bok%3A978-3-642-17922-8 (1)

SpringerBriefs in Applied Sciences and Technology

Computational Mechanics

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Prasanta Sahoo • Tapan BarmanJoão Paulo Davim

Fractal Analysisin Machining

123

Prasanta SahooDepartment of Mechanical EngineeringJadavpur UniversityKolkata 700032Indiae-mail: [email protected]

Tapan BarmanDepartment of Mechanical EngineeringJadavpur UniversityKolkata 700032Indiae-mail: [email protected]

João Paulo DavimDepartment of Mechanical EngineeringUniversity of AveiroCampus Universitário de Santiago3810-193 AveiroPortugale-mail: [email protected]

ISSN 2191-5342 e-ISSN 2191-5350ISBN 978-3-642-17921-1 e-ISBN 978-3-642-17922-8DOI 10.1007/978-3-642-17922-8Springer Heidelberg Dordrecht London New York

� Prasanta Sahoo 2011This work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcast-ing, reproduction on microfilm or in any other way, and storage in data banks. Duplication of thispublication or parts thereof is permitted only under the provisions of the German Copyright Law ofSeptember 9, 1965, in its current version, and permission for use must always be obtained fromSpringer. Violations are liable to prosecution under the German Copyright Law.The use of general descriptive names, registered names, trademarks, etc. in this publication does notimply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general use.

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Preface

The present book deals with fractal analysis of surface roughness in differentmachining processes. Surface roughness is an important attribute of any machinecomponent. Conventionally several statistical roughness parameters are used fordescribing surface roughness. But surface topography is a non-stationary randomprocess for which the variance of the height distribution of roughness features isrelated to the length of the sample. Consequently, instruments with different res-olutions and scan lengths yield different values of these statistical parameters forthe same surface. A logical solution to this problem is to use scale-invariantparameters to characterize rough surfaces. In this context, to describe surfaceroughness, the concept of fractals is considered. Fractals retain all the structuralinformation and are characterized by single descriptor, the fractal dimension,D. Fractal dimension is intrinsic property of the surface and independent of thefilter processing of measuring instrument as well as the sampling length scale.

Four machining processes viz. CNC end milling, CNC turning, electrical dis-charge machining and cylindrical grinding are considered for three differentmaterials. The generated machined surfaces are measured to find out fractaldimension (D) of the surfaces. The experimental results are further analyzed withresponse surface methodology (RSM) to consider the effects of process parameterson fractal dimension. Also the effect of work-piece material variation on fractaldimension of machined surfaces is considered. It is believed that the present bookwill prove to add significant contribution to the existing literature from the point ofview of both industrial importance and academic interest.

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Contents

1 Fundamental Consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Surface Metrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Fractal Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Fractal Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.2 Fractal Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.3 Self-Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.4 Self-Affinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.5 Fractal Description of Roughness . . . . . . . . . . . . . . . . . 91.3.6 Fractal Dimension Calculation . . . . . . . . . . . . . . . . . . . 111.3.7 Fractal Dimension Measurement in the Present Study . . . 12

1.4 Review of Roughness Study in Machining . . . . . . . . . . . . . . . . 121.5 Design of Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.5.1 Full Factorial Design . . . . . . . . . . . . . . . . . . . . . . . . . . 191.5.2 Central Composite Design . . . . . . . . . . . . . . . . . . . . . . 19

1.6 Response Surface Methodology. . . . . . . . . . . . . . . . . . . . . . . . 211.7 Closure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2 Fractal Analysis in CNC End Milling . . . . . . . . . . . . . . . . . . . . . . 292.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2 Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.1 Design of Experiments . . . . . . . . . . . . . . . . . . . . . . . . 302.2.2 Machine Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2.3 Cutting Tool Used. . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2.4 Work-Piece Materials . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3 Results and Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3.1 RSM for Mild Steel . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3.2 RSM for Brass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.3.3 RSM for Aluminium . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.4 Closure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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3 Fractal Analysis in CNC Turning . . . . . . . . . . . . . . . . . . . . . . . . . 453.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2 Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2.1 Design of Experiments . . . . . . . . . . . . . . . . . . . . . . . . 473.2.2 Machine Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2.3 Cutting Tool Used. . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2.4 Work-Piece Materials . . . . . . . . . . . . . . . . . . . . . . . . . 47


3.4 Closure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4 Fractal Analysis in Cylindrical Grinding . . . . . . . . . . . . . . . . . . . 574.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.2 Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2.1 Design of Experiments . . . . . . . . . . . . . . . . . . . . . . . . 584.2.2 Machine Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2.3 Work-Piece Materials . . . . . . . . . . . . . . . . . . . . . . . . . 58


4.4 Closure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5 Fractal Analysis in EDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.2 Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2.1 Design of Experiments . . . . . . . . . . . . . . . . . . . . . . . . 715.2.2 Machine Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.2.3 Work-Piece Materials . . . . . . . . . . . . . . . . . . . . . . . . . 725.2.4 Tool Electrode Used . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.3 Results and Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.3.1 RSM for Mild Steel . . . . . . . . . . . . . . . . . . . . . . . . . . 745.3.2 RSM for Brass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.3.3 RSM for Tungsten Carbide . . . . . . . . . . . . . . . . . . . . . 77

5.4 Closure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

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Chapter 1Fundamental Consideration

Abstract The importance and usefulness of fractal dimension in describing sur-face roughness over the conventional roughness parameters are presented in thischapter. The fundamental of fractal dimension and the methodology for evaluationof fractal dimension are also discussed. Literature survey is carried out for fourdifferent types of machining processes and shows that there is scarcity of litera-tures which deal with fractal description of surface roughness. Fundamentals ofdesign of experiments and response surface methodology are also discussed.

1.1 Introduction

Surfaces are irregular though they may look like very smooth. When the surfacesare magnified, the irregularities become prominent. This is true for the machiningsurfaces as well. In a material removal process such as machining, unwantedmaterial is removed and altered surface topography is obtained. The surfacegenerated consists of inherent irregularities left by the cutting tool, which arecommonly defined as surface roughness. Such a surface is composed of a largenumber of length scales of superimposed roughness that are generally character-ized by the standard deviation of surface peaks. Three statistical characteristics aregenerally used to describe the structure of machined surface topography: texture,waviness and roughness. The texture determines the anisotropic property of thesurface. The waviness reflects the reference profile (or surface). The surfaceroughness is formed by the micro deformation during the machining process.

Surface roughness plays an important role. It has large impact on themechanical 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

article, Benardos and Vosniakos (2003) have presented a fishbone diagram withparameters that affect surface roughness. As a case study, they have consideredtwo machining operations—turning and milling. They broadly classified the fac-tors as machining parameters, cutting tool properties, work-piece properties andcutting phenomena. Machining parameters may include process kinematics, depthof cut, cutting speed, feed rate, etc. Cutting tool properties may include toolmaterial, nose radius, tool shape, etc. Work-piece properties may include work-piece hardness, work-piece size etc. and cutting phenomena includes vibration,cutting force variations, chip formation, etc. It is obvious that for other machiningoperations also, there are several factors that affect surface roughness. Manyresearchers have attempted to model surface roughness but the developed modelsare far from complete as it is not possible to consider all the controlling factors in aparticular study. So, researchers always pay attention to model surface roughnessin a better way so that surface roughness modeling can be done more accurately.

Surface roughness is generally expressed by three types of conventionalroughness parameters viz. amplitude parameters, spacing parameters and hybridparameters. Amplitude parameters are the measure of vertical characteristics ofsurface deviation. Center line average roughness (Ra), root mean square roughness(Rq), etc. are the examples of these types of parameters. Spacing parameters aremeasures of the horizontal characteristics of surface deviations. Examples of suchparameters are mean line peak spacing (Rsm), high spot count, etc. On the otherhand, hybrid parameters are the combination of both vertical and horizontalcharacteristics of the surface deviations e.g. root mean square slope of the profile,root mean square wavelength, peak area, valley area, etc. Most commonly usedroughness 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 bea random process for which statistical parameters such as the variances of theheight, the slope and curvature are used for characterization (Nayak 1971).However, it has been found that the variances of slope and curvature dependstrongly on the resolution of the roughness-measuring instrument or any otherform 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 non-stationary random process for which the variance of the height distribution isrelated to the length of the sample (Sayles and Thomas 1978). Consequently,instruments with different resolutions and scan lengths yield different values ofthese statistical parameters for the same surface. The conventional methods ofcharacterization are therefore fraught with inconsistencies which give rise to theterm ‘parameter rash’ (Whitehouse 1982) commonly used in contemporary liter-ature. The underlying problem with the conventional methods is that althoughrough surfaces contain roughness at a large number of length scales, the charac-terization parameters depend only on a few particular length scales, such as theinstrument resolution or the sample length. A logical solution to this problem is touse scale-invariant parameters to characterize rough surfaces. In this context, todescribe surface roughness, the concept of fractals is applied. The concept is based

2 1 Fundamental Consideration

on the self-affinity and self-similarity of surfaces at different scales. Fractals retainall the structural information and are characterized by single descriptor, the fractaldimension, D. Fractal dimension is intrinsic property of the surface and inde-pendent of the filter processing. Roughness measurements on a variety of surfacesshow that the power spectra of the surface profiles follow power laws. This sug-gests that when a surface is magnified appropriately, the magnified image looksvery similar to the original surface. This property can be mathematically describedby the concepts of self-similarity and self-affinity. The fractal dimension, whichforms the essence of fractal geometry, is both scale-invariant and is closely linkedto the concepts of self-similarity and self-affinity (Mandelbrot 1982). It is thereforeessential to use fractal dimension to characterize rough surfaces and provide thegeometric structure at all length scales (Bigerelle et al. 2005). The possibleapplication of fractal geometry to tribology was explored (Ling 1990). Theinfluence of processing techniques on the fractal or non-fractal structure was alsoexamined (Majumdar and Bhushan 1990).

In a material removal process, mechanical intervention happens over lengthscales which extend from atomic dimensions to centimeters. The machine vibra-tion, clearances and tolerances affect the outcome of the process at the largest oflength scales (above 10-3 m). The tool form, feed rate, tool radius in the case ofsingle point cutting (Venkatesh et al. 1998) and grit size in multiple point cutting(Venkatesh et al. 1999), affect the process outcome at the intermediate lengthscales (10-6–10-3 m). The roughness of the tool or details of the grit surfacesinfluence 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 dischargemachining, milling, cutting or grinding and worn surfaces (Brown and Savary1991; Tricot et al. 1994; Hasegawa et al. 1996; He and Zhu 1997; Ge and Chen1999; Zhang et al. 2001; Jiang et al. 2001; Zhu et al. 2003; Jahn and Truckenbrodt2004; Kang et al. 2005; Han et al. 2005) have fractal structures, and fractalparameters can reflect the intrinsic properties of surfaces to overcome the disad-vantages of conventional roughness parameters. Thus, to characterize the rough-ness of machined surfaces in different machining processes fractal dimension isused 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 1.1shows a display of surface texture with unidirectional lay. Roughness is producedby fluctuations of short wavelengths characterized by asperities (local maxima)and valleys (local minima) of varying amplitudes and spacing. This occurs due tothe mechanism of the material removal process. Waviness is the surface irregu-larities of longer wavelengths and may result from such factors as machine or workpiece deflections, vibration, chatter, heat treatment or warping strains. Lay is the

1.1 Introduction 3

principal direction of the predominant surface pattern, usually determined by theproduction process. Flaws are unexpected and unintentional interruptions in thetexture. Apart from these, the surface may contain large deviations from nominalshape of very large wavelength, which is known as error of form. These are notconsidered as part of surface texture.

Any engineering surface composes of a vast number of peaks and valleys and itis not possible to measure the height and location of each of the peaks. So mea-surement of a surface is carried out on a sampling length where it is assumed thatthe surface outside and inside the sampling length is statistically similar. In orderto determine the numerical assessment of a sample’s surface texture, three char-acteristic lengths are associated with the profile (ISO 4287, 1997) viz. samplinglength, evaluation or assessment or cut off length and traverse length. The sam-pling length is the length over which the parameter to be measured will havestatistical significance. Cut off length is the length of the surface over which the

Fig. 1.1 Display of surfacetexture


measurement is made. This length may include several sampling lengths—typically five times. The measurement is the integration of the individual samplinglengths. The total length of the surface traversed by the stylus in making a mea-surement is called the traverse length. It will normally be greater than the eval-uation length, due to the necessity of allowing run-up and over-travel at each endof the evaluation length to ensure that any mechanical and electrical transients areexcluded from the measurement.

There are several methods to study the surface topography which are developedover the years. The most common method of studying surface texture is the surfaceprofilometer (Fig. 1.2). In this method, a fine, very lightly loaded, stylus is traveledsmoothly at a constant speed across the surface under examination. The transducerproduces an electrical signal, proportional to displacement of the stylus, which isamplified and fed to a chart recorder that provides a magnified view of the originalprofile. But this graphical representation differs from the actual surface profilebecause of difference in magnifications employed in vertical and horizontaldirections. Surface slopes appear very steep on profilometric record though theyare rarely steeper than 10� in actual cases. The shape of the stylus also plays a vitalrole in incorporating error in measurement. The finite tip radius (typically 1–2.5microns for a diamond stylus) and the included angle (of about 60� for pyramidalor conical shape) results in preventing the stylus from penetrating fully into deepand 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 avery delicate surface because of the load applied on it. In such cases non-contacting optical profilometer having optical heads replacing stylus may be used.Reflection of infrared radiation from the surface is recorded by arrays of photo-diodes and analysis of the same in a microprocessor result in the determination ofthe surface topography. Vertical resolution of the order of 0.1 nm is achievable

Motor and gearbox

converter

Data logger

Transducer

Skid

Specimen

Stylus

Amplifier

A-DrecorderChart

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

1.2 Surface Metrology 5

though maximum height of measurement is limited to few microns. This method isclearly 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 andcubes using integer dimensions of 0, 1, 2, and 3, respectively. A measure of theobject such as the length of a line, the area of a surface and the volume of a cubeare associated with each dimension. These measures are invariant with respect tothe unit of measurement. It means that the length of a line remains independent ofwhether a centimeter or a micrometer scale is used. However, a multitude ofobjects found in nature appear disordered and irregular for which the measures oflength, area and volume are scale-dependent. This suggests that the dimensions ofsuch objects cannot be integers. A generalized concept of a dimension and theorigins of fractal geometry are now discussed.

Mandelbrot (1967) founded fractal geometry when he showed that fordecreasing the unit of measurement, the length of a natural coastline does notconverge but, instead, increase monotonically. On plotting the length L as afunction of the unit of measurement [ on a log–log plot, he found a simple relationof the form L * [1-D. Mandelbrot finally made an interesting conclusion that thereal number D associated with every coastline is the dimension of the coastline.This study marked the origins of fractal geometry, which has now found numerousapplications in characterizing and describing disordered phenomena in science andengineering.

