NOISE AND VIBRATION DIAGNOSTICS IN ROTATING MACHINERY

Transcription

1 NOISE AND VIBRATION DIAGNOSTICS IN ROTATING MACHINERY Jiří TŮMA Faculty of Mechanical Engineering, VŠB Technical University of Ostrava, 17. listopadu, Ostrava-Poruba, CZECH REPUBLIC ABSTRACT The main topic of the paper deals with diagnostics of rotating and reciprocating machines based on RPM, noise and vibration measurements. Key words are a standard order tracking method, synchronised averaging and the Vold-Kalman order filtering. Examples of noise and vibration measurements are taken from the diagnostics of the truck, car and machining centre transmission units, namely the vibration and noise excited by teeth of gears in mesh. 1. INTRODUCTION The paper is focused on methods employed in noise and vibration diagnostics of rotating and reciprocating machines for condition monitoring in operation or quality control in production. The main sources of vibration in the low frequency range are unbalances of rotational machine parts, misalignments and bent shafts while in the audible frequency range, noise and vibration of geared axis systems can originated from parametric self-excitation due to the variation of tooth-contact stiffness in mesh cycle transmission error due to the inaccuracy of gears in mesh non-uniform load and rotational speed. Another source of noise and vibration are bearings. Typical faults in rolling bearing are cracks or pitting on the inner or outer race or on the rolling element itself. Vibration of large rotor systems fitted by journal bearings is the special field of diagnostics. Vibration of shafts is excited by fluid-induced instabilities in a fluid-film bearing known as a whip and whirl. The basic tool for signal processing in diagnostics is the Fourier transform (RT) of signals. As the input time record to be transformed in the frequency domain can be generally consisting of the complex values, the diagnostic signal can be one or two-dimensional. The twodimensional signal corresponds to two orthogonal co-ordinates of the shaft position during rotation in the bearing and the shaft trajectory forms an orbit. The resulting frequency 1

2 spectrum is called a Full spectrum. The full spectrum has the same relationship to a standard spectrum (based on the conjugate symmetric vector evaluated by use of the FT from a real input signal) in the frequency domain as the orbit and timebase waveform in the time domain. The base frequency of all these exciting forces is related to the machine rotation frequency. An extensive vibration is excited when the base frequency or its harmonics meet the structural resonance frequency of machines. The machine is tested during steady-state rotation or run up / coast down. It should be mentioned that any driven unit does not rotate at a purely constant speed but its speed slowly varies around an average value. Spectrum components of the diagnostic signal result from simultaneously amplitude and phase modulation of so called carrying harmonic components that correspond to the excitation at a purely steady-state rotation. The amplitude modulation of harmonic signals arises from the non-uniform periodic load while a phase modulation is there due to the non-uniform rotational speed. Rotation speed variations at the fixed signal sampling frequency cause the smearing of the dominating components in the frequency spectra. An 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 dimensionless revolutions rather than seconds and the corresponding FFT spectra are measured in dimensionless orders rather than frequency. This technique 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 the reciprocal value of the revolution number per a record corresponding to input data for the Fast Fourier Transform (FFT). Clear information about the origin of the extensive vibration cannot be given by a single frequency spectrum but by a multispectrum recorded during variation of the machine RPM. The signal/noise ratio is improved by averaging in both the frequency and time domains. Signal resampling is employed in order to eliminate a phase modulation effect at a tacho pulse frequency. A lot of practical mechanical systems contain multiple shafts that may run coherently through fixed transmissions, or partially related through belt slippage and control loops, or independently, as for instance a cooling fan in an engine compartment. For coherently running shafts it is possible to use the above mentioned order analysis technique. On the other hand, non-coherently running systems with multiple orders decoupling close and crossing order can be extracted by the Vold-Kalman order tracking. The standard methods based on FT enables only speed limited order tracking while the Vold-Kalman order tracking filtering is without slew rate limitation. 2. TACHO, RPM AND TORSIONAL VIBRATION MEASUREMENTS Rotational speed is measured in terms of the number of revolutions per minute (RPM) while the torsional vibration is measured in terms of the angle, angular velocity or angular acceleration. The uniform rotational speed of the constant RPM corresponds to the shaft angle as a linear function of time. The shaft angle as a non-linear time function is associated with the torsional vibration. The rotational speed is obtained as the first derivative of the angle. The torsional vibration can be measured by using shaft encoders giving usually a train of pulses, rather then a sinusoid. The simplest method for evaluation of the instantaneous rotational speed is the reciprocal value of the time between two consecutive pulses. The time interval is determined by interpolation some 5 times more accurately than indicated by the 2

