyses showed very different results between the component types. Figures 27 and 28 show the average number of drops to failure with the corresponding standard deviations of both components at the different test temperatures. The trends are clearly distinguishable and contradictory. The average number of drops to failure of the WL-CSP component boards increased about 300% when the testing temperature was increased from room temperature (RT) to 75◦ C. The difference between the average drops to failure was statistically significant at less than 1% risk level (Wilcoxon rank-sum test, 5% risk level). When the temperature was increased further from 75 to 100◦ C, the average drops to failure increased about 25%. However, this difference failed to be statistically significant (risk level 45%) due to the high standard deviation within the drops-to-failure data of the component boards tested at 100◦ C. Because of the large dispersion in drops-to-failure data of the component board drop tested at 100◦ C, additional boards were tested at 125◦ C in order to find out the tendency of change toward higher temperatures. Four boards were tested and all of them showed an increasing trend in the number of drops to failure (about 25% increase) as compared to those drop tested at 100◦ C. The average number of drops to failure of the CSP–BGA component boards decreased about 40% when the temperature was increased from room temperature to 70◦ C. When the temperature was increased further from 70 to 110◦ C the average number drops to failure decreased about 70%. The Wilcoxon rank-sum test showed that all differences in the average

4

Reliability under Combined Loading Conditions

401

drops to failure between the different test temperatures were statistically significant at less than 1% risk level. The failure modes were inspected with the help of optical and scanning electron microscopy from the cross sections prepared by standard metallographic methods. The primary failure mode, that is, the mode that first caused the electrical failure of the daisy chain networks, in the case of the WL-CSP component was cracking of the solder interconnections at the proximity of the component-side interfacial region. The micrographs shown in Fig. 29 illustrate the observed differences in the modes of failure in the component boards drop tested at elevated temperatures as compared to those drop tested at room temperature. The crack propagation path had changed with the increased temperature of the component: At room temperature, cracks propagated in the component-side (Cu,Ni)6 Sn5 intermetallic layer. However, as the temperature of the component was increased, the crack paths gradually changed from the interfacial layers into the bulk solder. Cracking of the intermetallic layers was rarely observed in the interconnections drop tested at 125◦ C. Small cracks were observed occasionally at the PWB-side interfacial regions of the solder interconnections at all temperatures, but they were very rare and had seldom propagated more than about one-third of the width of the interconnections. The nucleation site of cracks was the same regardless of the test temperature.

Figure 29 Failure modes of the WL-CSP assemblies: The failure mode changed gradually from cracking of the brittle intermetallic layers to cracking of the bulk solder with increasing temperature of the component.

402 Board-Level Reliability of Lead-Free Solder under Mechanical Shock and Vibration Loads

Figure 30 Failure mode of the CSP–BGA assemblies: Only one failure mode was identified after drop testing at the different temperatures—cracking of the component-side intermetallic layers. The micrographs in Fig. 30 show the failure modes of the CSP–BGA component boards drop tested at the different temperatures. Two different failure modes were identified, but only one of them was directly responsible for the electrical failure of the assembly: (1) cracking of the component-side intermetallic compound (IMC) layers and (2) cracking of the RCC layer underneath the copper-soldering pads. Only the IMC layer cracking had caused electrical failures of the daisy chain structure. In all of the samples inspected the cracks were located at the component-side interfacial region of the interconnection. Cracks in the RCC buildup layer were identified at all temperatures, but cracking of the Cu wiring on the PWB (that connects two solder interconnections) was not observed. RCC cracking is not directly seen in the measurements of electrical continuity of the daisy chain networks, but cracks change the stress distribution inside the interconnections and in this way they affect the lifetime of the interconnections. 4.2.3 Discussion As discussed above, under mechanical shock loading solder interconnections fail at the intermetallic layers instead of the bulk solder due to the fact that under very fast loading the ultimate tensile strength of Sn-rich solder interconnections is increased by strain rate hardening. As a consequence, the stresses concentrate on the corners of the solder interconnections in the vicinity of the interfacial reaction layers where they grow above the fracture strength of the brittle intermetallic compound layers. However, it is well known that mechanical properties of many materials are temperature dependent. Change of temperature can affect the strains and stresses produced during the drop test in three different ways: Increasing the test temperature above room temperature (i) decreases both the yield strength and the elastic modulus of solder interconnections, (ii) decreases the stiffness of the PWB, and (iii) introduces thermally induced stresses. In addition, the strain rate hardening at elevated temperatures can be different. The effects of these three factors in WL-SCP component boards are quantitatively discussed in more detail in [84] It is also well known that when interconnections are at high relative temperatures (at above half of the melting point expressed in Kelvin), the restoration processes operate effectively. Thus, one can anticipate that the stress relaxation takes place relatively fast and the effect of thermally induced stresses is negligible as compared to the first two. With increasing temperature the decreases of yield strength and elastic modulus of the solder make the solder interconnections deform plastically at lower stress. Figure 31 shows the ultimate tensile strength of Sn3.4Ag0.8Cu solder at room temperature and at 120◦ C. The

4

Reliability under Combined Loading Conditions

403

140 Sn3,4Ag0,8Cu (22°C) 120

Sn3,4Ag0,8Cu (120°C)

Stress [MPa]

100 80 60 40 20 0 0.000001

0.0001

0.01

1

100

10,000

Strain Rate [%/s]

Figure 31

Strain rate sensitivity of Sn3.4Ag0.8Cu solder at 22 and at 120◦ C [70].

strain rate of solder interconnections during the drop tests is in the range of 100–1000 % sec−1 [28, 70]. Because of the reduced strength of solder interconnections, the plastic deformation is more significant and the levels of the highest stresses in the interconnections are decreased [83, 84]. Thus, softer solder interconnections seem to be beneficial for the reliability of electronic products under accidental drops. The stiffness of the PWB, on the other hand, is also temperature dependent and therefore the strains on the PWB can change with increasing temperature. Figures 27 and 28 show the measured maximum strains on the PWB during the drop impact of the WL-CSP and CSP–BGA component boards, respectively, at different temperatures over the temperature region of the tests. The strains were measured with thin-sheet strain gauges attached on the PWBs at the location shown in Fig. 25 and 26 (see Figs. 32 and 33). The measurements show that the strain of the CSP/BGA component boards is increased while a similar trend 0.310 0.300

Strain, %

0.290 0.280 0.270 0.260 0.250 0.240 0.230 20

70 Temperature, °C

120

Figure 32 Measured longitudinal strain on the PWB of the WL-CSP component board at different temperatures (see Fig. 25 for gauge location).

404 Board-Level Reliability of Lead-Free Solder under Mechanical Shock and Vibration Loads 0.310 0.300

Strain, %

0.290 0.280 0.270 0.260 0.250 0.240 0.230 20

70

120

Temperature, °C

Figure 33 Measured longitudinal strain on the PWB of the CSP/BGA component board at different temperatures (see Fig. 26 for gauge location). in not detectable in the case of the WL-CSP component boards. The fact that the bending amplitude of the CSP/BGA component boards increases with increased temperature of the component (while that of the WL-CSP component board fails to do so) can be understood by comparing the infrared images of the component boards presented in Fig. 34 and 35. Due to the much larger size and greater heat dissipation from the large CSP/BGA component, the heat is conducted over a much larger area on the PWB as compared to the WL-CSP

100

T, °C

80 60 40 23 100 °C

23 °C T, °C 100 80 60 40 23

Figure 34 Temperature distribution of the WL-CSP component board at the 125◦ C target temperature.

4

Reliability under Combined Loading Conditions 100

405

T, °C

80 60 40 100 °C

23 °C T, °C 100 80 60 40 23

Figure 35 Temperature distribution of the CSP/BGA component board at the 100◦ C target temperature. board. The highly localized heating in the case of WL-CSP makes the effect of decreased stiffness of the PWB insignificant while it is highly significant in the case of the CSP/BGA component board. Therefore, in the case of WL-CSP component boards, the reduced strength of the solder is the most significant consequence of increased temperature [84]. Due to the decreased stress levels, the failure mode changes from cracking of the intermetallic layers to cracking of the bulk solder. The energy needed for the propagation of the cracks through the bulk of the solder is higher than that needed for propagation through the brittle intermetallic layers and, therefore, the lifetime under the test is increased. The case of CSP/BGA component boards is, however, more complicated because both the decreased strength of solder and decreased stiffness of PWB (i.e., increased strain of the PWB) are significant. The final outcome depends on which of these effects is more influential. The stresses and their distribution were evaluated with the help of finite element simulations [83]. Stresses during the loading at room temperature (23◦ C) were compared to those formed at 100◦ C. The results indicate that the effect of increased bending of the PWB is more significant than the reduced strength of solder. The results show that the concurrent decreased strength of solder and the decreased stiffness of the PWB increase the peeling stress (s33) at the component-side interfacial region about 10%, and this can explain the observed decrease in the reliability performance with increased temperatures. The results discussed above exemplify why modifications to the currently employed reliability tests are needed. Single-load reliability tests form an incomplete understanding of the failure mechanisms in real service environments, where the various loadings are taking place simultaneously. Understanding why the different tests yield different failure mechanisms and ultimately different lifetimes is therefore essential. This can be achieved by knowing how the stress states are produced, how the materials respond to different types of loading, and how

406 Board-Level Reliability of Lead-Free Solder under Mechanical Shock and Vibration Loads the microstructures of solder interconnections affect the failure mechanisms. Thus, potential reliability risks can be identified and solved only by understanding the physics of failure behind the observed failure modes in the failed devices.

5

CONCLUSIONS The ongoing trend toward ever smaller electronic products forces larger scales of integration and the use of smaller and finer pitch components, such as (wafer-level) chip scale packages and flip chips. Because of the small-scale interconnections, components become closer to the PWBs and the strains and stresses experienced by solder interconnections are increased. These miniaturized interconnections must be able to withstand a variety of different loadings out of which the sudden mechanical shocks are perhaps the most severe threat to their reliability. Hence, understanding why the different tests produce different failure mechanisms and ultimately different reliability performance is of outmost importance. This can only be achieved by knowing how the stress states are produced, how the materials respond to different types of loading, and how the microstructures of solder interconnections affect the failure mechanisms. Therefore the emphasis of this chapter was placed on describing the loading condition under mechanical shock loading and how it differs from that of thermal cycling. The failure modes under mechanical shock loads are different from those typically observed in the thermal cycling, where solder interconnections fail by cracking through the bulk solder assisted by the localized recrystallization. Under mechanical shock loading the strain rate hardening of the solder forces cracks to propagate in the intermetallic layers on either side of the interconnections instead of the bulk solder. However, the failure modes and mechanisms under single-load environments are most likely different from those taking place in real operation where many different loads act simultaneously. The examples discussed in this chapter showed how the drop impact reliability of electronic devices can be highly different depending on the loading history of the device before the drop impact or depending on the temperature of the device during the drop impact. Finally, because reliability of electronic devices has become an ever-important factor of customer satisfaction, more comprehensive and extensive reliability testing is needed before new products can be released into the markets. But longer testing time adds the cost of new products not only due to the direct expenses related to testing but also due to increased time to market of new products. Hence, the employment of more efficient testing methods has become important. When developing new accelerated reliability tests, one must pay attention to the failure modes and mechanisms in order to make sure that they are not changed by the increased acceleration factor of the test. The method of vibration testing of electronic component boards that was discussed in detail in this chapter is a very promising new method to significantly accelerate the drop reliability tests while, at the same time, preserve the same failure modes and mechanisms as take place in board-level drop tests.

APPENDIX Consider the two-dimensional case in Fig. A.1, which shows a schematic of a rigid component mounted in the center of a circuit board that is bent. In the figure hjoint is the height of the interconnections and hPWB is the thickness of the PWB. The following assumptions are made: (i) The curvature of the PWB is constant (constant bending moment and ideally soft interconnections), (ii) the number of interconnections is large, (iii) the interconnections are evenly distributed, and (iv) the vertical stress in an interconnection depends on its elongation only. Before the board is bent, the surface of the board is at level S0 = 0, and during bending the surface follows S (x ). Thus, the elongation of the interconnections is ε(x )hjoint = −S (x ) + S0 = −S (x ), where a positive sign of S (x ) means displacement upward. No external force is

Appendix y

L

component solder interconnection

rigid component S S0

x hPW B

circuit board plane of symmetry

Figure A.1

407

hjoint hjoint

R

circuit board

q = L R

Elongation of interconnections during bending.

acting on the  component and, therefore, the sum of vertical stresses in the interconnections must be zero, FyBGA = ∫ σy dx = 0. Hence, the elongation of the edge interconnections can be calculated with the following formulas: ε(x )hjoin = −S (x ) + S0 = −S (x ),  S (x ) = R 2 − x 2 + S1 ,  L  L  −S (x ) σy dx = E dx = 0 FyBGA = 0 ⇒ hjoin 0 0  L   L S (x ) dx = ( R 2 + x 2 + S1 ⇒ 0 0 ⎡

⎤  2  R⎣ L R L ⎦ 1 L2 ⇒ S1 = − 1− − arcsin , ≈− 2 R L R 3 R where

√ 1 + x = 1 + 12 x + 18 x 2 + · · · , 1 x3 1 · 3 x5 + + 2 3 2·4 5 −S (L) 1L ⇒ ε(L) = = hjoin 3R

arcsin x = x +

1 · 3 · 5 x7 (2n − 3)! x 2n−1 + 2·4·6 7 (2n − 2)! 2n − 1 1 , hjoin

where ε is the vertical tensile strain of the interconnections, S (x ) is the location of the PWB surface during bending, S1 is a constant that depends on the bending radius R, radius, FyBGA are the vertical loads acting on the component, L is the distance from the component center to the interconnection, and hjoint is the height of the interconnections. Furthermore, R  L. In addition to elongation there is also shear deformation γ . If the middle plane of the board does not elongate during bending, which is usually the case, then the upper surface elongates for L: L  = ⇒ L = R, R   L + L =  R + 12 hPWB 1 1L hPWB = hPWB , 2 2R L 1 L hPWB γ = = , hjoin 2 R hjoin ⇒ L =

408 Board-Level Reliability of Lead-Free Solder under Mechanical Shock and Vibration Loads where  is the bending angle and hPWB is the thickness of the board. Thus, the relation between the shear deformation and the elongation of the edge can be written as 3 hPWB γ = . ε 2 L

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412 Board-Level Reliability of Lead-Free Solder under Mechanical Shock and Vibration Loads 55a. Yaguchi, A., Tanaka, M., Naka, Y., Yamamoto, K., Kimoto, R., and Yamada, M., “Impact Strength Evaluation of Solder Joints in BGA by Dropping Steel Rod,” in Proceedings of The 56th Electronic Component and Technology Conference, San Diego, CA, May 30–June 2, 2006, IEEE/EIA CPMT, New York, 2006, pp. 55–63. 56. Wang, Y. Q., Low, K. H., Che, F. X., Pang, H. J. L., and Yeo, S. P., “Modeling and Simulation of Printed Circuit Board Drop Test,” in Proceedings of the 5th Electronic Packaging and Technology Conference, Dec. 10–12, 2003, Singapore, IEEE/CPMT, New York, 2003, pp. 263–268. 57. Amagai, M., Toyoda, Y., Ohnishi, T., and Akita, S., “High Drop Reliability: Lead-Free Solders,” in Proceedings of the 54th Electronic Components and Technology Conference, June 1–4, 2004, Las Vegas, NV, IEEE/EIA CPMT, New York, 2004, pp. 1304–1309. 58. Yu, Q., Watanabe, K., Tsurusawa, T., Shiratori, M., Kikano, M., and Fujiwara, N., “The Examination of the Drop Impact Method,” in Proceedings of The 9th Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems, June 1–June 4, 2004, Las Vegas, NV, IEEE/CPMT, New York, 2004, pp. 336–342. 59. Tee, T. Y., Luan, J., Pek, E., Limb, C. T., and Zhong, Z., “Novel Numerical and Experimental Analysis of Dynamic Responses Under Board Level Drop Test,” in Proceedings of the 5th International Conference on Thermal and Mechanical Simulation and Experiments in Micro-Electronics and Micro-Systems, May 9–12, 2004, IEEE, Leuven /Brussels, Belgium, 2004, pp. 133–140. 60. Kim, S. M., Shon, E. S., Kay, Y. H., Kim, Y. J., Kim, J. Y., and Kim, J. D., “Reliability Characterization of Organic Solderability Preservatives (OSP) of IC Packages by Drop and Cyclic Bend test,” Key Engineering Materials, Vol. 297–300, No. 2, pp. 893–898, 2005. 61. Reiff, D., and Bradley, E., “A Novel Mechanical Shock Test Method to Evaluate LeadFree BGA Solder Joint Reliability,” in Proceedings of the 55th Electronic Components and Technology Conference, May 31–June 3, 2005, Lake Buena Vista, FL, IEEE/EIA CPMT, New York, 2005, pp. 1519–1525. 62. Chen, Y. S., Wang, C. S., Wang, T. C., Chan, W. H., Chang, K. C., and Yuan, T. D., “Solder Joint Reliability Assessment for Flip Chip Ball Grid Array Components with Various Designs in Lead-Free Solder Materials and Solder Mask Dimensions,” Journal of Electronic Materials, Vol. 36, No. 1, pp. 6–16, 2007. 63. Keimasi, M., Azarian, M. H., and Pecht, M., “Comparison of Flex Cracking of Multilayer Ceramic Capacitors Assembled with Lead-Free and Eutectic Tin-Lead Solders,” in The Proceedings of the 26th Annual Capacitor and Resistor Technology Symposium, Apr. 3–6, 2006, Orlando, FL, ECA, 2006, pp. 15–25. 64. Keimasi, M., Azarian, M. H., and Pecht, M., “Isothermal Aging Effects of Flex Cracking of Multilayer Ceramic Capacitors with Standard and Flexible Terminatirs,” Microelectronics Reliability, Vol. 47, pp. 2215–2225, 2007. 65. IPC/JEDEC-9702, “Monotonic Bend Characterization of Board-Level Interconnects,” IPC and JEDEC Solid State Technology Association, 2007. 66. JESD22–B110, “Subassembly Mechanical Shock,” JEDEC Solid State Technology Association, 2004. 67. JESD22–B104, “Mechanical Shock,” JEDEC Solid State Technology Association, 2004. 68. Suhir, E., “Could Shock Tests Adequately Mimic Drop Test Conditions?” Journal of Electronic Packaging, Vol. 124, pp. 170–177, 2002. 69. Marjam¨aki, P., “Vibration Test as a New Method for Studying the Mechanical Reliability of Solder Interconnections Under Shock Loading Conditions, TKK-EVT-17, Doctoral dissertation, Helsinki University of Technology, Otamedia, Espoo, 2007.

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CHAPTER

18

DYNAMIC RESPONSE OF PCB STRUCTURES TO SHOCK LOADING IN RELIABILITY TESTS

1

Milena Vujosevic

Ephraim Suhir

Intel Corporation Folsom, California

University of California Santa Cruz, California University of Maryland, College Park, Maryland ERS Co. Los Altos, California

INTRODUCTION The recent advancements in wireless electronics (WiFi/WiMax), concurrent with the needs for weight reduction, have created new generations of electronic devices and systems in consumer electronics as well as in medical and military fields. At the same time, the handheld and portable computing, information, and communication devices have become exposed to harsh end-user handling conditions. Such conditions were not encountered in the past to such a significant extent as they are now. This has created a set of unique mechanical (“physical”) reliability concerns for electronic components. This is particularly true for ball grid array (BGA) and pad grid array (PGA) materials and structures. BGA systems are widely used in today’s electronic packages and devices. In addition, further miniaturization of electronic components and increased input/output (I/O) density have resulted in solder joints that are significantly smaller than solder joints employed in the past, thereby increasing the vulnerability of board-level solder joint interconnections. Moreover, the environmental concerns have driven the electronics industry transition to “green electronics” and to the introduction of environmentally friendly novel material systems. An important part of such a transition is replacement of leaded solder materials with lead-free solders. It has been established that the introduction of lead-free solder alloys, such as SAC (Sn–Ag–Cu), has, in general, an adverse effect on solder joint reliability, especially when subjected to a shock load. This is due to several factors. First, SAC alloys are significantly stiffer (i.e., have greater Young’s moduli) than tin–lead alloys, which result in higher solder joint loading under the same external deformation. Indeed, as follows from the simplest Hooke’s law formulation σ = E ε, the stress goes up for the given strain ε with an increase in the Young’s modulus E of the material, and the strain in the BGA structures is due primarily to the thermal expansion/contraction mismatch of the electronic package and the printed circuit board (PCB). Second one of the critical factors affecting the shock and vibration response of a metal is the nature of the intermetalic layers that form on either side of the solder joint. The intermetalic compounds that form during reflow of SAC solders are generally more brittle than the base metal. This circumstance has an adverse effect on solder joint reliability, especially

Structural Dynamics of Electronic and Photonic Systems Copyright © 2011 John Wiley & Sons, Inc.

Edited by Ephraim Suhir, David S. Steinberg and T. X. Yu

415

416 Dynamic Response of PCB Structures to Shock Loading in Reliability Tests under dynamic loading condition [1, 2]. Third, lead-free solder materials are characterized by higher melting temperatures than lead–tin solders, and therefore the stress level at lowtemperature conditions is higher than for lead–tin solders. Indeed, in the simplest formulation of the thermal stress level, the “external” thermal strain can be defined as ε = α t and is due primarily to the mismatch in the coefficients of thermal expansion (CTEs) of the package and the PCB “composites” and the change in temperature. In addition, lead-free solders are more prone to aging and degrade more rapidly than lead–tin solders. Finally, the general trend in miniaturization of structural elements in electronic systems leads to a smaller size of today’s solder joints compared to the joints employed a decade ago. It is noteworthy in this connection that lead-free solders are usually characterized by higher yield points than lead–tin solder eutectics, and therefore the inelastic strains in them might be lower than in lead–tin solders despite larger temperature variations. It is unclear yet, however, to what extent this generally favorable circumstance might compensate for the adverse effects of lead-free solders as far as the overall reliability of the BGA system is concerned. The general consensus today is that because the loading conditions are more severe, the size of the joints is smaller, and the material itself is more vulnerable, the new generation of solder joint systems is more prone to failure, particularly to a shock failure, than the solder joint systems of the past. It has been determined that drop/shock loading has become the primary cause of solder joint failure in portable electronic devices [3]. The ability to predict and, if necessary, minimize the adverse consequences of dynamic loading on BGAs is of obvious practical importance. Moreover, significantly reduced reliability margins call for better understanding of the deformation and failure mechanism taking place in board-level interconnects during shock events. In the analysis that follows we address several major factors of the dynamic response of PCB structures to shock loading.

2

MEASURED DEFORMATIONS AND PREDICTED FAILURES IN ASSEMBLY SUBJECTED TO SHOCK LOADING Experimental evaluation of strains and stresses in solder joints encounters plenty of difficulties and is not always practically feasible. Because of this, solder joint loading and its proximity to failure is usually established indirectly through measurements of the PCB response. The typical mechanical responses that can be measured directly on a PCB are acceleration, displacements, and strain. The acceleration measurements are carried out by placing accelerometers at the predetermined location on the board. Although from the experimental standpoint these measurements are easy to perform, it has been found that the acceleration of the board is not an adequate measure of solder joint (SJ) loading [4, 5]. Instead, board deformations (displacements and/or strains) are considered a more relevant measure of solder joint response to shock load. Board displacement measurements have been carried out through the application of high speed-camera and digital image correlation methods. These methods are capable of providing full-field deformation of the board as well as the displacements at its selected points. These measurements have been successfully applied by numerous investigators. They require, however, the use of expensive equipment and highly skilled technical work. Such requirements are too stringent for routine qualification tests. Measurements of the induced board strains, on the other hand, are much less costly. Thus, such measurements have become the method of choice in the electronic industry. Strain measurements are conducted by application of strain gages, which are being placed at one or more locations on the board [3, 7–11]. To develop a good understanding of the effect of the board deformations on the strains in and reliability of solder joints, one has to establish meaningful correlations between the strain gage reading(s) and the solder joint loading.

2 Measured Deformations and Predicted Failures in Assembly Subjected to Shock Loading

417

Accel.

The board-level tests represent an idealized case for a PCB assembly shock response. The geometry, the boundary conditions, the magnitude of loading, and the drop directions can all be well controlled. This is not the case with the product-level test, where many more different parameters can affect the response. Because of the tighter control of the tests, the board-level tests are more predictable than the product-level tests and are therefore more meaningful and more useful for generating practical insights into the physics of the deformation process. One should have in mind, however, that the board-level tests might not always adequately reproduce the deformation and failure mechanisms taking place and observed in the systemlevel tests. It is not easy to determine the product-level performance based on the board-level test data: the product boundary conditions and impact orientation might be quite different from those in the experimental setups. That is why establishing appropriate “transfer functions” (correlation functions) between different types of tests is important, so that the necessary physically meaningful correlations can be made. Here, again, developing a good understanding of the physics and mechanics of the shock deformation process is crucial [3, 12]. Deformation and failure of BGA components in the shock test condition and in actual system applications is a complex process that takes place at least on three different scales: the scale of the board, the scale of the solder joint, and the scale of the solder joint material and interfaces (Fig. 1). On the scale of the board, the PCB deformations occur due to the acceleration applied to its supports (support contour). The significant bending of the board that takes place creates stressing (deformations) of the solder joints of surface-mounted components. The amount of the board deformation depends on the magnitude and the duration of the shock input, the total mass of the PCB assembly, and the structural characteristics of the board: for example, size, thickness, material properties, boundary conditions (i.e., the structure and materials of the board attachments). At the level of the solder joint, the loading transferred from the board creates stresses and deformations, both elastic and inelastic, in the solder joint material. The resulting stress distribution is considerably nonuniform and depends on the size, properties of materials, and interfaces in the solder joint. The failure process that takes place in a solder joint can be in the form of plasticity combined with damage evolution and/or due to a single crack propagation. It is a function not only of the stress experienced by the solder joint but also of the strength of the solder material and the interfaces. Material strength is directly related to the composition of the material microstructure, its disorder, and micromechanical properties as well as the strain rates. When evaluating product capabilities in shock, direct measurements of deformations at the location of the solder joints are difficult to perform. The adopted method in industry is to determine the solder joint deformations/strains and the solder material proximity to failure

Input pulse G t0

(1)

Ip

(2)

(3)

p

d e Measured

Ib

tb

strain

ε = ε (G,BC,t0,Ib,Ip,p,d,materials)

D

σ Stress σ = σ (ε,materials, geometry)

Damage D = D (σ, material strength)

Figure 1 Deformation process in shock takes place at three different scales.

418 Dynamic Response of PCB Structures to Shock Loading in Reliability Tests indirectly by evaluating the board deformations. The full insight into the solder joint failure process during shock testing on the PCB level can be established only provided that the physics, mechanics, and metallurgy of the correlation between the measured deformations and the resulting failure characteristics on all the three scales are well understood. With that respect, in order for the test measurements to be appropriately interpreted, it is important to understand how the loading experienced by the solder joints is related to the measured deformations of the board. The relationship between the board deformations and solder joint response is a function of many parameters related not only to the board “global” characteristics (geometry and stiffness) but also to the local characteristics of the assembly. Complete and in-depth understanding of the deformation and failure at all three major scales as well as the influence of the relevant parameters on the major quantities of interest is a “prerequisite” for establishing the necessary correlations between the board deformation and the solder joint loading and failure. Such an understanding will enable one to determine the transfer functions between the BGA responses on the board level and the system level. Analysis of the drop impact should be comprehensive and integrated and should include goal-oriented design of the appropriate test vehicles (experimental setups) and the actual testing, predictive modeling (both analytical and computer-aided), and studies of the attributes of the mechanical behavior of the structures and materials of interest at the board and the system levels.

3

BOARD-LEVEL SHOCK TESTS Board-level shock tests have become the key qualification test for portable electronic products. This was the motivation for JEDEC (Joint Electronic Device Engineering Council) to issue a board-level drop test standard (JEDEC standard JESD22-B111, “Board Level Drop Test Method of Components for Handheld Electronic Products”). In addition to JEDEC tests [8, 9] different shock tests are being conducted in the industry [3, 7, 10, 11]. Characterization of surface-mounted (SM) components and their packages when subjected to shock tests on the board level has been carried out using shock test boards. These tests provide board strain-based design limits that are the basis for the specification documentation for the package subjected to shock tests [4–8]. The tested shock test board assembly consists of a board with one or more packages surface mounted on one side of the board (Fig. 2). Additional weights are often needed to be attached to the board to enhance its dynamic response in bending. Detailed procedures for a number of different component-level tests can be found in the literature. Here, only a short description of a typical test is provided. During the test, a PCB assembly (board with surface-mounted packages, added weights, strain gages, accelerometers, etc.) is mounted on the drop test fixture. This fixture is attached to a rigid drop table that can move vertically along the guide rods (Fig. 3). Drop testing is conducted by releasing the drop table from a predetermined (prespecified) height. When the table hits the strike surface on the rigid base, the acceleration shock pulse is transmitted to the board through the drop table. This causes the PCB to deform. The board experiences impact-induced vibrations until the vibrations fade away. Drop tests can be effectively substituted with shock tests if the shock tester is adequately tuned so that the drop conditions are appropriately mimicked (reproduced) [5]. In shock testing, different types of input pulse shapes can be defined, such as, say, trapezoidal and/or sinusoidal (see Fig. 4). Two parameters have to be specified for each profile: the duration (in microseconds) and the amplitude (acceleration in number of g’s, where g = 9.81 m/sec2 is the acceleration due to gravity). The type of the input pulse used should depend on the use conditions to be simulated [13–15]. Selection of the input pulse affects the dynamic characteristics of the board response. During the tests different mechanical parameters are measured,

4 Theoretical Considerations

VI

IV

II

IV

VI

V

III

I

III

V

VI

IV

II

IV

VI

419

Figure 2 Examples of shock test boards: one-component board (top), weights are added to facilitate bending; multicomponent (JEDEC) board (bottom).

such as strains, displacements, accelerations, and failure of solder joints by monitoring the electrical “opens” via the component’s daisy chain.

4

THEORETICAL CONSIDERATIONS The models that have been employed to simulate the PCB assembly response to a shock loading are based, as a rule, on a linear approximation [15–20]. The series of studies have provided first-order estimates of the board deformations when subjected to an impact loading as well as the stress in the board-level interconnects. Theoretical studies [17–21] have been particularly insightful. In the cases of very thick boards and low-level shock and/or inertia loads associated with small displacements, linear approaches provide a fairly accurate picture of the state of the board deformation. In the case of high-level shocks and significant inertia forces, the linear approximation is not adequate [22–26]. In this section, some theoretical principles and practical consequences of the linearity assumption are addressed. A dynamics problem is considered linear when (1) the material constitutive model is linear elastic (“material’s linearity”), (2) the deformations are small (“geometric linearity”), and (3) no change in the boundary conditions or contact issues are involved. The last criterion is not of a concern here, and the first two criteria will be addressed in more detail. A material’s constitutive model is represented by a relationship between stresses and strains. When this relationship is linear, the material is called linearly elastic. However,

420 Dynamic Response of PCB Structures to Shock Loading in Reliability Tests

Guide Rods PCB Assembly XXXXXXXX

SXXXXX

Base Plate

Base Place Drop Table SXXX Surface

Drop Table

Rigid Base

Accelaration (g)

Figure 3 Typical drop test setup and mounting scheme for PCB test assembly.

A0 tw

tw Time (sec)

Figure 4

A0

Time (sec)

Shock input profiles (left: half sine; right: trapezoidal).

when material behavior includes deviations from linearity such as, for example, plasticity, creep, and/or damage, the behavior becomes nonlinear, and it is characterized by a nonlinear stress–strain relationship (“material’s nonlinearity”). Deformations are considered small (“infinitesimal”) if both the displacements and the rotation angles of the cross sections are small. The usually adopted criterion of “small” (see, e.g., [27]) is that the displacement is smaller than half of the smallest structural dimension (height for a beam or thickness for a plate). The angles of rotation of the cross sections of the structural element are small if they are significantly smaller than unity. The important implication of the small-deformation approximation is that the state of the equilibrium can be established on the initial, undeformed configuration of the structural element. When this is the case, any influence of the in-plane forces, if any, on the lateral (bending) deformations and strains can be neglected. However, when the deformations are not small, establishing equilibrium on the undeformed configuration can lead to a significant error.

4 Theoretical Considerations

421

When the deflections are large, the “geometric” nonlinearity takes place. Mechanically (“mathematically”), this is reflected in the relationship between the displacements and the strains that are nonlinear. In this case, the use of linear relationships can lead to significant errors and, hence, be misleading. Equilibrium cannot be established on an initial (undeformed) configuration, because the influence of in-plane forces on the lateral displacements cannot be neglected. The in-plane and lateral deformations become coupled. There exist numerous studies of the problems of the geometric nonlinearity of thin-plate systems subjected to static or dynamic loading and used in various fields of engineering (see, e.g., [23–26]). The geometrically nonlinear response of PCBs was addressed, apparently for the first time, in [27] when the application of the periodic loading to the PCB support contour was considered and in [28] when the support contour of a flexible PCB was subjected to a constant, suddenly applied acceleration. A detailed analysis of the nonlinear dynamic response of a square, simply supported PCB used in an experimental setup has been recently addressed by the authors of this chapter [6]. Figure 5 illustrates the role of the in-plane (membrane) forces. The first observation is that the in-plane forces are negligible for very small displacements. Their impact rapidly increases, however, with an increase in the deflections. The relationships between the level of loading, the deflections, and the magnitude of the in-plane stresses are not linear: for example, a twofold increase in the deflection (from 0.5 h to 1.0 h, where h is the plate’s thickness) results in a fivefold increase in the in-plane (“membrane”) stress. In the linear case (when no external load is applied in the plate’s middle plane), stresses are caused by bending only [23–29]. In the nonlinear case, the stresses are due to both bending and “stretching”. The tensile membrane strains, if loading is significant, might be substantially higher than the bending strains. As shown in Fig. 6 [6], the role of the in-plane (“membrane”) stresses and strains, when the lateral displacements of the board are not very small, becomes very important and might result in a significant increase in the total maximum stress. For example, when the deflection is equal to the PCB’s thickness h, the maximum tensile stress in the PCB (this stress is caused now by both bending and tension) increases by 15% in comparison with the linear (small-displacement) case. An additional and an important manifestation of the role of the geometric nonlinearity is that the stiffness of the PCB structure does not remain constant during the deformation process. For example, the in-plane tension makes the PCB stiffer (with respect to lateral loading), while the in-plane compression makes it more compliant [29]. Another way to look

Max. membrane stress (nondimensional)

2.5 2 1.5 1 0.5

0.2

0.4

0.6

0.8

1

Amplitude/plate thickness

Figure 5 Influence of deflection on membrane (in-plane) stress.

422 Dynamic Response of PCB Structures to Shock Loading in Reliability Tests Max. stress-nonlinear/Max. stress-linear

1.16

1.1 Beam Square plate 1.06

1 0

0.2

0.4 0.6 0.8 Amplitude/thickness

1

Figure 6 Influence of large deflection on maximum stress (when amplitude is only equal to plate thickness, stress increases 15% in comparison to a very small amplitude) [28].

F

F

Δ

Figure 7

Δ

Stiffening (left) and softening (right) nonlinearity.

at it is to consider force–displacement curves (Fig. 7). Notice that its slope (which has the physical meaning of stiffness) does not remain constant: It increases in the case of “stiffening” nonlinearity and decreases in the case of “softening” nonlinearity. We would like to point out that the described nonlinear effects are due to the reactive nature of the in-plane stresses that increase in a nonlinear fashion with an increase in the external (e.g., impact) loading and the bending deformations (deflections). When the in-plane stresses are not of the reactive nature but are applied as independent external loads (i.e., in the problems of elastic stability), the effects of “stiffening” of a structural element (plate, beam, shell) subjected to tension and “softening” of this element, when subjected to compression, can be observed even in the linear case, that is, in the case described by linear differential equations [30]. This phenomenon indicates that in dynamic problems, when the deflections (amplitudes) are significant, the change in stiffness might cause a change in the dynamic properties, such as frequency (as opposed to the linear behavior of the PCB when the frequencies of its free vibrations are independent of the level of loading and the vibration amplitudes, remain unchanged). Figure 8 illustrates the impact that geometric nonlinearity can have on the frequency of free (actually shock-induced) vibrations of a structure, that is, how nonlinearity of deformation affects the frequency of a simple structure such as a square plate supported by immovable hinges along all edges. If the amplitude of vibration is large, the tensile force is introduced by immovable supports when the plate deflects (see Fig. 9).

4 Theoretical Considerations

423

1.45 1.40 1.35 1.30 ωn

1.25

ω0

1.20 1.15 1.10 1.05 1.00 0

0.2

0.4

0.6

0.8

1

Amplitude/plate thickness

Nonlinear Amplitude (mm)

Figure 8 Influence of deflection on frequency of a square plate with immovable hinges along the edge.

8 6 4 2 500 1000 1500 Input G (no.of g’s)

2000

Figure 9 Square simply supported plate (30 mm × 30 mm × 1.5 mm): nonlinear relationship between the impact load and the amplitude of the response. The presence of tensile force makes the system stiffer, and consequently results in the increase in the natural frequencies of lateral vibrations. For example, when the deflection becomes equal to the plate thickness, the frequency of the plate increases more than 40% from its linear value (that corresponds to the situation when the displacements are very small [26]). As a result of the elevated in-plane (“membrane”) stresses in the PCB that are generated in high impact and/or high inertia drop conditions, the relationship between the magnitude of the load (determined by the initial impact velocity) and the induced PCB deflections becomes non-linear, with a rigid cubic characteristic of the restoring force. Figure 8 is based on the equations for amplitude of vibration provided in [25] for the case of a simply supported plate with an impact load applied to its support contours. The above examples are theoretical solutions obtained for simple structures. They are used here merely to illustrate the effects that geometric nonlinearity has on structural properties, dynamic properties, stresses, and strains. These examples will be used in the next section to explain observations from test board shock experiments. In this section we show that the behavior of a typical flexible PCB test board subjected to a shock load is indeed highly nonlinear. The source of the experimental data presented here is a large database generated by various investigators and summarized in [3, 15, 31].

424 Dynamic Response of PCB Structures to Shock Loading in Reliability Tests 5

PCB RESPONSE IN SHOCK TESTS: EXPERIMENTAL DATA

5.1 Displacement Evolution One of the parameters that can be measured in a PCB shock test is board displacement. For example, Figure 10 depicts out-of-plane displacements of several points on a 1.57-mm (62-mil) board exposed to a 50 g 10-μsec trapezoidal shock input. The maximum displacement recorded is about 13 mm. This value is more than 8 times greater than the board thickness. This is a principal indication that the test board behavior is highly geometrically nonlinear. Moreover, the relative size (8x) of the lateral deformation for the relevant range of shock inputs places the PCB system in a category of systems with very large displacements.

5.2 Strain as a Function of Input Acceleration The third set of experimental data illustrates the relationship between the input acceleration and the measured strain. In Fig. 11, the data points represent strain readings for 40 mil and 62 mil boards exposed to half sine shocks of different magnitude [3, 31]. The experimental curves were here fitted through the data to obtain an understanding of the nature of the relationship between the acceleration input and the strain reading. The point (0, 0) represents the initial state of deformation, when zero strain is measured while board is at rest (applied acceleration equal to zero). All the other experimental points are recorded relative to this initial state. For the relevant magnitudes of input shock the behavior of strain as a function of acceleration amplitude is clearly nonlinear.

5.3 Quasi-Static Test To characterize the force–deflection behavior of the test board a quasi-static test was performed using an Instron mechanical tester [31]. The board was attached to the metal plate using eight standoffs (as shown in Fig. 1), so that the boundary conditions remain identical to those in the shock test, for which this board is designed. The load was then applied at the center of the board and the deflection of the center recorded. The board was allowed to deform

15

Displacement (mm)

10 5 0 −5 −10 −15

0

0.01

0.02

0.03

0.04

0.05

Time (s)

Figure 10 Displacement of shock test board (thickness 1.57 mm, shock input 50 g trapezoidal).

5

PCB Response in Shock Tests: Experimental Data

425

5 Peak strain (normalized)

4.5 4 3.5

40 mils

3

62 mils

2.5

Poly. (62 mils)

2

Poly. (40 mils)

1.5 1 0.5 0 0

1 2 3 4 5 Peak input acceleration (normalized)

Figure 11

6

Test data for 40- and 62-mil boards with appropriate fit.

70 60 F

50 F (kgf)

40 40 mils 62 mils

30 20 10 0 −10

0

2

4

6

8

10

12

14

16

Displacement (mm)

Figure 12 Stiffening nonlinearity of test boards (thin lines are added to illustrate the difference between the initial and final slope of force–displacement curve). up to the level of displacements usually observed in high-input shock tests: 13–15 mm. The force–deflection curves are plotted in Fig. 12 for 40- and 62-mil boards. The board behavior is nonlinear and exhibits stiffening of the PCB with increasing deformation. The stiffening is a consequence of the tensile in-plane (membrane) stresses that develop during the deformation, as explained in Section 3.

5.4 Impact of Shock Magnitude on Dynamic Properties The previous consideration shows that there is a change in stiffness when the structure undergoes large-displacement behavior. The quasi-static tests indicate that this change is of the stiffening type. This means that the PCB becomes stiffer when the displacements are larger. Since the stiffness changes, so will the dynamic properties, such as frequency and period of vibrations. The actual shock tests clearly confirm this conclusion. Figure 13 represents the time evolution of strain for a 40-mil-thick STB exposed to four different levels of input

426 Dynamic Response of PCB Structures to Shock Loading in Reliability Tests 50 G 80 G 100 G 120 G

4

Strain (normalized)

3 2 1 0 −1

0

0.005

0.01

0.015

0.02

−2 −3 −4

Time (s)

Figure 13 Dependence of frequency on input shock. Shock test on 40-mil STB; 2 μsec, half-sine input. Increase in input load causes decrease in period of vibration (i.e., increase in frequency). acceleration (2 μsec, half-sine shock). A change in shock level causes not only a change in the amplitude of the response but also a change in the period of vibration. More precisely, when shock input increases, the period becomes smaller and frequency higher. The response therefore becomes stiffer, confirming previous findings. In other words, in large-amplitude vibrations, the frequency we see in the strain and displacement data is not the natural frequency of the board, as is commonly assumed. The frequency we see is in fact the frequency of the response, which is a function not only of the intrinsic characteristics of the board (geometry, mass, materials) but also of the input load.

5.5 Experimental Evaluation of Bending and Membrane Effects The experimental evidence reported above shows the effect of membrane forces only in an implicit way. Although the test data imply that these forces are significant, the membrane effect has not been explicitly evaluated. It is of interest to find out what part of the measured board deformation can be attributed to bending and what part can be attributed to membrane forces. To determine the forces N and the moments M in the given cross section, two strain numbers are needed: strain on the bottom and strain on the top. The corresponding relationships are Eh εbottom + εtop N = , (1) 1−ν 2 M =

Eh 2 εbottom − εtop , 6(1 − ν 2 ) 2

(2)

where (Fig. 14) εbending = εM =

εbottom − εtop , 2

(3)

εmembrane = εN =

εbottom − εtop . 2

(4)

5

PCB Response in Shock Tests: Experimental Data N

M

M

427

N

A Bending

Total strain

εbottom

εM

Membrane

εN

Figure 14 Strain distribution in a typical cross section of a thin plate exposed to bending.

In (1) and (2), E is Young’s modulus of the board, ν is the Poisson ratio of the board material, h is the board thickness. Strain εmembrane represents a membrane component of the strain, while εbending is a part of the strain attributed to bending. Both εmembrane and εbending are related to the top and bottom strains, as shown in (3) and (4). Therefore, to determine forces in a board by using strain gages, two strain gages are needed in the given location (“cross section”): one placed on the top and one placed on the bottom. The only exceptions are the cases where either M = 0 or N = 0. Moment M = 0 when the board is exposed to stretching only. This is not the case, however, in shock tests when the board is being bent by the inertia forces acting in the direction perpendicular to the board’s plane. The other case, N = 0, corresponds to the case of so-called pure bending. In a dynamic event, the bending will be “pure” only when peak out-of-plane displacements of the board are smaller than half of the board thickness [31]. In all other cases, in-plane forces will be activated and bending will not be pure any more. Therefore, by using one strain reading (one strain gage), the bending moments can be estimated only when displacement of the board is small. In a typical PCB shock test, displacements are significantly greater than the board thickness. In the numerical example carried out in [6], the predicted amplitude of the nonlinear vibrations is as high as 8.4 mm, while the PCB is only 1.5 mm thick. In a system drop, displacements are also significant—several times greater than board thickness. Therefore, in the problems of practical interest both the bending moments M and the lateral forces N are present and should be accounted for. To estimate the role of the in-plane forces a special test was designed and carried out. Two strain gages were placed in the center of the board, one on the bottom and the other on the top (Fig. 15). Boards of different thicknesses were tested using different shock inputs, and both the top and bottom strain evolutions were recorded during the shock event (Fig. 16). Of interest are the instances when the induced strains are at their peak (in this case the first peak in strain was used as a reference). It is clear that the top and the bottom strains are not of the same magnitude, indicating the presence and the effect of the in-plane forces. The resulting bending and membrane components of the strain are computed from experimental data in Fig. 16 using expressions (3) and (4). They are depicted in Fig. 17. Plots in Fig. 17 clearly indicate that the strain due to the “membrane” effect cannot be considered insignificant. Moreover, it is of the same order of magnitude as the bending strain. Its exact magnitude relative to the bending strain is a function of the board thickness and the shock input. The principal conclusion from these tests is that the effects of membrane

428 Dynamic Response of PCB Structures to Shock Loading in Reliability Tests ε top

STB

Strain gages

h

h Shock table ε bottom

Figure 15 Test setup: bare board with two strain gages placed in the center: one on the top and one on the bottom. Left: typical strain distribution in the center cross section.

strain (normalized)

Strain: bottom vs. top (32 mil board; 40g trapez. 4 x 100 gr mass) 3 2.5 2 1.5 1 0.5 0 −0.5 7.14 −1 −1.5 −2

Bottom strain Top strain 7.16

7.18

7.2

7.22

Time (s)

Figure 16 Top strain and bottom strain.

Effect of tensile forces vs. effect of bending (32 mil board; 40g trapez. Input; 4 x 100g) Strain (normalized)

1.5 1 0.5 0 −0.5 7.14 −1 −1.5 −2

Figure 17 board.

Bending strain 7.16

7.18

7.2

7.22

Membrane

Time (s)

Bending and membrane components of strain measured in the center of a bare

effects cannot be disregarded. In the numerical example carried out in the [6], the following data were obtained for the predicted stresses in the center of a square, simply supported PCB: bending stress σb = ±3.2 kg/mm2 , in-plane stress σ0 = +5.4 kg/mm2 , that is, by a factor of 1.7 greater than the maximum bending stress. This means that the stress on the convex side of the plane is as high as 8.6 kg/mm2 , while the stress on the convex side is also a tensile one and is only 2.2 kg/mm2 . The above discussion is relevant for the correct understanding of the test metric as it relates to solder joint loading. In the literature it is usually stated that failure of solder joints in shock

6

Modeling of PCB Shock

429

is caused by the “differential bending” between the package and the board (see, e.g., [17]). It is usually considered that by measuring the strain the board bending (flexure) is determined. Moreover, the terms “strain” and “flexure” are often used interchangeably. Our experimental and modeling data indicate that significant membrane stretching occurs during PCB tests and that the induced in-plane stresses and strains can be even higher than the bending stresses and strains. Hence, the strain measured in the component tests is due to the combined action of the bending and in-plane (“membrane”) loading. When the same component is placed on a system board, strain measured there is also a combination (but a different combination) of the strains caused by the board’s bending and stretching. This is important to keep in mind when test–system transfer functions are being developed.

6

MODELING OF PCB SHOCK Numerical techniques such as finite element analysis (FEA) are in wide use in today’s engineering practice. This is due primarily to the availability of various general-purpose commercial computer packages and many other computer programs written for specific applications. These techniques are, to an increasing extent, being used for design optimization and margin assessments, risk assessments for design changes, feasibility analysis of design concepts, and the reduction in the number of physical tests [31–32]. The numerical (simulation) models are also often used to provide additional insights into the physics of a particular problem as well as for developing guidelines for new physical tests. In the case of drop reliability, drop tests are usually used in the industry to check drop impact performance of IC packages. These tests are expensive and time consuming and involve a lot of manpower. Modeling is therefore an efficient tool to obtain solutions faster and at a lower expense. Considering the important role that FEA methods now play in the field of electronic packaging, it is important to understand what FEA can and cannot do and whether the most appropriate approach was used to obtain a sufficiently accurate solution to a particular problem of interest. First of all, the FEA is used to develop an idealized (“mathematical”) model of a particular structure and then to obtain numerical information on the mechanical behavior of this model. The FEA models are based on a set of assumptions and approximations regarding, for example, the input loads, the boundary condition, the geometry of the structure, the material properties, and the expected behavior. The FEA models are represented by a set of equations that describe the given problem. The FEA solutions are based on a set of numerical procedures that solve those equations. An FEA solution solves only the mathematical model. If the model is inadequate, the solution is meaningless and misleading. Therefore, the key step in any mechanics modeling work is to select or develop a model that is based on the physics of the deformation process. The physics of the deformation process can be understood by evaluating and analyzing experimental evidence. The experimental data from impact shock tests of flexible PCBs unquestionably indicate that the behavior of a PCB assembly during shock tests is highly nonlinear. The geometric nonlinearity associated with the board behavior is of the stiffening type and places the board behavior into the category of geometrically nonlinear systems with very large displacements, significant rotation angles, and moderate to medium strains [31]. Material nonlinearity in the dynamics of BGA structures might also be present and most often is associated with solder behavior at high strain rates. An adequate material model for the drop strain rates of interest is yet to be developed. The model will be rate dependent and include nonlinearity due to plasticity and/or damage. It is important to make sure, however, that the stresses in the PCB itself are well below the level that might lead to the PCB material nonlinearity. Thus, the description of a high-impact shock event should be based on nonlinear theory.

430 Dynamic Response of PCB Structures to Shock Loading in Reliability Tests 6.1 Finite Element Formulation The system of equations of motion that is solved in the course of a dynamics FEA simulation is of the form [M ]{u} ¨ + [C ]{u} ˙ + [K ]{u} = {F (t)},

(5)

where {u}, {u}, ˙ and {u} ¨ are displacement, velocity, and acceleration vectors, respectively, {F (y)} is a load vector, and [M ], [C ], and [K ] are the mass, damping, and stiffness matrices, respectively. In nonlinear analyses the stiffness and damping matrices are not constant but are functions of {u}. Mathematically, (5) generally represents a set of nonlinear differential equations of second order. A nonlinear dynamics approach requires a direct time integration of the full system of coupled equations of motion. The term “direct” indicates that prior to the numerical integration no transformation of the equations into a different form is carried out (as opposed to the modal superposition analysis where the equations are first decoupled). In direct integration Eq. (5) are integrated using a numerical step-by-step procedure. A solution is sought at discrete time points t0 , t1 = t0 + t, t2 = t1 + t, . . . , tn = tn−1 + t, while a variation of the displacements, velocities, and accelerations within each time interval t is assumed. It is the form of these assumed variations of displacements, velocities, and accelerations within each time interval that determines the accuracy, stability, and cost of the solution. Usually two methods are considered for the direct integration of the equation of motion: explicit and implicit. The description of these methods can be found in the literature on numerical procedures (see, e.g., [33–35]).

