l18 array design

Application of Central Composite and Orthogonal Array Design ...F F

Journal of Metallurgical Engineering, 1(1-2) January-December 2011 17

Application of Central Composite andOrthogonal Array Designs for Predictingthe Cutting Force

Srinivasa Rao G.*1 and Neelakanteswara Rao A.2

1Mechanical Engineering Department, RVR&JC College of Engineering,Guntur-522019, A.P., India, [email protected] Engineering Department, National Institute of Technology,

Warangal-506004, A.P.,India, [email protected]

Abstract: Development of empirical models for cutting forcesin turning operation has been a major activity in metal cuttingresearch. This paper presents an application of central compositeface centered (CCF) and L18(2

1 × 37) orthogonal array designsfor developing cutting force model in case of turning AISI 1015steel with HSS tool. Three process parameters namely cuttingspeed, feed, depth of cut and three tool parameters namely sidecutting edge angle, inclination angle, normal rake angle wereconsidered in developing the cutting force model. Eachparameter was set at three levels. Based on the experimentaldata, a mathematical model in terms of process and toolparameters was developed for main cutting force using multiplelinear regression. Confirmation tests were performed to verifythe predictability of the developed model.

Keywords: Turning, cutting force, central composite, orthogonalarray, multiple regression.

1. INTRODUCTIONIn a machining process, turning operation plays an important rolein reducing a particular work piece from the original stock to thedesired shape and size. In order to achieve economic objective of

*Corresponding Author: [email protected]

mailto:[email protected]


Srinivasa Rao G. and Neelakanteswara Rao A.F F

18 Journal of Metallurgical Engineering, 1(1-2) January-December 2011

this process, optimal cutting conditions have to be determined.Although one can determine the desirable cutting conditions basedon experience or hand book data, it does not ensure that theconditions obtained will be optimal or near optimal for thatparticular work-tool combination. Thus, in order to determine theoptimal cutting conditions, reliable mathematical models need tobe established. To ensure the effectiveness of the models, the designof experimental techniques should be used to plan the machiningexperiments efficiently and multiple regression methods can thenbe used for the particular work-tool combination based on themachining data collected on a specific machine.

Force modeling in turning is important for a multitude ofpurposes including tool life estimation, chatter prediction, tool wearmonitoring, thermal analysis, etc. Empirical approach has been apopular approach in developing cutting force model. Cutting forcein turning has been found to be influenced in varying amounts by anumber of factors such as speed, feed, depth of cut, work materialcharacteristics, tool geometry, use of cutting fluids, etc [1, 2]. In theearly phases of empirical work, researchers used one-factor-at-a-time strategy, i.e., varying one factor while keeping all other factorsconstant. This strategy cannot provide generalized conclusions aboutfactor effects. Further, interactions between factors cannot be studied.To overcome these limitations, researchers shifted their strategy tofull factorial designs and orthogonal array designs. Full factorialdesigns provide more complete information, but require largenumber of experiments. Orthogonal array designs, which are in someway fractional factorial designs, require less number of experiments,but provide lesser information compared to full factorial designs.

Experiments based on 33 full factorial designs were conducted[3] for prediction of tool life, force and power in terms of speed,feed and depth of cut. Optimization of the cutting speed, feed rate,and depth of cut with considerations of multiple performancecharacteristics including tool life, cutting force and surface roughnesswas done [4] using L9 orthogonal array (OA) design experiments.




The effects of shape of cutting edge, work piece hardness, feed rateand cutting speed on surface roughness and resultant forces wereexperimentally investigated [5] by using a four-factor two-levelfactorial design. An experimental investigation was conducted todetermine the effects of cutting speed, feed, effective rake angle andnose radius on the surface roughness in the finish hard turning ofthe bearing steel based on a 34 full factorial deign [6]. An L9orthogonal array has been used [7] to determine the optimum levelsfor the parameters insert radius, feed rate, and depth of cut on surfaceroughness while turning AISI 1030 steel bars.

An abductive network technique was adopted [8] to construct aprediction model for surface roughness and cutting force and oncethe process parameters (cutting speed, feed rate and depth of cut)are given, the surface roughness and cutting force can be predictedby this network. Artificial neural network approach has beenproposed [9] for modeling cutting forces. By seeing the literature, itcan be observed that the study of factorial effects and the empiricalmodel building was performed based on OA or factorial designs,particularly in case of force modeling. Further, tool related factorsare excluded from most of the studies.