3 actual sampling interval. The accuracy is satisfying for the RPM measurement based on only one pulse per shaft rotation. In the case of the large number of pulses per revolution, another method based on the phase demodulation is employed. The pulse signal consists of several harmonics of the basic pulse frequency. Each of the harmonic components is the carrying component that can be modulated by varying rotational speed. The phase modulation signal can be derived from the phase of the analytical signal that is evaluated using the Hilbert Transform technique. The real sampled signal is extended by an imaginary part to the complex signal called an analytical signal. The imaginary part of the mentioned analytical signal is the Hilbert Transform of the real signal. As the angle of the complex values ranges from π to +π, the true angle of the analytical signal as the time function with jumps at π or +π must be obtained by unwrapping based on the fact that the absolute value of the difference between two consecutive angles is less than π/2. The phase modulation signal is a non-linear term of the phase as the time function. The first derivative of the linear term corresponds to the steady-state rotational speed. The number of pulses per revolution limits the multiple (called order) of the rotational speed in Hz that can be evaluated in the frequency spectrum. The sampled instantaneous angular speeds are measured at the sampling frequency evaluated in orders that are equal to the number of pulses per revolution. Taking into account the Shannon-Nyquist theorem, the frequency span is limited by half the sampling frequency in orders. The reciprocal of the order is the length of the period in the number of samples. If the RPM is measured by a tacho signal where a pulse corresponds to a shaft revolution the frequency spectrum is limited by.5 order. To take an example of the rotational speed variation [6], the RPM time history of a digitally controlled electric motor is shown in figure 1. The instantaneous RPM was changing three levels in a non-regular way, but not randomly or chaotic. The motor propels a mechanical gearbox in a lathe. While running at idle an extensive non-usual noise was emitted. RPM RPM Revolutions RPM Orders Figure 1. Rotational speed variation and order spectrum The vibration signal was recorded to examine a relationship between the RPMvariations and noise emitted by the gearbox,. A strip-plot of acceleration signal divided into 22 revolutions of the gearbox primary shaft is shown in figure 2. Bursts on this strip-plot are distributed quite randomly along the revolution axis. 3

4 Figure 2. Acceleration signal versus number of the primary-shaft revolutions 3. ORDER ANALYSIS OF GEARBOX NOISE AND VIBRATION As the most of the dynamic forces, exciting vibration and noise are related to the machine rotation frequency, rotation speed variations at the fixed sampling frequency cause the modulation effect. An advantage of the order spectra is that a harmonic order component remains in the same analysis line that is independent on the rotational speed of the machine. The order spectra are a tool to distinguish harmonics of fundamental frequency components from structural frequency components, using a run-up or coast-down of the machine. Car or track transmission units are examples of axis systems that shafts are running in a coherent way. The test of the gearbox quality is based on the noise and vibration measurements. The pass-by noise test involves driving a vehicle through a test site at full acceleration in accordance with ISO R362. A test stand is employed to simulate the above mentioned gearbox operating condition. The gearbox under test is placed in a semi-anechoic test room. The primary shaft speed is slowly increased from a minimal to maximal RPM, while the gearbox is under load. Other operating conditions corresponding to slow vehicle acceleration or deceleration can be simulated as well. In this case the engine driving torque is decreased and the engine noise does not mask the gearbox noise. The gearbox noise and vibration spectrum consists of sinusoidal components. A component corresponding to the rotational speed in Hz multiplied by the number of teeth is referred to as the base toothmeshing frequency. All the spectrum components are usually broken into a combination of the following effects: a) low harmonics of the shaft speed due to unbalance, misalignments, or bent shaft, b) harmonics of the base toothmeshing frequency and their sidebands due to modulation, c) ghost components due to errors in the teeth of the index wheel of the gearcutting machine, d) components resulting from faults in rolling-element bearings. To demonstrate the order analysis, a noise signal corresponding to the gear trains under load of a car gearbox (see figure 3) is analysed. Contour and waterfall plots of the noise signal order spectra (Hanning window and acoustic A-weighting) during the run-up test are shown in figure 4 and 5, respectively. The noise spectra were captured by Pulse, the Brűel & Kjæer signal analyser, at the step of 5 RPM while the primary shaft rotational speed was increasing from 1 RPM to 3 RPM. 4