6.2 Errors due to Linearization The validity of the nonlinear and linear models can be evaluated through the examination of how well the numerical data agree with the test data. A typical PCB is a composite structure which is both inhomogeneous (because of various copper densities present in different regions of the board) and strongly orthotropic (since FR4 contains glass cloth woven by glass fibers and epoxy resin). Therefore, it is not easy to obtain the precise PCB material properties needed as accurate input in the model. Because of this, the difference that is obtained between the modeling and tests results is often attributed in engineering practice to the lack of knowledge of the actual properties of the board’s material. Since the next set of data is focused on explaining the errors due to linearization, the tests used as a reference are done on board made of aluminum, which is an isotropic and homogeneous material with well-known properties [3, 29]. The thickness of these boards was selected so that the board’s overall bending stiffness is comparable to the bending stiffness of typically used PCBs. Material properties were well characterized in tests and used as input in the model. The first set of tests focused on displacement measurements using a high-speed camera and digital image correlation [36]. These tests generated a full-field deformation distribution of the board during shock as well as extracted the time evolution of displacement at the selected points. Test results for out-of-plane displacements in a center of the board are presented in Fig. 18 together with the output of the modeling simulations. The agreement between the test and the nonlinear model is very good. For comparison, the results of the linear analysis are also shown. As can be seen, linear analyses largely overestimate displacements and underestimate the response frequency. In the second set of validation tests the focus was on strain validation. Strain gages were placed in the center of the board on the top and bottom surfaces of the board (Fig. 17) and the strain evolution over time was recorded (Fig. 19). The test data were compared to the nonlinear model as well as to the linear model. Again, the nonlinear model captures the strain evolution very well, while the linear model largely overestimates strain.

6

Modeling of PCB Shock

431

Center Board Displacement (mm)

Model vs. test: displacements (Al board; shock input: 50g trapezoidal) 40 Test

Nonlinear

20

Test

0 −20

Nonlinear 0

0.01

0.02

0.03

Linear

Linear

−40 Time (sec)

Figure 18 Model validation: displacement tests. Very good agreement between nonlinear model and test data.

Strain (normalized)

Top strain: 35-mil board 2 1.5 1 0.5 0 −0.5 0 −1 −1.5 −2 −2.5

0.01

0.02

0.03

Test Nonlinear Linear

Time (s)

Strain (normalized)

Bottom strain: 35-mil board 2.5 2 1.5 1 0.5 0 −0.5 0 −1 −1.5 −2

0.01

0.02

0.03

Test Nonlinear Linear

Time (s)

Figure 19 Strain measurements. Very good agreement between nonlinear model and test data. Linear model produces large errors.

As described earlier, membrane forces that are activated in large-displacement shock events have not only a quantitative but also a qualitative effect on strain distribution. In Fig 20, a ratio of bottom and top strain was presented from test data, nonlinear analysis, and linear analysis. Nonlinear analysis captures this ratio very well, while linear analysis provides a very inaccurate result. In this case, the difference between linear and nonlinear analyses is not only quantitative but also qualitative. This difference is not surprising, because linear methods of structural mechanics by definition completely disregard coupling between membrane forces and out-of-plane deformation (and resulting strain), as discussed earlier.

432 Dynamic Response of PCB Structures to Shock Loading in Reliability Tests Model vs. test: bottom strain/top strain (Al 32 mil board; shock input: 40g trapezoidal)

Strain ratio

10 8

Nonlinear

Test

6

Nonlinear

4

Linear

2 0 0

0.01 Linear

0.02

0.03

Time (s)

Figure 20 Model validation: strain test. Nonlinear models capture the nature of strain distribution accurately. Linear models fail to represent the actual physics not only quantitatively but also qualitatively. The presented series of validation studies indicate that the nonlinear model captures very well displacements, strains, and dynamic properties of the shock test structures (frequency/period). Linear models, on the other hand, are not capable of capturing either magnitudes of strains and displacement or dynamic characteristics of the response. Moreover, linear models do not provide appropriate qualitative understanding of the board response in typical PCB shock tests Because of this, application of linear methods in modeling PCB shock response can lead to erroneous conclusions not only about board deformation but also about the relationships between board deformations and stress in solder joints. Also, their application on the correlation of system board and test board response cannot provide appropriate answers, as indicated in the previous section. The nonlinear model, on the other hand, is capable of reproducing deformation characteristics observed in experiments. Because of this, the confidence increases that quantities that cannot be measured, such as stress in solder joints, will also be captured adequately (provided that adequate material model of solder is available).

7

CONCLUSIONS Adequate understanding of the physics of the dynamic response of the package/PCB structure employing BGA solder joint interconnections is important for several major reasons: (1) establishing correlations between measured board deformation and solder joint loading, (2) developing physics-based models, and (3) developing test board–system board transfer functions. We have demonstrated that the mechanics and physics of board deformation in highamplitude displacement shock events belong to a class of geometrically nonlinear problems. Relevant experimental data and their appropriate theoretical interpretations overwhelmingly confirm this fact. The appropriate model for simulation of this class of problems is a nonlinear dynamics model. These models are capable of reproducing all important features of deformation measured in a typical PCB shock test, thus increasing the confidence that quantities that cannot be measured, such as the SJ stress, will also be captured adequately. The application of linear models on the physical problems of interest have serious limitations, since these models lead to significant errors not only in magnitude of relevant quantities but also, more importantly, in understanding the nature of strain distribution in the board.

References

433

REFERENCES 1. Shina, S. G., Green Electronics Design and Manufacturing, McGraw-Hill Professional, New York, 2008. 2. Chin, Y. T, Lam, P. K., Yow, H. K., and Tou, T. Y., “Investigation of Mechanical Shock Testing of Lead-Free SAC Solder Joint in Fine Pitch BGA package,” Microelectronics Reliability, Vol. 48, pp. 1079–1086, 2008. 3. Vujosevic, M., and Lucero, A., “Predictive Modeling and Drop/Shock Tests for Reliability Assessment of Lead-Free BGA Structures,” Keynote Presentation, IEEE ASTR 2008, Portland, OR, Oct. 1–3, 2008. 4. Suhir, E., “Is the Maximum Acceleration an Adequate Criterion of the Dynamic Strength of a Structural Element in an Electronic Product?”, IEEE CPMT Transactions, Part A, Vol . 20, No. 4, Dec. 1997. 5. Suhir, E. , “Could Shock Tests Adequately Mimic Drop Test Conditions?”, 52nd ECTC Proc., 2002. 6. Suhir, E., and Vujosevic, M., “Nonlinear Dynamic Response of a ‘Flexible-and-Heavy’ Printed Circuit Board (PCB) to an Impact Load Applied to its Support Contour,” Journal of Physics D: Applied Physics, Vol. 42, No. 4, 2009. 7. Mercado, L., Girouard, S., and Hsieh, G., “A Mechanics-Based Strain Gage Metrology for Solder Joint Reliability Assessment”, paper presented at the 5th International Conference on Thermal and Mechanical Simulation and Experiments in Micro-electronics and MicroSystems, EuroSimE2004, 2004, pp. 533–541. 8. JEDEC Standard JESD22-B104-B, “Mechanical Shock,” 2001. 9. JEDEC Standard JESD22-B111, “Board Level Drop Tests Method of Components for Handheld Electronic Products,” 2003. 10. Zhou, C. Y., Yu, T. X., and Lee, R. S. W., “Design of Shock Tests to Mimic Real-Life Drop Conditions of Portable Electronic Devices,” IEEE, New York, 2006. 11. Reiff, D., and Bradley, E., “A Novel Mechanicl Shock Test Method to Evaluate LeadFree BGA Solder Joint Reliability,” in Proc. IEEE Electronic Components and Technology Conference, 2005, pp. 1519–1525. 12. Dasgupta, A., “Drop Durability of BGA Assemblies,” IEEE ASTR 2008, Portland, OR, Oct. 1–3, 2008. 13. Suhir, E., “Dynamic Response of Microelectronics and Photonics Systems to Shocks and Vibrations,” paper presented at the International Conference on Electronic Packaging, INTERPack’97, Hawaii, June 15–19, 1997. 14. Suhir, E., and Burke, R., “Dynamic Response of a Rectangular Plate to a Shock Load, with Application to Portable Electronic Products,” IEEE Transactions on Components, Packaging, and Manufacturing Technology, Part B: Advanced Packaging, Vol. 17, No. 3, 1994. 15. Loh, W. K., and Garner, L. J., “Solder Joint Reliability Prediction of Flip Chip Packages Under Sock Loading Environment,” in Proc. ASME InterPACK Conf., San Francisco, CA, July 17–22, 2005. 16. Luan, J., Tee, T. Y., Ooeck, E., Lim, C. T., and Zhong, Z., “Modal Analysis and Dynamic Responses of Board Level Drop Test,” paper presented at the 2003 ELECTRONICS Packaging Technology Conference. 17. Wong, E. H., “Dynamics of Board-Level Drop Impact,” ASME Journal of Electronic Packaging, Vol. 127, pp. 200–207, Sept. 2005.

434 Dynamic Response of PCB Structures to Shock Loading in Reliability Tests 18. Wong, E. H., Mai, Y-W., and Seah, S. K. W., “Board Level Drop Impact-Fundamental and Parametric Analysis,” ASME Journal of Electronic Packaging, Vol. 127, pp. 496–502, Dec. 2005. 19. Suhir, E., and Vujosevic, M., “Interfacial Stresses in a Bi-Material Assembly with a Compliant Bonding Layer,” Journal of Physics D: Applied Physics, Vol. 41, No.11, 2008. 20. Suhir, E., “Predicted Fundamental Vibration Frequency of a Heavy Electronic Component Mounted on a Printed Circuit Board,” ASME Journal of Electronic Packaging, Vol. 122, No. 1, 2000. 21. Suhir, E., “Response of a Heavy Electronic Component to Harmonic Excitations Applied to Its External Electric Leads,” in Electrotechnik & Informationstechnik , Springer, Wien, 2007. 22. Suhir, E., “Dynamic Response of Micro-Electronic Systems to Shocks and Vibrations: Review and Extension,” in E. Suhir, C. P. Wong, and Y. C. Lee, Eds., Micro- and OptoElectronic Materials and Structures: Physics, Mechanics, Design, Packaging, Reliability, Springer, 2007. 23. Chu, H., and Herrmann, G., “Influence of Large Amplitude on Free Flexural Vibration of Rectangular Elastic Plates,” Journal of Applied Mechanics, Vol. 23, p. 532, 1956. 24. Timoshenko, S., and Woinowsky-Krieger, S., Theory of Plates and Shells, McGraw-Hill, New York, 1959. 25. Haris, C. M., and Piersol, A. G., Hariss Shock and Vibration Handbook , 5th ed., McGrawHill Professional, New York, 2001. 26. Worden, K., and Tomlison, G. R., Nonlinearity in Structural Dynamics, Institute of Physics Publishing, Bristol, UK, 2001. 27. Suhir, E., “Response of a Flexible Printed Circuit Board to Periodic Shock Loads Applied to Its Support Contour,” ASME Journal of Applied Mechanics, Vol. 59, No. 2, 1992. 28. Suhir, E., “Nonlinear Dynamic Response of a Flexible Thin Plate to a Constant Acceleration Applied to Its Support Contour, with Application to Printed Circuit Boards Used in Avionic Packaging,” International Journal of Solids and Structures, Vol. 29, No. 1, 1992. 29. Vujosevic, M., Hezeltine, W., Blue, K., and Pei, M., “Physics of PCB Deformation in Shock,” Intel Assembly and Test Technology Journal , Jan. 2009. 30. Suhir, E., “Lateral Compliance of a Compressed Cantilever Beam, with Application to Micro-Electronic and Fiber-Optic Structures,” Journal of Applied Physics D, Vol. 41, No.1, 2008. 31. Zhu, L., and Marcinkiewicz, W., “Drop Impact Reliability Analysis of CSP Packages at Board and Product System Levels through Modeling Approaches,” paper presented at the Intersociety Conference on Thermal and Thermo-Mechanical Phenomena, 2004, pp. 296–303. 32. Katahira, T., Fujita, M., Shibata, T., Shiraori, M., and Yu., Q., “Various Levels of Drop Analysis for BGA in Mobile Phones,” Proceedings of InterPACK2007, ASME InterPACK’07, Vancouver, BC, Canada, July 8–12, 2007. 33. Zienkiewicz, O. C., and Taylor, R. L., Finite Element Method , Vol. 2, Solid and Fluid Mechanics, Dynamics and Nonlinearity, McGraw-Hill, London, 1991. 34. Belytschko, T., Liu, W. K., and Moran, B., Nonlinear Finite Elements for Continua and Structures, Wiley New York, 2001. 35. Bathe, K.-J., Finite Element Procedures, Prentice-Hall, New York, 1997. 36. Hezeltine, W., Intel internal document, Mar. 2007.

CHAPTER

19

LINEAR RESPONSE OF SINGLE-DEGREEOF-FREEDOM SYSTEM TO IMPACT LOAD: COULD SHOCK TESTS ADEQUATELY MIMIC DROP TEST CONDITIONS? Ephraim Suhir University of California Santa Cruz, California University of Maryland College Park, Maryland ERS Co. Los Altos, California

1

INTRODUCTION Drop tests are often substituted in qualification testing of micro-electronic and photonic products by shock tests [1]. In shock tests, a specially designed and appropriately tuned shock tester is employed, and a short-term load of the given magnitude (say, a constant or an halfsine load with the maximum value of 500 g) and duration (say, acting for 0.001 sec) is applied to the support structure of a component or a device [2]. The objective of the analysis that follows is to develop simple analytical (“mathematical”) predictive models for the evaluation of the dynamic response of a structural element to an impact load applied during drop or shock tests (Fig. 1). Based on these models, we intend to find out how a shock tester should be tuned in order to adequately mimic the drop test conditions. We assume that the maximum induced curvature (responsible for the level of the dynamic bending stresses) and the maximum induced acceleration (supposedly responsible for the functional, electronic or optical, reliability of a micro-electronic or photonic product) can be used as suitable characteristics of the dynamic response of a structural element in an electronic component or an optical device. The case of a structural element that can be idealized as an elongated, rectangular, simply supported plate is used to illustrate the suggested concept.

2 ANALYSIS 2.1 Elongated Rectangular Plate Subjected to Drop Tests 2.1.1 Basic Equation The dynamic response of an elongated rectangular plate to a shock load occurring when the plate is dropped on a rigid floor can be found from the following equation of motion (see, e.g., [3–5]): D

∂ 4w ∂ 2w + m 2 = 0, 4 ∂x ∂t

Structural Dynamics of Electronic and Photonic Systems Copyright © 2011 John Wiley & Sons, Inc.

Edited by Ephraim Suhir, David S. Steinberg and T. X. Yu

(1)

435

436 Linear Response of Single-Degree-of-Freedom System to Impact Load

Figure 1 where w = w (x , t) is the deflection function, Eh 3 (2) 12(1 − ν 2 ) is the plate’s flexural rigidity, E is Young’s modulus of the material, ν is its Poisson’s ratio, h is the plate’s thickness, m is the plate’s mass per unit area, and t is time. The origin of the coordinate x is in the middle of one of the long sides of the plate. The solution to Eq. (1) can be sought in the form ∞  Ti (t)Xi (x ), (3) w (x , t) = D=

i =1

where Ti (t) is the principal coordinate of the i th mode of vibrations and Xi (x ) is the mode (coordinate) function. It is assumed that the functions Xi (x ) are mutually orthogonal, that is,  a a Xi (x )Xj (x ) dx = , for i = j , (4) 2 0 where a is the plate’s width. For i = j the above integral is equal to zero. Multiplying both parts of formula (3) by Xj (x ), integrating the obtained expression over the plate’s width a, and using the above conditions of orthogonality, we have a w (x , t)Xi (x ) dx Ti (x ) = o  a 2 . (5) o Xi (x ) dx The deflection function w (x , y) must satisfy the initial conditions:  w (x , 0) = 0, w˙ (x , 0) = 2gH , w¨ (x , 0) = g,

(6)

where H is the drop height and g is the acceleration due to gravity. Conditions (6) can be translated, using relationship (5), into the following initial conditions for the principal

2 Analysis coordinate Ti (t):

Ti (0) = 0,

 T¨ i (0) = ci 2gH , a

where the factor

ci =  oa o

T¨ i (0) = ci g,

Xi (x ) dx Xi2 (x ) dx

437 (7) (8)

considers the effect of the i th vibration mode on the initial velocity and the initial acceleration of this mode. For an elongated plate simply supported over its long edges iπx Xi (x ) = sin , (9) a and formula (8) yields 4 ci = , i = 1, 3, 5, . . . . (10) iπ Hence, because of the symmetry of the structure with respect to its mid–cross section, only the odd modes of vibration occur. Formulas (7) and (10) indicate also that the initial velocities T˙ i (0) and initial accelerations T¨ i (0) of the i th mode of vibration are inversely proportional to the number of this mode. 2.1.2 Solution to Basic Equation We use Galerkin’s method (see, e.g., [3, 4]) to solve Eq. (1). In accordance with the procedure of this method, we multiply this equation by Xj (x ) and integrate the obtained expression over the plate’s width a with consideration of the orthogonality condition (4). This results in the following homogeneous equation for the principal coordinate: T¨ i (t) + ωi2 Ti (t) = 0, where

 ωi =

D m

a

IV o Xi (x )Xi (x ) dx a 2 o Xi (x ) dx

  a

2  D dx o Xi (x ) a 2 = m o Xi (x ) dx

(11)

(12)

is the natural frequency of the i th mode. In the case of a simply supported plate, using formula (9), we obtain 2  iπ D ωi = , i = 1, 3, 5, . . . . (13) a m The equation of motion (11) is of second order, while the sought solution for the principal coordinate, Ti (t), should satisfy three initial conditions (7). In order to obtain an equation of motion of third order, we differentiate Eq. (11) with respect to time t: ... (14) T i (t) + ωi2 T˙ i (t) = 0, whose solution

Ti (t) = Ai + Bi sin ωi t + Ci cos ωi t

(15)

contains three constants of integration, Ai , Bi , and Ci . These constants can be found, using conditions (7), as follows: ci ci  Ai = −Ci = 2 g, Bi = 2gH , (16) ωi ωi so that the principal coordinate Ti (t) is expressed as  √ 2gH g Ti (t) = ci sin ωi t + 2 (1 − cos ωi t ) . ωi ωi

(17)

438 Linear Response of Single-Degree-of-Freedom System to Impact Load In the majority of practically important cases, the frequencies ωi are high and the drop height H is large. Then the second term in the brackets in formula (17) is small in comparison with the first term, and the following simplified formula can be used to evaluate the principal coordinate of the i th mode: ci  Ti (t) = 2gH sin ωi t. (18) ωi Then formula (3) results in the following expression for the deflection function: w (x , t) =



2gH

∞  ci Xi (x ) sin ωi t. ωi

(19)

i =1

In the case of a simply supported plate, using formula (9), we obtain  ∞ iπx 4a 2 2mgH  1 sin sin ωi t. w (x , t) = 3 π D i3 a

(20)

i =1,3,5,...

This series is rapidly converging. 2.1.3 Maximum Curvature and Maximum Acceleration From (19) we obtain, by differentiation, the following expressions for the induced curvature and the acceleration: ∞   ∂ 2w ci  κ(x , t) = = 2gH X (x ) sin ωi t, (21) ∂x 2 ωi i i =1

w¨ (x , t) =

∂ 2w ∂t 2

∞   = − 2gH ci ωi Xi (x ) sin ωi t .

(22)

i =1

In the case of a simply supported plate, using formula (9) for the mode (coordinate) function, we obtain:  ∞ iπx 4 2mgH  1 (23) sin sin ωi t, κ(x , t) = π D i a i =1,3,5,...  ∞ 4π 2DgH  iπx w¨ (x , t) = − 2 i sin (24) sin ωi t . a m a i =1,3,5,...

The series (20) for the deflection function is rapidly converging, the series (23) for the curvature is converging slowly, and the series (24) for the acceleration is diverging. It is clear, however, that if the first terms only are retained in the series (23) and (24), the corresponding expressions will somewhat underestimate the actual maximum curvatures and accelerations (decelerations):   4 2mgH mgH κmax = = 1.8006 , (25) π D D   4π 2DgH 17.7715 DgH w¨ max = 2 = . (26) a m a2 m Formulas that overestimate the maximum curvature and the acceleration can be obtained, assuming that the entire initial potential energy, mgaH , is transferred into the strain energy  a

 2 1 V = D w1 (x ) dx (27) 2 o

3 Elongated Rectangular Plate Subjected to Instantaneous Impulse

439

of the fundamental mode of vibrations only. Assuming that the mode function w1 (x ) can be sought as πx w1 (x ) = A sin , (28) a we have 2 2 D 2 π V = A = mgaH , (29) 2 a3 so that 2 A = 2 a2 π



mgH , D

and formula (28) yields a 

 π 2

(30) 

mgH . 2 a D The maximum acceleration can be found as   2π 2 DgH 19.7392 DgH 2 w¨ max = Aω1 = 2 = . a m a2 m From formulas (25) and (31) we conclude that  D < 2.000, 1.8006 < κmax mgH κmax = w 

=

A=2

and formulas (26) and (32) indicate that the following condition is fulfilled:  m < 19.7392. 17.7715 < w¨ max a 2 DgH

(31)

(32)

(33)

(34)

Thus, the maximum curvature and the maximum acceleration can be found within rather narrow limits.

3

ELONGATED RECTANGULAR PLATE SUBJECTED TO INSTANTANEOUS IMPULSE APPLIED TO ITS SUPPORT CONTOUR As has been established in the previous section, the following initial conditions can be used when evaluating the dynamic response of an elongated rectangular plate subjected to drop tests:  w (x , 0) = 0, w˙ (x , 0) = 2gH . (35) These conditions are equivalent to the case of an instantaneous impulse,  t0 w¨ o (t) dt So =

(36)

0

of a short-term acceleration, w¨ o (t), applied to the support contour of the plate during time to . Indeed, the deflection function w (x , t) of a plate that has experienced an instantaneous acceleration impulse applied to its support contour can be sought in the form w (x , t) = So t −

∞  i =1

Ti (t) Xi (x ).

(37)

440 Linear Response of Single-Degree-of-Freedom System to Impact Load At the initial moment t = 0 of time, w (x , 0) = 0,

w˙ (x , 0) = So .

(38)

These conditions are analogous to conditions (35) and therefore will result in the same maximum curvature and the same maximum acceleration if the initial velocity is chosen as  So = 2gH . (39) If, for instance, w¨ o = const, then So = w¨ o to. In the case of a half-sine impulse, πt w¨ o (t) = w¨ o sin , to we have 2 So = w¨ o to . π The magnitude of a suddenly applied constant acceleration w¨ o can be established as √ 2gH So w¨ o = = . to to The maximum value, w¨ o , of a half-sine acceleration, πt w¨ o (t) = w¨ o sin , to can be found as √ π 2gH w¨ o = . 2 to

4

(40)

(41)

(42)

(43)

(44)

ERROR FROM SUBSTITUTING AN IMPACT LOAD WITH AN INSTANTANEOUS IMPULSE The objective of the analysis carried out in this section is to determine the maximum acceptable duration of the shock impulse applied during the shock tests so that the dynamic response of the system would not be much different from its response to an instantaneous impulse. We examine the case of a constant suddenly applied and suddenly removed acceleration as well as the case of a half-sine acceleration.

4.1 Constant Suddenly Applied and Suddenly Removed Acceleration The dynamic response of an elongated plate experiencing suddenly applied constant acceleration acting on its support contour can be sought in the form ∞

w (x , t) =

 1 Ti (t)Xi (x ). w¨ o t 2 − 2

(45)

i =1

Introducing this solution into the equation of motion (1) and using Galerkin’s method, we obtain the following equation for the principal coordinate Ti (t): T¨ i (t) + ωi2 Ti (t) = ci w¨ o ,

(46)

where the natural frequencies ωi are expressed by formula (12) and the factors ci are expressed by formula (8). The initial conditions w (x , 0) = 0,

w˙ (x , 0) = 0

(47)

4

Error from Substituting an Impact Load with an Instantaneous Impulse

441

for the deflection function w (x , t) can be translated, using formula (5), into the following initial conditions for the principal coordinates Ti (t): Ti (0) = 0,

T˙ (0) = 0.

Then Eq. (46) has the following solution: ci Ti (t) = 2 w¨ 0 (1 − cos ωi t ) , ωi and expression (45) results in the following formula for the deflection function:   ∞ t 2  ci Xi (x ) (1 − cos ωi t) . w (x , t) = w¨ o − 2 ω2 i =1 i

(48)

(49)

(50)

By differentiation, we obtain the following formulas for the curvatures, velocities, and accelerations: ∞  ∂ 2w ci  κ(x , t) = = − w ¨ X (x ) (1 − cos ωi t) , (51) o 2 i ∂x 2 ω i =1 i   ∞  ci ∂w w˙ (x , t) = Xi (x ) sin ωi t , (52) = w¨ o t − ∂t ω i =1 i   ∞  ∂ 2w w¨ (x , t) = = w¨ o 1 − ci Xi (x ) cos ωi t . (53) ∂t 2 i =1

In the case of a simply supported plate, ∞  4ma 2 iπx 1 sin w ¨ (1 − cos ωi t), o 3 3 π D i a i =1,3,5,... ⎞ ⎛ ∞  1 i π x 4 w¨ (x , t) = −w¨ o ⎝1 − sin cos ωi t ⎠ . π i a

κ(x , t) = −

(54)

(55)

i =1,3,5,...

Unlike series (23) and (24) for the curvature and the acceleration in the case of an instantaneous impulse, series (54) and (55) are converging and can be used for the evaluation of the maximum curvature and acceleration, with consideration of the effect of the higher modes of vibration. Formula (49) leads to the following expressions for the principal coordinate and its time derivative at a certain moment t = to of time: ci Ti (to ) = 2 w¨ o (1 − cos ωi to ) , (56) ωi ci w¨ o sin ωi to. (57) T˙ i (to ) = ωi If the external acceleration w¨ o is removed at this moment of time, the further vibrations of the plate become free, and the expression for the principal coordinate Ti (t) is as follows: ci Ti (t) = 2 w¨ o [sin ωi t o sin ωi t + (1 − cos ωi to ) cos ωi t] ωi 

 (58) to ωi to 2ci sin ωi t + . = 2 w¨ o sin 2 2 ωi

442 Linear Response of Single-Degree-of-Freedom System to Impact Load The plate’s deflections are 

 ∞  ωi to ci to w (x , t) = − Ti (t)Xi (x ) = −2w¨ o Xi (x ) sin sin ωi t + . 2 2 ω2 i =1 i =1 i ∞ 

(59)

By differentiation, we find 

 ∞  ωi to ∂ 2w ci  to = −2 w ¨ X (x ) sin sin ω , t + o i i ∂x 2 2 2 ω2 i =1 i 

 ∞  ωi to to ∂ 2w = 2w¨ o ci Xi (x ) sin sin ωi t + . w¨ (x , t) = ∂t 2 2 2 κ(x , t) =

(60)

(61)

i =1

In the case of a simply supported plate, κ(x , t) = −

8 ma 2 w¨ o π3 D

8 w¨ (x , t) = w¨ o π

∞  i =1,3,5,...



 1 to iπx ωi to t + sin sin sin ω , i i3 a 2 2

∞  i =1,3,5,...



 1 to iπx ωi to sin sin sin ωi t + . i a 2 2

(62)

(63)

When the time to is very short, one can put sin

ωi to ωi to = , 2 2



 to sin ωi t + = sin ωi t, 2

(64)

and formulas (60) and (61) yield κ(x , t) = −w¨ o to

∞  ci  X (x ) sin ωi t, ωi i

(65)

i =1

w¨ (x , t) = w¨ o to

∞ 

ci ωi Xi (x ) sin ωi t.

(66)

i =1

Comparing formula (60) with formula (65) and formula (61) with formula (66), we conclude that the factor χi =

sin (ωi to /2) ωi to /2

(67)

reflects the effect of the finite duration to of loading on the maximum curvature and the maximum acceleration of the i th mode of vibrations. Note that this factor is the same for all modes of vibration. The computed values χ0 (ωi t0 ) =

sin (ωi t0 /2) ωi t0 /2

of the factor χi are shown in the lower line of the table in Fig. 1. If, for instance, to =

π Ti = , 4ωi 8

(68)

(Ti = 2π/ωi is the period of vibrations of the i th mode), then formula (67) yields χi = 0.9745. Hence, one can conclude that a suddenly applied and suddenly removed short-term load of

4

Error from Substituting an Impact Load with an Instantaneous Impulse

443

finite duration to can be substituted with sufficient accuracy by its impulse if the load’s duration does not exceed 0.125 of the period of vibrations: to ≤ Ti /8.

4.2 Half-Sine Acceleration Following the same procedure in the case of a half-sine acceleration, we conclude that the effect of the finite duration of the impact load can be accounted for by the factor (Fig. 1). χ1i =

cos (ωi to /2) 1 − (ωi to /π )

2

=

cos (π t0 /Ti ) 1 − (2to /Ti )2

.

The expression for the deflection function is  



 ∞ π/ (ωi to ) sin ωi t − sin (π t/to ) to to πt w (x , t) = w¨ o ci Xi (x ) t − sin − π π to (π/to )2 − ωi2

(69)

(70)

i =1

for t ≥ to and



∞ sin(ωi to /2) sin ωi (t + to /2) w¨ o  ci Xi (x ) w (x , t) = 2π to ωi (π/to )2 − ωi2

(71)

i =1

for t ≥ to . The curvatures and the accelerations can be found by differentiation as follows: 

∞  π/ (ωi to ) sin ωi t − sin(π t/to ) ∂ 2w  κ(x , t) = = −w¨ o ci Xi (x ) , (72) ∂x 2 (π/to )2 − ωi2 i =1  

 ∞ sin ωi t + π/ (ωi to ) sin (π t/to ) ∂ 2w π  πt w¨ (x , t) = = w¨ o sin + ci Xi (x ) , (73) ∂t 2 to to (π/to )2 − ωi2 i =1

for t ≥ to and

 ∞ ∂ 2w w¨ o  ci  sin (ωi to /2) sin ωi (t + to /2) κ(x , t) = = −2π X (x ) , ∂x 2 to ω i (π/to )2 − ωi2 i =1 i 

∞ sin (ωi to /2) sin ωi (t + to /2) w¨ o  ∂ 2w = −2π ci ωi (x ) w¨ (x , t) = ∂t 2 to (π/to )2 − ωi2

(74)

(75)

i =1

for t ≥ to . In the case of a simply supported plate, formulas (72) and (75) yield 

∞  i π xo π/ (ωi to ) sin ωi t − sin (π t/to ) 4π i sin , κ(x , t) = − 2 w¨ o a a (π/to )2 − ωi2

(76)

i =1

 w¨ (x , t) = w¨ o

πt 4π 2 sin + 2 to a to

for t ≤ to and



 

∞ i π x sin ωi t + π/ (ωi to ) sin (π t/to ) D i sin m a (π/to )2 − ωi2 i =1





 ∞ D 1 ωtx sin (ωi to /2) sin ωi t + to/2 , sin m i a (π/to )2 − ωi2 i =1  

 ∞ i π x sin (ωi to /2) sin ωi t + to/2 8π 2 D w¨ o  i sin w¨ (x , t) = − 2 a m to a (π/to )2 − ωi2 8w¨ o κ(x , t) = to

i =1

for t ≥ to .

(77)

(78)

(79)

444 Linear Response of Single-Degree-of-Freedom System to Impact Load 5

WHAT IF THE APPLIED ACCELERATION IS NOT SHORT ENOUGH?

5.1 Constant Acceleration If the constant acceleration is not a very short term one, that is, if its duration to is not very short, formulas (51) and (53) should be used to evaluate the curvature and the acceleration. Considering a simply supported plate and retaining the first term only in formulas (54) and (55), we obtain the following expressions for the maximum curvature and the maximum acceleration (deceleration):

8ma 2 4 κmax = 3 w¨ o , αmax = − 1 + (80) w¨ o . π D π The second formula in (80) indicates that the induced acceleration can be 2.27 fold higher than the applied acceleration, so that the results of such shock tests can be misleading. The maximum curvature, in the case of an instantaneous (or short enough) impulse, can be found on the basis of formula (65). Considering a simply supported plate and retaining the first term only in the expansion (65), we have  4 m κmax = (81) to w¨ o . π D The ratio of the curvature, expressed by the first formula in (80), and the curvature, expressed by formula (81), is  2a 2 m 1 T1 2 χκ = 2 = . (82) = π to D ω1 to π to Hence, the curvature (and, hence, the resulting bending stress) in the case of a constant and not very short applied acceleration can be significantly larger than the curvature in the case of an instantaneous (or short enough) external acceleration. The ratio χκ increases with an increase in the natural period of vibrations. The physics of such a situation is due to the fact that for low-frequency (large-period) vibration systems the maximum value of the acceleration, no matter how large it might be, does not actually matter if the total duration of the acceleration is short: The system simply “does not have enough time” to deflect to its maximum amplitude (which is necessary to respond to the maximum value of acceleration) and responds to the impulse of the applied acceleration. In the case of a durable enough loading (compared to the natural period of vibrations), the system has enough time to respond to the maximum acceleration and “perceives” this acceleration as a statically applied loading. It is noteworthy that in the situation in question low-frequency (“highly compliant”) systems might experience smaller curvatures and smaller accelerations than high-frequency (“stiffened”) structures. This will occur if the load of the given (finite) duration is “quasi-static” for a high-frequency system but is “dynamic” (of a “ballistic” type) for a low-frequency system. Dynamic response results in such a case in lower induced curvatures and in lower accelerations than static response.

5.2 Half-Sine Acceleration Considering the fundamental mode of vibrations only (i = 1) and using formulas (72) and (73) for the forced vibrations during the initial period of time t ≤ to and formulas (74) and (75) for the free vibrations during the time t ≥to , we have

5 What If the Applied Acceleration Is Not Short Enough? 445

 π/ (ω1 to ) sin ω1 t − sin (π t/to ) 4  , w ¨ χ (x ) o 2

π ω12 (π/ω1 to ) − 1  

 sin ω1 t + π/ (ω1 to ) sin (π t/to ) 4 πt w¨ (x , t) = w¨ o sin + χ (x ) 2

to ω1 to π/ (ω1 to ) − 1

κ1 (x , t) = −

for t ≤ to and

 8w¨ o  sin (ω1 to /2) sin ω1 (t + to /2) χ (x ) , [π/(ω1 to )]2 − 1 ω13 to 

sin (ω1 to /2) sin ω1 (t + to /2) 8w¨ o χ (x ) w¨ 1 (x , t) = − ω1 to [π/ (ω1 to )]2 − 1 κ1 (x , t) = −

(83)

(84)

(85) (86)

for t ≥ to . The maximum values of the induced curvature and the acceleration during the initial period of forced vibrations take place at the moment t = to /2 of time, and formulas (83) and (84) yield





π/ (ω1 to ) sin (ω1 to ) /2 − 1 to 4  κmax = κ1 x , w ¨ χ (x ) =− , (87) o 2 [π/ (ω1 to )]2 − 1 π ω12   to sin (ω1 to /2) + π/ (ω1 to ) 4 χ (x ) . (88) w¨ max = w¨ 1 (x , ) = −w¨ o 1 + 2 ω1 to [π/ (ω1 to )2 ] − 1 The maximum value of the curvature, κmax , and the acceleration, αmax , take place when to =

π T1 = . w1 2

(89)

Formulas (87) and (88) yield κmax = −

2 w¨ o χ  (x ), π ω12

αmax → ∞.

(90)

Thus, if the curvature κmax has still a finite value and for a simply supported plate is 2a 2 m (91) w¨ o , π 3D the maximum induced acceleration can be infinitely high if damping is not accounted for. If the time to is short, formulas (85) and (86) yield κmax =

4w¨ o to2  χ (x ) sin ω1 t, (92) π2 4w¨ o t 2 (93) w¨ 1 = − 2 o ω12 χ (x ) sin ω1 t. π In the case of a simply supported plate,

a π 4t 2 κmax = κ1 = 2o w¨ o , (94) , 2 2ω1 a

D a π 4π 2 = 4 to2 w¨ o . , (95) w¨ max = α1 2 2ω1 a m The curvature κmax expressed by formula (94) is significantly smaller than the curvature expressed by the formula (91) while the maximum acceleration has a finite value. κ1 = −

446 Linear Response of Single-Degree-of-Freedom System to Impact Load Thus, we conclude that in order to adequately mimic the drop test conditions, the loading, when a short-term external acceleration is applied, should be made very short, well below the fundamental period of the system’s free vibrations; otherwise the measured response of the system can result in substantially higher curvatures and accelerations than those occurring during drop tests.

6

ENERGY APPROACH The distribution of energy among different modes of vibration and the resulting curvatures and the induced accelerations can be easily assessed based on the following simple reasoning. The initial energy of the i th mode of vibrations caused by a drop can be found from the second condition in (7) as Ei (0) =

1 Mi T˙ i2 = di mgaH , 2

where

 Mi = m

(96)

a

Xi2 (x ) dx

o

is the generalized mass of the i th mode and the factor 2

 a Xi (x ) dx di = o a 2 a o Xi (x ) dx

(97)

(98)

considers the distribution of the initial energy among different vibrations modes. For a simply supported plate, ma , 2

Mi =

di =

8 . (i π )2

(99)

Hence, the initial energy is distributed among the vibration modes inversely proportional to their numbers squared. The following energy balance condition should be fulfilled: ∞ 

Ei (0) = mgaH .

(100)

i =1

The right part of this equation is the initial potential energy. From (96), (99), and (100) we find ∞  π2 1 = , (101) 2 i 8 i =1

which is a well-known relationship [6]. As long as a linear approach is used, different modes of vibrations do not “intermingle,” and the curvature of the i th mode can be found by equating the strain energy of this mode to the initial kinetic energy Ei (0):  1 2 a  2 Xi (x ) dx = di mgaH , (102) DSi 2 o where Si is the amplitude of the i th mode. From (102) we have Si2 =

di 2mgaH  a  2 D X (x ) dx o

i

(103)

7 For a simply supported plate, Si =

4a 2 (i π )3



Probabilistic Approach

2mgH . D

447

(104)

The distributed curvature is therefore

 4 2mgH iπx sin , (105) iπ D a which is consistent with (23). The maximum accelerations of the i th mode can be found as  4i π 2DgH iπx w¨ i (x ) = Si ωi2 = 2 sin , (106) a m a which is consistent with (24). κi (x ) = Si Xi (x ) =

7

PROBABILISTIC APPROACH The actual drop height is never known with certainty, and therefore a probalistic approach could be used to assess the probability that a certain drop height is exceeded and, as a result, a certain level of curvature and/or acceleration is exceeded. Let us assume, for instance, that the drop height is distributed in accordance with the Rayleigh law (see, e.g., [7]): 2 H H exp , (107) fh (H ) = H0 2Ho2 where H0 is the most likely drop height. The probability that the drop height H exceeds a certain value H∗ is

H2 (108) P = P (H > H∗ ) = exp − ∗ 2 . 2Ho Then the probabilites that the given maximum curvature κmax and the maximum acceleration wmax will exceed certain levels κ∗ and a∗ , respectively, is



κ∗ α∗ P = exp − 2 = exp − 2 , (109) 2κo 2αo where, in the case of a simply supported plate,   4 2mgHo 4π 2DgHo κo = , w¨ o = 2 . π D a m From (108) we find the following expression for the effective drop height: √ He = H∗ = Ho −2 ln P

(110)

(111)

In practice, it is either the most likely or the maximum possible drop height, or the allowable stress (curvature), or the allowable acceleration which is critical for the given design or operation conditions. In addition, depending on the consequences of failure and the certainty/uncertainty with which the particular level of curvature and acceleration is defined, the probability P should be established. Then the effective drop height could be found from formula (111). This height can be, of course, considerably larger than the most likely drop height. Indeed, if the effective drop height coincides with the most likely drop height, then the probability P is as high as P = e −1/2 = 0.6065. If, however, He = H∗ = 3Ho , then the probability that this level is exceeded is only P = 0.0111.

448 Linear Response of Single-Degree-of-Freedom System to Impact Load 8

CONCLUSIONS The following conclusions can be drawn from the analysis: • Simple analytical models have been developed for evaluation of the response of a

structural element in a micro-electronic and/or photonic device to a shock load as a result of drop or shock tests. • The developed models were used to find out if shock tests of the given duration and magnitude can adequately mimic the drop test conditions; it has been found that it can be done if the lower (fundamental) frequency of the vulnerable structural element is known and the duration of loading in shock tests is chosen sufficiently short so that loading conditions of the instantaneous impulse type are secured. • A probabilistic approach can be successfully used to determine an effective drop height when drop testing is conducted.

REFERENCES 1. Steinberg, D. S., Vibration Analysis of Electronic Equipment, 2nd ed, Wiley, New York, 1988. 2. Military Handbook , MIL-HDBK-SE, June 1987. 3. Timoshenko, S. P., and Young, D. J., Vibration Problems in Engineering, 3rd ed., Van Nostrand, Princeton, NJ, 1955. 4. Timoshenko, S.P., and Woinowski-Krieger, S., Theory of Plates and Shells, McGraw-Hill, New York, 1940. 5. Den-Hartog, J. P., Mechanical Vibrations, 4th ed., McGraw- Hill, New York, 1956. 6. Gradshteyn, I.S., and Ryzhik, I.M., Tables of Integrals, Sums, Series and Products, Academic New York, 1965. 7. Suhir, E., Applied Probability for Engineers and Scientists, Mc-Graw-Hill, New York, 1997.

CHAPTER

20

SHOCK ISOLATION OF MICROMACHINED DEVICE FOR HIGH-g APPLICATIONS

1

Sang-Hee Yoon

Jin-Eep Roh and Ki Lyug Kim

University of California Berkeley, California

Agency for Defense Development Yuseong, Republic of Korea

INTRODUCTION With rapid advances in micromachining technology, the use of micromachined devices (MMDs) has become more evident in commercial applications where the MMDs are exposed to mild and well-controlled conditions. In recent years, there has been an unceasingly increasing need of MMDs for high–g applications, especially military ones. This trend can be explained by the motive powers of micromachining technology: high performance, low-power consumption, and low price [1–3]. As a new area of MMDs, the high-g application was never achieved before and remains a long-cherished goal. In high-g applications such as car crash recorders [4, 5], shock recorder systems [6, 7], and hard target smart fuse systems [8–11], MMDs are supposed to be exposed to high mechanical shocks with an amplitude range up to 60,000g and a frequency spectrum up to 10 kHz that concentrate significant amounts of mechanical energy into small regions in a short time [12–14]. Direct exposure to mechanical shocks causes structural damages to MMDs or at least threatens their shock survivability in spite of them being miniaturized. The mechanical shocks with a frequency spectrum above 10 kHz also seriously deteriorate a linear operation of any spring–mass MMDs such as sensors and actuators [15]. These problems hinder the use of MMDs in high-g applications. An effective method for isolating MMDs from transient, high-amplitude, and high-frequency mechanical excitations is therefore a need of the times and will be recognized as one of the most remarkable achievements in the design of MMDs [14, 16, 17]. Shock isolation deals with physical phenomena to attenuate mechanical shocks. A shock isolator in its most simple form may be a resilient element connecting equipment with foundation. The main function of the shock isolator is to control unwanted mechanical excitations so that their adverse effects on the equipment are kept within acceptable limit. As a method of isolating MMDs from mechanical excitations, the active shock isolator needs an actuator with feedback control loop and has an external power source that is too bulky to be used for MMDs, although it can be rapidly recovered from incident shocks and has better performance than passive one. Among passive shock isolators, a viscoelastic material-based shock isolator [18–21] has shown low efficiency at low and high temperatures due to the degradation and embrittlememt of the viscoelastic material, thereby being greatly restricted in its applications in high-g conditions [22]. The fluid damping shock isolator [23, 24], which dissipates mechanical shocks through heat and acoustic energy, has also been generally too bulky to be applied in MMDs. Moreover, the fluid damping shock isolator is not suitable for harsh environments. The nonobstructive particle damping (NOPD) shock isolator [25–27], which Structural Dynamics of Electronic and Photonic Systems Copyright © 2011 John Wiley & Sons, Inc.

Edited by Ephraim Suhir, David S. Steinberg and T. X. Yu

449

450 Shock Isolation of Micromachined Device for High-g Applications absorbs mechanical shocks through friction and momentum exchange between particles and the wall, cannot be used for MMDs since making small-diameter holes in MMDs is contrary to reason. It has been reported that the microparticle shock isolator, which uses a microgranular bed inspired from a woodpecker [28], can protect MMDs in high-g conditions and shows considerable promise. Even though the microparticle shock isolator showed significant possibility, the direct application of the existing shock isolators in MMDs was unfeasible or failed to show satisfactory performance [29]. The issue of effective shock isolation for MMDs therefore needs to be solved for high-g applications. To solve the greatest crux in MMD design, many microelectromechanical system (MEMS) researchers have tried to design their devices in two ways: designing MMDs using an over-range mechanical stopper [30–33] and designing MMDs by optimizing their dimension to make the maximum stress less than fracture strength [34–37]. Although the previous methods have achieved the performance improvement to some extent, published results suggest that an additional shock isolator is necessary for MMDs to improve their shock survivability in the high-g environments. To our knowledge, an effective shock isolator compatible with MMDs sensitive to mechanical excitations has not been achieved before and remains a long-cherished goal. The main aim of this chapter is to introduce a fundamental of mechanical vibration system, a shock isolation phenomenon, and the structure and working principle of various shock isolators and then explain a high-g design of MMDs. Lastly, a biomimetic shock isolator inspired from a pileated woodpecker (Dryocopus pileatus) will be discussed.

2

FUNDAMENTALS OF MECHANICAL VIBRATION SYSTEM

2.1 Undamped Free Vibration An undamped free vibration composed of a spring and a single mass that moves vertically is one of the simplest mechanical vibration systems (see Fig. 1a), showing the interactions of mass and stiffness under no exciting force or motion. The mass m is supported by the spring of a spring constant k . The force created by the static deflection xi of the spring supports a weight W (=mg) of the mass. The free-body diagram of the undamped free vibration (see Fig. 1a) also shows a relationship between the weight of the mass and the static deflection of the spring: (1) W = kxi , where xi is the equilibrium position of the mass. If the spring is displaced downward a distance x from xi and released, the undamped free vibration system will oscillate up and down. The spring force Fs is written as Fs = −kx = m x¨ ,

(2)

where x¨ is the second derivative of the displacement x of the mass with respect to time t, that is, the acceleration of the mass. The equation of motion can therefore be expressed as m

d 2x + kx = 0. dt2

(3)

The motion of the mass is periodic in time and can be represented by x = x0 cos(ωt),

(4)

2 Fundamentals of Mechanical Vibration System

kx

k

m

m

m +x

+x

m

(b)

kx

m F(t)

F(t) (c)

m +x

F(t)

. cx

kx

c

k

m +x

+x

+x

(a)

k

. cx

kx

c

k

451

m +x

F(t)

+x

(d)

Figure 1 Mechanical vibration system (left) and its free-body diagram (right): (a) undamped free vibration system; (b) damped free vibration system; (c) undamped forced vibration system; (d ) damped forced vibration system. where x0 is the initial displacement of the mass and ω is the frequency of the oscillation. The first and second derivatives of (4) show the velocity and acceleration of the mass, respectively: dx (5) = −ωx0 sin(ωt), dt d 2x x¨ = 2 = −ω2 x0 cos(ωt). (6) dt Inserting (4) into (3) yields the natural frequency of the mass ωn ,  k ωn = . (7) m In theory, the undamped free vibration system vibrates incessantly. All free vibrations, however, get attenuated after time due to damping. The time and frequency responses of the undamped free vibration system are shown in Figs. 2 and 3, respectively. x˙ =

2.2 Damped Free Vibration A damping element is usually added to the spring–mass system in the mechanical vibration system. This system is the same with the undamped free vibration system except for a viscous damping element such as an air damper and an oil damper. The system therefore vibrates under the inertia force by the mass, the spring force by the spring, and the damping force by the damper. The dampers which continuously decrease the velocity of the mass result in

452 Shock Isolation of Micromachined Device for High-g Applications . tan−1(x0)

2π/ωn

x0

Amplitude, x(t)

2π/ωd

Time, t

Undamped (ζ = 0) Underdamped (ζ < 1) Critically damped (ζ = 1) Overdamped (ζ > 1)

Figure 2

Time responses of mechanical vibration system.

the oscillating motion of the mass. To describe the damping force which opposes the motion, a viscous damping model is widely used where the damping force is proportional to the damping coefficient c and the velocity of the mass. The free-body diagram of the damped free vibration (see Fig. 1b) shows the relationship among the mass, spring, and damper. In the damped free vibration system, there are three forces of the weight directed downward, mg; the viscous damping force acting opposite to the direction of the displacement, c x˙ ; and the spring force acting opposite to the direction of the displacement, kx. Based on Newton’s second law, the equation of motion can be described by m

d 2x dx + c + kx = 0. dt dt2

Using the natural frequency √ of the undamped free vibration (ωn = the damping ratio (ς = c/(2 km)), (8) can be rewritten as

(8) 

dx d 2x + 2ζ ωn + ωn2 x = 0. 2 dt dt The general solution of (9) is given by     √ √ 2 2 x (t) = Ae −ζ + ζ −1 ωn t + Be −ζ − ζ −1 ωn t ,

k /m) and introducing

(9)

(10)

where A and B are arbitrary constants to be determined from the initial conditions. The behavior of the solution depends on the magnitude of the damping. There are three different conditions of the damping: underdamping, critical damping, and overdamping. In the underdamping system where ζ < 1, (10) is written as   1 − ζ 2 ωn t + ϕ , (11) x (t) = Ce−ζ ωn t cos where C and ϕ are arbitrary constants to be determined from the initial conditions. The vibration of this system diminishes exponentially with time, known as aperiodic. In the critical damping system where ζ = 1, (10) is expressed as x (t) = (D + Et)e −ωn t ,

(12)

2 Fundamentals of Mechanical Vibration System 5.0

ζ = 0, no damping ζ = 0.1 ζ = 0.2 ζ = 0.3 ζ = 0.5 ζ = 1.0, critical damping

4.5 4.0 Amplification ratio, Xk/F0

453

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.0

0.5

1.0 1.5 Frequency ratio, ω/ωn

2.0

2.5

(a) 180 160

Phase angle, φ (degree)

140 120 100 80 ζ = 0, no damping ζ = 0.1 ζ = 0.2 ζ = 0.3 ζ = 0.5 ζ = 1.0, critical damping

60 40 20 0 0.0

0.5

1.0

1.5

2.0

2.5

Frequency ratio, ω /ωn (b)

Figure 3 Frequency response of forced vibration system: (a) amplification ratio as a function of frequency ratio; (b) phase angle as a function of frequency ratio.

where D and E are arbitrary constants to be determined from the initial conditions. The motion diminishes eventually to zero, which means it is aperiodic. In the overdamping system where ζ > 1, the solution is the same with (10). The motion of this system is also diminished exponentially with time (aperiodic), regardless of the initial conditions imposed on the system. The time and frequency responses of the damped free vibration system are shown in Figs. 2 and 3, respectively.