Central composite designs, which are a combination of 2-levelfactorials and one-factor-at-a-time experiments, are known to be bestfit for fitting a second-order model [10]. In recent past, researchershave been using central composite designs for developing surfaceroughness models [11-13]. One of the key issues in empirical basedresearch is selection of experimental designs. It has been observedthat researchers adopted different experimental designs for similarworks. It might be of curiosity to verify the performance oforthogonal array design and Central composite designs with respectto cutting force. In this work, an attempt has been made to studythe effectiveness of the two designs for the same work-toolcombination. Further, in the present work, both process parameters,namely, cutting speed, feed and depth of cut, and tool parameters,namely, side cutting edge angle, inclination angle and normal rake



angle are considered for the development of the cutting force modelwhile turning AISI 1015 steel with HSS tool. Experiments wereconducted based on central composite face centered design and L18orthogonal array design. Separate models were developed usingthe data obtained from each design. Confirmation experiments wereconducted to verify the adequacy of the models.

2. PROCEDURES AND METHODS2.1 Cutting Force ModelThe proposed relationship between the cutting force and machiningindependent variables can be represented by the following:

Force = kAm Bn Cp Dq Er Fs (1)

Where F is the main cutting force, k, m, n, p, q, r, s are theconstants, A (side cutting edge angle), B (inclination angle),C (normal rake angle) are tool parameters, and D (cutting speed),E (feed rate), F (depth of cut) are process parameters. To facilitatethe determination of constants and parameters, the mathematicalmodel is linearized by performing a logarithmic transformation. Thelogarithmic transformed mathematical model is given by:

ln (Force) = ln k + m ln A + n ln B + p ln C + q ln D + r ln E + s ln F

(2)

Though a multiplicative model like equation (1) implicitlyincorporates interaction effects, upon its logarithmic transformation,equation (2) becomes a simple linear form without any interactionterms.

The second order model also is useful when the second ordereffects and the two way interactions amongst the process parametersand tool parameters were significant. The general second ordermodel for six parameters is of the form shown below:

Y = c0 + c1x1 + c2x2 + c3x3 + c4x4 + c5x5 + c6x6 + c11x12 + c22x2

2 + c33x32 +

c44x42 + c55x5

2 + c66x62 + c12x1x2 + c13x1x3 + c14x1x4 + c15x1x5 + c16x1x6 +

c23x2x3 + c24x2x4 + c25x2x5 + c26x2x6 + c34x3x4 + c35x3x5 + c36x3x6 +c45x4x5 +c46x4x6 + c56x5x6 (3)



Where x1, x2, x3, x4, x5, x6 are the independent variables, c0, c1, c2,..c56 are the constants and y is the response. Equation (3) is usefulwhen the second order effects of variables and the two wayinteractions among the variables are significant. In the present study,the parameters of equations (2) and (3) have been estimated by themultiple linear regression using a SPSS software package.

2.2 Materials and Processes

The experiments were carried out on TMX-2030 engine lathe. TheHSS tools (Co- 10%, W-9.3%, Cr-4.0%, Mo-3.6%, C-1.26%) withrequired cutting angles were ground on a tool and cutter grindingmachine using the standard procedure [1]. Tool geometry used wasas follows: end cutting edge angle 10°, normal side clearance angle8°, and normal end clearance angle 8°, were fixed and remainingside cutting edge angle, inclination angle and normal rake angleswere changed during experimentation.

Table 1The Chemical Composition of AISI 1015 Steel

Element C Si Mn P S Cr Ni Mo Fe

% 0.157 0.155 0.598 0.037 0.021 0.012 0.006 0.001 99.0

Table 2Level of Control Factors

Factor Factor Level1(-1) Level2 (0) Level3 (+1)symbol

A Side cutting edge angle(degrees) 10 20 30B Inclination angle(degrees) 0 10 20C Normal rake angle (degrees) 5 15 25D Cutting speed(m/min) 40 60 80E Feed (mm/rev) 0.052 0.104 0.156F Depth of cut(mm) 0.2 0.4 0.6



The material used in the tests for controlled machining was AISI1015 steel. The chemical composition of the AISI 1015 steel with a137 HB hardness, is given in Table 1. The steel bar stock was 65 mmdiameter, 300 mm length and these bars were trued, centered andcleaned by removing a 2 mm depth of cut from the outside surface,prior to the actual machining tests. The main cutting force wasmeasured with a multi component digital force indicator (IEICOSmade and model 652). The force signals were amplified by a3-channel charge amplifier. The range of force measurement is0-100 kg.