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 smallsquares 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 ofthe objects. These measures of length and area have a unique property that they areindependent of the unit of measurement [ and in the limit [ ? 0 these measures


remain finite and non-zero. This concept of Euclidean dimension thus can begeneralized 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 themeasure M independent of the unit of measurement [ in the limit of [ ? 0, then Dis the dimension of an object.

Contrary to common understanding of dimension, this generalization allows thedimension of an object to take non-integer values. If, in this argument, it isassumed that an object is broken into N equal parts then Eq. 1.3 can be written asM = N[D. Since the measure is invariant with the unit of measurement, one canwrite N * [-D. Now if the length of an object is evaluated, then the length wouldvary as L = N[1 * [1-D, as was observed for the lengths of the coastlines. It canbe 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 issueof self-similarity. Let us consider a one-dimensional line of unit length and break itup into N equal segments. Each segment of the line, of size 1/m, is similar to thewhole 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 numberof 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 mag-nification of m to be an exact replica of the whole square. However, the number ofsmall squares in the whole is N * m2. In general, for an object of dimension D, itfollows that

N�mD ð1:4Þ

Thus the dimension of the object can be written as

D ¼ log Nlog 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-integerdimension looks like, one can follow the recursive construction in Fig. 1.3, whichyields the Koch curve of dimension 1.26.

In this construction the first step is to break a straight line into three parts andreplace the middle portion by two segments of equal lengths. In the subsequentstages each straight segment is broken into three parts and the middle portion ofeach segment is replaced by two parts. If this recursion is continued infinite timesthen the Koch curve is obtained. This curve has some unique mathematicalproperties. Firstly, the curve is continuous but it is not differentiable anywhere.

1.3 Fractal Characterization 7

The non-differentiability arises because of the fact that if the curve is repeatedlymagnified, more and more details of the curve keep appearing. This means thattangent cannot be drawn at any point and therefore the curve cannot be differ-entiated. Secondly, the curve is exactly self-similar. This is because if a smallportion of the curve is appropriately magnified, it will be an exact replica of thewhole Koch curve. Thirdly, the dimension of the curve remains constant at allscales, although the curve contains roughness at a large number of scales. Thisscale-invariance of the dimension is an important property, which is utilized tocharacterize rough surfaces. The coastline of an island is an example of a self-similar object found in nature. Although these objects are not exactly self-similar,they are statistically self-similar. Statistical self-similarity means that the proba-bility distribution of a small part of an object will be congruent with the probabilitydistribution of the whole object if the small part is magnified appropriately.However, not all fractal objects are self-similar. This leads to the more generalconcept of self-affinity.

1.3.4 Self–Affinity

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

Fig. 1.3 Formation of Kochcurve


scaling in different directions. Thus these are not self-similar but self-affine. Thedimension of self-affine fractals cannot be obtained from Eq. 1.5, which is basedon the self-similarity of an object. Mandelbrot showed that the lengths of self-affine fractal curves do not follow the relation L * [1-D for all values of [ andtherefore the dimension of self-affine curves cannot be obtained by measuring theirlengths. Instead, the dimension of self-affine functions can be obtained from theirpower 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 theheight, the slope and the curvature. But, it has been observed that surfacetopography is a non-stationary random process. It means the variance of the heightdistribution is related to the sampling length and hence is not unique for aparticular surface. Rough surfaces are also known to exhibit the feature of geo-metric self-similarity and self-affinity, by which similar appearances of the surfaceare seen under the various degrees of magnification as quantitatively shown inFig. 1.4. Since increasing amounts of detail in the roughness are observed atdecreasing length scale, the concepts of slope and curvature, which inherentlyassume the smoothness of the surface, cannot be defined. So the variances of slopeand curvature depend strongly on the resolution of the roughness-measuringinstrument or some other form of filter and are therefore not unique. In contem-porary literature such a large number of characterizations parameter occurs that theterm ‘parameters rash’ is aptly used. The use of instrument-dependent parametersshows different values for the same surface. Thus, it is necessary to characterizerough surfaces by intrinsic parameters, which are independent of all scales ofroughness. This suggests the use of fractal geometry in characterizing the surfaceroughness. The fractal dimension is an intrinsic property and should be used forsurface characterization. It is invariant with length scales and is closely linked tothe concept of geometric self-similarity.

The self-similarity or self-affinity of rough surfaces implies that as the unit ofmeasurement is continuously decreased, the surface area of the rough surface (atwo-dimensional measure) tends to infinity and the volume (a three-dimensional

X

Z

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


measure) tends to zero. Here, self-similarity implies the property of equal mag-nification in all directions while self-affinity refers to unequal scaling in differentdirections. Thus, the Hausdorff or fractal dimension, D ? 1, of rough surfaces is afraction between 2 and 3. The profile of a rough surface z(x), typically obtainedfrom stylus measurements, is assumed to be continuous even at the smallest scales.This assumption breaks down at atomic scale. But for engineering surfaces thecontinuum is assumed to exist down to the limit of a zero-length scale. Sincerepeated magnifications reveal the finer levels of detail, the tangent at any pointcannot be defined. Thus the surface profile is continuous everywhere but non-differentiable at all points. This mathematical property of continuity, non-differ-entiability and self-affinity (Berry and Lewis 1980) is satisfied by the modifiedWeierstrass–Mandelbrot (W–M) fractal function, which is thus used to charac-terize and simulate such profiles. The W–M function has a fractal dimension D,between 1 and 2, and is given by

zðxÞ ¼ GðD�1ÞXa

n¼n1

cos 2pcnx

cð2�DÞn 1\D\2; c[ 1 ð1:6Þ

where, G is a scaling constant. The parameter n1 corresponds to the low cut-offfrequency of the profile. Since surfaces are non-stationary random process thelowest cut-off frequency depends on the length L of the sample and is given bycn1 = 1/L.

The W–M function has the interesting mathematical property that the series forz(x) converges but that for dz/dx diverges. It implies that it is non-differentiable atall points. The power spectrum of this W–M function can be expressed by acontinuous function as

SðxÞ ¼ G2ðD�1Þ

2 ln c1

x5�2Dð1:7Þ

When this equation is compared with the power spectrum of a surface, thedimension 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 withrespect to all frequencies of roughness and determines the position of spectrumalong the power axis. In this characterization method both G and D are independentof the roughness scales of the surface and hence intrinsic properties. The constantsof the W–M function, G, D, and n1 form a complete and fundamental set of scale-independent parameters to characterize a rough surface. The physical significanceof D is the extent of space occupied by the rough surface, i.e., larger D valuescorrespond to denser profile or smoother topography (Yan and Komvopoulos 1998;Sahoo and Ghosh 2007).


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). Fractalcalculation is generally involved with computer assisted image analysis oftopography images of a surface obtained in analog or digital signals using profi-lometer or microscopy, etc. An effective method to convert these signals into therequired data for calculating fractal dimensions must therefore be sought. Profileinstruments can be used to obtain data, which are then directly used to calculatefractal dimension. The methods for calculating profile fractal dimension mainlyinclude the yardstick, the box counting, the variation, the structure function and thepower spectrum method (Sahoo 2005).

The yardstick method employs the technique of ‘walking’ around a profilecurve in a step length, r. A point on the profile curve is chosen as a starting point ofdivider, whilst another point at a distance r from the starting point is taken as itsend point. Repetitively, find the point-pair of dividers in the same way until theprofile curve is entirely measured. Then, the summing up of the step lengthsenables the curve length to be determined. The repetition of this calculationprocess 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 theslope 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 incalculation error.

The principle of box counting method mainly involves an iteration operation toan 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 timesoperations, the number of sub-squares, which contain the discrete points of theprofile 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 allprofiles (self-affine or not), and of giving quickly an estimation of the dimension ofmathematical profiles. A well-known technique used to analyze surfaces consistsin performing ‘slices’ through the surfaces, which allows one to transform a three-dimensional problem to two-dimensional problem. In other words, a surface isreplaced by profiles, taken at different places, and the fractal dimension estimatedover profiles is then related to the three-dimensional fractal dimension by theclassical result: dimension of surface = 1 ? dimension of profiles. Such a tech-nique obviously decreases the problem size. Accurate results are hard to obtain forthe surface dimension and the variation method gives the best approximations.The variation method algorithm is based on the local oscillation of the profilefunction Z.

The power spectrum method involves the evaluation of the power of the profilefunction. The modified Weierstrass–Mandelbrot (W–M) function for a roughsurface is described by Eq. 1.6. The multi-scale nature of z(x) can be characterized


by its power spectrum, which gives the amplitude of the roughness at all lengthscales. 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 astraight line if S(x) is plotted as a function of x on a log–log graph. Using fastFourier transform (FFT), the power spectrum of profile can be calculated and thenbe plotted verses the frequency on a log–log scale. Thereafter, the fractaldimension, 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 curveas a time sequence z(x) with fractal character. The structure function s(s) ofsampling 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 valueof difference square, and s is the random choice value of data interval. Different sand the corresponding s(s) can be plotted verses the s on a log–log scale. Then, thefractal dimension D can be related to the slope m of a fitting line on log–log plotas: 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-typeprofilometer, Talysurf (Taylor Hobson, UK). The profilometer is set to a cut-offlength of 0.8 mm, Gaussian filter, traverse speed 1 mm/sec and 4 mm traverselength. Roughness measurements, in the transverse direction, on the work piecesare repeated four times and average of four measurements of surface roughnessparameter values is recorded. The measured profile is digitized and processedthrough the dedicated advanced surface finish analysis software Talyprofile. Thenfractal 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 researchershave tried to study surface roughness in machining. Though the present studyfocuses on fractal dimension in describing surface roughness, both conventionalroughness parameters and fractal dimension are reviewed here. Four machiningprocesses viz. turning, grinding, milling and electrical discharge machining arefocused 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 chipand tool on surface roughness in finish turning of C45 carbon steel. Yang andTarng (1998) have showed that feed rate is the most significant factor affectingsurface roughness in S45C steel turning. Also, with increasing feed rate, surface


roughness decreases. Abouelatta and Madl (2001) have found a correlationbetween surface roughness and cutting parameters and tool vibrations in turningconsidering three conventional roughness parameters viz. center line averageroughness value, maximum height of the profile and skewness. Davim (2001) haspresented a study of the influence of cutting parameters on the surface roughnessobtained in turning of free machining steel using Taguchi design and shown thatthe cutting velocity has a greater influence on the roughness followed by the feedrate. Lin et al. (2001) have shown that in turning feed rate is the critical parameterto affect the surface roughness, where increasing the feed rate will increase thesurface roughness. Suresh et al. (2002) have shown that surface roughnessdecreases with an increase in cutting speed, and increases as feed increases inturning of mild steel. Arbizu and Perez (2003) have developed models to deter-mine surface quality of parts obtained through turning processes and shown thatsurface roughness increases with increase in depth of cut and feed rate. Feng andWang (2003) have presented a nonlinear multiple regression analysis to predictsurface roughness in finish turning of Steel 8620 and Al 6061T materials. Dabnunet al. (2005) have concluded that feed rate is the main influencing factor on theroughness in turning of machinable glass–ceramic (Macor). Sahin and Motorcu(2005) have developed a surface roughness model for turning of mild steel withcoated carbide tools and shown that feed rate is the main affecting factor onsurface roughness. Surface roughness increases with increase in feed rate butdecreases with increase in cutting speed and depth of cut. Kirby et al. (2006) haveshown that the feed rate and tool nose radius have the highest effects on surfaceroughness in a turning operation of 6061-T6 aluminium alloy. Palanikumar et al.(2006) have focused on the parametric influence of machining parameters on thesurface roughness in turning of glass fiber reinforced polymer (GFRP) and shownthat roughness increases with increase in feed rate but roughness decreases withincrease in cutting speed. Singh and Rao (2007) have developed a model todetermine the effects of cutting conditions and tool geometry on surface roughnessin the finish hard turning of the bearing steel (AISI 52100) and concluded that feedrate is the dominant factor determining surface finish followed by nose radius andcutting velocity. Ramesh et al. (2008) have found in their study that feed rate is themain influencing factor on surface roughness in turning of titanium alloy.Palanikumar (2008) has found that the most significant machining parameter forsurface roughness is feed followed by cutting speed in machining glass fiberreinforced (GFRP). For modeling surface roughness in turning different method-ologies are used viz. RSM (Suresh et al. 2002; Dabnun et al. 2005; Sahin andMotorcu 2005; Palanikumar et al. 2006; Singh and Rao 2007; Ramesh et al. 2008;Palanikumar 2008; Gupta 2010), Taguchi analysis (Yang and Tarng 1998; Davim2001; Kirby et al. 2006; Nalbant et al. 2007; Palanikumar 2008;), artificial neuralnetwork (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 Tajdari2010). 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

1.4 Review of Roughness Study in Machining 13

most of the studies (Yang and Tarng 1998; Davim 2001; Lin et al. 2001; Sureshet al. 2002; Arbizu and Perez 2003; Jiao et al. 2004; Dabnun et al. 2005; Sahin andMotorcu 2005; Bagci and Isik 2006; Ramesh et al. 2008; Palanikumar 2008;Karayel 2009).

Grinding is the most commonly used manufacturing process in the industry andthis is a complex machining process with many interactive parameters and surfacequality produced is influenced by various parameters. Several researchers havetried to model surface roughness in grinding and few of the recent literatures arereviewed here. Zhang et al. (2001) have developed the relationships between thefractal dimension and conventional roughness parameters (Ra or Rq or Rsm ofsurface roughness) of different ground surfaces and justified the usefulness offractal theory. They concluded that fractal dimension D is relative to verticalparameters and transverse parameters of surface topography. Zhou and Xi (2002)have developed a model for predicting surface roughness in grinding taking intoconsideration the random distribution of the grain protrusion heights. Maksoudet al. (2003) have used artificial neural network to achieve desired surfaceroughness under grinding wheel surface topography variations. Hassui and Diniz(2003) have developed a relation between the process vibration signals androughness in a plunge cylindrical grinding operation of AISI 52100 quenched andtempered steel. Hecker and Liang (2003) have presented the prediction of thearithmetic mean surface roughness based on a probabilistic undeformed chipthickness model. Bigerelle et al. (2005) have shown that grinding could be char-acterized with an elementary function and the worn profile can be modeled by afractal 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 thesurface roughness in plunge center less grinding operation of 9SMn28, free-cuttingunalloyed steel. The analysis of variance shows that the grinding wheel dressingcondition most significantly affects the ground surface roughness. The surfaceroughness is additionally affected by the geometrical grinding gap set-up factorand the control wheel speed. Kwak (2005) has investigated the various grindingparameters affected the geometric error in surface grinding process using com-bined Taguchi method and response surface method. Four grinding parameterssuch as grain size, wheel speed, depth of cut and table speed are selected forexperimentation. A second-order response model for the geometric error isdeveloped and the utilization of the response surface model is evaluated withconstraints of the surface roughness and the material removal rate. Fredj andAmamou (2006) have tried to establish a model combining the application ofdesign of experiments (DOE) and neural network method for ground surfaceroughness prediction. Kwak et al. (2006) have developed a model for grindingpower spent during the process and the surface roughness in the external cylin-drical grinding of hardened SCM440 steel using the response surface method.They have shown from the study that the grinding power seems to increase linearlywith increasing work-piece speed and the traverse speed and surface roughness isdominantly affected by the change of the work-piece speed. Choi et al. (2008) have


established the generalized model for power, surface roughness, grinding ratio andsurface burning for grinding of various steel alloys using alumina grinding wheelsbased on the systematic analysis and experiments. It is seen that steady-statesurface roughness is primarily dependent only on the effective chip thickness.Mohanasundararaju et al. (2008) have developed a neural network and fuzzy-basedmethodology for predicting surface roughness in a grinding process for work rollsused in cold rolling. This methodology predicts the most likely estimates of sur-face roughness along with lower and upper estimates using fuzzy numbers.Siddiquee et al. (2010) have investigated the optimization of an in-feed centrelesscylindrical grinding process performed on EN52 austenitic valve steel (DIN:X45CrSi93) considering dressing feed, grinding feed, dwell time and cycle time asprocess parameters. They have optimized the multiple responses viz. surfaceroughness, out of cylindricity of the valve stem and diametral tolerance using greyrelational Taguchi analysis.