454 Shock Isolation of Micromachined Device for High-g Applications 2.3 Undamped Forced Vibration The motion of the vibration system with any exciting force or motion is called forced vibration, whereas that of the vibration system without exciting force or motion is called free vibration, which illustrates how a system’s natural frequency depends on the mass, the spring constant, and the damping coefficient. In a undamped forced vibration (see Fig. 1c), the spring–mass system is excited by a periodic variation of external forces at any frequency, F0 sin(ωt), where F0 and ω are the amplitude and frequency of the external force, respectively. The equation of motion of the undamped forced vibration is represented by d 2x + kx = F0 sin(ωt). (13) dt2 Since the mass is driven by the function F0 sin(ωt), a particular solution is logically assumed to be x0 sin(ωt). Substituting it into (13) and performing mathematical manipulations yield xst sin(ωn t), (14) x (t) = A1 sin(ωn t) + B1 cos(ωn t) + 1 − (ω/ωn ) where xst is the static spring deflection under constant load F0 and A1 and B1 are arbitrary constants to be determined from the initial conditions. The motion of the undamped forced vibration, composed of two sine waves of different frequencies, is not a harmonic motion. The time and frequency responses of the undamped forced vibration system are shown in Figs. 2 and 3, respectively. m

2.4 Damped Forced Vibration In a damped forced vibration system (see Fig. 1d ), the motion of the system contains two motions: the motion of the damped free vibration at the damped natural frequency and the steady-state harmonic motion at the forcing frequency. The first vibration decays quickly, while the second motion associated with the external force remains as long as the energy is present. For the damped forced vibration system compared with the undamped one, the equation of motion is equal to F0 sin(ωt): d 2x dx + c + kx = F0 sin(ωt). (15) 2 dt dt Since the mass is driven by the function F0 sin(ωt), a particular solution can be logically assumed as x0 sin(ωt − ϕ), where ϕ is the phase angle. Substituting it into (15) and performing mathematical manipulations yield two equations: m

kx0 − mω2 x0 − F0 cos ϕ = 0,

(16)

cωx0 − F0 sin ϕ = 0.

(17)

Solving (16) and (17) provides the expressions for x0 and ϕ: F0 , (18) x0 =  2 (cω) + (k − mω2 )2 cω ϕ = tan−1 . (19) k − mω2 The time and frequency responses of the damped forced vibration system are shown in Figs. 2 and 3, respectively. In this the undamped   vibration, there are three different frequencies: 2 2 natural  frequency ωn = k /m; the damped natural frequency ωd = k /m − c /(4m ) = 2 ωn 1 − ς ; and the frequency of maximum forced amplitude, that is, the resonant frequency.

3 Shock Isolation

3

455

SHOCK ISOLATION An object or a dynamic system is oscillating about an equilibrium position under mechanical shocks. The oscillating motion is either periodic or transient. For any object or dynamic system, there always exists energy dissipation in motion, which is termed damping. The main source of the damping is an interaction of the moving object with its surrounding environments such as air, fluid, and rough surface. The incident mechanical energy of shocks is irreversibly converted into heat and acoustic energy. The presence of the damping makes the amplitude of free vibration delay and leads to the reduction in amplitude of forced vibration. The general damping, however, is not enough for MMDs to survive under high-g environments. Thus, it is essential to carry out the research on a shock isolator for MMDs to secure the shock survivability of the MMDs in the high-g conditions. The fundamental concept of the shock isolator is shown in Fig. 4. The simplest vibration model with single degree of freedom consists of a mass or an inertia element representing MMDs connected to a foundation by an isolator having elastic and energy-dissipating elements. There are two types of shock isolators: a shock attenuator, which dissipates mechanical shocks over all frequency spectrums, and a shock low-pass filter, which passes the lowfrequency mechanical shocks but attenuates high-frequency ones. The performance of the shock attenuator is generally evaluated by absolute transmissibility, relative transmissibility, and motion response [38]. Two transmissibilities are the characteristics to express the attenuation of the transmitted motion or force caused by the shock isolator. The absolute transmissibility is the ratio of the amplitude of MMDs to that of the foundation when the source of vibration is an oscillating motion of the foundation that is a common situation in high-g applications (see Fig. 4a). If the source is an oscillating force originating from MMDs (see Fig. 4b), the absolute transmissibility represents the ratio of the force amplitude transmitted to the foundation to that of the exciting force. The second one is the relative transmissibility. For the shock isolator which dissipates incident mechanical shocks by deflecting itself, the relative transmissibility is represented as the ratio of the relative deflection amplitude of the isolator to that imposed at the foundation. This characteristic is therefore significant only in the shock isolator which attenuates the mechanical shocks transmitted from the foundation. Last but not least, the motion response is the third characteristic, expressed as x /(F0 /k ), where x is the

F = F0 sin(wt) m

x

MMD

m

x

S/V isolator

u = u0 sin(wt) Foundation (a)

Ft (b)

Figure 4 Schematic diagram of shock isolation systems: (a) shock isolation where motion u is imposed at the foundation and motion x is transmitted to MMDs; (b) shock isolation where force F is applied to (or by) MMDs and force Ft is transmitted to the foundation [38].

456 Shock Isolation of Micromachined Device for High-g Applications displacement amplitude of MMDs, F0 is the excitation force amplitude, and k is the static stiffness of the shock isolator. In addition, the space requirement for the shock isolator is determined by the motion response of MMDs when a exciting force is transmitted into the MMDs. The performance of the shock low-pass filter is generally evaluated by cutoff frequency, vibration absorption, and rolloff steepness [14]. The shock low-pass filter is similar to an electrical low-pass filter that attenuates electrical signals with higher frequency than the cutoff frequency. The cutoff frequency is defined as the frequency at which attenuation starts, that is, the frequency at which the transmitted mechanical shocks are 0.707 of the input ones. The cutoff frequency is therefore very important in expressing the operation range of the shock low-pass filter. Vibration absorption is defined as 1 − (E0 /Ei ), where E0 and Ei are the energies of the output and input of mechanical shocks, respectively, and is a key parameter to show the efficiency of the shock isolator. The last characteristic is the rolloff steepness, which is the rate of attenuation as measured in decibels per octave, thereby determining the selectivity of the shock isolator.

4

CLASSIFICATION OF SHOCK ISOLATOR The shock isolator can be classified into active and passive ones according to the existence of external power source. The active shock isolator is a control unit that reacts to impact disturbances, thereby mitigating the impact transmission into the protected equipment [39]. The passive isolator is an impact-absorbing unit without external power source and feedback control loop. According to the operation characteristics of the shock isolator, it is either a shock attenuator or a shock low-pass filter. According to its material, the shock isolator is also sorted into viscoelastic material-based isolator, fluid damping isolator, NOPD isolator, and woodpecker-inspired microparticle isolator. The shock isolator can be classified as an external shock isolator and an internal shock isolator, as shown in Fig. 5, according to the configuration of shock isolator and MMD. The external shock isolator (see Fig. 5a) has its springs and dampers placed out of the MMD, while the internal one (see Fig. 5b) has them in the MMD. Figure 6 shows the classification of the shock isolator.

4.1 Classification of Shock Isolator Based on Existence of External Power Source The shock isolator for MMDs can be classified as an active shock isolator, which needs an actuator with a feedback control loop and an external power source, and a passive shock isolator, which dissipates incident mechanical shocks to protect MMDs. The passive one consists of springs and dampers (or dashpots). The springs are intended to soften the incident shocks and the dampers terminate the oscillation which is excited in the system. The active one generally depends on an accelerometer and an electrical actuator, which allow a higher degree of shock isolation. Although the active shock isolator, due to its rapid recovery from incident mechanical shocks, shows better performance than passive one, it is too bulky to be used in MMDs. The active shock isolator is therefore not considered in this chapter. Passive shock isolators are commonly classified according to its working mechanism or material into viscoelastic material-based shock isolator, fluid damping shock isolator, NOPD shock isolator, and microparticle isolator.

4.2 Classification of Shock Isolator Based on Combination of Spring and Damper The fundamental functions of the shock isolator are load supporting and energy dissipating. For example, in a shock isolator made of natural rubber, two functions may be performed

4 Classification of Shock Isolator

457

External isolation

MMD

Core element (a) Internal isolation

MMD Core element

(b)

Figure 5 Two kinds of shock isolators for micromachined devices: (a) external shock isolator; (b) internal shock isolator. POWERSOURCE

OPERATION

MATERIAL

Shock isolator over all frequency Active Shock isolator for high frequency Viscoelasticity

Shock isolator Shock isolator over all frequency Passive

Fluid damping NOPD Etc.

Shock isolator for high frequency

Microparticle

Figure 6 Classification of shock isolators. by a single element. However, in the other shock isolator, the load-supporting element is separated from the energy-dissipating one. The shock isolator composed of metal springs may usually may not dissipate the incident energy; then extra energy-dissipating element (damper) is provided. In this chapter, for the purposes of classification, the spring and damper are assumed as separate elements.

458

cc = 2 km ς = c/cc

√

 ω0 = k /m (c = 0) k /m (Ff = 0)

η = Ff /(ku0 ) ξ = Ff /(m u¨ 0 ) ξf = Ff /F0

ω=



Ft = (Ft )0 sin(ωt + θ)

Ff

Ft = (Ft )0 sin(ωt + θ)

Ft

x = x0 sin(ωt + θ) or δ = δ0 sin(ωt + θ) where δ = x − u

u

k

x = x0 sin(ωt + θ) or δ = δ0 sin(ωt + θ) where δ = x − u

c

x

u = u0 sin(ωt) F = F0 sin(ωt)

Ft

m

F

Shock Isolator Using Rigidly Connected Coulomb Damper

u = u0 sin(ωt) F = F0 sin(ωt)

k

m

F x

Type of Shock Isolators

Shock Isolator Using Rigidly Connected Viscous Damper

Table 1

ks = NK

Ft = (Ft )0 sin(ωt + θ)

x = x0 sin(ωt + θ) or δ = δ0 sin(ωt + θ) where δ = x − u

Response

u = u0 sin(ωt) F = F0 sin(ωt)

Excitation

Ft

c

x

u

Damping Parameters √ cc = 2 km ς = c/cc

Frequency Parameters  ω = k /m (c = 0)  ω∞ = (N + 1)(k /m) (c = ∞)

u

k

m

F

Shock Isolator Using Elastically Connected Viscous Damper

Ft

Ff Ks = Nk

x

u

η = Ff /(ku0 ) ξ = Ff /(m u¨ 0 ) ξf = Ff /F0

 ω = k /m (Ff = 0)  ω∞ = (N + 1)(k /m) (Ff = ∞)

Ft = (Ft )0 sin(ωt + θ)

x = x0 sin(ωt + θ) or δ = δ0 sin(ωt + θ) where δ = x − u

u = u0 sin(ωt) F = F0 sin(ωt)

k

m

F

Shock Isolator Using Elastically Connected Coulomb Damper

4 Classification of Shock Isolator

459

Based on various types of dampers in combination with ideal springs, the shock isolator is generally classified as rigidly connected viscous shock isolator, rigidly connected Coulomb shock isolator, elastically connected viscous shock isolator, and elastically connected Coulomb shock isolator, as illustrated in Table 1. These shock isolators are evaluated with a single-degree-of-freedom concept where the foundation is assumed rigid and the isolator is assumed massless. In the rigidly connected viscous shock isolator, a viscous damper with damping coefficient c and a load-carrying spring with stiffness k are connected rigidly between the core element (MMD) and the foundation (case or housing). The viscous damper has the mechanical characteristic of transmitting the force Fc that is proportional to the rela˙ This isolator is generally called a linear isolator. The tive velocity δ˙ across itself, Fc = c δ. performance of the rigidly connected viscous shock isolator is studied in terms of absolute transmissibility Ta , relative transmissibility Tr , and motion response. The absolute transmissibility of this shock isolator, a ratio of the absolute displacement amplitude across the damper to the displacement imposed at the foundation x0 /u0 , is given by   ω 2 1 + 2ς x0 Ft ω0 Ta = = = (20)

2  , u0 F0 2 ω 2

1− ω + 2ς ω0 ω02 where Ft and F0 are the excited (transmitted) and response forces, respectively. The relative transmissibility of this isolator, a ratio of the relative displacement amplitude across the damper to the displacement imposed at the foundation δ0 /u0 , is written as   4 ω δ0 ω 0 Tr = = (21) 2  . u0 2 ω 2

1− ω + 2ς ω0 ω02 Its motion response x /(F0 /k ) can be expressed as  x0 1 = 2  2 . F0 /k 2 ω ω

1− 2 + 2ς ω0 ω0

(22)

The rigidly connected Coulomb shock isolator is an isolation system with a rigidly connected Coulomb damper (see the schematic in Table 1). In this shock isolator, a force Ff exerted by the damper on its mass is constant independent of position or velocity but is always in the opposite direction to the relative velocity across the damper. A typical example of the Coulomb damping is the relative motion of two elements arranged to slide one on the other with a constant force holding them together. The absolute transmissibility, relative transmissibility, and motion response of the rigidly connected Coulomb shock isolator are represented, respectively, as    2 ω02 1 + 4 η 1−2 2 π ω x0 Ta = = , (23)

2 u0 2 ω

1− 2 ω0

460 Shock Isolation of Micromachined Device for High-g Applications   4  2 ω 4 ω0 − π η δ0 Tr = = , 2 u0 ω2

1− 2 ω0   2 4 1 − ξ x0 π =

2 . F0 /k 2

1− ω ω02

(24)

(25)

The elastically connected viscous shock isolator consists of one viscous damper with damping coefficient c and two springs with stiffnesses ks and k , respectively (see the schematic in Table 1). The first spring with stiffness ks is connected in series with the damper and the second load-carrying spring with stiffness k is connected in parallel with the damper. The second load-carrying spring is interacted with the first spring by the parameter N = ks /k . This isolator is generally called a viscous relaxation isolator. The absolute transmissibility, relative transmissibility, and motion response of the elastically connected viscous shock isolator are given, respectively, as   N + 1 2 2 ω2 1 + 4 ς 2 N ω0 x0 Ft Ta = = = 2

2 , u0 F0 ω2 ω2 4 2 ω2

1− 2 + 2ς 2 N +1− 2 N ω0 ω0 ω0

(26)

 ω2 ω6 4 + 2 ς2 6 2 N ω0 ω0 δ0 Tr = =

2

2 , u0 2 ω2 4 2 ω2

1− ω + 2ς 2 N +1− 2 N ω02 ω0 ω0

(27)

 ω2 4 1 + 2 ς2 2 N ω0 x0 = 2

2 . F0 /k 2 2 2 ω 4 ω ω

1− + 2 ς2 2 N + 1 − 2 N ω02 ω0 ω0

(28)

Last of all, the elastically connected Coulomb shock isolator is equipped with an elastically connected Coulomb damper, as shown in the schematic of Table 1. The friction element can only transmit the force developed in the damper and the spring with stiffness ks . Since this is a Coulomb damper, a force Ff exerted by the damper on its mass is independent of the position or velocity across the damper, but always opposite to the motion. The absolute transmissibility

4 Classification of Shock Isolator

461

and relative transmissibility of this shock isolator can be written, respectively, as   2    4 N + 1  ω0 2 N +2 1 + η − 2 x0 π N N ω Ta = = ,

2 u0 ω2

1− 2 ω0      ω0 4  4 2 2 ω2  N + 2 0 + − η ω π N ω2 N δ0 = . Tr =

2 u0 ω2

1− 2 ω0

(29)

(30)

4.3 Classification of Shock Isolator Based on Working Mechanism (or Material) All mechanical systems have their own damping, which is evidenced by the fact that there is no perpetual motion. The damping can be significantly modulated by changing the material and structure because all materials and structures have noticeably different damping characteristics. This is the reason why our engineers try to optimize the shock isolation with changing material and structure. One example is an aircraft cabin whose floor is covered with a carpet and the ceiling and wall are coated with acoustic material to minimize the external noise from engines and flight. The discussion below highlights various shock isolations with different working mechanisms or materials which have been used until now and are being developed for future use. 4.3.1 Viscoelastic Material-Based Shock Isolator The viscoelastic material-based shock isolator shown in Fig. 7a attenuates incident mechanical shocks from external or internal environments through a viscoelastic material such as rubber, polymer, urethane, epoxy, and enamel. The viscoelastic material dissipates mechanical shocks as heat when it is stressed by deformation, especially shearing ones. This is because the viscoelastic material has a low shear modulus of 100 kPa–100 MPa but high loss factor of 2 or more. Its material properties (e.g., shear modulus, loss factor) are, however, inherently temperature and frequency dependent. For instance, it shows significant degradation in efficiency under low and high temperatures due to its degradation and embrittlememt, as shown in Fig. 7b. The viscoelastic material-based shock isolator for MMDs therefore can show an excellent efficiency only at moderate temperature and frequency. In this shock isolator, the viscoelastic material is implemented with a constraint layer because the constraint layer application, in general, yields the most effective shock isolation. Without the constraint layer, the viscoelastic material should have high stiffness although it is good for extensional and bending modes. These shortcomings cause serious limitations in using the viscoelastic material-based shock isolator for MMDs in high-g environments. 4.3.2 Fluid Damping Shock Isolator The viscous fluid damping shock isolator induces fluid flow through a well-designed orifice or annulus to dissipate incident mechanical shocks, as shown in Fig. 8. In general, the amount

462 Shock Isolation of Micromachined Device for High-g Applications

k1

c1

k2 c2

Viscoelastic material (a) 100

FRF amplitude (g/N)

no damp

–20°C

0°C

22°C

10

1

0.1

0.01 0

500

1000

1500

2000 2500 3000 Frequency (Hz) (b)

3500

4000

4500

5000

Figure 7 Viscoelastic material-based shock isolator: (a) schematic (left) and its equivalent electrical circuit (right); (b) amplification response as a function of frequency at −20◦ , 0◦ , and 22◦ C, compared to that with no damping. (Reproduced from [40] with permission of the publisher).

Piston rod

Piston head

Fluid

Annular orifice

Figure 8

Viscous fluid damping shock isolator.

of energy damped by the viscous fluid damping shock isolator is proportional to the velocity of the fluid movement. An automobile shock absorber is a representative example of this shock isolator. The viscous fluids frequently used in this shock isolator are silicon oil, grease, and so on. The viscous fluid damping shock isolator is not suitable for shock isolation of MMDs because it has several inherent problems: The viscous fluids are very sensitive to temperature change and the shock isolator is too bulky.

4 Classification of Shock Isolator

463

4.3.3 Magnetic Damping Shock Isolator A magnetic damping shock isolator depends on an eddy current, also known as a Foucault current, in a moving conductor to dissipate incident mechanical shocks. When the conductor is exposed to a changing magnetic field due to the relative motion between field source and conductor or the variation of the field with time, the eddy current causing repulsive or drag forces between the conductor and the magnet is induced, thereby achieving the dissipation of incident mechanical shocks. The magnetic damping shock isolator excels other passive shock isolators in many aspects: compact design but large damping capacity, construction of great simplicity and robustness, and use of common materials which have linear viscous damping characteristics and are almost insensitive to temperature change. Although this shock isolator is widely used in tuned-mass damper applications, there are some critical limitations in applying this shock isolator into MMDs. For example, MMDs should be made of conductive material and their operation should be insensitive to the magnetic field generated by the magnet. 4.3.4 Nonobstructive Particle Damping Shock Isolator A NOPD shock isolator dissipates incident mechanical shocks through the friction caused by the motion of particles within a closed hole and the deformation of the particles. Since the hole is partially filled with the particles, as shown in Fig. 9a, incident mechanical shocks are absorbed due to the friction and momentum exchange between the particles and the wall of the hole in the NOPD shock isolator. This shock isolator is therefore a shock attenuator which damps incident mechanical shocks of all frequency spectrums. Figure 9b shows the measured amplitude response as a function of frequency with and without NOPD shock isolator [41]. Although this shock isolator provides effective damping, it cannot be used for MMDs due to the following drawbacks: first, making several small-diameter holes in an MMD is contrary to reason; second, the particles for shock dissipation add weight to the MMD, which may be undesirable; third, the shock isolating behaviors are dependent on the amplitude of incident mechanical shocks and on the direction of the isolator in the mechanical shock field; last but not least, this shock isolator should be designed in an empirical way since it is very difficult to investigate its effectiveness in an analytic way. 4.3.5 Smart Material-Based Shock Isolator Many “smart” materials such as piezoelectric material, magnetorheological (MR) fluid, electrorheological (ER) fluid, and magnetostrictive and electrostrictive, material. are usually used to dissipate incident mechanical shocks. These materials undergo a controlled transformation through physical interaction when they are exposed to the shock field. Piezoelectric material creates electrical charge under mechanical stress and conversely experiences its mechanical deformation under an electrical field. An oscillating electric field also makes the piezoelectric material resonate at its natural frequency, thus adding an extra mode. The mechanical stress of the piezoelectric material generated by incident mechanical shocks is converted into heat, which is dissipated to damp the incident shocks. However, it is virtually impossible to integrate the piezoelectric material with MMDs. Thus, to apply the piezoelectric material-based shock isolator into MMDs for high-g applications, it needs to be improved technically. The MR and ER fluids undergo a phase change from liquid to semi solid when exposed to a magnetic or electric field, respectively. Magnetostrictive and electrostrictive materials experience an elastic strain when subjected to a magnetic or electric field, respectively. Based on the phase change or elastic strain generated under mechanical shocks, the MR and ER fluids can dissipate incident mechanical shocks. These materials, however, also have technical restrictions in their applications in MMDs.

464 Shock Isolation of Micromachined Device for High-g Applications

Cavity

Particle

(a) 50 Without With

40

Amplitude (dB)

30 20 10 0

−10 −20 −30

0

2000

4000

6000 8000 Frequency (Hz) (b)

10000

12000

Figure 9 A NOPD shock isolator: (a) cross-sectional schematic; (b) amplitude response as a function of frequency with and without NOPD shock isolator. (Reproduced from [41] with permission of the publisher). 4.3.6 Woodpecker-Inspired Microparticle Shock Isolator This passive shock isolator is inspired from the shock-isolating mechanisms of a woodpecker, shown in Fig. 10a. The woodpecker-inspired microparticle shock isolator [28] shown in Fig. 10b consists of a hard case made of metal and a microgranular bed. The metal hard case provides space for the microgranular bed and prevents an internal free flow of the microgranular bed within itself while protecting the microgranular bed and the embedded

4 Classification of Shock Isolator A

465

A´ B

B´

A-A´ cross section (a)

B-B´ cross section (b)

Figure 10 Woodpecker-inspired microparticle shock isolator. (a) Pileated woodpecker (top) and its skull cut (bottom) showing the woodpecker’s unique mechanism to prevent brain damage using its skull and spongy bone from drumming shocks. (Reproduced from [42] with permission of the publisher.). (b) Microparticle shock isolator using microgranular bed (microglass bead size is exaggerated for clarity, top) and its cross-sectional view (bottom) to protect micromachined devices from incident mechanical shocks.

MMDs from incident mechanical shocks. The microgranular bed has a large number of air gaps and also packs tightly the MMDs which are supposed to be exposed to incident mechanical shocks. It absorbs the high-frequency shocks and makes the transmitted shocks detour. The microgranular bed composed of numerous close-packed microglass beads absorbs the shortduration mechanical shocks in a kinetic way. This physical phenomenon is involved with mechanical energy dissipation through particle-to-particle collision and particle-to-particle or particle-to-wall friction. The microgranular bed therefore works as a shock low-pass filter, similar to the spongy bone of the woodpecker. The woodpecker-inspired microparticle shock isolator has three new features: First, the microgranular bed of this isolator is biomimetically inspired from the shock isolating mechanisms of the woodpecker, fully verified in nature; second, these mechanisms are compatible with MMDs sensitive to incident mechanical shocks without design modification because the only critical design parameter of this isolator is a diameter of the microglass beads. Last but not least, these mechanisms are inexpensive approaches using microglass beads but very effective in isolating transmitted mechanical shocks. As stated above, there are many possible methods for adding passive damping to MMDs which are sensitive to transmitted mechanical shocks under high-g environments. Although all previous passive shock isolators show satisfactory performance in damping incident mechanical shocks, no passive shock isolator can be applied to protect MMDs from incident mechanical shocks except the woodpecker-inspired microparticle shock isolator. Thus, the woodpecker-inspired microparticle shock isolator will be discussed in detail.

466 Shock Isolation of Micromachined Device for High-g Applications 5

DYNAMIC RESPONSE OF MICROMACHINED DEVICE UNDER SHOCK AND VIBRATION The dynamic response of MMDs, which have micromachined structures composed of mass, spring, and damping, is similar to that of the macrosized mechanical system in its dynamic behavioral pattern. In general, MMDs have a resonant frequency in the range of kilohertz because their mechanical structures have very small mass but relatively high stiffness. Dynamic response in micromachined structures (e.g., MMDs) is also significantly influenced by the electrostatic field and interlaminar stress, which are associated with the working principle or fabrication processes of the microfabricated devices. Although these factors may considerably influence the dynamic behavior of MMDs, their dynamic response is mainly dominated by mechanical mass (capacitance), mechanical spring (inductance), and mechanical damping (resistance). A lump model of MMDs based on equivalent mass, spring, and damping is therefore usually used to understand the dynamic response of MMDs under mechanical shocks. A brief summary of the equivalent mass, spring, and damping of MMDs will be given below.

5.1 Equivalent Mass in Micromachined Device A mass (equal to capacitance, fluid capacitance, and thermal mass in electrical, fluid, and thermal domains, respectively) can be estimated assuming the structural geometry and density of MMDs. The microfabricated MMD usually has a little different shape than the designed one because the microfabrication technology shows inevitable limitations and errors in making MMDs. For example, in a wet etching process, an etchant etches even a structural layer at lower extent the moment it removes a sacrificial layer at higher rate, thereby yielding some distortions in the fabricated MMDs. Besides, the inner and outer walls differ in their degree of contact with the etchant. The inner wall is gradually exposed to the etchant, whereas the outer wall is constantly exposed. The inner wall of the structure is therefore sloped after the wet etching process. The density also needs to be considered because a thin film has slightly smaller density than a bulk structure made of the same material in certain conditions. To characterize the dynamic structural behavior of MMDs under mechanical shocks, an equivalent mass approach for concentrated and distributed masses is very effective [43]. Table 2 gives formulas of the equivalent mass of a cantilever and double-supported beams Table 2

Equivalent Mass of Structure in Micromachined Devices

Condition

Schematic

meq = M + 0.23m

Cantilever beam of mass m with mass M at end (massless)

Cantilever beam with three masses (M 1 , M 2 , M 3 )

m

M

l3

meq = M1 +

l2 l1 M3

Simply supported beam of mass m with mass M at middle

Equivalent Mass

M2

 2  2 l2 l3 M2 + M3 l1 l1

M1

meq = M + 0.5m m

M

5

Dynamic Response of Micromachined Device under Shock and Vibration

467

loaded with concentrated weight(s). These formulas can cover most MMD structures, which are simply supported or double-supported beams.

5.2 Equivalent Stiffness in Micromachined Device Stiffness (equal to inductance, fluid inertia, and thermal inertance in electrical, fluid, and thermal domains, respectively) is one of the most important parameters in evaluating the dynamic behavior of MMDs under mechanical shocks [43]. Assuming the microstructure of MMDs is uniform in geometry, the stiffness of the microstructure under a concentrated load km can be written as the ratio of applied load F to deflection δ, such as for a macrosized helical spring: dF km = for single concentrated force, (31) dδ dT km, θ = for single concentrated torque, (32) dθ where km,θ is the torsional stiffness of the microstructure when it experiences the angular deflection θ under a single concentrated torque T . These formulas are available not for distributed load but for concentrated load. For the microstructure subjected to distributed load, an equivalent loading method is usually used to calculate its equivalent stiffness. Table 3 gives formulas of the equivalent stiffness of micromachined beams with a concentrated or distributed load. When multiple concentrated loads are applied the microstructure, an energy method may be employed to evaluate the equivalent stiffness. The strain energy stored in the microstructure is the same as the equivalent energy stored in an ideal spring. For bending loads,   M 2 dx keq δ 2 b = , (33) 2 l 2EI where Mb is the equivalent bending moment of each structure of length l subjected to multiple concentrated loads; E , I , keq , and δ are the Young’s modulus, moment of inertia, equivalent bending stiffness, and equivalent deflection of the microstructure, respectively. For torsion,   M 2 dx keq, θ θ 2 t , (34) = 2 l 2GIp where Mt is the equivalent torsional moment of each structure of length l subjected to multiple concentrated torsions; G, Ip , keq,θ , and θ are the shear modulus, polar moment of inertia, equivalent torsional stiffness, and equivalent angle of the microstructure, respectively.

5.3 Equivalent Damping in Micromachined Device A damping (equal to resistance, fluid resistance, and thermal resistance in electrical, fluid, and thermal domains, respectively) is the most difficult part to evaluate in the microstructure of MMDs. The main factors which determine the damping are the structural geometry and the viscosity and pressure of a neighboring fluid. The microstructure of MMDs in general has a remarkably large value of damping coefficient due to the squeeze damping in the micro sized geometry of the flexural parts and the reduced gap between the released element and substrate. The damping phenomenon determines primarily the shock survivability of the microstructure. A large value of the squeeze damping, which is common in the microstructure, can dramatically increase the shock survivability of the microstructure. The large squeeze damping, however, reduces the dynamic working bandwidth of MMDs to unacceptable levels [43]. For example, in piezoelectric actuators and acoustic sensors, based on their oscillation,

468 Shock Isolation of Micromachined Device for High-g Applications Table 3

Equivalent Stiffness of Structure in Micromachined Devices

Schematic

Equivalent Stiffness l

keq =

6EI 2l 3 − 3l 2 a − a 3

keq =

3EI(l + 2a)2 2(l − a)2 a 3

keq =

48EI a

keq =

384EI 5l 3

keq =

8EI l3

F

a l

F

a l



1 3l 2 − 4a 2



F

a l q

0.5 l

l q

l

a very low intrinsic damping property of the material (quartz, LiNbO3 , ZnO, AlN, PZT, etc.) is required to ensure high efficiency of the devices by minimizing energy dissipation. Damping is of great concern for MMDs exposed to mechanical shocks. The shock survivability of MMDs can be improved by increasing the damping by the following methods: configuring no fluid flow around the microstructure; employing high-viscosity fluid; and applying high-pressure encapsulation. To optimize the dynamic structural behavior of MMDs under

5

Dynamic Response of Micromachined Device under Shock and Vibration

Table 4

469

Equivalent Damping of Structure in Micromachined Devices

Condition

Schematic

Relative motion of two parallel surfaces with frictional coefficient μ

Equivalent Mass

A

ceq =

μA h

ceq =

4μN π ωX

∗ = ceq

4μl 3 h3

ceq =

π V0 μ ωh

h

m

Vibrating mass contacting with fixed surface (Coulomb friction)

X ω

N

Squeeze film damping

m 2l

h

m

Couette-type damping

V0 cos(wt)

h

m

∗ Equivalent

damping per unit of length

mechanical shocks, information on the equivalent damping value for the microstructure of MMDs is required. Table 4 gives some general formulas of the equivalent damping of various micro sized mechanical structures.

5.4 Design of Micromachined Device for High-g Applications Over the past several years, low-cost and micromachined MMDs have steadily emerged in the marketplace. Some of these developmental MMDs are being considered for high-g applications. An effective method for isolating MMDs from external shock disturbances is therefore required to improve their performance and reliability even in harsh environments. The MMDs that depend on micromachined structures are susceptible to incident mechanical

470 Shock Isolation of Micromachined Device for High-g Applications shocks since the shocks can cause ultimate damage on their structure and degrade their performance [44]. Previous studies on MMD design for high-g applications have focused on shock survivability based on failure theory. Tanner et al. [34] have carried out high-g shock tests of up to 40000g to discuss the fracture strength of microsized cantilevers made of silicon, thereby demonstrating that the shock survivability of the microstructure is primarily determined by structural dimensions and material properties. Chu [15] has reported that the micromachined piezoresistive accelerometer manufactured by Endevco Corporation can survive under mechanical shocks with a mechanical filter made of viscoelastic material. To develop MMDs with high-shock survivability, many engineers have tried to design their MMDs by two methods: employing a mechanical shock stopper to limit the travel distance of the microstructure under mechanical shocks [30–33] and optimizing the dimension of the microstructure so that its stress remains smaller than the fracture strength [34–37]. In MMD designs using an over-range mechanical stopper, there have been three types of mechanical stoppers, as shown in Fig. 11: conventional hard stopper, nonlinear spring-type stopper, and soft and thin film coating stopper [44]. Wilner [30] has described the capacitive accelerometer using over-range shock stoppers made of capsulation wafers and has demonstrated its shock survivability up to 10000g. Huang et al. [33] have also shown a mechanical stopper attached to a floating mass. Although these mechanical stoppers can provide marginal protection, they cannot be the real solution for improving the shock survivability of MMDs. This is because these mechanical stoppers can be secondary sources of vibrations which lead to failure, delamination, or performance deterioration of MMDs. To maximize the shock survivability of MMDs using a mechanical stopper, the shock force delivered to the microstructure should be minimized and also the travel length of the moving microstructure should be minimized. Moreover, the energy dissipation capability of the structure under mechanical shocks has to be maximized because the microstructure will vibrate between two or more mechanical stoppers. Increase in damping can dramatically improve the shock survivability of MMDs but also will considerably decrease their performance. The other design approach to improve the shock survivability of MMDs is optimizing the structural dimension to make the maximum stress of the devices less than the fracture strength under mechanical shocks. Tanner et al. [34] have investigated the shock survivability of microstructures by adjusting the structural dimension for a given mechanical shock amplitude using the experimental method. Yee et al. [36] have shown that the shock survivability of the magnetometer structure can be improved by changing the beam length of the device. This approach is attractive because no additional element or treatment is needed [44]. However, this method protects MMDs under mechanical shocks at the expense of device performance, for example, sensitivity. Moreover, using this method, full information about incident mechanical shocks during the design process is needed. Although the design methods introduced before showed satisfactory performance, published experimental results show still much room for

Floating mass

Substrate (Si)

(a)

Soft thin film

(b)

(c)

Figure 11 Design method of micromachined device using over-range mechanical stopper to improve its shock survivability under high-g conditions [44]: (a) conventional hard stopper; (b) nonlinear spring-type stopper; (c) soft and thin-film coating stopper.

6

Woodpecker-Inspired Microparticle Shock Isolator

471

improvement and suggest that an additional passive shock isolator is necessary for sensitive MMDs to meet the requirements of high-g applications.

6

WOODPECKER-INSPIRED MICROPARTICLE SHOCK ISOLATOR Nature causes the heritable traits helpful for survival to become more common in the biological system through natural selection. It consequently makes the biological system highly efficient. The biological system is therefore regarded as a powerful tool to solve engineering problems. To invent an effective shock isolator for MMDs, the shock isolation mechanisms of a woodpecker (D. pileatus) need to be investigated. The woodpecker is an interesting bird whose exceptional anatomy allows it to take the special ecological characteristic of being able to drill holes in a tree. To prevent physical damage (e.g., brain damage) from the drumming motion which is accompanied by high-g and high-frequency vibrations, the woodpecker depends on its own shock isolation mechanisms: drumming pose, mandibles and hyoid, and skull with spongy bone. The bird’s shock isolation mechanisms are investigated in two ways: theoretical analysis using a simplified mechanical vibration model of the woodpecker and experimental characterization of the bird’s skull with spongy bone. With these analyses, a woodpeckerinspired microparticle shock isolator published by Yoon et al. [14, 28] is reviewed. This shock isolator, biomimetically inspired from the woodpecker, controls unwanted incident mechanical shocks within acceptable limit using the microgranular bed composed of close-packed microglass beads, thereby securing the linear operation of MMDs and improving the shock survivability of MMDs under mechanical shocks.

6.1 Shock Isolation Mechanism of Woodpecker Several decades ago, there was a famous animated cartoon character in the movie Woody Woodpecker which was full of its notorious laugh of “ha-ha-ha-HAA-ha!” In this movie, a lot of people might have been interested in the woodpecker and its talent of drumming a hardy wood. The woodpecker can bore holes in a tree due to its exceptional anatomy. There are several reasons for drumming its beak against the tree [42, 45–46]: eating worms and other prey in the tree, making its nest, maintaining its territory from other birds, and displaying its sexuality. It has been also reported that the woodpecker bangs away to relieve its excitement and tension. Based on the quantitative analysis of woodpecker drumming by Stark et al. [46], a ladder-backed woodpecker (Picoides scalaris) drums the tree without brain damage as fast as 28.4 beats/sec. Most woodpecker species drum the tree at an amazing speed of about 20 beats/sec. with a decelerating force of 1500g. Prior to a detailed discussion of the shock isolation mechanisms of the woodpecker, then evolution process of its hyoid needs to be addressed. The woodpecker’s hyoid is one of the most interesting examples of adaptation whereby an organism becomes better able to live in its habitat. Unlike the human’s tongue, which is primarily muscular, the woodpecker’s hyoid is rigidly supported by a cartilage-and-bone skeleton [45]. The hyoid serves as an attachment site for muscles [47]. The hyoid of the woodpecker was quite short at first. With adaptive sense, a long hyoid would be proper since the young woodpecker is fed by its parents [48]. Ancestral woodpecker species began to seek worms deeper in the tree. The woodpecker with mutation for increased hyoid horn growth had a fitness advantage, as it could extend its hyoid farther to reach a prey. The hyoid of the golden-fronted woodpecker shown in Fig. 12 extends from its usual position just ventral to the lower mandible and wraps posterior around the skull to end between the orbits immediately dorsal to the base of the upper mandible. This bony structure aids the woodpecker in extending its hyoid extremely long distances in order to spear insects as well as evenly distributing incident mechanical shocks from drumming [45]. A detailed review of the shock isolation mechanisms will be given.

472 Shock Isolation of Micromachined Device for High-g Applications

(a)

(b)

Figure 12 Golden-fronted woodpecker. (a) hyoid highlighted; (b) anatomy of skull. (Reproduced from [49] with permission of the publisher.) 6.1.1 Allometric Analysis The shock isolation Mechanism of the woodpecker is closely related to its brain size. A theoretical analysis based on allometry is carried out by calculating the deceleration endurance of the woodpecker’s brain compared to that of the human brain. From the literature [42, 45–48], the woodpecker is known to tolerate deceleration of up to 1500g over about 0.5 msec during drumming. Assuming the same stress acting on the brains of the woodpecker and human, the yield strength Sy can be expressed as Fw Fh mw aw mh ah = ⇒ = , (35) Aw Ah Aw Ah where F is the applied force into a cross section of the brain with cross-sectional area A and mass m when the brain experiences a deceleration a and the subscripts w and h mean woodpecker and human, respectively. Equation (35) and the approximate values of the brains of the woodpecker and human summarized in Table 5 yield the deceleration in the woodpecker brain compared to that of the human brain: Sy =

mh Aw 1400 gr × (7 mm)2 π ah = (36) ah = 9.53 ah ∼ 10 ah . mw Ah 2 gr × (60 mm)2 π This shows that the deceleration in the woodpecker brain is about 10 times larger than that in the human brain because of its mass and size. Moreover, the woodpecker can withstand much higher mechanical shocks than humans due to its amazing anatomy. aw =

Table 5

Specification of Woodpecker and Human Brains

Item Woodpecker Human

Mass (g)

Radius (mm)

2 1400

7 60

6

Woodpecker-Inspired Microparticle Shock Isolator

473

Figure 13 Drumming motion of red-headed woodpecker. The bird’s beak is perpendicular to the wood surface and its body is parallel. The body moves like a catapult with a heavy and rigid tail for balance during the drumming motion. (Reproduced from [50] with permission of the publisher.)

6.1.2 Anatomic Analysis The shock isolation mechanisms of the woodpecker were previously unveiled by many researchers [42, 45–47]. There are several known anatomical reasons for the excellent shock isolation of the woodpecker. First, its head and body always remain in a perfectly straight line and its beak drums entirely perpendicular to the tree. Figure 13 shows four still images of drumming motion of a red-headed woodpecker. Its beak is perpendicular to the wood surface (drain pipe in Fig. 13) and its body, which moves like a catapult with a heavy and rigid tail for balance, is parallel to the wood surface, thereby allowing only horizontal force to transmit into its brain. Many scientists have shown that this drumming pose greatly reduces a shear force on the brain which is very harmful to brain tissue and causes loss of consciousness. That is, a force along the horizontal plane without rotation is not lethal to brain tissue. The impact in the woodpecker can therefore be considered as a horizontal one rather than a shear one. Second, another shock isolation mechanism of the woodpecker is linked directly to with its mandibles and hyoid. The flexible cartilage on the mandibles and the unique tongue hyoid, which passes through the throat, goes behind the neck, and is divided into two parts, can help the shock isolation, acting like the damper, which minimizes incident mechanical shocks on the head moving back and forth. When the woodpecker hits its beak on the tree, the fluid coming from its hyoid also acts like a damper to minimize incident mechanical shocks. Third is the spongy-like bone structure of the skull in the woodpecker. It is light and frothy so it dissipates incident mechanical shocks before the shocks propagate to the woodpecker’s brain. Besides, the bone of the woodpecker has a very narrow subdural space and therefore relatively little cerebrospinal fluid. This might reduce the fluid transmission of the incident shock waves. 6.1.3 Vibration Analysis Using Simplified Mechanical Vibration Model A dynamic response of the woodpecker’s head under mechanical shocks is theoretically analyzed. Figure 14 shows a schematic of the woodpecker’s head with its simplified mechanical vibration model. In this model, the upper and lower mandibles are assumed as two parallel mechanical springs, and the hyoid is considered a viscous damper connected with two parallel mechanical springs, as shown in Fig. 14b. There is also the secondary vibration model between brain and exoskeleton, which occurs in the skull with spongy bone. The effect of the beak and hyoid on the head under mechanical shocks has been investigated using simplified dynamic analysis of skull displacement, x (t), in the woodpecker. This is

474 Shock Isolation of Micromachined Device for High-g Applications s

Hyoid

Brain

Mandible

db

Spongy bone lb (a)

k1 c k2

Spongy bone encompassing brain

Mandible and hyoid

Force

(b)

Figure 14 The Woodpecker’s head during drumming motion: (a) schematic of woodpecker’s head; (b) simplified mechanical vibration model of woodpecker’s head. done by calculating the response of the mechanical vibration system under impact force F (t) composed of two springs and one viscous damper arranged in a parallel. Since it is virtually impossible to measure the material properties of the mandibles and hyoid, their material properties are considered as almost the same with the skull and cartilage of the human in this analysis. This approach gives good approximation values because all vertebrates have almost the same material properties. A ladder-backed woodpecker whose length is 17 cm is considered. When the woodpecker drums the tree at 28 beats/with a beak whose length lb is approximately 3 cm and diameter db is about 3 mm, the working distance of the back-and-forth motion s is about 5 cm (see Fig. 14a) [42, 45–47]. Thus, the time for back-or-forth motion is 1/(2 · 28 × Hz) = 0.018 sec. Assuming the same velocity for back-and-forth motion, the average drumming velocity v avg is written as s 0.05 m vavg = = = 2.8 m/sec. (37) t 0.018 sec

6

Woodpecker-Inspired Microparticle Shock Isolator

475

The magnitude of the average drumming force F can be obtained from the linear momentum equation, expressed as 2mvavg P 2 × 0.002 kg × 2.8 m/sec F = = = = 0.62 N, (38) t t 0.018 sec where P is the change in momentum and m is the mass of the woodpecker’s brain. In the next step, a spring constant of the mandibles, k , is obtained by using the elastic modulus of the skull and cartilage of the human, E = 20 GPa [51]. The spring constant of the mandibles is represented as 2Ab E 2 × (0.0015 m)2 π × (20 × 109 Pa) = (39) = 9.42 × 106 N/m, lb 0.03 m where k 1 and k 2 are the spring constants of upper and lower mandibles, respectively, and Ab is the cross-sectional area of the beak. The viscous damping coefficient c of the hyoid in its fluidic motion is obtained with Stokes’s equation, which is described as Fd c=− = 6π μr, (40) vavg k = k1 + k2 =

where Fd is the damping force which a particle (woodpecker’s skull here) with radius r experiences when the particle moves a velocity v avg in the working fluid whose viscosity is μ. Since there is almost no information on the viscosity of the woodpecker’s body fluid, the viscosities of olive oil (0.081 Pa·sec), castor oil (0.985 Pa·sec), and glycerol (1.5 Pa·sec) are used for approximation. The estimated c values from (40) are 0.0458, 0.5567, and 0.8478, respectively. The dynamic response of the woodpecker’s head under mechanical shocks can therefore be described as a second-order ordinary differential equation, a mathematical model of a linear vibration system with one degree of freedom, which is given by m x¨ + c x˙ + kx = F (t),

(41)

where m is the mass of the woodpecker’s head. Transformation of (41) into the standard form yields x¨ + 2ς ωn x˙ + ωn2 x = f (t), (42) where f (t) = F (t)/m is a unit external excitation. Because 0 ≤ ς < 1, this system is underdamped. Complete solution for the impulse response of the second-order ordinary differential equation [52] is written as    ωn e −ςωn (t−τ ) sin ωn 1 − ς 2 (t − τ ) h(t − τ ), (43) x (t − τ ) = | f |  1 − ς2 where | f | is the magnitude of f (x ), τ is the impact time, and h(t − τ ) is the Heaviside step function that nullifies the solution for t < τ. Equation (43) shows the displacement of the woodpecker’s head under drumming shocks, as shown in Fig. 15. With no damping (c = 0), the woodpecker’s head vibrates back and forth about 4.5 mm, which is more than the radius of the head and does not stop. At the damping coefficient of 0.0458 where the viscous damping coefficient of the hyoid is assumed as that of olive oil, the head resonates back and forth about 4 mm after drumming and the vibrations die away after 2 sec. This is also unacceptable because some vibrations still remain at the next drumming of t = 0.036 sec. When the damping coefficient c is 0.5567 (castor oil) or 0.8478 (glycerol), the calculated displacements are moderate. However, there are still more than 0.5-mm vibrations at the next cycle. The amplitude of the total vibration will therefore increase as time goes on. In short, the mandibles and hyoid of the woodpecker isolate incident mechanical shocks effectively but insufficiently. These results suggest that the last shock isolation mechanism of the woodpecker, the spongy bone structure in its skull, must

476 Shock Isolation of Micromachined Device for High-g Applications 0.0050

Displacement (m)

0.0025

c=0 c = 0.0458 c = 0.5567 c = 0.8478

0.0000

−0.0025

−0.0050 0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

Time (sec)

Figure 15 Calculated displacement of woodpecker’s head during drumming motion at damping coefficient c of (a) 0 (without damping), (b) 0.0458 (olive oil), (c) 0.5567 (castor oil), and (d ) 0.8478 (glycerol). play a key role in dissipating incident mechanical shocks and thus prevent brain damage. The woodpecker’s brain within the skull is tightly packed with a dense, yet spongy bone made of porous material with resilience rigidity, as shown in Fig. 10a. Because an analytical approach on this spongy bone structure is too complicated, shock isolation by the spongy bone is characterized experimentally. 6.1.4 Experimental Analysis The woodpecker’s spongy bone has been known to absorb high-frequency mechanical shocks and make the transmitted shocks detour. Yoon et al. [14, 28] have used a granular bed composed of close-packed microglass beads, shown in Fig. 16a, as a substitute for the spongy bone to quantitatively investigate the shock isolation of the spongy bone. In this experimental study, the skull and spongy bone of the woodpecker were translated into a metal case and a microgranular bed, respectively, in terms of system analogy. An experimental setup composed of microglass beads, sample cell, reference accelerometer, measurement accelerometer, vibration exciter, power amplifier, signal generator, and two-channel data recorder was used, as shown in Fig. 16b. In this experimental characterization, five kinds of microglass beads (GB 1–5) with average diameters of 68, 120, 375, 500, and 875 μm were used. After close packing of GB into the aluminum sample cell, the sample cell was vertically excited by the vibration exciter (B&K 4809). There were two piezoresistive accelerometers (Endevco 7264C): one was a reference accelerometer to measure the applied vibration; the other, embedded in the granular bed within the sample cell, measures the transmitted vibration through the granular bed. Both output voltages were recorded by the two-channel data recorder (HP 35670A). Figure 17 shows the measured vibration transmissibility as a function of frequency for GB 1–5 where the vibration transmissibility is defined as the ratio of the transmitted vibration amplitude to the applied one at each frequency. In this experiment, the granular bed absorbs

6

477

Woodpecker-Inspired Microparticle Shock Isolator 50 mm 4 2

60 mm 3

8

1 Vibration

5 6

(a)

7

(b)

Figure 16 Experimental characterization of woodpecker’s spongy bone using microgranular bed composed of close-packed microglass beads: (a) scanning electron microphotograph of microglass beads whose diameter is 68 ± 33 μm; (b) experimental setup composed of (1) microglass beads, (2) sample cell, (3) reference accelerometer, (4) measurement accelerometer, (5) vibration exciter, (6) power amplifier, (7) signal generator, and (8) two-channel data recorder.