2.3 Experimental DesignDesign of experimental techniques was used for execution of theplan of experiments, for six variables at three levels, whereby thelevels are the values taken by the factors. The factors to be studied

Table 3The Experimental Layout: L18 Orthogonal Array

Exp.No. A B C D E F Force(kgf)

1 –1 –1 –1 –1 –1 –1 3.002 –1 0 0 0 0 0 8.503 –1 +1 +1 +1 +1 +1 15.504 0 –1 0 –1 +1 +1 18.505 0 0 +1 0 –1 –1 2.506 0 +1 –1 +1 0 0 11.007 +1 0 –1 –1 0 +1 16.008 +1 +1 0 0 +1 –1 7.009 +1 –1 +1 +1 –1 0 5.0010 –1 +1 +1 –1 0 –1 5.0011 –1 –1 –1 0 +1 0 15.0012 –1 0 0 +1 –1 +1 7.0013 0 0 +1 –1 +1 0 12.0014 0 +1 –1 0 –1 +1 9.0015 0 –1 0 +1 0 –1 5.0016 +1 +1 0 –1 –1 0 4.0017 +1 –1 +1 0 0 +1 12.0018 +1 0 –1 +1 +1 –1 8.00



and the level of each factor are given in Table 2. The levels of eachfactor were selected based on machining data hand book [14].Experiments were conducted as per central composite face centered(CCF) design [10] and orthogonal array based designs [15]. Thisresearch assumes that the three-four- and five-factor interactionsare negligible, because high order interactions are normally assumedhighly impossible in practice. For six factors, the CCF design consistsof 45 runs, which includes a 26-1 fractional factorial portion(32 experiments), 12 axial points and a central point. For six 3-levelfactors, the smallest (in terms of number of experiments) orthogonalarray design is L18 (21 × 37), which consists of eight columns, andone 2-level factor and seven 3-level factors can be accommodated.

For the present work, the factors side cutting edge angle (A),inclination angle (B), normal rake angle (C), cutting speed (D), feedrate (E), and depth of cut (F) were assigned to columns 2, 3, 4, 5, 7and 8, and columns 1 and 6 were kept empty. The empty columnsprovide the necessary degrees of freedom for error estimation. Theexperimental layouts along with the response (main cutting force)obtained are shown in Tables 3 and 4. In the tables, ‘–1’ indicateslevel 1 of the factor, ‘0’ indicates level 2 of the factor and ‘+1’ indicateslevel 3 of the factor. Experiments were conducted in random orderto avoid any bias.

Table 4The Experimental Layout : CCF (6) Design

Exp.No. A B C D E F Force(kgf)1 –1 –1 –1 –1 –1 –1 3.002 –1 –1 –1 –1 +1 +1 20.003 –1 –1 –1 +1 –1 +1 8.504 –1 –1 –1 +1 +1 –1 7.005 –1 –1 +1 –1 –1 +1 5.006 –1 –1 +1 –1 +1 –1 6.007 –1 –1 +1 +1 –1 –1 1.508 –1 –1 +1 +1 +1 +1 15.009 –1 +1 –1 –1 –1 +1 8.0010 –1 +1 –1 –1 +1 –1 7.00

Table Cont’d



3. DATA ANALYSIS AND DISCUSSION OF RESULTSThe plan of tests was developed aiming at determining the relationbetween the process parameters and tool parameters with the cuttingforce. The analysis of experiments was made into two phases. The

