Theory and praxis of synchronised averaging in the time domain

Transcription

1 J. Tůma 43 rd International Scientific Colloquium Technical University of Ilmenau September 21-24, 1998 Theory and praxis of synchronised averaging in the time domain Abstract The main topics of the paper are as follows: an analyse of the method used in resampling of the time records, an effect of averaging in the time domain on the signal frequency spectrum and an analysis of geared axis systems. The key words for resampling are oversampling, interpolation, antialiasing filtration and decimation while for synchronised averaging are a moving averaging and a signal enhancement. Introduction Analysis of signals from machines running in cyclic fashion is preferred in terms of order spectra rather than frequency spectra. The order spectra are evaluated using time records that are measured in revolutions [-] rather than seconds and the corresponding FFT spectra are measured in orders [-] rather than frequency. This techniques is called order analysis or tracking analysis as the rotation frequency is being tracked and used for analysis. The resolution of the order spectrum is equal to 1/rev, where rev [-] is revolutions per a record corresponding to input data for the Fast Fourier Transform (FFT). As the most of dynamic forces, exciting vibration and noise are related to the machine rotation frequency, rotation speed variations at fixed signal sampling frequency cause the smearing of components in frequency spectra. An advantage of the order spectra is that a harmonic order component remains in the same analysis line independent of the speed of the machine. The order spectra are a tool to isolate harmonics of fundamental frequency components from structural frequency components using a run-up or coast-down of the machine. Digital order tracking Signals to be sampled are obviously passed through the analogue antialiasing filters. The cut-off frequency, f c, of the antialiasing filter is equal to the 1/2.56-fraction of the sampling frequency, f s. For 14 bits A/D conversion, the filter transient part ranging from f c to 156. f c and the 80 db attenuation are required to avoid an aliasing problem. As it was mentioned, the tracking technique benefits from matching of a time record with the time interval of a machine periode, for instance a revolution. For revolution speed variations, it couldn t be reach at a fixed sampling frequency. The number of the samples per a rotation, N i, is varying in a certain interval ranging from N min to N max. Thereby the sampled signal should be resampled along the record in each revolution period determined by a tacho signal measurement. The Fast Fourier Transform (FFT) requires a number of samples to be equal to m a power of two, N = 2. The high resolution of all the order spectra requires the larger possible length of an FFT-record. For the aliasing problem, the sampling frequency after resampling cannot be greater than the original sampling frequency. It means that all what can be done is decimation. Both these requirements result in the following inequality

2 Fig. 1. Frequency response of order 120 FIR filter designed with REMEZ 2 m 2 m + N < 1 min for the length of the resampled FFT-record. If 2 m = N min then the less number of samples per a period of the highest frequency spectrum component is equal to 2.56 which results in large interpolation errors? A solution of this problem consists in artificial oversampling by a factor of two or four that increases the sampling frequency by the corresponding multiple. The oversampling is performed by adding one or three extra samples with a value of zero in between the samples from the ADC [2]. The oversampled signal must be then lowpass filtered to restore the original spectrum. The lowpass filter is an equiripple Finite Response Filter (FIR) with a linear phase. Experimental results presented in the paper are reached with oversampling by a factor of four and using the REMEZ Parks-McClellan optimal filter designed by MATLAB. A frequency response of this filter shown in Fig. 1 gives the 120 db attenuation. Four times oversampling of signals allows decimation by interpolation and resampling by a maximum factor of 5.92 as it is shown in Fig. 2. The f frequency ratio is equal to the same factor of smax f s min sample Fig. 2. No decimation and maximum allowable decimation without any aliasing problem This part of the paper is focused on assessment of interpolation errors. The Fourier transform of a periodic signal is based on decomposition of the signal to a number of (co-) sinusoidal components at various frequencies. To simplify and clear up the interpolation errors, the signal consisting of only one sinusoidal component at a certain frequency is analysed. As it is shown in following diagram the errors can be interpreted as an amplitude and phase modulation signal of the sinusoidal signal considered as a carrying component. In the mentioned diagram the quantity x designates an amplitude error and τ designates a phase error of interpolation (see Fig. 3). Both the amplitude and phase modulation signals can be separated and evaluated using the theory of analytical signals.

