Research Article Relevance Vector Machine Based …downloads.hindawi.com/archive/2013/170746.pdf · At the output, the corresponding model predicts both MRR and SR. ... e modelling

Hindawi Publishing CorporationInternational Journal of Manufacturing EngineeringVolume 2013 Article ID 170746 9 pageshttpdxdoiorg1011552013170746

Research ArticleRelevance Vector Machine Based Analysesof MRR and SR of Electrodischarge Machining Designed byResponse Surface Methodology

Kanhu Charan Nayak1 Rajesh Kumar Tripathy1 and Sudha Rani Panda2

1 National Institute of Technology Rourkela 769008 India2 Biju Patnaik University of Technology Rourkela 769004 India

Correspondence should be addressed to Kanhu Charan Nayak nayakkanhu83gmailcom

Received 26 June 2013 Accepted 19 November 2013

Academic Editors G Dessein J-Y Hascoet A Lockamy and H Yu

Copyright copy 2013 Kanhu Charan Nayak et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Relevance vector machine is found to be one of the best predictive models in the area of pattern recognition and machine learningThe important performance parameters such as the material removal rate (MRR) and surface roughness (SR) are influenced byvarious machining parameters namely discharge current (119868119901) pulse on time (119879on) and duty cycle (tau) in the electrodischargemachining process (EDM) In this communication the MRR and SR of EN19 tool steel have been predicted using RVMmodel andthe analysis of variance (ANOVA) results were performed by implementing response surface methodology (RSM) The number ofinput parameters used for the RVM model is discharge current (119868119901) pulse on time (119879on) and duty cycle (tau) At the output thecorresponding model predicts both MRR and SR The performance of the model is determined by regression test error which canbe obtained by comparing both predicted MRR and SR from model and experimental data is designed using central compositedesign (CCD) based RSM Our result shows that the regression error is minimized by using cubic kernel function based RVMmodel and the discharge current is found to be one of the most significant machining parameters for MRR and SR from ANOVA

1 Introduction

Electrodischarge machining (EDM) has tremendous poten-tials on account of its versatile application in industry Theapplications are like high precision machining of all types ofelectrical conductors and hard material such as metal andmetal alloys like tool steel and steels used for die making inmetal forming processes and manufacturing of molds auto-motive aerospace and surgical components It has enormousadvantages without any physical contact between tool andwork pieceThis nonconventionalmachining process uses thethermoelectric energy for machining and removes the mate-rial by thermal erosion process The EDM process involvesfinite discrete periodic electric sparks created by the electricpulse generator at short intervals between the tool electrode(anode) andwork electrode (cathode) separated by a thin filmof dielectric liquid that causes thematerial removal inmotionand vaporize form and these tiny molten vaporize particles

are flushed away from the gap by continuous flushing ofdielectric liquidThismachining process provides productivewith increasing strength of the work material and maintainsdesire shape accuracy and surface integrity requirements [1]Material removal rate (MRR) and surface roughness (SR) arethe most important responses which are influenced by EDMparameters such as voltage gap discharge current pulse ontime pulse off time electrode gap and duty factor So itis important to determine the relationship between variousresponses (MRR and SR) withmachining criteria (machiningprocess parameters) and explore the effects of these processparameters on the responses For this a collection of math-ematical and statistical techniques called response surfacemethodology (RSM) is used RSM is an empirical modellingapproach useful for modelling and analysis of problemsin which responses are influenced as several variables andobjectives to optimize these responses [2]

2 International Journal of Manufacturing Engineering

A number of studies have been investigated for materialremoval rate and surface roughness in EDM process andeffect of machining parameters on machining performanceThe modelling of material removal rate through RSM Rah-man et al [3] conclude that the MRR is influenced consid-erably by peak current and pulse on time Khan et al [4]showed a mathematical model for investigation of processparameters (peak current pulse on time and pulse off time)on response parameter surface roughness with Ti-6Al-4V aswork piece material and copper as an electrode in electro-discharge machining They conducted a complete analysis ofvariance (ANOVA) to know the optimal process parametersand prediction of roughness using RSM Central compositedesign based RSM is used is relatively efficient and used forto design the experiment and the investigation of machiningparameters on the performance of EDM process It helps themodelling of process parameters [5] RSM is applied formod-elling and prediction of roughness in the electrodischargemachining process for EN8 steel considering discharge cur-rent and other machining parameters The most influencingfactor in this study was found to be discharging current[6] Another notable study by Prabhu and Vinayagam [7]was on the modelling and characterization of EDM processparameters of AISI D2 toll steel material with multiwallcarbon nanotubes and developed the prediction of SR asresponse parameter and pulse current pulse duration andpulse voltage as model variables with copper electrode usingBox-Behnken in RSM Li and Lee [8] studied the differentoptimal machining parameters (voltage discharge currentpulse on time and flushing pressure) setting and electrodepolarity (copper tungsten electrode) on response parameterMRR and SR by implementing RSM Tsai andWang [9] havecarried out the works to study the neural network based asRBFN and ANFIS for the prediction of surface finish in EDMaluminum and iron as work piece and copper as an electrode

Those above mentioned investigations were restricted tostudying the effect of various discharge machining parame-ters on the MRR and SR in EDM with EN19 tool steel andother machining parameters called duty cycle Hence thepresent study attempts to investigate themachining perform-ance using EN19 as work piece material EN19 has tremen-dous application towards the components of mediums andlarge cross section requiring high tensile strength and tough-ness for automatic engineering and gear and engine construc-tion such as crane shafts steering knocking connecting rods

Although the response surface methodology and otherprediction tools are a powerful approach for the investigationofmaterial removal rate and the surface roughness during theEDM process it is time-consuming During recent decadesthe RVM was been a popular machine learning tool forsolving complex problems In this model a general Bayesianframework is used for obtaining the sparse solutions toregression (function estimation) and classification tasks byutilizing linear models in the parameters [10] The Bayesianapproach has the advantage that it can be seamlessly incor-porated into the RVM framework and thus it inherits thesubsequent advancement made towards the faster computa-tions from a practical standpoint [11] Candela and Hansen[12] summarized the Bayesian framework used to train RVM

Figure 1 Experimental setup (electrode work piece and flushingarrangement)

highlighted the importance of adapting the basic functionsand presented the improvement to the RVM Caesarendra etal [13] used RVM to train the kurtosis bearing data and targetvectors of failure probability estimated byLR and employed topredict the failure probability of the individual unit of bearingsample Valls et al [14] introduced a general class of RVM-based system identification algorithms and yield confidenceintervals for the predictions

It was concluded from the above study that limited workhas been done on EDM of EN19 (alloy steel) and use ofRVMmodel to predict EDM responses namely MRR and SRTherefore it is needed to work out the present research forEN19 as work piece material and RVMmodel as the machinelearning algorithim The present study was initiated todevelop a multiinput-multioutput RVMmodel to predict thevalues of MRR and SR resulting from an EDM process Thethree process parameters namely discharge current pulseon time and duty cycles were varied to investigate theireffect on response parameters such as material removal rateand surface roughness The process optimizations of theseparameters were done by response surface methodology Inthis paper we use the application of large scale multikernelRVM for the prediction of MRR and SR for EDM process

