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International Journal of JOURNAL OF MECHANICAL ENGINEERING INTERNATIONAL Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEME AND TECHNOLOGY (IJMET) ISSN 0976 – 6340 (Print) ISSN 0976 – 6359 (Online) IJMET Volume 3, Issue 3, September - December (2012), pp. 394-403 © IAEME: www.iaeme.com/ijmet.asp ©IAEME Journal Impact Factor (2012): 3.8071 (Calculated by GISI) www.jifactor.com RELEVANCE VECTOR MACHINE BASED PREDICTION OF MRR AND SR FOR ELECTRO CHEMICAL MACHINING PROCESS Kanhu Charan Nayak1,Rajesh Ku. Tripathy1,Sudha Rani Panda2 1 National Institute of Technology, Rourkela, India 2 Biju Pattnaik University of Technology, Rourkela, India nayakkanhu83@gmail.com,rajesh.nitr11@gmail.com,sanjimuni@gmail.com ABSTRACT Relevance vector machines (RVM) was recently proposed and derived from statistical learning theory. It is marked as supervised learning based regression method and based on Bayesian formulation of a linear model with prior to sparse representation. Not only it is used for Classification but also it can handle regression method very handsomely. In this research the important performance parameters such as the material removal rate (MRR) and surface roughness (SR) are affected by various machining parameters namely flow rate of electrolyte, voltage and feed rate in the electrochemical machining process (ECM). We use RVM model for the prediction of MRR and SR of EN19 tool steel. The experimental design was done by Taguchi technique. The input parameters used for the model are flow rate of electrolyte, voltage and feed rate. At the output, the model predicts both MRR and SR. The performance of the model is determined by regression test error which can be obtained by comparing both predicted and experimental output. Our result shows the regression error is minimized by using Laplace kernel function RVM. Key words: Electrochemical machining, EN19 tool steel, Material removal rate, Relevance vector machine, surface roughness 1. INTRODUCTION In the recent years there is an increasing demand for the industry in modern manufacturing process. The use of new materials having high strength, high resistance and better shape and size can increase the demand of product with better accuracy than non conventional machining process. Electro chemical machining is one of the recent methods used for working extremely hard materials which are difficult to machine using conventional methods. Extensive development of the process has taken place in the recent years mainly due to the need to machine harder and tough materials, the increasing cost of manual labour and the need to machine configurations beyond the capability of conventional machining methods. This method gives high material removal rate (about 1500mm3/min) and excellent surface 394 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEME finish (0.1 to 2.5 microns) with no residual stress and thermal damage due to low temperature during operation [1]. It has tremendous application in the aerospace industry, automotive, forging dies, and surgical component. So it is required to investigate the effect of machining parameters on machining performance (material removal and surface roughness) for alloy steel. Due to high production cost and high energy required for machining, the study of machining performance is difficult by conducting number of experiment with various machining parameter setting. To debug this difficulty, different types of mathematical modelling are used for prediction of machining performance considering different setting of input parameters. During recent decades a number of mathematical methods are used for regression analysis. The relevance vector machine has recently proposed by the research community as they have a number of advantages. This RVM is mainly based on a Bayesian formulation of linear models with prior to sparse representation. It is used for both classifications as well as regression problems. A General Bayesian framework for obtaining the sparse solutions to classification and regression tasks utilizing this RVM model is given by tipping [2]. The Bayesian approach has the extra advantage that it can be seamlessly incorporated into the RVM framework and requires much less computation time to optimize the regression error which is modelled as probabilistic distribution [3-5]. In this research the various machining parameter settings were done by using Taguchi technique which was a statistical method for designing high quality systems. This Taguchi method uses a special design of orthogonal array to study the entire parameter space with a small number of experiments [6]. Here this method is proposed to evaluate MRR and SR for ECM process. This present study initiated to development of a multi input multi output RVM regression model to predict the values of MRR and SR for the ECM process. The three process parameters namely feed, voltage and flow rate of electrolyte with different levels were designed for experiment by implementing the Taguchi method. After prediction using RVM model, the predicted value and experimental value is compared and root mean square error is calculated. 