5.5 Rolling Bearing Diagnostics
Much of the background on this topic has been given in Section 2.2.3. The most important thing is that the spectrum of the raw signal often contains very little diagnostic information about bearing faults, and it has been established for many years that the benchmark method for bearing diagnostics is envelope analysis, in which the signal is bandpass filtered at high frequency bands and the fault pulse is amplified by structural resonance. It is then amplitude modulated to form an envelope signal, the spectrum of which contains the required diagnostic information, including the repetition rate (spherical pass frequency or spherical spin frequency) and modulation by the appropriate frequency as it passes through the load region (or moves relative to the measurement point).
However, envelope analysis techniques were designed more than 30 years ago, for example [36], and use simulation techniques with inherent limitations. Significant improvements can be achieved by utilizing digital processing techniques rather than blindly following analog methods in digital form.
Amplitude modulation using the Hilbert transform technique brings a number of benefits, as described in Section 3.3. This process is illustrated in Figure 5.37 (extracted from [37]). This is similar to the amplitude modulation process shown in Figure 3.30(d) in Chapter 3.
A direct benefit is the efficient extraction of the portion of the spectrum to be amplitude modulated by an ideal filter, so it can be decoupled from adjacent components that may be more powerful, such as meshing frequencies. This is not always possible in analog filters, where real-time digital filters are subject to the same limitations related to filter characteristics.
Figure 5.37 represents the envelope as the modulus of the analytic signal obtained by the inverse transformation. In fact, it is the square of the envelope signal that is best analyzed in [38] rather than the envelope itself. The reason for this is briefly explained in Figure 5.38, which compares the spectrum of the rectifier and the square sine wave. It should be noted that mathematically, the envelope of the signal is the square root of the square envelope, and similarly, the rectified signal is the square root of the square signal. The square root operation introduces an external fraction that is not present in the original square signal, resulting in a masking of the required information. As can be seen in Figure 5.38, the rectifier signals have sharp inflection points that need to be extended to infinity harmonics to replicate them. Since the entire operation is carried out digitally, it is not possible to remove these high harmonics by means of a low-pass filter (e.g. analog rectifiers), which can be aliased into the measurement range, resulting in masking. Note that since squaring doubles the frequency content of the signal, the sampling frequency should be doubled before digitizing it squared or rectified, although, as will be seen in Figure 5.37, this is equivalent to zero filling when processing the parsed signal.
Finally, the benefits of using unilateral spectrum are illustrated in Figure 5.39 (extracted from [38]). If the analytic signal (from the unilateral spectrum) is named fa(t), its square envelope is formed by multiplying it by its complex conjugation, and the spectrum of the square envelope will be a convolution of the respective spectrum.
therefore
When this convolution operation is performed, as shown in Figure 5.39(a), the result gives only the difference frequency, such as the sideband interval, which contains the required modulation information. However, for the equivalent real signal f(t), the spectrum of its squared value is only the convolution of F(f)(f(t)) and itself. This is illustrated in Figure 5.39(b) and it is seen that it gives the same difference frequency component, but mixed with sum frequency (the difference between positive and negative frequencies), which does not contain any diagnostic information and only obscures the true results.
As shown in Figure 5.39(c), this interference can be avoided as long as the frequency is shifted to introduce zero padding around the zero frequency and zero padding around the Nyquist frequency. This effectively means that for the same demodulation band, the sampling frequency must be doubled, and therefore the transform size must be doubled for the same problem.
When [38] was written, the primary means of separating gear and bearing signals was the use of SANC (Section 3.6.5), which required a real-value signal, but a frequency-domain-based DRS approach (Section 3.6.6) avoids this complexity by utilizing unilateral spectrum.
[38] It is shown that even if the power of the masked noise (random or discrete frequency) reaches three times the power of the bearing signal in the demodulation band, it is still advantageous to analyze the square envelope. With SK, it is often possible to find spectral bands with higher signal-to-noise ratios (SNRs) for bearing signals.
In the early stages of envelope analysis, there has been a lot of discussion about how to choose the most suitable frequency band for demodulation, which many claim is difficult, and some suggest using a percussion test to find the resonance of the housing. By using SK and kurtogram to find the most pulsed band (after removing discrete frequency masking), this problem is now largely solved. As shown in Figure 5.6 of this chapter, this essentially provides the same information about the difference in the decibel spectrum before and after the bearing failure occurs. If the spectral variation is caused by a bearing failure, it is clear that the optimal SNR of the bearing signal relative to the background noise corresponds to the largest decibel difference, independent of the spectral level. However, this requires a reference signal from a bearing in good working order, which is not required by the SK method.