Table 4 Cont’d

11 –1 +1 –1 +1 –1 –1 3.0012 –1 +1 –1 +1 +1 +1 19.0013 –1 +1 +1 –1 –1 –1 3.0014 –1 +1 +1 –1 +1 +1 16.0015 –1 +1 +1 +1 –1 +1 6.0016 –1 +1 +1 +1 +1 –1 5.0017 +1 –1 –1 –1 –1 +1 7.0018 +1 –1 –1 –1 +1 –1 7.5019 +1 –1 –1 +1 –1 –1 2.5020 +1 –1 –1 +1 +1 +1 19.0021 +1 –1 +1 –1 –1 –1 2.5022 +1 –1 +1 –1 +1 +1 16.5023 +1 –1 +1 +1 –1 +1 6.0024 +1 –1 +1 +1 +1 –1 5.5025 +1 +1 –1 –1 –1 –1 2.5026 +1 +1 –1 –1 +1 +1 22.0027 +1 +1 –1 +1 –1 +1 7.0028 +1 +1 –1 +1 +1 –1 6.5029 +1 +1 +1 –1 –1 +1 6.0030 +1 +1 +1 –1 +1 –1 5.0031 +1 +1 +1 +1 –1 –1 2.0032 +1 +1 +1 +1 +1 +1 15.5033 –1 0 0 0 0 0 8.5034 +1 0 0 0 0 0 8.0035 0 –1 0 0 0 0 8.0036 0 +1 0 0 0 0 9.0037 0 0 –1 0 0 0 9.0038 0 0 +1 0 0 0 8.0039 0 0 0 –1 0 0 9.0040 0 0 0 +1 0 0 8.0041 0 0 0 0 –1 0 4.5042 0 0 0 0 +1 0 12.0043 0 0 0 0 0 –1 4.0044 0 0 0 0 0 +1 12.0045 0 0 0 0 0 0 8.0046 0 0 0 0 0 0 8.00



first one concerned the analysis of the effects of factors and of theInteractions. Models for cutting force in terms of process parametersand tool parameters were developed in second phase. Finally, thecomparison between the models has been made.

3.1 Analysis of the Factors and InteractionsFrom L18 orthogonal array design, factor affects at three levels canbe obtained and interaction affects cannot be studied. Since theexperimental design is orthogonal, it is then possible to separateout the effect of each parameter at different levels. The influence ofeach control factor on the response considered i.e. cutting force, hasbeen performed with level mean analysis. A level mean of a factoris the average of the response value of experiments in which thefactor is at the particular level. For example, the mean value of theresponse for the side cutting edge angle at level 1, 2 and 3 can becalculated by averaging the response for experiments 1-3 & 10-12,4-6 & 13-15 and 7-9 & 16-18 respectively. The mean of the responsefor each level of the other cutting parameters can be computed in asimilar manner. The control factor with the strongest influence isdetermined by the difference between mean values of the factor athigh and low levels.

From the factorial portion of CCF design, both factor andinteraction effects (at two levels) can be obtained. It can be observedfrom axial and central portion of CCF design, consideringexperiments from 33 to 45, factor effects (at three levels) of eachfactor can be obtained when all other factors are at 0 levels. Usingthe experimental data, level means have been calculated. The levelmeans obtained from L18 design, factorial portion of CCF design,and axial portion of CCF design are given in Table 5 to 7. Theinfluence of each control factor can be more clearly presented withresponse graphs. A response graph shows the change of the responsewhen the settings of the control factor are changed from one level tothe other. The slope of the line determines the power of influence ofa control factor. Corresponding response plots are presented in Figs.1 to 3.



Table 5Average Response for L18 Design

A B C D E FLevel –1 9.00 9.75 10.33 9.75 5.08 5.08Level 0 9.67 9.00 8.33 9.00 9.58 9.25Level +1 8.67 8.58 8.67 8.58 12.67 13.00

Table 6Average Response for Factorial Portion of CCF Design

A B C D E F

Level –1 8.31 8.28 9.34 8.56 4.59 4.34Level +1 8.31 8.34 7.28 8.06 12.03 12.28

Table 7Average Response for One Factor at a Time Analysis

A B C D E F

Level –1 8.50 8.00 9.00 9.00 4.50 4.00Level 0 8.00 8.00 8.00 8.00 8.00 8.00Level +1 8.00 9.00 8.00 8.00 12.00 12.00