5 The noise signal contains, among other components, the significant 29 th harmonic of the primary shaft rotational speed and its harmonics. These harmonics belong to the 29-tooth gear response. The constant frequency components appear on the hyperbolic curves in the rotation speed - harmonic order plane. The waterfall and contour plots illustrate various harmonics falling along lines parallel with the third axis representing the RPM, which can thus be separated from the constant frequency components due to excessive amplification by a structural resonance. The components of the order spectra in the multibuffers can be viewed individually as a function of revolution speed, which is called a slice. The slices, corresponding to the harmonics of the toothmeshing frequency of all the gear trains, are stored into a database Figure 3. Gears under load Figure 4. Contour plot of the noise autospectra (3th gear) Figure 5. Waterfall noise autospectra (3th gear) The overall noise level assesses all the gearbox parts, namely gears and bearings. The noise level in terms of the power unit excited individually by any gear train can be evaluated as the sum of the spectrum component amplitudes (in the same unit) corresponding to the harmonics of its toothmeshing frequency. The vehicle pass by noise test is based on the measurement of the maximum overall noise RMS level. Accordingly, the maximum overall noise level of the gearbox and the maximum noise RMS level of each of the gear trains can be chosen as a gear quality criterion. The plot consisting of the overall noise RMS level (Total) and the slices in db versus the primaryshaft rotational speed are shown on the diagram in figure 6. The slices correspond to the harmonics of the 29-tooth gear toothmeshing frequency. The curve designated as 29/38 gear train is the sum of the power contributions of the five harmonic components resulting from noise excited only by the train of the 29- and 38- gears [5]. The train of the 17- and 7-tooth gears is analysed by use of the diagram in figure 7. For the low rotational speed of the 17-tooth gear on the gearbox secondary shaft, the 17/7-tooth gear train is less noisy at least by 5 db than the train of the 29- and 38- gears. For purpose of the quality control, the maximal values of the noise levels of the individual harmonics are under control. 5

6 db RPM 3 Total 29/38 gear train 1st harm onic 2nd harmonic 3rd harm onic 4th harm onic Figure 6. Total level and slices of noise autospectra (3th gear) related to the train of the 29- and 38- gears 9 85 db RPM 3 Total 17/7 gear train 1st harm onic 2nd harmonic 3rd harm onic 4th harm onic Figure 7. Total level and slices of noise autospectra (3th gear) related to the train of the 17- and 7- gears 4. SYNCHRONISED AVERAGING The well-known method to reduce estimation errors of random signal spectra is averaging of instantaneous spectra in the frequency domain [1]. As opposed to frequency spectrum averaging, the time domain averaging process deals with time records. The main effect of this technique is saving spectrum components that are harmonics of the desired (synchronous) frequency and rejecting unwanted spectrum components. Time domain averaging is obviously used to analyse signals from coherently rotating axis systems. 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) and a related frequency response are as follow [4] ( π Nf f ) ( π f f ) N 1 1 yt ( ) = xt ( it), N G( f ) = 1 sin, i= N sin where N is a number of averaged records and T is a time interval of a revolution and f = 1 T. The plot of the magnitude of the frequency response against the dimensionless frequency f f for N = 2 is shown in figure 8. 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. If the time interval, T, of a revolution is equal to the time interval of the FFT-record, then these centre frequencies coincide with the frequency of the FFT lines. It must 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. 6

7 G( f ) 1,2,8 1,6,4,2,5 1 1,5 2 2,5 3 f / f o Figure 8. Frequency response of the averaging in the time domain (N = 2) The synchronised averaging is obviously carried out at a steady-state rotation that obviously varies in some amount. An example illustrating synchronised averaging of the acceleration signal is shown in figure 9. The averaged time record is a response of the 29-tooth helical gear on the gearbox primary shaft rotating at 218 RPM. The averaged time record in this figure corresponds to a revolution of the primary shaft. The synchronised averaging filters out all the spectrum components with a 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 primary shaft. As there is only the 29-tooth gear under load on the primary shaft, the response assesses the toothmesh uniformity ,,2,4,6,8 Revolution 1, Orders Figure 9. Synchronously averaged acceleration signal and corresponding frequency spectrum The theory of analytical signals can be employed to amplitude and phase demodulation of averaged time records [2]. After removing both the amplitude and phase modulation signals from enhanced time signal, the average tooth mesh response is obtained. This response gives a measure of dynamic forces acting between the teeth of the mating gears during each mesh cycle. Therefore, it is a useful tool for verifying the gear contact ratio [3]. The amplitude modulation signal referred as an envelope gives information about uniformity of the driving or loading torque. An example, illustrating the average tooth mesh and envelope of both the 29- and 38-tooth meshing gears, is shown in figure 1. Both of the average tooth mesh responses are almost identical (differences may be explained by the different number of samples) while the envelopes are evidently differing from each other. The phase modulation signals giving information about uniformity of the rotational speed of both of the mentioned meshing gears are shown in figure 11. 7