5.5.1 Signal model of bearing failure
The best way to analyze a faulty bearing signal depends on the type of fault present. The main difference is a small, localized failure in the initial period, resulting in a sharp shock when the rolling element comes into contact with the fault, and an extended flaky failure, especially if the latter becomes smooth.
5.5.1.1 Local Failures
For local faults, one problem involves how to properly model the random interval of shocks. Perhaps the first to model a bearing fault signal as periodic non-stationary was [39], but the results were not very convincing, probably because the main resonant frequency of the fault excitation may be outside the measurement range, up to about 6 kHz. For example, as shown in Figure 5.6, local faults on similarly sized bearings only manifest themselves at frequencies above 8 kHz. In [40], good results were achieved by modeling the vibration signal of a local bearing failure as periodic nonstationary. However, it was later found that the way in which the random variation of the pulse interval was modeled in Model 1 was incorrect, and a more correct model was proposed in [41] (Model 2). As shown in Figure 5.40, the variation in Model 1 is modeled as a random "jitter" around a known average period, whereas in the correct model, it is actually the interval itself that is a random variable.
In particular, this has implications for predicting the uncertainty of the location of future pulses. For model 1, this is constant and determined by jitter, whereas in the real world, the change is due to slippage, which the system has no memory of, so the uncertainty increases over the time of prediction (model 2). As pointed out in [41] and more mathematically based in [42], this means that the signal of a local failure in a bearing is not really periodic non-stationary, but better known as "pseudo-periodic nonstationary". Figure 5.41 (from [41]) shows the practical consequences for a signal with a small amount of random variation. From the point of view of the interpreted spectrum, especially on low-order harmonics, there is often little practical difference in treating pseudo-periodic nonstationary signals as periodic nonstationary signals.
5.5.1.2 Extension Failures
In the case of extended wear, where an impact typically occurs when each rolling element exits the wear zone, signal envelope analysis can often reveal and diagnose the fault and its type. However, there is a tendency for the wear area to be worn out, in which case the impact may be much smaller than in the early stages. There have been cases where the extended wear zone no longer produces a noticeable shock, but if the bearing supports a mechanical element such as a gear, the fault can still be detected and diagnosed, as the fault usually modulates an otherwise regular TM signal. Figure 5.42 shows a typical modulation signal due to bearing inner ring extension wear. Since the rolling elements are in different positions on rough wear surfaces during the rotation of each inner ring, the signal contains both first-order (local mean) and second-order (amplitude-modulated noise) periodic nonstationary components. In Figure 3.56 of Chapter 3, the spectral correlation of such mixtures with specific characteristics in the periodic direction, but discrete and continuous characteristics in the normal frequency direction is shown. This is because the periodic signal has a periodic autocorrelation function (in the time lag or τ direction), so the Fourier transform in this direction also produces discrete components. As shown in Figure 3.56, if the modulation of the gear meshing signal is caused by a gear failure, it will be periodic and will only produce discrete components in both directions (similar to the "bed-of-nails" effect). However, if the periodic components are removed by DRS or other methods in Section 3.6, only the second-order periodic non-stationary components are left, in which case they can only come from extended bearing failures.
Note that the continuous lines of spectral correlation in Figure 3.56 are located at the lower harmonics of the shaft velocity (), but in principle also at the harmonics of the BPFI and the sidebands spaced around them by the shaft velocity. For inner ring faults, the shaft velocity may be the best option for extracting this information, but for unmodulated outer ring faults, components may be found in the spectral correlation at the harmonics of the BPFO. It is important to note that if the shaft velocity is the modulation frequency, then the signal is indeed second-order periodic nonstationary (since the cycle frequency is fully determined), whereas if the modulation frequency is BPFO or BPFI, the signal will be pseudo-periodic nonstationary.