2 Experimental Details

21 Experimental Setup The experiments are carried oututilizing LEADER-1 ZNC electrical discharge machine TheEDM has provision of movement in three axes 119883- and 119884-axis movement of working table and 119885-axis movement ofthe quill In this effort EN19 was selected as the work piecematerial and cylindrical copper electrode were employed formachining the work piece material Three process param-eters discharge current pulse on time and duty factor atdifferent levels are taken to carry out the machining processkeeping other input parameters (voltage = 40 volt SEN = 6ASEN = 3 working cycle = 08 sec and quill-up time 03 Sec)as consultant for each machining operation and to evaluatethe output responses MRR and SR Impulse flushing is usedwith a flushing pressure of 02 Kgcm2 The experimentalsetup is shown in Figure 1 The chemical composition and

International Journal of Manufacturing Engineering 3

mechanical properties of the work piece is given in Tables 1and 2 respectively

The machining was carried out for a fixed depth ofmachining of 02mm and for each depth of machining thetime is measured by using a digital stopwatch The experi-mental schedules are shown in Table 4 which is designed byRSM The weight of work piece before and after machiningwas measured by digital balance The initial and final weightof the work piece after end of eachmachining and the densityof work piece are used for the calculation of MRR by using(1) Surface roughness was measured after each machiningby using a portable stylus type profile meter Talysurf withsample length 08mm filter 2 CR evaluation length 4mmand traverse speed 1mmSec All the response parametersMRR and SR are tabulated in Table 4

MRR =(1198820 minus1198821) times 10

minus3times 109

120588119908 times timein mm3min (1)

where 1198820 = initial weight in Kg 1198821 = weight after onemachining in Kg and 120588119908 = density of material in Kgm3

22 Design of Experiment The design of experiments forexploring the effect of various predominant EDM processparameters (eg Pulse on time discharge current and dutycycle) on the machining characteristics (eg the materialremoval rate and surface roughness) was modelledThemainobjective of experimental design is to study the relationsbetween the response as a dependent variable and the variousprocess parameter levels It provides an opportunity to studythe individual effects of each factor

In the present work experiments were designed on thebasis of experimental design technique using RSM that isa mathematical and statistical techniques which are usefulfor the modelling and analysis of an experiment in which aresponse of interest is influenced by several variables and theobjective is to optimize the response The coded levels for allprocess parameters used are displayed in Table 3 To obtainan optimal response a central composite design (CCD)type of design is implemented in RSM which is presentedin Table 4 A CCD is used since it gives a comparativelyaccurate prediction of all response variable averages relatedto quantitiesmeasured during experimentation and relativelyefficient with respect to the number of runs required [15]

3 RVM Modelling

Relevance vectormachine (RVM) is amachine learning tech-nique based on a Bayesian formulation of a linear model withProper selection of prior that result in a sparse representationRVM is a special type of a sparse linear model in which thebasis functions are formed by using kernel functions centeredat the different training points [10 16] For this model the119898119894rsquos are the input parameters 119908119894 are the weight vectors fromhidden feature space to output and 119873 is the dimension offeature space The required block diagram of RVM model isshown in Figure 2

Input 1

Input 2

Input 3

K(m11 m)

K(m12 m)

K(m21 m)

K(m2nm)

K(m1nm)

sum

sum

y1

y2

w11

w12

w1s

wk1

wks

Figure 2 Shows the block diagram of RVM 119870(11989811 119898) 119870(1198981119899 119898) and 119870(11989821 119898) 119870(1198982119899 119898) Kernel functions for 119898119899input features 119908 weight vector 1199101 and 1199102 output responses (forMRR and SR)

Table 1 Chemical composition of work piece (EN19)

Elements Percentage of weightC 038ndash043Mn 075ndash100P 0035S 004Si 015ndash03Cr 08ndash110Mo 015ndash025

Table 2 Mechanical properties of work piece (EN19)

Density (Kgm3) 77 times 103

Poissonrsquos ratio 027ndash03Elastic modulus (GPA) 190ndash210Hardness (HB) 197

Table 3 EDM process parameters and their levels assigned forexperiment

Process parameters Unit Levelsminus2 minus1 0 1 2

Discharge current (119868119901) Ampere 1 2 4 7 8Pulse on time (119879on) Ms 50 200 500 750 1000Duty cycle (tau) 45 55 65 80 85

The output function 119910(119898) is defined as follows

119910 (119898) =

119873

sum

119894=1

119908119894120601 (119898 minus 119898119894) (2)

Basically the RVM model is identical to that of supportvectormachines (SVM) but here the kernel function does notsatisfy Mercerrsquos condition [10] The kernel function ldquo120601rdquo is acontinuous symmetric positive integral operator

Let us assume the sparse Bayesian regression model andassociated inference procedures to predict both MRR and


Table 4 CCD for process parameters using RSM and experimental results

Sl no 119868119901 119879on tau MRR (mm3min) SR (120583m)1 200 75000 5500 031107 3722 700 75000 8000 462736 783 200 20000 8000 177819 4174 400 50000 6500 268492 5135 400 50000 6500 259718 4796 700 20000 5500 397481 11277 700 75000 5500 457803 9128 400 50000 6500 269626 529 400 50000 6500 265843 52110 200 75000 8000 039685 3811 700 20000 8000 406222 103312 200 20000 5500 18393 35313 400 50000 6500 278798 4714 800 50000 6500 5404 114315 400 5000 6500 243872 58316 400 100000 6500 076172 41317 400 50000 6500 279229 48718 100 50000 6500 06589 34719 400 50000 8500 269359 473

SR for EDM process The experimental data set of input-output pairs obtained from RSM analysis are given in form of119898119899 119905119899

119873

119899=1 where ldquo119898119899rdquo are the input features and ldquo119905119899rdquo are the

output features By considering only the scalar valued outputwe follow the standard probabilistic formulation and addadditive noise with output samples for better data overfittingwhich is described as follows

119905119899 = 119910 (119898119899 119908) + 120576119899 (3)

where 120576119899 independent samples from some noise are processedand assumed to be zero-mean Gaussian noise with variance1205902 Thus the probability function defines the noise as 119901(119905119899 |119898) = 119873(119905119899 | 119910(119898119899) 120590

2) This probability distribution spec-

ifies a Gaussian distribution over the output 119905119899 with meanvalue as 119910(119898119899) and variance 1205902 [17 18]

Due to the assumption of independence of the likelihoodfunction the complete data set can be written as follows

119901 (119905 | 119908 1205902) =

1

(21205871205902)minus(1198732)

exp minus 1

21205902

1003817100381710038171003817119905 minus 1206011199081003817100381710038171003817

2 (4)

where 119905 = (1199051 119905119873)119879 are the output vectors 119908 = (1199080

119908119873)119879 are weight vectors and 120601 is the 119873 times (119873 + 1) design

matrix The 120601 is

120601 (119898119899) = [1 119870 (119898119899 1198981) 119870 (119898119899 1198982) 119870 (119898119899 119898119873)]119879

(5)

Here the119870 specifies the kernel functionwhichmaps the inputfeatures to high dimensional feature space just alike as SVM

Here we used different kernel fuctions like Gaussian kernellaplace kernel and spline kernel The different parameters inthe model obtained from training examples are 119908 and 120590

2We expect a normalized value of both 119908 and 120590

2 for betterprediction of testing data

To modify this approach we should follow the Bayesianprior probability distribution At first we encode a preferencefor smoother functions by making the popular choice of azero-mean Gaussian prior distribution over 119908 The distribu-tion is given as follows