Fig. 1: Response Plot for L18 Design

Fig. 2: Response Plot for Factorial of CCF Design



From the tables of level mean analysis, it can be observed thatthe feed rate and depth of cut have been showing consistentbehavior. As the feed and depth of cut increases from ‘–1’ level to‘+1’ level, the force increases, whereas other factors do not show thesame consistency. For example, as the side cutting edge anglechanges from ‘–1’ level to ‘+1’ level in the L18 data, the cutting forceincreases from ‘–1’ level to ‘0’ level then decreases from ‘0’ level to‘+1’ level. The same is not the case with one-factor-at-a-time analysisportion of CCF design (Table 7). Perhaps, the reason might havebeen the presence of interaction effects. The analysis of interactionsgives additional information about the process. Interaction effectscan be obtained by calculating all combinations of two controlfactors. For example, the interaction Ax D has four possiblecombinations of control factor settings: A1D1, A1D2, A2D1 andA2D2. The interaction matrix enables the construction of interactiongraphs, which indicate the existence or non-existence of interactionbetween two control factors. If the lines in the interaction graph areparallel, it indicates non-existence of interaction. The interactionmatrix and interaction graphs are shown in Table 8 and Fig. 4. Fromthe interaction graphs, it can be observed that the interactionsbetween the feed and depth of cut, feed and normal rake angle,depth of cut and normal rake angle are quite prominent. Theinteraction graph of control factors speed and feed shows a lesserinfluence, and remaining interactions are not prominent. In essence,it can be concluded that feed and depth of cut are having strongimpact on cutting force followed by rake angle and speed.

Fig. 3: Response Plot for One Factor at a Time Analysis



Table 8Interaction Matrices for the Cutting Force

AXB B1 B2 AXC C1 C2 AXD D1 D2 AXE E1 E2A1 8.25 8.38 A1 9.43 7.19 A1 8.50 8.12 A1 4.75 11.89A2 8.31 8.32 A2 9.25 7.38 A2 8.63 8.00 A2 4.43 12.18

AXF F1 F2 BXC C1 C2 BXD D1 D2 BXE E1 E2A1 4.44 12.18 B1 9.31 7.25 B1 8.43 8.13 B1 4.50 12.06A2 4.25 12.38 B2 9.38 7.31 B2 8.69 8.00 B2 4.69 12.00

BXF F1 F2 CXD D1 D2 CXE E1 E2 CXF F1 F2B1 4.44 12.12 C1 9.63 9.06 C1 5.19 13.50 C1 4.88 13.81B2 4.25 12.44 C2 7.50 7.06 C2 4.00 10.56 C2 3.81 10.75

DXE E1 E2 DXF F1 F2 EXF F1 F2D1 4.63 12.50 D1 4.56 12.56 E1 2.50 6.69D2 4.56 11.56 D2 4.13 12.00 E2 6.19 17.87

Fig. 4: Interaction Graphs for Parameters



3.2 Model for L18 DesignAs stated earlier L18 design does permit estimation of interactioneffects, and accordingly L18 design data has been used to fit the firstorder model. The input data to SPSS software is provided in the

Fig. 4(a): Interaction Graphs (Continued)



logarithmic transformation of actual values of factors. Backwardslinear regression, which eliminates the insignificant factors one at atime, option of SPSS is used to estimate the parameters and the finalresults are shown in Table 9.

Table 9Parameter Estimates

Variable Parameter estimate Standard error t Sig.

Intercept 5.392 0.126 42.901 0.000ln(C) -0.154 0.027 -5.644 0.000ln(E) 0.873 0.040 21.632 0.000ln(F) 0.875 0.040 21.635 0.000

Analysis of variance for the model

Source df Sum of squares Mean Square F-value Sig.

Model 3 5.864 1.955 322.609 0.000Error 14 8.482E-02 6.059E-02Total 17 5.949

R-square = 0.986

Based on the above results, the cutting force model developedis given by:

ln (Force) = 5.392 – 0.154 ln (C) + 0.873 ln (E) + 0.875 ln (F)

or

Force = 219.64 C–0.154 E0.873 F0.875

The R-square value of 0.986 indicates that 98.60% of thevariability in cutting force was explained by the model. It can beobserved that normal rake angle, feed and depth of cut are onlycoming into model. Based on the mathematical model, it can beobserved that feed, depth of cut and normal rake angle are showinga prominent effect on the cutting force. The other factors have nosignificant effect on the cutting force. Perhaps it is consistent withthe level means analysis of L18 data.



3.3 Model for CCF DesignAs stated earlier, central composite designs are best for fitting asecond order model, and accordingly CCF data is used to fit a secondorder model. Procedure stated above is applied for developingcutting force model using CCF design. The input data to SPSSsoftware is provided in coded form of factors i.e. –1 to +1. To beprecise, value of the factor in coded scale is = (actual value of thefactor – central value in the range)/ (difference between maximumvalue and central value in the range). For example the cutting speedvalue of 40 m/min is coded as 40-60/20 = –1. The parameterestimates obtained from SPSS software are shown in Table 11.