Figure 5.43 illustrates the spectral correlation between the cycle frequency equal to the shaft velocity for two inner ring failure cases in the same type of bearing. For localized faults, the difference is manifested in the high frequencies, more than 1000 shaft orders, while for extended faults, the differences are concentrated in the low frequencies, up to 15 times the gear engagement frequency. In the former case, the fault is easily detected by envelope analysis, but in the latter case it is less apparent. Figure 5.44 shows a practical example of a helicopter transmission input pinion bearing, where the wear of the extended inner ring was not detected until very late stages. With no on-board vibration monitoring, the metal particles were trapped in the dam and failed to reach the metal chip detector. When these measurements are taken on a transmission test bench, the wear zone has been flattened and is not visible by the envelope analysis at the BPFI, only at the harmonics of the shaft velocity, so that if this analysis is not carried out (removing the discrete frequency components), it may be misinterpreted as a gear failure.
5.5.2 Semi-automatic bearing diagnostics
In [43], a method for successfully diagnosing bearing faults is proposed for a wide range of situations, from high-speed gas turbine engine bearings to radar tower main bearings with a 12-second rotation cycle. This can be described as a semi-automated approach, as only a few parameters need to be adjusted for each case, including the size and speed of the bearing. As shown in Figure 5.45, it combines multiple techniques described in this chapter and chapter 3. This method was further developed in Sawalhi's doctoral dissertation [44].
It is usually best to start with order tracking (Section 3.6), as it is not always possible to separate discrete frequencies and random components unless order tracking is performed. A case is described in [37] where it is not possible to separate the gear and bearing signals using DRS prior to order tracking. No speedometer or shaft encoder signal was available, but it was found that instantaneous velocity information could be extracted by phase modulation of several gear meshing frequencies that were synchronized with the shaft speed. The best mapping of axis angles to time is obtained by averaging small estimates with similar appearances. In this case, the random velocity change is only 0.5% (1203-1209 rpm).
For separating discrete frequencies and random components (e.g., gear and bearing signals), it is generally preferable to choose DRS (Section 3.6.6) because it has the least problems in selecting parameters. The magnitude of the transform N should span 10-20 cycles of the minimum frequency to be removed (e.g. minimum shaft speed) and the delay should be at least three times the relevant length of the bearing signal. Assuming a slip of 1%, this would be equivalent to about 300 cycles of the center frequency of the demodulation band. Determining the latter may require an iteration, as it is best to decide after the SK procedure.
MED is only used for high-speed bearings, where the impulse response of the bandpass filter resonance is comparable to the interval length of the bearing fault pulse (BPFI is usually the highest fault frequency and therefore the shortest interval). This may be best decided by trial and error, based on whether MED improves SK or not.
A fast kurtosis program should be used to select the best demodulation band. Note that the kurtosis is sensitive to large random pulses that may be present in some signal implementations. If the final envelope spectrum does not show a periodic component, even if the SK is high, it should be checked to see if such random pulses from external sources dominate in certain bands.
In the final envelope analysis, it should be recognized that modulation effects are important for diagnosis. In general, inner ring failures are down-regulated at shaft speed, while rolling element faults are down-regulated at cage speed. For unidirectional loads, outer ring faults are not modulated, but modulation may occur at shaft speed due to significant unbalance or error forces, while modulation at cage speed may be due to variation between rolling elements. Note that in planetary bearings, the inner ring is fixed relative to the load, so the inner ring fault is often not modulated, while the signal for the outer ring fault is modulated by its frequency through the load zone. Since planetary gears resemble rolling elements in bearings, the modulation frequency can be calculated using an equation similar to BSF (Equation (2.15)).
The general procedure is illustrated in [43] through its application in three very different case histories, so a brief summary of these results is provided here.
5.5.2.1 Case history 1 – Helicopter gearbox
A test of a helicopter gearbox was conducted at the Australian Defence Science and Technology Organisation (DSTO), which operated to failure under heavy load conditions. These signals were blindly analyzed and did not indicate the type of failure. Table 5.1 lists the frequencies corresponding to planetary bearings (which actually fail), although all other potential bearing frequencies must be calculated.
Even if the growth of wear particles at the end of the test indicated a bearing failure, a preliminary analysis of the signal showed no signs of failure in the time domain signal or in the frequency spectrum. The spectrum is dominated by the gear component (harmonics of the main gear frequency and its side lobes) over the entire frequency range (up to 20 kHz). The original signal has a kurtosis of -0.6, similar to noise.
Applying the procedure in Figure 5.45, the discrete frequency components are removed using the linear prediction method in this example. Figure 5.46 compares the residuals of the linear prediction process with the original signal, and it can be seen that the residuals are slightly more modulated (kurtosis increases to 2.2), but the three "bursts" are related to the passage of the planet (with a period of 58.1 ms).
