Prediction of surface roughness during hard turning of AISI 4340 steel (69 HRC)

Predcton of surface roughness durng hard turnng of AISI 4340 steel (69 HRC) Agrawal, A., Goel, S., Rashd, W. B., & Prce, M. (2015). Predcton of surface roughness durng hard turnng of AISI 4340 steel (69 HRC). Appled Soft Computng, 30, 279-286. DOI: 10.1016/j.asoc.2015.01.059 Publshed n: Appled Soft Computng Document Verson: Early verson, also known as pre-prnt Queen's Unversty Belfast - Research Portal: Lnk to publcaton record n Queen's Unversty Belfast Research Portal Publsher rghts Crown copyrght 2015 Publshed by Elsever B.V. All rghts reserved. Ths s a pre-prnt of an artcle fnally publshed n Appled Soft Computng n May 2015: http://dx.do.org/10.1016/j.asoc.2015.01.059 General rghts Copyrght for the publcatons made accessble va the Queen's Unversty Belfast Research Portal s retaned by the author(s) and / or other copyrght owners and t s a condton of accessng these publcatons that users recognse and abde by the legal requrements assocated wth these rghts. Take down polcy The Research Portal s Queen's nsttutonal repostory that provdes access to Queen's research output. Every effort has been made to ensure that content n the Research Portal does not nfrnge any person's rghts, or applcable UK laws. If you dscover content n the Research Portal that you beleve breaches copyrght or volates any law, please contact openaccess@qub.ac.uk. Download date:13. Jun. 2018

Regresson modellng for predcton of surface roughness durng hard turnng of AISI 4340 steel (69 HRC) Anupam Agrawal a, Saurav Goel b*, Waleed Bn Rashd c and Mark Prce b a Department of Busness Admnstraton, Unversty of Illnos at Urbana-Champagn, USA b School of Mechancal and Aerospace Engneerng, Queen's Unversty, Belfast, BT95AH, UK c Insttute of Mechancal, Process and Energy Engneerng, Herot-Watt Unversty, Ednburgh, UK *Correspondng author Tel.: +44-028-90975625, Emal address: s.goel@qub.ac.uk, Fax: +44-028-90974148 Abstract: In ths study, 39 sets of hard turnng (HT) expermental trals were performed on a Mor-Sek SL- 25Y (4-axs) computer numercal controlled (CNC) lathe to study the effect of cuttng parameters n nfluencng the machned surface roughness. In all the trals, AISI 4340 steel workpece (hardened up to 69 HRC) was machned wth a commercally avalable CBN nsert (Warren Toolng Lmted, UK) under dry condtons. The surface topography of the machned samples was examned by usng a whte lght nterferometer and a reconfrmaton of measurement was done usng a Form Talysurf. The machnng outcome was used as an nput to develop varous regresson models to predct the average machned surface roughness on ths materal. Three regresson models - Multple regresson, Random Forest, and Quantle regresson were appled to the expermental outcomes. To the best of the authors knowledge, ths paper s the frst to apply Random Forest or Quantle regresson technques to the machnng doman. The performance of these models was compared to each other to ascertan how feed, depth of cut, and spndle speed affect surface roughness and fnally to obtan a mathematcal equaton correlatng these varables. Keywords: Hard turnng; Random Forest regresson; Quantle regresson 1

Abbrevatons: AISI ANOVA HT HRC CBN CNC DOE MSE OOB GA NN RFR RPM RSM var Amercan Iron and steel nsttute Analyss of varance Hard turnng Hardness on Rockwell C Scale Cubc boron ntrde Computer numercally controlled lathe Desgn of experments Mean squared error Out of bag Genetc algorthm Neural Networks Random forest regresson Rotaton of spndle per mnute Response surface methodology Varaton Nomenclatures: α ε f a p t m β n R Constant (ntercept) Normally dstrbuted error Feed Depth of cut the number of trees n a Random Forest specfcaton number of varables to use at each tree splt n Random Forest Expected ncrement n the response Spndle speed (RPM) Tool nose radus 2