Vibration transmissibility

10 GB 1 GB 2 GB 3 GB 4 GB 5

1 fc5

fc4

fc3

fc2

fc1

0.1

0.01

0.001 1

10 Frequency (kHz)

Figure 17 Measured vibration transmissibility as function of frequency for five kinds of microglass beads whose average diameter is 68 μm (GB 1), 120 μm (GB 2), 375 μm (GB 3), 500 μm (GB 4), and 875 μm (GB 5). the vibrations with high frequency just like an electrical low-pass filter. This is because the mechanical shocks in the frequency band of 0.1–10 MHz is generally dissipated by resonant scattering and Rayleigh scattering [53] and those in the inertia regime (0.1–100 kHz) are susceptible to losses through momentum absorption of microglass bead, particle-to-particle friction, and particle-to-particle collision when the particle diameter is 10–1000 μm. In the inertia regime, the mechanical shocks with higher frequency have more loss. The dependences of cutoff frequency, vibration absorptivity, and roll-off steepness on the diameter of the GB are shown in Fig. 18. Microgranular beds composed of GB 1–5 have cutoff frequencies of

478 Shock Isolation of Micromachined Device for High-g Applications 20

1.00

0

Curve fit Roll-off steepness

−40

0.75

15

0.50

10

5

0.25

−60 −80 −100 −120

Roll-off steepness (dB/dec)

−20

Vibration absorptivity

Cut off frequency (kHz)

Cut off frequency Vibration absorptivity

−140 0 0

100

200

300

400

500

600

700

800

0.00 900 1000

−160

Microglass bead diameter (μm)

Figure 18 Measured cutoff frequency, vibration absorptivity, and roll-off steepness of the microgranular bed as a function of microglass bead diameter. 15.84±1.56, 7.68±0.45, 5.62±1.14, 3.08±0.40, and 2.21±0.45 kHz, respectively. The cutoff frequency is inversely proportional to the diameter of the GB. The vibration absorptivity α is defined as α = 1 − (E0 /Ei ) at the frequency band of 0–25 kHz where E 0 and Ei are the energy of transmitted and applied mechanical shocks, respectively. The measured vibration absorptivities corresponding to GB 1–5 are 0.23 ± 0.056, 0.44 ± 0.027, 0.68 ± 0.031, 0.75 ± 0.043, and 0.87 ± 0.030, respectively. The vibration absorptivity generally increases with the diameter of the GB. The roll-off steepness of the microgranular bed as a function of the diameter of GB is also illustrated in Fig. 18. The roll-off steepness is extracted from the measured vibration transmissibility. The smaller the diameter of the GB, the greater the roll-off steepness. The shock isolation effect of the microgranular bed under the random vibration condition shows that the spongy bone of the woodpecker absorbs the high-frequency mechanical shocks and makes the transmitted shocks detour, thereby preventing physical damage on the woodpecker’s head.

6.2 Woodpecker-Inspired Shock Isolator Living things are regarded as a power platform to solve the problems arising in engineering society because the living things adapt themselves to nature and consequently have high efficiency. Engineers therefore attempt to design new devices which draw the working principle from living things, thereby achieving a superb performance and improving the present technology innovatively. In the same context, a woodpecker-inspired shock isolator using a microgranular bed has been designed and characterized by Yoon et al. [14, 28]. This shock isolator has protected MMDs under high-g conditions where, as is well known, tremendous shocks to tens of thousands of g’s and to several hundreds of kilohertz are accompanied. The shock isolator has borrowed a shock-isolating idea from the woodpecker, especially the bony skull with spongy bone. This is because it is impossible to design a shock isolator equipped with all the shock isolation mechanisms of the woodpecker owing to technical and material restrictions.

6

Woodpecker-Inspired Microparticle Shock Isolator

479

6.2.1 Structure and Working Principle The shock isolation mechanisms in the woodpecker introduced before suggest that the bird’s brain has to be packed tightly with the spongy bone inside the skull. The thick bony skull prevents certain physical deformation of the spongy bone and brain from external mechanical shocks. The spongy bone absorbs the high-frequency mechanical shocks and makes the transmitted mechanical shocks detour; thus incident shocks are not concentrated in the brain. To translate the shock isolation mechanisms in the woodpecker into the shock isolator for MMDs, the analogical conversion summarized in Table 6 is used. In this conversion, the brain corresponds to MMDs. The skull and spongy bone of the woodpecker are replaced by a metal case and a microgranular bed, respectively. The woodpecker-inspired shock isolator shown in Fig. 10b consists of the metal case and the microgranular bed. The metal case provides the space for the microgranular bed and prevents the internal free flow of the microgranular bed within itself while protecting the microgranular bed and embedded MMDs from external shocks. The microgranular bed with a great number of air gaps packs the MMDs tightly. It absorbs the high-frequency mechanical shocks and makes the transmitted mechanical shocks detour. This physical phenomenon is involved with the mechanical energy dissipations through particle-to-particle collision and particle-to-particle or particle-to-wall friction. The isolated mechanical shocks by this shock isolator are instantly absorbed within the close-packed microglass beads and are released over a relatively long period of time by the momentum of the microglass beads. This shock isolator therefore effectively limits the unwanted mechanical shocks in the high-g environments, similar to the woodpecker. 6.2.2 Experimental Characterization (60-mm Air-Gun Test) The woodpecker-inspired shock isolator using a microgranular bed has been characterized by using a 60 mm air gun up to 60000g’s. The 60-mm air gun composed of a pressure source, a pressure gauge, a one-way valve, a barrel, and a catcher with an aluminum ingot was used to introduce high-g mechanical shocks, as shown in Fig. 19. To compare the woodpeckerinspired shock isolator and the conventional hard resin-based shock isolator, two kinds of Table 6 System Analogy between Woodpecker and Microparticle Shock Isolator Component

Woodpecker

Shock Isolator

Case Shock isolator Protected material

Skull Spongy bone Brain

Metal case Microgranular bed Micromachined device

Catcher Al ingot

Pressure gauge

1-way valve Pressure source

Figure 19

Barrel

Projectile

Characterization of the woodpecker-inspired shock isolator using 60-mm air gun.

480 Shock Isolation of Micromachined Device for High-g Applications projectiles—BIRDs I and II—were prepared. MMDs were embedded in the close-packed microglass beads in the BIRD I, while MMDs were embedded in the hard resin of 3M ScotchcastTM Electrical Resin 8 in the BIRD II. It is of great value to investigate the shock isolation effect of the woodpecker-inspired shock isolator under high-g conditions with amplitude range of 60,000g and frequency spectrum above 100 kHz. Previously hard resins were commonly used to protect MMDs in high-g applications. To demonstrate the superiority of the woodpecker-inspired shock isolator in highg conditions, the shock survivability of MMDs using the woodpecker-inspired shock isolator was examined by comparing it with that using hard resin. In the 60-mm air-gun test, commercial MMDs known to be sensitive to mechanical shocks—silicon-controlled rectifiers (SCRs), diodes, optocouplers, and capacitors—were tested. Detailed information on commercial MMDs follows. The two kinds of SCRs—lead type and surface mount devices (SMDs)—have part numbers 2N2324 and MCR703A, respectively. The two kinds of diodes—lead type SMD type—have part numbers 1N965 and MMBZ5221BLT1, respectively. The optocoupler, whose part number is 5962-8978501ZA, is packaged as a dual-inline package (DIP). The 120-μF capacitor, whose part number is 595D127X0020R7T, is packaged as an SMD. These MMDs were exposed to high-g conditions of 20,000g, 40,000g, and 60,000g. Table 7 shows the air-gun test results of the MMDs exposed to high-g mechanical shocks with a woodpecker-inspired shock isolator or a hard resin-based shock isolator. The hard resinbased shock isolator fails to protect the MMDs at mechanical shocks of more than 40,000g. The measured shock failure rate, the ratio of number of failed MMDs to total number of tested MMDs, shows that the woodpecker-inspired shock isolator is better than the hard resin-based shock isolator in improving the shock survivability of sensitive MMDs. Figure 20 shows photographs of MMDs embedded in the hard resin-based isolator after a 60-mm air-gun test. Many of the MMDs failed, with a maximum shock failure rate of 26.4% (lead-type diode). The main failures in the MMDs were a bonding wire open failure in the optocoupler, a substrate fracture or delamination in the SCR and capacitor, and a discontinuity in the diode. Almost all MMDs embedded in the woodpecker-inspired shock isolator, however, survived at high-g conditions up to 60,000g, thereby experimentally verifying the effectiveness of the woodpecker-inspired shock isolator. In the woodpecker-inspired shock isolator, the highfrequency mechanical shocks corresponding to the resonance frequencies of the MMDs are dissipated by the close-packed microglass beads and the transmitted ones are detoured around the MMDs, so that the above-mentioned failures are effectively prevented. There was also no crack in the close-packed microglass beads, while the hard resin in the high-g conditions usually experienced many cracks, which caused the adverse effects on the embedded MMDs.

Table 7 Air-gun Test Results of MMDs Embedded in Woodpecker-Inspired Shock Isolator Compared to those Embedded in Hard Resin-Based Shock Isolator Device

Package Type

Shock Failure Rate (%) 20,000g

SCR Diode Optocoupler Capacitor

Lead SMD Lead SMD DIP SMD

0 0 0 0 0 0

Microglass Bead 40,000g 60,000g

(0/36) (0/28) (0/144) (0/104) (0/48) (0/44)

0 0 0 0 0 0

20,000g

Hard Resin 40,000g

60,000g

(0/36) 0 (0/36) 0 (0/36) 0 (0/36) 13.9 (5/36) (0/28) 0 (0/28) 0 (0/28) 0 (0/28) 0 (0/28) (0/144) 0.7 (1/144) 0 (0/144) 5.6 (8/144) 26.4 (38/144) (0/104) 0 (0/104) 0 (0/104) 0 (0/104) 9.6 (10/104) (0/48) 0 (0/48) 0 (0/48) 6.3 (3/48) 22.9 (11/48) (0/44) 0 (0/44) 0 (0/44) 0 (0/44) 6.8 (3/44)

7 Conclusions Optocoupler

SCR

Capacitor

Diode

481

Figure 20 Photographs of MMDs embedded in the hard resin-based shock isolator after 60-mm air-gun test. The MMDs experience bonding wire open failure for the optocoupler, substrate delamination failure for the SCR, dielectric layer failure for the capacitor, and discontinuity failure for the diode in high-g conditions. The 60-mm air-gun test up to 60,000g verified that the woodpecker-inspired shock isolator using a microgranular bed is superior to the conventional hard resin-based shock isolator in improving the shock survivability of the MMDs.

7

CONCLUSIONS In this chapter, several common and unique shock isolation mechanisms for MMDs have been reviewed. As there is an increasing need for MMDs in high-g applications, especially military ones, an effective shock isolator to protect MMDs from high-amplitude and high-frequency mechanical shocks has been intensively researched. Shock isolators should be able to control unwanted mechanical shocks so that their adverse effects on MMDs are kept within acceptable limits in the high-g conditions where tremendous mechanical shocks to tens of thousands of g’s and to several hundreds of kilohertz are accompanied. Although many shock isolators have been designed for MMDs, an effective shock isolator for MMDs has yet to be developed. To advance our understanding of the shock isolator to protect MMDs under high-g conditions, this chapter has presented (1) fundamentals of a mechanical vibration system, (2) general concepts of shock isolation, (3) structure and working principle of various shock isolators, (4) dynamic response of MMDs under mechanical shocks, and (5) woodpecker-inspired shock isolator. Based on a literature study of existing shock isolators for MMDs, except for the woodpecker-inspired shock isolator, all isolation have shown inherent technical limitations as well as unsatisfactory performance. Therefore, the issue of designing an effective shock isolation of MMDs for high-g applications still exists. To our knowledge, the woodpecker (D. pileatus) has the most superb physical characteristics for isolating mechanical shocks in

482 Shock Isolation of Micromachined Device for High-g Applications nature, and therefore has been studied in designing a new class of shock isolator of MMDs for high-g applications. The woodpecker’s shock isolation mechanisms of (1) resilient and rigid mandibles and muscle to dissipate mechanical shocks, (2) slinglike hyoid to reduce mechanical shocks, and (3) bony skull with spongy bone which tightly packs the woodpecker’s brain to low-pass filter mechanical shocks can offer some clues for developing an effective shock isolator for MMDs. The woodpecker-inspired shock isolator has been designed on the basis of only the third shock isolation mechanism of the woodpecker, due to technical and material limitations. A profound study about previous shock isolators and their shock isolation mechanisms will lead engineers, in the near future, to develop an effective shock isolator which is compatible with MMDs, that is, a shock isolator equipped with all shock isolation mechanisms of the woodpecker which will show excellent shock isolation performance.

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CHAPTER

21

RELIABILITY ASSESSMENT OF MICROELECTRONICS PACKAGES USING DYNAMIC TESTING METHODS X. Q. Shi Applied Science and Technology Research Institute Hong Kong, People’s Republic of China

G. Y. Li South China University of Technology Guangzhou, People’s Republic of China

Q. J. Yang Bosch Chassis Systems Asia-Pacific, Ltd. East Bentleigh, Australia

1

INTRODUCTION The rapid development trend of hand-held devices, such as cellular phones, personal data assistants (PDAs), MP3 players, DVD players, and portable multimedia players (PMPs), has pushed the development of electronic packages toward smaller, thinner, lighter, and higher density configurations. These trends might weaken the package with regard to thermal and mechanical reliability. Thermal loading is caused primarily by the mismatch of the coefficient of temperature expansion (CTE) of the packaging materials, but could be due also to temperature gradients. Mechanical loading includes bending, twisting, key pressing, shock impact, and vibration caused by handling, shipping, dropping, field application, and so on. Solder joints, which serve, in surface mount technology (SMT), as mechanical, thermal, and electrical interconnections between the electronic packages and the printed circuit board (PCB), are the most vulnerable part in a portable product when the product is subjected to thermal and mechanical loading. A comprehensive study of the mechanics behavior of solder joints is indispensable for understanding the reliability of portable electronic products. Therefore, various dynamic mechanical tests have been developed to assess the reliability of portable products which includes dynamic mechanical cycling, bending, twisting, dropping or shock impacting, and vibrating tests. Traditional accelerated temperature cycling (ATC) testing for reliability assessment is time consuming. Therefore different accelerated isothermal mechanical cycling approaches were developed as an alternative to ATC. The direct strain measurement (DSM) method was proposed by Xie et al. [1] to obtain the fatigue performance data of solder joints for reliability assessment. This method has been applied for peripheral leaded quad flat package (QFP) devices. Lau et al. [2] established a three-point bending (TPB) methodology. This method can Structural Dynamics of Electronic and Photonic Systems Copyright © 2011 John Wiley & Sons, Inc.

Edited by Ephraim Suhir, David S. Steinberg and T. X. Yu

485

486 Reliability Assessment of Microelectronics Packages Using Dynamic Testing Methods be used to simulate manufacturing, handling, shipping, and rework conditions. Other research efforts [3, 4] have tried to develop an accelerated vibration method for automotive electronics related applications. Comparing the test methods developed, the isothermal mechanical twisting (ITMT) methodology has been shown to be a better potential alternative to ATC, because this method can mimic the solder joint failure mechanisms similar to those from ATC tests [5, 6]. In the first part of this chapter, the ITMT testing methodology, including the experimental testing technique, numerical modeling method, correlation approach, and its application to a plastic ball grid array (PBGA) assembly, will be addressed in detail. The critical failure cause in portable electronic products is accidental drop impact. To assess the reliability of portable products due to drop impact, both product- and board-level drop tests were carried out in recent years. The board-level drop test is viewed as a more efficient for research since the repeatability of the product-level drop test has been shown to be rather poor. Joint Electron Device Engineering Council (JEDEC) standards have recommended the procedure and conditions for the board-level drop test of portable electronic products [7, 8]. However, it is expensive and time consuming. Thus, manufacturers and researchers are trying to develop simpler and cheaper test methods. Those efforts include the ball impact test (BIT) proposed by Yeh et al. [9], the cold bump pull (CBP) developed by NXP Semiconductors [10], and the four-point board-level dynamic bending (4PDB) tests [11] reported by Motorola. Compared with the JEDEC procedure and the methods mentioned above, the four-point dynamic bending test has the advantages of good reproducibility, high test throughput, cheap equipment, simple testing procedure, and consistent mechanical performance of solder joints unlike the drop test prescribed by JEDEC. Accordingly, the second part of this chapter introduces the 4PDB testing methodology for the assessment of the board-level reliability of modern electronic products. Electronic equipment can be subjected to many different forms of vibration over wide frequency ranges and acceleration levels [12]. For example, the vibration frequency spectrum for airplanes varies from about 3 to 1000 Hz, with acceleration levels that can range from about 1G to 5G. Missiles have the highest frequency range, with values of up to 5000 Hz. The vibration frequency spectrum for ships and submarines changes from about 1 to 50 Hz, but the most common range is from about 12 to 33 Hz. The maximum acceleration level in this range is about 1G and appears to be due to vibrations generated by the engines and propellers. For typical vehicles, the vibration frequency varies from 2 to 4 Hz with 3G –5G at the springs and from 8 to 15 Hz with 1G at the body. It is probably safe to say that all electronic equipment is subjected to certain types of vibration during its service life. To access the fatigue failures due to vibration, the accelerated fatigue test is generally run at the higher acceleration levels to induce failures in components and assemblies in a much shorter time than under normal operating conditions. In the third part of this chapter, a dynamic modal testing method and a constant-amplitude vibration fatigue testing method are introduced.

2

ISOTHERMAL MECHANICAL TWISTING TESTING METHOD

2.1 ITMT Testing Methodology As shown in Fig. 1, the ITMT methodology consists of three key components: experimental testing, numerical simulation, and mechanism correlation. In an experimental testing methodology, a new testing system (introduced in the next section) has been developed. With this system, the testing method, including the temperature and loading profile, can be defined for the testing samples, the test data can be obtained to calculate the failure rate of the sample, and the failure analysis techniques are employed to analyze the failure location, failure mode, and failure mechanism. In order to minimize the experimental effort, a numerical modeling methodology that includes typical package numerical models, a material properties library, and life prediction

487

2 Isothermal Mechanical Twisting Testing Method Mechanism Correlation

Testing System (chamber, tooling, etc.) Experimental Testing

Testing Method (temp, loading, etc.)

Experimental Results (life, failure, etc.)

ITMT Failure ITMT

ATC

Testing Samples (design, size, etc.)

ATC

AF 63%

Numerical Models (for the whole package) Numerical Simulation

Materials Models (Si, MC, solder, etc.)

Numerical Results (life, failure, etc.)

ITMT ATC

MTTF (ITMT)

MTTF (ATC)

Life

Life Prediction Model (Si, MC, solder, etc.)

Figure 1 Diagram of ITMT accelerated testing and analysis methodology. models has been established to simulate the potential failure sites and calculate the fatigue life. From the experimental testing and numerical modeling, life data and failure mechanisms such as failure location and failure mode can be obtained for the package to be studied subject to either ITMT or ATC loading. Correlation between the ITMT and ATC methods is then established for the package. In the following sections, the PBGA assembly is used as a suitable illustrative example to describe how to use the ITMT testing methodology to assess the reliability of the solder joints and compare it with the ATC testing method.

2.2 Experimental Testing Methodology An automated ITMT system was designed and built for highly accelerated reliability testing of the PBGA assemblies, as illustrated in Fig. 2. It consists of a thermal test chamber, a PCB grip fixture, a torsion system, a data logger, and a personal computer. The thermal chamber (WEISS DU22) with a temperature range from −75 to 185◦ C and the servomotor with a maximum torque of 6.9 N·m are used to apply the isothermal environmental load and the mechanical twist cycling load, respectively, onto the test vehicle (i.e., a PCB assembly). The temperature and mechanical loading profile are controlled by a personal computer. During the ITMT test, the topside of the PCB is fixed and a ±θ degree of rotation is applied to the bottom of the PCB. The twisting deformation of the PCB is transferred to the solder joints, thereby causing the area array solder joints to deform as in an actual service conditions. A data logger (Hydra Series II, 2625A) is used to continuously monitor the resistance change throughout the test. Software (Trendlink) is employed to plot out the time history of the resistance. In the study, solder joint failure was defined as a completely cracked joint with a threshold of 300  resistance (2  at the start). A 256 input/output (I/O) PBGA component which has 1.5 mm pitch between solder joints was mounted to PCB as a test vehicle. A cross-sectional view of the ball grid array (BGA) component layout is illustrated in Fig. 3. The PBGA package consists of six constituent materials: the PCB test board, copper pads, solder joints, silicon die, bismaleimide triazine

488 Reliability Assessment of Microelectronics Packages Using Dynamic Testing Methods BGA Specimen

Data Logger

Chamber

Connector Monitor PCB Grip Fixture

Personal Computer Torsion Motor

Figure 2 Copper Pad

Schematic diagram of ITMT system experimental setup. Solder Ball

Figure 3 Table 1

Silicon Die

Overmold

BT Substrate

PCB

Cross-sectional view of plastic BGA assembly.

Dimensions of PBGA Assembly

Material PCB Copper pad, φ = 0.557225 Solder ball, φ = 0.79 BT substrate Silicon die Overmold compound

Length (mm)

Width (mm)

Height (mm)

150 — — 27 9.275 24

150 — — 27 9.275 24

1.58 0.038 0.56 0.35 0.43 1.15

(BT) resin substrate, and overmold compound. The dimensions of the BGA assembly are given in Table 1. As only wear-out failures can be accelerated, the failure mode and mechanism must not be altered in the acceleration reliability test. For the ITMT accelerated stress tests, four main acceleration parameters, that is, ramp rate, dwell time, twisting angle, and environmental temperature, must be optimized so as to ensure the validity of the acceleration test. To reduce the quantities of reliability tests, the mechanical loading profile, which was determined by a great number of ITMT tests and finite element (FE) simulations for plastic QFP devices in [5], was employed in this work. As shown in Fig. 4, the mechanical loading profile had a

2 Isothermal Mechanical Twisting Testing Method

489

Angle of twist (deg) θ

Time (sec) 0 30

180

30

−θ 480

Figure 4 Typical twist loading profile for accelerated reliability test. twisting angle of θ, a ramp time of 30 sec, a dwell time of 180 sec at two maximum twisting angles, and a total cycle time of 480 sec. With this loading profile, a series of ITMT reliability tests were carried out at four different twist angles (3◦ , 4◦ , 5◦ , and 7◦ ) for two temperatures (25 and 100◦ C) to investigate the effects of the testing temperature and twisting angle on the failure mechanism. For each test condition, six specimens were used. The number of cycles to failure for each of the specimens was recorded as the fatigue life. For each test condition, the fatigue lives were averaged to calculate the mean fatigue life. Subsequently, the mean fatigue life was plotted as a function of twist angle for two temperatures (25 and 100◦ C). The mean fatigue life was found to exhibit a good linear relationship with the twist angle for both temperatures, indicating that the fatigue life is scaleable between two temperatures for any given twist angle. Scanning electron microscopy (SEM) was used to investigate the failure site and failure mechanism in the solder joint. It was found that at small twist angles (3◦ , 4◦ , and 5◦ ), the failures occurred at the corner solder joints. The failure site where the crack initiated was at the upper corner of the joint and propagated in the solder along the copper pad, as shown in Fig. 5. It reveals that the ITMT tests gave a similar failure site (upper corner of solder joint) and mechanism (low-cycle fatigue crack propagation) as the ATC tests. However, at the large twist angle (7◦ ), some ITMT tests displayed different crack propagation paths (e.g., in the PCB). Therefore, it was concluded that the ITMT has a failure similar to the ATC for the two temperatures (25 and 100◦ C) only at smaller twist angles. To get a higher accelerated factor, more ITMT reliability tests were added to the two test conditions (i.e., 4◦ and 5◦ at 100◦ C) for the statistical analysis. The standard two-parameter Weibull distribution [13] was used to analyze the solder joint reliability: F (t) = 1 − e −(t/η)β ,

(1)

where β is the shape parameter, η is the characteristic life to reach 63.2% failures, t is the time typically expressed in cycles to failure, and F (t) is the fraction of the sample that has failed as a function of time. For better accuracy, a median ranking approach [13] was used to estimate the cumulative density function (CDF): i − 0.3 F (t) = , (2) j + 0.4 where i is the failure number and j is the sample size. To perform a linear regression analysis, Eq. (1) can be rewritten as    1 = β ln (t) − β ln (η) . (3) ln ln 1 − F (t)

490 Reliability Assessment of Microelectronics Packages Using Dynamic Testing Methods

(a)

(b)

Figure 5 SEM micrographs show the crack initiated at the upper corner of the solder joint and propagated along the upper copper pad for both (a) ATC and (b) ITMT loading.

2 Isothermal Mechanical Twisting Testing Method

491

2 Lines for linear regression curve fit Symbols for experimental results

1

ln{ln(1/(1 - F(T)))}

ATC (54 min./cycle) MDS (angle = 4 deg.)

0

MDS (angle = 5 deg.)

−1 −2

Y = 4.513 X − 21.885 Y = 4.675 X − 27.073

R-Square = 0.97219

−3

Y = 4.612 X − 39.156

R-Square = 0.98481

R-Square = 0.96953

−4

1

2

3

4

5

6

7

8

9

10

ln(T)

Figure 6

Weibull statistical plot for both ATC and ITMT accelerated tests.

As can be seen in Fig. 6, when the two-parameter Weibull was used to plot the fatigue data of ITMT reliability tests, the two sets of data showed a good fit. This means that the two-parameter Weibull function is adequate for the analysis. To calculate the acceleration factor (AF) for the two test conditions, a number of ATC reliability tests were carried out for the same PBGA package. A thermal cycling environmental chamber with three temperature zones was employed in the study. The thermal loading profile used had a temperature range from −40 to 125◦ C, a ramp rate of 13.75◦ C/min., a dwell time of 15 min at the two extreme temperatures, and a total cycle time of 54 min. Again, the two-parameter Weibull function was employed to plot the fatigue data of the ATC tests. As can be seen from Fig. 6, the fatigue data of the ATC are parallel to the two sets of ITMT data. It means that the ITMT has a similar failure distribution as the ATC. Thus, the shape parameter β and the characteristic life η can be determined for both ITMT and ATC reliability tests. The mean time to failure (MTTF) is then calculated as [13]   1 MTTF = η 1 + , (4) β where (x ) denotes the gamma function. The results of shape parameter β, characteristic life η, and MTTF are tabulated in Table 2. It can be seen from Table 2 that the inequalities β > 4 and MTTF < η are satisfied for both ITMT and ATC reliability tests. It indicates that ITMT Table 2 Shape Parameter, Characteristic Life, and MTTF from ITMT and ATC Reliability Tests Shape Parameter, β ITMT (angle 4◦ ) ITMT (angle 5◦ ) ATC (54 min/cycle)

4.675 4.513 4.612

Characteristic Life, η MTTF 327 128 4874

299 117 4455

492 Reliability Assessment of Microelectronics Packages Using Dynamic Testing Methods and ATC have the same failure mode of rapid wear-out [13]. In other words, ITMT accelerates the failure rate through the higher stress level but it does not change the failure mechanism of creep flow and creep fatigue interaction. It is therefore concluded that the ITMT is an accelerated method in terms of the wear-out failure mechanism.

2.3 Numerical Modeling and Simulation Methodology Although ITMT can be used to quickly evaluate the long-term reliability of the PBGA assembly, experimentally predicting the fatigue life of solder joints would be expensive and difficult due to the micronature of the actual solder joints. Therefore, it is necessary and desirable to establish an ITMT numerical approach. It can be used to give insight into the failure mechanism of solder joints, to obtain the controlled failure parameters, and to predict the long-term reliability of the PBGA assembly. The general-purpose finite element model (FEM) code ABAQUS version 6.5 was used to conduct the numerical simulation for the PBGA package. As the out-of-plane deflection mechanical load was applied to the package, a three-dimensional model was created. Because of the geometric symmetry along the centerline of the PCB, only half of the PBGA package was modeled, as shown in Fig. 7. In the model, an antisymmetry boundary condition was applied along the centerline of the PCB. All the nodes at the topside of the package were imposed with the boundary condition UX = UY = UZ = 0. For the nodes at the bottom of the package, only vertical displacement UZ was applied as a twist load with UX = UY = 0. Since the solder alloy displays the characteristics of viscoplasticity at high temperature, the total inelastic strain γin of the solder was given by the sum of the time-independent plastic strain γpl and the time-dependent creep strain γcr : γin = γpl + γcr .

(5)

From [14], it was found that the yield stress and ultimate tensile stress (UTS) increase with increasing strain rate for any given temperature. However, when the strain rate reaches a

z

y

x

Figure 7 Half FEM of 256 I/O plastic BGA assembly.

2 Isothermal Mechanical Twisting Testing Method

493

certain value (e.g., 2.78 × 10−1 ), the yield stress and UTS do not increase anymore. This indicates that the inelastic deformation is mainly contributed by the plastic deformation. Therefore, the stress–strain curves obtained at the strain rate of 2.78 × 10−1 were employed in current FEMs to calculate the time-independent plastic strain of the solder. The creep strain rate is given as [15]     τ  nc τ  nl Qc Ql G G γcr = Ac exp − exp − sinh α + Al sinh α , (6) T G RT T G RT where Ac and Al equal 2 × 10−5 and 2.5 × 10−1 , respectively; nc and nl equal 5 and 3, respectively; Qc and Ql equal 48.5 and 81.5 kJ/mol, respectively; α = 1289; G is the shear modulus; τ is the shear stress; T is temperature; and R is Boltzmann’s constant. Equation (6) was implemented into ABAQUS to calculate the time-dependent creep strain of the solder. The material properties, that is, Young’s modulus E , CTE, and Poisson’s ratio, of the rest materials in the package are listed in Table 3. In total, 50,684 nodes and 54,641 three-dimensional elements were used in constructing the numerical model of the PBGA assembly. In the study, three simulations were conducted for the three twist angles of 3◦ , 4◦ , and 5◦ at a temperature of 100◦ C. For each simulation, four cycles were run. Throughout the FE simulations, it was observed that the stress state in the solder joint was in multiaxial. Therefore, a relationship between shear and tensile deformation is needed to account for the multiaxial stress state in solder joints. In the study, the von Mises yield criterion was employed to work out the equivalent stress and strain. The relationship between the equivalent strain and equivalent shear strain is given by [16] γeq εeq = √ . (7) 3 It was found that the maximum cumulative equivalent inelastic strain appeared at the upper corner of the solder joint, as highlighted in Fig. 8. This observation is similar to that from the ITMT reliability tests. It indicates that the numerical model can be used confidently into give an insight into the understanding of the failure site and failure mechanism. When the equivalent stress was plotted against the equivalent strain, the hysteresis loop can be obtained. A typical result is shown in Fig. 9 for the node with the maximum cumulative equivalent inelastic strain in the corner solder joint. It is observed that when the BGA assembly was held at the two extreme twisting angles, the loop exhibited apparent creep stress relaxation

Table 3

Material Properties Used in Numerical Model

Material

Young’s Modulus (GPa)

CTE (ppm/◦ C)

Poisson’s Ratio

67.18 − 0.108T 141.92 − 0.0442T

21 15.64 + 0.0041T

0.40 0.35

24.42 − 0.0226T 10.56 − 0.00957T 132.46 − 0.00954T

17.6 64.1 2.113 + 0.00235T

0.11 0.39 0.28

18.45 − 0.01191T 8.05 − 0.00519T 27.55 − 0.0401T

12.82 57 10.16 + 0.0162T

0.11 0.39 0.25

63Sn/37Pb Copper PCB In plane Out of plane Silicon BT In plane Out of plane Overmold

Note: T is absolute temperature.

494 Reliability Assessment of Microelectronics Packages Using Dynamic Testing Methods

Figure 8 Total cumulative equivalent inelastic strain distribution for corner solder joint. 20

10

Stress (MPa)

Stress relaxation 0 Twist cycling of 63Sn/37Pb solder joint (100 °C, 5 deg., 30 sec. ramp and 8 min./cycle)

−10

−20 −0.08 −0.06 −0.04 −0.02 0.00 0.02 Strain (mm/mm)

Figure 9

0.04

0.06

0.08

Typical stress–strain hysteresis loop for weakest site of corner solder joint.

and strain increment. This means that the creep flow and creep fatigue interaction are major causes of the solder joint failure.

2.4 Correlation of ITMT with ATC Method Since the point of maximum cumulative equivalent inelastic strain has the highest possibility for cracking, its inelastic strain energy density was taken to predict the fatigue life of

3

Four-Point Dynamic Bending (4PDB) Testing Method

495

Table 4 Comparison of Fatigue Life Obtained from Simulation and Experiment at 100◦ C Twist Angle (deg)

3 4 5

Fatigue Life (cycles) Simulation Experiment 432 276 113

452 (MTTF) 299 (MTTF) 117 (MTTF)

the solder joint. A subroutine was written to implement the modified energy-based fatigue model [17]:  m W p = C, (8) Nf ν (k −1) 2σf ν (k1 −1) for 1 Hz ≥ ν ≥ 10−3 Hz,   (k −1) −1) (k

 2 ν ν (9) = (k −1) 10−3 1 for 10−3 Hz > ν ≥ 10−4 Hz, 10−3 where Nf is the fatigue life, ν is the frequency, k and m are the frequency exponents, Wp is the plastic strain energy density, σf is the flow stress, and C is a constant. The frequency was determined from the total cycle time of the ITMT reliability test, the flow stress was calculated from the maximum stress range, and the inelastic strain energy density was determined from the area of the stress–strain hysteresis loop. The results are listed in Table 4. It is noted that, although the simulation gives a little bit lower fatigue life, it is in good agreement with experimental results. Meanwhile the ITMT simulation shows the same trend of relationship between the fatigue life and the twist angle as the ITMT reliability experiment, i.e., a bigger twist angle gives a higher environmental stress acceleration and shorter reliability assessment time for the specified temperature and loading profiles. Therefore, it is demonstrated that the virtual numerical model is applicable for the failure mechanism analysis and long-term reliability prediction of the PBGA package. Compared to the fatigue life obtained from the ATC test listed in Table 2, it is concluded that the ITMT testing method (both experiment and simulation) can be correlated to the ATC testing method well. The acceleration factors were experimentally obtained to be SF1 ≈ 15 for the twist angle of 4◦ and to be SF2 ≈ 38 for the twist angle of 5◦ .

3

FOUR-POINT DYNAMIC BENDING (4PDB) TESTING METHOD

3.1 PDB Testing Methodology The 4PDB testing method is a quick and accurate method to characterize the failure location and mechanism of IC packages. The 4PDB testing methodology includes both an experimental testing methodology and a numerical modeling approach. The experimental testing procedures include three steps: strain drop height calibration, failure generation, and failure analysis. Based on data from the experiment, the failure rate as a function of strain can be obtained, and the failure mode and failure mechanism can be identified by failure analysis. The numerical modeling methodology is used to extract the stress and strain induced in the solder joints. In the following sections, a PBGA assembly was used as an example to demonstrate how to use the 4PDB testing method to assess the board-level reliability of the portable electronic products under drop/impact loading, and failure analysis was then made to correlate the 4PDB testing with the JEDEC drop testing method.

496 Reliability Assessment of Microelectronics Packages Using Dynamic Testing Methods 3.2 Experimental Testing Methodology The typical setup of the four-point dynamic bending test is shown in Fig. 10. The test apparatus consists of a four-point bending fixture, an impact ball/slug tower system, and a data acquisition system. In the experiment, the test board is set on the bottom span of the four-point test fixture with the package facing down and the top span placed on the unpopulated side of the test board. The bottom span consists of two rollers with stainless steel rounds 5 mm in diameter fixed on the base of the fixture. The top span consists of a rigid trestle table supported by two rollers with 3-mm stainless steel rounds. The bumper is a self-adhesive damper attached to the test board which allows the dynamic load induced by a special designed ball/slug dropped at specified heights to impact the bumper. In testing, the ball/slug must be controlled such that it strikes the center of the top span guided by the tower of the equipment. The test board and top span are well aligned to be centered relative to the bottom span. As the test is under displacement control, the load and support span distance can significantly affect the bending moments. In this test, the combination of the support span of 60 mm and the load span of 35 mm was selected. For test board preparation, the package was mounted at the center of the PCB and placed either diagonal to the board or perpendicular to the board. In the diagonal orientation, the package was located such that the diagonal line connecting the opposite corner joints/pads were aligned with the longest board dimension. In the perpendicular orientation, the package was aligned such that the longest edge was aligned with the longest board dimension. To monitor the strain induced in the PCB due to the impact of the ball/slug, two strain gauges were mounted on the unpopulated face of the PCB board directly under the package such that the axis of strain detection was parallel to the long axis of the test board and then connected to a data acquisition system. In testing, all wires and strain gauges must be assured to be clear of the fixture and the impact ball/slug is of the proper mass. In order to measure the deflection of the PCB, speckles were sprayed onto the side surface of the fixture, and a high-speed camera was used to capture the deformation images of the speckle area. The shooting area was placed near the center of the PCB so that the maximum deflection could be captured. The speed of the camera is at least 8000 frames/sec, which means the interval t between Strain Data Ball

Bumper Strain Gauge Top Span Strain Data Logger

Test PCB Bottom Span Package High-Speed Camera

Figure 10 Setup of 4PDB test.

3

Four-Point Dynamic Bending (4PDB) Testing Method

PCB under bending during test

Presprayed speckle on the side of PCB

497

Package

Figure 11 Speckle image on side of PCB taken with high-speed camera. two images is 0.125 msec. Figure 11 presents a typical speckle image of the deformed PCB. From the image, it is obvious that the maximum deflection occurred at the center of the PCB. The 4PDB testing includes two steps. The first step is the strain–drop height curve calibration and the second step is the failure generation. A high-rate data acquisition system with frequency resolution greater or equal to 20 kHz is required so that the strain may be monitored during the impact duration (usually 5–10 msec). The system should be able to monitor as many as two channels at that frequency. The strain signal conditioner should have less than a 0.5-dB drop at 20 kHz. Two signal conditioning channels shall be available. The peak strain is defined as the maximum absolute strain value during the strain measurement. As an example, a PBGA package 13 × 11 × 1.4 mm was mounted on a PCB board in the diagonal orientation as the specimen for the 4PDB tests. The parameters of the specimen are listed in Table 5 and the ball layout is shown in Fig. 12. In this experiment, 35 boards were prepared. The weight of the top span and drop ball used in the experiment were 98 ± 8 g and 136 ± 5 g, respectively. In order to obtain the drop height for the failure generation tests, five boards were first tested to calibrate the stain–drop height relationship. In this step, all boards were tested multiple times with heights of 10, 20, 40, 80, 160, 240, 320, and 400 mm, respectively. Based on the peak strain data from boards 1–5, the strain–drop height curve could be plotted, as shown in Fig. 13. Based on the calibrated strain–drop height curve, the drop heights corresponding to certain strain values could be identified and used in the failure generation step. The test was then performed on the remaining 30 boards with respective drop heights derived from the curve. Once the failure generation tests were completed, the failure analysis was performed on all 30 test boards. Both red-dye penetration and cross-sectioning methods were used to identify the failure of the solder joints. Typical failed solder joints captured from the red-dye method Table 5 Parameters of PBGA Package Used in Four-Point Dynamic Bending Test Package size Ball count Ball pitch Solder ball diameter Solder resist opening Substrate type Package surface finish

13 × 11 × 1.4 mm 165B 0.65 mm 0.35 mm ± 0.05 mm 0.35 mm BT NiAu

498 Reliability Assessment of Microelectronics Packages Using Dynamic Testing Methods

Strain (× 10−6)

Figure 12

Ball layout of PBGA package.

9000 8000 7000 6000 5000 4000 3000 2000 1000 0

Experimental Data Fitting Curve Simulation Data

0

Figure 13

100

200 300 Drop Height (mm)

400

500

Strain–drop height curve for calibration.

are shown in Fig. 14. The picture reveals that some joints appear red, meaning the dye has penetrated the joints through a crack and they are considered to be the failed joints. Figure 15 shows typical failed solder joints from cross-sectioning observation. It reveals that the crack of the solder joint and the fracture of the PCB are the main failure modes due to high stress induced by the drop ball, which are similar to the failure modes observed in the JEDEC drop tests. Based on the failure analysis through the red-dye method, the number and location of the failed solder joints for each of the test boards were recorded for statistical analysis. A typical distribution of the failed solder joints for test strains of 0.0050, 0.0060, 0.0070, 0.0080, and 0.0090 are shown in Fig. 16. It reveals that the failed joints mainly occur at the corner of the package in the diagonal orientation, meaning that the solder joints located at the corner of the package encountered the largest stress for the package mounted diagonally onto the

3

Figure 14

Four-Point Dynamic Bending (4PDB) Testing Method

499

Typical failed solder joint identified from red-dye method.

board. A statistical analysis was then performed using MINITAB. Based on the information of the total tested joints, the failed joints, and the maximum strains induced, the failure strain causing the first joint failure was calculated, and the failure rate could be obtained. Figure 17 shows the probability plot of the failed joints, the failure rate as a function of strain, which is assumed to follow a Weibull distribution. In the analysis, the failure criterion was derived from the first failure joint. In this experiment, each board has 165 balls, and the strain that caused the first failure should correspond to the failure rate of 0.606%. The 80% lower bound confidence interval at the failure of the first joint (0.606%) should meet or exceed the requirements set by Motorola, and the corresponding lowest strain value could be determined to be 6252.84 × 10−6 in this example.

3.3 Finite Element Modeling and Simulation For a bending test, it is difficult to precisely measure the stress and strain developed in the solder joints during the test. Therefore, a finite element analysis (FEA) is often used. A threedimensional (3D) FEA model was constructed, and a FEA using ABAQUS was performed to extract the stress and strain induced in the PCB and the solder joints. Due to the symmetry of the specimen geometry, only a quarter of the bending specimen was modeled. A 3D view of the model is shown in Fig. 18. The geometry and the ball layout of the PBGA package were the same as in the experiment. To achieve calculation efficiency, the high-stress solder joints were meshed finely and the remainder was meshed coarsely. The material properties used in the modeling are listed in Table 6. The solder alloy used has a yield stress of 35.1 MPa and a tangent modulus of 127 MPa. Since the stiffness of the rolling supporters, the trestle table, and the steel ball is much greater than the PCB, they were assigned as rigid bodies. The bumper was treated as an incompressible rubber like material and was modeled by the Mooney-Rivlin model [18]. In the modeling, the different drop heights or initial velocities were applied. For a dynamic load from a free-all steel ball at height h, the velocity v at the moment the ball contacted the bumper is  v = 2gh, (10) which is the initial velocity of the drop ball to impact the bumper.

500 Reliability Assessment of Microelectronics Packages Using Dynamic Testing Methods

(a)

(b)

Figure 15 Cross-sectional picture of typical failed components: (a) solder joint cracks; (b) PCB fracture. From the modeling, the maximum deflection and stain induced on the test board can be obtained for different modeling conditions. As an example for the drop height of 393 mm, the results showed that the maximum deflection and the maximum strain of the PCB were 10.1 mm and 0.007011, respectively, and occurred at 2.675 msec. The strain distribution in the solder joints is shown in Fig. 19. The maximum equivalent strain of solder joints was found to be around 0.674 occurring at the outermost solder joint. The strain distribution also reveals that the solder joints at the corner of the package in the diagonal orientation to the test board encountered the largest strain, which is consistent with the experimental results. Figure 20 shows a comparison of the PCB deflection and the equivalent plastic strain of the outermost solder joint. It can be seen that the equivalent plastic strain of the outermost solder joint has similar trend as the PCB deflection with an increase in time. It was interesting to note

3

Four-Point Dynamic Bending (4PDB) Testing Method

(a)

(b)

501

( c)

(d)

(e)

Figure 16 Typical distribution of failed solder balls on test board for different test stains: (a) 0.0050, (b) 0.0060, (c) 0.0070, (d ) 0.0080, and (e) 0.0090.

99 Lower 80% Confidence Level

Failure Rate

90 80 70 60 50 40 30

Standard Failure Rate

20 Upper 80% Confidence Level

10 5 3 2 1 0.606

0.1 5

6

7

8

9

10

15

20

Strain (× 103)

Figure 17

Probability plot for dynamic bending strain to failure.

502 Reliability Assessment of Microelectronics Packages Using Dynamic Testing Methods

(a)

(b)

Figure 18 Three-dimensional view of FE model: (a) full model and (b) zoom-in model. Table 6

Material Properties of Specimen

Material

Elastic Modulus (GPa)

Shear Modulus (GPa)

Poisson’s Ratio

Substrate

17 (x , y), 7.4 (z ) 4.8 130 20.6

7.59 (xz , yz ), 3.31 (xy) — — —

0.39 (xz , yz ), 0.11 (xy) 0.4 0.3 0.25

34 33 (x , y), 9.8 (z )

— 7.59 (xz , yz ), 3.31 (xy)

0.35 0.39 (xz , yz ), 0.11 (xy)

Die-attach film Die Molding compound Solder PCB

Density (kg/m3 ) 1938 2200 2330 1970 7500 1910

that the trends exhibited three small platforms at times of about 0.8, 1.5, and 2.2 msec. The details from the simulation showed that the steel ball impacted the package three times after the first contact; the details can be obtained from the experiment. The simulations were also performed for drop heights of 68 and 163 mm. All simulation results of the PCB deflection are plotted in Fig. 13 for comparison. As seen, the simulation results are in good agreement with experimental data, indicating that the numerical simulation methodology developed in this study can be used to extract the details of stress and strain of the critical solder joints under drop/impact loading.

3

Four-Point Dynamic Bending (4PDB) Testing Method

503

PCB Deflection (mm)

12

1 PCB Deflection

10

0.8

Equivalent Plastic Strain

8

0.6

6 0.4

4

0.2

2 0

0 0

0.5

1

1.5 Time (ms)

2

Equivalent Plastic Strain

Figure 19 Equivalent strain distribution of solder joints for drop height of 393 mm.

2.5

Figure 20 Comparison of PCB deflection in drop direction with equivalent plastic strain of outermost solder joint.

3.4 Correlation of 4PDB with JEDEC Drop Test Method In correlating 4PDB testing with JEDEC drop testing, the failure locations, failure modes, and failure mechanism should be the same and the failure rates should be similar for both testing methods. A JEDEC drop test with testing condition of 1500G was conducted for the correlation of 4PDB testing with JEDEC drop testing. The typical probability plot for the drops to fail is shown in Fig. 21. Checking the failed packages, it was found that almost all failed solder joints were located at the corner of the package, which is quite similar to 4PDB testing except for the number of failures a solder joints. It is verified that the failure location is the same for the two testing methods, indicating that the failure mechanism is the same for both 4PDB and JEDEC drop tests. Typical failed solder joints from JEDEC drop testing are shown in Fig. 22. It reveals that for solder joint failure the crack initiates at the corner of the solder joint and propagates along the solder joint near the package side. Comparing the failure components in Fig. 15a for the 4PDB test, it is noted that the failure modes are quite similar for both 4PDB and JEDEC drop tests. Based on the failure analysis, it is concluded that the failure location, failure mode, and failure mechanism are the same for both 4PDB and JEDEC drop testing methods.

504 Reliability Assessment of Microelectronics Packages Using Dynamic Testing Methods 99 Substrate 1 Substrate 2

Accumulative Failure Rate

90 80 70 60 50 40 30 20 10 5 3 2 1 100

1000

Number of Drops

Figure 21

Figure 22

Probability plot for drops to failure.

Typical failed solder ball in JEDEC drop test.

Therefore, the 4PDB test can be used as a fast testing method to assess the reliability of electronic packages subject to drop/impact loading.

4

HIGH-FREQUENCY DYNAMIC VIBRATION TESTING METHOD

4.1 Vibration Testing Methodology Similar to other methodologies, vibration testing consists of a experimental testing method and a numerical simulation approach. In this section, the PBGA assembly is used as an example. First, dynamic modal testing and constant-amplitude vibration fatigue testing are introduced,

4 High-Frequency Dynamic Vibration Testing Method

505

and then the FEA approach is introduced for the assessment of fatigue life of the solder joints in the assembly.

4.2 Dynamic Modal Testing Method As shown in Fig. 23, a specially designed PCB assembly with four PBGA modules symmetrically mounted on the board was used as the test vehicle. The PBGA module was a daisy-chained module with 256 I/O solder balls. The size of the module was 27 × 27 mm, the overall thickness of the module was 2.3 mm, the pitch of solder balls was 1.5 mm, and the PCB was 210 mm × 140 mm with a thickness of 1.5 mm. In conducting modal testing of the PBGA assembly, the impact hammer method is always used. To minimize the influence of boundary conditions and the mass of transducers, all test specimens were subjected to a free boundary condition. A miniaccelerometer whose weight was negligible was used to measure the vibration response. For the free boundary condition, the PBGA assembly was tied with some rubber bands at four corners and hung freely without any other support. As the rubber bands are very soft, their stiffness is very small and can be negligible when compared with the stiffness of the PCB. Under this situation, the PCB assembly can be regarded as being in a free boundary condition. Figure 24 shows the test setup for the free boundary condition. An impact hammer was used to excite the assembly, and an accelerometer was used to measure the acceleration response of the assembly. In the testing, the response measurement location was fixed, while the impact hammer was moved through all 25 locations. At each location, the excitation and response from the accelerometer and impact hammer were measured by a four-channel portable Fast Fourier transform (FFT) analyzer SD-390 simultaneously and the frequency response function (FRF) was estimated. The signals were collected at the sampling frequency of 2560 Hz for each of 2048 samples. For each location, the measurement was repeated eight times, and all data were collected and averaged to decrease the influence of noise in the signals. To make sure the test data were reliable, the coherence function was also estimated. When values of the coherence function at all FRF peaks were above 0.95, the FRF was regarded as satisfactory and saved for further analysis.

Figure 23 Photo of specially designed PBGA assembly for vibration test.

506 Reliability Assessment of Microelectronics Packages Using Dynamic Testing Methods

Figure 24 Experimental setup for testing the PCB assembly with free boundary condition. To extract the modal parameters form the measured FRFs, the STAR System from Structural Measurement Systems was used. The number of modes in the measurements was first determined, and several subfrequency bands were then chosen according to the mode distribution. Finally, a polynomial curve fitting was employed to fit the FRFs for each subfrequency band and to extract the modal parameters. In the modal testing, a bare PCB was tested with the free boundary condition. The mode numbers were determined by the modal peaks method. The method plots all the measured FRFs into one figure and counts the number of peaks as the number of modes that happened in the measurements. The results show that there were 14 modes within 1 kHz. The PCB assembly with the PBGA modules was then tested. In order to compare the dynamic properties, the modal frequencies, and modal shape vectors of the bare PCB and the PCB assembly were compared. It was found that the modal shapes are almost the same for bare PCB and PBGA assemblies, but the modal frequencies are slightly different. Table 7 compares the identified modal frequencies for the bare PCB and PBGA assemblies. It is noted that mounting the PBGA module to the bare PCB could increase the resonance frequency. The more PBGA modules are mounted onto the PCB, the higher the resonance frequency obtained.

4.3 Constant-Amplitude Vibration Fatigue Testing Method As shown in Fig. 25, the vibration testing system consists of a vibration shaker, a mounting fixture, an event detector, and a computer. The vibration shaker (model A395) was from Ling Electronics and the vibration control software was from Spectral Dynamic. The software could control the shaker running according to any specified profile by using a random signal, a sweep sinusoidal signal, or a mixed random and sweep sinusoidal signal. Because the fundamental resonant vibration mode with the lowest natural frequency usually generates the greatest displacement and stress in the vibration system, a test profile with a narrow frequency range around the fundamental natural frequency of the PBGA assembly at a constant G level was used in the PBGA assembly test [12].

