Fractal Analysis in Machining

SpringerBriefs in Applied Sciences and Technology Computational Mechanics

For further volumes: http://www.springer.com/series/8886

Prasanta Sahoo Tapan Barman João Paulo Davim •

Fractal Analysis in Machining

123

Prasanta Sahoo Department of Mechanical Engineering Jadavpur University Kolkata 700032 India e-mail: [email protected]

João Paulo Davim Department of Mechanical Engineering University of Aveiro Campus Universitário de Santiago 3810-193 Aveiro Portugal e-mail: [email protected]

Tapan Barman Department of Mechanical Engineering Jadavpur University Kolkata 700032 India e-mail: [email protected]

ISSN 2191-5342 ISBN 978-3-642-17921-1 DOI 10.1007/978-3-642-17922-8

e-ISSN 2191-5350 e-ISBN 978-3-642-17922-8

Springer Heidelberg Dordrecht London New York Ó Prasanta Sahoo 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: eStudio Calamar, Berlin/Figueres Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

The present book deals with fractal analysis of surface roughness in different machining processes. Surface roughness is an important attribute of any machine component. Conventionally several statistical roughness parameters are used for describing surface roughness. But surface topography is a non-stationary random process for which the variance of the height distribution of roughness features is related to the length of the sample. Consequently, instruments with different resolutions and scan lengths yield different values of these statistical parameters for the same surface. A logical solution to this problem is to use scale-invariant parameters to characterize rough surfaces. In this context, to describe surface roughness, the concept of fractals is considered. Fractals retain all the structural information and are characterized by single descriptor, the fractal dimension, D. Fractal dimension is intrinsic property of the surface and independent of the filter processing of measuring instrument as well as the sampling length scale. Four machining processes viz. CNC end milling, CNC turning, electrical discharge machining and cylindrical grinding are considered for three different materials. The generated machined surfaces are measured to find out fractal dimension (D) of the surfaces. The experimental results are further analyzed with response surface methodology (RSM) to consider the effects of process parameters on fractal dimension. Also the effect of work-piece material variation on fractal dimension of machined surfaces is considered. It is believed that the present book will prove to add significant contribution to the existing literature from the point of view of both industrial importance and academic interest.

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Contents

1

Fundamental Consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Surface Metrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Fractal Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Fractal Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Fractal Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Self-Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Self-Affinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.5 Fractal Description of Roughness . . . . . . . . . . . . . . . 1.3.6 Fractal Dimension Calculation . . . . . . . . . . . . . . . . . 1.3.7 Fractal Dimension Measurement in the Present Study . 1.4 Review of Roughness Study in Machining . . . . . . . . . . . . . . 1.5 Design of Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Full Factorial Design . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Central Composite Design . . . . . . . . . . . . . . . . . . . . 1.6 Response Surface Methodology. . . . . . . . . . . . . . . . . . . . . . 1.7 Closure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 1 3 6 6 6 7 8 9 11 12 12 18 19 19 21 22 22

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Fractal Analysis in CNC End Milling . 2.1 Introduction . . . . . . . . . . . . . . . . 2.2 Experimental Details . . . . . . . . . . 2.2.1 Design of Experiments . . . 2.2.2 Machine Used . . . . . . . . . 2.2.3 Cutting Tool Used. . . . . . . 2.2.4 Work-Piece Materials . . . . 2.3 Results and Discussion. . . . . . . . . 2.3.1 RSM for Mild Steel . . . . . 2.3.2 RSM for Brass . . . . . . . . . 2.3.3 RSM for Aluminium . . . . . 2.4 Closure. . . . . . . . . . . . . . . . . . . .

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viii

Contents

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Fractal Analysis in CNC Turning . 3.1 Introduction . . . . . . . . . . . . . 3.2 Experimental Details . . . . . . . 3.2.1 Design of Experiments 3.2.2 Machine Used . . . . . . 3.2.3 Cutting Tool Used. . . . 3.2.4 Work-Piece Materials . 3.3 Results and Discussion. . . . . . 3.3.1 RSM for Mild Steel . . 3.3.2 RSM for Brass . . . . . . 3.3.3 RSM for Aluminium . . 3.4 Closure. . . . . . . . . . . . . . . . .

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Fractal Analysis in Cylindrical Grinding 4.1 Introduction . . . . . . . . . . . . . . . . . . 4.2 Experimental Details . . . . . . . . . . . . 4.2.1 Design of Experiments . . . . . 4.2.2 Machine Used . . . . . . . . . . . 4.2.3 Work-Piece Materials . . . . . . 4.3 Results and Discussion. . . . . . . . . . . 4.3.1 RSM for Mild Steel . . . . . . . 4.3.2 RSM for Brass . . . . . . . . . . . 4.3.3 RSM for Aluminium . . . . . . . 4.4 Closure. . . . . . . . . . . . . . . . . . . . . .

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5

Fractal Analysis in EDM . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . 5.2 Experimental Details . . . . . . . . . . 5.2.1 Design of Experiments . . . 5.2.2 Machine Used . . . . . . . . . 5.2.3 Work-Piece Materials . . . . 5.2.4 Tool Electrode Used . . . . . 5.3 Results and Discussion. . . . . . . . . 5.3.1 RSM for Mild Steel . . . . . 5.3.2 RSM for Brass . . . . . . . . . 5.3.3 RSM for Tungsten Carbide 5.4 Closure. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Fundamental Consideration

Abstract The importance and usefulness of fractal dimension in describing surface roughness over the conventional roughness parameters are presented in this chapter. The fundamental of fractal dimension and the methodology for evaluation of fractal dimension are also discussed. Literature survey is carried out for four different types of machining processes and shows that there is scarcity of literatures which deal with fractal description of surface roughness. Fundamentals of design of experiments and response surface methodology are also discussed.

1.1 Introduction Surfaces are irregular though they may look like very smooth. When the surfaces are magnified, the irregularities become prominent. This is true for the machining surfaces as well. In a material removal process such as machining, unwanted material is removed and altered surface topography is obtained. The surface generated consists of inherent irregularities left by the cutting tool, which are commonly defined as surface roughness. Such a surface is composed of a large number of length scales of superimposed roughness that are generally characterized by the standard deviation of surface peaks. Three statistical characteristics are generally used to describe the structure of machined surface topography: texture, waviness and roughness. The texture determines the anisotropic property of the surface. The waviness reflects the reference profile (or surface). The surface roughness is formed by the micro deformation during the machining process. Surface roughness plays an important role. It has large impact on the mechanical properties like fatigue behavior, corrosion resistance, creep life, etc. It also affects other functional attributes of machine components like friction, wear, light reflection, heat transmission, lubrication, electrical conductivity, etc. Surface roughness may depend on various factors like machining parameters, work-piece materials, cutting tool properties, cutting phenomenon, etc. In a review P. Sahoo et al., Fractal Analysis in Machining, SpringerBriefs in Computational Mechanics, DOI: 10.1007/978-3-642-17922-8_1,  Prasanta Sahoo 2011

1

2

1 Fundamental Consideration

article, Benardos and Vosniakos (2003) have presented a fishbone diagram with parameters that affect surface roughness. As a case study, they have considered two machining operations—turning and milling. They broadly classified the factors as machining parameters, cutting tool properties, work-piece properties and cutting phenomena. Machining parameters may include process kinematics, depth of cut, cutting speed, feed rate, etc. Cutting tool properties may include tool material, nose radius, tool shape, etc. Work-piece properties may include workpiece hardness, work-piece size etc. and cutting phenomena includes vibration, cutting force variations, chip formation, etc. It is obvious that for other machining operations also, there are several factors that affect surface roughness. Many researchers have attempted to model surface roughness but the developed models are far from complete as it is not possible to consider all the controlling factors in a particular study. So, researchers always pay attention to model surface roughness in a better way so that surface roughness modeling can be done more accurately. Surface roughness is generally expressed by three types of conventional roughness parameters viz. amplitude parameters, spacing parameters and hybrid parameters. Amplitude parameters are the measure of vertical characteristics of surface deviation. Center line average roughness (Ra), root mean square roughness (Rq), etc. are the examples of these types of parameters. Spacing parameters are measures of the horizontal characteristics of surface deviations. Examples of such parameters are mean line peak spacing (Rsm), high spot count, etc. On the other hand, hybrid parameters are the combination of both vertical and horizontal characteristics of the surface deviations e.g. root mean square slope of the profile, root mean square wavelength, peak area, valley area, etc. Most commonly used roughness parameters are centre line average value (Ra), root mean square value (Rq), mean line peak spacing (Rsm), etc. Conventionally, the deviation of a surface from its mean plane is assumed to be a random process for which statistical parameters such as the variances of the height, the slope and curvature are used for characterization (Nayak 1971). However, it has been found that the variances of slope and curvature depend strongly on the resolution of the roughness-measuring instrument or any other form of filter and are hence not unique (Thomas 1982; Bhushan et al. 1988; Majumdar and Tien 1990). It is also well known that surface topography is a nonstationary random process for which the variance of the height distribution is related to the length of the sample (Sayles and Thomas 1978). Consequently, instruments with different resolutions and scan lengths yield different values of these statistical parameters for the same surface. The conventional methods of characterization are therefore fraught with inconsistencies which give rise to the term ‘parameter rash’ (Whitehouse 1982) commonly used in contemporary literature. The underlying problem with the conventional methods is that although rough surfaces contain roughness at a large number of length scales, the characterization parameters depend only on a few particular length scales, such as the instrument resolution or the sample length. A logical solution to this problem is to use scale-invariant parameters to characterize rough surfaces. In this context, to describe surface roughness, the concept of fractals is applied. The concept is based

1.1 Introduction

3

on the self-affinity and self-similarity of surfaces at different scales. Fractals retain all the structural information and are characterized by single descriptor, the fractal dimension, D. Fractal dimension is intrinsic property of the surface and independent of the filter processing. Roughness measurements on a variety of surfaces show that the power spectra of the surface profiles follow power laws. This suggests that when a surface is magnified appropriately, the magnified image looks very similar to the original surface. This property can be mathematically described by the concepts of self-similarity and self-affinity. The fractal dimension, which forms the essence of fractal geometry, is both scale-invariant and is closely linked to the concepts of self-similarity and self-affinity (Mandelbrot 1982). It is therefore essential to use fractal dimension to characterize rough surfaces and provide the geometric structure at all length scales (Bigerelle et al. 2005). The possible application of fractal geometry to tribology was explored (Ling 1990). The influence of processing techniques on the fractal or non-fractal structure was also examined (Majumdar and Bhushan 1990). In a material removal process, mechanical intervention happens over length scales which extend from atomic dimensions to centimeters. The machine vibration, clearances and tolerances affect the outcome of the process at the largest of length scales (above 10-3 m). The tool form, feed rate, tool radius in the case of single point cutting (Venkatesh et al. 1998) and grit size in multiple point cutting (Venkatesh et al. 1999), affect the process outcome at the intermediate length scales (10-6–10-3 m). The roughness of the tool or details of the grit surfaces influence the final topography of the generated surface at the lowest length scales (10-9–10-6 m). It has been shown that surfaces formed by electric discharge machining, milling, cutting or grinding and worn surfaces (Brown and Savary 1991; Tricot et al. 1994; Hasegawa et al. 1996; He and Zhu 1997; Ge and Chen 1999; Zhang et al. 2001; Jiang et al. 2001; Zhu et al. 2003; Jahn and Truckenbrodt 2004; Kang et al. 2005; Han et al. 2005) have fractal structures, and fractal parameters can reflect the intrinsic properties of surfaces to overcome the disadvantages of conventional roughness parameters. Thus, to characterize the roughness of machined surfaces in different machining processes fractal dimension is used as the roughness parameter.

1.2 Surface Metrology Surface texture is a complex condition resulting from a combination of roughness (nano and micro-roughness), waviness (macro-roughness), lay and flaw. Figure 1.1 shows a display of surface texture with unidirectional lay. Roughness is produced by fluctuations of short wavelengths characterized by asperities (local maxima) and valleys (local minima) of varying amplitudes and spacing. This occurs due to the mechanism of the material removal process. Waviness is the surface irregularities of longer wavelengths and may result from such factors as machine or work piece deflections, vibration, chatter, heat treatment or warping strains. Lay is the

4

1 Fundamental Consideration

Fig. 1.1 Display of surface texture

principal direction of the predominant surface pattern, usually determined by the production process. Flaws are unexpected and unintentional interruptions in the texture. Apart from these, the surface may contain large deviations from nominal shape of very large wavelength, which is known as error of form. These are not considered as part of surface texture. Any engineering surface composes of a vast number of peaks and valleys and it is not possible to measure the height and location of each of the peaks. So measurement of a surface is carried out on a sampling length where it is assumed that the surface outside and inside the sampling length is statistically similar. In order to determine the numerical assessment of a sample’s surface texture, three characteristic lengths are associated with the profile (ISO 4287, 1997) viz. sampling length, evaluation or assessment or cut off length and traverse length. The sampling length is the length over which the parameter to be measured will have statistical significance. Cut off length is the length of the surface over which the

1.2 Surface Metrology

5

Motor and gearbox

Transducer

Skid Amplifier

Stylus A-D converter Specimen

Chart recorder

Data logger

Fig. 1.2 Component parts of a typical stylus surface-measuring instrument

measurement is made. This length may include several sampling lengths— typically five times. The measurement is the integration of the individual sampling lengths. The total length of the surface traversed by the stylus in making a measurement is called the traverse length. It will normally be greater than the evaluation length, due to the necessity of allowing run-up and over-travel at each end of the evaluation length to ensure that any mechanical and electrical transients are excluded from the measurement. There are several methods to study the surface topography which are developed over the years. The most common method of studying surface texture is the surface profilometer (Fig. 1.2). In this method, a fine, very lightly loaded, stylus is traveled smoothly at a constant speed across the surface under examination. The transducer produces an electrical signal, proportional to displacement of the stylus, which is amplified and fed to a chart recorder that provides a magnified view of the original profile. But this graphical representation differs from the actual surface profile because of difference in magnifications employed in vertical and horizontal directions. Surface slopes appear very steep on profilometric record though they are rarely steeper than 10 in actual cases. The shape of the stylus also plays a vital role in incorporating error in measurement. The finite tip radius (typically 1–2.5 microns for a diamond stylus) and the included angle (of about 60 for pyramidal or conical shape) results in preventing the stylus from penetrating fully into deep and narrow valleys of the surface and thus some smoothing of the profile are done. Some error is also introduced by the stylus in terms of distortion or damage of a very delicate surface because of the load applied on it. In such cases noncontacting optical profilometer having optical heads replacing stylus may be used. Reflection of infrared radiation from the surface is recorded by arrays of photodiodes and analysis of the same in a microprocessor result in the determination of the surface topography. Vertical resolution of the order of 0.1 nm is achievable

6

1 Fundamental Consideration

though maximum height of measurement is limited to few microns. This method is clearly advantageous in case of very fine surface features.

1.3 Fractal Characterization 1.3.1 Fractal Geometry Euclidean geometry describes ordered objects such as points, curves, surfaces and cubes using integer dimensions of 0, 1, 2, and 3, respectively. A measure of the object such as the length of a line, the area of a surface and the volume of a cube are associated with each dimension. These measures are invariant with respect to the unit of measurement. It means that the length of a line remains independent of whether a centimeter or a micrometer scale is used. However, a multitude of objects found in nature appear disordered and irregular for which the measures of length, area and volume are scale-dependent. This suggests that the dimensions of such objects cannot be integers. A generalized concept of a dimension and the origins of fractal geometry are now discussed. Mandelbrot (1967) founded fractal geometry when he showed that for decreasing the unit of measurement, the length of a natural coastline does not converge but, instead, increase monotonically. On plotting the length L as a function of the unit of measurement [ on a log–log plot, he found a simple relation of the form L * [1-D. Mandelbrot finally made an interesting conclusion that the real number D associated with every coastline is the dimension of the coastline. This study marked the origins of fractal geometry, which has now found numerous applications in characterizing and describing disordered phenomena in science and engineering.