Ra Ra Average value of machned surface roughness per unt change n surface roughness for th experment 1. Introducton Hard turnng (HT) process has now become a vable method to machne automotve components made of ferrous alloys wth hardness above 45 HRC. On account of reduced lead tme and producton cost, HT elmnates some of the processng steps and procedures nvolved durng classcal machnng processes for hard ferrous alloy materals; ndeed, 80% of the cycle tme was saved when hard turnng a pnon shaft (59-62 HRC) [1]. AISI 4340 medum carbon (0.4%C) hgh strength martenstc steel s one such desrable materal used very frequently to manufacture crtcal components n aerospace engneerng and automotve transmssons, ncludng the manufacture of bearngs, gears, shafts, and cams, whch requre tghter geometrc tolerances, longer servce lfe, and good surface fnsh [2]. In order to carry out a hard turnng operaton n a determnstc fashon, a machne tool wth hgh rgdty, and a cuttng tool wth hgh toughness, hardness, and chemcal nertness supplemented wth approprate machnng condtons are necessary. In ts current state, hard turnng dffers from conventonal turnng on account of a number of factors ncludng the cuttng tool, workpece, or the process tself, all of whch may nfluence the machnng outcome. These varables are: 1. Cuttng tool: Tool rake angle, tool clearance angle, nose radus, tool materal 2. Workpece: Hardness, mcrostructure, gran sze, workpece materal, etc. 3. Machnng parameters: feed, depth of cut, cuttng speed Because of the many complextes nvolved, the task to machne a component wth a determnstc level of precson becomes a challengng one. In an attempt to understand the contrbuton of these varables durng the hard turnng of 69 HRC steel wth a CBN cuttng tool, 39 trals were performed n ths work. 3

2. Lterature revew Hard turnng owes ts popularty prmarly to the capablty of generatng complex geometrc surfaces wth better form accuracy and mproved tolerances n one sngle machnng pass [3]. Prevous decades of manufacturng research on hard turnng have focused on fndng out the nfluence of tool geometry [4-5], tool wear [6-9], cuttng temperature, and cuttng forces [10]. Based on the outcome of these studes, the suggested cuttng condtons for HT are cuttng speeds between 100 and 250 m/mn, a feed rate n the range 0.05 to 0.2 mm/rev, and a depth of cut of less than 0.25 mm [11]. A machnng tral performed by Lma and co-workers [12] on AISI 4340 steel (42 and 48 HRC) between the feed range of 0.1-0.4 mm/rev usng both carbde and a PCBN nsert revealed hgh magntude of cuttng forces and poor machned surface. Chou et al. [13-14] found that an ncrease n the tool nose radus results n an ncrease n the amount of specfc cuttng energy and thereby an mproved machned surface, but at the expense of tool wear. Surface fnsh s the most common tangble outcome of any machnng process that can be used to characterze the qualty of the machnng snce t dctates the functonal propertes of a machned component. Ths s because surface roughness changes the contact trbology whch s central to processes rangng from adheson to frcton, wear, lubrcaton, and coatng systems [15-16]. Ths, n turn, nfluences the corroson resstance, fatgue resstance, creep resstance, and servce lfe of the component. Therefore, manpulatng machned surface roughness to hgh level of precson s a key requrement of many ndustral applcatons. In an attempt to accomplsh ths task, a wde varety of soft computng tools have been appled to the doman of hard turnng. Chandrasekaran et al. [17] revewed number of soft computng tools vz. neural networks, fuzzy sets, genetc algorthms, smulated annealng, ant colony optmzaton, and partcle swarm optmzaton, all of whch can convenently be appled to the machnng process dependng on the complexty of the varable nvolved. Mtal et al. [18] have revewed a great deal of lterature concernng the applcaton of statstcal methods on fnsh turnng a varety of materals. The statstcal data appled to the expermental data n ther work suggest that surface fnsh s prmarly dependent on the type of 4

workpece, feed rate used, and nose radus of the cuttng tool. The prmary focus of ths work s to nvestgate the nfluence of varous machnng parameters affectng the machned surface roughness. Some of the major studes found n the lterature pertanng to the optmzaton of hard turnng are tabulated n Table 1. It can be seen from ths table that none of the studes has attempted to optmze the hard turnng of 69 HRC hardened AISI 4340 steel wth a CBN tool, whereas t s very clear from the lterature that workpece hardness could be an mportant varable n nfluencng the machned surface roughness. In contrast to the lterature detaled above, ths paper focuses on modelng the results of experments va three regresson models. Multple regresson modelng has been used n lterature, however the prevalent analyss s focused on descrbng the mean of the response varable for each fxed value of the regressors, usng the condtonal mean of the response. Ths paper adds to ths knowledge base by applyng the Quantle Regresson technque, whch fts regresson curves to other parts of the dstrbuton of the response varable (and not merely the mean) and the Random Forest regresson (RFR) whch seeks to achve hgher accuracy n predctng the outcomes. The Quantle Regresson method helps to model the possbltes of dfferent rates of change n dfferent parts of the probablty dstrbuton of the response varable. RFR has been shown to be superor to other soft computng methods such as partal least squares, neural networks, and other technques n the arena of speces dstrbuton predcton [19], bologcal actvty predcton [20], and genetc applcatons [21], whch was the motvaton to apply RFR to the doman of hard turnng n ths work. Table 1: Lterature revew of optmzaton studes on hard turnng Work materal Tool materal Optmzaton tools Varables studed Ceramc nserts of alumnum oxde and ANOVA + RSM Cuttng velocty, feed, effectve rake angle, and AISI 52100 ttanum carbontrde [22] nose radus CBN cuttng tool [6] ANOVA + NN Cuttng speed, feed, workpece hardness, 5