4 High-Frequency Dynamic Vibration Testing Method

507

Table 7 Modal Frequencies of Bare PCB and PCB Assemblies under Free Boundary Condition Mode No. Bare PCB One PBGA Four PBGA Freq. (Hz) Frequency (Hz) Differential (%) Frequency (Hz) Differential (%) 1 2 3 4 5 6 7 8 9 10 11 12 13 14

87.66 120.96 212.65 280.59 316.11 348.37 419.74 471.11 667.26 693.18 738.77 771.09 802.92 948.27

90.30 124.25 215.07 285.12 320.44 356.00 424.43 479.92 678.27 698.60 746.82 784.12 820.54 963.67

3.01 2.72 1.14 1.61 1.37 2.19 1.12 1.87 1.65 0.78 1.09 1.69 2.19 1.62

101.44 130.31 224.25 294.77 338.83 364.40 430.00 501.57 668.14 704.78 774.21 821.22

15.72 7.73 5.45 5.05 7.19 4.60 2.44 6.47 0.13 1.67 4.80 6.50

Figure 25 Experimental setup of vibration testing system. The test specimen was clamped on two opposite sides by using two rectangle steel blocks and mounted on a solid test fixture which was then bolted to the header of the shaker. Shield ribbon cables were used to connect the PBGA modules to a special designed monitoring device called an event detector (model 128-105) from Analysis Tech. The event detector features selectable short-duration sensitivity and variable resistance threshold over a range of selectable direct current (dc) connector currents. It can generate a constant current from 10 to

508 Reliability Assessment of Microelectronics Packages Using Dynamic Testing Methods 100 mA and monitor the resistance up to 128 channels simultaneously. Any resistance change exceeding a preset threshold with a minimum duration of 0.1μsec can be detected by the event detector. In the test, 4 PBGA modules, in total 32 rings, were monitored simultaneously. Before the test, the resistance of each ring within the PBGA module was about 2 , the same as the ITMT test method, and the solder joint failure was defined as a completely cracked joint with a threshold of 300  resistance. Before the reliability test, several calibration tests are needed. Since the event detector is a sensitive equipment, the influence of noise such as conducted alternating current (ac) line noise and radiated high-frequency noise may generate false events in the event detector. If these false events were presented in the test results, they would provide wrong information on the fatigue life of the PBGA assembly. In order to overcome the influence of noise, every reliability test began with a noise test. In the test, the PBGA assembly was clamped in place with the vibration shaker turned off, and the cables securely connected the assembly to the event detector. The event detector was run overnight. If no false event was reported, the test setup was considered acceptable and the test could proceed to the next step. The continuity of all rings in the PBGA modules was monitored overnight to ensure no failure occurred before the reliability test. Then, a resonance frequency scanning test was performed to determine the fundamental natural frequency. In the test, a sweep sine excitation with 0.5G acceleration from 20 to 2000 Hz was used to excite the PBGA assembly. The dynamic response of the assembly was measured by an accelerometer attached at the center of the assembly. From the resonance scanning test, the fundamental natural frequency of the PCB assembly used in this study was identified to be 104 Hz. After that, the dynamic response test was conducted. The G level of excitation was gradually increased, the maximum response in acceleration at the center of the PBGA assembly was measured and the maximum deflection at the center of the PBGA assembly was determined. For a vibration system subjected to a harmonic excitation, its response can be written as (11) Z = Zmax sin(t + ϕ) and

Z¨ = −Zmax 2 sin(t + ϕ),

(12)

where  is the radian frequency of external excitation, and Z max is the maximum displacement of the system. The maximum deflection of the PBGA assembly can be determined as 9.8 max(Gout ) . (13) Zmax = 2 Table 8 shows typical test results of the PBGA assembly at different levels of input excitations: 0.5G, 1.0G, 2.5G, 5G, 7.5G, 10G, and 20G. Table 8 Dynamic Responses of PCB Assembly Subject to Different Input G Levels G in (G) 0.5 1 2.5 5 7.5 10 20

G out (G)

Z max (mm)

27 39 107 177 232 252 293

0.79 1.03 1.82 2.44 2.81 2.94 3.31

4 High-Frequency Dynamic Vibration Testing Method

509

With determined fundamental natural frequency and dynamic deflection of the PCB assembly at different G levels of excitation, a profile of input excitation for reliability testing was determined. In the reliability test, the excitation began from a frequency smaller than the fundamental natural frequency of the assembly, gradually swept to a frequency which was larger than the fundamental frequency, and then swept back. This process was repeated until the preset time was reached. An accelerometer mounted on the assembly continuously monitored the dynamic deflection of the assembly. The event detector continuously monitored the resistance of each ring and reported any failure detected. Table 9 shows the fatigue lives of the first failure detected for ring 1 (most outer rows) and ring 2 (second outer rows). The standard two-parameter Weibull distribution, as given in Eqs. (1)–(4), was employed to analyze the test data. As seen from Fig. 26, the Weibull plots exhibited good fitting for both rings 1 and 2, indicating that the two-parameter (shape parameter β and characteristic life η) Weibull distribution could be applied to analyze the fatigue performance of the assembly. From the curve fitting, the parameters β and η and the values of MTTF and first time to failure (FTTF) could be obtained for both rings 1 and 2. The results are summarized in Table 10. Based on the parameters, the probability density functions and reliability functions of rings 1 and 2 were determined, as plotted in Figs. 27 and 28, respectively. As seen from Table 9 and Figs. 27 and 28, the fatigue life of ring 1 was shorter than that of ring 2. In other words, the solder joints in ring 1 showed a higher probability to fail or lower reliability. The cross sections were performed to check the failure location. The failures were found to occur mainly at the outer solder joints. Figure 29 shows the cross-sectional view of a typical failed PBGA solder joint. As seen, the crack initiated at the right corner of the solder joint and propagated across the solder ball. More failure analyses were performed and all the results showed that the vibration fatigue failures occurred in the outer solder joint and at the PCB side.

4.4 Finite Element Analysis Approach Unlike FE thermal stress analysis of a PCB assembly, where a strip model or partial model is usually used, vibration analysis needs a full model of the assembly. Detailed FE modeling of the assembly is very time consuming and even impossible due to the limitation of computer resources. The PBGA assembly is a very complex structure, as each PBGA module has 256 solder joints. In order to overcome this difficulty, different modeling techniques, including structural dynamic modification, submodeling, and the “smeared” property technique, were developed to evaluate the dynamic properties of PCB assemblies [19–21]. However, a detailed stress analysis of solder joints is not available now and is still under development. Table 9 Summary of Fatigue Lives of Assembly Under Vibration Fatigue Loading (×106 cycles) Specimen Number 1 2 3 4

Ring Number

PBGA Module 1

PBGA Module 2

PBGA Module 3

PBGA Module 4

1 2 1 2 1 2 1 2

12.32 21.12 17.47 17.94 0.57 0.70 7.70 7.29

7.16 14.81 4.78 3.33 8.87 17.21 5.45 10.06

1.38 7.63 0.46 5.50 5.85 6.89 1.78 1.43

8.17 8.57 2.10 1.99 9.48 11.86 5.92 6.93

510 Reliability Assessment of Microelectronics Packages Using Dynamic Testing Methods Weibull Probability Plot

Probability

0.96 0.90 0.75 0.50 0.25 0.10 0.05 0.02 106

(a) Vibration cycles for ring 1

107

0.96 0.90 0.75 Probability

0.50 0.25 0.10 0.05 0.02 106

107 (b) Vibration cycles for ring 2

Figure 26 Weibull distribution of fatigue life of assembly subject to 2.5G input: (a) ring 1, and (b) ring 2. Table 10 Shape Parameter, Characteristic Life, MTTF and FTTF of the PBGA Assembly Under Vibration Test Ring Number 1 2

Shape Parameter, β

Characteristic Life, η (×106 )

MTTF (×106 cycles)

FTTF (×106 cycles)

1.32 1.45

6.48 10.18

6.19 8.92

0.46 0.70

In characterizing the dynamic properties of the PBGA assembly, a two-level submodeling technique was developed. The first-level modeling was to simulate a single joint so as to establish its effective mechanical properties. The second-level modeling consisted of the whole assembly but modeled the solder joints as beam elements with the effective mechanical properties. This approach can dramatically reduce the size of the FEM and can save the modeling and analysis time. With this model, the modal analysis was conducted to determine the modal frequencies and shape vectors of the assembly.

4 High-Frequency Dynamic Vibration Testing Method 1.2

Probability Density Function (x 10−7)

Ring 1 1.0

Ring 2

0.8

0.6

0.4

0.2

0.0 0

10

20 30 Million Vibration Cycles

40

50

Figure 27 Probability density function of PBGA assembly subject to 2.5G input.

1.0 Ring 1

Reliability Function

0.8

Ring 2

0.6

0.4

0.2

0.0 0

10

20

30

40

Million Vibration Cycles

Figure 28

Reliability function of PBGA assembly subject to 2.5G input.

50

511

512 Reliability Assessment of Microelectronics Packages Using Dynamic Testing Methods

Figure 29 Typical failure of solder joint of PCB assembly subject to vibration fatigue loading.

Figure 30 First-level FEM of solder joint.

4 High-Frequency Dynamic Vibration Testing Method Table 11

513

Properties of Different Materials Used in Model

Parameter

PCB

Cu

Solder

E x (GPa) E y (GPa) G xy (GPa) ν ρ (kg/m3 )

30.0 25.0 5.5 0.12 1971

31.7

121.0

0.40

0.24

In the first-level modeling of the PBGA assembly, the solder joints were modeled in detail, as shown in Fig. 30. The mechanical properties of the materials used in the model are listed in Table 11. In order to get the effective stiffness matrix, a unit displacement in the X , Y , and Z directions and a unit rotation around X , Y , and Z were imposed as the deformation to the model at both ends. Each time, one displacement or one rotation was imposed. Figure 31 (the coordinates are the same as Fig. 30) shows the deformed solder balls when unit translation or rotation displacements in six directions was imposed. After imposing the deformation, the reaction forces at all degrees of freedom at two ends of the solder ball could be calculated. According to the basic formula of a spring, the estimated reaction force at one direction is equivalent to the stiffness coefficient along that direction. Since the two ends of the solder ball were the surfaces, which included a lot of nodes, the reaction forces at all nodes of each end were summed together to get the total reaction force; thus a 12 × 12 stiffness matrix could be formed, which was equivalent to that of a beam with six degrees of freedom at each end of the beam. In the second-level modeling of the PBGA assembly, the following simplifications were made: (1) the solder joints were modeled as the beam elements with the effective stiffness matrix determined from the first-level modeling; (2) the PBGA module was modeled as a solid structure, as the fundamental resonance frequency of the PBGA module is much bigger than the upper limit of the interested frequency range. The PCB was modeled as the shell elements. The model is shown in Fig. 32. With the model, the modal analysis of the PBGA assembly with the free boundary condition was conducted. The calculated modal frequencies are summarized in Table 12 and compared with test results. As seen, the simulation results matched well with

Figure 31 Deformation of solder joints subject to unit translation displacements (upper from left to right representing X , Y , and Z direction translation, respectively) and unit rotation displacements (lower from left to right representing X , Y , and Z axis rotation, respectively).

514 Reliability Assessment of Microelectronics Packages Using Dynamic Testing Methods

(a)

(b)

Figure 32 Finite element model of PBGA assembly: (a) whole model and (b) zoom-in model.

5 Conclusions

515

Table 12 Comparison of Modal Frequencies Obtained from Experimental Test and Numerical Simulation Mode Number

1 2 3 4 5 6 7 8 9 10 11 12

Modal Frequency (Hz) Test FEM 101.44 130.31 224.25 294.77 338.83 364.40 430.00 501.57 668.14 704.78 774.21 821.22

117.34 134.43 261.73 288.61 368.00 382.74 448.83 529.57 686.56 694.39 819.45 833.34

Error (%)

15.67 3.16 16.71 −2.09 8.61 5.03 4.38 5.58 2.76 −1.47 5.84 1.48

experimental data for different modes, indicating that the two-level modeling method is able to predict the dynamic performance of the PCB assembly with acceptable accuracy.

5

CONCLUSIONS The dynamic testing method plays an important role in the reliability assessment of microelectronic products. This chapter has introduced three recent developed dynamic testing methodologies: ITMT, 4PDB, and vibration fatigue for assessment of long-term reliability of the PBGA assembly. Each method consists of the experimental testing technique, the FEA modeling approach, and the correlation method to existing relevant thermal or mechanical testing methods. To alternate the time-consuming ATC test, an ITMT testing methodology was developed to evaluate the long-term reliability of the PBGA assembly within days instead of months; in other words, the method can accelerate the testing time maximum by 38 times as compared to the ATC method. Many experimental tests validated that the ITMT testing method accelerates the wear-out failure only if the twist angle is smaller than 5◦ and the environmental temperature is within the range of 25–100◦ C. The ITMT numerical modeling/simulation is in good agreement with the ITMT experimental testing, but it can give further insights into the failure mechanism and wear-out reliability characteristic of the solder joints. Therefore, the ITMT testing methodology is an effective alternative to the ATC method for quick evaluation of the long-term reliability of solder joints for the PBGA assembly. To understand the solder joint failures due to highly localized strain created by highstrain-rate mechanical shock at the package level, a 4PDB test methodology was developed and demonstrated to produce controllable high-strain-rate environments. This test provides a method to quickly and effectively evaluate solder joint reliability in dynamic environments. Taking PBGA package as an example, the test setup, test procedure, data analysis, failure analysis, numerical modeling, and failure criteria correlation to the drop/impact test were demonstrated. The strain induced in the test board and solder joints to cause the first joint failure can be easily and quickly estimated. This estimate agrees well with the strain level that caused the package failure after the first drop in JEDEC drop tests. Compared to the

516 Reliability Assessment of Microelectronics Packages Using Dynamic Testing Methods drop/impact test, the 4PDB test has the advantage of good reproducibility, simple testing setup and procedure, and consistent mechanical behavior of solder joints. A high-frequency vibration testing methodology was introduced to assess the reliability of the PCB assembly modules. In the experimental testing, dynamic modal testing was performed to study the frequency response and analyze the changes of resonance frequencies of the bare PCB and the PCB with PBGA modules under the free boundary condition. The maximum difference of the resonance frequency was found to be about 15.7% for the bare PCB and the assembly with four PBGA modules. Constant-amplitude vibration fatigue testing was performed to assess the reliability and estimate the lifetime of the PCB assembly. The failure analyses showed that all the failures occurred at the solder joint and the PCB side. Furthermore, the two-level FEM approach was introduced to model and simulate the dynamic performance of the bare PCB and the PCB assemblies. As an example, the modal analysis was performed and the first 12 modes were identified, and the results were found to be in good agreement with experimental data. It should be pointed out that the methodologies introduced in this chapter can be applied also to other types of packages, such as QFP, CSPs, flip-chip (FC) packages, and die-stacked 3D packages.

REFERENCES 1. Xie, D. J., Chan, Y. C., Lai, J. K. L., and Hui, I. K., “Fatigue Life Estimation of Surface Mount Solder Joints,” IEEE Transactions on Components, Hybrids, and Manufacturing Technology, Vol. 19, No. 3, pp. 669–678, 1996. 2. Lau, J., and Pao, Y. H., Solder Joint Reliability of BGA, CSP, Flip Chip and Fine Pitch SMT Assemblies, McGraw-Hill, New York, 1997. 3. Sealing, S., and Dasgupta, A., “Alternative Accelerated Testing Method for Localization of Solder Fatigue Failures on Electronic Circuit Cards,” ASME Advances in Electronic Packaging, Vol. 26, No. 2, pp. 1593–1598, 1999. 4. Sharma, P., Natarajan, R., and Dasgupta, A., “Reducing Accelerated Test Time: Use of Vibration Loading to Accelerate Ageing Damage,” ASME Advances in Electronic Packaging, Vol. 26, No. 2, pp. 1801–1805, 1999. 5. Pang, H. L. J., Ang, K. H., Shi, X. Q., and Wang, Z. P., “Mechanical Deflection System (MDS) Test and Methodology for PBGA Solder Joint Reliability,” IEEE Transactions on Advanced Packaging, Vol. 24, No. 4, pp. 507–514, 2001. 6. Shi, X. Q., Pang, H. L. J., Yang, Q. J., Wang, Z. P., and Nie, J. X., “Quick Assessment Methodology for Reliability of Solder Joints in Ball Grid Array (BGA) Assembly—Part II: Reliability Experiment and Numerical Simulation,” Acta Mechanica Sinca, Vol. 18, No. 4, pp. 356–367, 2002. 7. JEDEC standard JESD22-B111, “Board Level Drop Test Method of Components for Handheld Electronic Products,” 2003. 8. JEDEC standard JESD22-B110, “Subassembly Mechanical Shock,” 2001. 9. Yeh, C. L., and Lai, Y. S., “Design Guideline for Ball Impact Test Apparatus,” Journal of Electronic Packaging, Vol. 129, pp. 98–104, 2003. 10. Zaal, J. J. M., and Van Driel, W. D., “Testing Solder Layer Connect Reliabilities under Drop Impact Loading Conditions,” in Proceedings of International Symposium on High Density Packaging and Microsystem Integration, Shanghai, China, Aug. 14-17, 2007, pp. 234–239.

References

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11. Reiff, D., and Bradley, E., “A Novel Mechanical Shock Test Method to Evaluate LeadFree BGA Solder Joint Reliability,” in Proceedings of the 5th Electronic Components and Technology Conference, 2005, pp. 1519–1525. 12. Steinberg, D. S., Vibration Analysis for Electronic Equipment, Wiley New York, 1988. 13. Weibull, W., Fatigue Testing and Analysis of Results, MacMillan, New York, 1961. 14. Shi, X. Q., Zhou, W., Pang, H. L. J., and Wang, Z. P., “Effect of Temperature and Strain Rate on Mechanical Properties of 63Sn/37Pb Solder,” ASME Journal of Electronic Packaging, Vol. 121, No. 3, pp. 179–185, 1999. 15. Shi, X. Q., Wang, Z. P., Yang, Q. J., and Pang, H. L. J., “Creep Behaviour and Deformation Mechanism Map of Eutectic Solder Alloy,” ASME Journal of Engineering Materials & Technology, Vol. 125, No. 1, pp. 81–88, 2003. 16. Meyers, M. A., and Chawla, K. K., Mechanical Metallurgy, Prentice-Hall, Englewood Cliffs, NJ, 1984. 17. Shi, X. Q., Wang, Z. P., Zhou, W., Pang, H. L. J., and Yang, Q. J., “A New Creep Constitutive Model for Eutectic Solder Alloy,” ASME Journal of Electronic Packaging. 18. Swanson, J., Release 10.0 Documentation for ANSYS, Pennsylvanian: ANSYS Company, 2006. 19. Wong, T. L., Stevens, K. K., and Wang, G., “Experimental Modal Analysis and Dynamic Response Prediction of PC Boards with Surface Mount Electronic Components,” Journal of Electronic Packaging, Vol. 113, pp. 244–249, 1991. 20. Pitarresi, J. M., “Modeling of Printed Circuit Boards Subject to Vibration,” IEEE International Symposium on Circuits and Systems, Vol. 3, pp. 2104–2107, 1990. 21. Pitarresi, J. M., Caletka, D. V., Caldwell, R., and Smith, D. E., 1991, “The ‘Smeared’ Property Technique for the FE Vibration Analysis of Printed Circuit Cards,” Journal of Electronic Packaging, Vol. 115, pp. 250–257, 1991.

CHAPTER

22

THERMAL CYCLE AND VIBRATION/DROP RELIABILITY OF AREA ARRAY PACKAGE ASSEMBLIES Reza Ghaffarian Jet Propulsion Laboratory California Institute of Technology Pasadena California

1

SUMMARY Commercial off-the-shelf area array packaging (COTS AAP) technologies are considered for use in a number of high-reliability electronics systems. Understanding the process and quality assurance (QA) indicators for reliability, as well as developing methods to project life under thermal and mechanical environment, is important for the low-risk insertion of these newly available electronic packages. Extensive investigations were performed by NASA in collaboration with industry and universities to address key manufacturing and reliability issues on the continuously evolving AAP technology. The reliability results based on test data will facilitate use of various low-pitch and high–input/output (I/O) area array packages, with more processing power in a smaller board footprint. Topics discussed in this chapter are as follows: • Packaging technology trend for AAPs, including plastic ball grid array (PBGA), chip-

scale packages (CSPs), and 3D stack packages. • Summary of key parameters that affect assembly reliability of PBGAs and flip-chip

•

•

• •

BGA (FCBGA) based on a comprehensive literature search. The effect of a doublesided mirror image array package on solder joint reliability is also addressed. IPC 9701A specification was reviewed for lead-free solder alloy requirements. Test results for a test vehicle having various packages including a PBGA with 1156 I/Os subjected to various thermal cycling. X-ray and scanning electron microscopy (SEM) and X-sections after thermal cycling. Literature survey and test results for a COTS PBGA 676 I/O package assembly with various ball/solder alloys representing the current migration of industry to lead-free solders. A number of test vehicles, assembled and supplied by an engineering manufacturing service (EMS) company collaborator, were subjected to thermal cycling evaluation as part of collaboration activity. The test results are presented. The test results for thermal cycling and vibration of numerous CSPs soldered with tin–lead and with and without underfill are presented. Extensive vibration and thermal cycle data were generated by the consortium team members of the Joint Council on Aging Aircraft/Joint Group on Pollution Prevention (JCAA/JG-PP). Vibration test results for BGA 225 I/O package assemblies with tin–lead and lead-free solder were reviewed and compared with other models and

Structural Dynamics of Electronic and Photonic Systems Copyright © 2011 John Wiley & Sons, Inc.

Edited by Ephraim Suhir, David S. Steinberg and T. X. Yu

519

520 Thermal Cycle and Vibration/Drop Reliability of Area Array Package Assemblies

•

•

•

•

2

analyses. Based on the obtained information, Weibull plots comparing times to failure under vibration for assemblies with tin–lead and lead-free solder alloys are presented. The test results showing the effects of thermal cycle and random vibration for PBGA and ceramic ball grid array (CBGA) packages assembled with tin–lead are also presented. SEM photomicrographs showing failures for thermal cycle and vibration were also presented. The effects of a priori isothermal aging on subsequent drop behavior were determined for various array packages, including a PBGA 1156 I/O. Optical photomicrographs revealing fine microcracks and intermetallic features are also presented and discussed. The effects of thermal shock cycles on subsequent drop behavior of various lead-free and tin–lead solder for array packages including a stack package were characterized. Plots showing number of drops to failure comparing the effects of different board surface finishes, ball lead-free alloys, and package styles are presented. A summary of key recent literature test data on vibration and drop for tin–lead and lead-free solder assemblies is provided. This includes Minor’s rule, and test results on its conservatism or nonconservatism for low–high and high–low damage accumulation.

TECHNOLOGY TREND FOR AREA ARRAY PACKAGES

2.1 Background Examples of typical area array packages are shown in Fig. 1. These include PBGAs with ball eutectic compositions of Sn63 Pb37 alloy or a slight variation. The CBGA package uses a higher melting ball (Pb90 Sn10 ) with eutectic solder attachment to the die and board. Column grid array (CGA or CCGA) is similar to BGA except it uses column interconnects instead of solder balls. FCBGA is similar to BGA, except it is internal to the package and flip-chip die is used. Migration to lead-free solder alloys had brought numerous options for selection of materials for balls, columns, and solder paste for assembly. Ball grid arrays (BGAs) and CSPs are now widely used for many electronic applications, including portable and telecommunication products [1]. BGAs with 1.27 mm pitch are implemented for high-reliability applications, generally demanding more stringent thermal and mechanical cycling requirements. The PBGAs introduced in the late 1980s and implemented with great caution in the early 1990s further evolved in the mid-1990s to the CSP (also known as fine-pitch BGA) having a much finer pitch from 0.4 down to 0.3 mm. Because of these developments, it has become even more difficult to distinguish different area array packages by size and pitch. For high-reliability applications, surface mount leaded packages, such as ceramic quad flat packs (CQFPs), are now being replaced with CCGAs with 1.27 mm pitch (distance between adjacent ball centers) or lower. Replacement is especially appropriate for packages with higher than 300 I/O counts where CQFP pitches become fine, making them extremely difficult to handle and assemble. In addition to size reduction, CCGAs also provide improved electrical and thermal performance; however, their solder columns are prone to damage, and it is almost impossible to rework defective solder joints. CCGA packages are preferred to CBGA since they show better thermal solder joint reliability than their CBGA counterparts. Superior reliability is achieved for larger packages and for higher than 300 I/Os when resistance to thermal cycling is further reduced with increasing package size. All ceramic packages with more than about 1000 I/Os come in the CCGA style with 1 mm pitch or lower in order to limit growth of the package size. The key drawbacks of PBGA and fine-pitch array packages remain the same, that is, inspection capability for interconnection integrity (cracks) and individual solder ball

2

Technology Trend for Area Array Packages

Wire Bond

Flip Chip

63Sn/37Pb Lead free

I/O < 600

63Sn/37Pb Lead free

PWB Plastic BGA (PBGA)

I/O > 600

PWB Flip Chip BGA (FCBGA) Flip Chip or Wire Bond

Wire Bond or Flip Chip

90Pb/10Sn High Melt

521

63Sn/37Pb Eutectic

90Pb/10Sn High Melt

I/O < 600

PWB PWB Ceramic BGA (CBGA) Reza Ghofforian

Figure 1 Representative key area array package (AAP) configurations: BGA, FCBGA, CBGA, and CCGA. reworkability. In addition, most array packages are commercial-off-the-shelf (COTS) packages and are to be subjected to additional stringent screening with added cost at the package level prior to their acceptance for high-reliability applications. The issues with PBGA COTS packages are essentially the same as other COTS issues, including package die source and material variations from lot to lot, availability of packages with radiation hard die, and outgassing for materials. Acceptability of ball integrity attachment before and after burn-in and coplanarity is also an issue to be considered. Assembly/inspection-related issues are additional key aspects of such implementation. Other issues include challenges of manufacturing of fine-pitch array packages and extremely small passives. Extensive work has been carried out by industry and a number of consortia in order to understand technology implementation of AAPs for high-reliability applications. The work included process optimization, assembly reliability characterization, and the use of inspection tools, including X-ray and optical microscopy, for quality control and damage detection due to environmental exposures [2–11]. Figure 2 illustrates a few samples of electronic package assemblies from plated through-hole (PTH) to surface mount technologies (SMT) as well as recent package assemblies with plastic and ceramic BGAs.

2.2 Area Array Packages (AAPs) Area array packages, for example, BGAs with 1.27 mm pitch (distance between adjacent ball centers) and finer pitch versions with 1 mm pitch, are the only choice for packages with higher than 300 I/O counts, replacing leaded packages such as the quad flat pack (QFP). AAPs also provide improved electrical and thermal performance, more effective manufacturing, and ease of handling compared to conventional SMT leaded parts. Finer pitch area array packages (FPBGA), also called CSPs, are further miniaturized versions of BGAs, or smaller configurations of leaded and leadless packages with pitches generally less than 1 mm. Figure 3 shows examples of BGA and CSP packages.

522 Thermal Cycle and Vibration/Drop Reliability of Area Array Package Assemblies

Figure 2 Packaging trends from through-hole to SMT technologies including QFP, BGA, CBGA, and CSP.

cm 1

2

3

4

5

6

7

Figure 3 Area array plastic packages from high 1156 I/Os with 1 mm pitch to low 192 I/Os with 0.4 mm pitch. 2.2.1 Advantages of Area Array Packages (AAPs) Area array packages offer several distinct advantages over fine-pitch surface mount components having gull wing leads: • High-I/O capability (100s to approximately 3000 balls can be built and manufactured,

but gull wing leads are limited to less than 400 I/Os) • Higher packaging densities (this is achievable since the limit imposed by package

periphery for the gull wing leads is not applicable in the case of AAPs because area, rather than periphery, is used as an appropriate design characteristic; hence, it is possible to mount more packages per the same board area.) • Faster circuitry speed than in gull wing surface mount components (SMCs) because the terminations are much shorter and therefore less inductive and resistive

2

Technology Trend for Area Array Packages

523

• Better heat dissipation because of more connections with shorter paths • Ability to apply conventional SMT manufacturing and assembly technologies such as

stencil printing and package mounting. AAPs are also robust in processing. This stems from their higher pitch (typically, 0.8–1.27 mm), better lead rigidity, and self-alignment characteristics during reflow processing. This latter feature, self-alignment during reflow (attachment by heat), is very beneficial and opens the process window considerably. 2.2.2 Disadvantages of Area Arrays AAPs however, are not compatible with multiple solder processing methods, and individual solder joints cannot be inspected and reworked using conventional methods. In ultra-lowvolume SMT assembly applications, the ability to inspect the solder joints visually has been a standard inspection requirement and is a key factor for providing confidence in the solder joint reliability. Advanced inspection techniques, including X-ray, need development to provide such confidence for BGAs and FPBGAs. The four chief drawbacks of AAP technology are: • Lack of direct visual inspection capability • Lack of individual solder joint reworkability • Interconnect routing between the chip and the PWB (PCB) requiring a multilayer

PWB (PCB) • Reduced resistance to thermal cycling due to use of rigid balls/columns

2.3 Chip-Scale Packages (CSPs) The trend in micro-electronics has been toward ever-increasing I/Os on packages, which is in turn driving the packaging configuration of semiconductors. Key advantages and disadvantages of CSPs compared to bare die are listed in Table 1. CSP technology can combine the strengths of various packaging technologies, such as the size and performance advantage of bare die assembly and the reliability of encapsulated devices. Table 1

Pros and Cons of Chip-Scale Package

Pros

Cons

Near chip size Widely used Testability for known good die (KGD) Ease of package handling Robust assembly process Only for area array version Accommodates die shrinking or expanding Standards Infrastructure Rework/package as whole

Moisture sensitivity Thermal management Limits package to low I/Os Electrical performance Routability Microvia needed for high I/Os Pitch limited to use standard printed wiring board (PWB) Reliability poor in most cases Underfill required in most cases to improve reliability Array package version Inspectability Reworkability of individual balls

524 Thermal Cycle and Vibration/Drop Reliability of Area Array Package Assemblies CSPs have already made a wide appearance in commercial industry as a result of these advantages, and now even their three-dimensional (3D) packages are being widely implemented. Unlike conventional BGA technology at typically 0.8–1.27 mm pitch, CSPs utilize lower pitches (e.g., currently, 0.8–0.3 mm) and hence will result in smaller sizes with their own challenges. The advantages offered by CSPs include smaller size (reduced footprint and thickness), lesser weight, easier assembly process, lower overall production costs, and improvement in electrical performance. CSPs are also tolerant of die size changes, since a reduced die size can still be accommodated by the interposer design without changing the CSP’s footprint. In an effort to systematically characterize the CSP as a package group, CSPs may be classified into the following categories or types: (1) flex circuit interposer type, (2) rigid substrate interposer type, (3) custom lead frame type, and (4) wafer-level package (WLP) type. A typical CSP process starts with the mounting of the die on the interposer using epoxy, usually of nonconductive type (although conductive epoxy is also used when the die backside needs to be connected to the circuit). The die is then wire-bonded to the interposer using gold or aluminum wires. Wire bond profiles must be as low and as close to the die as possible in order to minimize the package height. Plastic encapsulation to protect the die and the wires then follows, usually by transfer molding. After encapsulation, solder balls are attached to the bottom side of the interposer, then the package is marked, and, finally, the parts are singulated from the lead frame. Figure 4 shows flip-chip die bond, chip-and-wire direct-chip attachment, and chip-scale package configurations. In summary, several different approaches are being employed by different companies to meet the packaging challenge of mounting high-pin-count integrated circuits (ICs) to substrates. Each of these approaches has its merits and drawbacks. Here are some of the approaches: • Mount the IC internally, wire bond or flip chip, on a flexible/rigid organic or ceramic

substrate and package the chip into a suitable package material. Apply small solder bumps to the bottom of the package, flip over, and mount onto suitable mounting pads on the PWB. This approach is commonly referred to as BGA technology. If the package dimensions are nearly the same as those of the IC, this technology is called chipscale packaging, or CSP. The principal advantages of BGA and CSP technologies are their ability to protect the IC (with package) and their close similarity to flip chip technologies. • Attach the IC die to the bare PWB and wire bond from the die bonding pads directly to bonding pads of the PWB. This is commonly referred to as chip-on-board (COB) or chip-and-wire direct-chip attachment (DCA) technology. • Permanently attach small solder bumps to the bottom of the IC die, flip it over, and then mount it onto suitable mounting pads on the PWB . This is commonly referred to as direct flip chip technology.

2.4 3D Stack Packages The demand for high-frequency operation, high I/O density, and low parasitic along with a package-level integration requirement with small form factors and extreme miniaturization has led to new 3D packaging configurations. The 3D packaging combines flip-chip and wire bond interconnection, builds up and laminates substrates, and brings about package-level integration of disparate device functions through 3D die and package stacking, to name a few trends. Stack packaging—more than Moore—has recently become very attractive for use in commercial electronics because of cost and limitation of die fabricating with finer features.

2

Technology Trend for Area Array Packages

525

Flip Chip On Board

Wire Bond

TAB COB

CSP

Figure 4 Chip-on-board/flip-chip die attachment and chip-scale package configurations.

Moore’s law has been substantiated—the number of transistors on a given chip doubles about every two years (now it is 18 months). The exponential growth pattern for die density has allowed computers to become both cheaper and more powerful simultaneously. Increase in the package density is also achieved using area array for interconnection rather than conventional periphery use for leaded packages, such as QFP (quad flat package). Figure 5 illustrates stack packaging trends for those technologies that are in the development stage to those that are now mainstream. Currently, 3D packaging consists of stacking of packaged devices, called package-on-package (PoP) technology, and stacking of die within a package called package-in-package (PiP) technology, and stacked wire-bonded die (primarily memory) technology. These technologies are used today with the promise of stacking die (without wire bonds) using through-silicon-via (TSV) technology. Of the existing 3D package technology options, wire bonding remains the most popular method for low-density connections of less than 200 I/Os per chip. In the near future, however, it will become difficult to meet the increasing frequency requirements and demands for wiring connectivity merely by increasing the number of the peripheral wire bonds. In

8 chip stack 560 µ

Increasing Technical Capability

Through-silicon Via SiP Stack w/ Embedded Passives

3D SiP

3D Wafer-level Packaging

Chip-on-chip

3 PKG

MP-FCBGA

Stacked Die Package-on-package

Chip-on-chip

System-in-package

Current

Through silicon vias

Far Term

Figure 5 3D stack package trend: current and future.

526 Thermal Cycle and Vibration/Drop Reliability of Area Array Package Assemblies order to overcome such wiring connectivity issues, 3D chip-stacking technology using TSV is particularly attractive because it offers the possibility of solving serious interconnection problems while offering integrated functions for higher performance. Package-on-package (PoP) concepts have proven popular, particularly for hand-held portable applications. These packages offer significant advantages, including increased density through stacking of logic and memory devices in the same component footprint, as well as flexibility because of the assembler’s ability to select different memory devices for inclusion in the stack. Next-generation versions of PoP devices are now emerging, which offers reduction in component warpage during reflow and increased pin count due to the pitch reduction on both the top and the bottom package. The through-mold via (TMV) structure is added to the lower stack of packages to reduce component warpage and improve assembly yield. Assembly and reliability of such PoP TMV packages are being characterized in Ref. 12. Dip flux and dip paste were assessed for assembly of the upper package, with different dip depth variations studied to optimize the process for each material in terms of assembly yield and thermal cycling reliability. Two different underfill materials were also assessed for use on these components—one selected for optimal shock test performance and another one to optimize thermal cycling reliability. All assembly variations were subjected to accelerated thermal cycling from −40 to 125◦ C for 2000 cycles. Shock testing was performed on a subset of the assembly variations to complete the reliability assessment.

3

PBGA THERMAL CYCLE SOLDER JOINT RELIABILITY

3.1 Background Reliability of plastic and ceramic ball/column grid arrays has been assessed for high-reliability applications. The most critical variables incorporated in the various investigations have been package types, ceramic column/ball and plastic; board materials, FR-4 and polyimide; surface finishes, organic solver preservative (OSP), hot-air solder level (HASL), and Ni/Au; solder volumes, low, standard, and high; and environmental conditions. Plastic BGAs with a variety of sizes and shapes are abundantly available and are used widely by commercial industry for a variety of applications from benign office environments to high-end server applications. Military and avionic industries are also using them selectively after they have reached an acceptable level of maturity. Because of their wider applications, reliability of these packages is generally characterized by suppliers and verified by industry. Numerous publications and references are available through the Institute of Electrical and Electronics Engineers (IEEE) CPMT/ECTC, SMTA, IMAPS, and Association Connecting Electronics Industries proceedings and journals such as the Microelectronic Reliability Journal and IEEE Electronics Packaging Technology/Manufacturing. This section starts with a brief discussion of the IPC 9701A specification, showing key requirements for thermal cycling. Then, it categorizes the second-level assembly reliability for a number of PBGAs based on various parameters. Reliability data from researchers as well as suppliers are tabulated for a number of packages from low to very high I/Os. Finally, it presents reliability test data recently generated for a test vehicle with numerous fine-pitch and high-I/O area array packages.

3.2 IPC Standard for Thermal Cycle The IPC-9701 specification [19] addresses how thermal expansion mismatch between the package and PWB affects solder joint reliability. In order to compare solder joint reliability for different package technologies, numerous materials and process parameters were specified, including the following:

3

PBGA Thermal Cycle Solder Joint Reliability

527

• Specifies 0.093 in. for the PWB (e.g., FR-4) thickness in order to minimize bending

and to achieve conservative values on cycles-to-failure data • For tin–lead solder, limits surface finish choices to OSP and HASL in order to eliminate

the potential of interfacial failure • Limits pad configuration to NSMD (non–solder mask defined or Cu defined) in order

to eliminate failure due to stress risers • For BGA, defines PWB pad size to be 80–100% of the package pad size in order to

have a more realistic failure A revision, IPC-9701A, includes guidelines for Pb-free solder alloys. Appendix B of this specification provides guidelines for modifications of Pb-free solder joints. Two thermal cycle profiles were recommended for SAC (Sn–Ag–Cu) solder attachments depending on the reliability approach and use conditions: • Condition D10 (10 min dwell) requiring dwells of 10 min at both the hot and cold

temperature extremes. This is possibly the most efficient, accelerated thermal cycle profile since it induces the most strain energy per unit of time (considering the entire cycle) or per unit dwell time. Cycles-to-failure data generated under this condition should generally be used as stand-alone only and only when the physics of damage accumulation is understood by modeling. The test results may be used for comparison to those of lead-based solder assemblies to find out whether their performance is better or worse. • Condition D30+ (30 min or higher dwell) requiring dwells for 30 min and longer (i.e., 60 min) at the hot and cold temperature extremes in order to experimentally induce creep damage somewhat comparable to lead-based solder. Modeling in conjunction with experimental data at different dwell times may be required to better define such a comparison. An OSP surface finish is recommended for the Pb-free base solder alloys even though the final version of the specification includes immersion silver (IAg) based on additional input by industry. For Sn–Pb solders the acceptable surface finish was HASL. For Pb free, and tin–lead solders HASL is not allowed, since it is not compatible with Pb-free solder interconnections. Other surface finishes can be used for the manufacturer’s internal data comparison. Electroless nickel/immersion gold (ENIG) surface finish can be also used for internal data comparison; however, there is a risk of introducing unintended immature failure, as documented by industry. In this specification, the thermal cycle (TC) test ranges, test profiles, and number of test cycles (NTC) reported were also standardized. These include the reference cycle in the range from 0 to 100◦ C (TC1) and harsh military cycle condition from –55 to 125◦ C (TC4). Three out of five total TC conditions are identical to the test conditions recommended by JEDEC 22, method A104, revision A. The NTCs varied from a minimum value of 200 cycles to a reference value of 6000 cycles for the TC1 condition.

3.3 Plastic Package Thermal Cycle Assembly Reliability Table 2 lists cycles to failure (CTF) for a number of plastic packages with different configurations selected from those reported in the literature [13–18]. Thermal cycle test results are for Sn37 Pb balls with Sn37 Pb solder only even though most recent data generated by industry are for lead-free balls/solder. Lead-free packages/solders are yet to be fully adopted for high-reliability applications. However, in order to update the literature survey and link the behavior of the two solder alloys, data for the full array PBGA 676 I/Os are compared in [10] and briefly discussed under test results. The following section presents a discussion of the effect of a few key parameters on solder joint reliability.

528

Package (I/O, Pitch)

PBGA-119-1.27

PBGA-256-1.27

FCBGA-1849-1.27 PBGA-256-1.0

PBGA-676-1.0

PBGA-900-1.0

FCBGA-1020-1.0

FCBGA-1020-1.0

PBGA-313-1.27

PBGA-256-1.27

PBGA-676-1.0

1

2

3 4

5

6

7

8

9

10

11

0◦ C/100◦ C (2 cycles/hr) −30◦ C to 100◦ C (25 min, 15 min, 0.75) −40◦ C–125◦ C (15 min, 15 min, 1) −40◦ C–125◦ C (15 min, 15 min, 1)

33 × 33 (17.9 × 16.7) 35 × 35 (13 × 13)

27 × 27 (17.8 × 17.8 × 0.3)

27 × 27 (10 × 10)

0◦ C/100◦ C (10 min, 5 min, 2) 0◦ C/100◦ C (2 cycles/hr)

2)

2)

2)

31 × 31.5 (17 × 17 × 0.3) 33 × 33 (22.6 × 19.9)

27 × 27 (17.8 × 17.8 × 0.3)

No Info 17 × 17 (8.80 × 7.9)

0◦ C/100◦ C (10 min, 5 min, 0◦ C/100◦ C (10 min, 5 min, 0◦ C/100◦ C 0◦ C/100◦ C (10 min, 5 min, 0◦ C/100◦ C (10 min, 5 min,

27 × 27 ? (17.8 × 17.8 × 0.3) 27 × 27 (10 × 10) 2)

Thermal Cycle Condition (Ramp, Dwell, cycles/hr)

Package Size (Die Size, mm)

∼2000 (1% failure) 1341

3310 (1% failure)

2770 (1% failure)

5670 (1% failure)

4405

4686

No failure to 9000 cycles 3095 (1% failure) 3687 (1% failure)

6260 (1% failure)

First Failure

1830

3164

4000

N/A

N/A

5344

6012

No failure to 9000 cycles 4710 N/A

12,215

Mean Life (N63.2% )

Cycles-to-Failure Data for Tin–Lead/Tin–Lead Ball/Paste Illustrating Effect of Selected Key Variables.

Case No.

Table 2

PWB 1.6 mm thick [14] 27/32 fail, PWB 1.6 mm thick [16]

PWB 1.6 mm thick [14] [17] Full array PWB, 2.3 mm [15] 30/30 fail, PWB, PWB 2.36 mm thick [16] 28/28 fail PWB 2.36 mm thick [16] PWB 2.3 mm thick, 6-layer build-up BT [15] PWB 2.3 mm thick, 6-layer build up BT+ Cu heat sink [14] 13 pkg., PWB 1.6 mm thick [18]

27 pkg. [13]

Comments

3

PBGA Thermal Cycle Solder Joint Reliability

529

Table 2 clearly shows the effect of thermal cycle range on CTFs: as T increases, CTF generally decreases. Maximum and minimum temperature and dwell time at these temperatures contribute to failures. For example, the CT1%F (cycles to 1% failure) for PBGA 256 I/O and 1.27 mm pitch in the range of 0–100◦ C was more than 9000 cycles (case 2); it was significantly reduced to approximately 2000 cycles when the temperature range increased to –40/125◦ C (case 10) Package size, thickness, configuration, internal die attach type, and I/Os also play a significant role in CTF. For example, in comparing case 2 to case 3, a significant decrease in CT1%F is shown when package I/O increased from the 256 I/Os, 1.27 mm, to 1849 I/Os (>9000 vs. 3095 cycles in the range of 0–100◦ C). Note that package configuration is also different for the higher I/O package; it has flip-chip die rather than wire bond die attachment. Die size and its relation to package size and ball configuration affect the performance of PBGA packages whereas it has a lower effect on ceramic ball grid array/ceramic column grid array (CBGA/CCGA) CTF. This is not apparent from the cases presented in Table 2, but the die and package sizes are listed for the purpose of identifying such a correlation. The comparison of cases 7 and 8 indicates an increase in CTF when die size is increased. These results, contrary to the general trends, may be due to the confounding effects of adding a heat sink in case 8. The PWB thickness/stiffness also affects PBGA CTF. Preferred thickness was defined as 0.093 in. (2.3 mm) in IPC 9701 [19] since it is known that, generally, packages assembled on thinner PWBs show higher CTFs. Comparison for PBGAs with 1.27 mm in the range of 0–100◦ C is difficult since most test data are generated using IPC PWB thickness values and also since most low-I/O packages survived a large number of cycles. For example, PBGA 256 I/O showed no failures to 9000 cycles (case 2); one reason may be the use of thinner PWB for assembly. Double-sided, mirror-image PBGA assemblies have significantly lower CTF compared to their single-sided version. Table 3 shows an example of the effect of double-sided mirror Table 3 Comparison of TV-2 FPBGA Thermal Cycle Test Results (–55/125◦ C) to Literature Data for 0–100◦ C Thermal Cycle Range TV ID

Board Thickness (NSMD Pad Size)

Via Location (Diameter)

Thermal Cycle Range (Total Time)

Weibull Scale (No.)

Weibull Shape (m)

Acceleration Ratio

175 I/O FPBGAenhanced, single-side Condition 1 Condition 2 175 I/O FPBGA-standard, double-side Condition 1 Condition 2 TV-2 175 I/O FPBGA 9 data points 8 data points

1.57 ± 0.2 (300 μm) (400 μm)

On pad (125 μm)

0–100◦ C (32 min)

4331 3525

11.1 9.1

N/A

1616 1163

17.6 10.5

N/A

1126 1134

6 11.9

3.8

(300 μm) (400 μm)

1.27 (300 μm)

On pad (100 μm)

–55–125◦ C (68 min)

530 Thermal Cycle and Vibration/Drop Reliability of Area Array Package Assemblies image on CTFs for a 175-I/O flip-chip package [5, 20, 21]. Recently, Chaparala et al. [22] performed experimental and modeling analyses to verify results presented by Ghaffarian in his 1999 article published in Chip Scale Magazine. In 1999, it was reported that the mean times to failure for mirror-imaged CSP assemblies in thermal cycling are 40–60% lower than that observed for single-sided CSP assemblies. He identified the factors differentiating double-sided assemblies from single-sided assemblies as an increase in assembly standoff due to a second reflow pass, an increase in assembly stiffness, and thermal disturbance due to the package on the other side of the PWB.

4

TEST RESULTS FOR TC RELIABILITY OF HIGH I/O PBGA ASSEMBLIES

4.1 PBGA with Sn37 Pb Ball Assembled with Sn37 Pb The purpose of this aspect of the investigation was to characterize the reliability of high I/O and low pitch of various package assemblies using tin–lead solder paste. Figure 6 shows photos of representative packages used in the investigation. A design-of-experiment (DOE) technique was used to cover various aspects of processing and packaging assembly reliability. For PBGAs with 1156 I/Os and 1 mm pitch, the accessible peripheral balls were inspected with no apparent evidence of opens. Figure 7 documents representative photomicrographs taken after assembly. Quality of solder joints was acceptable, which was also verified by daisy chain continuity. All PBGA assemblies showed acceptable daisy chain continuity resistances; therefore, it is assumed that they had acceptable quality with no solder joint opens. To reduce cost, quality is verified after thermal cycling by cross-sectioning. PBGAs with 1156 and 256 I/Os showed minimum voids, but no shorts or other anomalies were apparent (see Fig. 8). Real-time X-ray inspection of all solder balls during evaluation revealed no signs of anomaly or damage. Generally, it is difficult to detect solder joint microcracks/damage by X ray. Even though X ray revealed no significant damage, it clearly identified other parameters of packages that can help better analyze observed failure data. For example, it clearly showed

cm 1

2

3

4

5

6

7

8

Figure 6 Photograph of PBGA, CSP, QFN, and TSOP packages included in the test vehicle for evaluation.

4

Test Results for TC Reliability of High I/O PBGA Assemblies

531

PBGA 256

PBGA 1156

Figure 7

Representative solder joints for PBGA 1156 and 256 I/Os with 1 mm pitch.

Figure 8 X-ray inspection of PBGA 1156 I/Os. that internal structures of the two thin small outline packages (TSOPs) are different, which may provide the reason for observing earlier failures of TSOP 40 compared to TSOP 48 I/Os under various thermal cycling conditions. Figures 9 and 10 show representative SEM photomicrographs of the PBGA 1156-I/O assembly before and after X sectioning, respectively. X sectioning was performed diagonally through the center of the package in order to reveal the microstructure of internal interconnections. It specifically shows balls under the die that were expected to exhibit more severe

532 Thermal Cycle and Vibration/Drop Reliability of Area Array Package Assemblies

Figure 9 Representative SEM photomicrographs of a PBGA 1156-I/O assembly after thermal cycling and prior to cross sectioning.

Figure 10 Cross-sectional SEM photomicrographs of PBGA 1156-I/O assembly after thermal cycling.

stress conditions due to a larger local thermal cycling coefficient-of-thermal-expansion (CTE) mismatch. Except for very fine microcracks at the package interfaces, no severe degradation was observed, either under the die or throughout the X section. Minor separation apparent at the die attachment may be due to either die bond separation or edge effects due to mounting of the sample and subsequent polishing.

5

Cycles-to-failure of PBGA 676 I/Os

533

4.2 PBGA with Lead-Free and Tin–Lead Comparison Commercial industry moves to use various lead-free solder alloys. This requires understanding of many reliability aspects in high-reliability applications. IPC 97xx specifications, covering surface mount assembly reliability for both thermal and mechanical testing requirements, are now well adopted by the industry and are widely referenced. Specifically, the IPC-9701 specification addresses how thermal expansion mismatch between the package and the PWB affects solder joint reliability. Since the release of its new revision that includes Pb-free solder, additional data have been gathered to determine the impact of various thermal cycle parameters on accelerated test results. Recently, these data were reviewed and compared for a PBGA 676-I/O package assembly including data generated by Celesica [10, 23, 24]. In the following discussion, requirements defined in IPC 9701A for lead-free solder alloys are compared for various industry test data for assembly of a full array PBGA 676 I/O package. Thermal cycle data results generated for the same package and the same test vehicle in two different facilities are also presented.

5

CYCLES-TO-FAILURE OF PBGA 676 I/Os The effects of a few key parameters for PBGA 676-I/O package assemblies are presented below.

