dcf) (3-4) 7T Jo If*'. \u2014 \u2014 s'm2 ,) (2 aw0 \u2014 2?r \/ Jo 1 r**> au>2 = 1 cos2d)d > awk = \\ cos kd>dd>, 7T J 'o 1 bwk = 7T j I s'mk(pd(j). 'o Chapter 3. Theoretical Background 54 These simple integrals are evaluated and shown down below as: aw0 -aw\\ -bw\\ -aw2 -bw2 -awk -**

,) + ^ - ( s i n 0 , + r 2 ( l - cos 0,))]. (5.3) In a similar manner: FvT-ave = K \" ^ N S t [^(1 - cos20 a ) - (20, - s in20,) + ^ - ( r 2 sin 0 , - (1 - cos 0,))]. (5.4) The above relations are derived for the up milling case. A similar exercise can be carried out for down milling. Since the procedures are identical, only the results will be given below: FxT-ave = K ' a N S t [n(20, - sin 20,) - (1 -cos 20,) + ^ - ( r 2 ( l - cos 0 , ) - s i n 0,)] (5.5) 87T Jt and, i*Vr-at,e = K , a N S t [n ( l - cos 20,) 4- (20, - sin 20,) + ^ - ( (1 - cos 0,) - r 2 sin 0,)]. (5.6) 07T Ot The relationships shown in equations (5.3), (5.4), (5.5), (5.6) indicate that the average forces are linear functions of feedrate, St. Hence for a given range of feedrates and for a known immersion it is possible to find the cutting constants from the slope and intercept of equations (5.3) and (5.4). The force measurements are taken for various feedrates in Chapter 5. Experimental Work 86 st (mm\/tooth) FXT (N) FYT (N) 0.0254 125.5 -166.1 0.0508 213.5 -270.2 0.1016 328.3 -418.6 Table 5.1: Averages for full immersion and new insert full immersion for the up-milling case. Substituting the immersion angle (n radians) into the average force equations, simpler relationships are obtained: F X T - A V E = ^(St2nr1 + 8h*r2) (5.7) 07T and FYT-ave = ^ ^ ( ^ + 8fc*). (5.8) 5.2 Experimental Results In this section, the values of the cutting constants will be calculated using those equations and the experimental data. 5.2.1 Calculation for a New Insert A new chamfered insert (see figure (5.2)) is used for cutting at 360 rpm with feeds of 0.001, 0.002, and 0.004 inch\/tooth. The forces are then averaged over the full period of the tooth. The results are shown in table (5.1). A line is fitted through the points as shown in figure (5.3). From the graph the slope and intercept of the curve are obtained and used for the necessary calculations. For the new insert, the cutting constants are: Ks = 2565 N\/mm2, n = 0.8, Chapter 5. Experimental Work -400 -1 -000 0J02 0J04 OJOO 0J06 Feedrate (mm\/tooth) Figure 5.3: Average forces vs. feed for full immersion, sh Chapter 5. Experimental Work 88 st (mm\/tooth) FXT (Af> FYT (TV) 0.0254 250.4 -234.9 0.0508 334.2 -321.9 0.1016 448.3 -476.0 Table 5.2: Averages for full immersion and worn insert r2 = 0.74, h* = 0.022 mm. (5.9) 5.2.2 Calculation for a Worn Insert A chamfered insert with 0.008 inch flank wear is used as the worn insert. The cutting test information is included in table (5.2). The results of the cutting constants are shown below: Ks = 2477 TV\/mm 2, ri = 0.81, r2 = 1-22, h* = 0.0394 mm. (5.10) Comparing the values for a new insert to that of a worn one, one can see that Ka and T\\ are very similar, ln the case of r2 and h* these values are considerably different. This difference is attributed to the flank zone cutting forces that occur as the wear-land increases. Chapter 5. Experimental Work 89 5.3 Calculation of Experimental F.S. Coefficients The next step after the calculation of cutting variables is the estimation of F.S. coeffi-cients experimentally. The method of experimentally obtaining Fourier