cuttng edge geometry Alumnum alloy 390, Ductle case ron, Medum carbon steel, alloy steel, nconel Carbde cuttng tool [18] Correlaton analyss Cuttng speed, feed and nose radus (See reference stated theren) TC coated tungsten Rotatable desgn + Cuttng speed, feed, AISI 4140 steel carbde [23-24] Al 2 O 3 + TCN mxed ceramc [25] Multple regresson ANOVA +Taguch depth of cut, tme of cut Cuttng speed, feed, and depth of cut Mld steel TN-coated tungsten carbde (CNMG) [26] RSM + GA Speed, feed, depth of cut and nose radus SCM alloy 440 steel Al 2 O 3 + TC [27] ANOVA +Taguch SPK alloyed steel Sntered carbde [28] ANOVA + DOE Cuttng speed, feed, and depth of cut Cuttng speed, feed, and depth of cut AISI D2 Steel Ceramc wper nserts [29] TC/TCN/Al 2 O 3 coated carbde tpped [30] Multple Regresson + NN Multple Regresson + Taguch + RSM Cuttng speed, feed, and cuttng tme Cuttng speed, feed, and depth of cut AISI 4340 steel (below 60 HRC) Zrcona toughened alumna (ZTA) cuttng [31] RSM + ANOVA Cuttng speed, feed, and depth of cut CBN, ceramc and carbde tools [32] Taguch + ANOVA + Tukey- Kramer comparson, Cuttng speed, feed rate, depth of cut, workpece hardness, and tool types 6

correlaton tests Cuttng speed, feed rate, AISI H11 steel CBN tool [33] ANOVA + RSM depth of cut, workpece hardness 3. Expermental detals and analyss Longtudnal hard turnng trals were performed on a Mor-Sek SL-25Y (4-axs) CNC lathe. The workpece specmen used was AISI 4340 steel that was hardened up to 69 HRC through heat treatment process. CBN cuttng nserts (type CNMA 12 04 08 S-B) havng a rake angle of 0, clearance angle of 5, and a nose radus of 0.8 mm were procured from Warren Toolng Lmted, UK. Post-machnng non-contact measurement of the surface roughness was done through a whte lght nterferometer (Zygo NewVew 5000) and the measurements were cross checked usng Talysurf. In the subsequent secton, the outcomes of the machnng trals are dscussed and analysed n terms of the statstcal models. Machnng by mechancal means has long been a conventonal technque and unlke non-conventonal machnng processes t s applcable unversally on almost all the real world materals [34]. Turnng s one such basc machnng process n whch the workpece s rotated at a partcular speed (cuttng speed) and the tool s fed aganst the workpece (feed) at a certan level of engagement (depth of cut). Essentally, the combnaton matrx of these three parameters s of crtcal mportance n determnng the outcome of the process. Proper selecton of these three parameters s an essental step to make the process more accurate n terms of the machned qualty of the component and other favourable outcomes. Accordngly, the followng expermental trals were done (Table 2) whch became key nput to the optmsaton data. Snce pror lterature has shown feed (between 0.1 0.2 mm/rev) to be the domnant and lmtng crtera for surface roughness [2], we accordngly chose closer values to cover a range of feeds (0.08, 0.09, 0.1 and 0.15) at several depths of cut and cuttng speed combnatons [11]. 7