5.1 Effect of Dwell Time A team of companies collaborated to evaluate the reliability of lead-free SnAgCu (SAC) PBGA 676-I/O packages assembled with tin–lead and lead-free SAC solder alloys on 93-milthick test vehicles [25]. Reliability evaluations included accelerated thermal cycling (ATC) from 0 to 100◦ C with two dwell time durations at 100◦ C: the normal short dwell duration of 10 min defined for tin–lead and a long dwell duration of 60 min recently proposed by IPC for lead-free solder. Figure 11 shows the Weibull plots of test results for the two test conditions. As expected, shorter dwell time resulted in higher cycles to failures for lead-free SAC balls assembled with lead-free SAC solder paste.

5.2 Effect of Lead-free Solder Alloys Progressively more complex test vehicles were built to characterize manufacturing as well as solder joint reliability of various packages [23, 24]. Figure 12 shows cycles-to-failure test data for the PBGA 676-I/O packages assembled onto 0.093-in.-thick boards using two types of solder alloys. This plot includes Weibull lines for test data as well as confidence level lines. A thermal cycle that was performed per IPC 9701 was in the range of −55 to 125◦ C, which was considered to be representative of harsh environmental applications. Test data showed that there is a decrease in the number of cycles to failure for the lower silver SAC 305 compared to those assembled with SAC 405 solder alloy, even though some failures were in the same ranges.

5.3 Effects of Solder Paste Alloys for Preferred TC, 0–100◦ C The coauthors’ team partners working in an EMS facility, Celestica, supporting mostly commercial OEMs, were required to perform extensive characterization of lead-free assemblies. They performed extensive testing characterization using the preferred thermal cycle profile (0–100◦ C, short dwell) recommended by IPC 9701. Various conditions included in the evaluation are as follows: • Tin–lead solder paste with SAC405/305 balled 676-I/O packages (cells 1-1 to 1-6) • SAC305 on various surface finishes (cells 2-1 to 2-6)

534 Thermal Cycle and Vibration/Drop Reliability of Area Array Package Assemblies 99.00 FGBGA 676

Cumulative Failure %

90.00

50.00

10.00 5.00

1.00 1000

10000 Cycles to Failure b

N63

N1

SAC/SAC 10 min.

12.0

5797

3950

SAC/SAC 60 min.

12.6

4280

2970

SAC/SnPb 10 min.

6.12

4290

2040

SAC/SnPb 60 min.

6.92

4244

2200

Figure 11 Effect of dwell time on lead-free second-level assembly (Bath et al., SMTAI 2005). Probability Plot for PBGA676 Weibull - 95% CI Type 1 (Time) Censored at 3010 - LSXY Estimates

Percent

99 Variable CELL 4-1 CYCLES

90 80 70 60 50 40 30

CELL 4-2 CYCLES Table of Statistics Shape Scale Corr F C 3.05826 2134.11 0.796 13 15 3.55491 2703.35 0.848 7 22

SAC 305

20 10 5

SAC 405

3 2 1

100

1000

10000

Data Note: CELL 4.1 had one component fail at thermal shock - removed from sample

Figure 12 Effect of lead-free SAC silver content on solder joint reliability of PBGA 676 (- 55/125◦ C).

6 CSP-TC and Vibration Behavior (Sn37 Pb)

535

Cell

Event Plot for PBGA676 ALL CELLS (0–100 °C) 1-1 1-2 1-3 1-4 1-5 1-6 2-1 2-2 2-4 2-5 2-7 3-1 3-3 3-4 5-1 5-2 0

1000

2000

3000 Cycles

4000

5000

6000

Figure 13 Effects of various assembly parameters on cycles to failures of SAC405/SAC305 PBGA 676 I/Os subjected to 0–100◦ C. • SAC405 on various surface finishes (cells 3-1 to 3-6) • Thick board (0.125 in.), primary and reworked (5-1 and 5-3)

Test results for assemblies with various boards, solders, and ball alloys are shown in Fig. 13 where the y axis represents various testing conditions. It is apparent that most of these assemblies showed no failures to 6000 thermal cycles or failures were limited. Weibull’s plots could not be generated due to insufficient number of failed samples.

5.4 Test Results for High-Reliability Applications In collaboration with coauthors, the test vehicles built at the EMS facility were tested under thermal cycling conditions that are generally considered for high-reliability applications. The PBGA 676-I/O assemblies were subjected to three thermal cycle conditions. All assemblies passed 200 thermal cycles with the following ranges and number of samples tested: • (−130/85◦ C), 4 parts • (−55/125◦ C), 6 parts • (−55/100◦ C), 8 parts

6

CSP-TC AND VIBRATION BEHAVIOR (Sn37 Pb)

6.1 Background Area array packages, in general, and CSP and flip-chip dies specifically lack thermal and mechanical resistance generally observed for PTH and leaded package assemblies soldered with Sn37 Pb alloys [26–30]. Lack of reliability resistance is further aggravated with the use of lead-free solder alloys especially under harsh thermal cycling and dynamic loading, such

536 Thermal Cycle and Vibration/Drop Reliability of Area Array Package Assemblies as drop and vibration. For these reasons, underfills are used in industry to improve solder joint reliability of these types of package assemblies in the early implementation of finer pitch area array packages even for tin–lead solder alloys. Underfills are used both within packages and on the PWBs. They are intended to absorb the CTE mismatch and reduce stress by distributing more uniformly among the solder joints. Underfilling, however, is undesirable because of additional process requirements, increased cost, and reduced manufacturing throughput. Another drawback of underfill is the inability to rework defective parts. Progress has been made to reduce the negative impact of underfilling by shortening the process time through the use of snap cure polymers and enabling reworkability by the development of reworkable underfills. So, it is generally accepted that underfill improves reliability; then one thought might be that if everything else failed to improve reliability of array packages/chips, underfilling might be the ultimate but undesirable solution. CSPs indeed were undefilled when for the first time Sony introduced its passport-size camera in early 1997, possibly to eliminate potential early failures. In this section, the test results [26] showing the effect of underfilling on the long-life thermal cycle behavior of various CSPs assembled with Sn37 Pb solder was first presented followed by vibration test results for virgin and cycled assemblies. Cycles to failure to 3000 cycles in the range of –30 to 100◦ C and 1500 cycles in the range of –55 to 125◦ C are presented. The investigation aimed at answering some of the key questions on the interaction of package type and underfill on thermal cycle behavior at the time when these types of packages were just started to be used by industry. Further work was carried out on the same types of test vehicles to characterize random vibration behavior under two Grms levels (7.8 and 16.9) and three time intervals (2, 4, and 6 hr). The purpose of testing was to determine time to failure (TTF) and to develop an understanding of failure mechanisms of CSPs with and without underfills. TTF for virgin assemblies with numerous chip-scale packages and a control TSOP were also subjected to two vibration levels for up to 6 hr. Results of assemblies with and without underfill along with SEM cross-sectional photomicrographs of failed specimens are also presented.

6.2 CSP Test Matrix and Procedures Eleven packages from 28 to 275 I/Os, as listed in Table 4, are evaluated under vibration. A photo of an assembled test vehicle held in two sides in preparation for vibration is shown in Fig. 14. The vibration testing was performed at a consortium team member, Boeing. The PWBs were fabricated, mostly from FR-4, 4.5 × 4.5 in. and 0.062 in. thick. Three types of Sn37 Pb solder pastes were included: no clean; water soluble (WS); and rosin mildly activated (RMA). Packages had different pitches, solder ball sizes and compositions, and daisy chain patterns. A few were underfilled even though it was known that they might not require underfilling. This was done in order to better understand the impact of underfill on solder joint reliability for different CSP types. On the other hand, package O required underfill; a few were not underfilled in order to better understand the reliability consequence of not using underfill for this package. TSOP was used as a control one and was not underfilled. Two assemblies, SN 037 without underfill and SN 062 with underfill, were mounted on the vibration shaker table using simple support fixturing (translation is fixed, rotation is not). As shown in Fig. 14, for each test vehicle (TV), one response accelerometer was placed near the center of the board and the other accelerometer was placed near the ceramic package. This fixturing was chosen since modeling has shown that using fixed edge boundary conditions results in excessive restraint of curvature/displacement during vibration testing. If a fixed edge is used, it is analytically shown that an unreasonably severe acceleration level (> 50Grms ) may be needed to initiate failures in leaded components within the desired 3-hr timeframe. A force of 10 ft-lb was used to tighten the clamps around the TVs.

6 CSP-TC and Vibration Behavior (Sn37 Pb) Table 4

CSP Package Configuration Matrix

Package ID

Package Style

B C D E F G

Leadless-1 TAB CSP-2 TSOP44 Leadless-2 TAB CSP-1 Chip on Flex-1 (COF-1) Wire bond on Flex-1 Wire bond on Flex-2 Chip on Flex-2 (COF-2) Ceramic CSP Wafer level

J K M N O

Package Size

Pad Size

Pitch

7 × 13.6 7.43 × 5.80 18.61 × 10.36 7 × 12.3 7.87 × 5.76 0.3 × 0.3 in.

0.35 × 0.7 0.4 0.27 × 0.5 0.30 × 0.75 0.4 .010 in.

12.1 × 12.1

0.375

0.8

12 × 12

0.25

0.5

0.5 × 0.5

.010 in.

15 × 15 0.413 × 0.413

0.4 .010 in.

537

I/O Count

Package. Thickness

Ball Diameter

28 40 44 46 46 99

0.8 0.885 1.13 0.8 0.91 1.75

— 0.3 n/a n/a 0.3 0.3

144

1.4

0.5

176

0.5

0.3

.020 in.

206

1.75

0.3

0.8 .020 in.

265 275

0.8 —

0.5 0.3

0.8 0.75 0.8 0.5 0.75 .020 in.

Note: All measurements are in millimeters unless otherwise specified.

Figure 14 Details of wire connections and location of two accelerometers (near the center).

The TVs were monitored continuously for daisy chain open during the thermal cycle and random vibration test for electrical interruptions and opens. The criteria for an open solder joint specified in IPC 9701 [9] was used to interpret electrical interruptions for both thermal cycle and vibration testing. Failures detected by continuous monitoring were verified manually at room temperature (RT) after subsequent removal from a thermal cycle chamber or from the random vibration table. For vibration, checking was done at 2-hr intervals. Monitoring

538 Thermal Cycle and Vibration/Drop Reliability of Area Array Package Assemblies was performed using an Anatech event detector which has 2000-ns detection limits for opens, where opens are loop resistances of >1000 . Two thermal profiles with the following profiles were used: • Cycle A: The cycle A condition ranged from –30 to 100◦ C and had an increase/decrease

heating rate of 2–5◦ C/min and a dwell of about 20 min at the high temperature to assure near-complete creep of the solder. The duration of each cycle was 82 min. • Cycle B: The cycle B condition ranged from –55 to 125◦ C, with a very high heating/cooling rate. This cycle represents near-thermal shock since it utilized a three-region chamber: hot, ambient, and cold. Heating and cooling rates were nonlinear with dwells at the extreme temperatures of about 20 min. The total cycle lasted approximately 68 min. Two assemblies, SN 037, without underfill, and SN 062, with underfill, were exposed to 2 hr of random vibration testing at a level of 7.8Grms followed by 2 hr at 16.9Grms . After a total of 4 hr of exposure at the two levels, the assembly with no underfill (SN 037) was removed for failure analysis. The assembly with underfill (SN 062) was subjected to further vibration at the level of 16.9Grms for two additional hours.

6.3 Results 6.3.1 Thermal Cycle Results: Positive/Negative Effect of Underfill CTF test results for assemblies with underfill were analyzed and compared to standard assemblies. Three categories of CSP assemblies with underfill based on their impact on thermal cycle reliability were identified: (1) improvement by underfilling, (2) minimal effect, and (3) degradation due to underfilling. A summary of CTFs for the TAB CSP-1 assembly that underfilling had negative effect, that is, it reduced CTFs, is shown in Table 5. It includes test results to 3000 cycles under condition A (-30/100◦ C) and to 1500 cycles under condition B (-55/125◦ C). Under both cycle conditions, assemblies with underfill showed much lower CTFs than those with no underfill. Assemblies with no underfill showed no failure to 3000 cycles under condition A, whereas those with underfill failed at 996, 1385, 1727, and about 2000 cycles. Note that the TAB CSP package is specifically designed to decouple the die CTE mismatch, thus improving solder joint reliability by use of a stress dampening elastomeric materials layer and flexible TAB lead interconnects. Results for a leadless 28-I/O package with improved CTF for underfill condition are plotted in Fig. 15. As expected, cycles to failure increased as temperature cycling range decreased. CTF for condition B ranged from 372 to 546 cycles with the scale factor [N 63.2% (No)] of 465 cycles and the Weibull shape factor (m) value of 7.2. For condition A, the CTF ranged from 641 to 1007 cycles with N63.2% of 839 cycles and m value of 7.5. Assemblies with underfill showed only one failure at 1374 cycles, out of four assemblies, when they were subjected to a total of 1500 cycles under the B thermal cycle condition. Under condition A (30–100◦ C), no failure was observed for the underfilled assemblies to a total of 3000 cycles. These limited test results clearly indicate that significant improvement may be achieved by underfilling for this category of peripheral leadless package. 6.3.2 Random Vibration Test Results A representative spectrum of power spectral input versus time for the 7.8Grms is shown in Fig. 16. Response plots for both assemblies with and without underfill are shown in Fig. 17. Resonance frequencies are very close to each other. TTFs under random vibration for assemblies with and without underfills are listed in Table 6. Note that the first 2 hr

6 CSP-TC and Vibration Behavior (Sn37 Pb) Table 5

539

Comparison of Cycles to Failure for Assemblies with and without Underfill

Package and Thermal Cycle Condition

No Underfill Number and Cycles to Failure

With Underfill Number and Cycles to Failure

3 out of 3 failures at 32 (?), Package F, TAB CSP, 3 out of 10 failures at 142, 710 cycles -55–125◦ C, B, 1500 cycles 709, 896, and 1380 cycles Package F, TAB CSP-1, No failure (15 assemblies) 4 out of 4 failures at 996, 1385, -30–100◦ C, A, 3000 cycles 1727, and <2000 cycles Note: CTF decrease due to underfilling.

100

B28, Leadless (−55/125 °C, B) Weibull, m = 7.0, No = 465 B28, Leadless (−30/100 °C, A) Weibull, m = 7.5, No = 838

90

Cumulative Failure Percentage

80 70 60

One underfill assembly failed at 1374 B cycles Two no failure to 1500 cycles

50 40

No failure of 4 assemblies with underfill to 3000 A cycles

30 20 10 0

0

100

200

300

400

500

600

700

800

900

1000

1100

Number of Thermal Cycles

Figure 15 Cycles to failures for a 28-I/O leadless package without and with underfill and their Weibull distributions. CTF increased for underfilled assemblies. (120 min) random vibration was performed at 7.8Grms . Except for chip-on-flex with 206 I/Os, all assemblies survived 2 hr at this level of random vibration. The TSOP that was used as a control package was the only one that showed no failure to 6 hr of random vibration (2 hr at 7.8 and 4 hr at 16.9Grms ). Note that the TSOP was not underfilled in either assembly and both showed no failures. In contrast, the wafer-level CSP was undefilled on both assemblies and their TTFs were in the same range. Most non-underfilled assemblies failed within half an hour after the random vibration level increased from 7.8Grms to 16.9Grms . The one with the longest survival time was the TAB CSP, 46-I/O package; that one showed a negative effect of CTF with underfilling. In general, underfilled assemblies had higher TTF and survived a total of 3 hr of random vibration; three

540 Thermal Cycle and Vibration/Drop Reliability of Area Array Package Assemblies Test Level: 0.000 dB Test Time: 000.02.01

Reference RVS: 7.600 Clipping: 3.00 Sigma

Test Range: 10.000, 200.000 Hz Resolution: 2.500 Hz

100 Control 10

1 Log g2/Hz DOF 200 RMS: 7.785 g

0.1

0.01

0.001 10

100 Log

11:19:33 19-Dec-2002

1000

2000

Frequency (Hz)

JPL Circut Boards, Run #64 Z-axis Random 0.085G^2/hz 7.8 G'sRMS Test Name: in2011,F15_ACME_CIRC_CARDS.099

Figure 16 Random Vibration spectrum with 7.8Grms level used for first 2 hr of testing.

Test Level: 0.000 dB Test Time: 0C0-01:49

Reference FMS: 7.800 Clipping 3.00 Sigma

Test Range 10000, 2000 000Hz Resolution 2500 Hz

100 Auxiliary

3

Auxiliary

4

10

Refrence

1 Log g2/Hz

0.1

RMS: 12397 G

Test Range 10000, 2000 000Hz Resolution 2500 Hz

Reference FMS: 7.800 Clipping 3.00 Sigma

DOF 200

10217 G 7,200 G

0.01

0.001 10 11:19:23 19-Dec-2002

Log

Frequency (Hz)

100

JPL Circuit Boards Run #64 Z-axis Random 0.85G^2hz 7.8 G'sRMS

7.8 Grms no underfill

1000

2000

#3 JPL037 CENTER RESPONSE 53-317721 05876 #4 JPL037, TESSARA RESPONSE 53- 5737 5-866

0.01

0.001 10 11:19:26 19-Dec-2002

Log

Frequency (Hz)

100

1000

2000

#3 JPL037 CENTER RESPONSE 53-317721 05876 #4 JPL037, TESSARA RESPONSE 53- 5737 5-866

7.8 Grms With underfill

Figure 17 Random vibration responses to 7.8Grms level showing resonance frequencies for assemblies with and without underfill.

6 CSP-TC and Vibration Behavior (Sn37 Pb) Table 6 Underfill

541

Random Vibration Failure Times (min) for Assemblies with and without

Package and I/O

No Underfill (SN037) Time, min (total Time 4 hr, 2 hr at 7.8Grms , 2 hr at 16.9Grms )

With Underfill (SN62) (Total Time 6 hr, 2 hr at 7.8Grms , 4 hr at 16.9Grms )

TSOP, 44 I/Os

No Failure (increased resistance, 4 hr, RT ) 136.7 (120 + 16.7) 152.9 (120 + 32.9) 223.1 (120 + 113.1) 155.9 (120 + 35.9) 161.6 (120 + 41.6)

No failure

No part

No failure

No part No part 136.7 (120 + 16.7) 164.7 (underfill)

No failure 18.2 234.4 (120 + 114.4) 182 (120 + 62)

Leadless-1, 28 I/Os Leadless-2, 46 I/Os TAB CSP-1, 46 I/Os TAB CSP-2, 40 I/Os Wire bond on Flex-1, 144 I/Os Wire bond on Flex-1, 176 I/Os Chip on Flex-1, 99 I/Os Chip on Flex-1, 206 I/Os Ceramic CSP, 265 I/Os Wafer-level CSP, 275 I/Os

No failure 207.9 (120 + 87.9) 330.4 (120 + 110.4) 207.9 (120 + 87.9) No failure

assemblies showed no failures to a total of 6 hr of random vibration (2 hr at 7.8Grms , 4 hr at 16.9Grms ). TTF interruptions observed for the first time as well as the second and tenth electrical interruptions were documented as defined by the IPC9701 specification. It was found that time differences between the first and the tenth interruptions are generally small with a maximum difference of 11.6 min. One assembly, chip-on-flex, 206 I/Os, showed a significant difference between the first and the tenth electrical interruptions. The first failure occurred after only 18.2 min and it took another 163 min before the tenth interruption was detected during continuous electrical monitoring. 6.3.3 Scanning Electron Microscopy Scanning electron microscopy and cross-sectional microscopy photomicrographs for the 28I/O leadless assemblies after a total of 4 hr of random vibration (2 hr at 7.8Grms and an additional 2 hr at 16.9Grms ) are shown in Figs. 18–21. Even though this part failed after 137 min of testing, no significant microstructural damage was observed when cross-sectioned after 4 hr. There are, however, signs of hairline cracks that can visually be observed only at higher magnifications. These microcracks are observed only for the pins located on the top right corner, as shown in Fig. 18. From cross-sectional micrographs, fine cracks in solder are apparent under the pin close to body. The crack is propagated an angle under the package body toward the void formed on top of the microvia with further initiation away from the microvia and its extension to the toe of the solder joint. These cracks are different from those generally formed due to thermal cycles that show clear signs of microstrutural changes with widening between the crack surfaces. Comparison of crack formation in TSOP under thermal cycles and vibration is shown in Fig. 21. Location, size of crack, microstructural changes, and crack propagations are different for the two failure types.

542 Thermal Cycle and Vibration/Drop Reliability of Area Array Package Assemblies

Figure 18 SEM photomicrograph of pins 1–4, top left corner, away from the center of board, after 2 hr at 7.8Grms and an additional 2 hr at 16.9Grms . No apparent damage due to vibration.

Figure 19 SEM photomicrographs of pins 25–28, top right corner, away from center of board, after 2 hr of vibration at 7.8Grms and an additional 2 hr at 16.9Grms . Hairline cracks are apparent on the enlarged pin 28 (left) and pin 27 (right).

7 BGA 225-I/O Vibration: Sn37 Pb and SAC397

543

Figure 20 SEM cross-sectional photomicrographs of pins 1 and 28 (top left and right corners) with hairline cracks in solder joint under the pin extended to the toe.

7

BGA 225-I/O VIBRATION: Sn37 Pb AND SAC397

7.1 Introduction The JCAA/JG-PP Lead-Free Solder Team performed a comprehensive characterization of lead-free electronic package assemblies subjected to various thermal and mechanical conditions representing harsh environmental applications [31]. Vibration testing was conducted by Boeing Phantom Works in Seattle with a comprehensive analysis to project in-service life using accelerated vibration test data [32, 33]. Additional modeling and analyses were also performed by industry and university to project times to failure for PBGA 225-I/O and other packages at different locations and loading conditions [34, 35]. Key aspects of testing, data analyses, and modeling are discussed in the following. Using these data and analysis as a basis, two Weibull plots were generated in one graph comparing times to failure for the tin–lead/tin–lead and SAC/SAC BGA 225-I/O assemblies. The Weibull plots are also presented.

7.2 Testing and Vibration Analysis by Boeing The JCAA/JG-PP test vehicles were divided in two types: “manufactured” with a board glass transition (Tg ) temperature of 170◦ C representing a lead-free requirement and “rework” with a Tg of 140◦ C representing the standard board for tin–lead applications. A number of the rework TVs were also assembled using parts with lead-free solder alloys/surface finish. The surface finishes were silver and tin–lead HASL, respectively. The main board was a six-layer, 0.09-in.-thick, circuit board with 12.5 × 9 in. The board was populated with 55 daisy chain packages consisting of ceramic leadless chip carrier (CLCC), plastic leaded chip carriers (PLCCs), thin small outline packages (TSOPs), thin quad flat packs (TQFPs), dual in-line packages, (DIPs), and BGAs. The daisy chain package and board completed resistance circuits for subsequent failure monitoring. The solder alloys selected included standard Sn37 Pb

544 Thermal Cycle and Vibration/Drop Reliability of Area Array Package Assemblies

(3,000–30/100°C, 877Cycles–55/125°C) (a) Failure of TSOP under two ranges of thermal cycles

(b) Failure of TSOP under 4 hr of vibration (2/2 hr at 7.8/16.9Grms)

Figure 21 Comparison of crack formation in TSOP under thermal cycles (top) and under vibration (bottom). Location, size of crack, microstructural changes, and crack propagations are different. (a) Failure of TSOP under two ranges of thermal cycles. (b) Failure of TSOP under 4 hr of vibration (2/2 hr at 7.8/16.9Grms ). and lead-free SAC396 (Sn3.9 Ag0.6 Cu) for solder balls in BGA and solder paste for reflow assembly. Detail on assembly and rework procedures as well as use of other solder alloys are posted at the JCAA/JG-PP team website. Thirty (in batches of 15) out 205 TVs were subjected to random vibration. Prior to vibration testing, first, the modal behavior of a “Pathfinder” TV was characterized by a laser vibrometer. The laser vibrometer data allowed the determination of the resonant frequencies and the actual deflection shapes of the printed wiring assembly (PWA) during the test. In addition, the strains

7 BGA 225-I/O Vibration: Sn37 Pb and SAC397

1 *Auxiliary Ch 8 Fundamental Reference

Sweep Number: 1.00 Sweep Rate 1: 1.000 oct/min Compression: Variable

Elapsed Time: 000:08:39 Filter Type: Proportional Fundamental: 80.000%, BB RMS: 509.mcyc

545

Remaining Time: 000:00:00 Test Range: 5.000, 2000.000 Hz Ponts Per Sweep: 2000

0.1

0.01

Log Displacement in(pK-pK)

0.001

0.0001

1e-05

1e-06 5 06:45:52 02-Sep-2004

Figure 22

10 Log

100

1000

2000

Frequency (Hz)

TN: 2052 No lead Circuit Board, 1st Set of 15 Boards RUN#6 Z-Axis, Sine SWEEP 1.0 Gpk Operator:TDK Sine Test Name: In2052_NoLead_Circuit_bd:002

Z RESPONSE, Board #008

Displacement vs. frequency under 1G sin sweep (manufactured PWA).

generated during a 1G sine sweep were calculated for 1053 points on the Pathfinder PWA. Vibration performed in increasing steps, in 60 min, in an effort to fail as many packages as possible within a short time. The test started at 9.8Grms levels and continued at 12, 14, 16, 18, 20, and 28Grms exposure with a total of 420 min. Figure 22 shows the displacement of a test vehicle versus frequency, as follows from accelerometer data during a 1G sine sweep of a manufactured test vehicle in the z axis. This illustrates that the maximum displacement (and therefore the most solder joint damage) is associated with the first mode (at 72 Hz). Figure 23 shows a plot of deflection under a 1G sine sweep for the first mode of vibration. The time to failure (in minutes) for each package on a typical manufactured test vehicle is shown in Fig. 24. As noted earlier, the random vibration spectrum was increased in amplitude every 60 min and the total length of the test (in the z axis) was 420 min. Only the z -axis test minutes are shown since shaking in the x and y axes produced little damage. Any package that did not fail during the test is shown as having survived 420+ minutes. Note that the packages that failed first were those on the centerline of the vehicle and those down the sides of the vehicle (near the wedgelocks). Therefore, the packages that failed first coincide with the regions of highest curvature induced by the first bending mode. A large number of X sections were performed on failed specimens. Figure 25 gives representative SEM photomicrographs showing fine microcrack formation for BGA 225 I/Os. A commercially available software package developed for solder joint lifetime prediction under vibration was used by the software producer company [34] and by Dr. Woodrow [33] in order to model vibration test data. The software requires the slope of the S –N plot (stress vs. cycles to failure) for the solder in order to calculate the damage accumulation rates. The S –N slopes (−1/b) used for Sn37 Pb and SAC solders were assumed to be -0.100035 [36] and -0.11066, respectively. The author states that the S –N plot for SAC has not been definitively determined and that this is the best number available at this time. The summary of the predicted lifetime for Sn37 Pb and SAC BGA 225 I/Os subjected to a constant 3Grms until

546 Thermal Cycle and Vibration/Drop Reliability of Area Array Package Assemblies

X 10−4 1.5 1 0.5 0 −0.5 −1 −1.5 −2 0 −2

Figure 23

−4

−6

−8

−10

0

6

4

2

12

10

8

Full field peak strains at 72 Hz (1G sine dwell, Pathfinder PWA).

258 169

318

368

420+ 187

192

420+

420+ 147

111

189

420

126

420+ 143

302

127 189

420+ 295

420+

129 420+

389

420+

272

301 300 420+

375

420+

373

420+

420+

331

188

277 121

82

363

362

97

420+

393

399

420+

420+

182

369

113

Minutes to Failure (PSD Increased Every 60 Minutes) 1-58

61-120

121-180

181-240

241-300

301-360

361-420

Figure 24 Times to failure for package and location (Pathfinder PWA).

7 BGA 225-I/O Vibration: Sn37 Pb and SAC397

SN8-U18

Sn37Pb/Sn37Pb

SN77-U18

547

SAC/SAC

Figure 25 SEM X-sectional photomicrographs showing fine microcrack formation for Sn37 Pb/Sn37 Pb (left) and SAC/SAC assemblies subjected to vibration at different levels. Table 7 Comparison of Predicted Times to Failure (years) for Sn37 Pb/S37 Pb and SAC/SAC Assemblies Under Assumed Constant Vibration Level of 3Grms (0.0062G 2 Hz−1 ). TV Condition TV TV TV TV TV TV

5, Sn37 Pb/Sn37 Pb 7, Sn37 Pb/Sn37 Pb 8, Sn37 Pb/Sn37 Pb 75, SAC/SAC 77, SAC/SAC 7, Sn37 Pb/Sn37 Pb

U4 BGA225 I/O Years

U6 BGA225 Years

14.3 19.2 40 0.9 0.6 0.8

297 7010 330 47 13.3 1.6

U18 BGA 225 I/O Years 4,145 23,800 35,850+ 2,430 102 228

Note: See [33].

failure (0.0062G 2 Hz−1 input into the first resonance) is shown in Table 7. The predictions were obtained by extrapolation of accelerated vibration test data and the following equation: Field lifetime = equivalent life minutes (at 9.9Grms ) × [(9.9Grms )/(Grms in field)]b . The test data used for extrapolations were from the test vehicles that had similar transmissibilities so that fair comparisons could be made. The comparisons reveal that the predicted Sn37 Pb solder joint lifetimes are much longer than those predicted for SAC in the same

548 Thermal Cycle and Vibration/Drop Reliability of Area Array Package Assemblies vibration environment (approximately 20 times longer for U4). This could be the cause for concern for some Pb-free electronics depending on their use environments. Further test and analysis should be done before SAC is widely accepted for use in high-reliability electronics.

7.3 CALCE Modeling of Vibration Data Preliminary simulation work for the JG-PP test article was also conducted at CALCE [35]. The finite element array (FEA) model projected vibration natural frequencies and mode shapes. Analysis also showed that boundary stiffness for the rotational spring was required to be modified such that FEA results match experimental test results. Based on adjusted results for the first natural frequency, it was reported that the second mode frequency and shape (coordinate function) were projected to be the same as experimental results. The PSD response from experiment shows that the first mode contributes more than 90% to the total strain response. FEA provided an estimate of the corresponding strain response everywhere on the PWB. From the FEA strain response plot, the strain scaling factors at the location of each BGA225 I/O package was calculated. The strain scaling factors near each package relative to strains in the PWB near package U55 with the highest strain are shown in Table 8. The strain values were locally averaged over the length scale of the strain gage characteristic dimensions. It was stated that these are in qualitative agreement with experimental laser vibrometry results. The generalized strain life Coffin–Manson durability was used to estimate more complex step stress excitation experimental data. Different packages experience different loading histories based both on their locations on the PWB and the excitation history. Excitation level was normalized to 10Grms (lowest vibration level). The result shows that Sn37 Pb has a larger durability scale factor (larger acceleration factor for vibration testing) than SAC396(e.g., 434 factor for Sn37 Pb vs. 143 for SAC at 28G excitation). Table 9 lists slopes from the test data as well as the slopes of the high-cycle-fatigue (Basquin) curve and low-cycle-fatigue (Coffin–Manson) curve for comparison [35]. The experimental data agree approximately with the secant slope, thus providing some confidence in the fatigue exponent used in the study. The exponents (slopes) used in the fatigue model are slightly smaller than the test data for Sn37 Pb and slightly larger than the test data for SAC396. The test results are presented in plots with normalized scales so that the slopes (fatigue exponents) could be compared without having to compare the intercepts (fatigue coefficients). The equivalent durability for all BGA225 packages were calculated based on the experimental durability (times-to-failure) results, the step stress excitation history, and the durability scale factors. The results adapted from CALCE are shown in Fig. 26. The x axis is the equivalent durability cycles at 10Grms (times to failure in seconds converted to equivalent cycles by multiplying the frequency of the first mode). Similar normalization could have been done Table 8 Example of Durability Scale Factors for BGA225-U4 for Both Sn3Pb and SAC Assemblies Excitation (Grms ) Durability scale factors for package U4 Sn37 Pb SAC Note: From [35].

9.9G

12G

14G

16G

18G

20G

28G

1 1

4.26 3.70

11.09 8.76

21.14 15.27

27.63 18.11

45.12 28.19

434.21 143

7 BGA 225-I/O Vibration: Sn37 Pb and SAC397

549

Table 9 Durability Exponents from Fatigue Model and Experimental Data for Sn37 Pb and SAC Solders.

Basquin exponent Coffin–Manson exponent Secant exponent Experimental exponent

Sn37 Pb

SAC

-0.093 [37] -0.52 [37] -0.13 -0.158

-0.11[38] -0.49 [38] -0.35 -0.27

Note: From [35].

Normalized PCB flexure

10

U55U4

U4

U43

U5

U44

1 U5 U43

SnPb

U55

U21 U6

U6

U21

SAC

U2

U44 U2

0.1 1

10 100 1000 Normalized number of cycles to failure

10000

Figure 26 Normalized durability ranking for BGA225 components at various locations on the PWB test specimen.

for any other Grms level. The durability data were normalized with respect to durability of component U43 SAC, which failed first in the test. The y axis shows the magnitude of the PWB flextures, normalized with respect to the highest PWB flexture which is near the component U55. The author states that the plot provides a convenient way to demonstrate the reduction in vibration durability when making the transition from Sn37 Pb to SAC solder.

7.4 Discussion and Weibull Plots for Sn37 Pb and SAC396 In [32], Weibull plots of the data were used to help to determine the ranking of Sn37 Pb and SAC396 due to vibration at various levels. In these plots, no adjustment was made for times to failure due to acceleration Grms steps in an early analysis. The author believed that the test data would not be suitable for Weibull analysis since the test conditions were changed during the test [i.e., the power spectural density (PSD) was increased every 60 min]. However, even using raw times-to-failure data, Weibull plots of data gave excellent line fits for the z -axis vibration data. The author postulated that this “goodness of fit” suggests that the actual strains experienced by the test vehicles did not track the input PSD levels. The Grms for the first mode measured using the accelerometer showed nonlinearity with respect to the input PSD level.

550 Thermal Cycle and Vibration/Drop Reliability of Area Array Package Assemblies Wedgelock/fixture interface was considered to be a potential source for such nonlinearity. This unexpected nonlinearity does not have any real impact upon the results. Even though such Weibull plots were visually appealing for possible ease of data comparison, they lacked fundamental features for comparison of vibration test data. In such plots, each data point was not representative of an identical Grms -level failure. A data point on the Weibull plot at the same failure percentage for two conditions may come from significantly two different Grms levels depending on time to failure. In addition, these plots are only applicable for a specific location of each package and a universal comparison for the same type of solder assembly could not be made. To generate meaningful plots that also enable the comparison of a large set of failure data, approaches similar to those presented by Dr. Woodrow [32] in his later paper and by CALCE [35] are considered. Figure 27 shows three-parmeter Weibull plots comparing failures with equivalent times to failure for a number of SAC396/SAC396 and Sn37 Pb/Sn37 Pb BGA225 assemblies. Failures for U4, U6, U18, U43, and U55 were included in this plot. It clearly demonstrates the higher durability trend for Sn37 Pb compared to SAC396. This trend is consistent with those generated in [35] for all data and in [32] for each individual package and package location. This plot, however, is more comprehensive and shows the trend for more data sets. Equivalent times to failure were estimated relative to the component (U18) with the lowest strain because of its location on board, as well as scaling Grms to the lowest value of 9.9Grms . The relative values given to cover the effect of locations are U55=1, U4=0.97, U5=0.96, U43=0.85, U6=0.68, U44=0.68, U21=0.63, U56=0.31, U2=0.29, and U18=0.27. The approach indicates complexity and uncertainty associated with the results of a complex board under step vibration testing.

Random Vibration for SAC/SAC and SnPb/SnPb 99.000

Probability-Weibull BGA225 SAC/SAC- All Weibull-3P RRX SRM MED FM F=25/S=0 Adj Points Unadj Points Adjusted Line Unadjusted Line

90.000

SAC/SAC

Fai lure Percentage (%)

50.000

SnPb/SnPb

BGA225 SnPb/SnPb- All Weibull-3P RRX SRM MED FM F=27/S=0 Adj Points Unadj Points Adjusted Line Unadjusted Line

10.000

5.000

1.000 10.000

100.000

1000.000

10000.000

100000.000

Reza Ghaffarian JPL 5/21/2009 11:49:28 AM 1000000.000

Time (Minutes)

BGA225 SAC/SAC- All: BGA225 SnPb/SnPb- All:

β=0.6320, η=5637.0161, γ=443.1362, ρ=0.9877 β=0.5572, η=4.8876E+4, γ=2590.3124, ρ=0.9815

Figure 27 Weibull plots for times to failure of all BGA225 after adjustment for location and Grms levels comparing Sn37 Pb and SAC396 assemblies.

8 CBGA Assemblies: Vibration and Thermal Cycle Synergism

8

551

CBGA ASSEMBLIES: VIBRATION AND THERMAL CYCLE SYNERGISM

8.1 Introduction 8.1.1 Purpose Thermal cycling characteristics of BGA assemblies have been widely reported, including those discussed previously in this chapter. Thermal cycling represents the on–off environmental condition for most electronic products and therefore is a key factor that defines reliability. As a result, much data are available for accelerated thermal cycle conditions, but very limited data are available on vibration and shock representative of aerospace applications. Test vehicles with daisy chain plastic and ceramic BGAs (CBGAs) ranging from 256 to 625 I/Os were subjected to random vibration representative of a spacecraft launch environment. The effect of board rigidity on behavior was also investigated by adding aluminum ribs/strips onto the board or bonding the aluminum plate to the board. This section compares accelerated thermal cycles-to-failure data under four temperature ranges before and after thermal random vibration for CBGAs with 361 and 625 I/Os. Stress and strain projections by finite element analysis are also presented. 8.1.2 Background Cycles to failure after vibration of BGAs and CCGA were presented by M. Cole of IBM [39]. Balls and columns were high-melt solder (Pb90 /Sn10 ) with 0.89 mm (0.035 in.) in diameter for CBGA and for CCGA 0.5 mm (0.02 inch) in diameter with 2.2 mm (0.050 or 0.087 in.) in height. Packages were assembled onto FR-4 boards with single test specimens mounted on 110 × 90-mm board clamped by screws at 101 × 75-mm locations. MIL-STD 810E was used to generate impact and vibration data for the test vehicle. No failure of CBGAs and CCGAs was observed after vibration with a heat sink of 73 g. CCGAs with higher heat sink weights of 100 and 150 g failed whereas CBGAs did not. Cracks were induced in CBGAs in the eutectic solder either in package or board sites when subjected at 20–2000 Hz with 7.73Grms . It was reported that for CBGAs crack initiations were similar to those of accelerated thermal cycling (ATC), but with no deformation typically present in ATC. Also, thermal mismatch induces both shear and tensile deformations, but vibration induces primarily tensile and does not cause deformation. It was observed that CBGA assemblies with heat sinks of lower weight than 150 g, which were subjected to shock and vibration, did not show any degradation in thermal fatigue life; no statistical differences between those with initial shock and vibration and those without any a priori test were observed. In another study [40], CBGAs with 256, 625, and 1089 I/Os were subjected to thermal cycling and vibration evaluation. The 1089 was fabricated with balls rather commonly supplied in column since they were used on metal matrix restraining core rather than polymeric board such as FR-4. For restrained board with a CTE (6 ppm/◦ C) close to the ceramic package (7.2 ppm/◦ C), local mismatch between solder to package/board was considered to play a primary role rather than global mismatch. Assemblies on restrained board were subjected to thermal cycling in the range of -55◦ C/125◦ C (a 5◦ C/min ramp and a 30-min dwell at each extreme). They reported no failures to 1500 cycles. They also performed vibrations on three CBGAs at three levels: • Level I was the military avionics equipment workmanship vibration level of Environ-

mental Stress Screening (ESS). • Level II was an enveloped subassembly based on several avionics programs. • Level III was the same as level II with PSD increased by 3 dB.

552 Thermal Cycle and Vibration/Drop Reliability of Area Array Package Assemblies Table 10

Vibration and Thermal Cycling for CBGAs

Vibration Level

Duration (min)

Input Level (gr)

Response (gr)

CBGA 256 I/O

CBGA 625 I/O

CBGA 1089 I/O

I II III

10 60 120

6.06 12.8 18.07

21 37 53

Pass (4/4) Pass (4/4) Pass (1/4)

Pass (2/2) Pass (0/2) NA

Pass (3/3) Pass (0/3) NA

Test results are shown in Table 10. All assemblies passed the level I whereas only CBGA with 254 I/Os passed the level II vibration. Finite element prediction for level I agreed with experimental results but for other levels had a mixed agreement.

8.2 Why Acceleration Methods? There are many reasons to perform an accelerated environmental verification and testing program for electronic assemblies, including: • Qualification of design for in-service conditions • Modeling of in-service condition or to project life • Definition of manufacturing variables and their effects • Screening for manufacturing defects • Demonstration of quality and reliability of a design • Demonstration of suitability for the intended use

For electronics in commercial applications, thermal cycling tests are commonly performed to simulate on–off conditions. However, most electronic systems are exposed to other environments, including mechanical fatigue and random vibration. For example, vibration occurs during transportation and mechanical fatigue occurs by repeated use of key punching for portable electronics. Occasional high shock could occur due to accidental drops. Drop test and mechanical fatigue are now considered for qualification of electronic assemblies, especially for newer chip-scale package assemblies and for ranking of lead-free solder alloys. In addition to a much harsher thermal requirement for high-reliability applications, assemblies are generally required to meet severe dynamic loads and vibration fatigue cycling. Therefore, there is a strong need to understand assembly behavior under such stress conditions. To understand behavior in a harsher environment, several test vehicles with PBGAs and CBGAs with 1.27-mm pitches were subjected to dynamic testing representative of a launch condition, 3 min vibration/shock at each axis. Each test vehicle had four package types that were commercially available and had I/Os from 250 to 625. Synergistic stresses including thermal cycle representative of the following environmental conditions were considered: • Prelaunch thermal cycling due to manufacturing, repair, screening, storage, and

transportation • Dynamic testing representative of the launch environment included sinusoidal vibration,

transient vibration, pyroshock, and acoustic vibration • Thermal cycling representative of internal and external temperature change of a

spacecraft

8 CBGA Assemblies: Vibration and Thermal Cycle Synergism

553

Effects of several test variables were considered: (a) Thermal cycling before and after vibration/shock (b) Increase in rigidity of board by addition of thin rigid strips in one case and bonding to a rigid plate in another case (c) Change in thermal history condition by exposure to thermal cycles before vibration/shock This section presents experimental results, as well as analyses for two CBGA assemblies (361 and 625 I/Os) subjected to either thermal cycling and vibration or both, one followed by another [2].

8.3 Test Procedures 8.3.1 Test Vehicle The test vehicles in this investigation included both plastic and ceramic packages on either FR-4 or a polyimide printed circuit board (PWB) with six layers 0.062 in. thick. Ceramic packages with 625 and 361 I/Os were included in the investigation. Solder balls for CBGAs had high-melt-temperature composition (Pb90 /10Sn10 ) and were about 0.035 in. in diameter. The high-melt balls are attached to the ceramic package with eutectic solder (Sn63 /Pb37 ). During the reflow process, eutectic paste at the PWB side as well as low-melt solid solder at the package will be reflowed and then solidify at a lower temperature to provide the electromechanical interconnections. The CBGAs had internal daisy chains that made a closed loop with daisy chains on the PWB, enabling the monitoring of solder joint failures through a continuous electrical monitoring system. Daisy chains for CBGA 625 were in a ring form, center to peripheral, in order to identify failure sites with increased thermal cycles. The first failure is known to occur from the peripheral ring in the corner solder joints with the maximum distance to the neutral point (DNP). To improve assembly reliability, the supplier had removed package internal daisy chain connections among a few corner balls, excluding them from assembly failure detection during electrical monitoring. This means that cycles-to-first-failure data cannot be directly correlated to package diagonal dimensions that are usually assumed to be equal to the maximum DNP. This, however, is a realistic condition representing a package with an active die. 8.3.2 Thermal Cycling Conditions Four different thermal cycle profiles were used: • Cycle A (−30 to 100◦ C) and B (−55 to 100◦ C), same as before.

• Cycle C: The cycle C condition ranged from −55 to 125◦ C with 2–5◦ C/min heating/

cooling rate. Dwells at extreme temperatures were at least 10 min with duration of 159 min for each cycle. • Cycle D: The cycle D condition ranged from −55 to 125◦ C, the same as condition C, but with very high heating/cooling rate It could also be considered a thermal shock since it used a three-region chamber: hot, ambient, and cold. Heating and cooling rates were nonlinear and varied between 10 and 15◦ C/min. with dwells at extreme temperatures of about 20 min. The total cycle lasted approximately 68 min.

554 Thermal Cycle and Vibration/Drop Reliability of Area Array Package Assemblies The criteria for an open solder joint specified in IPC-SM-785, Section 7.8, were used as guidelines to interpret electrical interruptions. Generally, once the first interruption was observed, there were many additional interruptions within 10% of the cycle life. This was especially true for ceramic packages. 8.3.3 Vibration Three TVs with three levels of rigidity representing three loading conditions were stacked with spacers and subjected to random vibration. The bottom TV was bonded to an aluminum plate, the middle had center and edge stiffener strips, and the top had no stiffeners. The TVs were clamped from the two sides of the PWB, the sides with no connectors, as shown in Fig. 28. The block of three stacked TVs was mounted on a very stiff thick Al plate on the shaker table with their natural frequencies well above the TV assembly range. A very low sine sweep allowed us to determine natural frequencies for the TV. The boards were subjected to a vibration spectrum in the range of 200–2000 Hz.

8.4 Test Results 8.4.1 Natural Frequency Measurement/Projection Plots for natural frequencies were generated using a simplified finite element modeling analysis to predict also the stress/strain condition at the solder joints [4–41]. Ten percent model damping was used for the random response calculation. Projections for the first, second, and third natural frequencies are compared to measured values as listed in Table 11. 8.4.2 Damage Induced for Thermal Cycling Alone Interface cracking was observed both at the board and package sides which increased with increasing number of thermal cycles. Figure 29 shows typical failures for the two cycling conditions, A and D. Failure under condition A were generally from the PWB and for condition

Figure 28 Photos of random vibration test setup; left photo shows two stacks of three TVs on a vibration table and the right photo shows the enlarged TV with the CBGA361 on the top right corner.

8 CBGA Assemblies: Vibration and Thermal Cycle Synergism

555

Table 11 Measured and Finite Element Projection of Natural Frequencies and Stress/Strains for CBGA 361 TV Condition Measured values for no Stiffener FEA for TV with no stiffener FEA for TV with stiffeners FEA for TV bonded to Al plate

Frequency Mode 1 (mode 2,3)

Amplitude (g2 /Hz)

Maximum Stress (psi)

Maximum Strain (μStrain)

356 (688,1700)

190

N/A

N/A

381 (594,795)

170

945

315

593 (820,1625)

43

558

185

3155 (3520,4700)

N/A

25

8

Figure 29 Cross sections of failure sites for CBGA 625 after 350 cycles under condition A and CBGA 361 under condition D. D were from the package sides. Failure mechanism differences could be explained either by global or local stress conditions. Modeling indicates that the high-stress regions shifted from the board to the package when stress conditions changed from global to local. For the A cycling, with slow heating/cooling ramping, which allowed the system to reach uniform temperature, damages could indicate a global stress condition. For the D cycle with rapid heating/cooling, damages could indicate a local stress condition. 8.4.3 Damage Induced with a Priori Vibration Damage induced by vibration was different in some cases, as shown in Figs. 30 and 31. Appearance of tensile deformation from the center of high-melt balls as shown was significantly different from those observed for the thermal cycling condition. Similar to the thermal cycle condition, damage was more dominant for the balls with higher DNP, especially for

556 Thermal Cycle and Vibration/Drop Reliability of Area Array Package Assemblies

Figure 30 Thermally cycled samples after vibration. Note tensile failure for the corner ball and minimum damage for the center balls.

Figure 31 Thermal cycled samples after vibration. Note mechanical crack propagation at two locations and minimum damage at a center ball. the corner balls. However, additional microcracks in eutectic solder joints, different from the norm for thermal cycling, were induced by tensile and shear load during random vibration. 8.4.4 Cycles to Failure for Thermal Cycling Alone Figure 32 shows Weibull plots of cycles to first failure for CBGA 625 and CBGA 361 under four different thermal cycle conditions. To generate plots, cycles to failure for a population

8 CBGA Assemblies: Vibration and Thermal Cycle Synergism

557

100

Cumulative Failure Percentage

90

−55/125°C, Thermal Shock CBGA625

80

−55/125°C, Thermal Shock CBGA361 −30/100°C, CBGA625

70 60

−55/100°C, Vibration + Thermal Cycle CBGA361

50 40

−55/125°C, CBGA 625

30

−55/100°C, CBGA625

20 10 0 100

150

200

250 300 350 Number of Thermal Cycles

400

450

500

Figure 32 Cycles-to-failure data for two CBGAs under four thermal conditions with two assemblies with a priori vibration condition were ranked from low to high and failure distribution percentiles were approximated using median plotting position, Fi = (i − 0.3)/(n + 0.4) [42]. Then, a two-parameter Weibull distribution was used to characterize failure distribution. The Weibull cumulative failure distribution was used to fit cycles-to-failure data. The equation is   m  N F (N ) = 1 − exp − N0 where F (N ) = cumulative failure distribution function N = number of thermal cycles No = scale parameter commonly referred to as characteristic life, number of thermal cycles with 63.2% failure occurrence m = shape parameter This equation, in double-logarithm format, results in a straight line. The slope of the line will define the Weibull shape parameter. 8.4.5 Synergism of Vibration and Thermal Cycling Cycles to failures for three assemblies with levels of rigidity and a priori vibration condition are shown in Table 12. Two test data for cycles to failure after vibration for the most severe condition, that is, the test vehicles with no rigidity enhancement, were marked in Fig. 32 for comparison only. It is difficult to draw a statistically meaningful conclusion because of insufficient sample size.