Table 10Parameter Estimates

Variable Parameter Estimate Standard Error t Sig.

Constant 8.304 0.082 100.953 0.000C -1.000 0.096 -10.451 0.000D -0.265 0.096 -2.767 0.009E 3.721 0.096 38.886 0.000F 3.971 0.096 41.498 0.000CxE -0.437 0.099 -4.436 0.000Cx F -0.500 0.099 -5.070 0.000Dx E -0.219 0.099 -2.218 0.033Ex F 1.875 0.099 19.011 0.000

Analysis of variance for the model

Source df Sum of Squares Mean Square F-value Sig.

Model 8 1171.222 146.403 470.352 .000Error 37 11.517 0.311Total 45 1182.739

R-square=0.990

Table 11Cutting conditions used in confirmation tests

Test A B C D E F

2 20 10 25 40 0.156 0.63 30 20 15 60 0.104 0.44 20 10 5 40 0.156 0.6



The cutting force model developed using CCF design was givenby:

Force = 8.304 – 1.0C – 0.265D + 3.721 E + 3.971F – 0.437CxE –0.5CxF – 0.219DxE + 1.875ExF

Cutting force model in terms of actual factors can be expressedas:

Force = 8.304 – 1.0 normal rake angle – 0.265 cutting speed +3.721 feed + 3.971 depth of cut – 0.437 normal rake angle x feed –0.50 normal rake angle x depth of cut – 0.219 cutting speed x feed +1.875 feed x depth of cut.

The R-square value of 0.990 indicated that 99.00% of thevariability in cutting force was explained by the model. Incomparison to the previous model, it can be observed that the cuttingspeed apart from normal rake angle, feed and depth of cut is alsocoming into the model. Further, four interaction terms are also intothe model. This is consistent with the level means and interactioneffects analysis of CCF design.

3.4 Confirmation TestsConducting confirmation experiments has been the final step of thedesign of Experimental (DOE) process. The confirmation isperformed by conducting tests using combinations of the factorsand levels that are not previously evaluated. Table 12 shows theconditions used in the confirmation tests. Table 13 shows the resultsobtained where a comparison was done between the foreseen valuesfrom the models developed in the present work with the valuesobtained experimentally.

Table 12Confirmation Tests and their Comparison with the Results

Test CCF design L18 designExperiment Model Error (%) Experiment Model Error (%)

2 16.5 16.54 0.24 16.50 16.95 2.733 8.00 8.28 3.50 8.00 7.74 3.374 21.00 20.30 3.33 21.00 21.68 3.24



It can be observed that the error percentages associated withboth the models have been within the limits. Therefore, we canconsider the empirical models, which correlate the cutting force withthe process and tool parameters, with a reasonable degree ofapproximation within the given working conditions.

4. CONCLUSIONSIn this work, the cutting force models have been developed byconsidering both process and tool parameters. Central compositeand orthogonal array designs are used to develop the models. Basedon the work, the following conclusions may be drawn:

1. It can be observed from L18 design that normal rake angle,feed and depth of cut are only coming into model whereasin CCF design the cutting speed apart from normal rakeangle, feed and depth of cut is also coming into the model.Further, four interaction terms are also into the model.Interactions between feed and depth of cut, feed and normalrake angle, depth of cut and normal rake angle, andcutting speed and feed are having significant impact on thecutting force.

2. Both L18 and CCF (6) designs have been showing sameprediction accuracy for cutting force. From cutting forceequation, it can be observed the absence of quadratic effectsof factors, and the linear effect of factors has been onlyidentified. In such cases, the L18 design is sufficient to revealthe information regarding the process.

3. Significance of interaction terms of parameters has beenclearly predicted in CCF design, whereas none of them areconsidered in L18 design. This is owing to the fact that inTaguchi’s design interactions between control factors arealiased with their main effects.

4. By explicit incorporation of the interaction terms, the CCFdesign model gives better insight into the process. So, CCFdesign model developed for cutting force serves as a goodalternative to the popular multiplicative model.

5. The significance of normal rake angle stresses the need forinclusion of tool parameters in empirical model buildingstudies.



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