FRACTAL CHARACTERIZATION AND RESPONSE SURFACE OPTIMIZATION OF MACHINED SURFACE TOPOGRAPHY P Sahoo*, K N Saha and G Pohit Department of Mechanical Engineering, Jadavpur University, Kolkata – 700032, India. * Corresponding author (Email: [email protected], Telefax: +91 33 2414 6890) Abstract Conventionally, surface topography is characterized by parameters such as centerline average value, root mean square average, mean line peak spacing. As the surface topography is non-stationary and multi-scaled, these parameters are not sufficiently descriptive characteristics of the surface. Roughness can be described by fractal dimension. Fractal dimension describes surface roughness invariant with length scale. The present work describes the evaluation of fractal dimension of machined surface topography generated in turning and grinding of MS and Al work-pieces. ANOVA analysis is done to find out significant factors affecting fractal dimension. Using Response Surface Methodology (RSM), the relationships between fractal dimensions with cutting parameters are found out. Using response optimization, a combination of cutting parameters is obtained to optimize the fractal dimension. Keywords: Fractal Dimension, ANOVA, RSM, Optimization, Turning, Grinding

1.

Introduction

Three statistical characteristics are generally used to describe the structure of rough surface topography: texture, waviness and roughness. The texture determines the anisotropic property of the surface. The waviness reflects the reference profile or surface. The surface roughness is formed by the micro deformation during the machining process. The roughness influences physical phenomena, which are relevant to many engineering problems such as wear, contact resistance, wave scattering and pool boiling. Conventionally, by three different types of parameters, viz., amplitude parameters, spacing parameters and hybrid parameters, roughness is characterized. Observations show that the deviation of a surface from its mean plane is non-stationary random process [1]. Due to the multi-scale nature of the surface, the variances and derivatives of surface peaks and other roughness parameters strongly depends on the resolution and filter processing of the measuring instruments. Rough surface should be characterized in such a way that the structural information of roughness is retained at all scale. To do so, quantifying the multi-scale nature of surface roughness is essential. The similarity of a surface profile under different magnifications can be statistically characterized by fractal geometry since its topography is statistically self-affine. The ability to characterize surface roughness using scale independent parameters is a specific feature of fractal approach. Based on Mandelbrot’s work [2, 3], many researchers have attempted to describe and model rough surfaces using the fractal geometry theory [4-21]. Substantial investigation indicates that the surface topography has self-affine fractal characteristics [3, 6, 9, 12]. Initially, Gagnepain and Roques-Crames [4] approached the 3-D roughness surface using random walk noise and white noise. The fractal dimension was calculated using the Box

counting method. Majumdar and Bhushan [5] simulated the machining surface based on the modified Weirstrass-Mandelbrot function, which is called the Majumdar-Bhushan function. Majumdar and Bhushan [5] thought that the self-affine surface could be characterized using two fractal parameters; namely the fractal dimension and the amplitude coefficient. Based on the fractal characteristics of a random Cantor set, Thomas and Krajcinovic [13] established a new model for elastic-perfectly plastic contact between surfaces. Until now, many fractal models have been developed and applied for surface simulation [6], elastic-plastic contact [7, 13], tribology [8], surface texture [9], adhesion [15], friction [16], wear [10, 11, 17] and so on. In a material removal process, mechanical intervention happens over length scales, which extend from atomic dimensions to centimeters. The machine vibration, clearances and tolerances affect the outcome of the process at the largest of length scales (above 10-3 m). The tool form, feed rate, tool radius in the case of single point cutting and grit size in multiple point cutting, affect the process outcome at the intermediate length scales (10-6 to 10-3 m). The roughness of the tool or details of the grit surfaces influence the final topography of the generated surface at the lowest length scales (10-9 to 10-6 m). It has been shown that surfaces formed by electric discharge machining, cutting or grinding [9], and worn surfaces [10, 11] have fractal structures, and fractal parameters can reflect the intrinsic properties of surfaces to overcome the disadvantages of conventional roughness parameters. However, there is lack of information regarding the characteristics of roughness generated in turning and grinding particularly with respect to fractal dimension. Thus there is scope and need for further study in this respect. In the present work, the surface profiles generated by turning and grinding of mild steel and aluminium are measured, digitized and processed to evaluate the fractal dimension. The relation between the fractal dimension and cutting parameters such as cutting speed, depth of cut and feed rate etc. is investigated using response surface methodology. Finally an attempt is made to optimize the cutting parameters with respect to fractal dimension using response optimization. ANOVA test is used to find out significant factors affecting the response. In order to get a prescribed surface, a proper combination of cutting parameters is necessary. Response surface method is used to develop a relation between fractal dimension and cutting parameters. The response surface method (RSM) is practical, economical and relatively easy for use. The experimental data are used to build a second order mathematical model to predict the fractal dimension. This method has been used by some researchers [22-27] for tool life and surface roughness.