1.3.2 Fractal Dimension To measure the length of a line, let us break the line into small units of length [ and then add the number of units in the form L ¼ R 21

ð1:1Þ

Similarly to measure the area of surface, let us break up the surface into small squares of size [ 9 [ and then add the number of units as A ¼ R 22

ð1:2Þ

Here in Eqs. 1.1 and 1.2 the exponents 1 and 2 correspond to the dimensions of the objects. These measures of length and area have a unique property that they are independent of the unit of measurement [ and in the limit [ ? 0 these measures

1.3 Fractal Characterization

7

remain finite and non-zero. This concept of Euclidean dimension thus can be generalized in the form M ¼ R 2D

ð1:3Þ

Here M is the measure and D is a real number. If the exponent D makes the measure M independent of the unit of measurement [ in the limit of [ ? 0, then D is the dimension of an object. Contrary to common understanding of dimension, this generalization allows the dimension of an object to take non-integer values. If, in this argument, it is assumed that an object is broken into N equal parts then Eq. 1.3 can be written as M = N[D. Since the measure is invariant with the unit of measurement, one can write N * [-D. Now if the length of an object is evaluated, then the length would vary as L = N[1 * [1-D, as was observed for the lengths of the coastlines. It can be easily seen that the length will be independent of [ only when D = 1.

1.3.3 Self-Similarity The generalized concepts of measure and dimension are fundamental to the issue of self-similarity. Let us consider a one-dimensional line of unit length and break it up into N equal segments. Each segment of the line, of size 1/m, is similar to the whole line and needs a magnification of m to be an exact replica of the whole line. Since the length of the line remains independent of 1/m, it follows that the number of units is N * m. Now let us consider a square, which has a side of unit length. Each small square of side 1/m is similar to the whole square and needs a magnification of m to be an exact replica of the whole square. However, the number of small squares in the whole is N * m2. In general, for an object of dimension D, it follows that N  mD

ð1:4Þ

Thus the dimension of the object can be written as D¼

log N log m

ð1:5Þ

This definition of dimension, which is based on the self-similarity of an object, is called the similarity dimension. To perceive what an object of a non-integer dimension looks like, one can follow the recursive construction in Fig. 1.3, which yields the Koch curve of dimension 1.26. In this construction the first step is to break a straight line into three parts and replace the middle portion by two segments of equal lengths. In the subsequent stages each straight segment is broken into three parts and the middle portion of each segment is replaced by two parts. If this recursion is continued infinite times then the Koch curve is obtained. This curve has some unique mathematical properties. Firstly, the curve is continuous but it is not differentiable anywhere.

8

1 Fundamental Consideration

Fig. 1.3 Formation of Koch curve

The non-differentiability arises because of the fact that if the curve is repeatedly magnified, more and more details of the curve keep appearing. This means that tangent cannot be drawn at any point and therefore the curve cannot be differentiated. Secondly, the curve is exactly self-similar. This is because if a small portion of the curve is appropriately magnified, it will be an exact replica of the whole Koch curve. Thirdly, the dimension of the curve remains constant at all scales, although the curve contains roughness at a large number of scales. This scale-invariance of the dimension is an important property, which is utilized to characterize rough surfaces. The coastline of an island is an example of a selfsimilar object found in nature. Although these objects are not exactly self-similar, they are statistically self-similar. Statistical self-similarity means that the probability distribution of a small part of an object will be congruent with the probability distribution of the whole object if the small part is magnified appropriately. However, not all fractal objects are self-similar. This leads to the more general concept of self-affinity.

1.3.4 Self–Affinity The definition of self-similarity is based on the property of equal magnification in all directions. However, there are many objects in nature, which have unequal

1.3 Fractal Characterization

9

Z

X

Fig. 1.4 Qualitative description of statistical self-affinity for a surface profile

scaling in different directions. Thus these are not self-similar but self-affine. The dimension of self-affine fractals cannot be obtained from Eq. 1.5, which is based on the self-similarity of an object. Mandelbrot showed that the lengths of selfaffine fractal curves do not follow the relation L * [1-D for all values of [ and therefore the dimension of self-affine curves cannot be obtained by measuring their lengths. Instead, the dimension of self-affine functions can be obtained from their power spectra.

1.3.5 Fractal Description of Roughness The deviation of a surface from its mean plane is assumed to be a random process, which is characterized by the statistical parameters such as the variance of the height, the slope and the curvature. But, it has been observed that surface topography is a non-stationary random process. It means the variance of the height distribution is related to the sampling length and hence is not unique for a particular surface. Rough surfaces are also known to exhibit the feature of geometric self-similarity and self-affinity, by which similar appearances of the surface are seen under the various degrees of magnification as quantitatively shown in Fig. 1.4. Since increasing amounts of detail in the roughness are observed at decreasing length scale, the concepts of slope and curvature, which inherently assume the smoothness of the surface, cannot be defined. So the variances of slope and curvature depend strongly on the resolution of the roughness-measuring instrument or some other form of filter and are therefore not unique. In contemporary literature such a large number of characterizations parameter occurs that the term ‘parameters rash’ is aptly used. The use of instrument-dependent parameters shows different values for the same surface. Thus, it is necessary to characterize rough surfaces by intrinsic parameters, which are independent of all scales of roughness. This suggests the use of fractal geometry in characterizing the surface roughness. The fractal dimension is an intrinsic property and should be used for surface characterization. It is invariant with length scales and is closely linked to the concept of geometric self-similarity. The self-similarity or self-affinity of rough surfaces implies that as the unit of measurement is continuously decreased, the surface area of the rough surface (a two-dimensional measure) tends to infinity and the volume (a three-dimensional

10

1 Fundamental Consideration

measure) tends to zero. Here, self-similarity implies the property of equal magnification in all directions while self-affinity refers to unequal scaling in different directions. Thus, the Hausdorff or fractal dimension, D ? 1, of rough surfaces is a fraction between 2 and 3. The profile of a rough surface z(x), typically obtained from stylus measurements, is assumed to be continuous even at the smallest scales. This assumption breaks down at atomic scale. But for engineering surfaces the continuum is assumed to exist down to the limit of a zero-length scale. Since repeated magnifications reveal the finer levels of detail, the tangent at any point cannot be defined. Thus the surface profile is continuous everywhere but nondifferentiable at all points. This mathematical property of continuity, non-differentiability and self-affinity (Berry and Lewis 1980) is satisfied by the modified Weierstrass–Mandelbrot (W–M) fractal function, which is thus used to characterize and simulate such profiles. The W–M function has a fractal dimension D, between 1 and 2, and is given by zðxÞ ¼ GðD1Þ

a X cos 2pcn x 1\D\2; ð2DÞn n¼n1 c

c[1

ð1:6Þ

where, G is a scaling constant. The parameter n1 corresponds to the low cut-off frequency of the profile. Since surfaces are non-stationary random process the lowest cut-off frequency depends on the length L of the sample and is given by cn1 = 1/L. The W–M function has the interesting mathematical property that the series for z(x) converges but that for dz/dx diverges. It implies that it is non-differentiable at all points. The power spectrum of this W–M function can be expressed by a continuous function as SðxÞ ¼

G2ðD1Þ 1 2 ln c x52D

ð1:7Þ

When this equation is compared with the power spectrum of a surface, the dimension D is related to the slope of the spectrum on a log–log plot against x. The constant G is the roughness parameter of a surface, which is invariant with respect to all frequencies of roughness and determines the position of spectrum along the power axis. In this characterization method both G and D are independent of the roughness scales of the surface and hence intrinsic properties. The constants of the W–M function, G, D, and n1 form a complete and fundamental set of scaleindependent parameters to characterize a rough surface. The physical significance of D is the extent of space occupied by the rough surface, i.e., larger D values correspond to denser profile or smoother topography (Yan and Komvopoulos 1998; Sahoo and Ghosh 2007).

1.3 Fractal Characterization

11

1.3.6 Fractal Dimension Calculation Fractal calculation mainly includes the calculation of profile fractal dimension (1 \ D \ 2) and the calculation of surface fractal dimension (2 \ D \ 3). Fractal calculation is generally involved with computer assisted image analysis of topography images of a surface obtained in analog or digital signals using profilometer or microscopy, etc. An effective method to convert these signals into the required data for calculating fractal dimensions must therefore be sought. Profile instruments can be used to obtain data, which are then directly used to calculate fractal dimension. The methods for calculating profile fractal dimension mainly include the yardstick, the box counting, the variation, the structure function and the power spectrum method (Sahoo 2005). The yardstick method employs the technique of ‘walking’ around a profile curve in a step length, r. A point on the profile curve is chosen as a starting point of divider, whilst another point at a distance r from the starting point is taken as its end point. Repetitively, find the point-pair of dividers in the same way until the profile curve is entirely measured. Then, the summing up of the step lengths enables the curve length to be determined. The repetition of this calculation process at various step lengths allows all the curve length to be evaluated. Further, plotting of the curve lengths verses the step lengths on a log–log scale gives the slope m of a fitting line to be related to the fractal dimension D as D = 1 - m. It is possible that this method has abandoned some pivotal points, resulting in calculation error. The principle of box counting method mainly involves an iteration operation to an initial square, whose area is supposed to be 1 and which covers the entire graph. The initial square is divided into four sub-squares and so on. After the n times operations, the number of sub-squares, which contain the discrete points of the profile graph are counted and the length L of the profile is approximately obtained. Then the fractal dimension is calculated as D = 1 ? log L/(n.log2). The variation method has the advantage of being proven theoretically for all profiles (self-affine or not), and of giving quickly an estimation of the dimension of mathematical profiles. A well-known technique used to analyze surfaces consists in performing ‘slices’ through the surfaces, which allows one to transform a threedimensional problem to two-dimensional problem. In other words, a surface is replaced by profiles, taken at different places, and the fractal dimension estimated over profiles is then related to the three-dimensional fractal dimension by the classical result: dimension of surface = 1 ? dimension of profiles. Such a technique obviously decreases the problem size. Accurate results are hard to obtain for the surface dimension and the variation method gives the best approximations. The variation method algorithm is based on the local oscillation of the profile function Z. The power spectrum method involves the evaluation of the power of the profile function. The modified Weierstrass–Mandelbrot (W–M) function for a rough surface is described by Eq. 1.6. The multi-scale nature of z(x) can be characterized

12

1 Fundamental Consideration

by its power spectrum, which gives the amplitude of the roughness at all length scales. The parameters G and D can be found from the power spectrum of the W– M function given by Eq. 1.7. Usually, the power law behavior would result in a straight line if S(x) is plotted as a function of x on a log–log graph. Using fast Fourier transform (FFT), the power spectrum of profile can be calculated and then be plotted verses the frequency on a log–log scale. Thereafter, the fractal dimension, D, can be related to the slope m of a fitting line on a log–log plot as: D = ‘(5 ? m). The structure function method considers all points on the surface profile curve as a time sequence z(x) with fractal character. The structure function s(s) of sampling data on the profile curve can be described as s(s) = [z(x ? s) z(x)]2 = cs 4 - 2D where [z(x ? s) - z(x)]2 expresses the arithmetic average value of difference square, and s is the random choice value of data interval. Different s and the corresponding s(s) can be plotted verses the s on a log–log scale. Then, the fractal dimension D can be related to the slope m of a fitting line on log–log plot as: D = ‘ (4 - m).

1.3.7 Fractal Dimension Measurement in the Present Study In the present study, roughness profile measurement is done using a stylus-type profilometer, Talysurf (Taylor Hobson, UK). The profilometer is set to a cut-off length of 0.8 mm, Gaussian filter, traverse speed 1 mm/sec and 4 mm traverse length. Roughness measurements, in the transverse direction, on the work pieces are repeated four times and average of four measurements of surface roughness parameter values is recorded. The measured profile is digitized and processed through the dedicated advanced surface finish analysis software Talyprofile. Then fractal dimension is evaluated following the structure function method.

1.4 Review of Roughness Study in Machining As surface roughness is an important parameter in the industry, many researchers have tried to study surface roughness in machining. Though the present study focuses on fractal dimension in describing surface roughness, both conventional roughness parameters and fractal dimension are reviewed here. Four machining processes viz. turning, grinding, milling and electrical discharge machining are focused for this purpose and presented one by one. In turning, many researchers have modeled surface roughness. Grzesik (1996) has studied the effect of tribological interactions at the interface between the chip and tool on surface roughness in finish turning of C45 carbon steel. Yang and Tarng (1998) have showed that feed rate is the most significant factor affecting surface roughness in S45C steel turning. Also, with increasing feed rate, surface

1.4 Review of Roughness Study in Machining

13

roughness decreases. Abouelatta and Madl (2001) have found a correlation between surface roughness and cutting parameters and tool vibrations in turning considering three conventional roughness parameters viz. center line average roughness value, maximum height of the profile and skewness. Davim (2001) has presented a study of the influence of cutting parameters on the surface roughness obtained in turning of free machining steel using Taguchi design and shown that the cutting velocity has a greater influence on the roughness followed by the feed rate. Lin et al. (2001) have shown that in turning feed rate is the critical parameter to affect the surface roughness, where increasing the feed rate will increase the surface roughness. Suresh et al. (2002) have shown that surface roughness decreases with an increase in cutting speed, and increases as feed increases in turning of mild steel. Arbizu and Perez (2003) have developed models to determine surface quality of parts obtained through turning processes and shown that surface roughness increases with increase in depth of cut and feed rate. Feng and Wang (2003) have presented a nonlinear multiple regression analysis to predict surface roughness in finish turning of Steel 8620 and Al 6061T materials. Dabnun et al. (2005) have concluded that feed rate is the main influencing factor on the roughness in turning of machinable glass–ceramic (Macor). Sahin and Motorcu (2005) have developed a surface roughness model for turning of mild steel with coated carbide tools and shown that feed rate is the main affecting factor on surface roughness. Surface roughness increases with increase in feed rate but decreases with increase in cutting speed and depth of cut. Kirby et al. (2006) have shown that the feed rate and tool nose radius have the highest effects on surface roughness in a turning operation of 6061-T6 aluminium alloy. Palanikumar et al. (2006) have focused on the parametric influence of machining parameters on the surface roughness in turning of glass fiber reinforced polymer (GFRP) and shown that roughness increases with increase in feed rate but roughness decreases with increase in cutting speed. Singh and Rao (2007) have developed a model to determine the effects of cutting conditions and tool geometry on surface roughness in the finish hard turning of the bearing steel (AISI 52100) and concluded that feed rate is the dominant factor determining surface finish followed by nose radius and cutting velocity. Ramesh et al. (2008) have found in their study that feed rate is the main influencing factor on surface roughness in turning of titanium alloy. Palanikumar (2008) has found that the most significant machining parameter for surface roughness is feed followed by cutting speed in machining glass fiber reinforced (GFRP). For modeling surface roughness in turning different methodologies are used viz. RSM (Suresh et al. 2002; Dabnun et al. 2005; Sahin and Motorcu 2005; Palanikumar et al. 2006; Singh and Rao 2007; Ramesh et al. 2008; Palanikumar 2008; Gupta 2010), Taguchi analysis (Yang and Tarng 1998; Davim 2001; Kirby et al. 2006; Nalbant et al. 2007; Palanikumar 2008;), artificial neural network (Pal and Chakraborty 2005; Kohli and Dixit 2005; Bagci and Isik 2006; Abburi and Dixit 2006; Feng et al. 2006; Zhong et al. 2006; Zhong et al. 2008; Muthukrishnan and Davim 2009; Karayel 2009; Gupta 2010; Chavoshi and Tajdari 2010). Also, the literature survey shows that mainly three cutting parameters viz. cutting speed, feed rate and depth of cut are the common parameters considered for