119901(119908

120572) =

119873

prod

119894=0

119873(119908119894 | 0 120572minus1

119894) (6)

where 120572 is a vector of 119873 + 1 hyper parameters The hyperparameters are associated with every weight between hiddenfeature and output

The Bayesian inference obtained from Bayrsquos rule which isgiven by the following

119901 (119908 120572 1205902| 119905) =

119901 (119905 | 119908 120572 1205902) 119901 (119908 120572 120590

2)

119901 (119905) (7)

The new given test point119898lowast can be predicted with respect totarget 119905lowast in terms of predictive distribution as follows

119901 (119905lowast | 119905) = int119901 (119905lowast | 119908 120572 1205902) 119901 (119908 120572 120590

2| 119905) 119889119908119889120572119889120590

2

(8)


The second term in the integral called as posterior distribu-tion over weight which is given by

119901 (119908 | 119905 120572 1205902) =

119901 (119905 | 119908 1205902) 119901 (119908 | 120572)

119901 (119905 | 120572 1205902)

= (2120587)minus(119873+1)2

times10038161003816100381610038161003816sum

10038161003816100381610038161003816

minus12

exp minus12(119908 minus 120583)

119879summinus1

(119908 minus 120583)

(9)

The posterior covariance term is given by the following

sum = (120590minus2Φ119879Φ + 119860)

minus1

120583 = 120590minus2sumΦ119879119905

(10)

with 119860 = diag(1205720 1205721 1205722 120572119873)Relevance vectormachinemethod is a learning procedure

to search for the best hyper parameters posterior modethat is the maximization of 119901(120572 1205902 | 119905) proportional to119901(119905 | 120572 120590

2)119901(120572)119901(120590

2) with respect to 120572 and 120573 in case of

the uniform hyper priors we need to maximize the term119901(119905 | 120572 120590

2) [19] The maximization term can be computed

as follows

119901 (119905 | 120572 1205902) = int119901 (119905 | 119908 120590

2) 119901 (119908 | 120572) 119889119908

= (2120587)minus119873210038161003816100381610038161003816

1205902119868 + 120601119860

minus112060111987910038161003816100381610038161003816

minus12

times exp minus12119905119879(1205902119868 + 120601119860

minus1120601119879)minus1

119905

(11)

Values of 120572 and 1205902 which maximize (11) cannot be obtainedin closed form and here we summarize formulae for theiriterative estimation The detail information regarding theestimation of hyperparameters can be obtained using expec-tation maximization algorithm based approach

For 120572 differentiating (11) and equating to zero we get thevalue of 120572new

119894as follows

120572new119894

=120574119894

1205832119894

(12)

where 120583119894 is the 119894th posterior mean weightThus we can definethe quantities 120574119894 as follows

120574119894 equiv 1 minus 120572119894sum

119894119894

(13)

where the sum119894119894 is the 119894th diagonal element of the posteriorweight covariance from (8) computed with the current 120572 and1205902 values For the noise variance 1205902 differentiation leads to

the reestimate as follows

(1205902)new

=119905 minus Φ120583

2

119873 minus sum119894 120574119894

(14)

800

700

400

200

100

1000

00

750

00

500

00

200

00

500

0

850

0

800

0

650

0

550

0

450

0

6045301500

Main effects plot for MRR

6045301500

Mea

nM

ean

Ip Ton

120591

Figure 3 Main effects of machining parameters on MRR

where 119873 in the denominator refers to the number of dataexamples

During the convergence of the hyper parameter esti-mation procedure we have to make predictions based onthe posterior distribution over the weights in which theconditions of the maximizing values are 120572MP and 120590

2

MP [19]We can then compute the predictive distribution for a newdata119898lowast by using (8)

119901 (119905lowast | 119905 120572MP 1205902

MP)

= int119901 (119905lowast | 119908 1205902

MP) 119901 (119908 | 119905 120572MP 1205902

MP) 119889119908

(15)

As both terms in the integral areGaussian sowe can computethe probability as follows

119901 (119905lowast | 119905 120572MP 1205902

MP) = 119873(119905lowast | 119910lowast 1205902

lowast) (16)

with

119910lowast = 120583119879120601 (119898lowast)

1205902

lowast= 1205902

MP + 120601(119898lowast)119879sum120601(119898lowast)

(17)

So the prediction of the output values (MRR and SR) denotedby 119910(119898lowast 120583) in (2) can be computed by taking the normalizedvalues of 1205902

lowastand 120601(119898lowast)

4 Result and Discussion

ANOVA and main effect plots for MRR and SR are done byresponse surface methodology using MINTAB 16 And withthe help of 40 sets of experimental input-output patternsthe proposed RVM modelling are carried out The softwareprograms for RVM model are implemented using MATLABversion 101


Table 5 Analysis of variance (ANOVA) for MRR and SR

Source MRR (mm3min) SR (120583m)119865 119875 of contribution 119865 119875 of contribution

119868119901 1037767 0000 8434 214529 0000 8584119879on 37747 0000 413 12039 0000 384tau Insignificant 1487 0002 0224119868119901 lowast 119868119901 19947 0000 677119879on lowast 119879on 65371 0000 58 Insignificant119868119901 lowast 119879on 66265 0000 558 6075 0000 204119868119901 lowast tau Insignificant 2467 119868119901 lowast tau 084

119878 = 005698 1198772 = 9987 1198772(adj) = 9984 119878 = 020722 1198772 = 9956 1198772(adj) = 9936

119865 it is a test to determine whether the interaction and main effects are significant119875 probability of obtaining a test statistic that at least as extreme as the actual calculated value if the null hypothesis is true119878 standard deviation error1198772 the amount of variation seen in MRR and SR and is explained by input factor1198772(adj) modified 1198772 that has been adjusted for a number of terms in the model

Table 6 Regression test error for different kernel functions

Noise factor = 01 number of iterations = 100Kernel functions Regression test error for MRR Regression test error for SRGaussian kernel 0083636 0079863Laplace kernel 0077997 0074544Linear spline kernel 0088783 0078388Cubic kernel 0070611 0066244Distance kernel 0129122 0103465Thin-plate spline 0086694 0075625Neighborhood indication 0257006 0220205

41 ANOVA and Main Effect Plots The influences of variousmachining parameters on MRR are shown in Figure 3 Thedischarge current has a highly significant effect on MRR andit has directly proportional effect MRR is increasing withpulse on time up to approximately 200120583s and then decreaseswith increasing value of pulse on timeThat isMRR increaseswith pulse on time then afterwards it decreases howeveraccording to the abridge ANOVA Table 5 effects are less thanthe discharge current on MRR The surface roughness is notmuch more important in case of roughing operation andrequired high MRR For this point of view the pulse on time(about 200120583s) gives the advantages in saving the machiningtime in EDMultimately which reduce the cost of product Soit is not required to take the high value of 119879on which is morethan this optimum value (about 200120583s) But MRR decreaseswith further increasing of 119879on It may be due to decrease offlushing time with increasing 119879on which causes the debrisrecast on machining surface Hence MRR decreases So itis required to choose an optimum machining parameterThe percentages of contributions are 8434 and 413 fordischarge current and pulse on time respectively as shownin Table 5