2. EXPERIMENTAL DETAILS 2.1 Experimental set up The experiments are carried out utilizing Electro-chemical machine unit as shown in fig. 1. Before machining we take the initial weight of work piece (EN19 tool steel) and after machining of 5minutes again we take the final weight. Initial weight, final weight, machining time and density of work material give the material removal rate per unit time when put in equation (1). ( )× × MRR= × (1) Where, W0=initial weight in Kg, W1=weight after one machining in Kg, ρw= density of material in Kg/m3 395 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEME Figure 1 Electro-chemical machining unit for conducting Experiment And after the end of each machining we have measured the surface roughness. Material removal based on anodic dissolution and the electrolyte flows between the electrodes and carries away the dissolved metal. In this process, a low voltage is applied across two electrodes as per the settings given in table 3 with a small gap size (0.1 mm – 0.5 mm) and with a high current density around 2000 A/cm2. A cylindrical type with hexagonal head copper electrode is employed for conducting the experiment. An electrolyte, typically NaCl dissolved with water (0.25Kg/lt.) is supplied to flow through the gap with a required velocity setting as given in table 3. Surface roughness were measured after each machining by using a portable stylus type profile meter, Talysurf with sample length 0.8mm, filter 2 CR, evaluation length 4mm and traverse speed 1mm/Sec. The work piece (EN19) material composition and mechanical properties are shown in Table 1 and Table 2 respectively. All the response parameters, MRR and SR are tabulated after experiments. Table 1 Chemical composition of work piece material in percentage by weight Work Chemical proportion in percentage of weight piece C Mn P S Si Cr Mo material EN19 0.38-0.43 0.75-1.00 0.035 0.04 0.15-0.3 0.8-1.10 0.15-0.25 steel Table 2 The mechanical characteristic of work piece material Mechanical Properties Density (Kg/m3) 7.7×103 Poisson’s ratio 0.27-0.3 Elastic Modulus 190-210 (GPA) Hardness (HB) 197 396 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEME 2.2 Design of Experiment Taguchi approach was used for design of this experiment. Three input process parameters, feed rate, voltage and flow rate of electrolyte were varying and level of each parameter is shown in table 3. The experimental design was according to L18 orthogonal array as shown in table 4. Table 3 Factors and levels used in experiment Levels Factors Unit 1 2 3 Voltage (V) Volt 8 10 12 Feed rate (f) mm/min 0.1 0.3 0.5 Flow rate of Liter/min 10 15 ----- electrolyte Table 4 Design of experiment by L18 orthogonal array in coded form No. of Factors Experiment Flow rate of Voltage (V) Feed electrolyte(Lt./min) rate(mm/min) 1 1 1 1 2 1 1 2 3 1 1 3 4 1 2 1 5 1 2 2 6 1 2 3 7 1 3 1 8 1 3 2 9 1 3 3 10 2 1 1 11 2 1 2 12 2 1 3 13 2 2 1 14 2 2 2 15 2 2 3 16 2 3 1 17 2 3 2 18 2 3 3 3. RELEVENCE VECTOR MACHINE MODELING Relevance vector machine (RVM) is an artificial intelligence method based on a Bayesian formulation of a linear model with an appropriate prior to sparse representation. RVM is a special type of a sparse linear model, in which the basis functions are formed by using kernel functions and these functions further map input features to a higher dimensional feature space [2]. The required architecture of RVM is shown in figure 2. 397 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – Sep 6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEME Figure 2 Architechure of RVM for prediction of MRR and SR The output function or the responses y (m) is defined as: N P (n) = ∑ wiψ ( n − ni ) (2) i =1 Where Ψ (n11, n), … , Ψ (n1m , n) ,Ψ (n21 , n), … , Ψ (n2m , n) are the Kernel functions, nm functions input features, w is the weight vector and p1, p2 are the output responses. Here the output process. responses are MRR and SR of the EDM proce Here input output pairs of data are obtained from Taguchi based Experimental design. As we N use supervised learning methodology so we assign input output pair as {nm , tm }m=1 , where nm m is the input features and tm is