2.

Fractal Dimension Evaluation

Fractal calculation mainly includes the calculation of profile fractal dimension (1
of this calculation process at various step lengths allows all the curve length to be evaluated. Further, plotting of the curve lengths verses the step lengths on a log-log scale gives the slope m of a fitting line to be related to the fractal dimension D as D=1-m. It is possible that this method has abandoned some pivotal points resulting in calculation error. The principle of box counting method mainly involves an iteration operation to an initial square, whose area is supposed to be 1 and which covers the entire graph. The initial square is divided into four sub-squares and so on. After the n times operations, the number of sub-squares, which contain the discrete points of the profile graph are counted and the length L of the profile is approximately obtained. Then the fractal dimension is calculated as D=1+log L/(n.log2). The variation method has the advantage of being proven theoretically for all profiles (self-affine or not), and of giving quickly an estimation of the dimension of mathematical profiles. A well-known technique used to analyze surfaces consists in performing ‘slices’ through the surfaces, which allows one to transform a 3-D problem to 2-D. In other words, a surface is replaced by profiles, taken at different places, and the fractal dimension estimated over profiles is then related to the 3-D fractal dimension by the classical result: dimension of surface = 1 + dimension of profiles. Such a technique obviously decreases the problem size. Accurate results are hard to obtain for the surface dimension and the variation method gives the best approximations. The variation method algorithm is based on the local oscillation of the profile function Z. The power spectrum method involves the evaluation of the power of the profile function. The modified Weierstrass–Mandelbrot (W–M) function for a rough surface can be α

2πγ x described as z ( x) = G ( D −1) ∑ cos ; 11 γ ( 2− D ) n n

where D is the fractal dimension; γn the

n = n1

discrete frequency spectrum of the surface roughness; and n1 the low cut-off frequency of the profile and G the characteristic length scale of the surface. The multi-scale nature of z(x) can be characterized by its power spectrum, which gives the amplitude of the roughness at all length scales. The parameters G and D can be found from the power spectrum of the W–M G 2( D −1) 1 function by S (ω ) = where S(ω) is the power of the spectrum, and ω is the 2 ln γ ω 5−2 D frequency of the surface roughness profile. Usually, the power law behavior would result in a straight line if S(ω) is plotted as a function of ω on a log–log graph. Using fast fourier transform (FFT), the power spectrum of profile can be calculated and then be plotted verses the frequency on a log–log scale. Thereafter, the fractal dimension, D, can be related to the slope m of a fitting line on a log–log plot as: D = ½(5 +m). The structure function method considers all points on the surface profile curve as a time sequence z(x) with fractal character. The structure function s(τ) of sampling data on the profile curve can be described as s(τ) = [z(x + τ) - z(x)]2 = cτ 4-2D where [z(x+τ)-z(x)]2 expresses the arithmetic average value of difference square, and τ is the random choice value of data interval. Different τ and the corresponding s(τ) can be plotted verses the τ on a log– log scale. Then, the fractal dimension D can be related to the slope m of a fitting line on log– log plot as: D = ½ (4 - m).

3.