Milling also is a popular machining process in modern industry. There areseveral researchers who have tried to model the roughness in milling process. Inthis section, few available literatures on surface roughness modeling in milling arereviewed. Fuh and Wu (1995) have developed a model for prediction of surfacequality in end milling of 2014 aluminium alloy and shown that surface roughnessis mainly affected by the feed rate and tool nose radius. Alauddin et al. (1996) havepointed out that feed rate is the most significant factor and with increase in feed,surface roughness increases while with increase in cutting speed, surface rough-ness decreases in end milling Inconel 718 using uncoated carbide inserts. Lou et al.(1998) have used multiple regression models to develop a surface roughness modelto predict Ra in CNC end milling of 6061 aluminum and concluded that the feedrate is the most significant factor. Yang and Chen (2001) found out the optimumcutting parameters for milling of Al 6061 material using Taguchi design consid-ering cutting speed, feed rate, depth of cut and tool diameter as the cuttingparameters. Lee et al. (2001) presented a method for the simulation of surfaceroughness of the machined surface in high-speed end milling. Lin (2002) hasoptimized cutting speed, feed rate and depth of cut with consideration of multipleperformance characteristics including removed volume, surface roughness andburr height in face milling of stainless steel and shown that the most influence ofthe cutting parameters is the feed rate. Mansour and Abdalla (2002) have con-cluded that with increase in feed rate or in axial depth of cut, surface roughnessincreases whilst with increase in cutting speed, surface roughness decreases in endmilling operations of EN32 materials. Ghani et al. (2004) have studied surfaceroughness in end milling of hardened steel AISI H13 with TiN coated P10 carbideinsert tool and concluded that use of high cutting speed, low feed rate and lowdepth of cut leads to better surface finish. Wang and Chang (2004) have analyzedthe influence of cutting condition and tool geometry on surface roughness in slotend milling of AL2014-T6. Oktem et al. (2005) have developed an effectivemethodology to determine the optimum cutting conditions leading to minimumroughness in milling of Aluminum (7075-T6) molded surfaces considering feed,cutting speed, axial depth of cut, radial depth of cut and machining tolerance as


cutting parameters. Reddy and Rao (2005) have developed a model to see theeffects of tool geometry, cutting speed and feed rate on surface roughness in endmilling of medium carbon steel. The investigations of this study indicate that theparameters cutting speed, feed, radial rake angle and nose radius are the primaryfactors influencing the surface roughness of medium carbon steel during endmilling. Reddy and Rao (2006a) have investigated the role of solid lubricantassisted machining with graphite and molybdenum disulphide lubricants on sur-face quality, cutting forces and specific energy while milling AISI 1045 steel usingcutting tools of different tool geometry (radial rake angle and nose radius). Reddyand Rao (2006b) have studied the effect of various parameters such as cuttingspeed, feed rate, radial rake angle and nose radius on surface roughness in millingof AISI 1045 materials. They have shown that surface roughness decreases withincreasing cutting speed. Jesuthanam et al. (2007) have developed a hybrid neuralnetwork trained with GA and Particle Swarm Optimization (PSO) for the pre-diction of surface roughness in CNC end milling operation of mild steel materials.For the development of network, spindle speed, feed, depth of cut and vibrationdata are considered. Chang and Lu (2007) have applied a grey relational analysisto determine the cutting parameters for optimizing the side milling process withmultiple performance characteristics and concluded that feeding-direction roughness, axial-direction roughness and waviness are improvedsimultaneously through the optimal combination of the cutting parametersobtained from the proposed two-stage parameter design. El-Sonbaty et al. (2008)have developed artificial neural network (ANN) models for the analysis andprediction of the relationship between the cutting conditions and the correspondingfractal parameters of machined surfaces in face milling operation using rotationalspeed, feed, depth of cut, pre-tool flank wear and vibration level as inputparameters. Routara et al. (2009) have studied the influence of machiningparameters on conventional roughness parameters in CNC end milling of alu-minium, steel and brass materials using response surface method. Berglund andRose’n (2009) have evaluated the connection between surface finish appearanceand measured surface roughness using scale sensitive fractal analysis in milling.Öktem (2009) has developed an integrated study of surface roughness to modeland optimize the cutting parameters in end milling of AISI 1040 steel materialwith TiAlN solid carbide tools under wet condition using ANN and GA. He hasshown that the axial depth of cut is the most important cutting parameters affectingsurface roughness (Ra). Zain et al. (2010a) have carried out a study using GA toobserve the optimal effect of the radial rake angle of the tool, combined with speedand 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, theGA technique recommends the best minimum surface roughness value. For endmilling also, to modeling surface roughness different tools are used like RSM(Alauddin et al. 1996; Mansour and Abdalla 2002; Wang and Chang 2004; Oktemet 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 andAykut 2006), ANN (Tsai et al. 1999; Balic and Korosec 2002; Benardos and


Vosniakos 2002; El-Sonbaty et al. 2008; Öktem 2009; Zain et al. 2010b). From theliterature survey, it is seen that most of literatures deal with conventional rough-ness parameters to describe surface roughness and also in the study, threemachining parameters viz. spindle speed, feed rate and depth of cut are the mostcommon 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 machiningprocess that can be used all types of conductive materials. It can also be used formachining of difficult-to-machine shapes and materials. In this section, few of theavailable literatures on surface roughness modeling in EDM are reviewed. Zhanget al. (1997) have investigated the effects on material removal rate, surfaceroughness and diameter of discharge points in electro-discharge machining (EDM)on ceramics and shown that the material removal rate, surface roughness and thediameter of discharge point all increase with increasing pulse-on time anddischarge current. Lee and Li (2001) have shown that the negative tool polaritygives better surface finish in EDM of tungsten carbide. Also, surface roughnessincreases with increasing peak current and pulse duration. Ramasawmy and Blunt(2002) have illustrated the influencing process factors in modifying the surfacetextures using Taguchi method in EDM on M300 tool steel and shown that thedirect current is the most dominant factor in modifying the surface texture. Lin andLin (2002) have studied an approach for the optimization of the electrical dis-charge machining process (work-piece polarity, pulse on time, duty factor, opendischarge voltage, discharge current, and dielectric fluid) with multiple perfor-mance characteristics viz. MRR, surface roughness and electrode wear ratio usinggrey relational analysis. Lin and Lin (2005) have tried to optimize the electricaldischarge 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 surfaceroughness and there is a strong interaction between the current intensity and thepulse on time factors being advisable to work with high current intensity valuesand low pulse on time values. They have justified the fact of having to employ highcurrent intensity values to obtain a better surface roughness because a better arcstability causes a more uniform production of sparks and a narrow variationinterval of the Ra and Rq roughness parameters. Yih-fong and Fu-chen (2003) havepresented an approach for optimizing high-speed EDM using Taguchi methods.They have concluded that the most important factors affecting the EDM processrobustness have been identified as pulse-on time, duty cycle, and pulse peakcurrent. Ramasawmy and Blunt (2004) have quantified the effect of processparameters on the surface texture using Taguchi method in EDM of steel andconcluded that the pulse current is the most dominant factor in affecting thesurface texture. Puertas et al. (2004) have carried out a study on the influence ofthe factors of intensity, pulse time and duty cycle over surface roughness, material


removal rate, etc. in EDM of a cemented carbide and observed that in the case ofRa parameter the most influential factors are intensity, followed by the pulse timefactor. Petropoulos et al. (2004) have emphasized the interrelationship betweensurface texture parameters and process parameters in EDM of Ck60 steel plates.They have considered amplitude, spacing, hybrid, as well as random process andfractal parameters. Puertas et al. (2005) have carried out a study on the influence ofEDM parameters over two spacing parameters in machining of siliconised orreaction-bonded silicon carbide (SiSiC) and shown that intensity, pulse time andduty cycle are most influential factors affecting mean spacing between peaks andthe number of peaks per cm whereas the dielectric flushing pressure is not aninfluential factor. Amorima and Weingaertner (2005) have shown that the increaseof average surface roughness of the work-piece is directly related to the increase indischarge current and discharge duration on the EDM of the AISI P20 tool steelunder finish machining. Ramakrishnan and Karunamoorthy (2006) have proposeda multi objective optimization method in WEDM process using parametric designof Taguchi method and identified that the pulse on time and ignition currentintensity are the influential parameters. Keskin et al. (2006) have shown thatsurface roughness has an increasing trend with an increase in the dischargeduration in EDM on steel work-pieces. Sahoo et al. (2009) have investigated theinfluence of machining parameters, viz., pulse current, pulse on time and pulse offtime on the quality of surface produced in EDM of mild steel, brass and tungstencarbide materials using response surface methodology. It is seen that the pulsecurrent has the maximum influence on the roughness parameters while pulse ontime has some effect and pulse off time has no significant effect on roughnessparameters. Shah et al. (2010) have shown that the material thickness has littleeffect on the material removal rate and kerf but is a significant factor in terms ofsurface roughness in wire electrical discharge machining (WEDM) of tungstencarbide samples. Now-a-days, artificial neural network is used as a tool in mod-eling of EDM process (Spedding and Wang 1997; Tsai and Wang 2001; Sarkaret al. 2006; Mandal et al. 2007; Assarzadeh and Ghoreishi 2008).

From the literature survey, it is revealed that there are many researches onsurface roughness modeling in different machining processes. However, most ofthe literatures deal with conventional roughness parameters and there is scarcity ofliteratures 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 permitsto carry out the modeling and analysis of the influence of process variables on theresponse variables. The response variable is an unknown function of the processvariables, which are known as design factors. The purpose of running experimentsis to characterize unknown relations and dependencies that exist in the observeddesign or process, i.e., to find out the influential design variables and the response


to variations in the design variable values. A scientific approach to planning theexperiment must be employed if an experiment is to be performed most efficiently.The statistical design of experiments refers to the process of planning the exper-iment so that appropriate data that can be analyzed by statistical methods will becollected, resulting in valid and objective conclusions in a meaningful way. Whenthe problem involves data that are subject to experimental errors, statisticalmethodology is the only objective approach to analysis. Sometimes, experimentsare repeated with a particular set of levels for all the factors to check the statisticalvalidation and repeatability by the replicate data. This is called replication. To getrid of any biasness, allocation of experimental material and the order of experi-mental runs are randomly selected. This is called randomization. To arrange theexperimental material into groups, or blocks, that should be more homogeneousthan the entire set of material is called blocking. So, when experiments are carriedout these things should be remembered. There are several methodologies fordesign 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 combina-tions of the levels of the factors in each complete trial or replication of theexperiments. The experimental design points in a full factorial design are thevertices of a hyper cube in the n-dimensional design space defined by the mini-mum and the maximum values of each of the factors. These experimental pointsare 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 becarried out. If there are n replicates of complete experiments, then there will be ntimes of the single replication experiments to be conducted. In the experimenta-tion, it must have at least two replicates to determine a sum of squares due to errorif all possible interactions are included in the model.

1.5.2 Central Composite Design

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

1.5 Design of Experiments 19

points are located on the axes of the coordinate system symmetrically with respectto the central point at a distance a from the design center.

There are two main varieties of CCD namely Face centered CCD and RotatableCCD. In face centered CCD, a k factor 3-level experimental design requires2k ? 2k ? C experiments, where k is the number of factors, 2k points are in thecorners of the cube representing the experimental domain, 2k axial points are in thecenter of each face of the cube ½ð�a; 0; . . .0Þ; ð0;�a; . . .0Þ; ð0; 0; . . .� aÞ� and Cpoints are the replicates in the center of the cube that are necessary to estimate thevariability of the experimental measurements, it is to say the repeatability of thephenomenon which carry out the lack-of-fit or curvature test for the model. Thecentre points may vary from three to six. The example of 3-level three factor FCCdesign is shown in Fig. 1.5. In this figure, the deep black circles represent thefractional points at the corner of cube while the white circles represent axial pointsin 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 ofa 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 experimentalruns are conducted in random order.

xi ¼½2x� ðxmax þ xminÞ�ðxmax � xminÞ

ð1:8Þ

The rotatable central composite design is the most widely used experimentaldesign for modeling a second-order response surface. A design is called rotatablewhen the variance of the predicted response at any point depends only on the distanceof the point from the center point of design. The rotatable design provides theuniformity of prediction error and it is achieved by proper choice of a: In rotatabledesigns, all points at the same radial distance (r) from the centre point have the samemagnitude 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 inthe 2k factorial design. A rotatable CCD consists of 2k fractional factorial points,augmented by 2 k axial points ½ð�a; 0; . . .0Þ; ð0;�a; . . .0Þ; ð0; 0; . . .� aÞ� and nc

Fig. 1.5 Face centeredcentral composite design withthree factors


centre points (0, 0, 0, 0…,0). Here also, the centre points vary from three to six. Withproper choice of nc the CCD can be made orthogonal or it can be made uniformprecision design. It means that the variance of response at origin is equal to thevariance of response at a unit distance from the origin. Considering uniform preci-sion, 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 tech-niques which are useful for the modeling and analysis of problems in which aresponse of interest is influenced by several variables and the objective is tooptimize the response (Montgomery 2001). RSM helps in analyzing the influenceof the independent variables on a specific dependent variable (response) byquantifying the relationships amongst one or more measured responses and thevital input factors. The mathematical models thus developed relating themachining responses and their factors facilitate the optimization of the machiningprocess. In most of the RSM problems, the form of the relationship between theresponse and the independent variables is unknown. Thus the first step in RSM isto find a suitable approximation for the true functional relationship betweenresponse of interest ‘y’ and a set of controllable variables {x1, x2, …, xn}. Usuallywhen the response function is not known or non-linear, a second order model isutilized (Montgomery 2001) in the form:

y ¼ b0 þXn

i¼1

bixiþXn

i¼1

biix2i þ

XX

i\j

bijxixj þ e ð1:9Þ

where, e represents the noise or error observed in the response y such that theexpected 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 theinput variables by minimizing the residual error measured by the sum of squaredeviations between the actual and estimated responses. The calculated coefficientsor the model equations however need to be tested for statistical significance andthus 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 isthe ratio between the regression mean square and the mean square error. TheF-ratio, also called the variance ratio, is the ratio of variance due to the effect of afactor (the model) and variance due to the error term. This ratio is used to measurethe significance of the model under investigation with respect to the variance of allthe terms included in the error term at the desired significance level, a: If thecalculated value of F-ratio is higher than the tabulated value of F-ratio for

1.5 Design of Experiments 21

roughness, then the model is adequate at desired a level to represent the rela-tionship between machining response and the machining parameters.

