between the two signals, [12]. fR and fI are the average values of the reference and inverse filtered forces, respectively. The best fit between the signals is found at the time delay, τ, where the energy in the difference between the two signals has a minimum. If η ¼0 the two signals are identical. The simulation results show that the inverse filters are able to counteract the dynamic influences and reconstruct the reference cutting forces within a small error margin for all cutting speeds and feed rates listed in Table 1.1. As seen in Figs 1.5a and 1.6a the difference between reference and inverse filtered forces increases as the cutting speed increases. This is expected since the cutting speed affect ramp up times and thereby the frequency content in the reference force signal. Higher cutting speed leads to higher frequencies in the reference force. Thus, the frequency range of the inverse filter may not be enough to fully describe the transient behavior in the force signal at high cutting speeds. Comparisons between reference and inverse filtered forces using maximum feed rate and cutting speed are shown in Figs 1.5b and 1.6b. Additionally, simulations were performed without considering the effect of the cross FRFs on the dynamometer outputs. Neglecting the cross FRFs did not show any significant changes in the results confirming that the cross FRFs are negligible. 1.4 Experimental Tests The proposed method was evaluated in experimental cutting tests using different cutting speeds, feed rates and radial immersion. To be able to test the method over a large span of cutting speeds, the tests were performed in aluminum. Forces in both x- and y-directions were recorded and inverse filtered. The results are compared with unfiltered, low-pass filtered and simulated cutting forces. The simulated cutting forces are mechanistic modeled and the cutting coefficients are estimated from milling tests in the work piece used in the experimental tests, see Table 1.2, [13]. The test setup is illustrated in Fig. 1.7. Equipment and cutting data used are listed in Table 1.1. Table 1.2 Estimated cutting force coefficients from milling tests vc Ktc Kte Krc Kre Kac Kae 200 [m/min] 732.26 26.38 163.71 15.65 52.07 18.90 [N/mm2] 400 [m/min] 693.05 18.37 128.67 9.66 36.80 14.771 [N/mm2] 800 [m/min] 638.00 17.92 99.43 11.33 5.47 12.69 [N/mm2] 1200 [m/min] 649.03 21.78 42.14 14.67 23.76 4.61 [N/mm2] 1490 [m/min] 617.35 19.71 24.12 12.77 87.01 16.72 [N/mm2] 2000 [m/min] 597.66 17.40 24.44 11.61 105.45 15.61 [N/mm2] 500 0.05 0.1 0.15 Feed Rate, fz [mm/tooth] Force [N] 0.2 0.25 a b 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 1000 Cutting Speed, vc [m/min] 1500 2000 0 0.002 0.004 0.006 0.008 0.01 Time [s] 1500 1000 Reference force, fR(t) Inverse Filtered, fI(t) Dynamometer Output, fD(t) 500 0 Fig. 1.5 Simulation results with feed in x-direction; (a) The isolines of η [%], Eq. (1.7). (b) Comparison between dynamometer output, inverse filtered and reference input forces (Cutting speed, vc ¼2000 [m/min]; Feed rate, f z ¼0:25 [mm/tooth]) 1 Improved Cutting Force Measurements in Milling Using Inverse Filtering 7
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