Resonance Frequencies for 8202A10
6.4 Concluding Remarks
0 2 4 6 8 10 12 14 16 18 20 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Frequency [Orders]
Normalized Amplitude
SES BSF harmonics FTF harmonics Shaft harmonics
Fig. 6.10 Squared envelope spectrum in order domain, raw signal, faulty bearing.Blue triangles: harmonics of the cage frequency (FTF);red stars: harmonics of the ball spin frequency (BSF);green crosses: harmonics of the input drive shaft
Table 6.5 Comparison of jittering effects as predicted and as obtained using experimental data (ratio between SA harmonic amplitude computed using high-precision tacho signal and SA harmonic amplitude computed using lower precision tacho signal)
Harmonic Predicted ratio Measured ratio
GMF1 0.993 0.990
GMF2 0.971 0.967
frequency synchronous component due to the jitter would be such thatD1/fsD1/93750s. Computing the attenuation for the modulus of the first gearmesh frequency related to the pinion connected to the reference shaft, which isf D1900 Hz, we obtain in average (based on Eq.6.35) square root of the first cumulant) a value of about 0.9991/1. According to the analytical prediction, therefore, the SA should be able of recovering almost correctly such harmonic. Since no reference is available, it is complicated to verify directly the model. However, a tachometer signal sampled withfs2 D 23,438 Hz is available, hence we can validate our predictions based on the ratio between the amplitude of the harmonics of the SA signal obtained using the higher-precision tacho and the amplitude of the harmonics of the SA signal obtained using the lower-precision tacho. Still, available data for comparison is limited. However, it provides a first validation of the observations proposed in the paper. Table6.5reports the predicted and experimental attenuation ratios (amplitude of harmonic computed using low precision tacho divided by amplitude of harmonic computed using the higher precision tacho) obtained from the analysis of eight vibration records in healthy gearbox conditions and stable operating conditions. Attenuation ratios are computed for the first two gearmesh harmonics, since they have high signal to noise ratio. Although the values match quite well showing a remarkable ability of the model in explaining the difference in amplitude of the SA harmonics, further validation is required before confirming the capability of the model of well-predicting the jitter effects on experimental data. In particular, the design of a specific experiment would solve the issue. It can be of some interest to put an effort in actually characterizing the distribution of jittering errors and eventually validating the analysis proposed in this paper.
0 2 4 6 8 10 12 14 16 18 20 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Frequency [Orders]
Normalized Amplitude
SES
BSF harmonics FTF harmonics Shaft harmonics
Fig. 6.11 Squared envelope spectrum in order domain, input drive shaft harmonics removed through SA.Blue triangles: harmonics of the cage frequency (FTF);red stars: harmonics of the ball spin frequency (BSF);green crosses: harmonics of the input drive shaft
value and the phase of the synchronous components in the signal, considering the dependency on the number of averages and on the entity of the jitter error. The analysis considered uniform jitter variables, though the model lends itself to the introduction of any other random model for the jitter times. An analytical approximation was derived for the attenuation and the phase shift. Numerical simulations showed adequate performance of the analytical approximation model in predicting the distortion of the generic synchronous harmonic, validating the study. The proposed expressions provide a mean for quantifying the severity of random jitter effects and, therefore, help in defining design criteria for vibration diagnostics systems. The results were derived considering a single sinusoid. However, the model is general and allow to consider any harmonic. The leakage interaction between different synchronous components can be considered by solving the equations forf¤f*, though for multiple harmonics the mixing might be complex. An important remark is that the given general model for the synchronous averaging gives ground for extending the analysis to different cases of interest, e.g. it can be adapted for studying the uncertainty in tacholess synchronous averaging procedures. As for the analytic approximations, it is recognized that a better estimation of the variance of the functiong1iis required for high error values, hence adding additional terms to the proposed second-order expansion is recommended. Finally, an experimental case study was presented to show the relevance of jitter effects in a practical diagnosis case where it is necessary to remove parasitic harmonics from the signal to detect the signature of a faulty gearbox bearing. In the considered case, relatively high precision in the tacho pulse times was available, hence the attenuation factor for most of the important harmonics (specifically, the first and second harmonics of the gearmesh frequency) was not relevant. By making use of two different reference signals, it was shown a good agreement between predicted and experimentally observed attenuation of the synchronous components due to jittering. In the considered
case, the jittering effect was fairly neglectable in terms of affecting the ability of separating the deterministic part and the random part of the signal. Hence, the deterioration of the SA did not affect negatively the diagnostic capability. The modelled (and successively observed) attenuation effect due to the jitter might be important for less precise tacho signals than the one available for this experimental analysis. On one side, the attenuation of some frequency components might distort the residual signal in a region of interest; on the other, the variance in the estimated magnitudes of harmonics of interest may lead to some scattering in the results when analyzing the CS1 part of the signal. It is therefore recommended to keep into account the importance of acquiring a precise synchronization signal when designing a monitoring system.
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