3.1. Expermental data Table 2: Expermental data obtaned from the hard turnng trals Experment # Feed (f) (mm/rev) Depth of cut (a p ) (mm) Cuttng speed (n) (RPM) Expermental measurement of Ra (mcron) 1 0.08 0.1 1608 0.502 2 0.08 0.105 1250 0.532 3 0.08 0.2 858 0.5902 4 0.08 0.2 965 0.539 5 0.08 0.452 1850 0.592 6 0.08 0.542 1072 0.5693 7 0.08 0.935 1072 0.5821 8 0.09 0.083 2145 0.667 9 0.09 0.125 1000 0.735 10 0.09 0.144 1072 0.683 11 0.09 0.2 858 0.6776 12 0.09 0.2 965 0.6179 13 0.09 0.2 1072 0.742 14 0.09 0.542 965 0.718 15 0.09 0.542 1072 0.65 16 0.09 0.753 2050 0.764 17 0.09 0.935 1072 0.625 18 0.1 0.045 2145 0.77 19 0.1 0.048 2681 0.781 20 0.1 0.133 1608 0.773 21 0.1 0.2 858 0.6687 22 0.1 0.2 965 0.7029 23 0.1 0.234 2145 0.772 24 0.1 0.352 2220 0.784 25 0.1 0.542 1072 0.6769 26 0.1 0.558 1400 0.812 27 0.1 0.754 858 0.809 28 0.1 0.935 1072 0.6966 29 0.15 0.019 2681 1.251 30 0.15 0.06 1287 1.361 31 0.15 0.1 2681 1.193 32 0.15 0.2 858 1.134 33 0.15 0.2 965 1.0854 34 0.15 0.2 1072 1.316 35 0.15 0.278 1608 1.312 36 0.15 0.542 1072 1.1083 37 0.15 0.657 1600 1.345 38 0.15 0.906 2600 1.523 39 0.15 0.935 1072 1.1337 8

Table 2 present the results of the average surface roughness for varous combnatons of tool feed (f), depth of cut (a p ), and cuttng speed (n). It can be seen from Table 2 that the best value of the machned surface roughness obtaned was 0.502 µm at a feed rate of 0.08 mm/rev, depth of cut of 0.1 mm, and cuttng speed of 1608 RPM. A queston may be asked as to why the feed rate was not lowered below ths pont. Ths s because the lowerng the feed rate below a certan crtcal rate s governed by other factors nvolved n the machnng operaton. Below the crtcal feed rate, ploughng between the cuttng tool wth the workpece worsens the machned surface and hence produces an undesrable outcome. From prevous experence [35], 0.08 mm/rev was consdered to be the crtcal feed rate and n order to avod any loss to the useful lfe of the cuttng tool, ths feed was chosen as the mnmum feed rate for the experment detaled n ths partcular work. 3.2. Multple regresson model Frst, multple regresson was appled to the data obtaned from the experment to predct the performance parameters of hard turnng as well as for the optmzaton of the process. In the smplest formulaton, average surface roughness (Ra) was consdered to be the functon of three lnear predctors: feed (f), depth of cut (a p ), and RPM (n) whch was modelled for the th experment by assumng a lnear functon as follows: Ra 1 f 2a p 3n (1) Equaton (1) defnes a straght lne. The parameter α s the constant or ntercept, and represents the error of ths model estmaton. The parameters β 1, β 2, and β 3 represent the expected ncrement n the response Ra per unt change n f, a p, n respectvely. The lnear model n equaton (1) assumes that the three ncluded varables are the most mportant determnants of surface roughness, and that the error ε s normally dstrbuted and uncorrelated to the varables. Model A (shown later n Table 3) shows the results of the multple regresson model specfed by equaton (2). Standard errors that are robust to the assumptons outlned earler are reported. These can be used to make vald statstcal nferences about the coeffcents, even though the data are not dentcally dstrbuted. The 9

