Resonance Frequencies for 8202A10
6.3 Results
6.3.2 Experimental Case Study
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
−0.5
−0.4
−0.3
−0.2
−0.1 0 0.1 0.2 0.3 0.4
Δ Var[g2i] relative error [%]
Approximated analytical
Fig. 6.7 Variance values forg2i, comparison of analytical expressions and numerical simulations varying jitter entity. Relative error in percentage points between simulation (reference) and analytical solutions
Table 6.2 Parameters used for simulating the distribution of the absolute value
Parameter Value
Number of iterations 100,000
N 200
T 1
f* 1
1e-4*2000
6.3.1.2 Approximation of the Absolute Value Distribution
Although extensive simulations were performed to validate the model, for the sake of brevity here only one comparison is reported. The used parameters are reported in Table6.2. From the developed model, it is known that the distribution of the squared absolute value will be affected by the number of revolutions averaged. The more revolutions will be averaged, the less will be the variance. Seemingly, it is expected to have higher attenuation and higher variance when the jitter is of a higher entity. For the analyzed case, the number of averagesN was set to a typical value used in helicopter vibration diagnostic systems. The value of the jitter parameter is approximatively one that could have for the first harmonic of the gearmesh frequency (i.e. one prominent harmonic in geared transmission typical signature, computed as shaft rotational speed times number of teeth of a connected gear) when sampling withfKPD10 kHz. Figure6.8shows the predicted probability density function usingdD20 terms in the expansion, compared to the simulated one for the squared absolute value. Seemingly, Fig.6.9shows a comparison between the estimated distribution of the variable Rin and the simulated one for the same set of parameters. Fair agreement between prediction and simulations is observed. It can be concluded that the analytical approximations offer a viable way of predicting the attenuation and phase errors due to the jittering error in pulse arrival times according to the proposed model. Note that since the jitter effects are proportional to frequency, an attenuation of around 10% is reached for the 2 kHz harmonic only by considering the uncertainty due to the sampling of the keyphasor signal. Moreover, the values present themselves a certain variability. The approximated analytical and simulated expected value and variance for the considered simulation are given in Table6.3.
100 90 80 70 60 50 40 30 20 10
00.88 0.89 0.9 0.91
|XSA(f*)|2
0.92 0.93 0.94 0.95
Simulation Laguerre expansion approximation
Probability density function value
Fig. 6.8 Approximated p.d.f. of the squared absolute value computed through Laguerre expansion (continuous line), and the simulated distribution (histogram)
25
20
15
10
5
0
–0.08 –0.06 –0.04 –0.02 0 R(f*)
0.02 0.04 0.06 0.08 Simulation
Normal approximation
Probability density function value
Fig. 6.9 Approximated p.d.f. of the random variableRcomputed through Normal approximation (continuous line), and the simulated distribution (histogram)
to assessing the capabilities of state-of-the-art algorithms in detecting various kinds of faults. Vibration data acquired with systems compliant with those that can be installed on board of a helicopter were used. Vibration signals were recorded from accelerometers located at seven specific locations on the main gearbox case. Moreover, a tachometric signal was recorder as a reference for performing synchronous averaging, digitalized with the same sampling rate of the vibration signals. Based on the kinematic relationships of the gearbox, the reference can be used for performing SA signal extraction for each shaft via interpolation. The considered test consisted of disassembling a ball bearing component and damaging three of the ball
elements by scratching the surface. The characteristic defect frequencies of the considered ball bearing assembly are given in Table6.4, referring to the nominal shaft speed of 100 Hz. The defect frequenciesfo, fi, fb, fcare respectively the outer race defect frequency, the inner race defect frequency, the ball spin defect frequency and the cage defect frequency. Those defect frequencies are computed from the simplified kinematic relations of the bearing and correspond to the expected frequencies of impulses emitted from a damaged bearing when periodically striking a localized defect. The periodic impacts excite the high-frequency resonances of the component and can therefore be better detected when demodulating the signal around resonances. Further details can be found in the literature [4]. A comprehensive discussion about the signature of localized defects in bearings is given in [16]. In the context of this work, we focus on the diagnosis using the popular squared envelope spectrum5(SES) of the signal. The SES of a discrete signalxcan be computed as [8]:
SESxDˇˇˇDFT n
xfŒn 2oˇˇˇ2 (6.50)
Where DFT denotes discrete Fourier transform andxfis the filtered analytic signal through some filter f aimed to enhance some properties ofxf. A common filtering operation is, e.g., that of band-pass filtering the signal in a narrow region in which the impulsive content of the signal due to the damage is revealed. The reason for the successful performance of SES is thoroughly discussed in [4]. In fact, it is shown in [4] that the SES of the signal coincides with the integrated cyclic spectral density (CSD) when the random part of the signal is dominating. Moreover, conditions for the equivalence of the SES with the integrated cyclic coherence (ICC) are derived in [10]. The power of such a cyclostationary framework is in defining strong theoretical basis for the analysis of the CS2 part of a signal, including the derivation of proper statistical tests. In general, in order for the SES to perform optimally according to [10], a whitening of the signal should be performed before the analytic transformation. When the periodic components to be removed are known, such a whitening can be efficiently performed using the SA procedure. In this section, it is first shown that performing SA removal for signal pre-whitening effectively improves the diagnostic capability in the studied case. Once the importance of SA is proved on the experimental case, data from the same campaign is used in order to discuss the effect of jittering in SA extraction according to the results presented in Sect.6.3.1.
Figure6.10shows the squared envelope spectrum of a faulty acquisition from the closest sensor to the faulty bearing.
No band-pass filtering is performed and no whitening as well. Each vibration record in this analysis is 2 s long and the vibration signal is acquired with 93,750 Hz sampling frequency. The signal is analyzed in the angular domain synchronously with rotations of the shaft connected to the bearings of interest. Frequencies are expressed in orders of the fundamental shaft frequency (100 Hz in the considered case) and amplitudes are normalized to maximum value. It is readily clear that the harmonic content of the signal due to multiple CS components is completely hiding the fault signature. The dominant periodic components appear at integer multiples of the reference shaft. Hence, a first strategy for enhancing our envelope can be to separate the periodic mean with the period of the shaft from the rest of the signal. Using the SA procedure and 100 averages, we compute the periodic mean and remove it from the signal. Then, we show the squared envelope spectrum after this gross pre-whitening step in Fig.6.11. As it can be seen, most of the important periodic components that were masking the fault signature are removed, and the harmonics of the ball pass frequency (red squares) modulated from the cage frequency (blue squares) are now readily visible. Here the whitening was successful thanks to a precise extraction and removal of the SA part from the signal. Clearly, this would not be the case otherwise. It has been shown how extracting the periodic mean from a signal can improve the diagnosis using the tool of SA. The discussion of other techniques is out of the scope of the paper, and the case was presented as a support to the importance of extracting correctly the SA from a signal, even when CS2-based diagnosis is concerned. According to the model proposed for the jittering error, the random error introduced on the unitary
5Note that the SES quantity is a squared-amplitude spectrum (signal-units2) and not a power spectrum.
Table 6.3 Simulated and approximate analytic expected value and variance for the random variablesjXSA(f)j2andR
Value Simulation Analytic E[jXSAj2] 0.9159 0.9158 Var[jXSAj2] 2.21e-05 2.19e-05
E[R] 13.5e-05 0
Var[R] 33.6e-05 33.6e-05
Table 6.4 Characteristic frequencies of the considered bearing
fo[Hz] fi[Hz] fb[Hz] fc[Hz]
508.9 691.2 221.8 42.4
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.