In the ANOVA Table, there is a P-value or probability of significance for eachindependent variable in the model the value of which shows whether the variableis significant or not. If the P-value is less or equal to the selected a-level, then theeffect of the variable is significant. If the P-value is greater than the selecteda-value, then it is considered that the variable is not significant. Sometimes theindividual variables may not be significant. If the effect of interaction terms issignificant, then the effect of each factor is different at different levels of the otherfactors. ANOVA for different response variables are carried out in the presentstudy using commercial software Minitab (Minitab user manual 2001) withconfidence 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 withthe essence of fractal dimension to describe surface roughness. The basics ofsurface metrology including the different roughness parameters along with thesurface roughness measurement technique are presented. Basics of fractaldimension and its calculation are also discussed. Then the essence of design ofexperiments and different design of experiment techniques are presented.Response surface methodology (RSM) is discussed which is used to analyze theexperimental data in the subsequent chapters.

References

Abburi NR, Dixit US (2006) A knowledge-based system for the prediction of surface roughnessin turning process. Robotics Comput-Integr Manuf 22:363–372

Abouelatta OB, Madl J (2001) Surface roughness prediction based on cutting parameters and toolvibrations in turning operations. J Mater Process Technol 118:269–277

Alauddin M, El Baradie MA, Hashmi MSJ (1996) Optimization of surface finish in end millingInconel 718. J Mater Process Technol 56:54–65

Amorima FL, Weingaertner WL (2005) The influence of generator actuation mode and processparameters on the performance of finish EDM of a tool steel. J Mater Process Technol166:411–416

Arbizu IP, Pérez CJL (2003) Surface roughness prediction by factorial design of experiments inturning processes. J Mater Process Technol 143–144:390–396

Assarzadeh S, Ghoreishi M (2008) Neural-network-based modeling and optimization of theelectro-discharge machining process. Int J Adv Manuf Technol 39:488–500

Bagci E, Aykut S (2006) A study of Taguchi optimization method for identifying optimumsurface roughness in CNC face milling of cobalt-based alloy (stellite 6). Int J Adv ManufTechnol 29:940–947

Bagci E, Isik B (2006) Investigation of surface roughness in turning unidirectional GFRPcomposites by using RS methodology and ANN. Int J Adv Manuf Technol 31:10–17


Balic J, Korosec M (2002) Intelligent tool path generation for milling of free surfaces usingneural networks. Int J Mach Tools Manuf 42:1171–1179

Benardos PG, Vosniakos GC (2002) Prediction of surface roughness in CNC face milling usingneural networks and Taguchi’s design of experiments. Robotics Comput-Integr Manuf18:343–354

Benardos PG, Vosniakos GC (2003) Predicting surface roughness in machining: a review. Int JMach Tools Manuf 43(8):833–844

Berglund J, Rose’n BG (2009) A method development for correlation of surface finishappearance of die surfaces and roughness measurement data. Tribol Lett 36(2):157–164

Berry MV, Lewis ZV (1980) On the Weierstrass-Mandelbrot fractal function. Proc R Soc A370:459–484

Bhushan B, Wyant JC, Meiling J (1988) A new three-dimensional non-contact digital opticalprofiler. Wear 122:301–312

Bigerelle M, Najjar D, Iost A (2005) Multiscale functional analysis of wear a fractal model of thegrinding process. Wear 258:232–239

Brown CA, Savary G (1991) Describing ground surface texture using contact profilometry andfractal analysis. Wear 141:211–226

Chang CK, Lu HS (2007) Design optimization of cutting parameters for side milling operationswith multiple performance characteristics. Int J Adv Manuf Technol 32:18–26

Chavoshi SZ, Tajdari M (2010) Surface roughness modelling in hard turning operation of AISI4140 using CBN cutting tool. Int J Mater Form. doi:10.1007/s12289-009-0679-2

Choi TJ, Subrahmanya N, Li H, Shin YC (2008) Generalized practical models of cylindricalplunge grinding processes. Int J Mach Tools Manuf 48:61–72

Dabnun MA, Hashmi MSJ, El-Baradie MA (2005) Surface roughness prediction model by designof experiments for turning machinable glass-ceramic (Macor). J Mater Process Technol164–165:1289–1293

Davim JP (2001) A note on the determination of optimal cutting conditions for surface finishobtained in turning using design of experiments. J Mater Process Technol 116:305–308

El-Sonbaty IA, Khashaba UA, Selmy AI, Ali AI (2008) Prediction of surface roughness profilesfor milledsurfaces using an artificial neural network and fractal geometry approach. J MaterProcess Technol 200:271–278

Feng CX, Wang XF (2003) Surface roughness predictive modeling: neural networks versusregression. IIE Trans 35:11–27

Feng CXJ, Yu ZG, Kusiak A (2006) Selection and validation of predictive regression and neuralnetwork models based on designed experiments. IIE Trans 38:13–23

Fredj NB, Amamou R (2006) Ground surface roughness prediction based upon experimentaldesign and neural network models. Int J Adv Manuf Technol 31:24–36

Fuh KH, Wu CF (1995) A proposed statistical model for surface quality prediction in end-millingof Al alloy. Int J Mach Tools Manuf 35(S):1187–1200

Ge S, Chen G (1999) Fractal prediction models of sliding wear during the running–in process.Wear 231:249–255

Ghani JA, Choudhury IA, Hassan HH (2004) Application of Taguchi method in the optimizationof end milling parameters. J Mater Process Technol 145:84–92

Grzesik W (1996) A revised model for predicting surface roughness in turning. Wear194:143–148

Gupta AK (2010) Predictive modelling of turning operations using response surfacemethodology, artificial neural networks and support vector regression. Int J Prod Res48(3):763–778

Han JH, Ping S, Shengsun H (2005) Fractal characterization and simulation of surface profiles ofcopper electrodes and aluminum sheets. Mater Sci Eng A 403:174–181

Hasegawa M, Liu J, Okuda K, Nunobiki M (1996) Calculation of the fractal dimensions ofmachined surface profiles. Wear 192:40–45

Hassui A, Diniz AE (2003) Correlating surface roughness and vibration on plunge cylindricalgrinding of steel. Int J Mach Tools Manuf 43:855–862

References 23

http://dx.doi.org/10.1007/s12289-009-0679-2

He L, Zhu J (1997) The fractal character of processed metal surfaces. Wear 208:17–24Hecker RL, Liang SY (2003) Predictive model of surface roughness in grinding. Int J Mach Tools

Manuf 43:755–761ISO 4287:1997 (1997) Geometrical product specification (GPS)—surface texture: profile

method—terms, definitions and surface texture parameters. International Organization ofStandardization, Geneva

Jahn R, Truckenbrodt H (2004) A simple fractal analysis method of the surface roughness.J Mater Process Technol 145:40–45

Jesuthanam CP, Kumanan S, Asokan P (2007) Surface roughness prediction using hybrid neuralnetworks. Mach Sci Technol 11:271–286

Jiang Z, Wang H, Fei B (2001) Research into the application of fractal geometry in characterizingmachined surfaces. Int J Mach Tools Manuf 41:2179–2185

Jiao Y, Lei S, Pei ZJ, Lee ES (2004) Fuzzy adaptive networks in machining process modeling:surface roughness prediction for turning operations. Int J Mach Tools Manuf 44:1643–1651

Kang MC, Kim JS, Kim KH (2005) Fractal dimension analysis of machined surface depending oncoated tool wear. Surf Coat Technol 193(1–3):259–265

Karayel D (2009) Prediction and control of surface roughness in CNC lathe using artificial neuralnetwork. J Mater Process Technol 209:3125–3137

Keskin YH, Halkacı HS, Kizil SM (2006) An experimental study for determination of the effectsof machining parameters on surface roughness in electrical discharge machining (EDM). Int JAdv Manuf Technol 28:1118–1121

Kirby ED, Zhang Z, Chen JC, Chen J (2006) Optimizing surface finish in a turning operationusing the Taguchi parameter design method. Int J Adv Manuf Technol 30:1021–1029

Kohli A, Dixit US (2005) A neural-network-based methodology for the prediction of surfaceroughness in turning process. Int J Adv Manuf Technol 25:118–129

Krajnik P, Kopac J, Sluga A (2005) Design of grinding factors based on response surfacemethodology. J Mater Process Technol 162–163:629–636

Kwak JS (2005) Application of Taguchi and response surface methodologies for geometric errorin surface grinding process. Int J Mach Tools Manuf 45:327–334

Kwak JS, Sim SB, Jeong YD (2006) An analysis of grinding power and surface roughness inexternal cylindrical grinding of hardened SCM440 steel using response surface method. Int JMach Tools Manuf 46:304–312

Lee SH, Li XP (2001) Study of the effect of machining parameters on the machiningcharacteristics in electrical discharge machining of tungsten carbide. J Mater Process Technol115:344–358

Lee KY, Kang MC, Jeong YH, Lee DW, Kim JS (2001) Simulation of surface roughness andprofile in high-speed end milling. J Mater Process Technol 113:410–4125

Lin TR (2002) Optimisation technique for face milling stainless steel with multiple performancecharacteristics. Int J Adv Manuf Technol 19:330–335

Lin JL, Lin CL (2002) The use of orthogonal array with grey relational analysis to optimize theelectrical discharge machining process with multiple performance characteristics. Int J MachTools Manuf 42:237–244

Lin JL, Lin CL (2005) The use of grey-fuzzy logic for the optimization of the manufacturingprocess. J Mater Process Technol 160:9–14

Lin WS, Lee BY, Wu CL (2001) Modeling the surface roughness and cutting force for turning.J Mater Process Technol 108:286–293

Ling FF (1990) Fractals, engineering surfaces and tribology. Wear 136:141–156Lou MS, Chen JC, Li CM (1998) Surface roughness prediction technique for CNC end-milling.

J Ind Technol 15 (1), November 1998 to January 1999Majumdar A, Bhushan B (1990) Role of fractal geometry in roughness characterization and

contact mechanics of surfaces. Trans ASME J Tribol 112:205–216Majumdar A, Tien CL (1990) Fractal characterization and simulation of rough surfaces. Wear

136:313–327


Maksoud TMA, Atia MR, Koura MM (2003) Applications of artificial intelligence to grindingoperations via neural networks. Mach Sci Technol 7(3):361–387

Mandal D, Pal SK, Saha P (2007) Modeling of electrical discharge machining process using backpropagation neural network and multi-objective optimization using non-dominating sortingalgorithm-II. J Mater Process Technol 186:154–162

Mandelbrot BB (1967) How long is the coast of Britain? Statistical self-similarity and fractionaldimension. Science 156:636–638

Mandelbrot BB (1982) The fractal geometry of nature. W H freeman, New YorkMansour A, Abdalla H (2002) Surface roughness model for end milling: a semi-free cutting

carbon casehardening steel (EN32) in dry condition. J Mater Process Technol 124:183–191Minitab User Manual Release 13.2 (2001) Making data analysis easier. MINITAB Inc. State

College, PAMohanasundararaju N, Sivasubramanian R, Gnanaguru R, Alagumurthy N (2008) A neural

network and fuzzy-based methodology for the prediction of work roll surface roughness in agrinding process. Int J Comput Methods Eng Sci Mech 9:103–110

Montgomery DC (2001) Design and analysis of experiments. Wiley, New YorkMuthukrishnan N, Davim JP (2009) Optimization of machining parameters of Al/SiC-MMC with

ANOVA and ANN analysis. J Mater Process Technol 209:225–232Nalbant M, Gokkaya H, Sur G (2007) Application of Taguchi method in the optimization of

cutting parameters for surface roughness in turning. Mater Des 28:1379–1385Nayak PR (1971) Random process model of rough surfaces. Trans ASME J Lubr Technol

93:398–407Öktem H (2009) An integrated study of surface roughness for modeling and optimization of

cutting parameters during end milling operation. Int J Adv Manuf Technol 43:852–861Oktem H, Erzurumlu T, Kurtaran H (2005) Application of response surface methodology in the

optimization of cutting conditions for surface roughness. J Mater Process Technol 170:11–16Pal SK, Chakraborty D (2005) Surface roughness prediction in turning using artificial neural

network. Neural Comput Appl 14:319–324Palanikumar K (2008) Application of Taguchi and response surface methodologies for surface

roughness in machining glass fiber reinforced plastics by PCD tooling. Int J Adv ManufTechnol 36:19–27

Palanikumar K, Karunamoorthy L, Karthikeyan R (2006) Parametric optimization to minimisethe surface roughness on the machining of GFRP composites. J Mater Sci Technol22(1):66–72

Petropoulos G, Vaxevanidis NM, Pandazaras C (2004) Modeling of surface finish in electro-discharge machining based upon statistical multi-parameter analysis. J Mater Process Technol155–156:1247–1251

Puertas I, Luis CJ (2003) A study on the machining parameters optimisation of electricaldischarge machining. J Mater Process Technol 143–144:521–526

Puertas I, Luis CJ, Álvarez L (2004) Analysis of the influence of EDM parameters on surfacequality, MRR and EW of WC-Co. J Mater Process Technol 153–154:1026–1032

Puertas I, Luis CJ, Villa G (2005) Spacing roughness parameters study on the EDM of siliconcarbide. J Mater Process Technol 164–165:1590–1596

Ramakrishnan R, Karunamoorthy L (2006) Multi response optimization of wire EDM operationsusing robust design of experiments. Int J Adv Manuf Technol 29:105–112

Ramasawmy H, Blunt L (2002) 3D surface characterisation of elctropolished EDMed surface andquantitative assessment of process variables using Taguchi Methodology. Int J Mach ToolsManuf 42:1129–1133

Ramasawmy H, Blunt L (2004) Effect of EDM process parameters on 3D surface topography.J Mater Process Technol 148:155–164