Basics of ANOVA and RSM

To determine the significant factors or factor interactions in affecting the surface roughness, measured roughness parameters and fractal dimension are tested in analysis of variance (ANOVA). Fisher test factor (F) and Pooled factor for estimating error variance (p)

are used to see the significant factors affecting the response [28]. The analysis is done with the help of MINITAB software (Release 13.1) based on 95% confidence level. Statistical methods for the generation of the parameter values for several designs are known as Design of Experiments (DOE) methods. The real experiment or computer analysis results of interest are called response variables or simply responses and can be used to fit to them an approximate analytical model. Such an analytical model is known as a Response Surface approximation and the related concepts are known as the Response Surface methodology (RSM). Response surface modeling methods originally were developed to analyze experimental data and to create empirical models of the observed response values. The eventual objective of RSM is to determine the optimum operating conditions for the system or to determine a region of the factor space in which operating requirements are satisfied. Since the exact and explicit relationship between the design variables and the responses is impossible to be obtained, a possible strategy to resolve the problem is to approximate the relationship with an empirical mode of the form: n

n

i =1

i =1

y(x) = y(x1,x2,x3,….xn) = a o + ∑ ai xi + ∑

n

∑a j =1

ij

xi x j

where x= (x1, x2,…….xn) is the vector of design variables, n is the number of design variables and a0, ai, aij are the coefficients of the polynomial obtained using least square fitting techniques. There are many different designs to fit response surfaces viz. Central Composite Design (CCD), the Box-Behnken Design, Planckett-Burman Design etc. Box-Behnken has proposed some three-level designs for fitting response surfaces. These designs are formed by combining 2k factorials with incomplete block. The Box-Behnken design is a spherical design, with all points lying on a sphere of radius 2 and in the present study the same has been used for response surface analysis.

4.

Experimental Details

In the experimental study, work-pieces of MS and Al are turned and ground for different combination of cutting parameters (according to the requirement of ANOVA and RSM analysis). The machined surfaces are then measured with the help of Talysurf instrument and processed through the Talyprofile software to get the fractal dimension. These data are used for ANOVA and RSM analysis. Mild Steel (C-45 medium carbon steel equivalent to AISI 1045 grade) and aluminium are used as work-piece materials for turning and grinding operations. They are produced as cylindrical bars of 40 mm diameter and 48.94 mm diameter for turning and grinding operations respectively. Turning operation is done in an Engine Lathe (Make- Mysore Kirloskar Ltd.). A three-jaw chuck is used to hold the workpiece. The carbide tip cutting tool is rigidly held and supported in a tool post. The work-piece is produced as a cylindrical bar. For different combinations of rpm, feed and depth of cut machining is carried out. During machining, no chatter, which would create significant vibration, is identified. Taking carbide tip tool for turning operations, the effect of tool angle on surface roughness is eliminated. Grinding operation is carried out in a universal type cylindrical grinding machine (Make – HMT). For different combinations of rpm, longitudinal feed and radial infeed plunge-cut cylindrical grinding operation is carried out for both the material. Though the length of the cylindrical bar is 50 mm but grinding operation is carried out in 15 mm length. The rpm of the wheel is at a constant value of 2120 rpm. Oil-emulsion is used as a coolant at the time of grinding. The surface roughness parameters on the generated surfaces are measured with the Talysurf (Make – Taylor Hobson, UK). The Talysurf instrument (Surtronic 3+) is a portable, self-contained instrument for the

measurement of surface texture. The parameter evaluations are microprocessor based. The measurement results are displayed on an LCD screen and can be sent to an optional printer or another computer for further evaluation. The instrument is powered by non-rechargeable alkaline battery (9V). It is equipped with a diamond stylus having a tip radius 5 µm. The measuring stroke always starts from the extreme outward position. At the end of the measurement the pickup returns to the position ready for the next measurement. The selection of cut-off length determines the traverse length. Usually as a default, the traverse length is five times the cut-off length though the magnification factor can be changed. The measured profile of Talysurf is digitized and processed through the advanced surface finish analysis software Talyprofile for evaluation of the fractal dimension. Fractal dimension is evaluated using structure function method.

5.

Results and Discussion

ANOVA test results and RSM test results are presented below for different materials, viz., MS and Al and operations, viz., turning and grinding. 5.1.