14

1 Fundamental Consideration

most of the studies (Yang and Tarng 1998; Davim 2001; Lin et al. 2001; Suresh et al. 2002; Arbizu and Perez 2003; Jiao et al. 2004; Dabnun et al. 2005; Sahin and Motorcu 2005; Bagci and Isik 2006; Ramesh et al. 2008; Palanikumar 2008; Karayel 2009). Grinding is the most commonly used manufacturing process in the industry and this is a complex machining process with many interactive parameters and surface quality produced is influenced by various parameters. Several researchers have tried to model surface roughness in grinding and few of the recent literatures are reviewed here. Zhang et al. (2001) have developed the relationships between the fractal dimension and conventional roughness parameters (Ra or Rq or Rsm of surface roughness) of different ground surfaces and justified the usefulness of fractal theory. They concluded that fractal dimension D is relative to vertical parameters and transverse parameters of surface topography. Zhou and Xi (2002) have developed a model for predicting surface roughness in grinding taking into consideration the random distribution of the grain protrusion heights. Maksoud et al. (2003) have used artificial neural network to achieve desired surface roughness under grinding wheel surface topography variations. Hassui and Diniz (2003) have developed a relation between the process vibration signals and roughness in a plunge cylindrical grinding operation of AISI 52100 quenched and tempered steel. Hecker and Liang (2003) have presented the prediction of the arithmetic mean surface roughness based on a probabilistic undeformed chip thickness model. Bigerelle et al. (2005) have shown that grinding could be characterized with an elementary function and the worn profile can be modeled by a fractal curve defined by only two parameters (amplitude and fractal dimension) with an infinite summation of these elementary functions. Krajnik et al. (2005) have used response surface methodology to develop a model to minimize the surface roughness in plunge center less grinding operation of 9SMn28, free-cutting unalloyed steel. The analysis of variance shows that the grinding wheel dressing condition most significantly affects the ground surface roughness. The surface roughness is additionally affected by the geometrical grinding gap set-up factor and the control wheel speed. Kwak (2005) has investigated the various grinding parameters affected the geometric error in surface grinding process using combined Taguchi method and response surface method. Four grinding parameters such as grain size, wheel speed, depth of cut and table speed are selected for experimentation. A second-order response model for the geometric error is developed and the utilization of the response surface model is evaluated with constraints of the surface roughness and the material removal rate. Fredj and Amamou (2006) have tried to establish a model combining the application of design of experiments (DOE) and neural network method for ground surface roughness prediction. Kwak et al. (2006) have developed a model for grinding power spent during the process and the surface roughness in the external cylindrical grinding of hardened SCM440 steel using the response surface method. They have shown from the study that the grinding power seems to increase linearly with increasing work-piece speed and the traverse speed and surface roughness is dominantly affected by the change of the work-piece speed. Choi et al. (2008) have

1.4 Review of Roughness Study in Machining

15

established the generalized model for power, surface roughness, grinding ratio and surface burning for grinding of various steel alloys using alumina grinding wheels based on the systematic analysis and experiments. It is seen that steady-state surface roughness is primarily dependent only on the effective chip thickness. Mohanasundararaju et al. (2008) have developed a neural network and fuzzy-based methodology for predicting surface roughness in a grinding process for work rolls used in cold rolling. This methodology predicts the most likely estimates of surface roughness along with lower and upper estimates using fuzzy numbers. Siddiquee et al. (2010) have investigated the optimization of an in-feed centreless cylindrical grinding process performed on EN52 austenitic valve steel (DIN: X45CrSi93) considering dressing feed, grinding feed, dwell time and cycle time as process parameters. They have optimized the multiple responses viz. surface roughness, out of cylindricity of the valve stem and diametral tolerance using grey relational Taguchi analysis. Milling also is a popular machining process in modern industry. There are several researchers who have tried to model the roughness in milling process. In this section, few available literatures on surface roughness modeling in milling are reviewed. Fuh and Wu (1995) have developed a model for prediction of surface quality in end milling of 2014 aluminium alloy and shown that surface roughness is mainly affected by the feed rate and tool nose radius. Alauddin et al. (1996) have pointed out that feed rate is the most significant factor and with increase in feed, surface roughness increases while with increase in cutting speed, surface roughness decreases in end milling Inconel 718 using uncoated carbide inserts. Lou et al. (1998) have used multiple regression models to develop a surface roughness model to predict Ra in CNC end milling of 6061 aluminum and concluded that the feed rate is the most significant factor. Yang and Chen (2001) found out the optimum cutting parameters for milling of Al 6061 material using Taguchi design considering cutting speed, feed rate, depth of cut and tool diameter as the cutting parameters. Lee et al. (2001) presented a method for the simulation of surface roughness of the machined surface in high-speed end milling. Lin (2002) has optimized cutting speed, feed rate and depth of cut with consideration of multiple performance characteristics including removed volume, surface roughness and burr height in face milling of stainless steel and shown that the most influence of the cutting parameters is the feed rate. Mansour and Abdalla (2002) have concluded that with increase in feed rate or in axial depth of cut, surface roughness increases whilst with increase in cutting speed, surface roughness decreases in end milling operations of EN32 materials. Ghani et al. (2004) have studied surface roughness in end milling of hardened steel AISI H13 with TiN coated P10 carbide insert tool and concluded that use of high cutting speed, low feed rate and low depth of cut leads to better surface finish. Wang and Chang (2004) have analyzed the influence of cutting condition and tool geometry on surface roughness in slot end milling of AL2014-T6. Oktem et al. (2005) have developed an effective methodology to determine the optimum cutting conditions leading to minimum roughness in milling of Aluminum (7075-T6) molded surfaces considering feed, cutting speed, axial depth of cut, radial depth of cut and machining tolerance as

16

1 Fundamental Consideration

cutting parameters. Reddy and Rao (2005) have developed a model to see the effects of tool geometry, cutting speed and feed rate on surface roughness in end milling of medium carbon steel. The investigations of this study indicate that the parameters cutting speed, feed, radial rake angle and nose radius are the primary factors influencing the surface roughness of medium carbon steel during end milling. Reddy and Rao (2006a) have investigated the role of solid lubricant assisted machining with graphite and molybdenum disulphide lubricants on surface quality, cutting forces and specific energy while milling AISI 1045 steel using cutting tools of different tool geometry (radial rake angle and nose radius). Reddy and Rao (2006b) have studied the effect of various parameters such as cutting speed, feed rate, radial rake angle and nose radius on surface roughness in milling of AISI 1045 materials. They have shown that surface roughness decreases with increasing cutting speed. Jesuthanam et al. (2007) have developed a hybrid neural network trained with GA and Particle Swarm Optimization (PSO) for the prediction of surface roughness in CNC end milling operation of mild steel materials. For the development of network, spindle speed, feed, depth of cut and vibration data are considered. Chang and Lu (2007) have applied a grey relational analysis to determine the cutting parameters for optimizing the side milling process with multiple performance characteristics and concluded that feedingdirection roughness, axial-direction roughness and waviness are improved simultaneously through the optimal combination of the cutting parameters obtained from the proposed two-stage parameter design. El-Sonbaty et al. (2008) have developed artificial neural network (ANN) models for the analysis and prediction of the relationship between the cutting conditions and the corresponding fractal parameters of machined surfaces in face milling operation using rotational speed, feed, depth of cut, pre-tool flank wear and vibration level as input parameters. Routara et al. (2009) have studied the influence of machining parameters on conventional roughness parameters in CNC end milling of aluminium, steel and brass materials using response surface method. Berglund and Rose’n (2009) have evaluated the connection between surface finish appearance and measured surface roughness using scale sensitive fractal analysis in milling. Öktem (2009) has developed an integrated study of surface roughness to model and optimize the cutting parameters in end milling of AISI 1040 steel material with TiAlN solid carbide tools under wet condition using ANN and GA. He has shown that the axial depth of cut is the most important cutting parameters affecting surface roughness (Ra). Zain et al. (2010a) have carried out a study using GA to observe the optimal effect of the radial rake angle of the tool, combined with speed and feed rate in influencing the surface roughness result. With the highest speed, lowest feed rate and highest radial rake angle of the cutting conditions scale, the GA technique recommends the best minimum surface roughness value. For end milling also, to modeling surface roughness different tools are used like RSM (Alauddin et al. 1996; Mansour and Abdalla 2002; Wang and Chang 2004; Oktem et al. 2005; Reddy and Rao 2005; Reddy and Rao 2006b; Routara et al. 2009), Taguchi analysis (Yang and Chen 2001; Lin 2002; Ghani et al. 2004; Bagci and Aykut 2006), ANN (Tsai et al. 1999; Balic and Korosec 2002; Benardos and

1.4 Review of Roughness Study in Machining

17

Vosniakos 2002; El-Sonbaty et al. 2008; Öktem 2009; Zain et al. 2010b). From the literature survey, it is seen that most of literatures deal with conventional roughness parameters to describe surface roughness and also in the study, three machining parameters viz. spindle speed, feed rate and depth of cut are the most common machining parameters (Fuh and Wu 1995; Lou et al. 1998; Tsai et al. 1999; Yang and Chen 2001; Lin 2002; Ghani et al. 2004; Wang and Chang 2004; Bagci and Aykut 2006; Zhang and Chen 2007; Routara et al. 2009). Electrical discharge machining (EDM) is a non-conventional machining process that can be used all types of conductive materials. It can also be used for machining of difficult-to-machine shapes and materials. In this section, few of the available literatures on surface roughness modeling in EDM are reviewed. Zhang et al. (1997) have investigated the effects on material removal rate, surface roughness and diameter of discharge points in electro-discharge machining (EDM) on ceramics and shown that the material removal rate, surface roughness and the diameter of discharge point all increase with increasing pulse-on time and discharge current. Lee and Li (2001) have shown that the negative tool polarity gives better surface finish in EDM of tungsten carbide. Also, surface roughness increases with increasing peak current and pulse duration. Ramasawmy and Blunt (2002) have illustrated the influencing process factors in modifying the surface textures using Taguchi method in EDM on M300 tool steel and shown that the direct current is the most dominant factor in modifying the surface texture. Lin and Lin (2002) have studied an approach for the optimization of the electrical discharge machining process (work-piece polarity, pulse on time, duty factor, open discharge voltage, discharge current, and dielectric fluid) with multiple performance characteristics viz. MRR, surface roughness and electrode wear ratio using grey relational analysis. Lin and Lin (2005) have tried to optimize the electrical discharge machining process using grey-fuzzy logic considering pulse on time, duty factor and discharge current as process parameters. Puertas and Luis (2003) have modeled centre line average value (Ra) and root mean square roughness value (Rq) in terms of current, pulse on time and off time in EDM on soft steel (F-1110). It has been seen that the current intensity has the most influence on surface roughness and there is a strong interaction between the current intensity and the pulse on time factors being advisable to work with high current intensity values and low pulse on time values. They have justified the fact of having to employ high current intensity values to obtain a better surface roughness because a better arc stability causes a more uniform production of sparks and a narrow variation interval of the Ra and Rq roughness parameters. Yih-fong and Fu-chen (2003) have presented an approach for optimizing high-speed EDM using Taguchi methods. They have concluded that the most important factors affecting the EDM process robustness have been identified as pulse-on time, duty cycle, and pulse peak current. Ramasawmy and Blunt (2004) have quantified the effect of process parameters on the surface texture using Taguchi method in EDM of steel and concluded that the pulse current is the most dominant factor in affecting the surface texture. Puertas et al. (2004) have carried out a study on the influence of the factors of intensity, pulse time and duty cycle over surface roughness, material

18

1 Fundamental Consideration

removal rate, etc. in EDM of a cemented carbide and observed that in the case of Ra parameter the most influential factors are intensity, followed by the pulse time factor. Petropoulos et al. (2004) have emphasized the interrelationship between surface texture parameters and process parameters in EDM of Ck60 steel plates. They have considered amplitude, spacing, hybrid, as well as random process and fractal parameters. Puertas et al. (2005) have carried out a study on the influence of EDM parameters over two spacing parameters in machining of siliconised or reaction-bonded silicon carbide (SiSiC) and shown that intensity, pulse time and duty cycle are most influential factors affecting mean spacing between peaks and the number of peaks per cm whereas the dielectric flushing pressure is not an influential factor. Amorima and Weingaertner (2005) have shown that the increase of average surface roughness of the work-piece is directly related to the increase in discharge current and discharge duration on the EDM of the AISI P20 tool steel under finish machining. Ramakrishnan and Karunamoorthy (2006) have proposed a multi objective optimization method in WEDM process using parametric design of Taguchi method and identified that the pulse on time and ignition current intensity are the influential parameters. Keskin et al. (2006) have shown that surface roughness has an increasing trend with an increase in the discharge duration in EDM on steel work-pieces. Sahoo et al. (2009) have investigated the influence of machining parameters, viz., pulse current, pulse on time and pulse off time on the quality of surface produced in EDM of mild steel, brass and tungsten carbide materials using response surface methodology. It is seen that the pulse current has the maximum influence on the roughness parameters while pulse on time has some effect and pulse off time has no significant effect on roughness parameters. Shah et al. (2010) have shown that the material thickness has little effect on the material removal rate and kerf but is a significant factor in terms of surface roughness in wire electrical discharge machining (WEDM) of tungsten carbide samples. Now-a-days, artificial neural network is used as a tool in modeling of EDM process (Spedding and Wang 1997; Tsai and Wang 2001; Sarkar et al. 2006; Mandal et al. 2007; Assarzadeh and Ghoreishi 2008). From the literature survey, it is revealed that there are many researches on surface roughness modeling in different machining processes. However, most of the literatures deal with conventional roughness parameters and there is scarcity of literatures which deal with fractal dimension modelling in machining.

1.5 Design of Experiments The design of experiments technique (DOE) is a very powerful tool, which permits to carry out the modeling and analysis of the influence of process variables on the response variables. The response variable is an unknown function of the process variables, which are known as design factors. The purpose of running experiments is to characterize unknown relations and dependencies that exist in the observed design or process, i.e., to find out the influential design variables and the response

1.5 Design of Experiments

19

to variations in the design variable values. A scientific approach to planning the experiment must be employed if an experiment is to be performed most efficiently. The statistical design of experiments refers to the process of planning the experiment so that appropriate data that can be analyzed by statistical methods will be collected, resulting in valid and objective conclusions in a meaningful way. When the problem involves data that are subject to experimental errors, statistical methodology is the only objective approach to analysis. Sometimes, experiments are repeated with a particular set of levels for all the factors to check the statistical validation and repeatability by the replicate data. This is called replication. To get rid of any biasness, allocation of experimental material and the order of experimental runs are randomly selected. This is called randomization. To arrange the experimental material into groups, or blocks, that should be more homogeneous than the entire set of material is called blocking. So, when experiments are carried out these things should be remembered. There are several methodologies for design of experiments. Some of DOE methods are discussed below.

1.5.1 Full Factorial Design Full factorial design creates experimental points using all the possible combinations of the levels of the factors in each complete trial or replication of the experiments. The experimental design points in a full factorial design are the vertices of a hyper cube in the n-dimensional design space defined by the minimum and the maximum values of each of the factors. These experimental points are also called factorial points. For three factors having four levels of each factors, considering full factorial design, total 43 (64) numbers of experiments have to be carried out. If there are n replicates of complete experiments, then there will be n times of the single replication experiments to be conducted. In the experimentation, it must have at least two replicates to determine a sum of squares due to error if all possible interactions are included in the model.