Series coefficients involves dividing the force vs. time plot at 12 equidistant times per spindle revolution (equivalent to 30\u00b0 angular displacement of the spindle). The forces at those 12 instances can then be used to obtain the mean and fundamental coefficients as well as the first 2 frequency component coefficients after the fundamental. More detail concerning this method is included in the appendix B. The following equations give the necessary coef-ficients: OxtO -1 , 2^ (vo + yi + + 2\/11), Oxt3 = i , g(yo \u2014 3\/2 + 2\/4 \u2014 2\/6 + 2\/8 - 2\/10), Kt3 = 1, g(lh - 2\/3 + 2\/5 - 2\/7 + 2\/9 - 2\/11), a x t l = 7^(yo - 2\/6) - axt3 bxti = ^(2\/3 - 2\/9) + \u00b0xt3 axt2 \u2014 1, 4(2\/0 - 2\/3 + 2\/6 ~ 2\/9), Kt2 = 1, j ( 7 i - 72 + 73 - 74). Here, yo, 2\/i, 2\/n are the forces obtained experimentally at 0, 30\u00b0, 60\u00b0, 330\u00b0 of spindle revolution, and 71,72,73,74 corresponds to 45\u00b0, 135\u00b0, 225\u00b0, 315\u00b0 of spindle revolution. The experimental coefficients (especially their magnitudes) are now compared with the exact method of calculating the Fourier Series coefficients already shown in earlier sections (first for a new tool then for a worn tool). A second set of experiments were done with round inserts (i.e. inserts with radiused cutting edge) after the problems experienced with chamfered inserts. An insert with Chapter 5. Experimental Work F.S. coeff. theoretical experimental magnitudetheor. magnitudeexpeT. axto(N) 332.4 295.8 332.4 295.8 a\u00abti(JV) 424.5 269.7 699.3 530.6 bxti(N) 555.7 457.0 axt2(N) -309.9 -209.5 548.7 583.4 bxt2(N) 452.8 544.5 axt3(N) -168.5 -269.7 191.0 272.7 bxt3(N) -90.0 40.0 Table 5.3: F.S. coeff. comparison for a new tool F.S. coeff. theoretical experimental magnitudetheor. magnitudeexper. OxtO 451.0 427.1 451.0 427.1 519.2 329.2 905.5 761.5 bxti 741.9 686.7 -387.3 -302.5 656.4 709.1 bxt2 530.0 641.3 0-xt3 -162.7 -309.2 184.9 314.3 bxt3 -87.9 56.7 Table 5.4: F.S. coeff. comparison for a worn tool Chapter 5. Experimental Work 91 chamfer is used for improved surface finish quality. The clearance angle on the chamfered edge is nearly zero. This poses the difficulty of determining the effects of secondary edge cutting. It would be difficult to estimate the equivalent chip thickness for this situation, since cutting would take place on this nearly flat edge. Consequently, the estimation of the depth of cut would also be difficult. The second set of tests were done with aluminium as well as the steel from the same stock as before. Before the test results are shown, a second method of cutting constant calculations are presented below. 5.4 Cutting Constant Calculations Using FT and FR The tangential force and the radial force are linearly dependent on the uncut chip thick-ness. Exploiting this relationship, one could plot FT and FR VS. h and from this obtain directly the four constants. The only consideration is the realization that h is dependent on the immersion angle, (f>, through the relationship h = S tsin0. For a given ~~- 2\u00abPI ) PHI_S - PHI_S - 2 \u00bb P I ; aw[j] - s i n ( P H I _ S ) \/ ( P I * n ) ; bwjj] - (1 - c o s ( P H I _ S ) ) \/ | P I * n ) ; ( l \/ ( n + l | U \/ ( 2 \u00ab P I ) ; i f { ( j % z 0 ) \u00bb T [ j ] - a | j ) ; b T [ j ) - b | j ) ; awT| j ] - aw[j ) ; bwT(j] - bw[ j ] ; ) e l s e < a T ( j ) - 0 .0; b T | j ] - 0 .0; awT| j ] - 0 .0; bwT| j ) - 0 .0; } tempi \u00bb a T | j ) * a T [ j ) + b T | j ] * b T | j ] * tempi; temp2 - a w T j j ] \u00ab a w T | j ) + b w T | j ] \u00bb b w T l j ] + temp2; PHI - n ' p h i ; sumT - a T ( j ) \u00ab c o s ( P H I ) \u2022 bT] j ) * s i n ( P H I ) + sumT; SUHW - a w T ] j J \u00ab c o s ( P H I ) * b w T | j ) * s i n ( P H I ( + SUMW; f o r < if ( j - 2 ; j < - a r r - 2 ; j++) i i n - o i n - ( d o u b l e ) j ; a x | j ] - ( a | j - l ) - ( r ' b | j - l ) ) + a | j + l ] + ( r * b [ j + l J ) ) \/ 2 . 