regresson results of Model A show that ths model can explan 92.5% of varaton n the data, and the model s therefore a very reasonable predctor of surface roughness. Model A s as follows: Ra 5 0.279 9.455 f 0.0539a 5.61 10 n (2) p Among the three predctor varables, feed s the most sgnfcant predctor of surface roughness: the coeffcent of feed β 1 s sgnfcant at a greater than 99.999 level (ndcatng that there s more than a 99.999% chance that feed has a strong domnance on the surface roughness). Smlarly, cuttng speed s also found to be a sgnfcant predctor of surface roughness: the coeffcent β 3 s sgnfcant at a >99% level. The depth of cut s not found to be a sgnfcant predctor of surface roughness. In fgure 1, the relatve mportance of an ndvdual regressor s contrbuton to the multple regresson model A s analysed by usng four methods. Here, relatve mportance refers to each regressor s contrbuton (R 2 ) from unvarate regresson, and all unvarate R 2 values add up to the full model R 2. The four methods used are as follows: 1. Averagng over orderngs proposed by Lndeman, Merenda and Gold (LMG) [36] 2. Comparng what each regressor s able to explan n addton to all other regressors that are avalable by ascrbng to each regressor the ncrease n R 2 when ncludng ths regressor as the last of the 3 regressors n our dataset (LAST) 3. Comparng what each regressor alone s able to explan by comparng the R 2 values from 3 regresson models wth one regressor only (FIRST) 4. Usng the product of the standardzed coeffcent and the margnal correlaton, a measure proposed by Hoffman and detaled by Pratt (PRATT) [37]. In ths work, 1000 bootstraps were used for replcatons for creatng 95% confdence ntervals (depcted as vertcal lnes wthn the bars n fgure 1). The results show that rrespectve of the method used, feed s by far the most mportant predctor of surface roughness, followed by cuttng speed and depth of cut. 10

% of R 2 0 20 60 100 % of R 2 0 20 60 100 % of R 2 0 20 60 100 % of R 2 0 20 60 100 Relatve Importance on Surface Roughness wth 95% bootstrap confdence ntervals Method LMG Method Last feed rpm dept feed rpm dept Method Frst Method Pratt feed rpm dept feed rpm dept Fgure 1: Relatve Rmportance 2 93.13%, of ndvdual metrcs regressor s are normalzed contrbuton to sum tested 100%. by four methods Equaton (1) presupposes that the assocaton between dependent varable Ra and the ndependent varables f, a p, and n s addtve. However, the smultaneous nfluence of two ndependent varables (.e. feed and depth of cut) on surface roughness may not be addtve. For example, the mpact of feed may depend on the depth of cut. Such an effect s known as an nteracton effect, and these effects represent the combned effects of predctors on the dependent varable. In what follows, equaton (1) s modfed to nclude the nteracton of each par of ndependent varables, as well as the nteracton of all three varables. The equaton n (1) can be modfed as follows: Ra 1 f 2a p 3r 4 f * a p 5 f n 6n a p 7 f n a p (3) 11

Table 3: Multple Regresson models Dependent Varable : Surface Roughness Base Model Interacton Models A B C D E (better model) Feed (β 1 ) 9.455 9.127 7.786 9.345 9.886 (0.59) (0.94) (1.49) (0.51) (1.95) Depth of Cut (β 2 ) 0.0539-0.0452 0.0485-0.271 0.414 (0.06) (0.21) (0.05) (0.08) (0.31) RPM (β 3 ) 5.61 10-5 5.56 10-5 -8.1 10-6 -9.8 10-6 -1.9 10-6 (2.6 10-5 ) (2.5 10-5 ) (9.6 10-5 ) (2.2 10-5 ) (2.2 10-5 ) Feed Depth of Cut (β 4 ) 0.892-5.91 (2.21) (2.91) Feed RPM (β 5 ) 0.00116-5.0 10-5 (0.00) (0.00) Depth of Cut RPM (β 6 ) 0.000223-0.00019 (4.3 10-5 ) (0.00) Feed Depth RPM (β 7 ) 0.00335 (0.00) Constant -0.279-0.242-0.0849-0.164-0.223 (0.08) (0.08) (0.14) (0.05) (0.19) Adjusted R 2 0.925 0.924 0.928 0.947 0.95 No. of trals 39 39 39 39 39 Values n parentheses ndcate robust Standard Errors of the coeffcents Equaton (3) represents an extended model where the objectve s to explore whether or not the smultaneous effects of the three predctor varables (n pars and all three together) are sgnfcant. In Table 3, Models B, C, and D show the nteracton effect one par at a tme, and model E shows the nteracton effect of all three varables. Adjusted R-squares have been reported for all models these adjust for the number of explanatory terms n a model (the adjusted R-square value ncreases only f the new term mproves the model more than would be expected by chance). Model B shows that the coeffcent of β4 s not sgnfcant. Model C shows that the coeffcent of β5 s not 12