1.5.2 Central Composite Design A Box–Wilson Central Composite Design, commonly called ‘‘Central Composite Design (CCD)’’ is frequently used for building a second order polynomial for the response variables in response surface methodology without using a complete full factorial design of experiments. To establish the coefficients of a polynomial with quadratic terms, the experimental design must have at least three levels of each factor. In CCD, there are three different point viz. factorial points, central points and axial points. Factorial points are vertices of the n-dimensional cube which are coming from the full or fractional factorial design where the factor levels are coded to -1, +1. Central point is the point at the center of the design space. Axial

20

1 Fundamental Consideration

Fig. 1.5 Face centered central composite design with three factors

points are located on the axes of the coordinate system symmetrically with respect to the central point at a distance a from the design center. There are two main varieties of CCD namely Face centered CCD and Rotatable CCD. In face centered CCD, a k factor 3-level experimental design requires 2k ? 2k ? C experiments, where k is the number of factors, 2k points are in the corners of the cube representing the experimental domain, 2k axial points are in the center of each face of the cube ½ða; 0; . . .0Þ; ð0; a; . . .0Þ; ð0; 0; . . .  aÞ and C points are the replicates in the center of the cube that are necessary to estimate the variability of the experimental measurements, it is to say the repeatability of the phenomenon which carry out the lack-of-fit or curvature test for the model. The centre points may vary from three to six. The example of 3-level three factor FCC design is shown in Fig. 1.5. In this figure, the deep black circles represent the fractional points at the corner of cube while the white circles represent axial points in the center of each face of the cube and the star mark represents the centre points. For the three factor experiment, eight (23) factorial points, six axial points (2 9 3) and six centre runs, a total of 20 experimental runs can be considered. The value of a is chosen here as 1. The upper and lower limits of a factor are coded as +1 and -1 respectively using the following relations Eq. 1.8. Generally, the experimental runs are conducted in random order. xi ¼

½2x  ðxmax þ xmin Þ ðxmax  xmin Þ

ð1:8Þ

The rotatable central composite design is the most widely used experimental design for modeling a second-order response surface. A design is called rotatable when the variance of the predicted response at any point depends only on the distance of the point from the center point of design. The rotatable design provides the uniformity of prediction error and it is achieved by proper choice of a: In rotatable designs, all points at the same radial distance (r) from the centre point have the same magnitude of prediction error. For a given number of variables, the a required to achieve rotatability is computed as a ¼ ðnf Þ1=4 ; where nf is the number of points in the 2k factorial design. A rotatable CCD consists of 2k fractional factorial points, augmented by 2 k axial points ½ða; 0; . . .0Þ; ð0; a; . . .0Þ; ð0; 0; . . .  aÞ and nc

1.5 Design of Experiments

21

centre points (0, 0, 0, 0…,0). Here also, the centre points vary from three to six. With proper choice of nc the CCD can be made orthogonal or it can be made uniform precision design. It means that the variance of response at origin is equal to the variance of response at a unit distance from the origin. Considering uniform precision, for three factor experimentation, eight (23) factorial points, six axial points (2 9 3) and six centre runs, a total of 20 experimental runs may be considered and the value of a is ð8Þ1=4 ¼ 1:682.

1.6 Response Surface Methodology Response Surface Method (RSM) adopts both mathematical and statistical techniques which are useful for the modeling and analysis of problems in which a response of interest is influenced by several variables and the objective is to optimize the response (Montgomery 2001). RSM helps in analyzing the influence of the independent variables on a specific dependent variable (response) by quantifying the relationships amongst one or more measured responses and the vital input factors. The mathematical models thus developed relating the machining responses and their factors facilitate the optimization of the machining process. In most of the RSM problems, the form of the relationship between the response and the independent variables is unknown. Thus the first step in RSM is to find a suitable approximation for the true functional relationship between response of interest ‘y’ and a set of controllable variables {x1, x2, …, xn}. Usually when the response function is not known or non-linear, a second order model is utilized (Montgomery 2001) in the form: y ¼ b0 þ

n X i¼1

bi x i þ

n X i¼1

bii x2i þ

XX

bij xi xj þ e

ð1:9Þ

i\j

where, e represents the noise or error observed in the response y such that the expected response is (y -eÞ and b’s are the regression coefficients to be estimated. The least square technique is being used to fit a model equation containing the input variables by minimizing the residual error measured by the sum of square deviations between the actual and estimated responses. The calculated coefficients or the model equations however need to be tested for statistical significance and thus the following tests are performed. To check the adequacy of the model for the responses in the experimentation, Analysis of Variance (ANOVA) is used. ANOVA calculates the F-ratio, which is the ratio between the regression mean square and the mean square error. The F-ratio, also called the variance ratio, is the ratio of variance due to the effect of a factor (the model) and variance due to the error term. This ratio is used to measure the significance of the model under investigation with respect to the variance of all the terms included in the error term at the desired significance level, a: If the calculated value of F-ratio is higher than the tabulated value of F-ratio for

22

1 Fundamental Consideration

roughness, then the model is adequate at desired a level to represent the relationship between machining response and the machining parameters. In the ANOVA Table, there is a P-value or probability of significance for each independent variable in the model the value of which shows whether the variable is significant or not. If the P-value is less or equal to the selected a-level, then the effect of the variable is significant. If the P-value is greater than the selected a-value, then it is considered that the variable is not significant. Sometimes the individual variables may not be significant. If the effect of interaction terms is significant, then the effect of each factor is different at different levels of the other factors. ANOVA for different response variables are carried out in the present study using commercial software Minitab (Minitab user manual 2001) with confidence level set at 95%, i.e., the a-level is set at 0.05.

1.7 Closure In this chapter, different basic considerations are discussed. The chapter starts with the essence of fractal dimension to describe surface roughness. The basics of surface metrology including the different roughness parameters along with the surface roughness measurement technique are presented. Basics of fractal dimension and its calculation are also discussed. Then the essence of design of experiments and different design of experiment techniques are presented. Response surface methodology (RSM) is discussed which is used to analyze the experimental data in the subsequent chapters.

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Chapter 2

Fractal Analysis in CNC End Milling

Abstract This chapter deals with the fractal dimension modeling in CNC end milling operation. Milling operations are carried out for three different materials viz. mild steel, brass and aluminium work-pieces for different combinations of spindle speed, feed rate and depth of cut. The generated surfaces are measured with Talysurf instrument and analyzed to get fractal dimension. The experimental results are further processed to model fractal dimension using response surface methodology (RSM). It is seen that spindle speed and depth of cut are the significant factors affecting fractal dimension for mild steel. For brass material, the significant factors are spindle speed and feed rate but for aluminium the significant factor is depth of cut. In general, for mild steel and brass, with increase in spindle speed, D increases. Comparing the developed response surface models, it is concluded that the models are material specific and the tool-work-piece material combination plays a vital role in fractal dimension of the generated surface profile. Keywords Fractal dimension (D)  CNC End Milling  RSM  Mild steel  Brass  Aluminium

2.1 Introduction CNC milling is a popular machining process in the modern industry because of its ability to remove materials with a multi-point cutting tool at a faster rate with a reasonably good surface quality. In order to get specified surface roughness, selection of controlling parameters is necessary. There has been a great many research developments in modeling surface roughness and optimization of the controlling parameters to obtain a surface finish of desired level since only proper selection of cutting parameters can produce a better surface finish. But such studies

P. Sahoo et al., Fractal Analysis in Machining, SpringerBriefs in Computational Mechanics, DOI: 10.1007/978-3-642-17922-8_2, Ó Prasanta Sahoo 2011

29

30

2 Fractal Analysis in CNC End Milling

Table 2.1 Variable levels used in the experimentation Levels Aluminium Brass -1 -0.5 0 0.5 1

Mild steel

d

N

f

d

N

f

d

N

f

0.10 0.15 0.20 0.25 0.30

4,500 4,750 5,000 5,250 5,500

900 950 1,000 1,050 1,100

0.10 0.15 0.20 0.25 0.30

1,500 1,800 2,100 2,400 2,700

550 600 650 700 750

0.150 0.175 0.200 0.225 0.250

2,500 2,750 3,000 3,250 3,500

300 350 400 450 500

are far from complete since it is very difficult to consider all the parameters that control the surface roughness for a particular manufacturing process. In CNC milling there are several parameters which control the surface quality. The analysis of surface roughness on CNC end milling process is a big challenge for research development. Several factors involved in machining process have to be optimized to obtain a desired surface quality. In this study, three machining parameters are considered viz. spindle speed, feed rate and depth of cut. Also the study is conducted on three different materials, viz. mild steel, brass and aluminium to consider the effect of work-piece material variation on fractal dimension of machined surfaces. The experimental results are analyzed using RSM.

2.2 Experimental Details 2.2.1 Design of Experiments A full factorial design is used with five levels of each of the three design factors viz. depth of cut (d, mm), spindle speed (N, rpm) and feed rate (f, mm/min). Thus the design chosen was five level-three factor (53) full factorial design consisting of 125 sets of coded combinations for each work-piece material. Three cutting parameters are selected as design factors while other parameters have been assumed to be constant over the experimental domain. The upper and lower limits of a factor were coded as +1 and -1 respectively using Eq. 1.8. The process variables/design factors with their values on different levels are listed in Table 2.1 for three different work-piece materials.

2.2.2 Machine Used The machine used for the milling tests is a ‘DYNA V4.5’ CNC end milling machine having the control system SINUMERIK 802 D with a vertical milling head. The specification of CNC end milling machine has been shown in Table 2.2. For generating the milled surfaces, CNC part programs for tool paths were created with specific commands. The compressed coolant servo-cut was used as cutting environment.

2.2 Experimental Details

31

Table 2.2 Specification of CNC end milling machine Table size

450 9 250 mm

Table load capacity X Travel Y Travel Z Travel Spindle nose to table Spindle centre to column Taper of spindle nose Spindle speed Rapid on X and Y axis Rapid on Z axis Spindle motor X axis motor Y axis motor Z axis motor Contro system Power requirement Lubricating oil

200 Kgs 250 mm 175 mm 175 mm 300 mm 280 mm BT 30 9,000 rpm 15 m/min 10 m/min 3.7 kW 3 Nm 3 Nm 6 Nm 802 D SINUMERIK 7.5 kW/10 H.P. Tellus 33 or EN KLO 68

2.2.3 Cutting Tool Used Coated carbide tools are known to perform better than uncoated carbide tools. Thus commercially available CVD coated carbide tools were used in this investigation. The tools used were flat end mill cutters produced by WIDIA (EM-TiAlN). The tools were coated with TiAlN coating. For each material a new cutter of same specification was used. The details of the end milling cutters are given below: Cutter diameter = 8 mm Overall length = 108 mm Fluted length = 38 mm Helix angle = 30° Hardness = 1,570 HV Density = 14.5 g/cc Transverse rupture strength = 3,800 N/mm2

2.2.4 Work-Piece Materials The present study was carried out with three different materials, viz., 6061-T4 Aluminium, AISI 1040 steel and Medium leaded Brass UNS C34000. The chemical composition and mechanical properties of the work-piece materials are shown in Table 2.3. All the specimens were in the form of 100 9 75 9 25 mm blocks.

32

2 Fractal Analysis in CNC End Milling

Table 2.3 Composition and mechanical properties of work-piece materials Work material Chemical composition (W%t) Mechanical property Aluminium (6061-T4)

Brass (UNS C34000)

Mild Steel (AISI 1040)

0.2%Cr, 0.3%Cu, 0.85%Mg, 0.04%Mn, 0.5%Si, 0.04%Ti, Hardness—65 0.25%Zn, 0.5%Fe and balance Al BHN, Density—2.7 g/cc, Tensile Strength— 241 MPa 0.095%Fe, 0.9%Pb, 34%Zn and balance Cu Hardness—68 HRF, Density— 8.47 g/cc, Tensile strength— 340 MPa 0.42%C, 0.48%Mn, 0.17%Si, 0.02%P, 0.018%S, 0.1%Cu, Hardness—201 0.09%Ni, 0.07%Cr and balance Fe BHN, Density— 7.85 g/cc, Tensile strength— 620 MPa

2.3 Results and Discussion CNC milling operations are carried out on mild steel, brass and aluminium workpieces to get machined surfaces for different combinations of spindle speed, feed rate and depth of cut. The generated surfaces are measured using Talysurf instrument and further processed to get fractal dimension (D). Full factorial design of experiments is considered in the study and the experimental results are presented in Table 2.4. The influences of the cutting parameters (d, N and f) on the profile fractal dimension D have been assessed for three different materials. The second order model was postulated in obtaining the relationship between the fractal dimension and the machining variables using response surface methodology (RSM). The analysis of variance (ANOVA) was used to check the adequacy of the second order model. The results for the three different materials are presented one by one.

2.3.1 RSM for Mild Steel The second order response surface equation for the fractal dimension in mild steel milling is obtained in terms of coded values of design factors as: D ¼1:3836 þ 0:0136d þ 0:0115N þ 0:0069f  0:0063dN þ 0:0003df  0:0106Nf  0:0283d2 þ 0:0169N 2 þ 0:0032f 2

ð2:1Þ

2.3 Results and Discussion

33

Table 2.4 Experimental results for CNC milling considering full factorial design Sl Depth of Spindle Feed D for mild D for D for No cut(d) speed(N) rate(f) steel brass aluminium 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5

-1 -1 -1 -1 -1 -0.5 -0.5 -0.5 -0.5 -0.5 0 0 0 0 0 0.5 0.5 0.5 0.5 0.5 1 1 1 1 1 -1 -1 -1 -1 -1 -0.5 -0.5 -0.5 -0.5 -0.5 0 0 0 0 0 0.5 0.5 0.5 0.5

-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5

1.31 1.33 1.29 1.30 1.32 1.29 1.33 1.37 1.37 1.34 1.32 1.35 1.38 1.34 1.39 1.38 1.36 1.36 1.40 1.34 1.40 1.38 1.41 1.36 1.37 1.41 1.39 1.35 1.39 1.38 1.31 1.37 1.40 1.41 1.40 1.38 1.32 1.37 1.39 1.39 1.41 1.40 1.36 1.41

1.28 1.31 1.22 1.28 1.27 1.30 1.29 1.30 1.32 1.27 1.38 1.33 1.31 1.30 1.31 1.36 1.33 1.30 1.31 1.32 1.37 1.35 1.34 1.35 1.30 1.30 1.27 1.26 1.25 1.28 1.31 1.29 1.31 1.29 1.29 1.38 1.34 1.31 1.28 1.29 1.35 1.33 1.32 1.32

1.34 1.34 1.37 1.29 1.36 1.38 1.32 1.35 1.35 1.36 1.35 1.34 1.33 1.34 1.34 1.35 1.36 1.35 1.34 1.34 1.34 1.35 1.35 1.38 1.38 1.37 1.36 1.35 1.31 1.37 1.36 1.39 1.31 1.35 1.32 1.35 1.34 1.34 1.34 1.38 1.33 1.31 1.37 1.36 (continued)