3 Concerning a formulae for interpolation the simplest one is either linear or quadratic. Concerning the length of an FFT-record, there are two ways how to perform interpolation and resampling. The number of samples per the FFTrecord can reach either the greater possible value of 4 2 m or the final value of 2 m. Resampling to the number of 4 2 m samples per the FFT-record requires further decimation by four. For a limited filtration efficiency of the REMEZ filter in the stop band of the frequency spectrum, the lowpass digital antialiasing filtration is repeated. As the length of the FFT-record is a power of two this filtration is performed by using an FFT and an inverse FFT. Fig. 3. Interpolation errors Resampling to the number of 2 m samples is without decimation and further lowpass filtering, thereby this way of resampling is speeded up in comparison with resampling to the number of 4 2 m samples. The speed up resampling is paid by raising interpolation errors as can be seen in Fig. 5. This figure shows a magnitude of the order spectra corresponding to the reasmpled sinusoidal time signal of 150 Hz. The sampled time signal and its FFT are evaluated in floating point arithmetic operated with variables of the single type accuracy (4 bytes/ 7-8 digits). This accuracy of arithmetic operations is much better then the accuracy of the 12-, 14- or 16-bit ADC (less than or equal to two bytes) introducing a quantisation error. The FFT spectrum of the testing signal is shown in Fig. 4. The noise level in the spectrum gives a Signal/Noise ratio of 5 and a half decades. It is assumed that the original signal is sampled at 2048 Hz and the tacho signal frequency equals to 1.5 Hz. Therefore, the time signal frequency in ORD equals to 100. The number of samples, corresponding to a period of the tacho signal, equals to the non-integer value of 2048/1.5 = / 3. As it was mentioned above, the record of / 3 samples can be resampled without any aliasing problem to 1024 samples (0 to 400 ORD). Peaks except one of 100 ORD showed in the order spectra in Fig. 5 Fig. 4. FFT spectrum of a sinusoidal time signal (accuracy 7-8 digits) represent interpolation errors. The order spectra, corresponding to the speeded up evaluation, contain considerable aliasing components in spite of the antialiasing filter attenuation. It may be concluded that linear interpolation magnifies the aliasing components. The order analysis is demonstrated on gearbox vibration measurements during a run-up test. A car gearbox under test was driven from the input shaft rotational speed of 1800 RPM to 3500 RPM. Both the vibration signal (acceleration) at the input shaft bearing and the tacho signal from a trigger on the input shaft were recorded by a sound card (16-bit resolution / 44.1 khz sampling frequency) of a PC. The sound card includes a stereo pair of Σ analogue-todigital converters, which behave like a lowpass filter with the cut-off frequency of 04. f s and the 80 db attenuation in the stop band beginning at 06. f s.

4 Fig. 5. FFT-spectrum magnitude of the resampled 150 Hz sinusoidal time signal (f s = 2048 Hz). Tacho signal frequency of 1.5 Hz A spectral map (running autospectra in RMS) of the order spectra is shown in Fig. 6. The constant frequency components show up on hyperbolic curves in the number of revolutions - harmonic order plane. During the gearbox run-up test, two pairs of gears were under load. The gear train transmitting power from the input shaft to its main shaft of the 40/39-gear ratio and another one transmitting power from the main shaft to the axle of the 18/69-gear ratio. The basic toothmeshing component of the gear train 40/39 is the most significant in level. Fig. 6. Spectral map of vibrations during a gearbox run-up test Synchronised averaging The well-known method how to reduce estimation errors of random signal spectra is averaging of instantaneous spectra; therefore, averaging takes place in the frequency domain. As opposed to frequency spectrum averaging, the time domain averaging process deals with time records. The main effect of this technique is ability to enhance spectrum components that