0 ; b x | j ) - ( b [ j - 1 ) + < r + a [ j - 1 ) ) + b | j + 1 | - < r - a | j + 1 ) )> \/2 .0; e y [ j ) - ( ( r ' a | j - 1 ] ) + b [ j - 1 ] + ( r ' a ( j + 1 ] ) - b | j + 1 | ) \/ 2 . 0 ; b y | j ) - ( < r * b ( j - 1 ] ) - a | j -1 ]+a[ j + 1] + ( r * b [ j + 1 | ) 1 \/ 2 . 0 ; awx(j) - (aw(j -1] - ir2*bw|j -1 |>+aw]j+1)+(r2'bw| j+1]>) \/2 .0; bwxjj] - ( b w j j - 1 | + I r ? * a w [ j - 1 | ) + b w | j + 1 ) - ( r 2 ' a w ( j + 1 ] ) ) 1 2 . 0 ; awyjj] - ( ( r 2 ' a w | j - 1 | ) + b w | j - 1 ) + ( r 2 * a w [ j + 1 J ) - b w | j + 1 ] ) \/ 2 . 0 ; bwyjj] - ( ( r2*bw| j - I | ) -aw | j -1 )+aw[ j+1]+(r2*bw| j+1] ) ) \/ 2 .0 ; i f ( t \u2014 0 ) ( magnitude_x - C l * s q r t ( a x ( j J * a x [ j \\ + b x [ j ] * b x ( j ] | ; m a g n i t u d e y - C l * s q r t ( a y | j ] * a y | j ] + b y | j ] * b y [ j ) ) ; magnitude_wx - C 2 * s q r t ( a w x | j ) * a w x | j ] + b w x | j ] * b w x | j ] ) ; magnitude_wy - C 2 * s q r t < a w y | j j \u00ab a w y j j ] + b w y | j ] \u00ab b w y | j ] ) ; f p r i n t f ( f 3 , \"%lf %lf l l f l l f %d\\n\",magnitude_x, magnitude_y. magnitude wx, ) magnitude ax| j ] - 0 .0; bx| j ] - 0 .0; eyl j l - 0.0; by] j ) - 0.0; awx|j] - 0 .0; b w x | j | - 0 .0; awy[j] - 0 .0; bwy[j] - 0 .0; PHI - n ' p h i ; whi le ( PHI >- 2'P1 ) PHI - PHI - 2*PI; SUMX - (ax Ij J *cos(PHIt) + ( b x [ j | + s in (PHI) ) + SUMX; SUMY - (ay I j ) ' c o s ( P H I ) ) + ( b y j j j * s i n ( P H I ) ) + SUMY ; SUMWX - (awx] j | ' cos (PHI) ) + (bwx( j | * s in (PHI) ) < SUMWX; SUMNY - ( a w y j j ) \u00ab C O S ( P H I ) ) + (bwy\\ j ) \u00ab s i n ( P H I ) ) + SUMWY; \/ \u2022 i f (dphi 10) { dimax - Cl*ax[1]+C2'awx|1 | dimay - C l \u00ab a y | 1 | + C 2 * a w y | 1 j dimbx - Cl'bxj1]+C2*bwx[1] Appendix A. Listing of F.S. Modelling Program for Steady State Face Milling dimby - Cl*by[1)+C2*bwy[1); fund \u2022 pow(pov( climax, 2) + pow(dimay, 2) + pow(dimbx,2) + pow( dimby, 2), 2); r a t io - fund\/(pow(Cl*ax|0]+C2\u00abawx|0),2)+pow(Cl 4ay[0]+C2*awy[0),2)); dev \u2022 (dimax\"dimby-dimay*dimbx)\/sqrt(fund); p r i n t f < ' l l f %lf %lf\\n\",fund,ratio,dev); fp r in t f ( f4 , \"S l f \u00bblf %lf\\n\",fund,ratio,dev); ) *\/ t - l ; wear_torque - C2*R*SUMW; Torque - Cl*R*sumT; F_x - C1\u00abSUMX; F y - -C1*SUMY; F wx - C2*SUMWX; F~wy - -C2*SUMWY; fprintf(f2,\"%lf %lf\\n\",Torque,weartorque); fp r in t f ( f l , \"%l f %lf %lf Uf \\n\" ,F_x ,F_y ,F vx,F_wy); \/ \u2022 p r i n t f ( \" \u00bb l f \u00bblf * l f %lf\\n\",F x,F_y,F vx,F wy) ; ' \/ ) fc loael f1) ; fclose(f2); fe loself3) ; fclosef f4); Appendix B Approximate Method of Finding the First Three F.S. Coefficients Given the function, \/ (x) , it is possible to find its Fourier Series expansion coefficients, a 0 , a n , bn after integrating \/ \/(x)dx, J \/ (x) cos nxdx, J \/ (x) sin nxdx respectively. If, how-ever only a limited number of values for f(x) is known the above method cannot be used. The method explained below is suitable for experimental