The effects of various input parameters on SR are shownin Figure 4 The graph shows that the Ip is proportional toSR and it is highly significant and has more percentage of

contribution towards the SR Surface roughness also isaffected by pulse on time duty factor with less contributionthan discharge current Pulse on time influences the SR in thesame manner as it affects the MRR As discussed earlier thelarger value of pulse on time decreases the MRR but it mayimprove the surface quality by filling the crater with moltenmetal called recast layer However this recast layer affectsthe mechanical and metallurgical properties of machinedcomponent which is treated as one of the disadvantage [20]This is explained by the fact that tominimize the SR Tonmaybe taken as less than 200120583s and also for low value of 119868119901 (1Ato 2A)The percentages of contribution of 119868119901 119879on and tau are8584 384 and 0224 as shown in Table 5

42 Prediction of MRR and SR by RVM The proposed mod-elling is carried out by relevance vector machine (RVM) withthe help of 40 sets of experimental input-output patternsobtained from RSM (response surface methodologies) inEDM process As lesser amount of data available from theexperimental design we again sampled the data and furtherused those data for both training and testing of RVMmodelThedifferentmachining parameters such as discharge current(119868119875) pulse on time (119879on) and duty cycle (tau) are the inputto RVM model At the output the respective RVM modelpredicts parameters like material removal rate (MRR) and


Table 7 Regression test error for different noise factors

For cubic kernel function number of iterations = 100Noise factor Regression test error for MRR Regression test error for SR005 0037754 0035251001 0012026 0018637

Table 8 Regression test error for number of iterations

For cubic kernel function noise factor = 001No of iterations Regression test error for MRR Regression test error for SR100 0012026 0018637200 0010694 0015266250 001293 0015358

1210

864

1210

864


Mea

nM

ean

Ip Ton

120591

800

700

400

200

100

1000

00

750

00

500

00

200

00

500

0

850

0

800

0

650

0

550

0

450

0

Figure 4 Main effects of machining parameters on SR

surface roughness (SR) The performance of RVMmodellingis obtained from regression test error The regression testerror for RVMprediction using different kernel functionwiththe number of iterations (119873119905) and noise factor is given inTables 6 7 and 8 The lower regression error corresponds tothe better prediction of MRR and SR for the respective RVMmodel

From Table 6 it is quite clear that with number of iter-ations 119873 = 100 and noise factor as 01 the cubic kernelfunction is the optimized one and gives a lower value ofregression test error Further from Table 7 by decreasing thenoise factor from 01 to 001 we get better prediction ofboth MRR and SR with a low value of regression test errorFrom Table 8 it is followed that further by increasing thenumber of iterations to 200 at a constant noise factor therespective value of regression test error is minimized Fromthese above results it is observed that the better predictionperformance of model depends upon the regression errorThe different kernel functions and the nature of experimentperformed play an important role in designing the bestregression model From our previous paper that is for theprediction of MRR and SR for electrochemical machiningwe obtain the result as follows The minimum regression test

0 2 4 6 8 10 12 14 16 18 20No of experiments

6

5

4

3

2

1

0

MRR

(mm3m

in)

Experimental MRRMRR (predicted using RVM)

Figure 5The comparison between predictedMRR and experimen-tal MRR

error for both MRR and SR is minimized by the help oflaplace kernel function with 0001 noise factor [18] It maygive the idea to model RVM taking any experimental data forprediction of responses with the above said kernel functionand then other kernel which minimize the computationaltime Prediction of SR and MRR by RVM gives low erroras compared to the least square support vector machine andartificial neural network based predictive model for ECMprocess [21] Rather conducting large number of experimentswe can use proposed model for predicting the responsesThis can save the time and cost required for performing thenumber of experiments In Table 9 and Figures 5 and 6 wecompare the results of both experimental data and predicteddata of both MRR and SR

5 Conclusion

The experimental design of EDM was successfully imple-mented using CCD based on RSM From the RSM analysisthe most significant parameter towards the response wasfound to be the discharging current The proper selection ofpulse time can save the machining time as well as cost of


Table 9 Experimental and predicted values of MRR and SR

119868119901 119879on tau MRR(mm3min) SR (120583m) MRR (predicted using

RVM) in mm3minSR (predicted

using RVM) in 120583m20 75000 5500 0311073 372 033439221 3710796635700 75000 8000 4627355 78 4602598191 790183655200 20000 8000 177819 417 1775926522 4169651963400 50000 6500 2684917 513 2700207547 4943501339400 50000 6500 259718 479 2700207547 4943501339700 20000 5500 3974812 1127 4014896139 1120501356700 75000 5500 4578032 912 4540240762 8886500507400 50000 6500 269626 52 2700207547 4943501339400 50000 6500 2658429 521 2700207547 4943501339200 75000 8000 039685 38 0379393774 3585126803700 20000 8000 4062215 1033 4005057172 1032266034200 20000 5500 1839296 353 1868379133 356482374400 50000 6500 278798 47 2700207547 4943501339800 50000 6500 5404 1143 5343913023 1143558399400 5000 6500 243872 583 2426300736 5825788079400 100000 6500 0761718 413 0881463669 4105567142400 50000 6500 2792288 487 2700207547 4943501339100 50000 6500 06589 347 0672458618 3782897979400 50000 8500 269359 473 2745193125 4986478772400 50000 4500 26557 543 262671966 5343644844

13121110

9876543210


Experimental SRSR (predicted using RVM)

SR (120583

m)

Figure 6 The comparison between predicted SR and experimentalSR

component in manufacturing industry High value of pulseduration decreases the MRR Roughing operation ldquo119879onrdquo maybe taken as 200120583s to 400120583s The predicted values of MRRand SR from RVMmodel were found to be good and agreedwith experimental results The performance of the modelis determined by using regression test error The value ofregression test error is found to be 0010694 and 0015266 forMRR and SR This optimized result obtained by using cubickernel RVM with the noise factor as 001 and the numberof iterations as 200 This shows RVM as the best prediction

method with low regression test error which mainly leadsto reduce the time and cost of the EDM process for betterproduct design in the industry Regression test error may befurther reduced by considering more numbers of inputs tothe RVMmodel as training and also better selection of modelparameters

Acknowledgments

The authors would like to thank Department of MechanicalEngineering National Institute of Technology RourkelaIndia for providing the facility to conduct the experimentAlso they are thankful to anonymous reviewers and theirsuggestions for helping them to improve the quality of thepaper

References

[1] P K Mishra Nonconventional Machining The Institution ofEngineers Textbook Series Narosa Publishing House 2011

[2] D C Montgomery Design and Analysis of Experiments JohnWiley amp Sons New York NY USA 6th edition 2005

[3] MM RahmanM A R Khan K KadirgamaMM Noor andR A Bakar ldquoModeling of material removal on machining ofTi-6Al-4V through EDM using copper tungsten electrode andpositive polarityrdquo World Academy of Science Engineering andTechnology vol 71 pp 576ndash581 2010

[4] M A R Khan M M Rahman K Kadirgama M A Malequeand M Ishak ldquoPrediction of surface roughness of TI-6AL-4V


in electrical discharge machining a regression modelrdquo Journalof Mechanical Engineering and Sciences vol 1 pp 16ndash24 2011

[5] M K Pradhan and C K Biswas ldquoModelling and analysis ofprocess parameters on surface roughness in EDM of AISID2 tool steel by RSM approachrdquo World Academy of ScienceEngineering and Technology vol 57 pp 814ndash819 2009

[6] S S Baraskar S S Banwait and S C Laroiya ldquoMathematicalmodeling of electrical discharge machining process throughresponse surfacerdquo International Journal of Scientific and Engi-neering Research vol 2 no 11 pp 1ndash10 2011