34 Table Sl No 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88

2 Fractal Analysis in CNC End Milling 2.4 (continued) Depth of Spindle cut(d) speed(N) -0.5 0.5 -0.5 1 -0.5 1 -0.5 1 -0.5 1 -0.5 1 0 -1 0 -1 0 -1 0 -1 0 -1 0 -0.5 0 -0.5 0 -0.5 0 -0.5 0 -0.5 0 0 0 0 0 0 0 0 0 0 0 0.5 0 0.5 0 0.5 0 0.5 0 0.5 0 1 0 1 0 1 0 1 0 1 0.5 -1 0.5 -1 0.5 -1 0.5 -1 0.5 -1 0.5 -0.5 0.5 -0.5 0.5 -0.5 0.5 -0.5 0.5 -0.5 0.5 0 0.5 0 0.5 0

Feed rate(f) 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0

D for mild steel 1.36 1.38 1.38 1.32 1.39 1.41 1.38 1.42 1.43 1.43 1.41 1.38 1.41 1.38 1.38 1.40 1.38 1.37 1.34 1.41 1.38 1.40 1.39 1.36 1.40 1.38 1.43 1.41 1.40 1.39 1.43 1.40 1.39 1.38 1.43 1.38 1.39 1.35 1.37 1.40 1.41 1.35 1.32 1.37

D for brass 1.29 1.37 1.37 1.36 1.32 1.31 1.26 1.25 1.26 1.27 1.29 1.36 1.27 1.32 1.27 1.29 1.35 1.35 1.32 1.31 1.31 1.36 1.34 1.32 1.32 1.36 1.37 1.33 1.35 1.34 1.36 1.24 1.27 1.23 1.26 1.28 1.27 1.27 1.33 1.25 1.28 1.38 1.33 1.32

D for aluminium 1.35 1.36 1.35 1.34 1.34 1.38 1.41 1.35 1.37 1.34 1.35 1.36 1.36 1.35 1.39 1.31 1.34 1.34 1.37 1.35 1.31 1.28 1.34 1.34 1.37 1.35 1.36 1.35 1.37 1.36 1.36 1.38 1.32 1.29 1.33 1.32 1.38 1.38 1.33 1.33 1.34 1.33 1.36 1.36 (continued)

2.3 Results and Discussion Table Sl No 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125

2.4 (continued) Depth of Spindle cut(d) speed(N) 0.5 0 0.5 0 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1 0.5 1 0.5 1 0.5 1 0.5 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -0.5 1 -0.5 1 -0.5 1 -0.5 1 -0.5 1 0 1 0 1 0 1 0 1 0 1 0.5 1 0.5 1 0.5 1 0.5 1 0.5 1 1 1 1 1 1 1 1 1 1

35

Feed rate(f) 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1

D for mild steel 1.39 1.41 1.36 1.38 1.37 1.39 1.38 1.44 1.43 1.44 1.43 1.42 1.30 1.42 1.38 1.38 1.39 1.35 1.38 1.33 1.36 1.40 1.39 1.37 1.35 1.39 1.41 1.40 1.38 1.38 1.36 1.39 1.41 1.41 1.40 1.40 1.36

D for brass 1.29 1.31 1.39 1.33 1.32 1.31 1.36 1.36 1.37 1.37 1.34 1.34 1.29 1.28 1.26 1.24 1.26 1.31 1.28 1.31 1.27 1.30 1.37 1.33 1.34 1.27 1.33 1.39 1.37 1.31 1.29 1.35 1.37 1.37 1.36 1.33 1.31

D for aluminium 1.31 1.28 1.33 1.36 1.33 1.37 1.34 1.34 1.34 1.3 1.3 1.36 1.34 1.32 1.32 1.29 1.36 1.37 1.24 1.33 1.33 1.22 1.36 1.34 1.34 1.32 1.31 1.32 1.34 1.32 1.35 1.33 1.35 1.33 1.31 1.3 1.32

The developed model is checked for adequacy by ANOVA and F-test. Table 2.5 presents the ANOVA table for the second order model proposed for fractal dimension, D given in Eq. 2.1. It can be seen that the P-value is less than 0.05 which means that the model is significant at 95% confidence level. Also the

36

2 Fractal Analysis in CNC End Milling

Table 2.5 ANOVA for second order model for D in CNC milling of mild steel Source Degrees of freedom Sum of squares Mean squares Fcalculated Regression Residual error Total

9 115 124

0.051657 0.080004 0.131661

0.005740 0.000696

8.25

Table 2.6 ANOVA for model coefficients for D in CNC milling of mild steel Source Degrees of freedom Sum of squares Mean squares Fcalculated d N f d*N d*f N*f Error Total

4 4 4 16 16 16 64 124

0.0293648 0.0146848 0.0052688 0.0232112 0.0075072 0.0159072 0.0357168 0.1316608

0.0073412 0.0036712 0.0013172 0.0014507 0.0004692 0.0009942 0.0005581

13.15 6.58 2.36 2.60 0.84 1.78

F0.05

P

1.96

0

F0.05

P

2.52 2.52 2.52 1.82 1.82 1.82

0.000 0.000 0.063 0.004 0.636 0.054

Fig. 2.1 Main effect plot for mild steel

calculated value of the F-ratio is more than the standard value of the F-ratio for D. It means the model is adequate at 95% confidence level to represent the relationship between the machining response and the considered machining parameters of the CNC end milling process on mild steel. Table 2.6 represents the ANOVA table for individual model coefficients where it can be seen that there are three effects with a P-value less than 0.05 which means that they are significant at 95% confidence level. These significant effects are: depth of cut, spindle speed and the interaction between spindle speed and depth of cut. Figure 2.1 depicts the main effects plot for the fractal dimension and the design factors considered in the present study. From this figure also, it is seen that spindle speed and depth of cut have the significant effect on fractal dimension. To see the effects of process parameters on fractal dimension in the experimental regime, three dimensional surface as well as contour plots are presented at high level and low level of the parameters (Figs. 2.2, 2.3, 2.4).

2.3 Results and Discussion

37

Fig. 2.2 Surface and contour plot of fractal dimension for mild steel: a at high level of spindle speed, b at low level of spindle speed

Fig. 2.3 Surface and contour plot of fractal dimension for mild steel: a at high level of depth of cut, b at low level of depth of cut

Fig. 2.4 Surface and contour plot of fractal dimension for mild steel: a at high level of feed rate, b at low level of feed rate

38

2 Fractal Analysis in CNC End Milling

Table 2.7 ANOVA for second order model for D in CNC milling of brass Source Degrees of freedom Sum of squares Mean squares Fcalculated Regression Residual Error Total

9 115 124

0.138293 0.048614 0.186907

0.015366 0.000423

36.35

Table 2.8 ANOVA for model coefficients for D in CNC milling of brass Source Degrees of freedom Sum of squares Mean squares Fcalculated d N f d*N d*f N*f Error Total

4 4 4 16 16 16 64 124

0.0006512 0.1095792 0.0264432 0.0043968 0.0092528 0.0196048 0.0169792 0.1869072

0.0001628 0.0273948 0.0066108 0.0002748 0.0005783 0.0012253 0.0002653

0.61 103.26 24.92 1.04 2.18 4.62

F0.05

P

1.96

0

F0.05

P

2.52 2.52 2.52 1.82 1.82 1.82

0.654 0.000 0.000 0.433 0.015 0.000

Fig. 2.5 Main effect plot for brass

2.3.2 RSM for Brass The second order response surface equation for fractal dimension in brass milling is obtained in terms of coded values of design factors as: D ¼1:3130 þ 0:0015d þ 0:0408N  0:0175f þ 0:0071dN þ 0:0014df  0:0098Nf  0:0008d2  0:0142N 2 þ 0:0163f 2

ð2:2Þ

The developed model is checked for adequacy by ANOVA and F-test. Table 2.7 presents the ANOVA table for the second order model proposed for D given in Eq. 2.2. It can be seen that the P-value is less than 0.05 which means that the model is significant at 95% confidence level. Also the calculated value of the F-ratio is more than the standard value of the F-ratio for D. It means the model is adequate at 95% confidence level to represent the relationship between the

2.3 Results and Discussion

39

Fig. 2.6 Surface and contour plot of fractal dimension for brass: a at high level of spindle speed, b at low level of spindle speed

Fig. 2.7 Surface and contour plot of fractal dimension for brass: a at high level of depth of cut, b at low level of depth of cut

machining response and the considered machining parameters of the CNC end milling process on brass. Table 2.8 represents the ANOVA table for individual model coefficients where it can be seen that spindle speed, feed rate, the interaction between spindle speed and feed rate and the interaction of depth of cut and feed rate are significant factors at 95% confidence level. Figure 2.5 depicts the main effects plot for the fractal dimension and the design factors considered in the present study. From this figure also, it is seen that spindle speed and feed rate have the significant effect on fractal dimension. Figures 2.6, 2.7, 2.8 show the estimated three-dimensional surface as well as contour plots for fractal dimension as functions of the independent machining parameters. All these figures clearly depict the variation of fractal dimension with controlling variables within the experimental regime.

40

2 Fractal Analysis in CNC End Milling

Fig. 2.8 Surface and contour plot of fractal dimension for brass: a at high level of feed rate, b at low level of feed rate

Table 2.9 ANOVA for second order model for D in CNC milling of aluminium Source Degrees of freedom Sum of squares Mean squares Fcalculated Regression Residual error Total

9 115 124

0.025241 0.0717 0.096941

0.002805 0.000624

4.5

F0.05

P

1.96

0

2.3.3 RSM for Aluminium The second order response surface equation has been fitted using Minitab software for the response variable D. The equation can be given in terms of the coded values of the independent variables as: D ¼1:3433  0:0128d þ 0:0013N  0:0062 f  0:0011dN  0:0095 df þ 0:0122Nf  0:0135d 2 þ 0:0041 N 2 þ 0:0056 f 2

ð2:3Þ

Table 2.9 presents the ANOVA table for the second order model proposed for D given in Eq. 2.3. It can be appreciated that the P-value is less than 0.05 which means that the model is significant at 95% confidence level. Also the calculated value of the F-ratio is more than the standard value of the F-ratio for D. It means the model is adequate at 95% confidence level to represent the relationship between the machining response and the considered machining parameters of the CNC end milling process. Table 2.10 represents the ANOVA table for individual model coefficients where it can be seen that depth of cut and the interaction between spindle speed and feed rate are significant at 95% confidence level. Figure 2.9 depicts the main effects plot for the fractal dimension and the design factors considered in the present study. From this figure also, it is seen that depth of cut has the significant effect on fractal dimension. Figures 2.10, 2.11, 2.12 show the estimated three-dimensional surface as well as contour plots for fractal

2.3 Results and Discussion

41

Table 2.10 ANOVA for model coefficients for D in CNC milling of aluminium Source Degrees of freedom Sum of squares Mean squares Fcalculated F0.05 d N f d*N d*f N*f Error Total

4 4 4 16 16 16 64 124

0.0146608 0.0004928 0.0032048 0.0110272 0.0102352 0.0226432 0.0346768 0.0969408

0.0036652 0.0001232 0.0008012 0.0006892 0.0006397 0.0014152 0.0005418

6.76 0.23 1.48 1.27 1.18 2.61

2.52 2.52 2.52 1.82 1.82 1.82

P 0.000 0.922 0.219 0.243 0.307 0.003

Fig. 2.9 Main effect plot for aluminium

Fig. 2.10 Surface and contour plot of fractal dimension for aluminium: a at high level of spindle speed, b at low level of spindle speed

dimension as functions of the independent machining parameters. All these figures clearly depict the variation of fractal dimension with controlling variables within the experimental regime.

42

2 Fractal Analysis in CNC End Milling

Fig. 2.11 Surface and contour plot of fractal dimension for aluminium: a at high level of depth of cut, b at low level of depth of cut

Fig. 2.12 Surface and contour plot of fractal dimension for aluminium: a at high level of feed rate, b at low level of feed rate

2.4 Closure For three different work-piece materials, fractal dimension models are developed in CNC end milling using response surface method. The second order response models have been validated with analysis of variance. A comparison of the response surface models for fractal dimension in different materials reveals the fact that these models are material specific or in other words, the tool-work-piece material combination plays a vital role in fractal dimension of the generated surface profile. Also the effect of the cutting parameters on fractal dimension is different for different materials as evidenced from Tables 2.6, 2.8 and 2.10. Accordingly, optimum machining parameter combinations for fractal dimension depend greatly on the work-piece material within the experimental domain.

2.4 Closure

43

However, it can be concluded that it is possible to select a combination of spindle speed, depth of cut and feed rate for achieving the surface topography with desired fractal dimension within the constraints of the available machine. Thus with the known boundaries of desired fractal dimension and machining parameters, machining can be performed with a relatively high rate of success.

Chapter 3

Fractal Analysis in CNC Turning

Abstract Modeling of fractal dimension in CNC turning of mild steel, brass and aluminium work-pieces are presented in this chapter. Spindle speed, feed rate and depth of cut are considered as the process parameters. The generated surface in CNC tuning operations are measured and processed to calculate fractal dimension. The experimental results are then analyzed with RSM. From the analysis, it is seen that the work-piece speed is the most significant factor affecting the fractal dimension for mild steel turning whereas feed rate is the significant factor for both brass and aluminium materials. It can be concluded from the analysis that for all the materials, with increase in feed rate, fractal dimension, D decreases. So, to get smoother surface, feed rate should be at low level. With increase in spindle speed, fractal dimension increases giving smoother surface for mild steel turning. Keywords Fractal dimension (D) Aluminium

 CNC turning  RSM  Mild steel  Brass 

3.1 Introduction Turning operation is an old and very common machining process in the industry. In recent times, uses of computer numerically controlled (CNC) machines have become popular to minimize the operator input and to get higher surface finish. Turning operations are carried out on a lathe. In turning, there are several machining parameters which control the surface quality of the machined work-piece which include cutting conditions, tool variables and work-piece variables. Cutting conditions include speed, feed and depth of cut where as tool variables include tool material, nose radius, rake angle, cutting edge geometry, tool vibration, tool overhang, tool point angle, etc. and work-piece variables include material hardness and other mechanical properties. It is very difficult to consider all the parameters that

P. Sahoo et al., Fractal Analysis in Machining, SpringerBriefs in Computational Mechanics, DOI: 10.1007/978-3-642-17922-8_3, Ó Prasanta Sahoo 2011

45

46

3 Fractal Analysis in CNC Turning

Table 3.1 Process parameters levels used in the experimentation for all the three materials Process variables Unit Levels A Depth of cut(d) B Spindle speed(N) C Feed rate(f)

mm rpm mm/rev

-1.682

-1

0

1

1.682

0.032 528 0.0224

0.1 800 0.07

0.2 1,200 0.14

0.3 1,600 0.21

0.368 1,872 0.2576

Table 3.2 Design matrix of the rotatable CCD design with coded and actual value Std. order Run order Coded values Actual values 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

20 1 9 11 7 8 13 3 10 6 5 14 12 19 2 4 17 16 18 15

d

N

f

d

N

f

-1 1 -1 1 -1 1 -1 1 -1.682 1.682 0 0 0 0 0 0 0 0 0 0

-1 -1 1 1 -1 -1 1 1 0 0 -1.682 1.682 0 0 0 0 0 0 0 0

-1 -1 -1 -1 1 1 1 1 0 0 0 0 -1.682 1.682 0 0 0 0 0 0

0.1 0.3 0.1 0.3 0.1 0.3 0.1 0.3 0.032 0.368 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

800 800 1,600 1,600 800 800 1,600 1,600 1,200 1,200 528 1,872 1,200 1,200 1,200 1,200 1,200 1,200 1,200 1,200

0.07 0.07 0.07 0.07 0.21 0.21 0.21 0.21 0.14 0.14 0.14 0.14 0.0224 0.2576 0.14 0.14 0.14 0.14 0.14 0.14

control the surface quality. In a turning operation, it is the vital task to select the cutting parameters to achieve the high quality performance. For this, modeling of the surface roughness is necessary to predict and control the desired level of surface roughness. In this study, CNC turning operations are carried out varying the machining parameters, viz., depth of cut (mm), spindle speed (rpm) and feed rate (mm/rev). Machining surfaces are further analyzed to find out the profile fractal dimension. These experimental results are further analyzed using response surface methodology.