situations where only the first three values of the Fourier Series coefficients are needed. The insight to the method becomes clear when it is assumed that the Fourier Series expansion of f(x) contains no terms beyond the third harmonics. So the Fourier Series expansion of f(x) can be written as: \/ (x ) = ao + ai cos(x) + bi s'm(x) + a2 cos(2x) \u2022+ b2 sin(2x) + a3 cos(3x) + b3 sin(3x) (B. l ) Next, it is assumed that 12 values of \/ (x) , y0, yi, y 2 , y u are available experimentally at intervals of 30\u00b0 (for one period of the function). Now taking the mean values for the 12 given values of f(x), it is possible to obtain ao as: a0 = Y^iVo + 2\/1 + V2 + 2\/3 + 2\/4 + y& + Ve + Vi + 2\/8 + 2\/9 + J\/io + Vw) (B.2) Similarly, a 3 and 63 can be obtained by adding the influence of a3 and b3 terms at each 30\u00b0 interval. Hence, a 3 = \\ Y] y cos 3x = \\(y0 - y2 + yA - y6 + ys - yi0) (B.3) o o b2 = I V y sin 3x = ^ (T\/J - y3 + y5 - y? + 2\/9 - 2\/n) ( B - 4 ) o o 115 Appendix B. Approximate Method of Finding the First Three F.S. Coefficients 116 Now putting x = 0,180,90,270 into equation (B. l ) , it is possible to obtain aj, bi, a 2 : ai = ^(2\/0 - 2\/6) ~ a-3 (B.5) * i = ^(2\/3-2\/9) + 63 (B.6) 0.2 - ^(2\/0 - 2\/3 + 2\/6 - 2\/9) (B- 7 ) The value of 62 remains to be found. The method to obtain b2 involves obtaining the values of y corresponding to x = 45,135, 225, 315 and call these values of j\/'s 71,72,73,74-Choosing the eight equidistant ordinates corresponding to x = 0 , 4 5 , 9 0 , 3 1 5 it is pos-sible to write b2 as: 2 1 b2 = g \u00a3 2 \/ s i n 2 x = ^(7i -72 + 73 -74) (B.8) Appendix C Workpiece Dimensions 117 Appendix C. Workpiece Dimensions Bibliography [1] T6nshoff, H. K . , Wulfsberg, J . P., Kals, H . J. J. , Konig, W., van Luttervelt, C. A . , \"Developments and Trends in Monitoring and Control of Machining Processes\", Annals of the CIRP, Vol. 37\/2\/1988, pp. 611-622. [2] Tonshoff, H .K. , Stanske C, \"Material Aspects in Machining of Forged Parts\",Proc. of Int. Conf. on High Productivity Machining, Materials and Processing, May 1985, pp. 207-222. [3] Sarin, V . K . , Buljan, S.T., \"Coated Ceramic Cutting Tools\",Proc. of Int. Conf. on High Productivity Machining, Materials and Processing, May 1985, pp. 105-111. [4] Armarego, E.J .A. , Brown R.H. , The Machining of Metals. Prentice Hall, Inc., En-glewood Cliffs, New Jersey, 1969. [5] Suzuki, H . , Weinmann, K. J., \" A n On-Line Tool Wear Sensor for Straight Turning Operations\", Journal of Engineering for Industry, Vol. 107, November 1985, pp. 397-399. [6] Stoferle, T. H. , Bellmann, B. , \"Continuous Measuring of Flank Wear\", 16th. Int. Mach. Tool Des. Res. Conference, 1976, pp. 647-651. [7] Iwata, K . , Moriwaki, T., \"An Application of Acoustic Emission Measurement to In-Process Sensing of Tool Wear\", Annals of the CIRP, Vol. 26\/1\/1977, pp. 21-26. [8] Ramalingam, S., Shi, T., Frohrib, D.A. , Moser, T., \"Acoustic Emission Sensing with an Intelligent Insert and Tool Fracture Detection in Multi-Tooth Milling\",Proc. of 119 Bibliography 120 16th NAMRC, May 1988, pp. 245-255. [9] Giusti, F., Santochi, M . , Tantussi, G., \"On-Line Sensing of Flank and Crater Wear of Cutting Tools\", Annals of the CIRP, Vol. 36\/1\/1987, pp.' 41-44. [10] Spur, G. , Leonards, F., \"Sensoren zur Erfassung von Prozesskenngrossen bei der Drehbearbeitung\", Annals of the CIRP, Vol. 24\/1\/1975, pp. 