[7] S Prabhu and B K Vinayagam ldquoModeling the machiningparameters of AISID2 tool steelmaterial withmulti wall carbonnano tube in electrical discharge machining process usingresponse surfacemethodologyrdquo International Journal of PhysicalSciences vol 7 no 2 pp 297ndash305 2012

[8] X P Li and S H Lee ldquoStudy of the effect of machining param-eters on the machining characteristics in electrical dischargemachining of tungsten carbiderdquo Journal of Materials ProcessingTechnology vol 115 no 3 pp 344ndash358 2001

[9] K M Tsai and P J Wang ldquoPredictions on surface finish in elec-trical dischargemachining based upon neural networkmodelsrdquoInternational Journal of Machine Tools andManufacture vol 41no 10 pp 1385ndash1403 2001

[10] M E Tipping ldquoSparse Bayesian learning and the relevance vec-tor machinerdquo Journal of Machine Learning Research vol 1 no3 pp 211ndash244 2001

[11] M NWernick A S Lukic D G Tzikas et al ldquoBayesian Kernelmethods for analysis of functional neuroimagesrdquo IEEE Transac-tions on Medical Imaging vol 26 no 12 pp 1613ndash1624 2007

[12] J Quinonero-Candela and L K Hansen ldquoTime series pre-diction based on the relevance vector machine with adaptivekernelsrdquo in Proceedings of the IEEE International Conference onAcustics Speech and Signal Processing pp I985ndashI988 May2002

[13] W Caesarendra A Widodo P H Thom and B S YangldquoMachine degradation prognostic based on RVM and ARMAGARCHmodel for bearing fault simulated datardquo in Proceedingsof the IEEE Prognostics and System Health Management Confer-ence (PHM rsquo10) IEEE China January 2010

[14] G Camps-Valls M Martınez-Ramon J L Rojo-Alvarez and JMunoz-Marı ldquoNonlinear system identification with compositerelevance vector machinesrdquo IEEE Signal Processing Letters vol14 no 4 pp 279ndash282 2007

[15] R L Mason R F Gunst and J L Hess Statistical DesignedAnalysis of Experiments with Applications to Engineering andScience A John Wiley amp Sons Publication John Wiley amp SonsNew York NY USA 2nd edition 2003

[16] W Caesarendra A Widodo and B S Yang ldquoApplication ofrelevance vector machine and logistic regression for machinedegradation assessmentrdquo Mechanical Systems and Signal Pro-cessing vol 24 no 4 pp 1161ndash1171 2010

[17] M Saarela T Elomaa and K Ruohonen ldquoAn analysis ofrelevance vector machine regressionrdquo Studies in ComputationalIntelligence vol 262 pp 227ndash246 2010

[18] K C Nayak R K Tripathy and S R Panda ldquoRelevance vectormachine based prediction of MRR and SR for electrochemicalmachining processrdquo International Journal of Mechanical Engi-neering and Technology vol 3 no 3 pp 394ndash403 2012

[19] C M Bishop andM E Tipping ldquoBayesian regression and clas-sificationrdquo inAdvances in LearningTheoryMethodsModels andApplications vol 190 of Nato Science Series III Computer andSystems Sciences pp 267ndash285 2003

[20] Y Zhang Y Liu R Ji and B Cai ldquoStudy of the recast layer ofa surface machined by sinking electrical discharge machiningusing water-in-oil emulsion as dielectricrdquo Applied Surface Sci-ence vol 257 no 14 pp 5989ndash5997 2011

[21] K C Nayak and R K Tripathy ldquoTaguchi integrated least squaresupport vector machine an alternative to artificial neural net-work analysis of electrochemical machining processrdquo IOSRJournal of Mechanical and Civil Engineering vol 1 no 3 pp 01ndash10 2012

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



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Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



A number of studies have been investigated for materialremoval rate and surface roughness in EDM process andeffect of machining parameters on machining performanceThe modelling of material removal rate through RSM Rah-man et al [3] conclude that the MRR is influenced consid-erably by peak current and pulse on time Khan et al [4]showed a mathematical model for investigation of processparameters (peak current pulse on time and pulse off time)on response parameter surface roughness with Ti-6Al-4V aswork piece material and copper as an electrode in electro-discharge machining They conducted a complete analysis ofvariance (ANOVA) to know the optimal process parametersand prediction of roughness using RSM Central compositedesign based RSM is used is relatively efficient and used forto design the experiment and the investigation of machiningparameters on the performance of EDM process It helps themodelling of process parameters [5] RSM is applied formod-elling and prediction of roughness in the electrodischargemachining process for EN8 steel considering discharge cur-rent and other machining parameters The most influencingfactor in this study was found to be discharging current[6] Another notable study by Prabhu and Vinayagam [7]was on the modelling and characterization of EDM processparameters of AISI D2 toll steel material with multiwallcarbon nanotubes and developed the prediction of SR asresponse parameter and pulse current pulse duration andpulse voltage as model variables with copper electrode usingBox-Behnken in RSM Li and Lee [8] studied the differentoptimal machining parameters (voltage discharge currentpulse on time and flushing pressure) setting and electrodepolarity (copper tungsten electrode) on response parameterMRR and SR by implementing RSM Tsai andWang [9] havecarried out the works to study the neural network based asRBFN and ANFIS for the prediction of surface finish in EDMaluminum and iron as work piece and copper as an electrode

Those above mentioned investigations were restricted tostudying the effect of various discharge machining parame-ters on the MRR and SR in EDM with EN19 tool steel andother machining parameters called duty cycle Hence thepresent study attempts to investigate themachining perform-ance using EN19 as work piece material EN19 has tremen-dous application towards the components of mediums andlarge cross section requiring high tensile strength and tough-ness for automatic engineering and gear and engine construc-tion such as crane shafts steering knocking connecting rods

Although the response surface methodology and otherprediction tools are a powerful approach for the investigationofmaterial removal rate and the surface roughness during theEDM process it is time-consuming During recent decadesthe RVM was been a popular machine learning tool forsolving complex problems In this model a general Bayesianframework is used for obtaining the sparse solutions toregression (function estimation) and classification tasks byutilizing linear models in the parameters [10] The Bayesianapproach has the advantage that it can be seamlessly incor-porated into the RVM framework and thus it inherits thesubsequent advancement made towards the faster computa-tions from a practical standpoint [11] Candela and Hansen[12] summarized the Bayesian framework used to train RVM

Figure 1 Experimental setup (electrode work piece and flushingarrangement)

highlighted the importance of adapting the basic functionsand presented the improvement to the RVM Caesarendra etal [13] used RVM to train the kurtosis bearing data and targetvectors of failure probability estimated byLR and employed topredict the failure probability of the individual unit of bearingsample Valls et al [14] introduced a general class of RVM-based system identification algorithms and yield confidenceintervals for the predictions

It was concluded from the above study that limited workhas been done on EDM of EN19 (alloy steel) and use ofRVMmodel to predict EDM responses namely MRR and SRTherefore it is needed to work out the present research forEN19 as work piece material and RVMmodel as the machinelearning algorithim The present study was initiated todevelop a multiinput-multioutput RVMmodel to predict thevalues of MRR and SR resulting from an EDM process Thethree process parameters namely discharge current pulseon time and duty cycles were varied to investigate theireffect on response parameters such as material removal rateand surface roughness The process optimizations of theseparameters were done by response surface methodology Inthis paper we use the application of large scale multikernelRVM for the prediction of MRR and SR for EDM process