the output features. Considering only the scalar valued response, we follow the standard probabilistic formulation and adding additive noise with output samples for better data over fitting, which is described in (eq. 3). tm = P(nm ; w) + ε m (3) Where ε n independent samples of zero mean Gaussian noise with variance as σ 2 .Thus the probability function defines the noise as p (tm | n ) = N (tm | P (nm ), σ 2 ) .This probability his distribution indicates a Gaussian distribution over the response tm with mean P(nm ) and variance σ 2 . Now we can identify our general basis function with the kernel as parameterized by the training vectors. likelihood, Due to the assumption of independence of the likelihood, the complete data set can be written as 1 1 2 p(t | w, σ 2 ) = M exp{− 2 t −ψ w } 2 − 2σ (2πσ ) 2 398 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEME Where t = (t1 ,..........tM )T are the output vectors, w = ( w0 ,.......wM )T are weight vectors and ψ is the M × ( M + 1) design matrix. ψ ( nm ) = [1, K (nm , n1 ), K ( nm , n2 ).............K ( nm , nM )]T The different process parameters in the RVM model are obtained from training examples are w and σ 2 however we expect a optimise value for both w and σ 2 for better prediction of MRR and SR for testing data. In RVM model, we follow the Bayesian prior probability distribution to modify previous probabilistic approach. First, we must encode a preference for smoother functions by making the popular choice of a zero-mean Gaussian prior distribution over w . The distribution given in (eq- 4) M p ( w / α ) = ∏ N ( wi | 0, α i−1 ) i =0 (4) Where α is the vector of N+1 hyperparameters. These hyperparameters are mainly associated with every weight between hidden feature and output. The Bayesian inference proceeds by calculating from Bay’s rule, which is given by p(t | w, α , σ 2 ) p( w, α , σ 2 ) p( w, α , σ 2 | t ) = p(t ) (5) The new test point from testing data n* can be predicted with respect to target t* in terms of predictive distribution as p(t* | t ) = ∫ p(t* | w,α , σ 2 ) p(w,α ,σ 2 | t )dwdα dσ 2 (6) The second term in the integral in eq-6 is called as posterior distribution over weight which is given by p ( t|w,σ 2 ) p(w | α) −1/2 1 ∑ exp − ( w − µ ) ∑ (w − µ) −1 p ( wt,α,σ ) = T | 2 2 = (2π )−(M +1)/2 (7) p(t | α,σ ) 2 The posterior covariance term obtained as ∑ = (σ −2 T + A) − 1 (8) µ = σ −2 ∑ T t (9) With A = diag (α 0 , α1 , α 2 ,………..α M ) Relevance vector machine method is a machine learning procedure to search for the best hyperparameters in posterior mode i.e. the maximization of p (α , σ 2 | t ) Proportional to p (t | α , σ 2 ) p (α ) p (σ 2 ) with respect to α and β , in case of the uniform hyper priors we need to maximize the term p (t | α , σ 2 ) .The maximization term can be computed as: 399 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEME 1 p(t | α,σ2) = ∫ p(t | w,σ2) p(w| α)dw= (2π)−N/2 | σ2I +ψ A−1ψT |−1/2 exp{− tT (σ2I +ψ A−1ψT )−1t} (10) 2 2 The Values of α and are obtained by maximize the (eq-10) For α, we differentiate the (eq.10) and then equating to zero. Finally we got γi α in ew = (11) µ i2 Where µi is the ith posterior mean weight and we can define the quantities γ i as γ i ≡ 1 − α i ∑ii (12) Where the ∑ ii the ith diagonal element of the posterior weight covariance from (eq-8) computed with the current α and σ 2 values. For the noise variance σ 2 , this differentiation leads to the re- estimate the variance as t − µ2 (σ 2 ) new = (13) M − ∑ iγ i Where M in the denominator refers to the number of data examples. For this convergence of the hyperparameter estimation procedure, we have to make predictions based on the posterior distribution over the weights, in which the condition of the 2 maximizing values are α MP and σ MP .We can then compute the predictive distribution, from (eq-6), for a new data m* by using(7): p ( t* |t , α MP , σ MP ) = ∫ p ( t* |w , σ M P ) p ( w|t , α MP ,σ M P ) dw 2 2 2 (14) As both terms in the integral are Gaussian, so we can compute p ( t* |t , α MP , σ MP ) = N (t* | y* , σ *2 ) 2 (15) With final value as P* = µ Tψ ( n* ) (16) σ *2 = σ MP + ψ ( n* )T ∑ψ ( n* ) 2 (17) So the prediction of the normalized valued as P(n* ; µ ) and which can be computed by taking the normalized value of σ *2 and ψ (n* ) . 