ANOVA Test

For turning operation of mild steel, the results are presented in Fig. 1 and Fig. 2. This analysis considers all the three cutting parameters, viz., RPM, feed rate, depth of cut and their two-factor interaction terms. Based on a 95% confidence interval, RPM, feed rate and interaction of feed rate and RPM have a statistically significant impact on fractal dimension, since their p-values are smaller than 5%. All the other factors have no significant impact on fractal dimension. This can be observed very clearly from the main effects plot given in Fig. 1 and interaction plot for D given in Fig. 2.

Fig. 1 Main effects plot for turning of MS

For grinding of mild steel, at 95% confidence level, longitudinal feed has the significant effect on fractal dimension. With increasing longitudinal feed rate, fractal dimension (D) decreases (Fig. 3 and Fig. 4). Though, RPM and radial infeed have no statistical significance on D but from main effects plot (Fig. 3) it can be said that while D increases with RPM, it decreases with radial infeed.

Fig. 2 Interaction plots for turning of MS

Fig. 3 Main effects plot for grinding of MS

Fig. 4 Interaction plots for grinding of MS

For turning operation of Al, analysis considers full factorial design. Based on a 95% confidence interval, feed rate and interaction of feed rate and RPM have a statistically significant impact on fractal dimension, since their p-values are smaller than 0.05. All the other factors have no significant impact on fractal dimension. This can be verified clearly from the main effects plot given in Fig. 5 and interaction plot for D given in Fig. 6.

Fig. 5 Main effects plot for turning of Al

Fig. 6 Interaction plots for turning of Al

Fig. 7 Main effects plot for grinding of Al

Fig. 8 Interaction plots for grinding of Al

ANOVA test result for grinding of Al shows that at 95% confidence level, longitudinal feed and interaction of RPM and radial infeed have the significant effect on fractal dimension (Fig. 7 and Fig. 8). 5.2.

Response Surface Methodology

For turning operation of mild steel, the MINITAB outputs are given below. A general equation among fractal dimension (D), RPM, feed rate (mm/rev) and depth of cut (mm/cut) is found out. The coefficients of the second order response surface are given in Table 1. The analysis of variance (ANOVA) test for the whole regression model is done that reveals that at 95% confidence level, the linear and interaction terms are not significant but square terms are significant. Also, overall regression model is significant. Fig. 9 shows the three dimensional response surface plots for the fractal dimension in terms of the process variables. The surface plots show how the fractal dimension relates to two other process variables based on the model equation. These represent the functional relationship between the response and the experimental variables. For holding value of 1000 RPM, it is seen that response is maximum at about feed rate of 0.75 mm/rev and depth of cut of 0.2 mm/cut. But for holding value of 270 RPM, the maximum is at 0.056 mm/rev feed rate and 0.1 mm/cut depth of cut. One can estimate the response optimization/minimization in the studied range in the same way from other surface plots. These surface plots also show that the response surface (D) exhibits curvature. In the studied range, it is seen that the D increase first with the feed rate and RPM but decreases beyond a certain limit keeping depth of cut constant. Response optimization has been done based on the experimental data. For maximization of D, global solution occurs when rpm is 1000, feed rate is 0.08 mm/rev and depth of cut is 0.2 mm/cut and the optimized response is D = 1.52.

Table 1 Coefficients of response surface for turning of MS

Term Coefficients Constant 1.50692 RPM -0.000388968 Feed 13.3728 Depth -4.31541 RPM*RPM 2.111090E-08 Feed*Feed -152.663 Depth*Depth 10.3750 RPM*Feed 0.00507583 RPM*Depth 0.000890411 Feed*Depth 10.7143

Fig.9 Surface plot for turning of MS

For grinding of mild steel, the coefficients of the second order response surface are given in Table 2. The ANOVA test for the whole regression model is performed and at 95% confidence level, the linear and square terms of the RSM are significant. Also, overall regression model is significant. Surface plots are shown in Fig 10. Using response optimization, general optimized cutting parameters are chosen. It is seen that the optimum fractal dimension occurs at RPM = 160, Long. Feed = 11.33 mm/s, Radial Infeed = 0.020 mm and the predicted response is 1.48271. Table 2 Coefficients of response surface for grinding of MS