2 Experimental Details

21 Experimental Setup The experiments are carried oututilizing LEADER-1 ZNC electrical discharge machine TheEDM has provision of movement in three axes 119883- and 119884-axis movement of working table and 119885-axis movement ofthe quill In this effort EN19 was selected as the work piecematerial and cylindrical copper electrode were employed formachining the work piece material Three process param-eters discharge current pulse on time and duty factor atdifferent levels are taken to carry out the machining processkeeping other input parameters (voltage = 40 volt SEN = 6ASEN = 3 working cycle = 08 sec and quill-up time 03 Sec)as consultant for each machining operation and to evaluatethe output responses MRR and SR Impulse flushing is usedwith a flushing pressure of 02 Kgcm2 The experimentalsetup is shown in Figure 1 The chemical composition and




MRR =(1198820 minus1198821) times 10

minus3times 109





3 RVM Modelling


Input 1

Input 2

Input 3

K(m11 m)

K(m12 m)

K(m21 m)

K(m2nm)

K(m1nm)

sum

sum

y1

y2

w11

w12

w1s

wk1

wks











119910 (119898) =

119873

sum

119894=1

119908119894120601 (119898 minus 119898119894) (2)







119873



119905119899 = 119910 (119898119899 119908) + 120576119899 (3)





119901 (119905 | 119908 1205902) =

1

(21205871205902)minus(1198732)

exp minus 1

21205902

1003817100381710038171003817119905 minus 1206011199081003817100381710038171003817

2 (4)




120601 (119898119899) = [1 119870 (119898119899 1198981) 119870 (119898119899 1198982) 119870 (119898119899 119898119873)]119879

(5)






119901(119908

120572) =

119873

prod

119894=0

119873(119908119894 | 0 120572minus1

119894) (6)



119901 (119908 120572 1205902| 119905) =

119901 (119905 | 119908 120572 1205902) 119901 (119908 120572 120590

2)

119901 (119905) (7)



2| 119905) 119889119908119889120572119889120590

2

(8)



119901 (119908 | 119905 120572 1205902) =

119901 (119905 | 119908 1205902) 119901 (119908 | 120572)

119901 (119905 | 120572 1205902)

= (2120587)minus(119873+1)2

times10038161003816100381610038161003816sum

10038161003816100381610038161003816

minus12


119879summinus1

(119908 minus 120583)

(9)


sum = (120590minus2Φ119879Φ + 119860)

minus1

120583 = 120590minus2sumΦ119879119905

(10)



2)119901(120572)119901(120590




as follows

119901 (119905 | 120572 1205902) = int119901 (119905 | 119908 120590

2) 119901 (119908 | 120572) 119889119908

= (2120587)minus119873210038161003816100381610038161003816

1205902119868 + 120601119860

minus112060111987910038161003816100381610038161003816

minus12

times exp minus12119905119879(1205902119868 + 120601119860


119905

(11)



119894as follows

120572new119894

=120574119894

1205832119894

(12)


120574119894 equiv 1 minus 120572119894sum

119894119894

(13)



(1205902)new

=119905 minus Φ120583

2

119873 minus sum119894 120574119894

(14)

800

700

400

200

100

1000

00

750

00

500

00

200

00

500

0

850

0

800

0

650

0

550

0

450

0

6045301500


6045301500

Mea

nM

ean

Ip Ton

120591




2


119901 (119905lowast | 119905 120572MP 1205902

MP)

= int119901 (119905lowast | 119908 1205902

MP) 119901 (119908 | 119905 120572MP 1205902

MP) 119889119908

(15)


119901 (119905lowast | 119905 120572MP 1205902

MP) = 119873(119905lowast | 119910lowast 1205902

lowast) (16)

with

119910lowast = 120583119879120601 (119898lowast)

1205902

lowast= 1205902


(17)









119878 = 005698 1198772 = 9987 1198772(adj) = 9984 119878 = 020722 1198772 = 9956 1198772(adj) = 9936













1210

864

1210

864


Mea

nM

ean

Ip Ton

120591

800

700

400

200

100

1000

00

750

00

500

00

200

00

500

0

850

0

800

0

650

0

550

0

450

0





6

5

4

3

2

1

0

MRR

(mm3m

in)




5 Conclusion






using RVM) in 120583m20 75000 5500 0311073 372 033439221 3710796635700 75000 8000 4627355 78 4602598191 790183655200 20000 8000 177819 417 1775926522 4169651963400 50000 6500 2684917 513 2700207547 4943501339400 50000 6500 259718 479 2700207547 4943501339700 20000 5500 3974812 1127 4014896139 1120501356700 75000 5500 4578032 912 4540240762 8886500507400 50000 6500 269626 52 2700207547 4943501339400 50000 6500 2658429 521 2700207547 4943501339200 75000 8000 039685 38 0379393774 3585126803700 20000 8000 4062215 1033 4005057172 1032266034200 20000 5500 1839296 353 1868379133 356482374400 50000 6500 278798 47 2700207547 4943501339800 50000 6500 5404 1143 5343913023 1143558399400 5000 6500 243872 583 2426300736 5825788079400 100000 6500 0761718 413 0881463669 4105567142400 50000 6500 2792288 487 2700207547 4943501339100 50000 6500 06589 347 0672458618 3782897979400 50000 8500 269359 473 2745193125 4986478772400 50000 4500 26557 543 262671966 5343644844

13121110

9876543210



SR (120583

m)




Acknowledgments


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












MRR =(1198820 minus1198821) times 10

minus3times 109





3 RVM Modelling


Input 1

Input 2

Input 3

K(m11 m)

K(m12 m)

K(m21 m)

K(m2nm)

K(m1nm)

sum

sum

y1

y2

w11

w12

w1s

wk1

wks











119910 (119898) =

119873

sum

119894=1

119908119894120601 (119898 minus 119898119894) (2)







119873



119905119899 = 119910 (119898119899 119908) + 120576119899 (3)





119901 (119905 | 119908 1205902) =

1

(21205871205902)minus(1198732)

exp minus 1

21205902

1003817100381710038171003817119905 minus 1206011199081003817100381710038171003817

2 (4)




120601 (119898119899) = [1 119870 (119898119899 1198981) 119870 (119898119899 1198982) 119870 (119898119899 119898119873)]119879

(5)






119901(119908

120572) =

119873

prod

119894=0

119873(119908119894 | 0 120572minus1

119894) (6)



119901 (119908 120572 1205902| 119905) =

119901 (119905 | 119908 120572 1205902) 119901 (119908 120572 120590

2)

119901 (119905) (7)



2| 119905) 119889119908119889120572119889120590

2

(8)



119901 (119908 | 119905 120572 1205902) =

119901 (119905 | 119908 1205902) 119901 (119908 | 120572)

119901 (119905 | 120572 1205902)

= (2120587)minus(119873+1)2

times10038161003816100381610038161003816sum

10038161003816100381610038161003816

minus12


119879summinus1

(119908 minus 120583)

(9)


sum = (120590minus2Φ119879Φ + 119860)

minus1

120583 = 120590minus2sumΦ119879119905

(10)