4. RESULT AND DISCUSSION design for the experiment was carried out with the help of machining parameters The like flow rate of electrolyte, voltage and feed rate by Taguchi technique using MINTAB 16. The experimental result for material removal rate and surface roughness were tabulated in table 5 with process parameters at different level in coded form. 400 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEME Table 5 Experimental result for MRR and SR No. of Factors Experimental result Experiment Flow rate of Voltage (V) Feed MRR(mm3/ SR(µm) electrolyte(Lt./min) rate(mm/min) min) 1 1 1 1 14.2078 3.27 2 1 1 2 29.6364 3.6 3 1 1 3 38.8766 4.4 4 1 2 1 21.5526 3.60 5 1 2 2 33.7838 4.67 6 1 2 3 40.7643 5.10 7 1 3 1 35.3227 4.20 8 1 3 2 38.8617 4.63 9 1 3 3 48.4136 5.43 10 2 1 1 26.9610 3.61 11 2 1 2 36.5981 4.63 12 2 1 3 46.1429 4.97 13 2 2 1 30.0955 4.33 14 2 2 2 39.2448 5.67 15 2 2 3 52.7840 5.40 16 2 3 1 39.4786 4.61 17 2 3 2 45.8097 5.87 18 2 3 3 64.5909 6.28 4.1. Prediction of MRR and SR using Relevance Vector Machine The Relevance Vector Machine (RVM) based mathematical modelling was carried out, with the help of 20 sets of experimental input-output patterns in MATLAB. These patterns were obtained from Taguchi based ECM process. The various machining parameters such as flow rate of electrolyte, voltage and feed rate are the input to RVM regression model. At the output, the model predicts both Material removal rate (MRR) and Surface Roughness (SR). The performance of the model was given in terms of regression test error. The regression test error for different kernel function with the number of iterations (Nt) and noise factor is given in table 6. The lower value of regression test error indicates better accuracy for the prediction of MRR and SR. Table 6 Regression test error for different kernel functions used for RVM model analysis Noise factor= 0.001, number of iteration=200 Kernel functions Regression test error for MRR Regression test error for SR Laplace kernel 0.00959 0.02050 Bubble kernel 0.15117 0.12380 Cubic kernel 0.02805 0.04056 Spline kernel 0.03839 0.03282 Gaussian kernel 0.03985 0.03042 401 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – Sep 6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEME From table 6 it is quite obvious that with N=200 and noise factor as 0.001, the Finally, table Laplace kernel function gives a lower value of regression test error. Finally, in table-7 and figure 3, 4 we compare experimental and predicted results of both MRR and SR for ECM process. The optimized regression error found to be 0.00959 and 0.02050 for MRR and SR respectively. Figure 3 Comparison between predicted MRR and experimental MRR Figure 4 Comparison between predicted SR and experimental SR Table 7 Experimental and pre predicted values of MRR and SR No. of Experimental result Prediction result using RVM Experiment MRR(mm3/min) SR(µm) MRR(mm3/min) SR(µm) 1 14.2078 3.27 14.40729 3.18747 2 29.6364 3.6 30.02487 3.667704 3 38.8766 4.4 39.52196 4.385094 4 21.5526 3.60 21.82517 3.655535 5 33.7838 4.67 33.69092 4.422571 6 40.7643 5.10 40.88906 5.049707 7 35.3227 4.20 34.79946 4.123016 402 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEME 8 38.8617 4.63 38.87723 4.880175 9 48.4136 5.43 47.97928 5.225895 10 26.9610 3.61 27.21022 3.763317 11 36.5981 4.63 36.01104 4.620697 12 46.1429 4.97 46.0731 4.856562 13 30.0955 4.33 30.71084 4.286162 14 39.2448 5.67 39.92134 5.334332 15 52.7840 5.40 53.3762 5.587596 16 39.4786 4.61 38.56387 4.60727 17 45.8097 5.87 45.26154 5.923428 18 64.5909 6.28 64.33993 6.224371 CONCLUSION The design of experiment in ECM process was successfully implemented using Taguchi based L18 orthogonal array. The RVM based regression model was simulated using MATLAB for the prediction of MRR and SR. This optimized result in terms of regression test error obtained from the RVM model under Laplace kernel with the noise factor as 0.001 and the number of iterations as 200.This promising result confirms the RVM as better prediction tool for ECM and other industrial machining process. REFERENCE [1] Pandey PC, Shan HS (2009) Modern Machining Process, 36th Reprint. Tata McGraw-Hill Publishing Company Limited, New Delhi. [2] Tipping M E (2001) Sparse Bayesian Learning and the Relevance Vector Machine. Journal of Machine Learning Research 1:211-244. [3] Wernick MN, Lukic AS, Tzikas G, Chen X, Likas A, Galatsanos NP, Yang Y, Zhao F, Strother SC (2007) Bayesian Kernel Methods for Analysis of Functional Neuroimages. IEEE Transactions on Medical Imaging 26(12):1613-1624. [4] Candela JQ, Hansen LK (2002) Time Series Prediction Based on the Relevance Vector Machine with Adaptive Kernels. IEEE 985-988. [5] Caesarendra W, Widodo A, Thom PH, Yang BS (2010) Machine Degradation Prognostic based on RVM and ARMA/GARCH Model for Bearing Fault Simulated Data. IEEE, Prognostics & Syst EM Health Management Conference (PHM2010 Macau). [6] Montgomery D C, (2005) Design and analysis of experiments, 6th edition. Wiley, New York 405-423. 403

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