558 Thermal Cycle and Vibration/Drop Reliability of Area Array Package Assemblies Table 12

9

Cycles to Failures After Random Vibration

TV Condition

ID- Vibration (3 axes) and Thermal Cycle Type

Thermal Cycles to failure

No stiffener under vibration

#4 Vib. + cycle B

With stiffeners

#50 Vib. + cycle B #34 Vib. + cycle A #33 Vib. + cycle A

Bonded to Al plate

#13 Vib. + cycle A #31 Vib.+ cycle A

#4, Failed between 292 and 326 cycles #50, Failed at 400 cycles #34, No failure to 434 #33, Failed between 292 and 326 cycles #13, Failed at 330 cycles #31, No apparent change to virgin test vehicles

DROP SHOCK BEHAVIOR WITH/WITHOUT ISOTHERMAL AGING OF Sn37 Pb AND LEAD-FREE BGA/CSPs

9.1 Background The primary objective of this part of the experimental research, carried out in collaboration with the Rochester Institute of Technology (RIT), is to characterize isothermal aging followed by drop integrity of surface mount mixed (Sn37 Pb and lead-free) assembly solder joints [11, 42–49]. The investigation involves both forward and backward compatibility. Forward compatibility (FC) refers to the compatibility of lead-free solder paste with lead-containing component finish. On the other hand, backward compatibility (BC) refers to the compatibility of tin–lead solder paste with lead-free component finish. A full factorial design is used in the investigation, with three factors—solder paste, component finish, and PWB surface finish. The Sn37 Pb paste and SAC305 (Sn–3.0% Ag 0.5% Cu) paste are used as Pb-containing and Pb-free levels, respectively. The PWB was assembled with a variety of packages, including flip chip (0.4-mm pitch), ultra-chip-scale package (UCSP), micro–lead frame (MLF) or quad flat pack no-lead (QFN), TSOP (0.5 mm pitch), and PBGA (1156 and 256 I/Os, 1 mm pitch). After assembly, the TVs were subjected to isothermal aging followed by drop tests to establish a relationship between the intermetallic growth at the solder/PWB interface and the mechanical integrity of solder joints. The compounding of tests, unlike singular test methods, provides a more realistic estimate of the reliability and life of the joint in the field. Representative assemblies were cross sectioned after tests to determine microstructural changes such as intermetallic growth due to isothermal aging and damage due to drops such as microcracks and growth paths.

9.2 Mixed Assembly (Tin–Lead and Lead-Free) Reliability Both backward and forward compatibility have issues and their reliability is not yet completely established because of the interplay of temperature and solder alloy metallurgy. Forward compatibility suffers from issues of low stand-off in area array packages due full melting of solder balls leading to poor compliance on the solder joints while backward-compatibility assemblies have issues with microstructural homogeneity of solder joints. The mixed assemblies

9

Drop Shock Behavior with/without Isothermal Aging of Sn37 Pb and Lead-free BGA/CSPs

559

are prone to voiding, especially in the case of FC assemblies. The effect of voids on solder joint reliability has not been established yet even though numerous studies have been carried out by industry. In the assembly of lead-free PBGA packages with tin–lead paste, solder joint metallurgical uniformity and reliability become major concerns [43]. Snugovsky et al. [44] have observed notable issues on similar lines with BC assemblies when subjecting them to thermomechanical fatigue (thermal cycling/shock). The authors observed weakening of the interface on the board side owing to increased lead concentration. This led to cracking of PBGA solder joints on the board-side interface, along the intermetallic region in BC assemblies. These cracks required lower energy to propagate than the ones through the solder matrix. Improper mixing, resulting in lead-rich regions in the solder joint, provides an easy path for cracks to propagate, especially when lead forms a continuous layer close to the interface. Prior experimental research [46] on the compatibility of solder joints revealed a better performance of BC and SAC assemblies when subjected to thermal shock loads. The work utilized a small ball grid device (PBGA169) to compare performances. Also, the work used HASL and OSP finishes only for the boards. The test methods utilized a lesser dynamic shear response to quantify the reliability and performance as opposed to dynamic methods like the drop test used by the industry to qualify and compare assembly performances. Prior work by Mattila et al. [49] on compounded loads for UCSP devices suggests that the location of failure can be one of the following: (a) the two interfaces (component–solder or solder–PWB), (b) bulk solder matrix, and (c) void-assisted crack propagation of pad side Cu3 Sn layer. The failure type was shown as a function of the load application. Failure mode (a) was observed in assemblies not subjected to either thermal cycling or aging, while those subjected to aging failed through void-assisted cracking, mode (c). The work also reveals that the aging phenomenon, as in the case of isothermal aging, produced a deteriorating effect on the solder joint while a cyclic thermal load improved the resistance to mechanical shock. The use of different surface finishes on PWB yields distinct intermetallic formation. Especially when using ENIG pad surface finish, a layer of Ni barrier is added between the gold coating and the copper pad in order to avoid the diffusion of copper and gold. This coating of Ni reacts with tin in the solder during reflow, forming a terminal intermetallic Ni3 Sn4 when using a Sn37 Pb eutectic alloy. This intermetallic is self-limiting in the sense that the growth of this phase stops beyond a particular width of its formation. The Ni/Ni3 Sn4 layers form a barrier between the copper in the pad and tin from the solder, inhibiting their reaction. When using a copper-containing paste like the SAC alloy system, the copper from the alloy interacts with the tin forming Cu6 Sn5 . Formation of this intermetallic has been observed to suppress the reaction of the nickel with tin. There are a large number of issues with FC and BC. This experiment was designed to answer a few key questions, including determining the effect of a priori thermal aging on subsequent drop test behavior of various combinations of solder alloys and surface finish as well as microstructural changes. Optimization of manufacturing parameters using large I/O area array packages along with fine-pitch CSP/flip chip and passive as low as 0201 sizes also performed.

9.3 Experimental Procedures 9.3.1 Design of Experiment and Tests The experimental design used three factors at two levels each for the study. The factors and levels of the DOE are shown in Table 13. The Sn–Pb finish components were assembled

560 Thermal Cycle and Vibration/Drop Reliability of Area Array Package Assemblies Table 13

Experimental Factors and Levels

Factor

Level 1

Level 2

Solder alloy Component finish PWB finish

Sn–Pb Sn–Pb ImAg

SAC305 Sn / SAC 405 ENIG

Table 14

Mechanical Shock Parameters

Parameter

Setting

Peak load Time period Height of drop Max no. of drop cycles

450G 2 msec 24 in. 30

using Sn37 Pb solder paste as tin–lead control. On the other hand, tin finish components were assembled using SAC305 paste as the lead-free control in the statistical analysis. In order to study the effect of growth of the interfacial intermetallic phase on the mechanical strength, the shock test was performed before and after a simulated intermetallic growth using isothermal aging at 150◦ C for 1000 hr. Table 14 lists the parameters used in the drop test. The functionality of the individual daisy chain in the components was continuously monitored to detect electrical failures during mechanical drop tests. 9.3.2 Test Vehicle and Test The custom-designed TV incorporated the following components as shown in Fig. 33: 1-PB 18 (flip chip); 2-PBGA 256; 3-PBGA 1156; 4-TSOP 48; 5-TSOP 40; 6-UCSP 192; 7-MLF/QFN 72; 8-MLF/QFN 36; 9-0603, 0402, and 0201 resistors (15 each); and 10-37 pin through-hole connected to daisy chains of devices.

Figure 33 Test vehicle used for isothermal aging followed by drops.

9

Drop Shock Behavior with/without Isothermal Aging of Sn37 Pb and Lead-free BGA/CSPs

561

The experimental design had eight combinations (two alloys x two component finishes x two board finishes). The experimental analysis will evaluate 24 circuit boards each of ENIG and ImAg surface finish using six replicates for each combination of alloy, component finish, and board finish. Among the six replicates, three samples are tested at the as-build condition and the other three tested upon isothermal aging at 150◦ C for 1000 hr. Among the three replicates at each test condition, two are subjected to mechanical drop testing and one sample is cross sectioned to determine the interfacial intermetallic thickness and other microstructural detail. Failure considered the daisy chain open for each individual package assembly.

9.4 Results and Discussion The assemblies based on failures observed can be broadly divided into four classes: high-I/O area array package (PBGA1156) assembled using solder paste, very fine pitch area array devices assembled with flux-dip alone (PB18 & UCSP), low-I/O PBGAs, and leaded/leadless device assemblies. Table 15 provides a summary of failures observed in the PBGA1156 device when subjected to mechanical shock before and after isothermal aging at 150◦ C. The failures in this device are random and deterioration in the mechanical strength of the joints was observed with isothermal aging. The overall number of failures (before and after the isothermal aging) in assemblies that used SAC paste (lead-free reflow profile) is higher than assemblies that used Sn–Pb paste (and Sn–Pb reflow profile). This could be due to the higher temperature of the lead-free profile and/or better mechanical shock resistance of the Sn37 Pb alloy. With the fine-pitch packages assembled with flux dip, severe degradation of solder joints was observed (Table 16). This is very apparent for flip chip, which had only peripheral bumps. Almost all assemblies failed after isothermal aging. The poor performance of this type of device assemblies could be attributed to their fine pitch/balls and shorter stand-off from the PWB, resulting from not using solder paste for the reflow process. In leaded devices (TSOP), the failures observed were not significant and were more random to draw any performance-related inferences. No failures were observed in the leadless devices either. In PBGA256 assemblies, only failures were observed in aged Sn37 Pb paste assemblies on ENIG finish PWB. The reasons for this behavior are yet to be determined. Location plays a role that needs to be consider in an overall evaluation.

Table 15 Cyclesa

Failures Observed in PBGA1156 Device When Subjected to Mechanical Shock

Combination (Solder ImAg ENIG Paste/Bump Composition) As Soldered 1000 hr at 150◦ C As Soldered 1000 hr at 150◦ C Sn–Pb/Sn–Pb Sn–Pb/Sn–Pb Sn–Pb/SAC Sn–Pb/SAC SAC/Sn–Pb SAC/Sn–Pb SAC/SAC SAC/SAC a

Failed (6) Failed (9) — — Failed (16) Failed (16) Failed (3) —

Failed Failed Failed Failed Failed Failed Failed Failed

(8) (5) (3) (5) (9) (3) — (1) — (3) —

— —

Failed (4) Failed (2)

Failed (22)

Failed (14)

Failed (5) Failed (1) Failed (1)

Failed (2) Failed (20) Failed (1)

Number of shock cycles the assemblies survived before failure is provided in parentheses.

562 Thermal Cycle and Vibration/Drop Reliability of Area Array Package Assemblies Table 16 Failures Observed in Flip-Chip (PB18) and UCSP192 Devices When Subjected to Mechanical Shock Cyclesa Reflow Component Profile Bump

Sn–Pb Sn–Pb Sn–Pb Sn–Pb SAC SAC SAC SAC a Number

Sn–Pb Sn–Pb SAC SAC Sn–Pb Sn–Pb SAC SAC

ImAg As Soldered UCSP PB18 — — — — — — — — — — Failed (15) Failed (21) — — Failed (12) —

ImAg 1000 hr at 150◦ C UCSP PB18 Failed Failed Failed Failed Failed Failed Failed Failed

(3) (1) (3) (9) (2) (2) (8) (4)

Failed — Failed Failed Failed Failed Failed Failed

(9) (1) (5) (4) (2) (1) (1)

ENIG As Soldered UCSP PB18 — — — — — — — —

— Failed (27) Failed (4) — Failed (9) Failed (16) Failed (4) Failed (1)

ENIG 1000 hr at 150◦ C UCSP PB18 Failed — Failed — Failed Failed Failed Failed

(3) Failed Failed (3) Failed Failed (2) Failed (2) Failed (2) Failed (2) Failed

(1) (1) (6) (4) (1) (1) (1) (1)

of shock cycles the assemblies survived before failure is provided in parentheses.

9.5 Microstructural Analysis The components that failed when subjected to mechanical shock/drop were cross sectioned in order to determine crack formation and microstructural changes due to mechanical shocks. The cross-sectional images are shown in Figs. 34 and 35. No signs of crack formation were observed for the leaded part (TSOP), as evident from Fig. 34. Also, no apparent gross cracking was observed for a PBGA shown in the figure. However, fine cracks were observed across the height of the balls/joints at much higher optical magnification, as shown in Fig. 35. No specific patterns were apparent from the analysis of these microstructures. The isothermally aged samples exhibited intermetallic growth of at least twice more than the initial thickness from the as-soldered samples (Fig. 36). Almost all assemblies with IAg surface finish exhibited stacked intermetallic growth due to the formation of Cu3 Sn between the Cu6 Sn5 and the Cu layer. ENIG assemblies did not show a similar structure. This is due to the fact that the nickel (barrier layer) in the ENIG finish prevents the formation of Cu–Sn intermetallic.

Figure 34 TSOP and PBGA1156 SAC/Sn–Pb(ball) after shock with no apparent signs of cracks.

10 Drop Behavior with/without a Priori Thermal Cycle SnPB/SnPb

563

SnPb/SnPb

SAC/SAC SAC/SnPb (ball)

Figure 35 Evidence of microcracks at higher magnifications for various solder alloy combinations with no specific apparent trend in their formation.

As build

Aged

Figure 36 Microstructure at interface before and after isothermal aging. Evidence of stacked intermetallic microstructure for aged sample.

10

DROP BEHAVIOR WITH/WITHOUT A PRIORI THERMAL CYCLE

10.1 Background This experimental study in collaboration with RIT University partners [50, 51] evaluated the interaction of a thermal cycle followed by drop of TVs assembled with Sn37 Pb and leadfree solder alloys. The test vehicle had a number of packages such as thin chip array BGA (CVBGA), UCSP, PoP, plastic ball grid array (PBGA-676 and 1156), very thin small outline package (TSOP-40 and -48), dual-row microlead frame (DRMLF), microlead frame (MLF-36 and -72), and chip resistors (0201, 0402, 0603). The scope of this section is limited to the performance evaluation for area array packages only.

564 Thermal Cycle and Vibration/Drop Reliability of Area Array Package Assemblies The PWB assemblies were subjected to a drop test before and after 200 and 500 thermal shock cycles in the range of -55 to 125◦ C using a dual chamber. For the drop test, the assemblies were subjected to 30 drops from a height of 3 ft generating an average G force of 485 N. After each drop, the individual package assembly was checked for end-to-end daisy chain continuity. The number of drops to the first failure was used to analyze the performance of each component for various DOE combination parameters and their relative drop resistances. Since each component had many independent daisy chains, the failure of the individual daisy chains was verified after test completion in order to further narrow the location(s) of the failures and how they progressed. Performance of assemblies with full underfill and corner staking was also evaluated but not reported.

10.2 Experimental Design and Reliability Test Many variables were considered in the experimental design: PWB surface finish (ENIG, HASL, ImAg), underfill type (no underfill, corner underfill, and full underfill), solder ball alloys (SAC405, 305, 105, see Table 17), and two thermal cycle/shock ranges with different number of cycles followed by a constant 30 drops. A large number of test vehicles were assembled under various conditions. Assembly of PoP packages was found to be challenging. Figure 37 shows an assembled TV with the following Table 17 Component

Component Solder Ball Alloys Alloy Combination

UCSP SAC305 PBGA1156 SAC305 POP Top—SAC105, Bottom—*SAC125Ni CVBGA SAC105, SAC305, SAC405 PBGA676 SnAg, SAC305, SAC405 *SAC125Ni (1.2%Sn/0.5%Ag/0.05%Cu/98.25%Ni)

Figure 37 Test vehicle assembly configuration with array and PoP packages for thermal shock (-55/125◦ C) followed by drops.

10 Drop Behavior with/without a Priori Thermal Cycle Table 18

565

Component Specifications

Component

I/O Count

Ball Diameter (mm)

Pitch (mm)

Body Size (mm)

UCSP PBGA CVBGA PBGA POP—top POP—bottom

192 676 432 1156 152 353

0.25 0.63 0.25 0.63 0.3 0.5

0.4 1.0 0.4 1.0 0.65 0.5

7 27 13 35 14 14

packages as identified in the photo: (1) UCSP-192, (2) POP, (3) PBGA676, (4) PBGA1156, (5) CVBGA432, (6) TSOP40, (7) TSOP48, (8) DRMLF, (9) MLF36, (10) MLF72, (11) 0201, (12) 0402, (13) 0603, and (14) LCC. Table 18 provides details pertaining to area array package components considered within this study. To evaluate the solder joint reliability, PWAs were subjected to thermal shock (-55 to 125◦ C) followed by 30 drops, each with the average peak load of 485 N. The cold chamber of the thermal shock equipment had to undergo defrosting after completion of every 75 cycles for proper functioning. During this defrost period, the test vehicles were maintained at +125◦ C. For the drop tests, the PWB assemblies were mounted with components facing downward as recommended by JEDEC 22-B111. Some secondary impacts were observed during the drop testing.

10.3 Test Results Figure 38 compares percentage drops to failure for fine-pitch CVBGA 432 I/Os (no failure) to those for PBGA 676 I/Os assembled on board with various surface finishes. In order to be able to better compare failure data, failure percentages are plotted on the y axis of the bar chart plots by using the maximum of 30 drops to be equivalent to 100 percentage points. Note that because of sensitivity of package failure to its location on the PWB, the failure based on the number of drops cannot be directly compared for different packages at different locations. The effect of PWB surface finish for each package category can be compared. For CVBGAs, SAC105 showed slightly better resistance to SAC405 under mechanical shock for PWB with HASL and ImAg finishes for both the control TVs and after 200 thermal shocks. With ENIG PWB finish, however, SAC405 performed better than SAC105 after 200 cycles. It is interesting to note that for CVBGAs and all surface finishes improvements were achieved due to 500 thermal shocks since there were no failures after 30 drops for both SAC105 and SAC405 assemblies. Note that the trend was not the same for PBGA676 since they failed after a few number of cycles for both SAC105 and SAC405 alloys. They showed much better resistance for the assemblies with SnAg solder alloys. Figure 39 compares failure data for the effect of PWB surface finish for high-I/O PBGA 1156 and fine-pitch UltraCSP package assemblies. UltraCSP components with SAC305 solder balls showed marked improvement in drop test after 200 cycles when compared to the as-build condition but failed within 500 cycles. For PBGA1156, a slight improvement was observed in drop resistance for all solder alloys after 500 cycles. There were PoP fall-outs in three out of nine cases; however, if the PoPs were successfully assembled, it was found that PoPs survived the 30 subsequent drops. Note that the package is located at the corner of the PWB.

100 80 60 40 20

4 S SA 05 C 10 5

5

100 80 60 40 20 0

05 05 /Ag C3 AC4 Sn S l So As

50

0T

SA

05 05 /Ag C3 AC4 Sn S S 0T 50

SA

40 20

SA C 3 SA 05 50 C 0T 40 S 5 Sn /A g

l So

SA C 3 SA 05 20 C 0T 40 S 5 Sn /A g

40 5 Sn /A g

C

5

0

SA

SA C 3 SA 05 50 C 0T 4 S SA 05 C 10 5

0

60

As

20

80

30

40

100

C

60

Comparison of Termination Alloy for PBGA 676 Board Finish - ENIG, Test Condition - No Underfill

SA

80

Drop Test Survival Percentage

100

SA C 3 SA 05 20 C 0T 4 S SA 05 C 10 5

05 05 /Ag C3 AC4 Sn S S 0T 20

SA

BGA676- ImAg

Comparison of Termination Alloy for CVBGA432 Board Finish - ENIG, Test Condition - No Underfill

SA C 3 SA 05 As C So S 40 l A 5 C 10 5

Drop Test Survival Percentage

CVBGA- ImAg

CVBGA- ENIG

50 0T S

As

Comparison of Termination Alloy for PBGA 676 Board Finish - ImAg, Test Condition - No Underfill

C SA

30 C SA

4 S SA 05 C 10 5

5

C 20

0T

SA

30 C SA

C 3 SA 05 As C So S 40 l A 5 C 10 5

0

Drop Test Survival Percentage

BGA676- HASL

Comparison of Termination Alloy for CVBGA432 Board Finish - ImAg, Test Condition - No Underfill

SA

Drop Test Survival Percentage

CVBGA- HASL

C 3 SA 05 C 40 5 Sn /A g

0

SA

C 3 SA 05 50 C 0T 4 S SA 05 C 10 5

20

SA

C 3 SA 05 20 C 0T 4 S SA 05 C 10 5

As

SA

C 3 SA 05 C So S 40 l A 5 C 10 5

0

40

C 3 SA 05 C 40 5 Sn /A g

20

60

20 0T S

40

80

SA

60

100

C 3 SA 05 C 40 So l S 5 n/ Ag

80

Comparison of Termination Alloy for PBGA 676 Board Finish - HASL, Test Condition - No Underfill

SA

100

Drop Test Survival Percentage

Comparison of Termination Alloy for CVBGA432 Board Finish - HASL, Test Condition - No Underfill

SA

Drop Test Survival Percentage

566 Thermal Cycle and Vibration/Drop Reliability of Area Array Package Assemblies

BGA676- ENIG

Figure 38 Alloy comparison for CVBGA and PBGA 676 with HASL/ImAg/ENIG.

20

IG AS L Im Ag 0T S

EN

H

50

0T S

EN

H

IG AS L Im Ag

0

20

EN

IG AS L Im Ag 0T S

0T S

H

50

IG AS L Im Ag

EN

H

20

As

So

EN

IG AS L l Im Ag

0

40

IG AS L l Im Ag

20

60

So

40

80

EN

60

100

H

80

Summary 567

Comparison of Board Finish Performance for PBGA 1156 Termination Alloy - SAC 305, Test Condition - No Underfill

As

100

Drop Test Survival Percentage

Comparison of Board Finish Performance for UCSP 192 Termination Alloy - SAC 305, Test Condition - No Underfill

H

Drop Test Survival Percentage

11

Figure 39 Effect of surface finish for fine-pitch UltraCSP192 and standard-pitch PBGA 1156-I/O package assemblies.

11

SUMMARY Even though large environmental test and field data are now gathered by industry for Sn37 Pb and limited for lead-free solder alloys, concerns still exist for use of lead-free alloys for high-reliability applications because of solder alloy proliferation and new area array packages, which generally have lower reliability resistance within the package and at the package assembly levels. A few key findings are as follows: • The “change” is the only thing that is constant in the AAP technology development,

•

• •

• • • •

•

so be ready for contradicted results from different authors, analyze them thoroughly, and then consider their applicability for a specific application. Array packages now come in all types, variations both within package and externally with different configurations. Internal variations include die attachment onto substrate with wire bond, flip chip, and stack die, externally through mold via. Solder ball alloys and pitches are continuously changing in some cases different array pitches for a package. Plastic ball grid arrays are generally more resistant to thermal cycling than their ceramic ball grid array counterparts. IPC 9701A provides requirements for performing thermal cycling for both Sn37 Pb and lead-free solder alloy assemblies. Guidelines for electronic bend testing and shock/drop tests are given in the recently published IPC 9702 to IPC9706. Double-sided mirror image of plastic ball grid array assemblies showed about 40–60% reduction in thermal cycle resistance. FCBGA are less robust than their wire bond version. For lead-free SAC solder alloys, generally as tin amount decreases, resistance to thermal cycle decreases, for example, SAC405 shows better cycles to failure than SAC105. The effects of underfill on thermal cycles to failures may be positive, neutral, or negative depending on package types. It improved the reliability of leadless package, was neutral for chip-on-flex, and had negative effects on the TAB CSP reliability. All packages with or without underfill survived 2 hr of random vibration at 7.8Grms and most failed within 4 hr of additional vibration at 16.9Grms . Underfilling improved resistance to vibration. TSOP with no underfill was the only package that survived 6 hr

568 Thermal Cycle and Vibration/Drop Reliability of Area Array Package Assemblies

•

• •

•

•

• • • •

•

•

of vibration. Location of the package on the board is another key factor that should be considered when such results are analyzed. Hairline cracks in solder joints induced by vibration are significantly more difficult to detect visually compared to more pronounced cracks with gross microstructural changes formed by thermal cycling. Under vibration, BGA 225 I/O assemblies showed much longer lifetime than those for a SAC396 lead-free solder. A near-thermal shock in the range of -55 to 125◦ C induced the most damage on CBGA assemblies, up to 50% reduction in cycles to failure compared to a thermal cycle condition with the same temperature range. Assemblies with three different levels of rigidity passed a launch random vibration/shock condition. However, cycles to failure after vibration were affected by the rigidity of the board, a significant reduction (>50%) for a less rigid and minimum reduction for highly rigid assemblies, bonded to an aluminum plate after vibration. Failures for CBGA assemblies subjected to thermal cycling were from either board or package sites in the eutectic solder joints. Those subjected to vibration showed tensile deformation in high-melt balls and tensile and shear failures in eutectic solders. Leaded and leadless component assemblies performed better than area array package assemblies under repeated drop shocks before and after isothermal aging. Area array assemblies with solder paste showed much higher mechanical shock resistance than those assembled with only flux. Assemblies with ImAg PWB pad surface finish showed slower deterioration due to isothermal aging than the ENIG finish assemblies. Repeated drop at G level of 485 N of thermally shock lead-free solder joint assemblies revealed that SnAg solder ball alloy outperformed SAC305 and SAC105 solder balls, irrespective of the PWB surface finish. In comparing SAC105 and SAC405 solder balls, the SAC105 component performed better than SAC405 with HASL and ImAg finishes but performed poorly when assembled with PWB with ENIG pad finish. UltraCSP component with SAC305 solder balls was found to show marked improvement in drop at G level of 485 N test resistance after 200 cycles of thermal shock; when compared to the as-build condition but were found to fail within 500 cycles of thermal shock. PoP assemblies survived the 30 drops at G level of 485 N with all three PCB finishes. Location plays a role in no-failure observation.

In addition to investigation generated by this author alone and in collaboration with industry and university partners, there are a number of excellent resources in the literature for mechanical testing of the evolving area array packaging technologies and lead-free solder alloys. For example, a recent paper by Dasgupta at CALCE [54] provided the vibration durability of both SAC305 and Sn37 Pb interconnect. The tests were performed under narrow-band harmonic vibration conducted at the first natural frequency of the PWB using constant-amplitude excitation. The SAC305 interconnects; were found to have lower fatigue durability than comparable Sn37 Pb interconnects; the trend is in agreement with results of a few other investigators [33, 55] while it counters others [56]. Failure analyses revealed two competing failure modes, one in the solder and another in the copper trace under the component. The authors concluded that even in the low-cycle fatigue (LCF) region where the SAC305 material shows more durability, the actual interconnects assembled with SAC305 solders may not outlast corresponding Sn37 Pb assemblies under similar excitation levels. These are due to competing failure mechanisms in the intermetallic compound (IMC) layer and in the PWB that often make the SAC305 assemblies

Acknowledgments

569

more vulnerable to PWB flexure. Also, SAC305 interconnects experience higher strain level than Sn37 Pb interconnects under the same excitation level because of different stress strain properties. Ahmer Syed and his collaborators at Amkor [57] presented a collection of test data comparing the effects of package design and solder materials under thermal cycling, drop, and cyclic bend performed on 0.4-mm-pitch area array packages. Test results indicate that lower silver content solder balls (SAC105 and SAC125Ni) performed better under drop conditions compared to high silver content (SAC305 and Sn3.5Ag), while temperature cycling reliability suffers as silver content decreases. The die size also showed a reverse trend depending on the loading condition. While larger die showed worse performance for temperature cycle and drop, bend cycling performance was worse for smaller die. A European consortium [58] was formed with the objectives of establishing test methodologies at PWB and at package levels and exploring for a universal failure model independent of the test methods. Various mechanical tests, including drop, cyclic bend, and ball impact, were performed to establish stress–strain properties, S –N fatigue relationship, crack propagation behavior, and damage accumulation characteristic of lead-free solder joints. The authors concluded that the high-speed cyclic bend test (HSCBT) is capable of replicating the boardlevel drop-shock test (BLDST) and offers better reproducibility and higher test throughput rate and its suitable both for materials system characterization and for detecting manufacturing variables. On the other hand, the package-level ball impact shear test (BIST) does not correlate well with the BLDST. Countering other investigators, they showed that lowamplitude followed by high-level (low–high) fatigue is more severe than high–low. It was shown that the propagation rate of cracks in the solder joint accelerated upon transitioning from low-amplitude to high-amplitude cyclic bending. The authors concluded that Miner’s rule is overly simplistic and its use will lead to overestimation of fatigue life. Borgesen at Binghamton University and his collaborators from Unovis [59] showed that a priori low-cycle fatigue or low-level G value drop improved the subsequent drop behavior of assemblies with lead-free solder joints. Andrew Perkin and coauthors at Georgia Tech showed that a priori temperature cycle followed by vibration had more damaging effects than vibration followed by temperature cycling [60] for ceramic column grid array assemblies. Although investigators are in agreement that the linear Miner rule is not accurate in projecting solder joint damage accumulation due to various damage levels of either the same or different environmental loadings; however, they disagree on the direction of nonlinearity in synergism. Our investigations on the effect of isothermal aging and thermal shock/cycling on subsequent drop behavior showed also similar unpredictable trends. This uncertainty adds additional cost to the use of lead-free solder alloys since modeling projections are only possible for previously known solder alloys and environmental sequence loading conditions. Continuously additional tests may be required for new lead-free solder alloys, especially for applications where the life of assemblies is estimated to be marginally acceptable.

ACKNOWLEDGMENTS This research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, and was sponsored by the National Aeronautics and Space Administration Electronic Parts and Packaging (NEPP) Program. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not constitute or imply its endorsement by the U. S. Government or the Jet Propulsion Laboratory, California Institute of Technology. The author would like to acknowledge colleagues from industry and JPL who were critical to the progress of many activities presented here for several years of collaboration and being true partners. Special thanks to Professor Ramkumar and his students at RIT University; Irene

570 Thermal Cycle and Vibration/Drop Reliability of Area Array Package Assemblies Sterian, Simin Bagheri, and others at Celestica; Dr. Namsoo Kim at Boeing; and Kurt Kessel and team members of JCAA/JG-PP consortium. The author further extends his appreciation to program managers of the NASA Electronic Parts and Packaging Program (NEPP), including Michael Sampson, Ken Label, Dr. Charles Barnes, and Phillip Zulueta, for their continuous support and encouragement.

ACRONYMS AAP BGA CBGA CCGA CEMA CGA COB COTS CSP CTE CTF Cu DCA DOE EDX EMS ENIG FCBGA FC/DCA FPBGA HASL HDI IAg IC ImAg iNEMI I/O IPC IR JPL KGD LCCC LGA MER MIP MLF MRO MSL NASA

Area array packaging Ball grid array Ceramic ball grid array Ceramic column grid array Center for Electronics Manufacturing and Assembly Column grid array Chip on board Commercial off the shelf Chip scale (size) package Coefficient of thermal expansion Cycles to failure Copper Direct chip attachment Design of experiment Energy dispersive X ray Engineering manufacturing service Electroless nickel immersion gold Flip chip ball grid array Flip-chip direct chip attach Fine pitch BGA, also called chip scale package (CSP) Hot-air solder leveling High-density interconnect Immersion silver Integrated circuit Immersion silver International Electronics Manufacturing Initiative Input/output Association Connecting Electronics Industries Infrared Jet Propulsion Laboratory Known good die Leadless ceramic chip carrier Land grid array Mars Exploration Rover Mandatory inspection point Microlead frame Mars Reconnaissance Orbiter Mars Science Laboratory National Aeronautics and Space Administration

References NEPP NGST Ni NSMD OSP PBGA PCB PTH PTHV PWA PWB QA QFN QFP RIT RMA RT SAM SEM SIP SMC SMD SMT Sn TC TCE Tg TMA TSOP TV UCSP WLP XRF

571

NASA Electronic Parts and Packaging Next Generation Space Telescope Nickel Nonsolder mask defined Organic solder preservative Plastic ball grid array Printed circuit board Plated-through hole PTH via Printed wiring assembly Printed wiring board Quality assurance Quad flat no lead Quad flat pack Rochester Institute of Technology Rosin mildly activated Room temperature Scanning acoustic microscopy Scanning electron microscopy Systems in a package Surface mount components Solder mask defined Surface mount technology Tin Thermal cycle Also CTE, thermal coefficient of expansion Glass transition temperature Thermal mechanical analysis Thin small outline package Test vehicle Ultra chip scale package Wafer-level package X-ray fluorescence

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574 Thermal Cycle and Vibration/Drop Reliability of Area Array Package Assemblies 45. Nelson, D., Pallavicini, H., Zhang, Q., Friesen, P., and Dasgupta, A., “Manufacturing and Reliability of Pb-Free and Mixed System Assemblies (Sn-Pb/Pb-free) in Avionics Environments,” Journal of SMT, Vol. 17 No. 1, 2004. 46. Kannabiran, A., Manian Ramkumar, S., Elavarasan, T. P., “Compatibility of Sn-Pb and Lead-Free Alloy and Component Finish on HASL and OSP Board Finishes,” Proceedings of SMTAI , Chicago, IL, 2006. 47. Chiu, T.-C., Zeng, K., Stierman, R., Edwards, D., and Ano, K., “Effect of Thermal Aging on Board-Level Drop Reliability,” in Proceedings of Electronics Components & Technology Conference, 2004. 48. Ramkumar, S. M., Ghaffarian, R., and Varanasi, A., “Lead-Free 0201 Manufacturing, Assembly and Reliability Test Results,” Journal of Microelectronics Reliability, 2006. 49. Matila, T. T., Kivilahti, J. K., “Reliability of Lead-Free Interconnection under Consecutive Thermal and Mechanical Loadings,” Journal of Electronic Materials, Vol. 35, No. 2, 2006. 50. Sakthivelan, S., Chheda, B. V., Ramkumar, S. M., and Ghaffarian, R., “Thermal and Mechanical Behavior of Lead-Free Area Array Packages with Full/Corner/No Underfill,” Proceeding IPACK2009, ASMEInterPack09 , San Francisco, CA, July19–23, 2009. 51. Sakthivelan, S., Chheda, B. V., Ramkumar, S. M., and Ghaffarian, R., “Thermal Cycle and Drop Reliability of Lead-Free Assemblies with No- and Corner-Underfill,” in Proceedings of SMTAI 2009 . 52. Weston, K., Sack, T., Shangguan, D., Yi, S., Wable, G., Cipielewski, T., Barthel, B., Jean, D., Kinyanjui, R., and Abtew, M., “EMS Forum on Lead-Free Assembly,” Apr. 2007. 53. Kim, H., Zhang, M., Kumar, C. M., Suh, D., Liu, P., Kim, D., Xie, M., and Wang, Z., “Improved Drop Reliability Performance with Lead Free Solders of Low Ag Content and Their Failure Modes,” Intel Corporation. 54. Zhou, Z., Al-Bassyiouni, M., and Dasgupta, A., “Vibration Durability Assessment of Sn3.0Ag0.5Cu and Sn37Pb Solders Under Harmonic Excitation,” Jounal of Electronic Packaging, Vol. 131, Mar. 2009. 55. Zhou, Y., Scanff, E., and Dasgupta, A., “Vibration Durability Comparison of Sn37Pb vs. SnAgCu Solders,” in Proceedings of ASME International Mechanical Engineering Congress and Exposition, Chicago, IL, Paper No. 13555. 56. Cuddalorepatta, G., and Dasgupta, A., “Cyclic Mechanical Durability of Sn3.0Ag0.5Cu Pb-Free Solder Alloy,” in Proceedings of the ASME International Mechanical Engineering Congress and Exposition, Orlando, FL, 2005. 57. Syed, A., Scanlan, J., Cha, S. W., Kang, W. J., Sohn, E. S., Kim, T. S., and Ryu, C. G., “Impact of Package Design and Materials on Reliability for Temperature Cycling, Bend, and Drop Loading Conditions,” in Electronic Components and Technology Conference, 2008, ECTC 2008, 58th, Lake Buena Vista, FL, May 27–30, 2008, pp. 1453–1461. 58. Wonga, E. H., Seah, S. K. W., van Driel, W. D., Caers, J. F. J. M., Owens, N., and Lai, Y.-S., “Advances in the Drop-Impact Reliability of Solder Joints for Mobile Applications,” Microelectronic Reliability Journal , Vol. 49, No. 2, pp. 139–149. 59. Yang, L., Raghavan, V., Roggeman, B., Yin, L., and Borgesen, P., “On the Complete Breakdown of Miner’s Rule for Lead Free BGA Joints,” in SMTAI Proceedings, San Diego, CA, 2009. 60. Perkins, A. E., and Sitaraman, S. K., Solder Joint Reliability Prediction for Multiple Environments, Springer, 2009 61. http://www.geia.org/.

CHAPTER

23

COULD AN IMPACT LOAD OF FINITE DURATION BE SUBSTITUTED WITH AN INSTANTANEOUS IMPULSE?

1

Ephraim Suhir

Luciano Arruda

University of California Santa Cruz, California University of Maryland College Park, Maryland ERS Co. Los Altos, California

Instituto Nokia de Technologia Manaus, Brazil

INTRODUCTION

1.1 The Problem We show that a single-degree-of-freedom system with a rigid cubic characteristic of the nonlinearity (restoring force) is a suitable analytical (“mathematical”) model that can be used to evaluate the dynamic response of a printed circuit board (PCB) subjected to a drop or a shock impact when appreciable reactive in-plane (“membrane”) forces occur. The inplane reactive forces arise because the PCB’s short edges (supports) cannot get closer during the band’s impact-induced vibrations. The model in question is known in the physical and mechanical literature on nonlinear vibrations as a “Duffing oscillator.” When modeling the dynamic response of either the PCB itself or a surface-mounted device (SMD) package, including ball-grid-array (BGA) or pad-grid-array (PGA) systems, on a board level, there is an obvious incentive in trying to simplify the modeling by substituting an impact load of finite duration with an instantaneous impulse. On the other hand, when the intent is to replace drop tests with shock tests, one has to properly “tune” the shock tester so as to adequately mimic the drop test conditions. In this analysis we obtain exact solutions to the Duffing equation, no matter how significant the level of nonlinearity might be, for the cases of an instantaneous impulse and of a suddenly applied and suddenly removed constant loading and use these solutions to determine the error (in terms of the predicted amplitudes and accelerations) from substituting an impact load of finite duration with an instantaneous impulse. We consider an elongated board which is currently employed in an accelerated test vehicle (experimental setup). Its short edges are simply supported, while its long edges are support free. So, in effect, we examine a long and narrow strip simply supported at the ends (short edges) when these ends experience an impact load.

1.2 Objective The dynamic response of a PCB to shocks and vibrations has been addressed in numerous investigations [1–9] with different objectives in mind. In the analysis that follows we consider Structural Dynamics of Electronic and Photonic Systems Copyright © 2011 John Wiley & Sons, Inc.

Edited by Ephraim Suhir, David S. Steinberg and T. X. Yu

575

576 Could an Impact Load of Finite Duration Be Substituted with an Instantaneous Impulse? the dynamic response of an elongated flexible PCB to a drop impact load applied to its supports. The board is being used in an accelerated test vehicle (experimental setup). Our objective is to evaluate the response of a PCB to a constant load of finite duration suddenly applied to and then suddenly removed from its supports. Particularly we intend to assess the error from substituting such a load with an instantaneous impulse. We consider a simplified situation when the board is simply supported at its short edges and is support free at its long edges. When modeling the dynamic response of a PCB or a particular SMD system on a board level (such as, say, a ball grid array or a pad grid array structure) to a shock load applied to the PCB supports during drop or shock tests, there is an obvious incentive in trying to simplify the analysis by substituting the impact load of finite duration with an instantaneous impulse. On the other hand, when there is intent to substitute drop tests with shock tests, one has to “tune” the shock tester, in terms of the magnitude and duration of the applied impact load, so that physically meaningful results are obtained. In either situation, it is important to find out the error (in terms of the predicted maximum displacements, accelerations, stresses and strains, etc.) from substituting the load of finite duration with an instantaneous impulse. When the impact load is significant and the PCB supports cannot get closer during the induced vibrations, appreciable in-plane reactive forces (stresses) arise, and the PCB vibrations become essentially nonlinear. Intuitively it is felt that a nonlinear system with a rigid characteristic of nonlinearity (such as a Duffing oscillator) should be less sensitive to the duration of the applied load than a linear system. Indeed, the physics of why there is a distinction between static and dynamic response of a structural element to a load of the given duration is governed by the relationship between the duration of loading and the natural period(s) of the element’s vibrations. If the period(s) of free vibrations are significantly shorter than the duration of loading, no inertia forces arise and the load can be assumed as a static one. In another extreme case, when the loading is negligibly short compared to the period(s) of vibrations, the dynamic response of the system can be evaluated assuming that the load acts as an instantaneous impulse. If the duration of the load is comparable with the period(s) of the system’s vibrations (it is naturally the most general case), then the dynamic response of the system should be evaluated using one of the methods of dynamic structural analysis. The nonlinear frequency of the dynamic response of a nonlinear system with a rigid characteristic of the restoring force (i.e., a PCB with significant reactive in-plane stresses) can be very high compared to the frequency of linear vibrations. This means that such a system, if the magnitude of the loading is high, has a very short period of vibrations (although it is not a “natural” period, because it is due to both the properties of the system and the level of loading). This circumstance might “convert” the load of given duration from a dynamic load, in the case of a linear oscillator, to a more or less strongly pronounced quasi-static load, in the case of a nonlinear system, so that the nonlinear oscillator might be much less sensitive to the dynamic nature of the impact than a linear system. Our analysis is, in effect, an attempt to quantify such an intuitive anticipation.

2

ANALYSIS

2.1 Nonlinear Response The impact-induced vibrations of an elongated PCB after the load is removed are characterized by the following equation of motion (equilibrium): Dw IV (x , t) − T (t)w  (x , t) + m w¨ (x , t) = 0,

(1)

where w (x ,t) is the deflection function, T (t) is the reactive tensile force, D = Eh3 /  12(1 − ν 2 ) is the flexural rigidity of the PCB, E and ν are the elastic constants of the material, h is the PCB thickness, t is time, and m is the PCB mass per unite area. The origin of the coordinate x is in the PCB’s mid–cross section. Assuming that the vibrations

2 Analysis

577

are caused by the initial velocities w˙ (x , 0) applied instantaneously to the PCB short edges, x = ±a, we seek the solution to Eq. (1) in the form πx w (x , t) = f (t) cos , (2)   2a where f (t) is the principal coordinate, X (x ) = cos π x / (2a) is the coordinate function, and a is half the PCB length. The solution (2) satisfies the zero boundary conditions w (±a, t) = 0, w  (±a, t) = 0. From (2) we have πx w˙ (x , t) = f˙ (t) cos . (3) 2a If the shock load is due to a drop impact, the vertical velocities w˙ (x , 0) of all the PCB points are the same at the initial moment of time, t = 0:  πx πx w˙ (x , 0) = f˙ (0) cos (4) = f˙0 cos = 2gH, 2a 2a where H is the drop height and g is the acceleration due to gravity. Using Fourier transform, we obtain 4 f˙0 = 2gH, π where the factor 4/π reflects the role of the coordinate function. The physical meaning of this factor is that different cross-sections of the board are accelerated differently depending on this location x . Introducing the sought solution (2) into Eq. (1), we obtain the following equation for the principal coordinate f (t):  π 2 T (t) f¨ (t) + ω2 f (t) + f (t) = 0, (5) 2a m where  π 2  D ω= 2a m is the linear frequency. Linear vibrations take place when the reactive tensile force T (t) is zero or small and could be neglected. This force can be found based on the following simple reasoning. The length s of the deflected elastic curve can be determined as  a  a   2 s=2 w (x , t) dx 1 + [w  (x , t)]2 dx ≈ 2a + 0

0

 a  π 2 πx π2 2 f 2 (t) sin2 = 2a + dx = 2a + f (t). 2a 2a 8a 0 The force T (t) that results in an elongation π2 2 s = s − 2a = f (t) 8a can be found, in accordance with Hooke’s law, as  π 2 h T (t) = E s = Ehf 2 (t). (6) 2a 4a Then Eq. (5) results in the following (Duffing-type) nonlinear equation for the principal coordinate f (t) : f¨ (t) + ω2 f (t) + μf 3 (t) = 0, (7) where μ=

 π 4 Eh 2a 4m

is the parameter of nonlinearity. In the case of forced vibrations, Eq. (1) should be replaced by the following inhomogeneous equation: Dw IV (x , t) − T (t)w  (x , t) + m w¨ (x , t) = q(t), (8)

578 Could an Impact Load of Finite Duration Be Substituted with an Instantaneous Impulse? where the load q(t) is equal to zero at x = ±a. In this case the function f (t) could be found from the equation Q(t) f¨ (t) + ω2 f (t) + μf 3 (t) = , (9) m where Q(t) = (4/π) q(t) is the generalized force.

2.2 Vibrations Caused by Instantaneous Impulse: Vibration Amplitudes In the case of an instantaneous impulse, the induced vibrations are free, and Eq. (7) should be used to evaluate the response. This equation can be written as  d ˙2 f (t) + ω2 f 2 (t) + 12 μf 4 (t) = 0, dt so that f˙ 2 (t) + ω2 f 2 (t) + 12 μf 4 (t) = C , (10) where C = const. With the zero displacement f0 = f (0) and nonzero velocity 4 f˙0 = f˙ (0) = 2gH, π that is, for typical drop impact initial conditions, we have 32 C = 2 gH. π On the other hand, when the displacement f (t) reaches its maximum value, AS , it is the velocity f˙ (t) that is zero. Then Eq. (10) results in the following biquadratic equation for the amplitude AS :

2 ω2 2 8 gH 4 AS + 2 AS − = 0. (11) μ π μ The solution to this equation can be written as AS = ηA A0S , where √ 4 2gH A0S = π ω is the linear amplitude (μ = 0) and the factor √ 1 + 2μ − 1 ηA = μ considers the effect of the nonlinearity on the amplitude of vibrations. In the last formula,

2 A0S μ=μ ω is the dimensionless parameter of nonlinearity. Thus, the vibration amplitudes could be found without obtaining the complete solution to Eq. (7), that is, without being able to establish the time-dependent function f (t).

2.3 Vibrations Caused by Suddenly Applied Constant Load: Maximum Displacement In the case of a suddenly applied stepwise load, Eq. (9) should be used to determine the principal coordinate f (t). This equation can be written as   d ˙2 1 Q f (t) + ω2 f 2 (t) + μf 4 (t) − 2 f (t) = 0. dt 2 m

2 Analysis

579

so that

Q (12) f˙ 2 (t) + ω2 f 2 (t) + 12 μf 4 (t) − 2 f (t) = C . m If the initial conditions f (0) and f˙ (0) are zero, the constant C is zero as well, and the velocity f˙ (t) can be expressed through the displacement f (t) as  Q 1 (13) f˙ (t) = 2 f (t) − ω2 f 2 (t) − μf 4 (t). m 2 The obtained equation determines the phase diagram for the nonlinear system in question. The displacement f (t) reaches its maximum value fmax at the end of the first quarter period of vibrations, when the velocity f˙ (t) is zero. This results in the equation ω2 4Q fmax − = 0. μ μm The maximum linear (μ = 0) displacement is 2Q 0 = , fmax mω2 and the solution to the cubic equation (14) can be written as 3 fmax +2

(14)

0 , fmax = ηQ fmax

where the factor

ηQ =

3

1+

 1 + 8/(27μf ) μf

+

3

1−

 1 + 8/(27μf ) μf

(15)

considers the effect of the nonlinearity on the maximum displacement. This factor changes from unity to zero, when the dimensionless parameter of nonlinearity

0 2 f μf = μ max ω changes from zero to infinity. Thus, we were able to find the expression for the maximum displacement without obtaining the time-dependent solution to the basic equation (9).

2.4 Dynamic Factor If one puts the acceleration f¨ (t) in Eq. (9) equal to zero and considers that the force Q(t) = Q is time independent, that is, remains constant during the time of loading, then the corresponding static displacement fst can be found from the equation ω2 Q fst3 + fst − = 0. (16) μ μm   The maximum static linear (μ = 0) displacement is fst0 = Q/ mω2 and turns out to be half of the maximum dynamic displacement that occurs when the load Q is suddenly applied to and remains on the system. The solution to Eq. (16) can be written as fst = ηst fst0 , where the factor        

0 2   1 + 4/ 27μst 1 + 4/ 27μst 3 1+ 3 1−   f ηst = + , μst = μ st (17) μst μst ω considers the effect of the nonlinearity on the maximum static displacement. The factor ηst changes from unity to zero when the parameter μst changes from zero to infinity. Calculations indicate that the factor ηz (that considers the effect of nonlinearity on the maximum dynamic

580 Could an Impact Load of Finite Duration Be Substituted with an Instantaneous Impulse? displacement) decreases more rapidly with an increase in the parameter μ of nonlinearity than the factor ηst (that considers the effect of nonlinearity on the maximum static displacement), and therefore the dynamic factor Kd =

fmax ηz =2 fst ηst

(which is equal to 2 for a linear system) trends to 1 for a highly nonlinear system. Hence, the effect of the dynamic application of the loading Q(t) is less strongly pronounced in a nonlinear system than in a linear one.

2.5 Vibrations Caused by Instantaneous Impulse: Solution to Basic Equation Equation (7) lends itself to the following exact solution: f (t) = AS cn(σS t, ε) = AS cnu,

(18)

where u = σs t, cnu is an elliptic cosine [14–17], σS is the parameter of the nonlinear frequency, and ε is the initial phase angle. Using the formulas (cnu) = −snudnu,

(snu) = cnudnu,

(dnu) = −kS2 snucnu,

for differentiating the elliptic functions, and the formulas sn2 u + cn2 u = 1, of the “elliptic geometry,” we obtain

dn2 u + kS2 sn2 u = 1,   f¨ (t) = −AS σS2 cnu 1 − 2kS2 sn2 u .

f˙ (t) = −AS σS snudnu,

(19)

Here snu is the elliptic sine, dnu is the function of delta amplitude, and kS is the modulus of the elliptic function. Introducing (18) and the second formula in (19) into Eq. (7), we conclude that this equation is fulfilled if the following relationships take place:     1 ω A μ ω2 S 2  2 (20) , kS = σS = ω + μAS = = 1− 2 . σS 2 2 σS 1 − 2k 2 S

The period of the function cn(t, ε) is 4K (kS ), where  π/2 dξ K (kS ) = 0 1 − kS2 sin2 ξ is the complete elliptic integral of the first kind. Hence, the period of the function cn(σS t, ε), that is, the period of the nonlinear vibrations, is (4/σS )K (k ), and the frequency is π σS pS = . 2K (kS ) The maximum acceleration (deceleration) can be found directly from Eq. (7): f¨S = −ω2 AS − μA3S = −σS2 AS . Thus, we were able to obtain the exact solution, given by formula (18), to the basic equation (7).

