Mathematical Modelling of Cutting Force As the Most Reliable Information Bearer on Cutting

34th INTERNATIONAL CONFERENCE ON PRODUCTION ENGINEERING 28. - 30. September 2011, Niš, Serbia University of Niš, Faculty of Mechanical Engineering

MATHEMATICAL MODELLING OF CUTTING FORCE AS THE MOST RELIABLE INFORMATION BEARER ON CUTTING TOOLS WEARING PHENOMENON

Obrad SPAIĆ, Zdravko KRIVOKAPIĆ, Rade IVANKOVIĆ Production and Management Faculty in Trebinje, East Sarajevo University, Trg palih boraca 1, Trebinje, Republika Srpska, Bosna i Hercegovina [email protected], [email protected], [email protected]

Abstract: Being one of their prominent exploitative characteristics, cutting tools durability depends on the character, intensity and the speed of wearing. Identification of tool wearing is of great significance for the purpose of avoiding sooner or later replacement of tools. The parameters of tool wearing can be measured by out-process and in-process-measuring systems. Given the extremely limiting role of the former in modern production lines, development of the latter (the indirect measuring systems) has gained prominence. The basis of indirect measuring systems comprises a set of various signals originating from the units of the system under treatment which stand in certain correlations with the wearing parameters. The paper presents mathematical models of axial force designed on the basis of experimental research in drilling tempered steel by twist drills made of high-speed steel manufactured by powder metallurgy.

Key words: tool, durability, wear, cutting force, mathematical model

1. INTRODUCTION Chin [4] showed that modified regression equations could be used to asses the value of tool wear. Identification of cutting elements wear is of high practical 2. AXIAL CUTTING FORCE IN DRILLING importance because, apart from allowing for timely replacement of tools, it also allows for management of Axial cutting force in drilling (the auxiliary movement wearing processes as well as for automation of treatment resistance), F3, along with other conditions unaltered, is in and technological processes. Production lines, in the function of cutting regime (the twist drill nominal particular those of mass and large scale automated diameter, the spindle speed and feed): production benefit from timely replacement of cutting F= f (D,n,s) (1). tools as it eliminates the low quality of final products and reduces production costs arising from sooner and later By way of experimental-analytical method, i.e. the theory replacement of tools. of experiment planning and the theory of regression As in modern production lines the out-process methods of analysis, the cutting force can be expressed in the form of measuring tool wear have become a significantly limiting a degree function: factor, development of online process measuring systems b1F b 2F b3F are gaining prominence. The process methods most F= CF 鬃 D n s (2), frequently applied are the indirect ones whose basis comprises a set of various signals originating from the where: units of the system under treatment which stand in certain CF, b1F, b2F, b3F – are the constants dependent on the type correlations with the wear parameters. The information of material, bearers (signals) of tools wear in cutting process that are D [mm] – is the drill nominal diameter, most often used by researchers are cutting force and n [rev/min] – is the spindle speed, and resistance. Thus, J. Sheikh-Ahmad and R. Yadav [1] s [mm/rev] – is the feed. designed a force model at milling composite materials by applying regression analysis. B. Lotfi, Z. W. Zhong and 3. EXPERIMENT PLANNING L. P. Khoo [2] designed a model to predict cutting force at milling dependent on tool orbit. J. T. Lin, D. With the aim to design a mathematical model of axial Bhattacharyya and V. Kecman [3] showed that measuring force as the information bearer on the wear phenomenon, cutting force enables for tool wear to be monitored the paper contains experimental research on the basis of without interruption of cutting process. Based on the which the constants CF, b1F, b2F, b3F were determined comparative analysis of the assessment of wear of a 8 mm within the assumed mathematical model (2). Given that diameter twist drill, C. Sanjay, M. L. Neema and C. W. the axial force is in the function of three parameters (D, n, s), the experiment was conducted in line with the Box-Wilson’s plan, repeating the experiment four times in complete three-factor orthogonal first-order plan, i.e. the the central plan point (n0=4) as illustrated in Table 1. Table 1. The three-factor plan matrix Coded values Real values Experi- The output n s mental d vector x0 x1 x2 x3 x1x2 x1x3 x2x3 x1x2x3 [rev/min [mm/rev [mm/min points [mm] [F] ] ] ]