3.2 Experimental Details

47

3.2 Experimental Details 3.2.1 Design of Experiments In a turning operation, there are many factors that can affect the surface roughness. But, the review of literature shows that the depth of cut (d, mm), spindle speed (N, rpm) and feed rate (f, mm/rev) are the most widespread machining parameters taken by the researchers. In the present study these are selected as design factors while other parameters have been assumed to be constant over the experimental domain. The process variables with their values are listed in Table 3.1. For the experimentation, a rotatable central composite design (Sect. 1.5.2) is selected and the experimental plan consists of experiment run order, standard order, coded values and actual values of process parameters as shown in Table 3.2.

3.2.2 Machine Used The machine used for the turning is a JOBBERXL CNC lathe having the control system FANUC Series Oi Mate-Tc and equipped with maximum spindle speed of 3,500 rpm, feed rate 15–20 m/rev and KVA rating-16 KVA. For generating the turned surfaces, CNC part programs for tool paths were created with specific commands.

3.2.3 Cutting Tool Used Coated carbide tools are known to perform better than uncoated carbide tools. Thus commercially available CVD coated carbide tools were used in this investigation. The tool holder is used as the PTGNR-25-25 M16 050, WIDIA and insert used as the TNMG 160404–FL, WIDIA. The tool is coated with titanium nitride coating having hardness, density and transverse rupture strength as 1,570 HV, 14.5 g/cc and 3,800 N/mm2. The compressed coolant WS 50–50 with a ratio of 1:20 with water was used as cutting environment.

3.2.4 Work-Piece Materials The present study was carried out with three different workpiece materials, viz., 6061-T4 aluminium, mild steel (AISI 1040) and medium leaded brass UNS C34000. All the specimens were in the form of bar with diameter 20 mm and

48

3 Fractal Analysis in CNC Turning

Table 3.3 Experimental results for CCD Std. order D for mild steel

D for brass

D for aluminium

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1.435 1.437 1.395 1.420 1.300 1.292 1.297 1.297 1.355 1.367 1.375 1.355 1.380 1.257 1.350 1.375 1.375 1.362 1.377 1.377

1.417 1.315 1.392 1.440 1.300 1.302 1.292 1.262 1.377 1.360 1.397 1.347 1.485 1.252 1.362 1.370 1.385 1.367 1.300 1.297

1.370 1.300 1.362 1.410 1.267 1.282 1.390 1.417 1.320 1.370 1.300 1.420 1.360 1.290 1.397 1.400 1.415 1.415 1.402 1.412

length 60 mm. The chemical and mechanical properties of the materials are already given in Table 2.3 (Chap. 2).

3.3 Results and Discussion To get machined surfaces, CNC turning operations are carried out for different combinations of spindle speed, feed rate and depth of cut. Three different workpiece materials are considered viz. mild steel, brass and aluminium. The generated surfaces are measured using Talysurf instrument (Sect. 1.3.7) and further processed to get fractal dimension (D). The experimental results are used for further analyses using response surface methodology (RSM) to model fractal dimension. For RSM, a rotatable central composite design of experiment is considered and the experimental results are presented in Table 3.3. The influences of the machining parameters on fractal dimension have been assessed for three different materials using RSM. The whole analyses are done using Minitab software. The results of RSM analyses are presented below.

3.3 Results and Discussion

49

Table 3.4 ANOVA for second order model for mild steel Source DF SS MS Regression Residual Error Total

9 10 19

0.049 0.0032 0.052

0.005,409 0.000319

F

F0.05

P

16.96

3.02

0

Table 3.5 Full ANOVA table for mild steel model Source Sum of squares DF Mean square Model A–d B–N C–f AB AC BC A2 B2 C2 Residual Lack-of-fit Pure error Cor total

0.049 0.0007933 0.023 0.003009 0.00211 0.0005281 0.003003 0.005608 0.002998 0.010 0.00318 0.002871 0.0003177 0.05186

9 1 1 1 1 1 1 1 1 1 10 5 5 19

0.005409 0.0007933 0.023 0.003009 0.00211 0.0005281 0.003003 0.005608 0.002998 0.010 0.0003189 0.0005742 0.00006354

F value

P value

16.96 2.49 72.48 9.44 6.62 1.66 9.42 17.59 9.40 32.46

0.0001 0.1458 0.0001 0.0118 0.0277 0.2271 0.1190 0.0018 0.0119 0.0002

9.04

0.0152

3.3.1 RSM for Mild Steel The second order response model is developed using Minitab in terms of coded values of the independent machining parameters, viz., work-piece speed, feed rate and depth of cut. The response model for mild steel material is given in the following equation. D ¼ 1:40674 þ 0:00453 d þ 0:02446 N  0:00883 f þ 0:005745 dN þ 0:002873 df þ 0:006850Nf  0:00697 d 2  0:00510 N 2  0:00947 f 2

ð3:1Þ

The developed model is also checked for adequacy. Table 3.4 represents the ANOVA table for the second order response model developed for D. It is clear that the developed model is significant at 95% confidence level. The calculated value of F ratio is greater than the tabulated value of F ratio and it can be concluded that the model is adequate at 95% confidence level. ANOVA table for mild steel (Table 3.5) shows that work speed, feed rate, interaction of depth of cut with work-piece speed are significant factors at 95% confidence level. The main effects plots for fractal dimension are shown in Fig. 3.1. From the main effect plots, it is seen that work-piece speed and feed rate are significant. It can also be concluded that with increase in work speed, D increases but with increase in feed rate, D decreases in mild steel turning. Response surface plots are also generated using

50

3 Fractal Analysis in CNC Turning

Fig. 3.1 Main effect plots for mild steel

Minitab. Figs. 3.2, 3.3 and 3.4 show the estimated three dimensional surface as well as contour plots for fractal dimension as functions of two independent machining parameters while the third machining parameter is held constant. All these figures clearly depict the variation of fractal dimension with controlling variables within the experimental regime.

3.3.2 RSM for Brass The second order response model for brass material is presented in terms of coded values of work-piece speed, feed rate and depth of cut in Eq. 3.2. D ¼ 1:36919 þ 0:00179 d  0:00386 N  0:03074 f þ 0:00133 dN  0:00155 df þ 0:00265 Nf  1:21695  1004 d 2 þ 0:00035 N 2  0:00543 f 2

ð3:2Þ

The developed model is checked for adequacy and ANOVA result for the model is presented in Table 3.6. From the ANOVA table, it is seen that the model is significant and adequate at 95% confidence level. From the full ANOVA table (Table 3.7), it is seen that feed rate is the main significant factor affecting fractal dimension in brass turning. The calculated F-value of the lack-of-fit for D is much lower than the tabulated value of the F-distribution (tabulated value 5.05) found from the standard table at 95% confidence level. It implies that the lack-of-fit is not significant relative to pure error. From the main effect plot (Fig. 3.5), it is seen that only feed rate is significant and the other parameters are insignificant. It is also

3.3 Results and Discussion

51

Fig. 3.2 Surface and contour plot of fractal dimension, D for mild steel: a at high level of spindle speed, b at low level of spindle speed

Fig. 3.3 Surface and contour plot of fractal dimension, D for mild steel: a at high level of depth of cut, b at low level of depth of cut

Fig. 3.4 Surface and contour plot of fractal dimension, D for mild steel: a at high level of feed rate, b at low level of feed rate

52

3 Fractal Analysis in CNC Turning

Table 3.6 ANOVA for second order model for brass Source DF SS MS Regression Residual Error Total

9 10 19

0.041 0.0034 0.045

Table 3.7 Full ANOVA table for brass model Source Sum of squares DF Model A–d B–N C–f AB AC BC A2 B2 C2 Residual Lack-of-fit Pure Error Cor Total

0.041 1.232E-4 5.753E-4 0.036 1.125E-4 1.531E-4 4.500E-4 1.707E-6 1.389E-5 3.405E-3 3.393E-3 2.775E-3 6.177E-4 0.045

9 1 1 1 1 1 1 1 1 1 10 5 5 19

0.004603 0.000319

F

F0.05

P

13.56

3.02

0

Mean square

F value

P value

4.603E-3 1.232E-4 5.753E-4 0.036 1.125E-4 1.531E-4 4.500E-4 1.707E-6 1.389E-5 3.405E-3 3.189E-4 5.551E-4 1.235E-4

13.56 0.36 1.70 107.57 0.33 0.45 1.33 5.032E-3 0.041 10.03

0.0002 0.5602 0.2221 0.0001 0.5775 0.5169 0.2763 0.9448 0.8437 0.0100

4.49

0.0624

Fig. 3.5 Main effect plots for brass

seen that with increase in feed rate, D decreases. The estimated three dimensional surface as well as contour plots for fractal dimension are presented in Figs. 3.6, 3.7 and 3.8. To draw these surface plots, fractal dimension is plotted as functions of two independent machining parameters while the third machining parameter is

3.3 Results and Discussion

53

Fig. 3.6 Surface and contour plot of fractal dimension, D for brass: a at high level of spindle speed, b at low level of spindle speed

Fig. 3.7 Surface and contour plot of fractal dimension, D for brass: a at high level of depth of cut, b at low level of depth of cut

Fig. 3.8 Surface and contour plot of fractal dimension, D for brass: a at high level of feed rate, b at low level of feed rate

54

3 Fractal Analysis in CNC Turning

Table 3.8 ANOVA for second order model for aluminium Source DF SS MS Regression Residual Error Total

9 10 19

0.041 0.0032 0.052

0.005409 0.000319

F

F0.05

P

16.96

3.02

0

Table 3.9 Full ANOVA table for aluminium model Source Sum of squares df Mean square Model A–d B–N C–f AB AC BC A2 B2 C2 Residual Lack-of-fit Pure error Cor total

0.052 9.174E-4 7.307E-5 0.047 1.726E-3 9.453E-5 2.720E-3 1.751E-5 8.496E-5 1.751E-5 0.017 9.952E-3 7.293E-3 0.070

9 1 1 1 1 1 1 1 1 1 10 5 5 19

5.841E-3 9.174E-4 7.307E-5 0.047 1.726E-3 9.453E-5 2.720E-3 1.751E-5 8.496E-5 1.751E-5 1.724E-3 1.990E-3 1.459E-3

F value

P value

3.37 0.53 0.042 27.08 1.000 0.055 1.580 0.010 0.049 0.010

0.0359 0.4825 0.8410 0.0004 0.3407 0.8196 0.2377 0.9217 0.8288 0.9217

1.36

0.3707

held constant at high and low levels. All these figures clearly depict the variation of fractal dimension with controlling variables within the experimental regime.

3.3.3 RSM for Aluminium The second order response model for aluminium material is presented in terms of coded values of the independent machining parameters, viz., work-piece speed, feed rate and depth of cut in Eq. 3.3. D ¼ 1:34809  0:00487 d  0:00138 N  0:03477 f þ 0:00519 dN þ 0:00122 df  0:00652 Nf þ 0:000390 d2 þ 0:00086 N 2 þ 0:00039 f 2

ð3:3Þ

Table 3.8 presents the ANOVA table for the second order model proposed for D of aluminium material. It is observed that the model is significant and adequate at 95% confidence level. From the full ANOVA table (Table 3.9), it is seen that feed rate is the main significant factor affecting fractal dimension in aluminium turning. The calculated F-value of the lack-of-fit for D is much lower than the tabulated value of the F-distribution (tabulated value 5.05) found from the standard table at 95% confidence level. From the main effects plot (Fig. 3.9), it is seen

3.3 Results and Discussion

55

Fig. 3.9 Main effect plots for aluminium material

Fig. 3.10 Surface and contour plot of fractal dimension, D for aluminium: a at high level of spindle speed, b at low level of spindle speed

that only feed rate is significant. It is also seen that with increase in feed rate, fractal dimension, D decreases. Response surface plots are also generated using Minitab. Figs. 3.10, 3.11 and 3.12 show the estimated three dimensional surface as well as contour plots for fractal dimension as functions of two independent machining parameters. The third machining parameter is held constant at high and low levels. From these figures, variations of fractal dimension with machining parameters can be observed within the experimental regime.

56

3 Fractal Analysis in CNC Turning

Fig. 3.11 Surface and contour plot of fractal dimension, D for aluminium: a at high level of depth of cut, b at low level of depth of cut

Fig. 3.12 Surface and contour plot of fractal dimension, D for aluminium: a at high level of feed rate, b at low level of feed rate

3.4 Closure Response surface models for three materials viz. mild steel, brass and aluminium are developed in CNC turning. All the developed second order models are adequate at 95% confidence level. From the analysis, it is seen that the work-piece speed is the most significant factor affecting the fractal dimension for mild steel turning whereas feed rate is the significant factor for both brass and aluminium materials. It can be concluded from the analysis that for all the materials, with increase in feed rate, fractal dimension, D decreases. So, to get smoother surface, feed rate should be at low level. With increase in spindle speed, fractal dimension increases giving smoother surface for mild steel turning.

Chapter 4

Fractal Analysis in Cylindrical Grinding

Abstract This chapter presents the fractal dimension modeling in cylindrical grinding of mild steel, brass and aluminium work-pieces. The experimentations are carried out for different combinations of work-piece speed, longitudinal feed and radial infeed. The generated surfaces are measured and processed to calculate fractal dimension. The experimental results are then analyzed with RSM. The longitudinal feed rate is the most significant factor affecting the fractal dimension for mild steel, whereas for brass, work-piece speed and longitudinal feed rate are the most significant factors. For aluminium materials, all the three process parameters are the significant factors affecting fractal dimension. Keywords Fractal dimension (D) Brass  Aluminium

 Cylindrical grinding  RSM  Mild steel 

4.1 Introduction Grinding is one of the common machining processes. In today’s production, finishing of components is done by grinding due to the fact that it has the great potential to replace other machining processes and to achieve significant reduction in production time and cost. The acceptance of grinding as a finishing process is connected with a high form and size accuracy, high surface finish and surface integrity of the work-piece. In grinding there are several parameters which control the surface quality. It is very difficult to consider all the parameters that control the surface roughness for a particular manufacturing process. In this study, only three machining parameters are considered viz. work-piece speed, longitudinal feed and radial infeed. Also the study is conducted on three different materials, AISI 1040 mild steel, UNS C34000 brass and 6061-T4 aluminium to consider the effect of workpiece material variation. The experimental results are analyzed using

P. Sahoo et al., Fractal Analysis in Machining, SpringerBriefs in Computational Mechanics, DOI: 10.1007/978-3-642-17922-8_4, Ó Prasanta Sahoo 2011

57

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4 Fractal Analysis in Cylindrical Grinding

Table 4.1 Process variables and their levels Parameters Unit Notation

1

2

3

4

Work-piece speed Long feed Radial infeed

56 11.33 0.02

80 17.00 0.04

112 22.66 0.06

160 28.33 0.08

rpm mm/rev mm

N f d

response surface modeling (RSM). The experimental details and the results are discussed below.

4.2 Experimental Details 4.2.1 Design of Experiments The process parameters chosen here are work-piece speed (N) in rpm, longitudinal feed (f) in mm/rev and radial infeed (d) in mm. The process variables/design factors with their values on different levels are listed in Table 4.1 for three different work-piece materials. The selection of the values of the variables is limited by the capacity of the machine used in the experimentation as well as the recommended specifications for different work-piece-tool material combination. Four levels, having nearly equal spacing, within the operating range of the parameters are selected for each of the factors. By selecting four levels, the curvature or nonlinearity effects can be studied. In the present investigation, full factorial design of experiment is considered for the experimentation and for four level three factors total 64 experimental trials are carried out for each of the work materials.