Term Constant RPM LongFeed RadialIn RPM*RPM LongFeed*LongFeed RadialIn*RadialIn RPM*LongFeed RPM*RadialIn LongFeed*RadialIn

Coefficients 1.35632 0.000455344 0.0161841 0.0316506 -1.07865E-06 -0.000616712 -16.6667 -1.90950E-05 0.00120192 0.0606796

Fig.10 Surface plot for grinding of MS

The second order model is found out using RSM for the relationship between the fractal dimension and the cutting parameters for turning of Al. The coefficients of response surface are given in Table 3. From these coefficients of the equation, it is seen that the feed rate has the most dominant effect on fractal dimension value in the studied range. It is also seen from the ANOVA test that the square terms are the most significant factors for predicting fractal dimension. Surface plots are also plotted for holding values of RPM, feed rate and depth of cut (Fig 11). Using response optimization, general optimized cutting parameters is chosen: It is seen that the optimum fractal dimension occurs at RPM = 270, Feed Rate = 0.028 mm/rev, Depth of cut = 0.135 mm/cut, and the predicted response is 1.35. Table 3 Coefficients of response surface for turning of Al

Term Constant RPM Feed Depth RPM*RPM Feed*Feed Depth*Depth RPM*Feed RPM*Depth Feed*Depth

Coefficients 2.38628 0.000975515 -16.4678 -10.020 -6.77113E-07 --28.9647 11.6667 0.00244618 -.000126712 105.8

Fig.11 Surface plot for turning of Al

For grinding of Al, the relationship between the fractal dimension and the cutting parameters is also obtained. Based on the analysis, the regression coefficients for the estimation of fractal dimension are given in Table 4. Table 4 Coefficients of response surface for grinding of Al

Term Constant RPM LongFeed RadialIn RPM*RPM LongFeed*LongFeed RadialIn*RadialIn RPM*LongFeed RPM*RadialIn LongFeed*RadialIn

Coefficients 1.2114 0.000569527 0.0216792 2.52764 9.245562E-7 -0.000525828 -20.3125 -2.75816E-05 -0.00300481 -0.0661959

Fig.12 Surface plot for grinding of Al

It can be seen that the fractal dimension increases with the increase of RPM, longitudinal feed and radial infeed. The radial infeed has the most dominant effect on surface roughness value in the studied range. It is also seen from the ANOVA test that the linear and square terms are the most significant factors while the interaction of the parameters are not significant at 95% confidence level. Surface plots are also done for high and low values of RPM, longitudinal feed and radial infeed (Fig. 12). It is seen that the fractal dimension exhibits curvature. From the surface plot, a combination of cutting parameters can be chosen for a desired fractal dimension. Using response optimization, general optimized cutting parameters is chosen: It is seen that the optimum fractal dimension occurs at RPM = 108, Long Feed = 16.995 mm/s, Radial Infeed = 0.040 mm and the predicted response is D = 1.37.

6.

Conclusions

The experiments are carried out by generating surfaces using different machining processes and materials with a motivation to study the effect of machining parameters on fractal dimension and also to find our a relationship between machining parameters and fractal dimension. Using ANOVA test, the significant factors affecting the fractal dimension for different machining processes are determined. For turning of MS, RPM and feed rate are the main parameters affecting the fractal dimension. In case of turning of Al, feed rate and the interaction of feed rate and RPM are the main affecting factor. In case of grinding of both MS and Al, longitudinal feed rate is the most significant factor. In general, it may be concluded that feed rate in case of turning and longitudinal feed in case of grinding are the significant factors that affect the fractal dimension of the surfaces generated. Relationship among fractal dimension (D), RPM, feed rate and depth of cut is found out using RSM. For turning of MS and Al, it is seen that the feed rate is the main part of the second order response equation and tally with ANOVA tests. For grinding of MS and Al, the response surfaces show that the linear terms as well as square terms are significant to predict the fractal dimension.

7.

Acknowledgement

The first author gratefully acknowledges the financial assistance of Department of Science and Technology, Govt. of India through a Science and Engineering Research Council (SERC) Fast Track Project for young scientists vide Ref. No. SR/FTP/ETA-11/2004 dated 28.06.2004.

8.

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