46 Estimation of Instantaneous Speed for Rotating Systems: New Processing Techniques 525 Bayesian RPM estimate Accel 3 BS = 2048 - 87.2% overlap Instantaneous RPM (rpm) Time (sec) Frequency (Hz) Overlapping blocks Accel 3 Bayesian on Time Spectral map with BS = 2048 - 87.5% overlap 3 x 104 2 1 0 0 5 10 15 20 30 25 0.5 1.5 2.5 600 500 400 400 300 300 200 200 100 0 0 100 -10 -20 -30 -40 -50 -60 -70 -80 b c a b Fig. 46.4 (a) User input, (b) choosing a range to reprocess, (c) reprocessed estimate and (d) reprocessed estimate over the time spectral map To understand the problem, it is necessary to revisit the formula used for Instantaneous RPM estimation [1]. Instantaneous RPMD 60 t2 – t1 .nC1/ (46.8) Here ‘n’ denotes the number of pulses between zero crossings considered. The values t2 and t1 are the zero crossings of two consecutive pulses that are approximated by a linear interpolation between two points which are ‘ t’ apart in time. Hence the error that can occur in the estimation of a zero crossing ranges across ˙ t=2. When this error is a significant percentage of the absolute value of time difference, the corresponding speed values can fluctuate across an unacceptable range. This is basically a sampling problem and the RPM curve when observed closely shows similar characteristics to bit noise. Figure 46.7 shows a zoomed in version of the data from the vacuum cleaner and the corresponding fluctuations. Consider a set of consecutive time instances recorded at zero crossings at around 24,500 RPM. t.2725/ – t.2724/ D 0:00244 s, t.2726/ t.2725/ D 0:00238 s and so on. Corresponding RPM estimates are 24,599 RPM and 25,249 RPM assigned at 15.2629 s and 15.2653 s respectively. Hence, in 0.002408 s, there has been a change of about 650 RPM which corresponds to a rate of change of 269,933 RPM/s. This is impossible considering the inertia of the system. While increasing the sampling rate is the best solution to reducing the error, increasing the absolute time difference between the recordings of zero crossings is less complicated and does not require sophisticated equipment. When the time
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