Ramesh S, Karunamoorthy L, Palanikumar K (2008) Surface roughness analysis in machining oftitanium alloy. Mater Manuf Process 23:174–181

Reddy NSK, Rao PV (2005) Selection of optimum tool geometry and cutting conditions using asurface roughness prediction model for end milling. Int J Adv Manuf Technol 26:1202–1210

References 25

Reddy NSK, Rao PV (2006a) Experimental investigation to study the effect of solid lubricants oncutting forces and surface quality in end milling. Int J Mach Tools Manuf 46:189–198

Reddy NSK, Rao PV (2006b) Selection of an optimal parametric combination for achieving abetter surface finish in dry milling using genetic algorithms. Int J Adv Manuf Technol28:463–473

Routara BC, Bandyopadhyay A, Sahoo P (2009) Roughness modeling and optimization in CNCend milling using response surface method: effect of workpiece material variation. Int J AdvManuf Technol 40:1166–1180

Sahin Y, Motorcu AR (2005) Surface roughness model for machining mild steel with coatedcarbide tool. Mater Des 26:321–326

Sahoo P (2005) Engineering tribology. Prentice Hall of India, New DelhiSahoo P, Ghosh N (2007) Finite element contact analysis of fractal surfaces. J Phys D Appl Phys

40:4245–4252Sahoo P, Routara BC, Bandyopadhyay A (2009) Roughness modeling and optimization in EDM

using response surface method for different workpiece materials. Int J Mach Mach Mater5(2–3):321–346

Sarkar S, Mitra S, Bhattacharyya B (2006) Parametric optimisation of wire electrical dischargemachining of c titanium aluminide alloy through an artificial neural network model. Int J AdvManuf Technol 27:501–508

Sayles RS, Thomas TR (1978) Surface topography as a non-stationary random process. Nature271:431–434

Shah A, Mufti NA, Rakwal D, Bamberg E (2010) Material removal rate, kerf, and surfaceroughness of tungsten carbide machined with wire electrical discharge machining. J MaterEng Perform. doi:10.1007/s11665-010-9644-y

Siddiquee AN, Khan ZA, Mallick Z (2010) Grey relational analysis coupled with principalcomponent analysis for optimisation design of the process parameters in in-feed centrelesscylindrical grinding. Int J Adv Manuf Technol 46:983–992

Singh D, Rao PV (2007) A surface roughness prediction model for hard turning process. Int J AdvManuf Technol 32:1115–1124

Spedding TA, Wang ZQ (1997) Parametric optimization and surface characterization of wireelectrical discharge machining process. Precis Eng 20:5–15

Suresh PVS, Rao PV, Deshmukh SG (2002) A genetic algorithm approach for optimization ofsurface roughness prediction model. Int J Mach Tools Manuf 42:675–680

Thomas TR (1982) Defining the microtopography of surfaces in thermal contact. Wear 79:73–82Tricot C, Ferlans P, Baran G (1994) Fractal analysis of worn surfaces. Wear 172:127–133Tsai KM, Wang PJ (2001) Predictions on surface finish in electrical discharge machining based

upon neural network models. Int J Mach Tools Manuf 41:1385–1403Tsai YH, Chen JC, Lou SJ (1999) An in-process surface recognition system based on neural

networks in end milling cutting operations. Int J Mach Tools Manuf 39:583–605Venkatesh K, Bobji MS, Biswas SK (1998) Some features of surface topographical power spectra

generated by conventional machining of a ductile metal. Mater Sci Eng A A252:153–155Venkatesh K, Bobji MS, Gargi R, Biswas SK (1999) Genesis of workpiece roughness generated

in surface grinding and polishing of metals. Wear 225–229:215–226Wang MY, Chang HY (2004) Experimental study of surface roughness in slot end milling

AL2014–T6. Int J Mach Tools Manuf 44:51–57Whitehouse DJ (1982) The parameter rash, is there a cure? Wear 83:75–78Yan W, Komvopoulos K (1998) Contact analysis of elastic-plastic fractal surfaces. J Appl Phys

84(7):3617–3624Yang JL, Chen JC (2001) A systematic approach for identifying optimum surface roughness

performance in end-milling operations. J Ind Technol 17, 2 February 2001 to April 2001Yang WH, Tarng YS (1998) Design optimization of cutting parameters for turning operations

based on the Taguchi method. J Mater Process Technol 84:122–129Yih-fong T, Fu-chen C (2003) A simple approach for robust design of high-speed electrical

discharge machining technology. Int J Mach Tools Manuf 43:217–227


http://dx.doi.org/10.1007/s11665-010-9644-y

Zain AM, Haron H, Sharif S (2010a) Application of GA to optimize cutting conditions forminimizing surface roughness in end milling machining process. Expert Syst Appl37:4650–4659

Zain AM, Haron H, Sharif S (2010b) Prediction of surface roughness in the end millingmachining using artificial neural network. Expert Syst Appl 37:1755–1768

Zhang JZ, Chen JC (2007) The development of an in-process surface roughness adaptive controlsystem in end milling operations. Int J Adv Manuf Technol 31:877–887

Zhang JH, Lee TC, Lau WS (1997) Study on the electro-discharge machining of a hot pressedaluminum oxide based ceramic. J Mater Process Technol 63:908–912

Zhang Y, Luo Y, Wang JF, Li Z (2001) Research on the fractal of surface topography of grinding.Int J Mach Tools Manuf 41:2045–2049

Zhong ZW, Khoo LP, Han ST (2006) Prediction of surface roughness of turned surfaces usingneural networks. Int J Adv Manuf Technol 28:688–693

Zhong ZW, Khoo LP, Han ST (2008) Neural-network predicting of surface finish or cuttingparameters for carbide and diamond turning processes. Mater Manuf Process 23:92–97

Zhou X, Xi F (2002) Modeling and predicting surface roughness of the grinding process. Int JMach Tools Manuf 42:969–977

Zhu H, Ge S, Huang X, Zhang D, Liu J (2003) Experimental study on the characterization ofworn surface topography with characteristic roughness parameter. Wear 255:309–314

References 27

Chapter 2Fractal Analysis in CNC End Milling

Abstract This chapter deals with the fractal dimension modeling in CNC endmilling operation. Milling operations are carried out for three different materialsviz. mild steel, brass and aluminium work-pieces for different combinations ofspindle speed, feed rate and depth of cut. The generated surfaces are measuredwith Talysurf instrument and analyzed to get fractal dimension. The experimentalresults are further processed to model fractal dimension using response surfacemethodology (RSM). It is seen that spindle speed and depth of cut are the sig-nificant factors affecting fractal dimension for mild steel. For brass material, thesignificant factors are spindle speed and feed rate but for aluminium the significantfactor is depth of cut. In general, for mild steel and brass, with increase in spindlespeed, D increases. Comparing the developed response surface models, it isconcluded that the models are material specific and the tool-work-piece materialcombination 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 itsability to remove materials with a multi-point cutting tool at a faster rate with areasonably good surface quality. In order to get specified surface roughness,selection of controlling parameters is necessary. There has been a great manyresearch developments in modeling surface roughness and optimization of thecontrolling parameters to obtain a surface finish of desired level since only properselection of cutting parameters can produce a better surface finish. But such studies


29

are far from complete since it is very difficult to consider all the parameters thatcontrol the surface roughness for a particular manufacturing process. In CNCmilling there are several parameters which control the surface quality. The analysisof surface roughness on CNC end milling process is a big challenge for researchdevelopment. Several factors involved in machining process have to be optimizedto obtain a desired surface quality. In this study, three machining parameters areconsidered viz. spindle speed, feed rate and depth of cut. Also the study is con-ducted on three different materials, viz. mild steel, brass and aluminium to con-sider the effect of work-piece material variation on fractal dimension of machinedsurfaces. 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 factorsviz. depth of cut (d, mm), spindle speed (N, rpm) and feed rate (f, mm/min). Thusthe design chosen was five level-three factor (53) full factorial design consisting of125 sets of coded combinations for each work-piece material. Three cuttingparameters are selected as design factors while other parameters have beenassumed to be constant over the experimental domain. The upper and lower limitsof a factor were coded as +1 and -1 respectively using Eq. 1.8. The processvariables/design factors with their values on different levels are listed in Table 2.1for three different work-piece materials.

2.2.2 Machine Used

The machine used for the milling tests is a ‘DYNA V4.5’ CNC end millingmachine having the control system SINUMERIK 802 D with a vertical millinghead. 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 createdwith specific commands. The compressed coolant servo-cut was used as cuttingenvironment.

Table 2.1 Variable levels used in the experimentation

Levels Aluminium Brass Mild steel

d N f d N f d N f

-1 0.10 4,500 900 0.10 1,500 550 0.150 2,500 300-0.5 0.15 4,750 950 0.15 1,800 600 0.175 2,750 3500 0.20 5,000 1,000 0.20 2,100 650 0.200 3,000 4000.5 0.25 5,250 1,050 0.25 2,400 700 0.225 3,250 4501 0.30 5,500 1,100 0.30 2,700 750 0.250 3,500 500

30 2 Fractal Analysis in CNC End Milling

http://dx.doi.org/10.1007/978-3-642-17922-8_1

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 inves-tigation. The tools used were flat end mill cutters produced by WIDIA(EM-TiAlN). The tools were coated with TiAlN coating. For each material a newcutter of same specification was used. The details of the end milling cutters aregiven below:

Cutter diameter = 8 mmOverall length = 108 mmFluted length = 38 mmHelix angle = 30�Hardness = 1,570 HVDensity = 14.5 g/ccTransverse rupture strength = 3,800 N/mm2

2.2.4 Work-Piece Materials

The present study was carried out with three different materials, viz., 6061-T4Aluminium, AISI 1040 steel and Medium leaded Brass UNS C34000. Thechemical composition and mechanical properties of the work-piece materials areshown in Table 2.3. All the specimens were in the form of 100 9 75 9 25 mmblocks.

Table 2.2 Specification of CNC end milling machine

Table size 450 9 250 mm

Table load capacity 200 KgsX Travel 250 mmY Travel 175 mmZ Travel 175 mmSpindle nose to table 300 mmSpindle centre to column 280 mmTaper of spindle nose BT 30Spindle speed 9,000 rpmRapid on X and Y axis 15 m/minRapid on Z axis 10 m/minSpindle motor 3.7 kWX axis motor 3 NmY axis motor 3 NmZ axis motor 6 NmContro system 802 D SINUMERIKPower requirement 7.5 kW/10 H.P.Lubricating oil Tellus 33 or EN KLO 68

2.2 Experimental Details 31

2.3 Results and Discussion

CNC milling operations are carried out on mild steel, brass and aluminium work-pieces to get machined surfaces for different combinations of spindle speed, feed rateand depth of cut. The generated surfaces are measured using Talysurf instrument andfurther processed to get fractal dimension (D). Full factorial design of experiments isconsidered 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 fractaldimension D have been assessed for three different materials. The second ordermodel was postulated in obtaining the relationship between the fractal dimensionand the machining variables using response surface methodology (RSM). Theanalysis of variance (ANOVA) was used to check the adequacy of the secondorder 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 steelmilling is obtained in terms of coded values of design factors as:

D ¼1:3836þ 0:0136d þ 0:0115N þ 0:0069f � 0:0063dN þ 0:0003df

� 0:0106Nf � 0:0283d2 þ 0:0169N2 þ 0:0032f 2 ð2:1Þ

Table 2.3 Composition and mechanical properties of work-piece materials

Work material Chemical composition (W%t) Mechanicalproperty

Aluminium(6061-T4)

0.2%Cr, 0.3%Cu, 0.85%Mg, 0.04%Mn, 0.5%Si, 0.04%Ti,0.25%Zn, 0.5%Fe and balance Al

Hardness—65BHN,Density—2.7g/cc,TensileStrength—241 MPa

Brass(UNSC34000)

0.095%Fe, 0.9%Pb, 34%Zn and balance Cu Hardness—68HRF,Density—8.47 g/cc,Tensilestrength—340 MPa

Mild Steel(AISI1040)

0.42%C, 0.48%Mn, 0.17%Si, 0.02%P, 0.018%S, 0.1%Cu,0.09%Ni, 0.07%Cr and balance Fe

Hardness—201BHN,Density—7.85 g/cc,Tensilestrength—620 MPa


Table 2.4 Experimental results for CNC milling considering full factorial design

SlNo

Depth ofcut(d)

Spindlespeed(N)

Feedrate(f)

D for mildsteel

D forbrass

D foraluminium

1 -1 -1 -1 1.31 1.28 1.342 -1 -1 -0.5 1.33 1.31 1.343 -1 -1 0 1.29 1.22 1.374 -1 -1 0.5 1.30 1.28 1.295 -1 -1 1 1.32 1.27 1.366 -1 -0.5 -1 1.29 1.30 1.387 -1 -0.5 -0.5 1.33 1.29 1.328 -1 -0.5 0 1.37 1.30 1.359 -1 -0.5 0.5 1.37 1.32 1.35

10 -1 -0.5 1 1.34 1.27 1.3611 -1 0 -1 1.32 1.38 1.3512 -1 0 -0.5 1.35 1.33 1.3413 -1 0 0 1.38 1.31 1.3314 -1 0 0.5 1.34 1.30 1.3415 -1 0 1 1.39 1.31 1.3416 -1 0.5 -1 1.38 1.36 1.3517 -1 0.5 -0.5 1.36 1.33 1.3618 -1 0.5 0 1.36 1.30 1.3519 -1 0.5 0.5 1.40 1.31 1.3420 -1 0.5 1 1.34 1.32 1.3421 -1 1 -1 1.40 1.37 1.3422 -1 1 -0.5 1.38 1.35 1.3523 -1 1 0 1.41 1.34 1.3524 -1 1 0.5 1.36 1.35 1.3825 -1 1 1 1.37 1.30 1.3826 -0.5 -1 -1 1.41 1.30 1.3727 -0.5 -1 -0.5 1.39 1.27 1.3628 -0.5 -1 0 1.35 1.26 1.3529 -0.5 -1 0.5 1.39 1.25 1.3130 -0.5 -1 1 1.38 1.28 1.3731 -0.5 -0.5 -1 1.31 1.31 1.3632 -0.5 -0.5 -0.5 1.37 1.29 1.3933 -0.5 -0.5 0 1.40 1.31 1.3134 -0.5 -0.5 0.5 1.41 1.29 1.3535 -0.5 -0.5 1 1.40 1.29 1.3236 -0.5 0 -1 1.38 1.38 1.3537 -0.5 0 -0.5 1.32 1.34 1.3438 -0.5 0 0 1.37 1.31 1.3439 -0.5 0 0.5 1.39 1.28 1.3440 -0.5 0 1 1.39 1.29 1.3841 -0.5 0.5 -1 1.41 1.35 1.3342 -0.5 0.5 -0.5 1.40 1.33 1.3143 -0.5 0.5 0 1.36 1.32 1.3744 -0.5 0.5 0.5 1.41 1.32 1.36