2 Analysis

581

2.6 Vibrations Caused by Suddenly Applied Constant Load: Solution to Basic Equation Using formula (13), one can represent the solution to Eq. (9) in the form  f u df t= =  , 2 2 4 σ Q 0 2 (Q/m) f (t) − ω f (t) − (1/2) μf (t) where  θ dθ u = u(σQ t, ε) = F (θ, kQ ) = 0 1 − kQ2 sin2 θ

(21)

is the elliptic integral of the first kind, [14–17], kQ is the modulus of the elliptic function, θ = amu is the amplitude of this function, and σQ is the parameter of the nonlinear frequency. In order to express the modulus kQ and the parameter σQ through the characteristics of system (9), we seek the inversion of the integral (21) as 1 − cnu , (22) f (t) = fmax 1 + δ − (1 − δ)cnu 0 is the maximum displacement and δ is the parameter of the modulus where fmax = ηQ fmax kQ . From (22) we obtain snudnu f˙ (t) = 2δσQ fmax , (23) [δ + 1 − (δ − 1)cnu]2

f¨ (t) = 2δσQ2 fmax

(δ + 1) (1 − 2kQ2 sn2 u)cnu + (δ − 1)[1 + (1 − 2kQ2 )sn2 u]

. (24) [δ + 1 − (δ − 1)cnu]3 Introducing formulas (22) and (24) into Eq. (9), we conclude that this equation is fulfilled if the relationships

   μm 3 ω2 m δ2 − 1 ω 2 μ = 8 , δ = 1+ fmax = 3 − fmax = 3 − ηQ , 3 0 2Q Q fmax 3 − δ2 (25)   (δ − 1)(3 − δ) 2δ kQ = , σQ = ω 8δ 3 − δ2 take place. From the first formula in (25) we have

0 2 f δ2 − 1 μf = μ max = 8 3 . ω 3 − δ2 When this parameter changes from√zero (linear system) to infinity (highly nonlinear system), the parameter δ changes from 1 to 3 = 1.7321, the modulus kQ changes from zero to  √  k = 12 2 − 3 = 0.2588, the maximum dynamic displacement fmax changes from 2Q/(f ω2 ) to zero, the angle α = arcsin kQ changes from zero to 15◦ , and the parameter σQ of the nonlinear frequency changes from ω (linear frequency) to infinity (frequency of a highly nonlinear, “next-tostatic” system). The formula for the amplitude θ = amu of the elliptic function can be obtained from (22) by putting cnu = cos θ. This yields

 1 fmax θ = 2 arcctg −1 . δ f

582 Could an Impact Load of Finite Duration Be Substituted with an Instantaneous Impulse? The amplitude θ reaches its maximum value θmax = π/2 when the displacement f (t) reaches its maximum value fmax . In this case the elliptic integral  θ dθ F (θ, kQ ) = 0 1 − kQ2 sin2 θ becomes the complete elliptic integral of the first kind: π  , kQ . u = K (kQ ) = F 2 Since the time required for the angle θ to change from zero to π/2 is equal to the quarter of the period of vibrations, we conclude that this period is 4K (k )/σQ , so that the frequency of the nonlinear vibrations can be found as π σQ pQ = . 2 K (kQ ) At the moment t = t0 of time solution (21) yields  f0 df u(σQ t0 , ε) = t0 =  2 2 4 σQ 2(Q/m)f (t) − ω f (t) − (1/2)μf (t) 0 F (θ0 , kQ ) 1 = = σQ σQ where



θ0

0



dθ 1 − kQ2 sin2 θ

(26) ,



 1 fmax θ0 = 2 arcctg −1 , δ f0

so that f0 =

fmax

θ0 2 After the displacement f0 is found, the velocity f˙0 can be determined by formula (13). 1 + δ cotan2

2.7 Free Vibrations with Nonzero Initial Conditions Let us revisit Eq. (7). With the initial conditions f (0) = f0 and = f˙ (0) = f˙0 formula (10) yields C = f˙02 + ω2 f02 + 12 μf04 . If the initial conditions f0 and f˙0 are due to the removal of the previously applied constant impact loading, then we have: C = 2(Q/m)f0 , and Eq. (13) yields Q (27) f˙ 2 (t) + ω2 f 2 (t) + 12 μf 4 (t) = 2 f0 . m The maximum displacement fmax = AQ takes place at the moment of time when the velocity f˙ (t) is zero, and relationship (27) yields ω2 2 Q AQ − 4 f0 = 0. μ μm This equation has the following solution:  

   ω2 Qf0  1 + 4μ − 1 = ηQ A0Q , AQ = μ mω4 A4Q + 2

(28)

(29)

2 Analysis 

where A0Q

=

2

583

Q f0 mω2

is the linear amplitude and the factor

 1 + 2μQ − 1 ηQ = μQ

considers the effect of the nonlinearity. This factor changes from unity to zero when the dimensionless parameter Qf μQ = 2μ 04 mω of nonlinearity changes from zero to infinity. The maximum acceleration (deceleration) can be found as f¨Q = −ω2 AQ − μA3Q = −σQ2 AQ , where σQ =



ω2 + μA2Q

is the parameter of the nonlinear frequency.

2.8 Error from Substituting Impact Load of Finite Duration with Instantaneous Impulse Equation (11) can be written as ω2 2 2 A − V02 = 0, μ S μ

(30)

ω2 2 2 Q 2 t02 = 0. AS − μ μ m2

(31)

A4S + 2

√ where V0 = (4/π ) 2gH is the initial velocity. This velocity can be determined as V0 = S /m when the applied load is caused by an instantaneous impulse S . In a situation when the actual impulse S is due to a suddenly applied and suddenly removed constant force Q acting during time t0 , we have S = Qt0 , so that Eq. (30) can be written as A4S + 2

This biquadratic equation has the following solution:  ⎞ ⎛   2 Q 2 t02 ω ⎝ AS =  1 + 2μ 2 4 − 1⎠. μ m ω Comparing solutions (29) and (32) we conclude that the ratio    2   1   1+8 δ −1 −1  2  1 + 4μ Qf0 − 1  2   3 − δ 2 1 + δ cotan (θ0 /2) AQ mω4   =  χA = = AS   2t 2 δ2 − 1 2 Q   0 1 + 2 2 F (θ0 , kQ ) − 1  1 + 2μ 2 4 − 1 δ 3 − δ2 m ω

(32)

(33)

584 Could an Impact Load of Finite Duration Be Substituted with an Instantaneous Impulse? characterizes the error, as far as the impact-induced amplitudes are concerned, from substituting a constant impact load Q applied for a short duration t0 of time (and removed when the displacement of the system is equal to f0 ) with an instantaneous impulse S = Qt0 . The ratio of the corresponding maximum accelerations (decelerations) can be found as   1 + χA2 (μ/ω2 )A2S f¨Q χf¨ = = χA . (34) 1 + (μ/ω2 )A2S f¨S This factor characterizes the error, as far as the impact-induced accelerations are concerned, from substituting a constant impact load Q applied for a short duration t0 of time with an instantaneous impulse S = Qt0 . The function χA = χA (δ, θ0 ) is tabulated in Table 1. This table enables one to determine, for the given degree of nonlinearity (characterized by the parameter δ of the modulus kQ0 of the elliptic function during the time of forced vibrations, that is, prior to the removal of the constant load Q) and the given duration of loading (characterized by the phase angle θ0 ), the effect of nonlinearity on the maximum displacements. Both the numerator and the denominator under the square root in formula (33) reduce with an increase in the degree of the nonlinearity, which is considered by the angle α. This situation, as evident from the Table 1 data, results in the fact that, for short loadings (small θ0 values), the factor χA has a minimum at certain α values. For δ values close to 1 (linear system), formula (33) yields sin(θ0 /2) χA = χA0 = (35) , θ0 = ωt0 . θ0 /2 The factor χA changes from 1 to 0.9004 when the phase angle θ0 changes from zero (instantaneous impulse) to 90◦ √ (the duration of loading is equal to a quarter of the period of vibrations). For δ values close to 3 = 1.7321, formula (34) yields   2 1.4142  χA =  = . (36)      θ0 2 (θ /2) 2 F , 0.2588) 0.5774 + cotan (θ 0 0 F θ0 , kQ 1/δ + cotan 2 The factor χA changes from 1 to 0.9972 when the phase angle θ0 changes from zero (instantaneous impulse) to 90◦ (the duration of loading is equal to a quarter of the period of vibrations). Table 1 Error in Predicted Amplitudes from Substituting Impact Load of Finite Duration with an Instantaneous Impulse

kQ0 δ θ0 0 5 10 20 30 40 50 60 70 80 90

α=0

α=5

α = 10

α = 14

α = 14.5

0 1

0.0872 1.0319

0.1736 1.1498

0.2504 1.5046

1 0.9998 0.9988 0.9950 0.9887 0.9799 0.9686 0.9550 0.9390 0.9208 0.9004

1 0.9992 0.9989 0.9949 0.9888 0.9804 0.9696 0.9567 0.9416 0.9245 0.9055

1 0.9994 0.9988 0.9962 0.9918 0.9855 0.9775 0.9680 0.9570 0.9445 0.9305

0.2419 1.4218 χA 1 0.9996 0.9995 0.9983 0.9961 0.9934 0.9900 0.9860 0.9815 0.9762 0.9700

Note: α = arcsin kQ0

1 0.9996 0.9996 0.9988 0.9973 0.9955 0.9932 0.9905 0.9874 0.9837 0.9792

α = 15 √

0.2588 3 = 1.7321 1 1 1 0.9999 0.9998 0.9997 0.9995 0.9993 0.9989 0.9983 0.9972

4 Conclusions

3

585

NUMERICAL EXAMPLE

3.1 Input Data PCB length 2a = 101 mm, thickness h = 1.00 mm, Young’s modulus E = 2020 kg/mm2 , Poisson’s ratio ν = 0.3, mass per unit area m = 2.95 × 10−10 kg × sec2 × mm−10 , phase angle when the constant load is removed θ0 = π/2.

3.2 Computed Data Flexural Rigidity: D=

Eh3 2020 × 1.003 = = 184.982 kg × mm 2 12(1 − ν ) 12(1 − 0.32 )

Linear frequency:

 π 2  D  π 2  184.982 = 766.143 sec−1 ω= = 2a m 101 2.95 × 10−10 Parameter of nonlinearity:  π 4 2020 × 1.00  π 4 Eh = 1602440.418 mm−2 sec−2 = μ= 2a 4m 101 4 × 2.95 × 10−10 The computed data in Table 2 is obtained for different drop heights. The duration t0 of loading in line 14 is accepted in such a way that the phase angle θ0 = σQ t0 is equal to π/2 = 90◦ . Table 3 illustrates how this duration was obtained for drop height H = 10 mm. The parameters of the nonlinear frequencies and the frequencies themselves, as well as the quarters of the corresponding times (periods of vibrations) in milliseconds, are shown for the assumed drop heights in Table 4. As evident from this table the frequencies of the vibrations caused by constant loads are approximately twice as high as the frequencies of vibrations due to instantaneous impulses. Comparing the corresponding times (vibration periods) with the loading durations t0 that result in phase angles θ0 = 90◦ , we conclude that an impact load of finite duration can be substituted by an instantaneous impulse if the duration does not exceed the time t0 = π/(2σS ), that is, a quarter of the “period” that corresponds to the parameter σS of the nonlinear frequency of vibrations caused by an instantaneous impulse.

4

CONCLUSIONS The Duffing oscillator is a suitable analytical (“mathematical”) model that can be used to analyze the nonlinear dynamic response of a printed circuit board (PCB) subjected to a drop or a shock impact. Simple and physically meaningful solutions are obtained for this oscillator when the excitation force can be idealized as an instantaneous impulse or a constant suddenly applied load of finite duration. The obtained solutions and the calculation procedures can be used to model the dynamic response of a PCB or a particular surface-mounted device (SMD) package, including ball grid array (BGA) and pad grid array (PGA) systems, on a board level to an impact load applied to the PCB supports during drop or shock tests. We have determined that the nonlinear system in question is, in general, less sensitive to the duration of the applied load than a linear system and that this sensitivity decreases with an increase in the degree of nonlinearity. Since the nonlinear frequency is strongly dependent on the magnitude of the applied load, we suggest that the nonlinear analysis be carried out prior to the assessment of

586 Could an Impact Load of Finite Duration Be Substituted with an Instantaneous Impulse? Table 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Response Characteristics for Different Drop Heights H , mm √ V0 = (4/π ) 2gH, mm/sec S = mV0 , ×107 kg × sec × mm−2 V0 A0S = , mm ω  0 2 μ = μ AS /ω √ 1 + 2μ − 1 ηA = μ AS = ηA A0S , mm  σS = ω 1 + μ (AS /ω)2 , sec−1 f¨S = −σS2 AS , mm/sec2    2 1 ω kS =  1− 2 2 σS αS = arcsin kS ,0 K (kS ) π σS pS = , sec−1 2K (kS ) t0 , msec Q = (S /t0 )104 , kg/mm2 0 = 2Q/(mω2 ), mm fmax  0 2 μf = μ fmax /ω ηQ 0 , mm fmax = ηQ fmax  δ = 3 − ηQ χA = χA (δ, θ0 ) AQ = χA AS , mm χf¨ = f¨Q /f¨S

10 563.688 1.6629

50 1260.444 3.7183

500 3985.874 11.7583

900 5347.611 15.7754

0.7358

1.6452

5.2026

6.9800

1.4780

7.3892

73.8930

133.0066

0.8180

0.6342

0.3893

0.3396

0.6019

1.0434

2.0253

2.3705

1080.0022

1526.9334

2675.8044

3097.0181

−71.5673g

−248.2361g

−1479.6943g

−2320.0715g

0.4984

0.6117

0.6775

0.6851

29.893 1.6858

37.709 1.7637

42.649 1.8262

43.246 1.8351

1006.325

1359.925

2301.577

2650.962

0.7183 2.3151 2.6740

0.5010 7.4218 8.5722

0.2890 40.8083 47.1343

0.2492 63.3142 73.1289

19.5203 0.3956 1.0578 1.6138 0.9878 0.5946 0.9759

200.6100 0.1999 1.7133 1.6733 0.9925 1.0356 0.9814

6065.0821 0.0675 3.1816 1.7125 0.9956 2.0165 0.9876

14599.6083 0.0507 3.7076 1.7173 0.9960 2.3611 0.9885

Table 3 Example of How Duration of Impact Was Computed for Given Phase Angle and Given Degree of Nonlinearity t0 , msec Q = (S /t0 ) × 104 , kg/mm2 0 = 2Q/(mω2 ), mm fmax  0 2 μf = μ fmax /ω ηQ 0 , mm fmax = ηQ fmax  δ = 3 − ηQ  σQ = ω 2δ/(3 − δ 2 ) θ0 = σQ t0

π/(8σS ) = 0.3636 4.5734 5.2824 76.1772 0.2679 1.4150 1.6529 2691.2 0.9785

0.7183 2.3150 2.6739 19.5191 0.3956 1.0578 1.6138 2188.2 π/2 = 1.5708

π/(4σS ) = 0.7272 2.2867 2.6412

π/(2σS ) = 1.4544 1.1434 1.3206

19.0443 0.3983 1.0520 1.6130

4.7611 0.5667 0.7484 1.5599

2180.6 1.5857

1797.6 2.6144

References

587

Table 4 Parameter of Nonlinear Frequency and Nonlinear Frequency for Instantaneous Impulse and (Suddenly Applied and Suddenly Removed Constant) Impact Load of Finite Duration

Instantaneous impulse

Parameter of nonlinear frequency, σS , sec−1 Nonlinear frequency, pS , sec−1 Parameter of the nonlinear frequency, σQ, sec−1 Nonlinear frequency, pQ, sec−1

Constant load

Drop Height, H = 10 mm

50 mm

500 mm

900 mm

1080 (1.4544 msec) 1006 (1.5614 msec) 2188 (0.7179 msec)

1527 (1.0287 msec) 1360 (1.1550 msec) 3133 (0.5014 msec)

2676 (0.5870 msec) 2302 (0.6824 msec) 5464 (0.2875 msec)

3097 (0.5072 msec) 2651 (0.5925 msec) 6295 (0.2495 msec)

2079 (0.7556 msec)

2977 (0.5276 msec)

5193 (0.3025 msec)

5982 (0.2626 msec)

the possible error in a modeling or a testing effort. We conclude that, as a rule of thumb, one can substitute an impact load of finite duration with an instantaneous impulse if this duration does not exceed the quarter of the “effective period” of the vibrations and that such period should be computed based on the parameter of the nonlinear frequency rather than on the nonlinear frequency itself. This should be done prior to carrying out the tests on the basis of the developed models. Although the analysis is carried out for to a simplified case of a simply supported elongated PCB employed in a specific accelerated test vehicle, whose ultimate goal is to predict the physical behavior of a SMD package and especially the performance of the BGA solder joint interconnections, it can be easily generalized for a PCB with other boundary conditions and/or with a finite aspect ratio and/or with different boundary conditions at the support contour.

REFERENCES 1. Suhir, E., “Response of a Flexible Printed Circuit Board to Periodic Shock Loads Applied to Its Support Contour,” ASME Journal of Applied Mechanics, Vol. 59, No. 2, 1992. 2. Suhir, E., “Nonlinear Dynamic Response of a Flexible Thin Plate to a Constant Acceleration Applied to Its Support Contour, with Application to Printed Circuit Boards Used in Avionic Packaging,” International Journal of Solids and Structures, Vol. 29, No. 1, 1992. 3. Suhir, E., “Predicted Fundamental Vibration Frequency of a Heavy Electronic Component Mounted on a Printed Circuit Board,” ASME Journal of Electronic Packaging, Vol. 122, No. 1, pp. 1–3, 2000. 4. Seah, S. K. W. Lim, C. T., Wong, E. H., Tan, V. B. C., and Shim, V. P. W., “Mechanical Response of PCBs in Portable Electronic Products during Drop Impact,” in Proceedings 4th Electronics Packaging Technology Conference (EPTC 2002), Singapore, Dec. 10–12, 2002, pp. 120–125. 5. Wang, Y. Q., “Modeling and Simulation of PCB Drop Test,” in Proc. 5-th EPTC , Singapore, 2003, pp. 263–268. 6. Luan, J. E., Tee, T. Y., Pek, E., Lim, C. T., and Zhong, Z. W., “Modal Analysis and Dynamic Responses of Board Level Drop Test,” in 5-th EPTC Conference4 Proc., Singapore, 2003, pp. 233–243.

588 Could an Impact Load of Finite Duration Be Substituted with an Instantaneous Impulse? 7. Tee, T. Y., Luan, J. E., Pek, E., Lim, C. T., and Zhong, Z. W., “Novel Numerical and Experimental Analysis of Dynamic Responses under Board Level Drop Test,” in EuroSime Conference Proc., Berlin, Germany, 2004, pp. 671–677. 8. Marjamaki, P., Mattila, T., and Kivilahti, J., “FEA of Lead-Free Drop Test Boards,” in 55-th ECTC Proc., Buena Vista, FL, 2005, pp. 1653–1658. 9. Suhir, E., “Response of a Heavy Electronic Component to Harmonic Excitations Applied to Its External Electric Leads,” in Electrotechnik & Informationstechnik , Springer, Wien, 2007, pp. 309–314. 10. Timoshenko, S. P., and Gere, J. M., Theory of Elastic Stability, 2nd ed., McGraw-Hill, New York, 1988. 11. Suhir, E., Structural Analysis in Microelectronics and Fiber Optics, Van Nostrand, 1997. 12. Pars, L. A., A Treatise of Analytical Dynamics, Heinemann, London, 1965. 13. Kauderer, H., Nichtlineare Mechanik , Springer, Berlin, 1958 (in German). 14. Sneddon, I. N., Special Functions of Mathematical Physics and Chemistry, 3rd ed., Longman, New York, 1980. 15. Appell, P., and Lacour, E., Principes de la theorie des functions elliptiques et applications, 2nd ed., Gauthier-Villars, Paris, 1922 (in French). 16. Oberhettinger, F., and Magnus, W., Anwendung der elliptischen Funktionen in Physiknund Technik , Springer Verlag, Berlin, 1949 (in German). 17. Spanier, J., and Oldham, K. B., An Atlas of Functions, Hemisphere, 1987. 18. Gradshteyn, I. S., and Ryzhik, I. M., Tables of Integrals, Series, and Products, Academic, New York, 1980.

Index

Accelerated temperature cycling (ATC) testing, 485, 494–495 Acceleration: and displacements, 7–9 dynamic response of elongated plate: constant suddenly applied/removed acceleration, 440–444 half-sine acceleration, 443–446 as dynamic strength criterion, 357–359 input, strain as function, 424 mean-squared acceleration density, 110–113 Active isolation systems, 322 Actuators (hard disk drives), 337 Air–spring isolation systems, 321–322 Amplitudes, complex, 59 AOW, see Array of wires Area array package assemblies, 519–569 advantages of, 522–523 BGA 225-I/O vibration, 543–550 CALCE modeling of vibration data, 548–549 testing and vibration analysis by Boeing, 543–548 Weibull plots of data, 549–550 CBGA assemblies, 551–558 chip-scale packages, 523–524 CSP-TC and vibration behavior, 535–543 pros and cons of, 523 disadvantages of, 523 drop behavior and a priori thermal cycle, 563–567 drop shock behavior and isothermal aging of solder joints, 558–563 PBGA thermal cycle solder joint reliability, 526–535 cycles-to-failure, 533–535 high I/O assemblies, 530–533 ICP standard for thermal cycle, 526–527 plastic packages, 527–530 technology trend for, 520–526 3D stack packages, 524–526 Arms (hard disk drives), 337 Array of wires (AOW), 357–366 compressed cantilever beam on elastic foundation, 362–363 Structural Dynamics of Electronic and Photonic Systems Copyright © 2011 John Wiley & Sons, Inc.

in portable electronic devices, 357–362 unembedded cantilever wire (beam) subjected to axial compression, 363–366 ATC (accelerated temperature cycling) testing, 485, 494–495 Ball grid arrays (BGAs): BGA 225-I/O vibration, 543–550 CALCE modeling of vibration data, 548–549 testing and vibration analysis by Boeing, 543–548 Weibull plots of data, 549–550 ceramic, 551–558 CCGA vs., 520 configuration, 521 vibration and thermal cycle synergism, 551–558 configuration, 521 plastic: key drawbacks of, 520, 521 thermal cycle solder joint reliability, 526–535 Ball impact test (BIT), 486 Ball pull test, 196. See also High-speed ball shear and pull tests Ball shear test, 196. See also High-speed ball shear and pull tests Beam structures: dynamic impact model, 153–157 natural frequencies, 76–83 response of cantilever beam to random vibrations, 117–118 Benchtop platforms (vibration isolation), 322 BGAs, see Ball grid arrays BIT (ball impact test), 486 Bivariate correlation functions, in PCBs, 66–68 Bivariate spectral densities, in PCBs, 66–68 BLDT, see Board-level drop tests Board-level drop tests (BLDT), 229–249 correlation of high-speed ball shear/pull tests and, 246–249 drop test setup, 230–231 effects of thermal aging on drop test, 231–237 experimental procedures, 230 failure mode comparison, 235, 237–245 test vehicles, 230 Edited by Ephraim Suhir, David S. Steinberg and T. X. Yu

589

590 Index Board-level dynamic bending test, four-point (4PDB), 486, 495–504 correlation with JEDEC drop test method, 503–504 experimental methodology, 496–499 finite element modeling and simulation, 499–503 PDB testing methodology, 495 Board-level reliability, 195 Board-level shock tests: lead-free solders under mechanical shock and vibration loads, 371–408 PCB structures, 415–432 experimental data, 424–429 measured deformation and predicted failures, 416–418 modeling of, 429–432 theoretical considerations, 419–423 Boundary conditions, in finite element analysis, 330 Case histories, in testing, 192–193 CBGAs, see Ceramic ball grid arrays CBT (cold bump pull) test, 486 CCGA (column grid arrays), 520, 521 Ceramic ball grid arrays (CBGAs): CCGA vs., 520 configuration, 521 vibration and thermal cycle synergism, 551–558 CGA (column grid arrays), 520, 521 Chapman–Kolmogorov equation, 45 Characteristic frequency, 345 Chip-scale packages (CSPs), 523–524 CSP-TC and vibration behavior, 535–543 results, 538–543 text matrix and procedures, 536–538 pros and cons of, 523 Cold bump pull (CBP) test, 486 Column grid arrays (CGA, CCGA), 520, 521 Commercial, off-the-shelf area array packaging (COTS AAP) technologies, 519. See also Area array package assemblies Complex amplitudes, 59 Complex-frequency characteristics, 60–62 Complex-frequency spectrum, 59 Complex spectrum (complex spectral density, complex spectral characteristic of the function), 60 Component-level shock tests, 374–376 Connector fretting corrosion, in random vibration, 11–13 Constant-amplitude vibration fatigue testing method, 506–509 Contact nonlinearity, 328–329 Correlation theory of random processes, 63

COTS AAP (commercial, off-the-shelf area array packaging) technologies, 519. See also Area array package assemblies Counterweights, 189 CSPs, see Chip-scale packages Damped forced vibration, 454 Damped free vibration, 451–453 Damping, 2–3, 185 defined, 185 and induced accelerations, 359 optimized, 71–73 viscoelastic damping materials: problems caused by, 185–186 for severe sine vibration environment, 3–4 Delta-correlated process, 65 Dendrites, 2 Deterministic approaches, 53 Dimensional tolerances, 178 Displacements: in PCB shock tests, 424 related to frequency and acceleration, 7–9 in sine vibration environment, 91–92, 102–104 Drop tests: bending of component board under, 376–380 board-level, 156, 229–249 correlation of high-speed ball shear/pull tests and, 246–249 drop test setup, 230–231 effects of thermal aging on drop test, 231–237 experimental procedures, 230 failure mode comparison, 235, 237–245 test vehicles, 230 drop/impact, 36, 135–158 dynamic model, 147–157 experimental set-up, 136–138 impacts in different orientations, 140–142 impacts under different velocities, 142–147 necessity for, 159–160 repeatability of impact tests, 138–140 for dynamic robustness, 358 free fall tests, 160–161, 164–171 hard disk drives, 337–355 loading of solder interconnections during, 380–384 modeling as substitute for, 429 product-level, 156 reliability of lead-free solder, 371–372 shock tests substituted for, 435–448 analysis, 435–439 energy approach, 446–447 error from substituting impact load with instantaneous impulse, 440–443 instantaneous impulse applied to support contour, 439–440

Index probabilistic approach, 447 when applied acceleration is not short enough, 444–446 vibration tests as replacements for, 386–391 Duffing oscillator, 575–587 dynamic factor, 579–580 error from substituting impact load of finite duration with instantaneous impulse, 583–584 free vibrations with nonzero initial conditions, 582–583 nonlinear response, 576–578 numerical example, 585 vibration caused by instantaneous impulse, 578 vibrations caused by instantaneous impulse, 580 vibrations caused by suddenly applied constant load, 578–579, 581–582 Duhamel integral, 20–21 Duration of process, width of spectrum and, 62 Dynamic impedance, 60–61 Dynamic loading, 35. See also Shock-excited vibrations Dynamic stability of systems, electronics for control of, 17–18 Dynamic vibration testing (microelectronics packages), 504–515 constant-amplitude vibration fatigue testing method, 506–509 dynamic modal testing method, 505–506 finite element analysis, 509–515 vibration testing methodology, 504–505 Electrodynamic shaking equipment, 186–187 External loading (random vibrations): dynamic response of multi-degree-of-freedom linear system to, 57–61 elongated PCB subjected to, 54–55 Failure-free operating period (FFOP), 180–181 Failure modes and mechanisms, 1–17 and companies’ attempts to save money, 14–15 of components boards under fast deformation rates, 384–386 connector fretting corrosion in random vibration, 11–13 difficulty in solving failure problems, 13–14 displacements related to frequency and acceleration, 7–9 high-speed ball shear and pull tests, 199, 200 and manufacturing methods and material properties, 184–185 in opto-electronic fiber-optic systems, 17 physics of, 1–3 plug-in PCBs tied together in 5g peak sine vibration environment, 4–5

591

resonance coupling of outer housing enclosure with internal PCBs, 9–11 and shipping bonuses, 15 snubbers for severe environments, 5–7, 11 thermal and vibration cycling, 183–184 and thermal cycling, 183–184 thick aluminum casting, 16 and vibration cycling, 183–184 viscoelastic damping materials, 3–5, 185–186 Fatigue life: and bending curvature of PCB, 104 isolation systems, 96 Miner’s cumulative fatigue damage ratio, 118–119 snubbers, 5, 11 solder joints, 176–177 three-band technique, 119–121, 298–300 vibration fatigue damage evaluation, 277–305 experiment, 285–291 failure analysis, 291–294 finite element analysis, 296–298 methodology development, 294–296 model development and validation, 298–304 test vehicle design, 278–285 viscoelastic damping on PCBs, 3–4 FEA, see Finite element analysis FEMs (finite element models), 102–103, 341–342 FFOP (failure-free operating period), 180–181 Finer pitch area array packages (FPBGAs), 521. See also Chip-scale packages (CSPs) Finite element analysis (FEA), 327–336 commercial FEA codes, 329 harmonic (frequency domain) simulation, 333–334 history of, 327 linear, 327–328 microelectronics package reliability, 499–503, 509–515 modal analysis, 331–332 modeling PCB shock, 429–430 nonlinear, 328–329 power spectral density analysis, 335–336 response spectrum simulations, 334–335 steps in FEA simulation, 330–331 transient (time domain) simulation, 332–333 vibration fatigue damage, 296–298 Finite element models (FEMs), 102–103, 341–342 Fluid damping shock isolators, 461, 462 Forced vibration: damped, 454 undamped, 454 Fourier integral, 59 Fourier integral formula, 59 Fourier series, 58–59 Fourier transforms, 60

592 Index Four-point board-level dynamic bending (4PDB) test, 486, 495–504 correlation with JEDEC drop test method, 503–504 experimental methodology, 496–499 finite element modeling and simulation, 499–503 PDB testing methodology, 495 FPBGAs (finer pitch area array packages), 521. See also Chip-scale packages (CSPs) Free fall tests, 160–161, 164–171 Free vibration: damped, 451–453 heavy electronic components, 33 undamped, 450–451 Frequency, and acceleration and displacement, 9 Frequency domain (harmonic) simulation, 333–334 Frequency response function: of the amplitudes, 61 for the phase angles, 61 Gain factor, 61 Gaussian distribution curve, 113–114 Geometric nonlinearity, 328, 421 Half-sine shock pulse, 124–126 Half-sine shock pulse drop test, 125–128 Hard disk drives (HDDs), 337–355 development of, 337–338 drop test simulation, 343–344 finite element modeling, 341–342 power spectrum analysis, 350–354 pseudoresonance phenomenon, 344–348 pulse shape effect, 348–350 shock robustness of, 338–341 Harmonics, random vibrations and, 70–71 Harmonic excitation: heavy electronic components subjected to, 26–31 single-degree-of-freedom systems, 21–22 Harmonic (frequency domain) simulation, 333–334 HDDs, see Hard disk drives Heads (hard disk drives), 337 Heavy electronic components: elements subjected to harmonic excitation, 26–31 natural vibration frequency of, 31–34 Hertz contact theory, 151 High-frequency dynamic vibration testing (microelectronics packages), 504–515 constant-amplitude vibration fatigue testing method, 506–509 dynamic modal testing method, 505–506 finite element analysis, 509–515 vibration testing methodology, 504–505

High-speed ball shear and pull tests, 197–249 brittle failure of lead-free vs. lead-tin solder balls, 205–215 correlation of board-level drop test and, 229–249 experimental procedures, 197–198 failure modes, 199, 200 IMC growth after thermal aging, 215–229 testing speed effects, 199, 201–205 Holding fixtures, 186 Impact tests, see Drop tests, drop/impact Impedance: dynamic, 60–61 static, 60 Impulse response: complex-frequency characteristics as spectrum of, 60–62 external loading of multi-degree-of-freedom system, 58 Inertia blocks (vibration isolation), 324–326 Isolated workstations (vibration isolation), 322–323 Isolation cradles (vibration isolation), 323–324 Isolation systems, 96. See also Shock isolation; Vibration isolation fatigue life, 96 for sensitive facilities/equipment, 312 mass-spring-damper model, 318–320 system configurations, 322–326 system types, 321–322 Isothermal mechanical twisting (ITMT) testing, 486–487 correlation of ATC with, 494–495 experimental methodology, 487–492 methodology, 486–487 microelectronics package reliability, 486–495 numerical modeling and simulation methodology, 492–494 Lead-free solders, 255–272 board-level reliability under mechanical shock and vibration loads, 371–408 brittle failure of lead-tin solder balls vs., 205–215 dynamic material properties of, 268 mechanical properties of, 268–272 microstructure of, 257–261 problems with, 176 quasi-static material properties of, 262 reliability under mechanical drop loading, 229 short-term loading effect on, 36–37 specimen preparation, 258 tangential modulus of, 266–268 yield of, 264–266 Young’s modulus, 262–264

Index Lead-tin solders: brittle failure of lead-free solder vs., 205–215 hazardous effects of, 255 mechanical properties of, 268–272 microstructure of, 258–259 tangential modulus of, 266–268 yield of, 264–266 Young’s modulus, 262–264 Lead wires: effects of sine vibrations on, 83–89 thermal cycling on, 179–180 Linear finite element analysis, 327–328 Linearly elastic materials, 419–420 Linear response (shock-excited vibrations), 37–39 Linear systems, transformation of stationary random processes by, 64–65 Linear vibrations, 19–31 damping, 2–3 harmonic excitation, 21–22, 26–31 heavy electronic components: natural vibration frequency of, 31–34 subjected to harmonic excitation, 26–31 lateral oscillations of PCB support contour, 23–25 multiple-degrees-of-freedom systems, 22–23 single-degree-of-freedom system, 19–22 Logic board (hard disk drives), 337 Magnetic damping shock isolators, 463 Manufacturing methods: failures from, 184–185 and reliability of solder joints, 177–178 Markovian processes, 45 Mass–spring–damper model (vibration isolation), 318–320 Material nonlinearity, 328 Material properties: failures from, 184–185 in finite element analysis, 330 of lead-free solders: dynamic, 268 quasi-static, 262 MDOF systems, see Multiple-degrees-of-freedom systems Mean-squared acceleration density, 110–113 Mean time between failures (MTBF), 180–181 Mechanical vibration system: damped forced vibration, 454 damped free vibration, 451–453 undamped forced vibration, 454 undamped free vibration, 450–451 Meshing (FEA), 330 Microelectronics package reliability, 485–516 four-point dynamic bending testing, 495–504 correlation of 4PDB with JEDEC drop test method, 503–504

593

experimental methodology, 496–499 finite element modeling and simulation, 499–503 PDB testing methodology, 495 high-frequency dynamic vibration testing, 504–515 constant-amplitude vibration fatigue testing method, 506–509 dynamic modal testing method, 505–506 finite element analysis, 509–515 vibration testing methodology, 504–505 isothermal mechanical twisting testing, 486–495 correlation of ITMT with ATC method, 494–495 experimental methodology, 487–492 ITMT methodology, 486–487 numerical modeling and simulation methodology, 492–494 Micromachines devices (MMDs), 449–482 classification of shock isolators, 456–465 design for high-g applications, 469–471 dynamic response under shock and vibration, 466–471 equivalent damping in, 467–469 equivalent mass in, 466–467 equivalent stiffness in, 467 mechanical vibration system fundamentals, 450–454 shock isolation, 455–456 woodpecker-inspired microparticle shock isolator, 471–481 Microparticle shock isolator, 450, 471–481 experimental characterization, 479–481 mechanism in woodpecker, 471–478 structure and working principle, 479 Miner’s cumulative fatigue damage ratio, 118–119 MMDs, see Micromachines devices Modal analysis (FEA), 331–332 Money-saving efforts, failures related to, 14–15 MTBF (mean time between failures), 180–181 Multiple-degrees-of-freedom (MDOF) systems: dynamic response to external loading, 57–61 linear vibrations, 22–23 random loading, 66–68 shock protection, 36 Natural frequencies: beam structures, 76–83 heavy electronic components, 31–34 nonlinear systems, 57 PCB housings, 2–3 PCBs, 2, 97–104, 140–131 sine vibration environment, 102–104 uniform flat plates, 97–101 Negative-stiffness isolation systems, 322 Nonlinear finite element analysis, 328–329

594 Index Nonlinearity of materials, 420–421 Nonlinear response (shock-excited vibrations), 39–42 Nonlinear systems, natural frequencies in, 57 Nonobstructive particle damping shock isolators, 463 Octave rule: alternatives to, 95–96 sine vibration-induced failures, 10–11 sine vibration reduction, 89–90 Oil film slider table, 188–189 Optical tables (vibration isolation), 323 Optimized damping, 71–73 Opto-electronic fiber-optic systems, failure modes in, 17 Oscilloscopes, 75–76 Package-level reliability, 195 PBGAs, see Plastic ball grid arrays PCBs, see Printed circuit boards PCB housings, 2 natural frequency of, 2–3 resonance coupling of PCBs and, 9–11 PCB support contour: harmonic oscillations of, 23–25 harmonics and dynamic response to shock applied to, 70–71 optimized damping, 71–73 periodic impulses applied to, 42 vibrations transmitted to heavy components, 26–31 Periodic impulses, 35–47 linear response, 37–39 nonlinear response, 39–42 Smoluchowski’s equation, 45–47 stochastic instability, 43–44 stochastic phase approximation, 44–45 Phase factor, 61 Plastic ball grid arrays (PBGAs): key drawbacks of, 520, 521 thermal cycle solder joint reliability, 526–535 cycles-to-failure, 533–535 high I/O assemblies, 530–533 ICP standard for thermal cycle, 526–527 plastic packages, 527–530 Plug-in PCBs, 95 loose edge guides on, 101–102 snubbers bonded to, 4–5, 11 tied together in 5g peak sine vibration environment, 4–5 Pneumatic isolators, 321–322 Power spectral density (PSD), 63–64, 110–113 Power spectral density (PSD) analysis, 335–336 Power spectrum, 63–64

Power spectrum analysis, 350–354 Predictive models, probabilistic, 54 Printed circuit boards (PCBs), 1 bending curvatures of, 104 bivariate correlation functions and bivariate spectral densities, 66–68 designed to operate in random vibrations, 121–122 desired natural frequency for shock environments, 140–131 dynamic impact model, 152–157 dynamic response to shock loading, 415–432 board-level shock tests, 418–419 experimental data, 424–429 measured deformation and predicted failures, 416–418 modeling of, 429–432 theoretical considerations, 419–423 dynamic response when subjected to drop/shock with reactive in-plane forces, 575–587 dynamic factor, 579–580 error from substituting impact load of finite duration with instantaneous impulse, 583–584 free vibrations with nonzero initial conditions, 582–583 nonlinear response, 576–578 numerical example, 585 vibration caused by instantaneous impulse, 578 vibrations caused by instantaneous impulse, 580 vibrations caused by suddenly applied constant load, 578–579, 581–582 electronic components on, 95 elongated PCB subjected to external loading, 54–55 mounting components on, 2 natural frequencies, 2, 97–100, 102–104 PCB housings, 2 natural frequency, 2–3 resonance coupling of PCBs and, 9–11 PCB support contour: harmonic oscillations of, 23–25 harmonics and dynamic response to shock applied to, 70–71 optimized damping, 71–73 periodic impulses applied to, 42 vibrations transmitted to heavy components, 26–31 plug-in: loose edge guides on, 101–102 snubbers bonded to, 4–5, 11 tied together in 5g peak sine vibration environment, 4–5 resonance coupling of housings and, 9–11

Index responses to shock pulses, 128–133, 192 in severe sine vibration environments, 3–4, 89–90 sine vibrations and components soldered to, 83–89 snubbers, 95–96 solder joint vibration with components mounted on, 175 types of, 95 viscoelastic damping materials for severe sine vibration environments, 3–4 Probabilistic methods and approaches, 53–54. See also Random vibrations Probabilistic risk management, 54 Protective coatings, 2 PSD (power spectral density), 63–64, 110–113 PSD (power spectral density) analysis, 335–336 Pseudoresonance: defined, 347 hard disk drives, 344–348 Pulse shape effect (hard disk drives), 348–350 Pulse shocks, 124, 191, 192 Q factor, 8–9 Quasi-static test, 424–425 Random processes: correlation theory of, 63 log-log input and response curves for, 110 PSD (mean-squared acceleration density), 110–113 spectral theory of, 63–64 Random vibrations, 53–73 approximate fatigue life using three-band technique, 119–121 bivariate correlation functions and bivariate spectral densities in PCBs, 66–68 complex-frequency characteristics as spectrum of impulse response, 60–62 connector fretting corrosion in, 11–13 correlation theory of random processes, 63 designing PCBs to operate in, 121–122 deterministic vs. probabilistic approaches to, 53–54 duration of process and width of its spectrum, 62 effects of, 107–108 external loading: dynamic response of multi-degree-offreedom linear system to, 57–61 elongated PCB subjected to, 54–55 Gaussian and Rayleigh probability distribution functions for fatigue life estimation, 113–115 and higher modes (harmonics), 70–71 history of analysis and testing methods, 119

595

linear dynamic systems: multi-degree-of-freedom response to external loading, 57–61 transformation of stationary random processes, 64–65 optimized damping, 71–73 probability of exceeding given levels, 68–70 in reliability evaluations, 358 response of cantilever beam to, 117–118 sine vibrations vs., 108–110 single-degree-of-freedom system and fatigue life in, 115–117 solution using Duhamel integral, 58 solution using Fourier integral, 58–61 spectral theory of random processes, 63–64 transformation of stationary random processes by linear dynamic systems, 64–65 white noise, 65–66 Random vibration analysis, 335 Rayleigh distribution, 113–114 Reliability: defined, 180, 195 lead-free solders: under mechanical drop loading, 229 under mechanical shock and vibration loads, 371–408 levels of, 195 MTBF compared with FFOP, 180–181 random vibrations in evaluations, 358 of solder joints, 177–178. See also Solder joint reliability Resonance coupling, of housing and PCBs, 9–11 Response spectrum simulations, 334–335 Rigid-body dynamics, impact impulse and, 147–151 Root-mean-square (RMS), 110 SDOF systems, see Single-degreeof-freedom systems Sensitive research and production facilities, 309–326 design considerations, 309, 312, 315–316 sources of vibration: building services, 309–311 external, 310–311 internal, 310–311 testing and evaluation, 316–318 vibration criteria, 312–315 vibration isolation, 318–326 isolation system types, 321–322 mass-spring-damper model, 318–320 system configurations, 322–326 Severe environments: octave rules for reducing sine vibration, 89–90 plug-in PCBs tied together in, 4–5

596 Index Severe environments: (continued) snubbers for, 5–7, 11 viscoelastic damping materials for, 3–4 Shaft failure (spinning gyros), 193–194 Shipping bonuses, 15–16 Shock(s), 122–133 common types of, 124 defined, 190 desired PCB natural frequency for, 140–131 failures due to, 190–191 half-sine shock pulse, 124–126 half-sine shock pulse drop test, 125–128 PCB responses to shock pulses, 128–133, 192 pulse, 124, 191, 192 responses to common shock pulses, 128 shock response spectra, 124 specifying shock motion and environment, 191 velocity, 124, 191 Shock-excited vibrations, 35–47 linear response, 37–39 nonlinear response, 39–42 Smoluchowski’s equation, 45–47 stochastic instability, 43–44 stochastic phase approximation, 44–45 Shock isolation: dynamic response under shock and vibration, 466–471 mechanical vibration system fundamentals, 450–454 for micromachines devices, 449–482 classification of shock isolators, 456–465 design for high-g applications, 469–471 dynamic response under shock and vibration, 466–471 equivalent damping in, 467–469 equivalent mass in, 466–467 equivalent stiffness in, 467 mechanical vibration system fundamentals, 450–454 shock isolation, 455–456 woodpecker-inspired microparticle shock isolator, 471–481 microparticle, 471–481 Shock isolators: classification of, 456–465 based on combination of spring and damper, 456–461 based on existence of external power source, 456 based on working mechanism or material, 461–465 function of, 449–450 fundamental concept, 455 microparticle, 450, 471–481 experimental characterization, 479–481

mechanism in woodpecker, 471–478 structure and working principle, 479 Shock motion, specifying, 191 Shock response spectra, 124, 191 Shock table tests, 161, 165–171 Shock tests, 159–172 comparison of methods, 163–164, 167–171 component- vs. product level, 160 criteria and specifications, 160 drop (free-fall) tests, 160–161, 164–171 dynamic response of PCB structures to shock loading, 415–432 board-level shock tests, 418–419 experimental data, 424–429 measured deformation and predicted failures, 416–418 modeling of, 429–432 theoretical considerations, 419–423 for dynamic robustness, 358 hard disk drives, 338–341 methods of, 374–386 reliability of lead-free solder, 372 shock table tests, 161, 165–171 shock test machine, 161–163 substituted for drop tests, 435–448 analysis, 435–439 energy approach, 446–447 error from substituting impact load with instantaneous impulse, 440–443 instantaneous impulse applied to support contour, 439–440 probabilistic approach, 447 when applied acceleration is not short enough, 444–446 Shock test machine, 161–163 Side wedge clamps, 95 Sine vibrations, see Sinusoidal (sine) vibrations Single-degree-of-freedom (SDOF) systems: dynamic response of PCB subjected to drop/shock with reactive in-plane forces, 575–587 dynamic factor, 579–580 error from substituting impact load of finite duration with instantaneous impulse, 583–584 free vibrations with nonzero initial conditions, 582–583 nonlinear response, 576–578 numerical example, 585 vibrations caused by instantaneous impulse, 578, 580 vibrations caused by suddenly applied constant load, 578–579, 581–582 dynamic response to external loading, 57–58 and fatigue life in random vibrations, 115–117 harmonic excitation, 21–22

Index linear response: to impact load, 435–448 to periodic impulses, 37–39 linear vibrations, 19–21 nonlinear response periodic impulses, 39–42 Sinusoidal (sine) vibrations, 75–95 box structures and frame structures, 105–106 effects on lead wires and solder joints, 83–89 large ball grid array vibration fatigue life, 92–93 large dynamic displacements in PCBs, 91–92 natural frequencies for beam structures, 76–83 octave rule, 10–11, 89–90 plug-in PCBs tied together in 5g peak environment, 4–5 random vibrations vs., 108–110 sine sweeps through a resonance, 93–95 viscoelastic damping materials, 3–4 Site assessments (vibration-sensitive facilities), 317–318 Sliders (hard disk drives), 337 Smart material-based shock isolators, 463–464 Smoluchowski’s equation, 45–47 Snubbers, 11, 95–96 Solders: attachment of, 175 lead-free, 255–272 brittle failure of lead-tin solder balls vs., 205–215 brittle failure of lead-tin vs., 205–215 dynamic material properties of, 268 mechanical properties of, 268–272 microstructure of, 257–261 problems with, 176 quasi-static material properties of, 262 reliability under mechanical drop loading, 229 short-term loading effect on, 36–37 specimen preparation, 258 tangential modulus of, 266–268 yield of, 264–266 Young’s modulus, 262–264 lead-tin: brittle failure of lead-free solder balls vs., 205–215 brittle failure of lead-free vs., 205–215 hazardous effects of, 255 mechanical properties of, 268–272 microstructure of, 258–259 tangential modulus of, 266–268 yield of, 264–266 Young’s modulus, 262–264 plasticity of, 184 short-term loading effect on, 36–37 types of, 175 Solder creep, 183–184, 257 Soldering, 1–2

597

Solder joints, 175–181 attachment of solders, 175 dimensional tolerances, 178 effects of sine vibrations on, 83–89 evolving use of, 195–196 failure of, 183–184 lead-free solders, problems with, 176 manufacturing/assembly quality and reliability of, 177–178 MTBF compared with FFOP, 180–181 in portable products, 358 predicting fatigue life of, 176–177 reliability of, see Solder joint reliability thermal cycling on, 179–180 vibration with components mounted on PCBs, 175 Solder joint reliability, 195–252 area array package assemblies: BGA 225-I/O, 543–550 CSP-TC, 535–543 drop behavior and a priori thermal cycle, 563–567 drop shock behavior and isothermal aging of joints, 558–563 PBGA, 526–535 board-level, under mechanical shock and vibration loads, 371–408 under combined loading conditions, 391–406 methods of shock impact testing, 374–386 vibration tests as replacement for drop tests, 386–391 high-speed ball shear and pull tests, 197–249 brittle failure of lead-free vs. lead-tin solder balls, 205–215 correlation of board-level drop test and, 229–249 experimental procedures, 197–198 failure modes, 199, 200 IMC growth after thermal aging, 215–229 testing speed effects, 199, 201–205 and push for miniaturization, 196–197 Spectral densities: bivariate, 66–68 complex, 60 power, 63–64 Spectral theory of random processes, 63–64 Spindle (hard disk drives), 337 Spindle motor (hard disk drives), 337 Spinning gyros, failures in shafts of, 193–194 Static impedance, 60 Stationary random processes: transformation of, 64–65 white noise, 65–66 Stochastic instability (shock-excited vibrations), 43–44

598 Index Stochastic phase approximation (shock-excited vibrations), 44–45 Strain, as function of input acceleration, 424 Structural evaluations (vibration-sensitive facilities), 318 Suspension (hard disk drives), 337 System-level reliability, 195 Testing, 183–194. See also specific tests case histories, 192–193 effects of shock, 190–191 equipment and fixtures, 186–190 failures: from manufacturing methods and material properties, 184–185 from thermal and vibration cycling, 183–184 of viscoelastic damping materials, 185–186 pulse shocks, 192 shafts of spinning gyros, 193–194 specifying shock motion and environment, 191 Theory of probability, 53 Thermal aging: effects on board-level drop tests, 231–237 IMC growth after, 215–229 Thermal cycling: failures from, 183–184 on solder joints and lead wires, 179–180 3D stack packages, 524–526 Three-band technique (fatigue life), 119–121, 298–300 Three-point bending (TPB) methodology, 485 Time domain (transient) simulation, 332–333 TPB (three-point bending) methodology, 485 Transient (time domain) simulation, 332–333 Transmissibility (Q) factor, 8–9 Ultrasmall ping-pong balls, 96 Undamped forced vibration, 454 Undamped free vibration, 450–451 Uniform flat plates, natural frequencies of, 97–101 Velocity shocks, 124, 191 Vibration cycling, failures from, 183–184

Vibration fatigue damage, 277–305 experiment, 285–291 failure analysis, 291–294 finite element analysis, 296–298 methodology development, 294–296 model development and validation, 298–304 test vehicle design, 278–285 Vibration isolation, 312 isolation system types, 321–322 mass-spring-damper model, 318–320 system configurations, 322–326 Vibration machines, 186–187 Vibration surveys, 316–317 Vibration tests: microelectronics package reliability, 504–515 constant-amplitude vibration fatigue testing method, 506–509 dynamic modal testing method, 505–506 finite element analysis, 509–515 methodology, 504–505 as replacement for drop tests, 386–391 Vibration test fixtures: counterweights, 189 design considerations, 188–190 oil film slider table, 188–189 types of, 187–188 Viscoelastic damping materials: failure of, 185–186 for severe sine vibration environment, 3–4 Viscoelastic material-based shock isolators, 461, 462 Viscous damping, 359

White noise, 65–66 Wiener–Khinchin formulas, 63 Woodpecker-inspired microparticle shock isolator, 464–465, 471–481 experimental characterization, 479–481 shock isolation mechanism, 471–478 allometric analysis, 472 anatomic analysis, 473 experimental analysis, 476–478 vibration analysis, 473–476 structure and working principle, 479