5 are harmonics of the desired (synchronous) frequency and reject unwanted spectrum components. Therefore, the technique requires an accurate knowledge of the repetition frequency of the desired signal. Time domain averaging is obviously used to analyse signals from rotating machines (engines, gearboxes, etc.). This analysis is focused on levels of the harmonic components of the rotational frequency. A mathematical model of the time domain averaging of an input time signal, x(t), is a moving average N 1 1 yt ( ) = xt ( it), N i= 0 where N is a number of averaged records and T is a time interval of a revolution. A frequency response of the moving average filtration is as follow ( Nf f ) G( f ) = 1 sin π 0, N sin( π f f0) where f0 = 1 T. The plot of the magnitude of the frequency response against the dimensionless frequency f f 0 for N = 20 is shown in Fig. 7. This figure illustrates that the time domain averaging corresponds to comb filtering with centre frequencies coinciding with the integer multiples of the rotational frequency, f 0. If the time interval, T, of a revolution is equal to the time interval of the FFT-record, than these centre frequencies coincides with the FFT lines. It should be noted that the number of side lobes to the main lobes in the comb filter frequency response increases with N, as does the sharpness of the main lobes. G ( f ) 1 0,8 0,6 0,4 0, ,5 1 1,5 2 2,5 3 f / f o Fig. 7. Frequency response of the averaging in the time domain (N = 20) The synchronised averaging is obviously carried out at a steady-state rotation that obviously varies in some amount. The mentioned run-up test lasted nearly 40 seconds while the total number of input shaft revolutions was equal to Because increasing of the rotational frequency by 1 RPM per a revolution which is comparable with the steady-state rotation, the synchronised averaging of the acceleration signal in the course of 180 input shaft revolutions has been performed. The result is seen in Fig. 8. The time record in this figure corresponds to a revolution of the input shaft. The synchronised averaging filters out all the spectrum components with frequency which is a non-integer multiple of a synchronous frequency. Therefore, the averaged time record is only a response of gears under load on the input shaft. As there is only the 40-tooth gear under load on the input shaft, the response assesses the uniformity of the tooth mesh. The theory of analytical signals can be employed to amplitude and phase demodulation of averaged time records [1,3]. After removing both the amplitude and phase modulation signals, the average toothmesh response is obtained. This response gives a measure of dynamic forces acting between the teeth of the mating gear pair during each mesh cycle. Therefore, it is a useful tool for verifying the gear contact ratio. An example, illustrating 3 out of 40 identical averaged toothmeshes corresponding to 3 pitch rotations, is shown in Fig. 9.

6 Fig. 8. Synchronously averaged acceleration signal recorded during a run-up test Fig. 9. Average toothmesh signal (3 out of 40 teeth) Conclusion The order analysis and the signal enhancement are a powerful tool not only for diagnostics of gear mesh dynamics but for all the mechanical systems running in cyclic fashion. The averaged time records give a more clear indication of the operating condition of their rotating parts than the frequency spectrum. References [1] Tůma, R. - Kuběna, R. - Nykl, V.: Assesment of gear quality considering the time domain analysis of noise and vibration signals. In: Proceedings of the 1994 Internationale Gearing Conference, Newcastle, UK, 1994, pp ISBN , pp [2] Gade, S., Herlufsen, H., Konstantin-Hansen, H., Wismer, N.J Order Tracking Analysis. Brüel & Kjaer Technical Review, No. 1, [3] Tůma, J.: Zpracování signálů získaných z mechanických systémů. Sdělovací technika, Praha 1997, s, 174. ISBN Author: Doc. Ing. Jiří Tůma, CSc. (Dr.-Ing, Associate Professor) Technical University of Ostrava, Faculty of Mechanical Engineering, Department of Control Systems & Instrumentation, tř. 17. listopadu, Ostrava Poruba, Czech Republic Tel (0) , Fax-Nr (0) , jiri.tuma@vsb.cz