The magnitudes of Hxx(ω) and Hyy(ω) matches well with their respective minimum-phase FRFs Hxx(min)(ω) and Hyy (min)(ω). The combined FRFs indicate a good amplitude correction over the entire frequency range, even at dominant modes. Due to the minimum-phase transformation, the phase responses of measured and minimum-phase FRFs differ, which is clearly visible in the phase responses of the combined FRFs. However, the phase responses of the combined FRFs are dominated by linear trends which can be related to constant delays in the time domain and will not affect the amplitude responses of the inverse filtered signals. Thus, the phase responses affecting the inverse filtered signals are obtained by removing the linear phase trends from the combined FRFs. As seen in Figs 1.3 and 1.4, the phases of the combined FRFs have deviated approximately 7 degrees from zero at 2500 Hz. Therefore, it is expected that frequencies above 2500 Hz will be out of range for the inverse filter. To avoid influences from the resonances at 2550 Hz in the x-direction and 2700 Hz in the y-direction, the frequency limit of the inverse filters was set to 2400 Hz in both directions; higher frequencies are removed by low-pass filtering. Low-pass filtering also removes any high-frequency noise present in the inverse filtered signal. High frequency noise is common when performing inverse filtering due to the nature of the filter. The reason is that mechanical systems normally act as low-pass filters, attenuating high frequencies. When these systems are inverted they will instead act as high-pass filters and therefore respond badly to high frequency noise. The low-pass filter design used in the simulations is a Butterworth filter of order 3. To remove any phase distortion caused by the low-pass filter, zero-phase filtering is performed by filtering the data in both forward and reverse directions. To test the behavior of the inverse filters, simulations are carried out for the cutting speeds and feed rates listed in Table 1.1. Down milling and 50 % radial immersion is used to excite the dynamometer with a high frequency transient signal and thereby clear effects from the dynamometer dynamics appear. The simulations are carried out using bothx andy as feed directions. Mechanistic modeled cutting forces are used as input (see Table 1.2), these are filtered through ETFEs of the measured mixed-phase transfer functions, hxx(n), hxy(n) andhyy(n), hyx(n). Each output is then inverse filtered and compared to the reference cutting force (simulated input force), Figs 1.5 and 1.6. The simulated output forces have contributions both from the point FRFs and the cross FRFs. Due to the phase distortion caused by the minimum-phase transformation, 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 0.2 0.4 0.6 0.8 1 Frequency [Hz] Coherence 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 2 4 6 8 10 Frequency [Hz] Magnitude [N/N] H xx H xy H xx H xy Fig. 1.1 Measured FRFs of the force dynamometer from impulse excitation. The dynamometer is excited in the x-direction and responses collected in x- and y-directions 4 M. Magnevall and T. Beno
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