8 29-tooth gear 38-tooth gear Average tooth mesh ,25,5,75 1 Tooth Pitch Rotation,25,5,75 1 Tooth Pitch Rotation Envelope ,,2,4,6,8 1, Revolution,,2,4,6,8 1, Revolution Figure 1. Average tooth mesh and envelope signals as a response of the 29- and 38- gears 29-tooth gear 38-tooth gear deg deg,5 -,5 -,5,,2,4,6,8 1, Revolution,5,,2,4,6,8 1, Revolution Figure 11. Phase modulation signals as a response of the 29- and 38- gears As the frequency of the modulation signal carryingcomponent is equal to the toothmeshing frequency, the variation of the transmission error cannot be extracted. 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. 5. VOLD-KALMAN ORDER TRACKING FILTERING Employing of the Vold-Kalman tracking filtering is demonstrated on analysis of the acceleration signal that is measured on the table of the horizontal machining centre [7]. Vibration is excited by the natural unbalance of the spindle. The RPM run-up ranges to 1 8

9 RPM and it takes 18s. Results are shown in figure 12, 13 and 14. The analytical instrumentation used for the Vold-Kalman tracking filtering was the Brűel & Kjæer origin and comprised the PULSE signal analyser. Figure 12. Vibration time signal of the run-up Figure 13. An short time Fourier transform of the acceleration signal Figure 14. Time slice at 18.68s to detect harmonics of the rotational speed,42th order 2,45th order,781th order 3rd order 1st order RPM profile of run-up Figure 14. Overlapped waveforms (table acceleration signal in X-direction) of the.42 th,.781 th, 1 st, 2.45 th and 3 rd order of the rotational speed of the spindle extracted using a two-pole Vold-Kalman filter with 1% bandwidth 9

10 An short time Fourier transform of the acceleration signal in figure 13 shows that dominating orders of the spindle rotational speed are equal to the.42 th,.781 th, 1 st, 2.45 th and 3 rd order of the rotational speed of the spindle. The unbalance of a driving motor excites the.781 th order. The unbalance or misalignment of the spindle excites the 1 st and 3 rd orders. Faults in rolling-bearings probably excite the.42 th and 2.45 th orders. 6. CONCLUSION The paper gives an overview of analytical tools for the diagnostics of rotating and reciprocating machines that are well suited to the frequency of the exciting force. This frequency is equal to the integer multiple of the base rotational frequency of the machine. The people that are not skilled in the frequency analysis appreciate the averaged time records giving very clear indication of the operating condition of the rotating machinery. 7. REFERENCES [1] Tůma, J. Analysis of Gearbox Vibration in Time Domain. In Proceedings of the Euronoise 92, Imperial College London. 1 st ed. London (UK) : Institute of Acoustics, Vol 14 Part 4, Book 2, pp. [2] Tůma, J. Analysis of Periodic and Quasi-Periodic Signals in Time Domain. In Proceedings of the Noise 93, St. Petersburg (Russia). 1 st. ed. Auburn (USA) : Auburn University, Volume 6, pp [3] Tůma, R. & Kuběna, R. & Nykl, V. Assessment of gear quality considering the time domain analysis of noise and vibration signals. In Proceedings of the 1994 International Gearing Conference. 1 st ed. Newcastle (UK) : Technical University, pp [4] Tůma, J. Theory and experience in synchronised averaging in the time domain. In Proceedings of the 43. Internationales Wissenschaftliches Kolloquium. Technische Universitat Ilmenau Thuringen, st ed. Ilmenau : Technical University of Ilmenau, vol. 1, pp [5] Tůma, J. & Růna, B. Order analysis and averaging in the time domain of gearbox noise and vibration signals. In Proceedings of the 6 th International Congress on Sound and Vibration, Lyngby, Lyngby (Denmark) : Technical University of Denmark, vol. 1, pp [6] Tůma, J. & Kočí, P. Rotational Vibrations of Digitally Controlled Units running ad Idle. In Red. B. Katalinic. Proceedings 1 th International DAAAM Symposium. 1 st ed. Vienna (Austria) : DAAAM, pp [7] Gade, S. & Herlufsen, H. & Vold, H. Characteristics of the Vold-Kalman Order Tracking Filter. Brüel & Kjær Technical Review, No This research has been conducted at the Department of Control Systems and Instrumentation as a part of the research project No. 11/98/683 Gearing Design Centre and Laboratory for Transmission Noise Reduction Research and has been supported by the Czech Grant Agency. 1