Gear Transmission Error Measurements based on the Phase Demodulation

Gear Transmission Error Measurements based on the Phase Demodulation JIRI TUMA Abstract. The paper deals with a simple gear set transmission error (TE) measurements at gearbox operational conditions that means under load and during rotation. The analysis method is focused on the processing of pulse signals generated by encoders attached to both the gears in mesh. The analysis technique benefits from demodulation of a phase-modulated signal. The theory is illustrated by experiments with a car gearbox and measurement errors are discussed. Keywords: Gears, gearboxes, transmission error, encoders, phase demodulation, synchronised averaging, average tooth mesh, measurement accuracy, angular vibration. 1. Introduction Noise and vibration problems in gearing are mainly concerned with the smoothness of the drive. The parameter that is employed to measure smoothness is Transmission Error (T. E.). This parameter can be expressed as a linear displacement at a base circle radius defined by the difference of the output gear s position from where it would be if the gear teeth were perfect and infinitely stiff. Many references have attested to the fact that a major goal in reducing gear noise is to reduce the transmission error of a gear set. The basic equation for T.E. of a simple gear set is given as TE n = 2 r n (1) 1 2 ( m) Θ Θ 1 2 where n 1, n2 - teeth numbers of pinion and wheel respectively, Θ 1, Θ 2 are angles of rotation of the mentioned gears and r 2 is a wheel radius. T.E. results not only from manufacturing inaccuracies such as profile errors, tooth pitch errors and run-out, but from a bad design. The pure tooth involute deflects under load due to the finite mesh stiffness caused by tooth deflection. A gearcase and shaft system deflects due to load as well. To complicate meters, a gear axis system is a very complicate dynamic system. Department of Control Systems and Instrumentation, VŠB - Technical University of Ostrava, 17.listopadu 15, Ostrava-Poruba, 78 33, Czech Republic, tel.: 596993482 jiri.tuma@vsb.cz

While running under load one of very important parameters, tooth contact stiffness, is varying what excites the parametric vibration and consequently noise. There are many possible approaches to measuring T.E., but, as Derek Smith points out [1], in practice, measurements based on the use of encoders dominates. A possible alternative is the use of tangentially mounted accelerometers to measure the torsional accelerations of the two gears, which causes the problem of signal transfer from the rotational parts and its double integration. As it will shown later, the common encoders generating a string of pulses in hundreds per rotation are satisfying the sever requirements for measurement accuracy for the mentioned quantity in microns. The National Engineering Laboratory at East Kilbride developed a first workable system for laboratory use while Dr. R. G. Munro introduced the redesigned system for industrial use in the 196s. The sketch of the gear set consisting of the 21- and 44-tooth gears under test and attached incremental rotary encoders, designated by E1 and E2 is shown in figure 1. Both the encoders are of Heidenhain origin, the ERN 46-5 type. A perfectly uniform rotation of gear produces an encoder signal having in its frequency spectrum a single component at the frequency that is a multiple of the gear rotational frequency. As both the encoders generate 5 impulses per encoder rotation, the frequency of the single components in orders (a multiple of the encoder rotational frequency) is equal to the same number as the number of the impulses. A gearbox is a machine running in cyclic fashion, which is the reason to prefer frequency spectra of signals in term of dimensionless orders rather than frequency in Hz. 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 record corresponding to input data for the Fast Fourier Transform (FFT). As the measurement is focused on one complete gear revolution, the spectrum resolution is one order. Pulse signals from encoders are recorded by PULSE, the n 1 =21T Brüel & Kjær signal analyser. To simplify the phase pinion demodulation an Order Analyser instrument was employed Θ 1 which resulted in time history records corresponding to one E 1 complete gear revolution. A method of synchronised averaging in the time domain was employed for reducing random noise in Θ 2 wheel the measured data. As it is known, the order analysis is based on E 2 data resampling in such a way, that sampling frequency follows the mean frequency of shaft rotational speed during one complete shaft revolution. The mean rotational frequency is n 2 =44T evaluated by means a train of pulses generated once per a shaft revolution. The time interval between two consecutive pulses is determined by interpolation some 5 times more accurate than indicated by the actual sampling interval. Therefore the pulses Fig. 1. Measurement arrangement distribution inside this time record gives information about the instantaneous rotation angle of each of the gears under test. As a consequence of Shannon s sampling theorem a few pulses must be recorded during each mesh cycle. It means, that the number of pulses produced per encoder revolution must be a multiple of the tooth number. If five harmonics of toothmeshing frequency are required then the number of pulses per gear revolution must be at least ten times