(continued)

2.3 Results and Discussion 33

Table 2.4 (continued)

SlNo

Depth ofcut(d)

Spindlespeed(N)

Feedrate(f)

D for mildsteel

D forbrass

D foraluminium

45 -0.5 0.5 1 1.36 1.29 1.3546 -0.5 1 -1 1.38 1.37 1.3647 -0.5 1 -0.5 1.38 1.37 1.3548 -0.5 1 0 1.32 1.36 1.3449 -0.5 1 0.5 1.39 1.32 1.3450 -0.5 1 1 1.41 1.31 1.3851 0 -1 -1 1.38 1.26 1.4152 0 -1 -0.5 1.42 1.25 1.3553 0 -1 0 1.43 1.26 1.3754 0 -1 0.5 1.43 1.27 1.3455 0 -1 1 1.41 1.29 1.3556 0 -0.5 -1 1.38 1.36 1.3657 0 -0.5 -0.5 1.41 1.27 1.3658 0 -0.5 0 1.38 1.32 1.3559 0 -0.5 0.5 1.38 1.27 1.3960 0 -0.5 1 1.40 1.29 1.3161 0 0 -1 1.38 1.35 1.3462 0 0 -0.5 1.37 1.35 1.3463 0 0 0 1.34 1.32 1.3764 0 0 0.5 1.41 1.31 1.3565 0 0 1 1.38 1.31 1.3166 0 0.5 -1 1.40 1.36 1.2867 0 0.5 -0.5 1.39 1.34 1.3468 0 0.5 0 1.36 1.32 1.3469 0 0.5 0.5 1.40 1.32 1.3770 0 0.5 1 1.38 1.36 1.3571 0 1 -1 1.43 1.37 1.3672 0 1 -0.5 1.41 1.33 1.3573 0 1 0 1.40 1.35 1.3774 0 1 0.5 1.39 1.34 1.3675 0 1 1 1.43 1.36 1.3676 0.5 -1 -1 1.40 1.24 1.3877 0.5 -1 -0.5 1.39 1.27 1.3278 0.5 -1 0 1.38 1.23 1.2979 0.5 -1 0.5 1.43 1.26 1.3380 0.5 -1 1 1.38 1.28 1.3281 0.5 -0.5 -1 1.39 1.27 1.3882 0.5 -0.5 -0.5 1.35 1.27 1.3883 0.5 -0.5 0 1.37 1.33 1.3384 0.5 -0.5 0.5 1.40 1.25 1.3385 0.5 -0.5 1 1.41 1.28 1.3486 0.5 0 -1 1.35 1.38 1.3387 0.5 0 -0.5 1.32 1.33 1.3688 0.5 0 0 1.37 1.32 1.36

(continued)


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 forfractal dimension, D given in Eq. 2.1. It can be seen that the P-value is less than0.05 which means that the model is significant at 95% confidence level. Also the


SlNo

Depth ofcut(d)

Spindlespeed(N)

Feedrate(f)

D for mildsteel

D forbrass

D foraluminium

89 0.5 0 0.5 1.39 1.29 1.3190 0.5 0 1 1.41 1.31 1.2891 0.5 0.5 -1 1.36 1.39 1.3392 0.5 0.5 -0.5 1.38 1.33 1.3693 0.5 0.5 0 1.37 1.32 1.3394 0.5 0.5 0.5 1.39 1.31 1.3795 0.5 0.5 1 1.38 1.36 1.3496 0.5 1 -1 1.44 1.36 1.3497 0.5 1 -0.5 1.43 1.37 1.3498 0.5 1 0 1.44 1.37 1.399 0.5 1 0.5 1.43 1.34 1.3

100 0.5 1 1 1.42 1.34 1.36101 1 -1 -1 1.30 1.29 1.34102 1 -1 -0.5 1.42 1.28 1.32103 1 -1 0 1.38 1.26 1.32104 1 -1 0.5 1.38 1.24 1.29105 1 -1 1 1.39 1.26 1.36106 1 -0.5 -1 1.35 1.31 1.37107 1 -0.5 -0.5 1.38 1.28 1.24108 1 -0.5 0 1.33 1.31 1.33109 1 -0.5 0.5 1.36 1.27 1.33110 1 -0.5 1 1.40 1.30 1.22111 1 0 -1 1.39 1.37 1.36112 1 0 -0.5 1.37 1.33 1.34113 1 0 0 1.35 1.34 1.34114 1 0 0.5 1.39 1.27 1.32115 1 0 1 1.41 1.33 1.31116 1 0.5 -1 1.40 1.39 1.32117 1 0.5 -0.5 1.38 1.37 1.34118 1 0.5 0 1.38 1.31 1.32119 1 0.5 0.5 1.36 1.29 1.35120 1 0.5 1 1.39 1.35 1.33121 1 1 -1 1.41 1.37 1.35122 1 1 -0.5 1.41 1.37 1.33123 1 1 0 1.40 1.36 1.31124 1 1 0.5 1.40 1.33 1.3125 1 1 1 1.36 1.31 1.32


calculated value of the F-ratio is more than the standard value of the F-ratio forD. It means the model is adequate at 95% confidence level to represent the rela-tionship between the machining response and the considered machining parame-ters of the CNC end milling process on mild steel. Table 2.6 represents theANOVA table for individual model coefficients where it can be seen that there arethree effects with a P-value less than 0.05 which means that they are significant at95% confidence level. These significant effects are: depth of cut, spindle speed andthe interaction between spindle speed and depth of cut. Figure 2.1 depicts the maineffects plot for the fractal dimension and the design factors considered in thepresent study. From this figure also, it is seen that spindle speed and depth of cuthave the significant effect on fractal dimension. To see the effects of processparameters on fractal dimension in the experimental regime, three dimensionalsurface as well as contour plots are presented at high level and low level of theparameters (Figs. 2.2, 2.3, 2.4).

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 F0.05 P

Regression 9 0.051657 0.005740 8.25 1.96 0Residual error 115 0.080004 0.000696Total 124 0.131661

Table 2.6 ANOVA for model coefficients for D in CNC milling of mild steel


d 4 0.0293648 0.0073412 13.15 2.52 0.000N 4 0.0146848 0.0036712 6.58 2.52 0.000f 4 0.0052688 0.0013172 2.36 2.52 0.063d*N 16 0.0232112 0.0014507 2.60 1.82 0.004d*f 16 0.0075072 0.0004692 0.84 1.82 0.636N*f 16 0.0159072 0.0009942 1.78 1.82 0.054Error 64 0.0357168 0.0005581Total 124 0.1316608

Fig. 2.1 Main effect plot formild steel


Fig. 2.2 Surface and contour plot of fractal dimension for mild steel: a at high level of spindlespeed, 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 ofcut, 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


2.3.2 RSM for Brass

The second order response surface equation for fractal dimension in brass millingis obtained in terms of coded values of design factors as:

D ¼1:3130þ 0:0015d þ 0:0408N � 0:0175f þ 0:0071dN þ 0:0014df

� 0:0098Nf � 0:0008d2 � 0:0142N2 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 forD given in Eq. 2.2. It can be seen that the P-value is less than 0.05 which meansthat the model is significant at 95% confidence level. Also the calculated value ofthe F-ratio is more than the standard value of the F-ratio for D. It means the modelis adequate at 95% confidence level to represent the relationship between the

Table 2.7 ANOVA for second order model for D in CNC milling of brass


Regression 9 0.138293 0.015366 36.35 1.96 0Residual Error 115 0.048614 0.000423Total 124 0.186907

Table 2.8 ANOVA for model coefficients for D in CNC milling of brass



Fig. 2.5 Main effect plot forbrass


machining response and the considered machining parameters of the CNC endmilling process on brass. Table 2.8 represents the ANOVA table for individualmodel coefficients where it can be seen that spindle speed, feed rate, the interactionbetween spindle speed and feed rate and the interaction of depth of cut and feedrate are significant factors at 95% confidence level. Figure 2.5 depicts the maineffects plot for the fractal dimension and the design factors considered in thepresent study. From this figure also, it is seen that spindle speed and feed rate havethe significant effect on fractal dimension. Figures 2.6, 2.7, 2.8 show the estimatedthree-dimensional surface as well as contour plots for fractal dimension as func-tions of the independent machining parameters. All these figures clearly depict thevariation of fractal dimension with controlling variables within the experimentalregime.

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


2.3.3 RSM for Aluminium

The second order response surface equation has been fitted using Minitab softwarefor the response variable D. The equation can be given in terms of the coded valuesof the independent variables as:

D ¼1:3433� 0:0128d þ 0:0013N � 0:0062 f � 0:0011dN � 0:0095 df

þ 0:0122Nf � 0:0135d2 þ 0:0041 N2 þ 0:0056 f 2 ð2:3Þ

Table 2.9 presents the ANOVA table for the second order model proposed forD given in Eq. 2.3. It can be appreciated that the P-value is less than 0.05 whichmeans that the model is significant at 95% confidence level. Also the calculatedvalue of the F-ratio is more than the standard value of the F-ratio for D. It meansthe model is adequate at 95% confidence level to represent the relationshipbetween the machining response and the considered machining parameters of theCNC end milling process. Table 2.10 represents the ANOVA table for individualmodel coefficients where it can be seen that depth of cut and the interactionbetween 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 designfactors considered in the present study. From this figure also, it is seen that depthof cut has the significant effect on fractal dimension. Figures 2.10, 2.11, 2.12 showthe estimated three-dimensional surface as well as contour plots for fractal

Table 2.9 ANOVA for second order model for D in CNC milling of aluminium


Regression 9 0.025241 0.002805 4.5 1.96 0Residual error 115 0.0717 0.000624Total 124 0.096941

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


dimension as functions of the independent machining parameters. All these figuresclearly depict the variation of fractal dimension with controlling variables withinthe experimental regime.

Fig. 2.9 Main effect plot for aluminium

Table 2.10 ANOVA for model coefficients for D in CNC milling of aluminium



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


2.4 Closure

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

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

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


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

2.4 Closure 43

Chapter 3Fractal Analysis in CNC Turning

Abstract Modeling of fractal dimension in CNC turning of mild steel, brass andaluminium work-pieces are presented in this chapter. Spindle speed, feed rate anddepth of cut are considered as the process parameters. The generated surface inCNC tuning operations are measured and processed to calculate fractal dimension.The experimental results are then analyzed with RSM. From the analysis, it is seenthat the work-piece speed is the most significant factor affecting the fractaldimension for mild steel turning whereas feed rate is the significant factor for bothbrass and aluminium materials. It can be concluded from the analysis that for allthe materials, with increase in feed rate, fractal dimension, D decreases. So, to getsmoother 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) � CNC turning � RSM � Mild steel � Brass �Aluminium

3.1 Introduction

Turning operation is an old and very common machining process in the industry. Inrecent times, uses of computer numerically controlled (CNC) machines havebecome 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 machiningparameters which control the surface quality of the machined work-piece whichinclude cutting conditions, tool variables and work-piece variables. Cutting condi-tions include speed, feed and depth of cut where as tool variables include toolmaterial, nose radius, rake angle, cutting edge geometry, tool vibration, tool over-hang, tool point angle, etc. and work-piece variables include material hardness andother mechanical properties. It is very difficult to consider all the parameters that


45

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

Table 3.1 Process parameters levels used in the experimentation for all the three materials

Process variables Unit Levels

-1.682 -1 0 1 1.682

A Depth of cut(d) mm 0.032 0.1 0.2 0.3 0.368B Spindle speed(N) rpm 528 800 1,200 1,600 1,872C Feed rate(f) mm/rev 0.0224 0.07 0.14 0.21 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

d N f d N f

1 20 -1 -1 -1 0.1 800 0.072 1 1 -1 -1 0.3 800 0.073 9 -1 1 -1 0.1 1,600 0.074 11 1 1 -1 0.3 1,600 0.075 7 -1 -1 1 0.1 800 0.216 8 1 -1 1 0.3 800 0.217 13 -1 1 1 0.1 1,600 0.218 3 1 1 1 0.3 1,600 0.219 10 -1.682 0 0 0.032 1,200 0.14

10 6 1.682 0 0 0.368 1,200 0.1411 5 0 -1.682 0 0.2 528 0.1412 14 0 1.682 0 0.2 1,872 0.1413 12 0 0 -1.682 0.2 1,200 0.022414 19 0 0 1.682 0.2 1,200 0.257615 2 0 0 0 0.2 1,200 0.1416 4 0 0 0 0.2 1,200 0.1417 17 0 0 0 0.2 1,200 0.1418 16 0 0 0 0.2 1,200 0.1419 18 0 0 0 0.2 1,200 0.1420 15 0 0 0 0.2 1,200 0.14

46 3 Fractal Analysis in CNC Turning



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 parameterstaken by the researchers. In the present study these are selected as design factorswhile other parameters have been assumed to be constant over the experimentaldomain. The process variables with their values are listed in Table 3.1. For theexperimentation, a rotatable central composite design (Sect. 1.5.2) is selected andthe experimental plan consists of experiment run order, standard order, codedvalues 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 controlsystem FANUC Series Oi Mate-Tc and equipped with maximum spindle speed of3,500 rpm, feed rate 15–20 m/rev and KVA rating-16 KVA. For generating theturned surfaces, CNC part programs for tool paths were created with specificcommands.

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 inves-tigation. The tool holder is used as the PTGNR-25-25 M16 050, WIDIA and insertused as the TNMG 160404–FL, WIDIA. The tool is coated with titanium nitridecoating 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 of1:20 with water was used as cutting environment.


The present study was carried out with three different workpiece materials, viz.,6061-T4 aluminium, mild steel (AISI 1040) and medium leaded brass UNSC34000. All the specimens were in the form of bar with diameter 20 mm and


http://dx.doi.org/10.1007/978-3-642-17922-8_1

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


To get machined surfaces, CNC turning operations are carried out for differentcombinations of spindle speed, feed rate and depth of cut. Three different work-piece materials are considered viz. mild steel, brass and aluminium. The generatedsurfaces are measured using Talysurf instrument (Sect. 1.3.7) and further pro-cessed to get fractal dimension (D). The experimental results are used for furtheranalyses using response surface methodology (RSM) to model fractal dimension.For RSM, a rotatable central composite design of experiment is considered and theexperimental results are presented in Table 3.3.

The influences of the machining parameters on fractal dimension have beenassessed for three different materials using RSM. The whole analyses are doneusing Minitab software. The results of RSM analyses are presented below.