sgnfcant. Hence, models B and C are not sgnfcant mprovements over model A. However, model D shows that the coeffcent of β6 s sgnfcant, and therefore t can be asserted that model D s a better model to predct surface roughness than model A. Fnally, model E shows that the coeffcent of β7 s sgnfcant at 99.99%, and therefore model E s also a better model to predct surface roughness. Snce Model E can explan a larger varaton of data than model D (adjusted R 2 s hgher), Model E can therefore be chosen as the preferred model. Overall, multple regresson results, along wth the nteracton terms, suggest that the followng model (E) s a better predctor of data than model A of equaton (2). Ra 0.223 9.886 f 0.00335 f n a p 0.414a p 1.93 10 5 n 5.91f a p 5.02 10 5 f n 0.00188n a p (4) Equaton (4) explans 95% of the varaton n the data, and therefore s a very good ft wth the expermental data. Overall, Multple regresson analyss helps n dentfyng two models that can be used for predctng surface roughness. Model A n equaton (2) s a smpler model, whch can be used for qucker predcton of the surface roughness, and can explan 92.5% of varaton n the expermental data. Model E n equaton (4) s a more complex model, but can explan 95% of varaton n the expermental data. 3.3. Random Forest Regresson Model Random Forest [38] s an ensemble or dvde-and-conquer approach that s smlar to nearest neghbour predctor and s used to mprove the performance of predcton whle usng regresson. Ths decson tree methodology s based on machne learnng technque [39] whch asserts that t s possble to acheve hgher predcton accuracy by usng ensembles of trees, where each tree n the ensemble s grown n accordance wth the realzaton of a random vector. Predctons are generated by aggregatng over the ensemble. Aggregaton over the ensemble results n a reducton of varance, 13

and therefore the accuracy of the predcton s enhanced. Random Forests seek to reduce the correlaton between the aggregated quanttes by drawng a subset of the covarates at random. In a Random Forest, each node s splt among a subset of predctors randomly chosen at that node. A Random Forest algorthm for regresson s as follows: 1. Draw t bootstrap samples from the orgnal data. 2. For each of the bootstrap samples, grow a regresson tree by random samplng m of the predctors and choose the best splt among those varables. 3. Predct new data by aggregatng the average predctons of the t trees. The Random Forest regresson needs nput data (the three predctors - feed, depth of cut, spndle speed, and the response varable of surface roughness), the number of trees (t), and the number of varables to use at each splt (m). The random property arses out of two factors: (a) each of the t trees s based on a random subset of the observatons, and (b) each splt wthn each tree s created based on a random subset of m canddate varables. Random Forests can be used to rank the mportance of varables n a regresson problem n a natural way. Essentally, a Random Forest Model tres to predct the outcome varable (surface roughness) from a group of potental predctor varables (feed, depth of cut, and cuttng speed). If a predctor varable s "mportant" n makng the predcton accurate, then by gvng t random values, we must be able to obtan a larger mpact on how well a predcton can be made, compared to a varable that contrbutes lttle. The varable mportance score tres to capture ths phenomenon. More formally, the mportance of a gven varable s ncreasng n mean square error for regresson n the forest when the observed values of ths varable are randomly permuted n the samples not consdered for that tree (known as out of bag or OOB [38]). So, for each tree t of the forest, consder the assocated OOB sample. Let error1 denote the mean squared error of a sngle tree t on ths OOB (t) sample. Now, randomly permute the values of predctor x n the OOB (t) sample to get a perturbed sample 14

and compute the error of predctor x on the perturbed sample. Denote ths by error2. Then, the varable mportance of predctor x can be denoted as mp 1 ( error2 error1). t Random Forest Regresson on the data was run for t = (500, 1000, 1500) and m = (1, 2, 3) to ascertan the senstvty of the predcton to the number of trees and the number of splts. The number of trees (t) was ncreased untl there was no ncrease n the varaton explaned by the model. Table 4 provdes the mportance scores for the three regressors for nne sets of regressons. A measure of the goodness-of-ft for Random Forest Regresson Models s the pseudo-r 2 value, calculated from the OOB mean squared error (MSE) of the trees and the varaton (var) of the response varable (surface roughness) explaned by the model as follows: 2 MSE( oob) pseudor 1. Table 4 also reports the pseudo-r 2 values, and the model wth t=500 var and m=3 provded the best ft. Table 4: Importance scores of the three regressors for RFR (seed =99) t= 300 t= 500 t= 1000 m 3 2 1 3 2 1 3 2 1 Feed 2.665 2.347 1.749 2.672 2.344 1.757 2.678 2.347 1.754 t Depth of Cut 0.067 0.149 0.0361 0.066 0.138 0.346 0.064 0.139 0.351 RPM 0.116 0.298 0.488 0.118 0.302 0.468 0.116 0.311 0.463 Varaton (var) MSE (oob) 89.12% 89.04% 77.85% 89.36% 88.99% 78.81% 89.23% 89.03% 80.14% 0.0083 0.0084 0.0169 0.0081 0.0084 0.0162 0.0082 0.0084 0.0151 Pseudo R 2 0.991 0.991 0.978 0.991 0.991 0.979 0.991 0.991 0.981 The mportance scores measure how much more helpful than random a partcular predctor varable s n successfully predctng the outcome varable (surface roughness). The best ft estmaton (t=500 and m=3) shows that feed s the best predctor of surface roughness, followed by spndle 15