higher than the number of teeth. The encoder generating 5 pulses per revolution seems to be an optimum. The length of resampled time record equals to 248 samples per gear revolution. The sample number is a power of two, which is required by FFT and in corresponding order spectrum, ranging to 8 orders, there is a space for ± 3 sideband components around the carrying component of 5 orders in a frequency spectrum. The frequency range of the mentioned Order Analyser in the described conditions limits the gear rotational speed to the value of 19 RPM. Gear loading has not any influence on the discussed sampling problem. As the pulse signal is not a sinusoid, its frequency spectrum consists of several harmonics of the basic pulse frequency. Variation of angular velocity during rotation causes the phase modulation of the carrying signal that can be any of the harmonic components. The phase modulation gives rise to sidebands around the carrying frequency in the frequency spectrum of the modulated harmonic signal. The phase modulation signal can be derived from the phase of the analytical signal that is evaluated using the Hilbert Transform technique, which is the topic of the following paper section. 2. Phase demodulation using the Hilbert transform An analytic time signal is a useful tool for demodulation. The analytic signal, z () t, is generally combined from real and imaginary parts z () t x() t + j y( t) = z( t) ( jϕ( t) ) = exp, (2) where y () t - the Hilbert transform of x ( t). The relationship between the FFT of the y ( t) and x( t) i ( N i) X i of the length, N, is given by Y = jsign 2. (3) As the angle of the complex values ranges from π to +π () t = arctan( y( t) x( t) ) ϕ, (4) the true angle of the analytical signal as the time function with jumps at π or +π must be obtained by unwrapping that is based on the fact that the absolute value of the difference between two consecutive angles is less than π. The principle of phase unwrapping for a harmonic signal modulated by another harmonic signal is shown in figure 2. The relationship between the phase of the analytical ϕ t, is as follows signal and the phase modulation signal, ( ) M () t = ω t + ϕ ( t) ϕ, (5) M where ω - an angular frequency of the carrier component. The phase modulation signal is a fluctuation of the phase angle around the linear term, ω t.

2π +π π 4 2 Unit -2-4,1,2,3,4,5,6,7,8,9 1 Discontinuities removing (2f f samp ϕ π) ϕ < π ϕ + 2π ϕ, ϕ > +π ϕ 2π ϕ 7 6 rad 5 4 3 2 1,2,4,6,8 1,15,1,5 -,5 -,1 -,15 rad,2,4,6,8 1 Fig. 2. Principle of phase unwrapping 3. Encoder accuracy To evaluate errors in pulses distribution against the angle of rotation, both the encoders were mounted on a shaft what ensured the same rotational speed of them. As the running was not perfectly uniform from the point of the measurement method sensitivity, both the pulse signals were under influence of phase modulation. Using the analytical method described above, the difference between modulation signals gives the error in pulse distribution. RMS of Error in radians + 168 RPM x 635 RPM o 139 RPM Circle part E 2 E 1 1/order order Fig. 3. Encoder accuracy

The frequency spectrum of the resulting error is shown in figure 3. The frequency axis is in orders. The quantity order determines a part of a circle related to the error level in the spectrum. The error level at the distance corresponding to the tooth pitch of the adjacent teeth determines the final accuracy of the T.E. measurement. As it is evident the magnitude of an error at 21 and 44 order is less than 1-5 radians, i.e. approximately 2 angular seconds. 4. T.E. measurements A perfectly uniform rotation of gear produces an encoder signal having in its frequency spectrum a single component at the frequency that is a multiple of the gear rotational frequency. As both the encoders generate 5 impulses per a shaft rotation, the frequency of the single components in order is equal to the same number as it is the number of the impulses, i. e. 5 orders. The gear speed variation as an effect of loaded teeth deflection results in the phase modulation of the impulse signal base frequency. The phase-modulated signal contains sideband components around the carrying component. The distance of the dominating sideband components from the carrying components equals to the integer multiple of the tooth number as it is shown in figures 4 and 5. The frequency axis of both the frequency spectra is in order; it means the multiples of the gear rotational frequency. The frequency of the carrying component is equal to 5 orders while the sideband component associated with the corresponding gear is at the distance of ± 21k or ± 44k (where k = 1, 2,... ) order units from the mentioned carrying component frequency. Take notice of the fact that the dominating components in both the sidebands exceed the background noise level at least 1 times or even more. Both the spectra were evaluated from time signals that are a result of synchronised averaging of 1 revolutions of gears under test. Phase demodulation, which is based V 1 1,1,1 1 1 V,1,1 1 1 Enhanced Spectrum - 21-tooth gear 395 416 437 458 479 5 521 542 563 584 65 Orders Fig. 4. Frequency spectrum of phase modulated signal generated by the E1 encoder 1 1 Enhanced Spectrum - 44-tooth gear 28 324 368 412 456 5 544 588 632 676 72 Orders on the theory of the Hilbert transform and phase unwrapping, enable the evaluation of the angular vibration of both the gears in Fig. 5. Frequency spectrum of phase modulated signal generated by the E2 encoder