1 +1 -1 -1 -1 +1 +1 +1 -1 6.0 250 0.027 6.67 F1

2 +1 +1 -1 -1 -1 -1 +1 +1 10.0 250 0.027 6.67 F2

3 +1 -1 +1 -1 -1 +1 -1 +1 6.0 500 0.027 13.33 F3

4 +1 +1 +1 -1 +1 -1 -1 -1 10.0 500 0.027 13.33 F4

5 +1 -1 -1 +1 +1 -1 -1 +1 6.00 250 0.107 26.67 F5

6 +1 +1 -1 +1 -1 +1 -1 -1 10.0 250 0.107 26.67 F6

7 +1 -1 +1 +1 -1 -1 +1 -1 6.0 500 0.107 53.33 F7

8 +1 +1 +1 +1 +1 +1 +1 +1 10.0 500 0.107 53.33 F8

9 +1 0 0 0 0 0 0 0 7.75 355 0.053 18.67 F9

10 +1 0 0 0 0 0 0 0 7.75 355 0.053 18.67 F10

11 +1 0 0 0 0 0 0 0 7.75 355 0.053 18.67 F11

12 +1 0 0 0 0 0 0 0 7.75 355 0.053 18.67 F12

3.1. Experiment conditions Axial force measuring was conducted by means of a The experiment was conducted by means of twist drills three-component dynamometer manufactured by (TD) DIN 338, nominal diameter Ø6.0; Ø7.75 and Ø10.0 “Kistler“, TYP 8152B2, with the measurement range mm, made of high-speed steel with 8% of Co, from 100 to 900 kHZ, integrated with the universal manufactured by powder metallurgy, used for drilling milling machine and Global Lab software. blind hole in tubes made of chrome-molybdenum alloy For the data acquisition during the experiment, the steel for enhancement, Č.4732, thermally treated to 43-45 KISTLER high-frequency amplifier, type AE-Piezotron HRc hardness. Coupler 5125B was used while a two-channel DAQ The chemical composition, thermal treatment conditions Scope PCI-5102 was used as an AD convertor. The and microstructure of the TD steel and test tubes are acquired signals were processed by means of a virtual shown in References [5]. instrument, aided by Global Lab software. The data Construction of twist drills followed the recommendations acquisition schema is illustrated in Figure 1. in References pertaining to drilling hard treatable materials (tempered steel) as well as previous experience. The drills were manufactured by grinding technology. The geometrical elements of TD are shown in References [5]. The experiment was conducted using test tubes, Ø60 mm in diameter, with the thickness adjusted to the blind hole drilling depth 1=3xd. Test tubes’ hardness was evenly distributed along longitudinal and cross cut and within the prescribed limits. Cutting regimes were defined in line with the recommendations stated in References [5], adhering to the interval limits of variations of influential factors Figure 1. Data acquisition schema during the experiment 2 , 2 ( n sr  n min n max and ssr  s mins max ) and are shown in the three-factor plan matrix (Table 1). 4. EXPERIMENT RESULTS The experiment was conducted in the laboratory of the Faculty of Mechanical Engineering in Podgorica, Measurement of axial force was conducted in line with University of Montenegro, on the universal milling the stated Plan matrix in five measuring points. The first machine Typ: FGU-32. Integrated with the milling measurement was conducted during drilling of a blind machine was the equipment for measuring axial force and hole, l=3d, with sharp twist drills while the fifth (the last) torque manufactured by KISTLER. measurement was conducted subsequent to the achieved During the experiment, the 8% solution of Teolin H/VR lengths of drilling (in mm) during which the twist drills in the quantity of 1 l/min was used for cooling and wear reached the following maximally allowed (defined lubrication. in advance) values: for TD Ø6.0 mm – 0.25 mm 3.2. Axial force measuring for TD Ø7.75 mm – 0.30 mm for TD Ø10.0 mm – 0.35 mm ln( s) ln( smax ) The mean value of the wear band width of the back X3 = 2 + 1 ln( smax) ln( s min ) surfaces was taken to represent the maximum wear value (B ≈ 0.04D), at the edge of regular area which is at By applying regression analysis, the parameter of the k 0.025mm distance from outer fibres (Figure 2). model b0 is determined on the basis of results of all N = 2 At different cutting regimes (the nominal diameter, the + n0 plan points, in line with the pattern [6]: spindle speed and the feed), TD reached the maximally 1 N b0= x 0i y i (5), N i= 0

while parameters bj are determined on the basis of results of N = 2k points arranged along the vertexes of hypercube according to the pattern:

1 N b= x y ,( j = 1,2,3) (6). jN ji i Figure 2. Twist drill wear i= 0 Assessment of the significance of the first-order model was done by means of application of F-criteria, for the allowed wear value at different drilling lengths. The adopted level of significance q=0.05, according to the measured axial force value at maximal wear is shown in pattern: Table 2. b2 F= Nj , j = 1,2,3 Table 2. Axial force values at maximal TD wear rj 2 ( ) (7). SE Exp. D n s lmax Fmax points [m [rev/min [mm/rev [mm] [N] Dispersion of experiment results in the multi-factor space m] ] ] is as follows:

1. 6.0 250 0.027 1265.40 956.50 n0 1 2 2. 10.0 250 0.027 2700.00 594.50 S2 = y - y (8). Ef ( 0i 0 ) 3. 6.0 500 0.027 1533.00 1090.50 E i= 0 2 4. 10.0 500 0.027 2340.00 1139.00 SE = 0.007 , 5. 6.0 250 0.107 999.00 955.50 6. 10.0 250 0.107 4800.00 638.50 where: 7. 6.0 500 0.107 1800.00 1925.50 fE = n0-1 = 3 – is the degree of freedom of experimental error. 8. 10.0 500 0.107 756.00 1579.70 The degrees of freedom of parameters b , (j = 0, 1, 2, 3) 9. 7.75 355 0.053 2569.13 783.30 j are: 10. 7.75 355 0.053 2336.63 682.90 f = 1, which results in the tabled value of the dispersion 11. 7.75 355 0.053 2569.13 655.50 bj relation Fri,1-q/1, 3 = 10.10 [7]. 12. 7.75 355 0.053 2801.63 754.80 The parameters of the model and assesments of significance are listed in Table 3, which shows that all 5. MATHEMATICAL MODELLING OF parameters of the first-order model (b0, b1, b2 i b3) are AXIAL FORCE significant.

The pattern of axial cutting force (2) subsequent to Table 3. Model parameters linearisation can be expressed in the following form: Disper. relations Assesmen Model parameters y= b0 + b 1 x 1 + b 2 x 2 + b 3 x 3 (3), (Fri) ts b 6.820 F 76831.617 sign. where: 0 b0 b1 0.121 Fb1 16.813 sign. y = lnF, b0 = lnCF, x1 =ln(D), x2 =ln(n) and x3 =ln(s). The orthogonal first-order plan with constant members b2 -0.129 Fb2 19.112 sign. can be applied to the above pattern with coding performed b3 0.298 Fb3 102.457 sign. by means of the equations of transformation: b12 -0.021 Fb12 0.575 not sign. b13 0.103 Fb13 12.214 sign.

ln( D) ln( Dmax ) (4). b23 0.091 Fb23 9.484 not sign. X1 = 2 + 1, ln( Dmax) ln( D min ) b123 -0.039 Fb123 1.724 not sign. On the basis of listed parameters the empirical model was ln( n) ln( nmax ) deduced of axial force at maximal TD wear: X2 = 2 + 1, and ln( nmax) ln( n min ) yˆ = 6.820 + 0.1215x1- 0.121x 2 + 0612x 3 (9). Revisiting the original coordinates by means of the 3 535.15 -9.98 7 948.21 -16.75 equations of transformation (4) provided for deduction of 4 554.3 -13.19 8 1483.64 -6.08 a concrete empirical axial force model: 9-12 907.81 26.24 5.358 D0.822 s 0.433 If we compare the modelled and experimental values of F = (10). n0.372 axial force in the experimental points, we can see that maximal deviation of experimental points from modulled Testing of adequacy of the defined model was done surface is 16.75%, thus providing for the model (14) to be according to Fisher’s criterium [6]: considered a mathematical interpretation of the goal function. However, in the central plan point the deviation S2 F = LF of modulled results from the experimental ones is rLF 2 (11). SE 26.24%, which means that mathematical model fails to properly descibe the response function within the FrLF = 15.061. boundaries of the covered multi-factor space. Therefore, Dispersion of experiment results in the multi-factor space the null hypothesis that the effects of squared elements in is as follows: the model equal zero must be jettisoned.