Table 3.3 Experimental results for CCD

Std. order D for mild steel D for brass D for aluminium

1 1.370 1.435 1.4172 1.300 1.437 1.3153 1.362 1.395 1.3924 1.410 1.420 1.4405 1.267 1.300 1.3006 1.282 1.292 1.3027 1.390 1.297 1.2928 1.417 1.297 1.2629 1.320 1.355 1.377

10 1.370 1.367 1.36011 1.300 1.375 1.39712 1.420 1.355 1.34713 1.360 1.380 1.48514 1.290 1.257 1.25215 1.397 1.350 1.36216 1.400 1.375 1.37017 1.415 1.375 1.38518 1.415 1.362 1.36719 1.402 1.377 1.30020 1.412 1.377 1.297


http://dx.doi.org/10.1007/978-3-642-17922-8_2

http://dx.doi.org/10.1007/978-3-642-17922-8_2

http://dx.doi.org/10.1007/978-3-642-17922-8_1


The second order response model is developed using Minitab in terms of codedvalues of the independent machining parameters, viz., work-piece speed, feed rateand depth of cut. The response model for mild steel material is given in thefollowing equation.

D¼1:40674þ0:00453dþ0:02446N�0:00883 f þ0:005745dNþ0:002873df

þ0:006850Nf �0:00697d2�0:00510N2�0:00947 f 2 ð3:1Þ

The developed model is also checked for adequacy. Table 3.4 represents theANOVA table for the second order response model developed for D. It is clear thatthe developed model is significant at 95% confidence level. The calculated valueof F ratio is greater than the tabulated value of F ratio and it can be concluded thatthe 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 withwork-piece speed are significant factors at 95% confidence level. The main effectsplots for fractal dimension are shown in Fig. 3.1. From the main effect plots, it isseen that work-piece speed and feed rate are significant. It can also be concludedthat 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

Table 3.4 ANOVA for second order model for mild steel

Source DF SS MS F F0.05 P

Regression 9 0.049 0.005,409 16.96 3.02 0Residual Error 10 0.0032 0.000319Total 19 0.052

Table 3.5 Full ANOVA table for mild steel model

Source Sum of squares DF Mean square F value P value

Model 0.049 9 0.005409 16.96 0.0001A–d 0.0007933 1 0.0007933 2.49 0.1458B–N 0.023 1 0.023 72.48 0.0001C–f 0.003009 1 0.003009 9.44 0.0118AB 0.00211 1 0.00211 6.62 0.0277AC 0.0005281 1 0.0005281 1.66 0.2271BC 0.003003 1 0.003003 9.42 0.1190A2 0.005608 1 0.005608 17.59 0.0018B2 0.002998 1 0.002998 9.40 0.0119C2 0.010 1 0.010 32.46 0.0002Residual 0.00318 10 0.0003189Lack-of-fit 0.002871 5 0.0005742 9.04 0.0152Pure error 0.0003177 5 0.00006354Cor total 0.05186 19


Minitab. Figs. 3.2, 3.3 and 3.4 show the estimated three dimensional surface aswell as contour plots for fractal dimension as functions of two independentmachining parameters while the third machining parameter is held constant. Allthese figures clearly depict the variation of fractal dimension with controllingvariables within the experimental regime.

3.3.2 RSM for Brass

The second order response model for brass material is presented in terms of codedvalues of work-piece speed, feed rate and depth of cut in Eq. 3.2.

D¼1:36919þ 0:00179 d� 0:00386 N � 0:03074 f þ 0:00133 dN � 0:00155 df

þ 0:00265 Nf � 1:21695� 10�04 d2þ 0:00035 N2� 0:00543 f 2 ð3:2Þ

The developed model is checked for adequacy and ANOVA result for themodel is presented in Table 3.6. From the ANOVA table, it is seen that the modelis 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 fractaldimension in brass turning. The calculated F-value of the lack-of-fit for D is muchlower than the tabulated value of the F-distribution (tabulated value 5.05) foundfrom the standard table at 95% confidence level. It implies that the lack-of-fit is notsignificant relative to pure error. From the main effect plot (Fig. 3.5), it is seen thatonly feed rate is significant and the other parameters are insignificant. It is also

Fig. 3.1 Main effect plots for mild steel


Fig. 3.2 Surface and contour plot of fractal dimension, D for mild steel: a at high level of spindlespeed, 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 depthof 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 feedrate, b at low level of feed rate


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

Table 3.6 ANOVA for second order model for brass



Table 3.7 Full ANOVA table for brass model

Source Sum of squares DF Mean square F value P value

Model 0.041 9 4.603E-3 13.56 0.0002A–d 1.232E-4 1 1.232E-4 0.36 0.5602B–N 5.753E-4 1 5.753E-4 1.70 0.2221C–f 0.036 1 0.036 107.57 0.0001AB 1.125E-4 1 1.125E-4 0.33 0.5775AC 1.531E-4 1 1.531E-4 0.45 0.5169BC 4.500E-4 1 4.500E-4 1.33 0.2763A2 1.707E-6 1 1.707E-6 5.032E-3 0.9448B2 1.389E-5 1 1.389E-5 0.041 0.8437C2 3.405E-3 1 3.405E-3 10.03 0.0100Residual 3.393E-3 10 3.189E-4Lack-of-fit 2.775E-3 5 5.551E-4 4.49 0.0624Pure Error 6.177E-4 5 1.235E-4Cor Total 0.045 19

Fig. 3.5 Main effect plots for brass


Fig. 3.6 Surface and contour plot of fractal dimension, D for brass: a at high level of spindlespeed, 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 ofcut, 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


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


The second order response model for aluminium material is presented in terms ofcoded 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 N2 þ 0:00039 f 2 ð3:3Þ

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

Table 3.9 Full ANOVA table for aluminium model

Source Sum of squares df Mean square F value P value

Model 0.052 9 5.841E-3 3.37 0.0359A–d 9.174E-4 1 9.174E-4 0.53 0.4825B–N 7.307E-5 1 7.307E-5 0.042 0.8410C–f 0.047 1 0.047 27.08 0.0004AB 1.726E-3 1 1.726E-3 1.000 0.3407AC 9.453E-5 1 9.453E-5 0.055 0.8196BC 2.720E-3 1 2.720E-3 1.580 0.2377A2 1.751E-5 1 1.751E-5 0.010 0.9217B2 8.496E-5 1 8.496E-5 0.049 0.8288C2 1.751E-5 1 1.751E-5 0.010 0.9217Residual 0.017 10 1.724E-3Lack-of-fit 9.952E-3 5 1.990E-3 1.36 0.3707Pure error 7.293E-3 5 1.459E-3Cor total 0.070 19

Table 3.8 ANOVA for second order model for aluminium




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 usingMinitab. Figs. 3.10, 3.11 and 3.12 show the estimated three dimensional surface aswell as contour plots for fractal dimension as functions of two independentmachining parameters. The third machining parameter is held constant at high andlow levels. From these figures, variations of fractal dimension with machiningparameters can be observed within the experimental regime.

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 ofspindle speed, b at low level of spindle speed


3.4 Closure

Response surface models for three materials viz. mild steel, brass and aluminiumare developed in CNC turning. All the developed second order models are ade-quate at 95% confidence level. From the analysis, it is seen that the work-piecespeed is the most significant factor affecting the fractal dimension for mild steelturning whereas feed rate is the significant factor for both brass and aluminiummaterials. It can be concluded from the analysis that for all the materials, withincrease 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 dimensionincreases giving smoother surface for mild steel turning.

Fig. 3.11 Surface and contour plot of fractal dimension, D for aluminium: a at high level ofdepth 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 feedrate, b at low level of feed rate


Chapter 4Fractal Analysis in Cylindrical Grinding

Abstract This chapter presents the fractal dimension modeling in cylindricalgrinding of mild steel, brass and aluminium work-pieces. The experimentations arecarried out for different combinations of work-piece speed, longitudinal feed andradial infeed. The generated surfaces are measured and processed to calculatefractal dimension. The experimental results are then analyzed with RSM. Thelongitudinal feed rate is the most significant factor affecting the fractal dimensionfor mild steel, whereas for brass, work-piece speed and longitudinal feed rate arethe most significant factors. For aluminium materials, all the three processparameters are the significant factors affecting fractal dimension.

Keywords Fractal dimension (D) � Cylindrical grinding � RSM � Mild steel �Brass � Aluminium

4.1 Introduction

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


57

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



The process parameters chosen here are work-piece speed (N) in rpm, longitudinalfeed (f) in mm/rev and radial infeed (d) in mm. The process variables/designfactors with their values on different levels are listed in Table 4.1 for three dif-ferent work-piece materials. The selection of the values of the variables is limitedby the capacity of the machine used in the experimentation as well as the rec-ommended specifications for different work-piece-tool material combination. Fourlevels, having nearly equal spacing, within the operating range of the parametersare selected for each of the factors. By selecting four levels, the curvature or non-linearity effects can be studied. In the present investigation, full factorial design ofexperiment is considered for the experimentation and for four level three factorstotal 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 machineequipped with maximum wheel speed of 1910 rpm. The wheel signature of themachine is A70K5V10 and wheel diameter of 270 mm. The maximum grindinglength is about 340 mm. The compressed coolant WS 50–50 with a ratio of 1:20was used as cutting environment. The details of the machine used in this study areshown in the Table 4.2.


The present study is carried out with three different materials, viz., AISI 1040steel, medium leaded brass UNS C34000 and 6061-T4 aluminium. The chemicalcomposition and mechanical properties of the work-piece materials are alreadydiscussed in the Chap. 2 (Table 2.3). All the specimens are in the form of roundbars of diameter 48 mm and length 50 mm.

Table 4.1 Process variables and their levels

Parameters Unit Notation 1 2 3 4

Work-piece speed rpm N 56 80 112 160Long feed mm/rev f 11.33 17.00 22.66 28.33Radial infeed mm d 0.02 0.04 0.06 0.08

58 4 Fractal Analysis in Cylindrical Grinding

http://dx.doi.org/10.1007/978-3-642-17922-8_2

http://dx.doi.org/10.1007/978-3-642-17922-8_2#Tab3


Cylindrical grinding operations are carried out on mild steel, brass and aluminiumwork-pieces to get machined surfaces for different combinations of work-piecespeed, longitudinal feed and radial infeed. The generated surfaces are measuredusing Talysurf instrument and further processed to get fractal dimension (D). Theexperimental results are used for further analyses using response surface meth-odology (RSM) to model fractal dimension. For RSM, full factorial design ofexperiments is considered and the design matrix and the experimental results ofcylindrical grinding of mild steel, brass and aluminium work-pieces are presentedin 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 (AISI1040), brass and aluminium grinding are presented below.


The second order response surface equation has been fitted using Minitab softwarefor the response variable D. The equation can be given in terms of the coded valuesof 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 0:00219N2 þ 0:00375f 2 þ 0:00375d2 ð4:1Þ

Table 4.2 Specification of the cylindrical grinding machine used in the experiment

Make HMT Model K130U MachineNo

57169

Maximum grinding length 340 mmMaximum distance between

centers340 mm

Maximum travel of the table 310 mmMaximum swivel of the table 200 mmGrinding wheelWheel speed 1910 and 2120 rpmWheel Signature A70K5V10Wheel Diameter 270 mmFace width 40 mmBore diameter 50 mmWork headNumber of speed 8 (56-80-112-160-224-315-

450-630)Swivel 90� towards wheel and 30�

away from wheelMorse taper 3


Table 4.3 Design matrix of process variables and the experimental results

Stdorder

Runorder

N Workpiecespeed (rpm)

f LongitudinalFeed (mm/rev)

d Radialinfeed (mm)

D formildsteel

D forbrass

D foraluminium

1 22 1 1 1 1.46 1.390 1.392 45 1 1 2 1.48 1.408 1.353 7 1 1 3 1.40 1.415 1.384 37 1 1 4 1.46 1.415 1.375 54 1 2 1 1.47 1.413 1.376 38 1 2 2 1.44 1.435 1.347 26 1 2 3 1.45 1.420 1.348 42 1 2 4 1.43 1.433 1.379 13 1 3 1 1.47 1.453 1.35

10 43 1 3 2 1.43 1.445 1.3411 63 1 3 3 1.42 1.420 1.3612 59 1 3 4 1.42 1.445 1.3513 32 1 4 1 1.41 1.455 1.3514 5 1 4 2 1.45 1.468 1.3715 40 1 4 3 1.44 1.450 1.3616 34 1 4 4 1.45 1.450 1.3517 51 2 1 1 1.49 1.413 1.4018 10 2 1 2 1.47 1.415 1.4119 14 2 1 3 1.45 1.428 1.3620 21 2 1 4 1.46 1.428 1.3821 1 2 2 1 1.43 1.440 1.3722 64 2 2 2 1.42 1.445 1.3723 48 2 2 3 1.44 1.430 1.3524 61 2 2 4 1.41 1.425 1.3625 23 2 3 1 1.47 1.455 1.3526 31 2 3 2 1.45 1.455 1.3427 53 2 3 3 1.35 1.448 1.3428 29 2 3 4 1.39 1.455 1.3529 12 2 4 1 1.45 1.460 1.3530 56 2 4 2 1.45 1.430 1.3631 2 2 4 3 1.43 1.443 1.3432 46 2 4 4 1.44 1.448 1.3533 25 3 1 1 1.47 1.415 1.3534 3 3 1 2 1.45 1.430 1.3935 19 3 1 3 1.46 1.420 1.3536 33 3 1 4 1.48 1.425 1.3737 8 3 2 1 1.48 1.393 1.3438 49 3 2 2 1.47 1.430 1.3739 17 3 2 3 1.44 1.428 1.3340 6 3 2 4 1.47 1.425 1.3741 18 3 3 1 1.42 1.455 1.35

(continued)


The analysis of variance (ANOVA) technique has been used to check theadequacy 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 thestandard tabulated value of table (F-table) for 95% confidence level, then themodel is considered adequate within the confidence limit. From Table 4.4, it isobserved that the developed model is adequate at 95% confidence level. From theANOVA table of individual parameters (Table 4.5), it can be concluded that thelongitudinal feed rate is the most significant factor affecting the fractal dimensionat 95% confidence level. The main effect plots of fractal dimension D is presentedin Fig. 4.1. From this figure, it is seen that longitudinal feed rate and radial infeedhave influences on fractal dimension. The estimated three dimensional surface as


Stdorder

Runorder

N Workpiecespeed (rpm)

f LongitudinalFeed (mm/rev)

d Radialinfeed (mm)

D formildsteel

D forbrass

D foraluminium

42 44 3 3 2 1.47 1.450 1.3443 15 3 3 3 1.43 1.453 1.3244 57 3 3 4 1.45 1.468 1.3545 36 3 4 1 1.43 1.463 1.3646 35 3 4 2 1.39 1.468 1.3647 28 3 4 3 1.45 1.428 1.3748 41 3 4 4 1.42 1.455 1.3949 11 4 1 1 1.45 1.440 1.4150 47 4 1 2 1.48 1.408 1.3851 30 4 1 3 1.44 1.435 1.3852 27 4 1 4 1.47 1.430 1.3853 39 4 2 1 1.47 1.453 1.3954 16 4 2 2 1.45 1.445 1.3555 9 4 2 3 1.46 1.460 1.3556 52 4 2 4 1.47 1.455 1.3557 24 4 3 1 1.47 1.470 1.3558 50 4 3 2 1.41 1.453 1.3459 4 4 3 3 1.46 1.450 1.3460 20 4 3 4 1.42 1.465 1.3561 62 4 4 1 1.45 1.472 1.4062 60 4 4 2 1.45 1.465 1.3963 58 4 4 3 1.45 1.470 1.4064 55 4 4 4 1.44 1.445 1.38