speed (rpm) and depth of cut. 3.4. Quantle Regresson Model Quantle Regresson [40] s a method for estmatng relatonshp between varables for all portons of a probablty dstrbuton. Whle multple regressons provdes a summary for the means of the dstrbutons correspondng to the set of regressors, Quantle regresson helps to compute several dfferent regresson curves correspondng to the varous percentage ponts of the dstrbutons and thus provdes a complete pcture of the data. The τ th quantle could be thought of as splttng the area under the probablty densty nto two parts: one wth area below the τ th quantle and the other wth area 1-τ above t [40]. For example, 10% of the populaton les below the 10th quantle. Thus, equaton (1) for the τ th quantle wll reduce to the followng equaton (5): Ra f 2d 3 1 n (5) Whle the Multple Regresson Model specfes the change n the condtonal mean of the dependent varable (surface roughness) assocated wth a change n the regressors (feed, depth of cut, and spndle speed), the Quantle Regresson Model specfes changes n the condtonal quantle. Thus, the Quantle Regresson model can be consdered a natural extenson of the Multple Regresson model. Ths model can help n nspectng the rate of change of surface roughness by quantles. Thus, whle equaton (1) addresses the queston how does feed, depth of cut, and spndle speed affect surface roughness?, t does not and cannot answer a more nuanced queston: does feed, depth of cut, and spndle speed nfluence surface roughness dfferently for samples wth low surface roughness than for samples wth average surface roughness? The latter queston can be answered by (for example) comparng the regresson for the 50 th quantle wth that for the 10 th quantle of surface roughness. Table 5 and fgure 2 show the estmated effect of feed, depth of cut, and spndle speed on surface roughness for the 10 th, 25 th, 50 th, 75 th and 90 th quantles. The estmates shown here used bootstrapped standard errors [41] wth 1000 replcatons. 16

Table 5: Quantle Regresson Dependent varable : surface roughness Quantle 10 th 25 th 50 th 75 th 90 th Feed Depth of cut RPM Constant 8.218 8.21 9.201 10.53 10.47 (0.42) (0.57) (1.10) (0.91) (1.04) 0.0207 0.0281 0.052 0.0626 0.0362 (0.04) (0.04) (0.04) (0.04) (0.04) 6.39 10-5 6.04 10-5 4.51 10-5 7.29 10-7 0.0001 (1.9 10-5 ) (1.73 10-5 ) (3.1 10-5 ) (4.12 10-5 ) (6.7 10-5 ) -0.213-0.203-0.251-0.277-0.34 (0.06) (0.06) (0.10) (0.11) (0.10) Observatons 39 39 39 39 39 Bootstrapped Standard errors n parentheses (1000 replcatons) Accordng to the Multple Regresson model A (shown earler n Table 3), for each change of one unt n feed rate, the average change n the mean of surface roughness s about 9.455 unts. The quantle regresson results ndcate that the effect of feed on surface roughness has a lower mpact for lower quantles of surface roughness. For the 10 th quantle of surface roughness, for each change of one unt n feed rate, the average change n the mean of surface roughness s about 8.218 unts. The Multple Regresson model overestmates ths effect at the 10 th quantle. Smlarly, for the 75 th quantle of surface roughness, for each change of one unt n feed rate, the average change n the mean of surface roughness s about 10.53 unts. The Multple Regresson model underestmates ths effect at the 75 th quantle. Overall, quantle regresson estmates suggest that the effect of feed on surface roughness s lower at lower levels of surface roughness and hgher as surface roughness ncreases. The effect of spndle speed s n the opposte drecton,.e. the effect of spndle speed on surface roughness s hgher at lower levels of surface roughness and reduces as surface roughness ncreases. However, t agan becomes mportant as a varable at very hgh levels of surface roughness. 17