2 For the purpose of designing a mathematical model in 轾N骣 k n0 2 1 犏 2 2 order to properly decribe axial force within the boundaries SLF =邋琪yu - N b i - y 0u - y 0 f 犏琪 ( ) (12). of the covered multi-factor space it is necessary to apply LF 臌u= 1桫 i =0 u = 1 the second-order polynomial model which necessitates 2 additional amount of information on the diffusion system, SLF = 0.104 8 . i.e. additional number of experiments. The degree of freedom of model adequacy is: fLF=N-k-1-(n0-1)=5, thus entailing the tabled value of 6. CONCLUSION dispersion relation of Ft(5%;5;3) = 9.0 [7]. Given the FrLF > Ft(5%;5;3) the mathematical model fails to The mathematical models designed by means of describe the correctly observed function of response, orthogonal first-order plan and regression analysis fail to which implies existence of effects of mutual action among properly describe the response function, i.e., axial cutting the model parameters. force within the boundaries of the covered multi-factor Three-factor orthogonal first-order plans allow for space due to mutual action of the model parameters, the assessment of basic effects as well as the effects of mutual squared elements action effects, as well as the action of a action of the first and second order on the empirical series of parameters whose source, nature and span of model of the response function. Should the complete action are unknown. three-factor first-order model be applied, the pattern to This points to the fact that highly complex processes that describe the (axial force) response function becomes non- take place in the zone of drilling tempered steel and which linear and can be expressed in the form of the equation: are conditioned by actions of numerous influential and mutually collinear factors, cause difficulties for y = b0 +b1x1+b2x2+b3x3+b12x1x2+b13x1x3+ mathematical models to be applied so as to describe the b23 x2x3+b123 x1x2x3 (13). behaviour of mechanical, thermo-dynamical, tribological, On the basis of values of the model parameters chemical and other phenomena in the cutting zone. corresponding to mutual action effects (see Table 4) and taking into account the assessment of significance, while also revisting the original coordinates by means of the REFERENCES equations of transformation (4), the following non-linear empirical model of axial force is deduced: [1] Sheikh-Ahmad J., Yadav R. (2008) Model for lnF = 6.82+2.187∙ln(D)+0737∙ln(n)-2.994∙ln(s)+ predicting cutting forces in machining CFRP, 0.586ln(D)∙ln(s)+0.38∙ln(n)∙ln(s)-7.536 (14). International Journal of Materials and Product Technology, Vol. 32, Number 2-3, 152 – 167 Given that the dispersion relation value of the parameter [2] Lotfi B., Zhong Z. W., Khoo L. P. (2009) Prediction b13 (9.484) is approximate to the tabled value (10.10), its of cutting forces along Pythagorean-hodograph influence has been included in the model. The modelled curves, The International Journal of Advanced values of axial force, as well as the relative error in Manufacturing Technology, Vol. 43, Numbers 9-10, relation to the experimental results are shown in Table 4. 872-882 [3] Lin J. T., Bhattacharyya D., Kecman V. (2003) Table 4. Modelled values of axial force Multiple regression and neural networks analyses in Exp. Modelle Error Exp. Modelle Error composites machining, Composites Science and points d values % point d values % Technology, Vol. 63, 539–548 s [4] Sanjay C., Neema M. L., Chin C. W. (2005) 1 830.47 -13.19 5 1024.26 -6.07 Modelingof tool wear in drilling by statistical 2 860.2 -9.97 6 1602.63 -16.75 analysis and artificial neural network, Journal of Materials Processing Technology, Vol. 170, 494–500 [5] Spaić O. (2006) Uporedna analiza habanja zavojnih burgija od brzoreznog čelika proizvedenog konvencionalnom metalurgijom i metalurgijom praha, MSc thesis, University of East Sarajevo, Faculty of Production and Management Trebinje [6] Stanić J. (1986) Metod inženjerskih mjerenja, Faculty of Mechanical Engineering Beograd [7] Laković R., Nikolić B. (1999) Primijenjena statistika 2. dio – eksperiment, University of Montenegro, Faculty of Electrical Engineering, Podgorica

CORRESPONDENCE

Obrad SPAIĆ, PhD, Assoc. prof., University of East Sarajevo, Faculty of Production and Management, Trebinje, Trg palih boraca br. 1, Trebinje, [email protected] Zdravko KRIVOKAPIĆ, PhD, Prof., University of Montenegro, Faculty of Mechanical Engineering, Podgorica, Džordža Vašingtona bb, Podgorica, [email protected] Rade IVANKOVIĆ, PhD., Prof., University of East Sarajevo, Faculty of Production and Management Trebinje, Trg palih boraca br. 1, Trebinje, [email protected]