4.2.2 Machine Used The machine used for grinding is a HMT made, K130U grinding machine equipped with maximum wheel speed of 1910 rpm. The wheel signature of the machine is A70K5V10 and wheel diameter of 270 mm. The maximum grinding length is about 340 mm. The compressed coolant WS 50–50 with a ratio of 1:20 was used as cutting environment. The details of the machine used in this study are shown in the Table 4.2.

4.2.3 Work-Piece Materials The present study is carried out with three different materials, viz., AISI 1040 steel, medium leaded brass UNS C34000 and 6061-T4 aluminium. The chemical composition and mechanical properties of the work-piece materials are already discussed in the Chap. 2 (Table 2.3). All the specimens are in the form of round bars of diameter 48 mm and length 50 mm.

4.2 Experimental Details

59

Table 4.2 Specification of the cylindrical grinding machine used in the experiment Make HMT Maximum grinding length Maximum distance between centers Maximum travel of the table Maximum swivel of the table Grinding wheel Wheel speed Wheel Signature Wheel Diameter Face width Bore diameter Work head Number of speed Swivel Morse taper

Model K130U Machine No

57169

340 mm 340 mm 310 mm 200 mm 1910 and 2120 rpm A70K5V10 270 mm 40 mm 50 mm 8 (56-80-112-160-224-315450-630) 90° towards wheel and 30° away from wheel 3

4.3 Results and Discussion Cylindrical grinding operations are carried out on mild steel, brass and aluminium work-pieces to get machined surfaces for different combinations of work-piece speed, longitudinal feed and radial infeed. The generated surfaces are measured using Talysurf instrument and further processed to get fractal dimension (D). The experimental results are used for further analyses using response surface methodology (RSM) to model fractal dimension. For RSM, full factorial design of experiments is considered and the design matrix and the experimental results of cylindrical grinding of mild steel, brass and aluminium work-pieces are presented in Table 4.3. The influences of the machining parameters viz. work-piece speed, longitudinal feed, radial infeed on the profile fractal dimension for mild steel (AISI 1040), brass and aluminium grinding are presented below.

4.3.1 RSM for Mild Steel The second order response surface equation has been fitted using Minitab software for the response variable D. The equation can be given in terms of the coded values of the independent variables as the following: D ¼ 1:53125  0:01081N  0:02637f  0:03162d  0:00095Nf þ 0:00255Nd þ 0:00055fd þ 0:00219N 2 þ 0:00375f 2 þ 0:00375d 2

ð4:1Þ

60

4 Fractal Analysis in Cylindrical Grinding

Table 4.3 Design matrix of process variables and the experimental results Std Run N Workpiece f Longitudinal d Radial D for D for order order speed (rpm) Feed (mm/rev) infeed (mm) mild brass steel

D for aluminium

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

1.39 1.35 1.38 1.37 1.37 1.34 1.34 1.37 1.35 1.34 1.36 1.35 1.35 1.37 1.36 1.35 1.40 1.41 1.36 1.38 1.37 1.37 1.35 1.36 1.35 1.34 1.34 1.35 1.35 1.36 1.34 1.35 1.35 1.39 1.35 1.37 1.34 1.37 1.33 1.37 1.35

22 45 7 37 54 38 26 42 13 43 63 59 32 5 40 34 51 10 14 21 1 64 48 61 23 31 53 29 12 56 2 46 25 3 19 33 8 49 17 6 18

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3

1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 1 1 1 1 2 2 2 2 3

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1

1.46 1.48 1.40 1.46 1.47 1.44 1.45 1.43 1.47 1.43 1.42 1.42 1.41 1.45 1.44 1.45 1.49 1.47 1.45 1.46 1.43 1.42 1.44 1.41 1.47 1.45 1.35 1.39 1.45 1.45 1.43 1.44 1.47 1.45 1.46 1.48 1.48 1.47 1.44 1.47 1.42

1.390 1.408 1.415 1.415 1.413 1.435 1.420 1.433 1.453 1.445 1.420 1.445 1.455 1.468 1.450 1.450 1.413 1.415 1.428 1.428 1.440 1.445 1.430 1.425 1.455 1.455 1.448 1.455 1.460 1.430 1.443 1.448 1.415 1.430 1.420 1.425 1.393 1.430 1.428 1.425 1.455

(continued)

4.3 Results and Discussion

61

Table 4.3 (continued) Std Run N Workpiece order order speed (rpm)

f Longitudinal Feed (mm/rev)

42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

3 3 3 4 4 4 4 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4

44 15 57 36 35 28 41 11 47 30 27 39 16 9 52 24 50 4 20 62 60 58 55

3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

d Radial D for infeed (mm) mild steel 2 1.47 3 1.43 4 1.45 1 1.43 2 1.39 3 1.45 4 1.42 1 1.45 2 1.48 3 1.44 4 1.47 1 1.47 2 1.45 3 1.46 4 1.47 1 1.47 2 1.41 3 1.46 4 1.42 1 1.45 2 1.45 3 1.45 4 1.44

Table 4.4 ANOVA for the response model of D for mild steel Source DF Seq SS Adj MS F Regression Residual error Total

9 54 63

0.012291 0.029903 0.042194

0.001366 0.000554

2.47

D for brass

D for aluminium

1.450 1.453 1.468 1.463 1.468 1.428 1.455 1.440 1.408 1.435 1.430 1.453 1.445 1.460 1.455 1.470 1.453 1.450 1.465 1.472 1.465 1.470 1.445

1.34 1.32 1.35 1.36 1.36 1.37 1.39 1.41 1.38 1.38 1.38 1.39 1.35 1.35 1.35 1.35 1.34 1.34 1.35 1.40 1.39 1.40 1.38

F0.05

P

2.04

0.020

The analysis of variance (ANOVA) technique has been used to check the adequacy of the developed model at 95% confidence level. As per this technique, if the calculated value of the F-ratio of the regression model is more than the standard tabulated value of table (F-table) for 95% confidence level, then the model is considered adequate within the confidence limit. From Table 4.4, it is observed that the developed model is adequate at 95% confidence level. From the ANOVA table of individual parameters (Table 4.5), it can be concluded that the longitudinal feed rate is the most significant factor affecting the fractal dimension at 95% confidence level. The main effect plots of fractal dimension D is presented in Fig. 4.1. From this figure, it is seen that longitudinal feed rate and radial infeed have influences on fractal dimension. The estimated three dimensional surface as

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4 Fractal Analysis in Cylindrical Grinding

Table 4.5 ANOVA for individual parameter of D for mild steel Source DF SS MS Fcalculated N f d N*f N*d f*d Error Total

3 3 3 9 9 9 27 63

0.0021187 0.0074563 0.0034062 0.0059187 0.0034187 0.0050312 0.0148437 0.0421937

0.0007062 0.0024854 0.0011354 0.0006576 0.0003799 0.0005590 0.0005498

1.28 4.52 2.07 1.20 0.69 1.02

F0.05

P

2.76 2.76 2.76 2.04 2.04 2.04

0.300 0.011 0.128 0.337 0.711 0.452

Fig. 4.1 Main effect plots for D in cylindrical grinding of mild steel

well as contour plots for D as function of the independent machining parameters are presented in Figs. 4.2, 4.3, 4.4.

4.3.2 RSM for Brass The second order response surface equation has been fitted using Minitab software for the response variable D. The equation can be given in terms of the coded values of the independent variables as the following: D ¼ 1:36367 þ 0:00134N þ 0:03714f þ 0:00450d  0:00114Nf  0:00060Nd  0:00300fd þ 0:00172N 2  0:00289f 2 þ 0:00094d 2 ð4:2Þ

4.3 Results and Discussion

63

Fig. 4.2 Surface and contour plots of D for mild steel: a at high level of radial infeed, b at low level of radial infeed

Fig. 4.3 Surface and contour plots of D for mild steel: a at high level of work-piece speed, b at low level of work-piece speed

Fig. 4.4 Surface and contour plots of D for mild steel: a at high level of longitudinal feed, b at low level of longitudinal feed

64

4 Fractal Analysis in Cylindrical Grinding

Table 4.6 ANOVA for the response model of D for brass Source DF Seq SS Adj MS

F

F0.05

P

12.46

2.04

0.000

Table 4.7 ANOVA for individual parameter of D for brass Source DF SS MS Fcalculated

F0.05

P

2.76 2.76 2.76 2.04 2.04 2.04

0.000 0.000 0.632 0.034 0.116 0.051

Regression Residual error Total

N f d N*f N*d f*d Error Total

3 3 3 9 9 9 27 63

9 54 63

0.016541 0.007968 0.024509

0.00305352 0.01316367 0.00016602 0.00210742 0.00153633 0.00191367 0.00256836 0.02450898

0.001838 0.000148

0.00101784 0.00438789 0.00005534 0.00023416 0.00017070 0.00021263 0.00009512

10.70 46.13 0.58 2.46 1.79 2.24

Fig. 4.5 Main effect plots for D in cylindrical grinding of brass

From ANOVA analysis of the second order model at 95% confidence level, it is seen that the model is adequate (Table 4.6). From ANOVA table of individual parameters (Table 4.7), it can be concluded that the work-piece speed, longitudinal feed rate and interaction between work-piece speed and longitudinal feed are the most significant factors affecting the fractal dimension. The main effect plots of fractal dimension D is presented in Fig. 4.5. From this figure also, it is seen that work-piece speed and longitudinal feed are significant while the radial infeed is insignificant on fractal dimension in the studied range. The estimated three dimensional surface as well as contour plots for D as function of the independent machining parameters are presented in Figs. 4.6, 4.7, 4.8. It is seen that with

4.3 Results and Discussion

65

Fig. 4.6 Surface and contour plots of D for brass: a at high level of radial infeed, b at low level of radial infeed

Fig. 4.7 Surface and contour plots of D for brass: a at high level of work-piece speed, b at low level of work-piece speed

Fig. 4.8 Surface and contour plots of D for brass: a at high level of longitudinal feed, b at low level of longitudinal feed

66

4 Fractal Analysis in Cylindrical Grinding

Table 4.8 ANOVA for the response model of D for aluminium Source DF Seq SS Adj MS F

F0.05

P

5.97

2.04

0.000

Table 4.9 ANOVA for individual parameter of D for aluminium Source DF SS MS Fcalculated

F0.05

P

2.76 2.76 2.76 2.04 2.04 2.04

0.009 0.000 0.028 0.007 0.015 0.431

Regression Residual error Total

N f d N*f N*d f*d Error Total

9 54 63

3 3 3 9 9 9 27 63

0.012886 0.012950 0.025836

0.0019672 0.0095922 0.0014672 0.0041516 0.0036266 0.0013016 0.0037297 0.0258359

0.001432 0.000240

0.0006557 0.0031974 0.0004891 0.0004613 0.0004030 0.0001446 0.0001381

4.75 23.15 3.54 3.34 2.92 1.05

increase in work-piece speed and longitudinal feed, the fractal dimension increases i.e. the surface gets smoother while the radial infeed is kept constant at middle level.

4.3.3 RSM for Aluminium The second order response surface equation has been fitted using Minitab software for the response variable D. The equation can be given in terms of the coded values of the independent variables as the following: D ¼ 1:48062  0:01616N  0:07047f  0:02253d þ 0:00282Nf  0:00097Nd þ 0:00202fd þ 0:00297N 2 þ 0:01078f 2 þ 0:00359d 2

ð4:3Þ

From the ANOVA analysis of the second order model at 95% confidence level, it is seen that the model is adequate (Table 4.8). From the ANOVA table of individual parameters (Table 4.9), it can be concluded that the work-piece speed, longitudinal feed rate and radial infeed are the significant factors affecting the fractal dimension at 95% confidence level. Also the interaction between workpiece speed and longitudinal feed and between work-piece speed and radial infeed are significant at 95% confidence interval. The main effect plots of fractal dimension D is presented in Fig. 4.9. From this figure also, it is seen that workpiece speed, longitudinal feed and radial infeed are significant in the studied range. The variations of fractal dimension with two machining parameters are presented in Figs. 4.10, 4.11, 4.12 while the third machining parameter is kept constant.

4.3 Results and Discussion

67

Fig. 4.9 Main effect plots for D in cylindrical grinding of aluminium

Fig. 4.10 Surface and contour plots of D for aluminium: a at high level of radial infeed, b at low level of radial infeed

Fig. 4.11 Surface and contour plots of D for aluminium: a at high level of work-piece speed, b at low level of work-piece speed

68

4 Fractal Analysis in Cylindrical Grinding

Fig. 4.12 Surface and contour plots of D for aluminium: a at high level of longitudinal feed, b at low level of longitudinal feed

4.4 Closure Response surface models for three materials viz. mild steel, brass and aluminium are developed in cylindrical grinding. All the developed second order models are adequate at 95% confidence level. For mild steel, the longitudinal feed rate is the most significant factor affecting the fractal dimension whereas for brass materials, the work-piece speed, longitudinal feed rate and interaction between work-piece speed and longitudinal feed are the most significant factors. For brass materials, with increase in work-piece speed and longitudinal feed, the fractal dimension increases i.e. the surface gets smoother while the radial infeed is kept constant at middle level. For aluminium materials, it is seen that the work-piece speed, longitudinal feed rate and radial infeed are the significant factors affecting the fractal dimension.

Chapter 5

Fractal Analysis in EDM

Abstract In this chapter fractal dimension modeling in electrical discharge machining is discussed. Machining operations are carried out for different combinations of pulse current, pulse-on time and pulse-off time on mild steel, brass and tungsten carbide materials. The generated machined surfaces are measured to calculate fractal dimension. The experimental results are then analyzed to model fractal dimension using response surface methodology. From the response surface models, it is seen that the effect of the cutting parameters on fractal dimension is different for different materials. For tungsten carbide and brass, both pulse current and pulse on time play a significant role in determining the fractal dimension while for mild steel it is only the pulse current that plays the significant role. A comparison of the response surface models for fractal dimension in different materials reveals the fact that these models are material specific. Keywords Fractal dimension (D)  EDM  RSM  Mild steel  Brass  Tungsten carbide

5.1 Introduction Electrical discharge machining (EDM) is a widespread machining technique used for all types of conductive materials including metals, metallic alloys, graphite, composites and ceramic materials. It is a non-conventional machining process used for machining of difficult-to-machine materials and shapes with high degree of accuracy (El-Hofy 2005). It is based on removing material from a part by means of a series of repeated electrical discharges created by electric pulse generated at short intervals between two electrodes; a tool electrode and a work-piece electrode. The electrodes are separated by a dielectric fluid that makes it possible to flush eroded particles from the gap between the electrodes. The electric spark P. Sahoo et al., Fractal Analysis in Machining, SpringerBriefs in Computational Mechanics, DOI: 10.1007/978-3-642-17922-8_5, Ó Prasanta Sahoo 2011

69

70

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Table 5.1 Variable levels used in the experimentation Levels Current (I, amp) Pulse on time (ti, ls)

Pulse off time (to, ls)

-1 0 1

50 75 100

3.125 6.250 9.375

50 100 150

Table 5.2 Design matrix of the FCC design (coded values and actual value of the factors) Std. order Run order Coded value 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

5 2 8 12 18 16 14 1 9 11 6 13 19 3 7 20 10 15 17 4

Current (I)

Pulse on time (ti)

Pulse off time (t0)

-1 1 -1 1 -1 1 -1 1 -1 1 0 0 0 0 0 0 0 0 0 0

-1 -1 1 1 -1 -1 1 1 0 0 -1 1 0 0 0 0 0 0 0 0

-1 -1 -1 -1 1 1 1 1 0 0 0 0 -1 1 0 0 0 0 0 0

raises the surface temperature of both the tool and work-piece to a point that is in excess of the melting or even boiling points of the substances. Thus material is mainly removed in the liquid and vapor phases, and the surface generated consists of debris either been melted or vaporized during machining. Since the tool does not physically contact the work-piece, no mechanical stress is exerted on the workpiece and the characteristics of the EDM process are thus not governed by the mechanical properties of the work-piece material. Instead, the thermal and electrical properties play a significant role in the process performance. The EDM performance is characterized by three parameters, viz., material removal rate (MRR), electrode wear rate (EWR) and surface roughness. In this study, surface roughness is modeled based on fractal dimension for three different materials viz. mild steel, brass and tungsten carbide materials in EDM using response surface methodology (RSM). The experimental details and the results for different materials are presented below.