mesh individually. The unwrapped phase of the frequency modulated signal that is produced by the encoder E1 is shown in figure 6. The diagram forms almost a straight-line function. After subtracting a linear term from unwrapped phase, the dependence of the phase variation on time is obtained as it is shown in figure 7. Take note of the diagram scale. The relationship between the unwrapped phase and phase modulation signal results from the formulae (5). Unwrap Angular Vibration in 36 27 18 9,,3,5,8 1, Time t /T Angular Vibration in,3,25,2,15,1,5 -,5 -,1 -,15 -,2,,2,4,6,8 1, Fig. 6. Unwrapped phase of E1 encoder signal versus time Fig. 7. Phase modulation signal versus time As it is shown in figure 8 the frequency spectrum of phase modulation signal contains a family of harmonics components to the basic 21-order component excited by toothmeshing of the 21-tooth gear. The low frequency components in the frequency spectrum result from nonuniform driving torque and overlap the toothmeshing response. The phase modulation signal in rees during the pinion revolution is shown in figure 9. The enhanced signal contains five harmonics of the toothmeshing frequency, each of them with 3 pairs of sidebands that cause the amplitude modulation of angle variation. When all these sidebands are removed a purely periodic signal is obtained. The filtration in the frequency domain can be considered as an averaging of the second stage. Therefore, one of these periods corresponding to the gear tooth pitch rotation can be taken as a representative to characterize angular vibration in average. The result of mentioned averaging is called the average toothsmesh. The term averaged toothmesh was introduced to associate vibration and noise measurement with a gear design [2]. 1,1,1 1 1 Spectrum of Ang. Vibration 21 42 63 84 15 Orders Fig. 8. Frequency spectrum of phase modulation signal

Toothmeshing frequency harmonics with 3 sideband components Enhanced Time Signal,2,1 -,1 -,2,,2,4,6,8 1, One out of 21 periods Enhanced Time Signal Average without sidebands Av erage Toothmesh,2,2,1,1 de g -,1 -,1 -,2 -,2,5 1,,2,4,6,8 1, Tooth Pitch Rotation Gear revolution Fig. 9. The second stage of angular vibration averaging for the 21-tooth gear The same average toothmesh in angular variation can be evaluated for the 44-tooth gear as it is shown in figure 1. Angle variation can be easily transformed into the arc length variation. The difference between both the angle variations gives the transmission error. The only problem consists in the true phase delay between these periodic signals because the signals from the encoders are recorded separately. Solving of this problem is based on the similarity of responses both the gears to dynamic forces acting between mating teeth, for instance in acceleration some point on the gearcase. Both the encoder pulse signals are sampled together with the acceleration signal. Two-stage averaging of the twice-measured acceleration signal gives average toothmesh responses that are delayed. The lag for the maximum correlation gives the relative delay.,2,1 -,1 Enhanced Time Signal -,2,,2,4,6,8 1, Average Toothmesh,1,5 de g -,5 -,1,5 1,5 Tooth Pitch Rotation Gear revolution Fig. 1 The second stage of angular vibration averaging for the 44-tooth gear

T.E. is given as the difference between the angular vibration signals in the arc length produced by the mating gears. The result is shown in Figure 11. All the experimental data was taken from a car gearbox. The results correspond to the rotational speed of 5 RPM at the input shaft and almost full load. The measurement method was tested at the maximum rotational speed of 125 RPM. µm 5 4 3 2 1-1 - - 2 - - 3 - - 4 - - 5,2,4,6,8 1 Tooth pitch rotation 21 21 -- tooth tooth gear gear 44-tooth 44-tooth gear gear T.E. T.E. Fig. 11. Transmission error against rotation angle in range of the tooth pitch Conclusion The paper is focused on the problem of the simple gear set transmission error measurement (T.E.). Variation of T.E. is the cause of angular vibration of both the mating gears and consequently gearcase vibration and noise. This paper deals with only one measurement method that is based on the use of encoders generating a string of 5 pulses per gear revolution and reviews the phase demodulation for evaluating of the gear angular vibration. The theory is illustrated by experimental data. Acknowledgements This research has been conducted at the Technical University of Ostrava as a part of the research project No. CEZ 3212 and has been supported by the Czech Ministry of Education. The author benefits from the research work done for the SKODA Auto Company. References [1] Derek Smith J. Gear Noise and Vibration, 1 st ed. New York Basel : Marcel Dekker Inc., 1999. ISBN: -8247-65- [2] Tůma, Jiří. Analysis of Periodic and Quasi-Periodic Signals in Time Domain, In: Proceedings of the Noise 93, St. Petersburg (Russia). Auburn (USA) : Auburn University, 1993, Volume 6, pp. 245-25.