Accuracy of speed determination of a machine from frequency demodulation of response vibration signals
Abstract
In the diagnostics of variable speed machines, it is often important to know the instantaneous speed very accurately. This might be to perform computed order tracking, which requires a map of time to rotation angle, the latter being the integral of the instantaneous frequency.
The speed can be determined very accurately by frequency demodulation of a tacho signal, and in some cases, it can be determined by frequency demodulation of a response vibration signal. However, it has recently been realised that the result of the frequency demodulation depends on the vibration parameter being demodulated (displacement, velocity or acceleration) since differentiating a signal changes the mixture of amplitude modulation (AM) and frequency modulation (FM).
This paper uses simulation and actual measurements to compare the accuracy of different methods of frequency demodulation, namely differentiation of the instantaneous phase, the frequency domain energy operator (FDEO), which can be shown to be the squared envelope of the derivative of a signal, and the Teager Kaiser energy operator (TKEO), which is similar to the FDEO, but slightly less accurate. A good estimate of squared frequency can be obtained as the ratio of the squared envelope of the derivative of the signal (ie the FDEO) to the squared envelope of the signal, while a somewhat poorer estimate can be obtained from the ratio of the TKEO of the derivative to the TKEO of the signal (which will thus be seen to give an estimate of the squared frequency of the derivative of the signal).
The paper then shows that the best response parameter to demodulate is that for which the FRF is uniform within the demodulation frequency range, ie displacement if it is on a spring line, and acceleration if it is on a mass line. Simple estimates are given for the errors involved.
The speed can be determined very accurately by frequency demodulation of a tacho signal, and in some cases, it can be determined by frequency demodulation of a response vibration signal. However, it has recently been realised that the result of the frequency demodulation depends on the vibration parameter being demodulated (displacement, velocity or acceleration) since differentiating a signal changes the mixture of amplitude modulation (AM) and frequency modulation (FM).
This paper uses simulation and actual measurements to compare the accuracy of different methods of frequency demodulation, namely differentiation of the instantaneous phase, the frequency domain energy operator (FDEO), which can be shown to be the squared envelope of the derivative of a signal, and the Teager Kaiser energy operator (TKEO), which is similar to the FDEO, but slightly less accurate. A good estimate of squared frequency can be obtained as the ratio of the squared envelope of the derivative of the signal (ie the FDEO) to the squared envelope of the signal, while a somewhat poorer estimate can be obtained from the ratio of the TKEO of the derivative to the TKEO of the signal (which will thus be seen to give an estimate of the squared frequency of the derivative of the signal).
The paper then shows that the best response parameter to demodulate is that for which the FRF is uniform within the demodulation frequency range, ie displacement if it is on a spring line, and acceleration if it is on a mass line. Simple estimates are given for the errors involved.