5.2 Experimental Details

71

Table 5.3 Specification of the equipment used in the experimentation Particulars Specification Trade name Type of construction Worktable Fixed work tank Table longitudinal movement Table cross movement Maximum dielectric level over table Maximum work piece height Maximum work piece weight Servo head Servo system Quill travel Electrode platen size Accuracy of quill movement Dielectric system Filtration flushing Flushing Flushing pressure Generator Models Working current Pulse on time setting Pulse off time setting Power source connection

TOOL CRAFT A 25 ‘C’ type 300 mm 9 200 mm 465 mm 9 270 mm 9 200 mm 100 mm 175 mm 140 mm 90 mm 45 kg Stepped drive 150 mm 100 mm sq 0.01 mm over 200 mm better than 10 l side, 1.23 l/min (max) 15 kPa A 25 25 A maximum through current selector 2–2,000 ls 2–2,000 ls 400/440 V, 50 Hz, 3-ph supply

5.2 Experimental Details 5.2.1 Design of Experiments There are a large number of factors that can be considered for machining of a particular material in EDM. It is very difficult to conduct the experiment with considering all the process variables. However, the review of literature shows that the following three machining parameters are the most widespread among the researchers and machinists to control the EDM process: pulse current (I, amp), pulse-on time (ti, ls) and pulse-off time (to, ls). In the present study these are selected as design factors while other parameters have been assumed to be constant over the experimental domain. A face-centered central composite (FCC) design is used with three levels of each of the three design factors. Considering three factors and six replicates at the center point, the present design contains 20 experiments, which were performed in a random order. The upper and lower limits of a factor are coded as +1 and -1 respectively, the coded value being calculated using Eq. 1.8. The process variables with their values on different levels are listed in Table 5.1. The selection of the values of the variables is limited by the capacity of

72

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Table 5.4 Composition and electrical/thermal properties of work-piece materials Work Material Composition (%Wt) Electrical and thermal property Tungsten carbide

Mild Steel (AISI 1040)

Brass (UNS C34000)

Electrical resistivity: 6 9 10-5 ohm-cm Thermal conductivity: 84 W/m-K Melting point: 2850°C 0.42%C, 0.48%Mn, 0.17%Si, 0.02%P, Electrical resistivity:1.7 9 10-5 ohm-cm 0.018%S, 0.1%Cu, 0.09%Ni, 0.07%Cr Thermal conductivity: 52 W/m-K and balance Fe Melting point:1515°C 0.095%Fe, 0.9%Pb, 34%Zn and Electrical resistivity:6.6 9 10-6 ohm-cm balance Cu Thermal conductivity:115°W/m-K Melting point: 900°C

94%WC–6%Co

the machine used in the experimentation as well as the recommended specifications for different workpiece–tool material combinations. Table 5.2 shows the experimental matrix of the FCC design employed in the present study.

5.2.2 Machine Used The machine used for carrying out the machining operations is a ‘Toolcraft A25’ EDM machine having the stepped drive servo system and filtration flushing capability. It is capable of generating maximum pulse current of 25 A, pulse on time of 2,000 ls and pulse off time of 2,000 ls. The specification of the machine is presented in Table 5.3.

5.2.3 Work-Piece Materials The present study was carried out with three different work-piece materials, viz., tungsten carbide, AISI 1040 mild steel and medium leaded brass UNS C34000. The chemical composition and electrical/thermal properties of the work-piece materials are shown in Table 5.4. All the specimens were in the form of 20 mm 9 20 mm 9 4 mm blocks.

5.2.4 Tool Electrode Used Electrolytic copper having 99.9% copper in composition and density 8,904 kg/m3 was used as tool electrode since it worked better in combination with the workpiece materials considered in the present study. The tool electrode was in the form of cylinder of diameter 15.9 mm and 50 mm in length mounted axially in line with work-piece. The tool electrode was given negative polarity where as work-piece is

5.2 Experimental Details

73

Table 5.5 Electrode material properties Particulars

Specifications

Material Composition Density Melting point Conductivity Tensile strength

Electrolytic copper 99.09% copper 8 904 kg/mm3 1083 C° 101.41% IACS 23.47 kg/mm2

Table 5.6 Experimental results Std. order Run order

D for WC

D for MS

D for Brass

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1.383 1.350 1.356 1.250 1.410 1.313 1.356 1.216 1.390 1.270 1.323 1.250 1.320 1.386 1.263 1.313 1.310 1.343 1.310 1.293

1.413 1.310 1.330 1.276 1.426 1.306 1.426 1.283 1.333 1.286 1.346 1.343 1.346 1.316 1.356 1.306 1.363 1.306 1.330 1.316

1.440 1.406 1.430 1.400 1.453 1.423 1.420 1.386 1.413 1.410 1.440 1.420 1.430 1.406 1.420 1.423 1.410 1.400 1.416 1.416

5 2 8 12 18 16 14 1 9 11 6 13 19 3 7 20 10 15 17 4

positive polarity (Puertas et al. 2005). The properties of the tool electrode have been given in Table 5.5. Kerosene was used as dielectric because of its high flash point, good dielectric strength, transparent characteristics and low viscosity and specific gravity.

5.3 Results and Discussion As mentioned earlier, according to FCC design of experiments, machining operations are carried out to generate machined surfaces (EDMed). The generated machined surfaces are measured with Talysurf and further processed to calculate

74

5

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Table 5.7 ANOVA for second order model for D in EDM of mild steel Source DF Seq SS Adj SS Adj MS F

F0.05

P

4.66

3.02

0.012

Table 5.8 ANOVA for machining parameters for D in EDM of mild steel Source DF Seq SS Adj SS Adj MS F

F0.05

P

3.81 3.81 3.81

0 0.139 0.556

Regression Residual Error Total

I ti t0 Error Total

2 2 2 13 19

9 10 19

0.029628 0.007057 0.036685

0.021962 0.004156 0.000912 0.009656 0.036685

0.029628 0.007057

0.022226 0.003419 0.000912 0.009656

0.003292 0.000706

0.011113 0.001709 0.000456 0.000743

14.96 2.3 0.61

fractal dimension. Experimental results of fractal dimension for tungsten carbide, mild steel and brass materials are presented in Table 5.6. The influences of the machining parameters (I, ti and t0) on the profile fractal dimension D have been assessed for three different materials using RSM. The second order model was postulated in obtaining the relationship between the fractal dimension and the machining variables. The analysis of variance (ANOVA) was used to check the adequacy of the second order model. The results for the three different materials are presented one by one.

5.3.1 RSM for Mild Steel The second order response surface equation for fractal in EDM of mild steel is obtained in terms of coded values of design factors as: D ¼ 1:33  0:0467 I  0:0143 ti þ 0:0083 to þ 0:0033 Iti  0:0133 Ito þ 0:0117 ti to  0:0127 I 2 þ 0:0223 ti2 þ 0:0089 to2

ð5:1Þ

The developed model is checked for adequacy by ANOVA and F-test. Table 5.7 presents the ANOVA table for the second order model proposed for D given in Eq. 5.1. The developed model is significant at 95% confidence level as the P-value is less than 0.05. Also the model is adequate at 95% confidence level to represent the relationship between the machining response and the considered machining parameters as the calculated value of the F-ratio is more than the standard value of the F-ratio for D. Table 5.8 represents the ANOVA table for individual machining parameters where it can be seen that only pulse current is the significant parameter at 95% confidence level. Figure 5.1 shows the main effects plot for the fractal dimension. From this figure also, it is seen that pulse current has

5.3 Results and Discussion

75

Fig. 5.1 Main effect plot of fractal dimension for mild steel

Fig. 5.2 Surface and contour plot of fractal dimension for mild steel: (a) at high level of pulse on time, (b) at low level of pulse on time

Fig. 5.3 Surface and contour plot of fractal dimension for mild steel: (a) at high level of current, (b) at low level of current

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Fig. 5.4 Surface and contour plot of fractal dimension for mild steel: (a) at high level of pulse off time, (b) at low level of pulse off time

Table 5.9 ANOVA for second order model for D in EDM of brass Source DF Seq SS Adj SS Adj MS Regression Residual Error Total

9 10 19

0.003635 0.000982 0.004616

0.003635 0.000982

0.000404 0.000098

F

F0.05

P

4.11

3.02

0.019

significant effect on fractal dimension while pulse on time and pulse off time have no effect on fractal dimension of the surface topography generated in EDM of mild steel. The estimated three-dimensional surface as well as contour plots for fractal dimension are presented in Figs. 5.2, 5.3, and 5.4. To draw these surface plots, fractal dimension is plotted as functions of two independent machining parameters while the third machining parameter is held constant. All these figures clearly depict the variation of fractal dimension with controlling variables within the experimental regime.

5.3.2 RSM for Brass The second order response surface equation for the fractal dimension of brass surfaces machined in EDM is also obtained in terms of coded values of design factors as: D ¼ 1:42  0:013 I  0:0107 ti  0:0017 to  0:0067 ti to  0:0068 I 2 þ 0:0115 ti2  0:0002 to2

ð5:2Þ

The developed model is checked for adequacy by ANOVA and F-test. Table 5.9 presents the ANOVA table for the second order model proposed for D given in Eq. 5.2. It is seen that the developed model is significant at 95%

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77

Table 5.10 ANOVA for machining parameters for D in EDM of brass Source DF Seq SS Adj SS Adj MS F I ti t0 Error Total

2 2 2 13 19

0.001693 0.001557 2.83E - 05 0.001338 0.004616

0.00182 0.001502 2.83E - 05 0.001338

0.00091 0.000751 1.42E - 05 0.000103

8.84 7.3 0.14

F0.05

P

3.81 3.81 3.81

0.004 0.008 0.873

Fig. 5.5 Main effect plot of fractal dimension for brass

confidence level. Also the calculated value of the F-ratio is more than the standard value of the F-ratio for D which implies the model is adequate at 95% confidence level to represent the relationship between the machining response and the considered machining parameters of the EDM process on brass. Table 5.10 represents the ANOVA table for individual machining parameters where it can be seen that pulse current and pulse on time are the significant factors affecting fractal dimension. Figure 5.5 depicts the main effects plot for the fractal dimension and the design factors considered. From this figure also, it is seen that pulse current and pulse on time have the significant effect on fractal dimension. Figures 5.6, 5.7, and 5.8 shows the estimated three-dimensional surface as well as contour plots for fractal dimension. To draw these surface plots, fractal dimension is plotted as functions of two independent machining parameters while the third machining parameter is held constant. All these figures clearly show the variation of fractal dimension with controlling variables within the experimental regime.

5.3.3 RSM for Tungsten Carbide The second order response surface equation has been fitted using Minitab software for the response variable D. The equation can be given in terms of the coded values of the independent variables as:

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Fig. 5.6 Surface and contour plot of fractal dimension for brass: (a) at high level of pulse on time, (b) at low level of pulse on time

Fig. 5.7 Surface and contour plot of fractal dimension for brass: (a) at high level of current, (b) at low level of current

Fig. 5.8 Surface and contour plot of fractal dimension for brass: (a) at high level of pulse off time, (b) at low level of pulse off time

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79

Table 5.11 ANOVA for second order model for D in EDM of tungsten carbide Source DF Seq SS Adj SS Adj MS F F0.05 Regression Residual error Total

9 10 19

0.046159 0.006621 0.05278

0.046159 0.006621

0.005129 0.000662

7.75

3.02

Table 5.12 ANOVA for machining parameters for D in EDM of tungsten carbide Source DF Seq SS Adj SS Adj MS F F0.05 I ti t0 Error Total

2 2 2 13 19

0.026341 0.01304 0.003839 0.00956 0.05278

0.025185 0.014658 0.003839 0.00956

0.012592 0.007329 0.00192 0.000735

17.12 9.97 2.61

3.81 3.81 3.81

P 0.002

P 0 0.002 0.111

Fig. 5.9 Main effect plot of fractal dimension for tungsten carbide

D ¼ 1:31  0:0497 I  0:035ti þ 0:0023 to  0:0146Iti  0:0121 Ito  0:0029 ti to þ 0:0138 I 2  0:0296 ti2 þ 0:0371 to2

ð5:3Þ

The analysis of variance (ANOVA) and the F-ratio test have been performed to check the adequacy of the developed model. Table 5.11 presents the ANOVA table for the second order model proposed for D given in Eq. 5.3. It is seen that the developed model is significant at 95% confidence level. Also the calculated value of the F-ratio is more than the standard value of the F-ratio for D. It means the model is adequate at 95% confidence level to represent the relationship between the machining response and the considered machining parameters of the EDM process. Table 5.12 represents the ANOVA table for individual machining parameters where it can be seen that pulse current and pulse on time are significant at 95% confidence level. Figure 5.9 shows the main effects plot for the fractal dimension and the design factors considered in the present study. From this figure also, it is seen that both pulse current and pulse on time have the significant effect on fractal dimension while the effect of pulse off time is insignificant.

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Fig. 5.10 Surface and contour plot of fractal dimension for tungsten carbide: (a) at high level of pulse on time, (b) at low level of pulse on time

Fig. 5.11 Surface and contour plot of fractal dimension for tungsten carbide: (a) at high level of current, (b) at low level of current

Fig. 5.12 Surface and contour plot of fractal dimension for tungsten carbide: (a) at high level of pulse off time, (b) at low level of pulse off time

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81

Figures 5.10, 5.11, and 5.12 shows the estimated three-dimensional surface as well as contour plots for fractal dimension. To draw these surface plots, fractal dimension is plotted as functions of two independent machining parameters while the third machining parameter is held constant. These figures clearly depict the variation of fractal dimension with controlling variables within the experimental regime.

5.4 Closure Response surface models are developed for fractal dimension in EDM of three different materials. A comparison of the response surface models reveals the fact that these models are material specific or in other words, the tool–workpiece material combination plays a vital role in fractal dimension modeling. Also the effect of the cutting parameters on fractal dimension is different for different materials as evidenced from Table 5.8, Table 5.10 and Table 5.12. For tungsten carbide and brass, both pulse current and pulse on time play a significant role in determining the fractal dimension while for mild steel it is only the pulse current that plays the significant role. Accordingly, optimum machining parameter combinations for fractal dimension depend greatly on the workpiece material within the experimental domain. However, it can be concluded that it is possible to select a combination of pulse current, pulse on time and pulse off time for achieving the surface topography with desired fractal dimension within the constraints of the available machine.

References El-Hofy HAG (2005) Advanced machining processes. McGraw-Hill, New York Puertas I, Luis CJ, Villa G (2005) Spacing roughness parameters study on the EDM of silicon carbide. J Mater Process Technol 164–165:1590–1596