2)119901(120572)119901(120590




as follows

119901 (119905 | 120572 1205902) = int119901 (119905 | 119908 120590

2) 119901 (119908 | 120572) 119889119908

= (2120587)minus119873210038161003816100381610038161003816

1205902119868 + 120601119860

minus112060111987910038161003816100381610038161003816

minus12

times exp minus12119905119879(1205902119868 + 120601119860


119905

(11)



119894as follows

120572new119894

=120574119894

1205832119894

(12)


120574119894 equiv 1 minus 120572119894sum

119894119894

(13)



(1205902)new

=119905 minus Φ120583

2

119873 minus sum119894 120574119894

(14)

800

700

400

200

100

1000

00

750

00

500

00

200

00

500

0

850

0

800

0

650

0

550

0

450

0

6045301500


6045301500

Mea

nM

ean

Ip Ton

120591




2


119901 (119905lowast | 119905 120572MP 1205902

MP)

= int119901 (119905lowast | 119908 1205902

MP) 119901 (119908 | 119905 120572MP 1205902

MP) 119889119908

(15)


119901 (119905lowast | 119905 120572MP 1205902

MP) = 119873(119905lowast | 119910lowast 1205902

lowast) (16)

with

119910lowast = 120583119879120601 (119898lowast)

1205902

lowast= 1205902


(17)









119878 = 005698 1198772 = 9987 1198772(adj) = 9984 119878 = 020722 1198772 = 9956 1198772(adj) = 9936













1210

864

1210

864


Mea

nM

ean

Ip Ton

120591

800

700

400

200

100

1000

00

750

00

500

00

200

00

500

0

850

0

800

0

650

0

550

0

450

0





6

5

4

3

2

1

0

MRR

(mm3m

in)




5 Conclusion






using RVM) in 120583m20 75000 5500 0311073 372 033439221 3710796635700 75000 8000 4627355 78 4602598191 790183655200 20000 8000 177819 417 1775926522 4169651963400 50000 6500 2684917 513 2700207547 4943501339400 50000 6500 259718 479 2700207547 4943501339700 20000 5500 3974812 1127 4014896139 1120501356700 75000 5500 4578032 912 4540240762 8886500507400 50000 6500 269626 52 2700207547 4943501339400 50000 6500 2658429 521 2700207547 4943501339200 75000 8000 039685 38 0379393774 3585126803700 20000 8000 4062215 1033 4005057172 1032266034200 20000 5500 1839296 353 1868379133 356482374400 50000 6500 278798 47 2700207547 4943501339800 50000 6500 5404 1143 5343913023 1143558399400 5000 6500 243872 583 2426300736 5825788079400 100000 6500 0761718 413 0881463669 4105567142400 50000 6500 2792288 487 2700207547 4943501339100 50000 6500 06589 347 0672458618 3782897979400 50000 8500 269359 473 2745193125 4986478772400 50000 4500 26557 543 262671966 5343644844

13121110

9876543210



SR (120583

m)




Acknowledgments


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119873



119905119899 = 119910 (119898119899 119908) + 120576119899 (3)





119901 (119905 | 119908 1205902) =

1

(21205871205902)minus(1198732)

exp minus 1

21205902

1003817100381710038171003817119905 minus 1206011199081003817100381710038171003817

2 (4)




120601 (119898119899) = [1 119870 (119898119899 1198981) 119870 (119898119899 1198982) 119870 (119898119899 119898119873)]119879

(5)






119901(119908

120572) =

119873

prod

119894=0

119873(119908119894 | 0 120572minus1

119894) (6)



119901 (119908 120572 1205902| 119905) =

119901 (119905 | 119908 120572 1205902) 119901 (119908 120572 120590

2)

119901 (119905) (7)



2| 119905) 119889119908119889120572119889120590

2

(8)



119901 (119908 | 119905 120572 1205902) =

119901 (119905 | 119908 1205902) 119901 (119908 | 120572)

119901 (119905 | 120572 1205902)

= (2120587)minus(119873+1)2

times10038161003816100381610038161003816sum

10038161003816100381610038161003816

minus12


119879summinus1

(119908 minus 120583)

(9)


sum = (120590minus2Φ119879Φ + 119860)

minus1

120583 = 120590minus2sumΦ119879119905

(10)



2)119901(120572)119901(120590




as follows

119901 (119905 | 120572 1205902) = int119901 (119905 | 119908 120590

2) 119901 (119908 | 120572) 119889119908

= (2120587)minus119873210038161003816100381610038161003816

1205902119868 + 120601119860

minus112060111987910038161003816100381610038161003816

minus12

times exp minus12119905119879(1205902119868 + 120601119860


119905

(11)



119894as follows

120572new119894

=120574119894

1205832119894

(12)


120574119894 equiv 1 minus 120572119894sum

119894119894

(13)



(1205902)new

=119905 minus Φ120583

2

119873 minus sum119894 120574119894

(14)

800

700

400

200

100

1000

00

750

00

500

00

200

00

500

0

850

0

800

0

650

0

550

0

450

0

6045301500


6045301500

Mea

nM

ean

Ip Ton

120591




2


119901 (119905lowast | 119905 120572MP 1205902

MP)

= int119901 (119905lowast | 119908 1205902

MP) 119901 (119908 | 119905 120572MP 1205902

MP) 119889119908

(15)


119901 (119905lowast | 119905 120572MP 1205902

MP) = 119873(119905lowast | 119910lowast 1205902

lowast) (16)

with

119910lowast = 120583119879120601 (119898lowast)

1205902

lowast= 1205902


(17)









119878 = 005698 1198772 = 9987 1198772(adj) = 9984 119878 = 020722 1198772 = 9956 1198772(adj) = 9936













1210

864

1210

864


Mea

nM

ean

Ip Ton

120591

800

700

400

200

100

1000

00

750

00

500

00

200

00

500

0

850

0

800

0

650

0

550

0

450

0





6

5

4

3

2

1

0

MRR

(mm3m

in)




5 Conclusion






using RVM) in 120583m20 75000 5500 0311073 372 033439221 3710796635700 75000 8000 4627355 78 4602598191 790183655200 20000 8000 177819 417 1775926522 4169651963400 50000 6500 2684917 513 2700207547 4943501339400 50000 6500 259718 479 2700207547 4943501339700 20000 5500 3974812 1127 4014896139 1120501356700 75000 5500 4578032 912 4540240762 8886500507400 50000 6500 269626 52 2700207547 4943501339400 50000 6500 2658429 521 2700207547 4943501339200 75000 8000 039685 38 0379393774 3585126803700 20000 8000 4062215 1033 4005057172 1032266034200 20000 5500 1839296 353 1868379133 356482374400 50000 6500 278798 47 2700207547 4943501339800 50000 6500 5404 1143 5343913023 1143558399400 5000 6500 243872 583 2426300736 5825788079400 100000 6500 0761718 413 0881463669 4105567142400 50000 6500 2792288 487 2700207547 4943501339100 50000 6500 06589 347 0672458618 3782897979400 50000 8500 269359 473 2745193125 4986478772400 50000 4500 26557 543 262671966 5343644844

13121110

9876543210



SR (120583

m)




Acknowledgments


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119901 (119908 | 119905 120572 1205902) =

119901 (119905 | 119908 1205902) 119901 (119908 | 120572)

119901 (119905 | 120572 1205902)

= (2120587)minus(119873+1)2

times10038161003816100381610038161003816sum

10038161003816100381610038161003816

minus12


119879summinus1

(119908 minus 120583)