4. Comparson of Multple Regresson wth Random Forest Regresson In ths secton, Multple Regresson and Random Forest Regresson results are compared wth each other to evaluate ther effectveness n predctng the value of surface roughness (the Quantle Regresson methodology s not compared snce that technque s used to understand how the effect of predctor varables s dfferent at dfferent quantles of surface roughness, and therefore one-onone comparson wth other technques s not possble). The values of the surface roughness obtaned from the 39 expermental trals, and the predcted values of the three models presented n the work.e. Model A (smplfed multple regresson model), Model E (complex multple regresson model) and Random Forest Regresson Model are correspondngly plotted n Fgure 2, Fgure 3, and Fgure 4 to hghlght the dfferences of each model wth respect to expermental values. Fgure 2: Comparson of expermental surface roughness wth Multple Regresson Model A 18

Fgure 3: Comparson of expermental surface roughness wth Multple Regresson Model E Fgure 4: Comparson of expermental surface roughness wth Random Forest Regresson Model 19

From Fgure 2, Fgure 3, and Fgure 4, t appears that whle all three proposed models were good at predctng the surface roughness, however they were more accurate only when the surface roughness was below an average value of 1 mcron. As the surface roughness tends to worsen beyond 1 mcron, Model E becomes more accurate than Model A because t takes nto consderaton the parng of the nput varables. In general, the trend of the plot predcted by the Random Forest Regresson Model shows a lot more consstency n the values n contrast to Model E and Model A. Fnally, the standard devatons of the dfferences of the predcted values from the three models versus the actual values from experments are shown n Table 6. Table 6: Standard devaton of the model wth respect to experments Standard devaton of expermental values vs. predcted values for the whole experment Standard devaton of expermental values vs. predcted values for Ra below 1 mcron Model A Model E RFR 0.0740 0.0565 0.0465 0.0479 0.0447 0.0298 It can be seen that both for the surface roughness measurement below 1 mcron and for the whole set of experments, the Random Forest Regresson Model exhbts the least standard devaton compared to the Multple Regresson Models (Model A and Model E). Also, Model E shows lower standard devaton than Model A for the whole experment, but for lower measure of the surface roughness ether Model A or Model E can relably be used. 5. Conclusons Ths study presents an approach of modellng comprehensve expermental trals (39 trals) to predct the average value of machned surface roughness durng hard turnng of AISI 4340 steel (69 HRC) wth a CBN cuttng tool. For the frst tme, a novel approach, namely the Random Forest Regresson Model has been appled to the machnng doman and an excellent correlaton has been found between the model and the expermental results, as the standard devaton of the predcted values from the 39 expermental result sets was only 0.0465. Among the other trals, the best value 20

of the machned surface roughness obtaned was 0.502 µm at a feed rate of 0.08 mm/rev, 0.1 mm depth of cut, and cuttng speed of 1608 RPM. Based on the comprehensve models developed and proposed n ths work, the followng conclusons could be made: 1. Qute smlar to other precson machnng processes, the expermental outcome of 39 sets of trals of hard turnng of AISI 4340 steel (69 HRC) showed that the value of machned surface roughness s most sgnfcantly mpacted by the feed rate followed by the cuttng speed and depth of cut. Although the feed rate was found to play a domnant role compared to the other two parameters, t cannot be lowered beyond a certan crtcal extent due to ploughng phenomena. 2. Multple Regresson Models appled to the 39 expermental datasets obtaned from n-house trals revealed the followng mathematcal equatons whch could provde 92.5% and 95% accurate predctons of machned surface roughness compared wth the expermental results: Ra 0.279 9.455 f 0.0539a p 5.61 10 5 n Ra 0.223 9.886 f 0.00335 f n a p 0.414a p 1.93 10 5 n 5.91f a p 5.02 10 5 f n 0.00188n a p 3. Whle Multple Regresson Models were found suted to addresses the queston how does feed, depth of cut, and spndle speed affect surface roughness?, further robustness check was performed usng the Quantle Regresson Model proposed n ths work whch answers the queston does feed, depth of cut, and spndle speed nfluence surface roughness dfferently for samples wth low surface roughness than for those samples wth average surface roughness? It was found that the effect of feed on surface roughness s lower at lower levels of surface roughness and hgher as surface roughness ncreases. The effect of spndle speed s n the opposte drecton. 4. A novel modellng approach,.e. Random Forest Regresson, has been presented and appled to the machnng process for the frst tme and s found to be more accurate than Multple regresson models n predctng surface roughness. 21

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