349-354. [11] Aoyama, H. , Kishinami, T., Saito, K. , \" A Study on a Throw Away Tool Equipped with a Sensor to Detect Flank Wear\", Bull. Japan Soc. of Prec. Eng., Vol. 21\/3\/1987, pp. 203-208. [12] Yellowley, I., Hosepyan, Y . , 1990, \"Tool Wear Sensor\", US patent # 5,000,036. [13] Colwell, L. V . , Mazur, J. C , Devries, W. R., \"Analytical Strategies for Automatic Tracking of Tool Wear\", 6th. North American Metalworking Research Conference, 1978, pp. 276-282. [14] Uehara, K . , Kiyosawa, F., Takeshita, H. , \"Automatic Tool Wear Monitoring in N.C. Turning\", Annals of the CIRP, Vol. 28\/1\/1979, pp. 39-42. [15] Lai, C.T., \"The Influence of Tool Wear and Breakage on Forces in Bar Turning\", M . Eng. Thesis, McMaster University, Hamilton, Ontario, Canada. [16] Jiang, C. Y . , Zhang, Y . Z., Xu , H. J., \"In-Process Monitoring of Tool Wear Stage by the Frequency Band-Energy Method\", Annals of the CIRP, Vol. 36\/1\/1987, pp. 45-48. [17] Lister, P. M . , Barrow, G., \"Tool Condition Monitoring Systems\", 26th. Int. Mach. tool Des. Res. Conference, 1986, pp. 271-288. Bibliography 121 [18] Barrow, G., \"A Review of Experimental and Theoretical Techniques for Assessing Cutting Temperatures\", Annals of the CIRP, Vol. 22\/2\/1973, pp. 203-211. [19] Colwell, L. V., \"Cutting Temperature versus Tool Wear\", Annals of the CIRP, Vol. 24\/1\/1975, pp. 72-76. [20] Shaw, M. C , Metal Cutting Principles. Clarendon Press, Oxford, 1984. [21] Yellowley, I., \"The Utilization of Numerically Controlled Machine Tools\", Depart-ment of Regional Industrial Expansion Report No. 102, June 1985. [22] Juneja, B. L., Sekhon, G. S., Fundamentals of Metal Cutting and Machine Tools. John Wiley and Sons, New Delhi, 1987.. [23] Yellowley, I., \"A Note on the Significance of the Quasi-Mean Resultant Force and the Modelling of Instantaneous Torque and Forces in Peripheral Milling Operations\", Journal of Engineering for Industry, Vol. 110, August 1988, pp. 301-303. [24] Fu, H. J., DeVor, R. E., Kapoor, S. G., \"A Mechanistic Model for the Prediction of the Force Signal in Face Milling Operations,\" ASME Journal of Engineering for Industry, Vol. 106, 1984, pp. 81. [25] Yellowley, I., \"Observations on the Mean Values of Forces, Torques and Spe-cific Power in the Peripheral Milling Process\", Int. J. Mach. Tool Des. Res., Vol. 25\/4\/1985, pp. 337-346. [26] Papazafiriou, T., Elbestawi, M.A., \"Development of a Wear Related Feature for Tool Condition Monitoring in Milling\", U.S.A.-Japan Symposium on Flexible Au-tomation, July 1988, pp. 1009-1016. Bibliography 122 [27] Elbestawi, M . A . , Papazafiriou, T., Du, R .X . , \"In-Process Monitoring of Tool Wear in Milling Using Cutting Force Signature\", Int. J. Mach. Tools Manuf act., Vol. 31\/1\/1991, pp. 55-73. [28] Colwell, L. V . , \"Predicting the Angle of Chip Flow for Single Point Cutting Tools\", Trans. AS ME, Vol. 76\/1954, pp. 199-203. [29] Altinta\u00a7, Y . , Yellowley, I., \"The Identification of Radial Width and Axial Depth of Cut in Peripheral Milling\", Int. J. Mach. Tools Manufact., Vol. 27\/3\/1987, pp. 367-381. [30] Altinta\u00a7, Y . , \"Process Monitoring and Tool Breakage Recognition in Mill ing\", Ph.D. thesis, 1987, McMaster University, Hamilton, Ontario, Canada. [31] Pandy, P.C., Shan, H.S., \"Analysis of cutting forces in peripheral and face milling operations\",\/^. J. Prod. Res., Vol. 10, NO. 4, 1972, pp. 379-391. 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