Table 4.4 ANOVA for the response model of D for mild steel

Source DF Seq SS Adj MS F F0.05 P

Regression 9 0.012291 0.001366 2.47 2.04 0.020Residual error 54 0.029903 0.000554Total 63 0.042194


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

4.3.2 RSM for Brass


D ¼ 1:36367þ 0:00134N þ 0:03714f þ 0:00450d � 0:00114Nf � 0:00060Nd

� 0:00300fd þ 0:00172N2 � 0:00289f 2 þ 0:00094d2

ð4:2Þ

Table 4.5 ANOVA for individual parameter of D for mild steel

Source DF SS MS Fcalculated F0.05 P

N 3 0.0021187 0.0007062 1.28 2.76 0.300f 3 0.0074563 0.0024854 4.52 2.76 0.011d 3 0.0034062 0.0011354 2.07 2.76 0.128N*f 9 0.0059187 0.0006576 1.20 2.04 0.337N*d 9 0.0034187 0.0003799 0.69 2.04 0.711f*d 9 0.0050312 0.0005590 1.02 2.04 0.452Error 27 0.0148437 0.0005498Total 63 0.0421937

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


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

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


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

Table 4.7 ANOVA for individual parameter of D for brass



Table 4.6 ANOVA for the response model of D for brass



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


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

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

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


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



D ¼ 1:48062� 0:01616N � 0:07047f � 0:02253d þ 0:00282Nf � 0:00097Nd

þ 0:00202fd þ 0:00297N2 þ 0:01078f 2 þ 0:00359d2 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 ofindividual parameters (Table 4.9), it can be concluded that the work-piece speed,longitudinal feed rate and radial infeed are the significant factors affecting thefractal dimension at 95% confidence level. Also the interaction between work-piece speed and longitudinal feed and between work-piece speed and radial infeedare significant at 95% confidence interval. The main effect plots of fractaldimension D is presented in Fig. 4.9. From this figure also, it is seen that work-piece speed, longitudinal feed and radial infeed are significant in the studied range.The variations of fractal dimension with two machining parameters are presentedin Figs. 4.10, 4.11, 4.12 while the third machining parameter is kept constant.

Table 4.8 ANOVA for the response model of D for aluminium



Table 4.9 ANOVA for individual parameter of D for aluminium




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 lowlevel 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


4.4 Closure

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

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


Chapter 5Fractal Analysis in EDM

Abstract In this chapter fractal dimension modeling in electrical dischargemachining is discussed. Machining operations are carried out for different com-binations of pulse current, pulse-on time and pulse-off time on mild steel, brass andtungsten carbide materials. The generated machined surfaces are measured tocalculate fractal dimension. The experimental results are then analyzed to modelfractal dimension using response surface methodology. From the response surfacemodels, it is seen that the effect of the cutting parameters on fractal dimension isdifferent for different materials. For tungsten carbide and brass, both pulse currentand pulse on time play a significant role in determining the fractal dimension whilefor mild steel it is only the pulse current that plays the significant role. A com-parison of the response surface models for fractal dimension in different materialsreveals the fact that these models are material specific.

Keywords Fractal dimension (D) � EDM � RSM � Mild steel � Brass � Tungstencarbide

5.1 Introduction

Electrical discharge machining (EDM) is a widespread machining technique usedfor all types of conductive materials including metals, metallic alloys, graphite,composites and ceramic materials. It is a non-conventional machining processused for machining of difficult-to-machine materials and shapes with high degreeof accuracy (El-Hofy 2005). It is based on removing material from a part by meansof a series of repeated electrical discharges created by electric pulse generated atshort intervals between two electrodes; a tool electrode and a work-piece elec-trode. The electrodes are separated by a dielectric fluid that makes it possible toflush eroded particles from the gap between the electrodes. The electric spark


69

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

Table 5.1 Variable levels used in the experimentation

Levels Current (I, amp) Pulse on time (ti, ls) Pulse off time (to, ls)

-1 3.125 50 500 6.250 100 751 9.375 150 100

Table 5.2 Design matrix of the FCC design (coded values and actual value of the factors)

Std. order Run order Coded value

Current (I) Pulse on time (ti) Pulse off time (t0)

1 5 -1 -1 -12 2 1 -1 -13 8 -1 1 -14 12 1 1 -15 18 -1 -1 16 16 1 -1 17 14 -1 1 18 1 1 1 19 9 -1 0 0

10 11 1 0 011 6 0 -1 012 13 0 1 013 19 0 0 -114 3 0 0 115 7 0 0 016 20 0 0 017 10 0 0 018 15 0 0 019 17 0 0 020 4 0 0 0

70 5 Fractal Analysis in EDM



There are a large number of factors that can be considered for machining of aparticular material in EDM. It is very difficult to conduct the experiment withconsidering all the process variables. However, the review of literature shows thatthe following three machining parameters are the most widespread among theresearchers 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 areselected as design factors while other parameters have been assumed to be constantover the experimental domain. A face-centered central composite (FCC) design isused with three levels of each of the three design factors. Considering three factorsand 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 factorare coded as +1 and -1 respectively, the coded value being calculated usingEq. 1.8. The process variables with their values on different levels are listed inTable 5.1. The selection of the values of the variables is limited by the capacity of

Table 5.3 Specification of the equipment used in the experimentation

Particulars Specification

Trade name TOOL CRAFT A 25Type of construction ‘C’ typeWorktable 300 mm 9 200 mmFixed work tank 465 mm 9 270 mm 9 200 mmTable longitudinal movement 100 mmTable cross movement 175 mmMaximum dielectric level over table 140 mmMaximum work piece height 90 mmMaximum work piece weight 45 kgServo headServo system Stepped driveQuill travel 150 mmElectrode platen size 100 mm sqAccuracy of quill movement 0.01 mm over 200 mmDielectric systemFiltration flushing better than 10 lFlushing side, 1.23 l/min (max)Flushing pressure 15 kPaGeneratorModels A 25Working current 25 A maximum through current selectorPulse on time setting 2–2,000 lsPulse off time setting 2–2,000 lsPower source connection 400/440 V, 50 Hz, 3-ph supply


http://dx.doi.org/10.1007/978-3-642-17922-8_1

the machine used in the experimentation as well as the recommended specifica-tions for different workpiece–tool material combinations. Table 5.2 shows theexperimental 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 flushingcapability. It is capable of generating maximum pulse current of 25 A, pulse ontime of 2,000 ls and pulse off time of 2,000 ls. The specification of the machineis presented in Table 5.3.


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-piecematerials are shown in Table 5.4. All the specimens were in the form of20 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 work-piece materials considered in the present study. The tool electrode was in the formof cylinder of diameter 15.9 mm and 50 mm in length mounted axially in line withwork-piece. The tool electrode was given negative polarity where as work-piece is

Table 5.4 Composition and electrical/thermal properties of work-piece materials

Work Material Composition (%Wt) Electrical and thermal property

Tungsten carbide 94%WC–6%Co Electrical resistivity: 6 9 10-5

ohm-cmThermal conductivity: 84 W/m-KMelting point: 2850�C

Mild Steel (AISI1040)

0.42%C, 0.48%Mn, 0.17%Si, 0.02%P,0.018%S, 0.1%Cu, 0.09%Ni,0.07%Crand balance Fe

Electrical resistivity:1.7 9 10-5

ohm-cmThermal conductivity: 52 W/m-KMelting point:1515�C

Brass (UNSC34000)

0.095%Fe, 0.9%Pb, 34%Zn andbalance Cu

Electrical resistivity:6.6 9 10-6

ohm-cmThermal conductivity:115�W/m-KMelting point: 900�C


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


As mentioned earlier, according to FCC design of experiments, machining oper-ations are carried out to generate machined surfaces (EDMed). The generatedmachined surfaces are measured with Talysurf and further processed to calculate

Table 5.6 Experimental results

Std. order Run order D for WC D for MS D for Brass

1 5 1.383 1.413 1.4402 2 1.350 1.310 1.4063 8 1.356 1.330 1.4304 12 1.250 1.276 1.4005 18 1.410 1.426 1.4536 16 1.313 1.306 1.4237 14 1.356 1.426 1.4208 1 1.216 1.283 1.3869 9 1.390 1.333 1.413

10 11 1.270 1.286 1.41011 6 1.323 1.346 1.44012 13 1.250 1.343 1.42013 19 1.320 1.346 1.43014 3 1.386 1.316 1.40615 7 1.263 1.356 1.42016 20 1.313 1.306 1.42317 10 1.310 1.363 1.41018 15 1.343 1.306 1.40019 17 1.310 1.330 1.41620 4 1.293 1.316 1.416

Table 5.5 Electrode material properties

Particulars Specifications

Material Electrolytic copperComposition 99.09% copperDensity 8 904 kg/mm3

Melting point 1083 C�Conductivity 101.41% IACSTensile strength 23.47 kg/mm2


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


The second order response surface equation for fractal in EDM of mild steel isobtained 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 tito � 0:0127 I2 þ 0:0223 t2i þ 0:0089 t2

o 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 forD given in Eq. 5.1. The developed model is significant at 95% confidence level asthe P-value is less than 0.05. Also the model is adequate at 95% confidence levelto represent the relationship between the machining response and the consideredmachining parameters as the calculated value of the F-ratio is more than thestandard value of the F-ratio for D. Table 5.8 represents the ANOVA table forindividual machining parameters where it can be seen that only pulse current is thesignificant parameter at 95% confidence level. Figure 5.1 shows the main effectsplot for the fractal dimension. From this figure also, it is seen that pulse current has

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

Regression 9 0.029628 0.029628 0.003292 4.66 3.02 0.012Residual Error 10 0.007057 0.007057 0.000706Total 19 0.036685

Table 5.8 ANOVA for machining parameters for D in EDM of mild steel


I 2 0.021962 0.022226 0.011113 14.96 3.81 0ti 2 0.004156 0.003419 0.001709 2.3 3.81 0.139t0 2 0.000912 0.000912 0.000456 0.61 3.81 0.556Error 13 0.009656 0.009656 0.000743Total 19 0.036685


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

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


significant effect on fractal dimension while pulse on time and pulse off time haveno effect on fractal dimension of the surface topography generated in EDM of mildsteel. The estimated three-dimensional surface as well as contour plots for fractaldimension 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 parameterswhile the third machining parameter is held constant. All these figures clearlydepict the variation of fractal dimension with controlling variables within theexperimental regime.

5.3.2 RSM for Brass

The second order response surface equation for the fractal dimension of brass sur-faces 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 tito � 0:0068 I2

þ 0:0115 t2i � 0:0002 t2

o 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 forD given in Eq. 5.2. It is seen that the developed model is significant at 95%

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

Table 5.9 ANOVA for second order model for D in EDM of brass


Regression 9 0.003635 0.003635 0.000404 4.11 3.02 0.019Residual Error 10 0.000982 0.000982 0.000098Total 19 0.004616


confidence level. Also the calculated value of the F-ratio is more than the standardvalue of the F-ratio for D which implies the model is adequate at 95% confidencelevel to represent the relationship between the machining response and the con-sidered machining parameters of the EDM process on brass. Table 5.10 representsthe ANOVA table for individual machining parameters where it can be seen thatpulse current and pulse on time are the significant factors affecting fractaldimension. Figure 5.5 depicts the main effects plot for the fractal dimension andthe design factors considered. From this figure also, it is seen that pulse current andpulse on time have the significant effect on fractal dimension. Figures 5.6, 5.7, and5.8 shows the estimated three-dimensional surface as well as contour plots forfractal dimension. To draw these surface plots, fractal dimension is plotted asfunctions of two independent machining parameters while the third machiningparameter is held constant. All these figures clearly show the variation of fractaldimension 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 softwarefor the response variable D. The equation can be given in terms of the coded valuesof the independent variables as:

Table 5.10 ANOVA for machining parameters for D in EDM of brass


I 2 0.001693 0.00182 0.00091 8.84 3.81 0.004ti 2 0.001557 0.001502 0.000751 7.3 3.81 0.008t0 2 2.83E - 05 2.83E - 05 1.42E - 05 0.14 3.81 0.873Error 13 0.001338 0.001338 0.000103Total 19 0.004616

Fig. 5.5 Main effect plot offractal dimension for brass


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 offtime, (b) at low level of pulse off time

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


D ¼ 1:31� 0:0497 I � 0:035ti þ 0:0023 to � 0:0146Iti � 0:0121 Ito

� 0:0029 tito þ 0:0138 I2 � 0:0296 t2i þ 0:0371 t2

o ð5:3Þ

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

Table 5.11 ANOVA for second order model for D in EDM of tungsten carbide


Regression 9 0.046159 0.046159 0.005129 7.75 3.02 0.002Residual error 10 0.006621 0.006621 0.000662Total 19 0.05278

Table 5.12 ANOVA for machining parameters for D in EDM of tungsten carbide


I 2 0.026341 0.025185 0.012592 17.12 3.81 0ti 2 0.01304 0.014658 0.007329 9.97 3.81 0.002t0 2 0.003839 0.003839 0.00192 2.61 3.81 0.111Error 13 0.00956 0.00956 0.000735Total 19 0.05278

Fig. 5.9 Main effect plot offractal dimension for tungstencarbide


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

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

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


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

5.4 Closure

Response surface models are developed for fractal dimension in EDM of threedifferent materials. A comparison of the response surface models reveals the factthat these models are material specific or in other words, the tool–workpiecematerial combination plays a vital role in fractal dimension modeling. Also theeffect of the cutting parameters on fractal dimension is different for differentmaterials as evidenced from Table 5.8, Table 5.10 and Table 5.12. For tungstencarbide and brass, both pulse current and pulse on time play a significant role indetermining the fractal dimension while for mild steel it is only the pulse currentthat plays the significant role. Accordingly, optimum machining parameter com-binations for fractal dimension depend greatly on the workpiece material withinthe experimental domain. However, it can be concluded that it is possible to selecta combination of pulse current, pulse on time and pulse off time for achieving thesurface topography with desired fractal dimension within the constraints of theavailable machine.

References

El-Hofy HAG (2005) Advanced machining processes. McGraw-Hill, New YorkPuertas I, Luis CJ, Villa G (2005) Spacing roughness parameters study on the EDM of silicon

carbide. J Mater Process Technol 164–165:1590–1596


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