(9)


sum = (120590minus2Φ119879Φ + 119860)

minus1

120583 = 120590minus2sumΦ119879119905

(10)



2)119901(120572)119901(120590




as follows

119901 (119905 | 120572 1205902) = int119901 (119905 | 119908 120590

2) 119901 (119908 | 120572) 119889119908

= (2120587)minus119873210038161003816100381610038161003816

1205902119868 + 120601119860

minus112060111987910038161003816100381610038161003816

minus12

times exp minus12119905119879(1205902119868 + 120601119860


119905

(11)



119894as follows

120572new119894

=120574119894

1205832119894

(12)


120574119894 equiv 1 minus 120572119894sum

119894119894

(13)



(1205902)new

=119905 minus Φ120583

2

119873 minus sum119894 120574119894

(14)

800

700

400

200

100

1000

00

750

00

500

00

200

00

500

0

850

0

800

0

650

0

550

0

450

0

6045301500


6045301500

Mea

nM

ean

Ip Ton

120591




2


119901 (119905lowast | 119905 120572MP 1205902

MP)

= int119901 (119905lowast | 119908 1205902

MP) 119901 (119908 | 119905 120572MP 1205902

MP) 119889119908

(15)


119901 (119905lowast | 119905 120572MP 1205902

MP) = 119873(119905lowast | 119910lowast 1205902

lowast) (16)

with

119910lowast = 120583119879120601 (119898lowast)

1205902

lowast= 1205902


(17)









119878 = 005698 1198772 = 9987 1198772(adj) = 9984 119878 = 020722 1198772 = 9956 1198772(adj) = 9936













1210

864

1210

864


Mea

nM

ean

Ip Ton

120591

800

700

400

200

100

1000

00

750

00

500

00

200

00

500

0

850

0

800

0

650

0

550

0

450

0





6

5

4

3

2

1

0

MRR

(mm3m

in)




5 Conclusion






using RVM) in 120583m20 75000 5500 0311073 372 033439221 3710796635700 75000 8000 4627355 78 4602598191 790183655200 20000 8000 177819 417 1775926522 4169651963400 50000 6500 2684917 513 2700207547 4943501339400 50000 6500 259718 479 2700207547 4943501339700 20000 5500 3974812 1127 4014896139 1120501356700 75000 5500 4578032 912 4540240762 8886500507400 50000 6500 269626 52 2700207547 4943501339400 50000 6500 2658429 521 2700207547 4943501339200 75000 8000 039685 38 0379393774 3585126803700 20000 8000 4062215 1033 4005057172 1032266034200 20000 5500 1839296 353 1868379133 356482374400 50000 6500 278798 47 2700207547 4943501339800 50000 6500 5404 1143 5343913023 1143558399400 5000 6500 243872 583 2426300736 5825788079400 100000 6500 0761718 413 0881463669 4105567142400 50000 6500 2792288 487 2700207547 4943501339100 50000 6500 06589 347 0672458618 3782897979400 50000 8500 269359 473 2745193125 4986478772400 50000 4500 26557 543 262671966 5343644844

13121110

9876543210



SR (120583

m)




Acknowledgments


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119878 = 005698 1198772 = 9987 1198772(adj) = 9984 119878 = 020722 1198772 = 9956 1198772(adj) = 9936













1210

864

1210

864


Mea

nM

ean

Ip Ton

120591

800

700

400

200

100

1000

00

750

00

500

00

200

00

500

0

850

0

800

0

650

0

550

0

450

0





6

5

4

3

2

1

0

MRR

(mm3m

in)




5 Conclusion






using RVM) in 120583m20 75000 5500 0311073 372 033439221 3710796635700 75000 8000 4627355 78 4602598191 790183655200 20000 8000 177819 417 1775926522 4169651963400 50000 6500 2684917 513 2700207547 4943501339400 50000 6500 259718 479 2700207547 4943501339700 20000 5500 3974812 1127 4014896139 1120501356700 75000 5500 4578032 912 4540240762 8886500507400 50000 6500 269626 52 2700207547 4943501339400 50000 6500 2658429 521 2700207547 4943501339200 75000 8000 039685 38 0379393774 3585126803700 20000 8000 4062215 1033 4005057172 1032266034200 20000 5500 1839296 353 1868379133 356482374400 50000 6500 278798 47 2700207547 4943501339800 50000 6500 5404 1143 5343913023 1143558399400 5000 6500 243872 583 2426300736 5825788079400 100000 6500 0761718 413 0881463669 4105567142400 50000 6500 2792288 487 2700207547 4943501339100 50000 6500 06589 347 0672458618 3782897979400 50000 8500 269359 473 2745193125 4986478772400 50000 4500 26557 543 262671966 5343644844

13121110

9876543210



SR (120583

m)




Acknowledgments


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














1210

864

1210

864


Mea

nM

ean

Ip Ton

120591

800

700

400

200

100

1000

00

750

00

500

00

200

00

500

0

850

0

800

0

650

0

550

0

450

0





6

5

4

3

2

1

0

MRR

(mm3m

in)




5 Conclusion






using RVM) in 120583m20 75000 5500 0311073 372 033439221 3710796635700 75000 8000 4627355 78 4602598191 790183655200 20000 8000 177819 417 1775926522 4169651963400 50000 6500 2684917 513 2700207547 4943501339400 50000 6500 259718 479 2700207547 4943501339700 20000 5500 3974812 1127 4014896139 1120501356700 75000 5500 4578032 912 4540240762 8886500507400 50000 6500 269626 52 2700207547 4943501339400 50000 6500 2658429 521 2700207547 4943501339200 75000 8000 039685 38 0379393774 3585126803700 20000 8000 4062215 1033 4005057172 1032266034200 20000 5500 1839296 353 1868379133 356482374400 50000 6500 278798 47 2700207547 4943501339800 50000 6500 5404 1143 5343913023 1143558399400 5000 6500 243872 583 2426300736 5825788079400 100000 6500 0761718 413 0881463669 4105567142400 50000 6500 2792288 487 2700207547 4943501339100 50000 6500 06589 347 0672458618 3782897979400 50000 8500 269359 473 2745193125 4986478772400 50000 4500 26557 543 262671966 5343644844

13121110

9876543210



SR (120583

m)




Acknowledgments


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













using RVM) in 120583m20 75000 5500 0311073 372 033439221 3710796635700 75000 8000 4627355 78 4602598191 790183655200 20000 8000 177819 417 1775926522 4169651963400 50000 6500 2684917 513 2700207547 4943501339400 50000 6500 259718 479 2700207547 4943501339700 20000 5500 3974812 1127 4014896139 1120501356700 75000 5500 4578032 912 4540240762 8886500507400 50000 6500 269626 52 2700207547 4943501339400 50000 6500 2658429 521 2700207547 4943501339200 75000 8000 039685 38 0379393774 3585126803700 20000 8000 4062215 1033 4005057172 1032266034200 20000 5500 1839296 353 1868379133 356482374400 50000 6500 278798 47 2700207547 4943501339800 50000 6500 5404 1143 5343913023 1143558399400 5000 6500 243872 583 2426300736 5825788079400 100000 6500 0761718 413 0881463669 4105567142400 50000 6500 2792288 487 2700207547 4943501339100 50000 6500 06589 347 0672458618 3782897979400 50000 8500 269359 473 2745193125 4986478772400 50000 4500 26557 543 262671966 5343644844

13121110

9876543210



SR (120583

m)




Acknowledgments


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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