Next, a small wavelet kurtosis plot (Section 5.3) of the residual signal (Figure 5.46(b)) was generated using a series of filter banks (3, 6, 12, 24 filters per octave), and the results are shown in Figure 5.47. The maximum kurtosis obtained using 12 filters per octave (center frequency of 18,800 Hz and bandwidth of 1,175 Hz) is 12.
Finally, Figure 5.48 shows the squared envelope spectra over two frequency ranges for the last measurement. Figure 5.48(a) shows a strong harmonic pattern spaced by the cage velocity of the planetary bearing. This basically indicates that there is a variation in the rotation for each cage. This may be a cage failure, but it is also usually an indicator of variation between rolling elements. Figure 5.48(b) shows a strong component corresponding to BPFI in a slightly higher frequency range. Since it is a planetary bearing, no inner ring failure is expected to cause modulation, and no modulation sidelobes can be found in the envelope spectrum. When disassembling the gearbox, serious lamellae were found on the inner ring of one planetary bearing, with slight lamellae on the three rollers, which explained the modulation on the cage speed. The final damage is shown in Figure 5.49.
The analysis parameter trends for this case are discussed in Chapter 6 Forecasts.
5.5.2.2 Case history 2 – High-speed bearings
Measurements were carried out on a bearing test bench from FAG Bearing, Germany, which tested the bearings for failure. At several moments in its service life, the bearings are disassembled and inspected. The bearings being tested are used for high-speed applications and are therefore tested at 12,000 rpm, which is the typical rotational speed for gas turbine bearings. The accelerometer used to capture the data is mounted via magnets, which is reasonably doubted since spectrum examination shows that the installed resonant frequency is about 12 kHz.
This is the case depicted in Figure 5.4, where the MED technique significantly improves the impulse characteristics of the signal due to the overlap between the individual impulse responses. The small wavelet kurtosis plot of the signal before and after the MED technique (Figure 5.50) shows that the kurtosis increases from 2.5 to 20, making the pulse characteristics obvious.
Figure 5.51 shows the envelope spectrum in the later stages of fault development, and the demodulation bandwidth is shown in Figure 5.50(b). This is a typical inner ring fault envelope spectrum with a range of harmonics at BPFI (1398 Hz) and low harmonics and side lobes at bearing speed (196 Hz).
The trends in the analytical parameters of this case study are also discussed in the projections in Chapter 6.
5.5.2.3 Case history 3 – Radar tower bearings
Measurement data was received before and after the replacement of the main bearing of the radar tower. The radar drive system consists of a motor, a gearbox, and a spur pin/ring gear combination. The motor rotates at 1800 rpm (30 Hz) and is connected to a three-stage reduction gearbox. The rotation period of the final tower is 12 seconds (0.082 Hz). The ring teeth used to drive the tower are an integral part of the bearing, so they are replaced at the same time, but the drive pinion remains the same. The reason for the replacement was increased noise (possibly due in part to gear wear), but the current analysis clearly shows evidence of bearing failure in the old bearing. Compared to the first two cases, the difference in speed is huge, but even then, the same analysis procedure can be intervened with very few operators. Due to the slower speed, the fault shock is well separated, so there is no need to use MED filtering.
The spectrum before and after the replacement did not show any indication of bearing failure, but the original time signal did show a failure in the form of a series of pulses protruding from the background signal. Applying DRS shows that the latter is dominated by deterministic components, mainly from gears (which dominate the spectrum). The local shocks visible from the time signal of the old bearing (before and after the application of DRS) are absent in the signal of the new bearing. Removing the gear signal increased the kurtosis from 8.4 to 64.9. Even without the use of SK, the envelope analysis of the signal in Figure 5.52(c) shows the bearing failure frequency (4.79 Hz) with modulated sidelobes with a rotational speed of 0.082 Hz. Interestingly, it is not possible to determine whether the fault is in the inner or outer ring, partly because this is a thrust bearing, so BPFI and BPFO are the same (in Equations (2.12) and (2.13) in Chapter 2, φ = 90◦).
In addition, there is a reason why faults in both fixed and moving tracks are modulated in shaft speed, the former because the load is a bit eccentric and therefore rotates around the bearing, and the latter because the length of the path to the sensor is constantly changing.
Figure 5.53 shows the results of applying the optimal bandpass filter using a small wavelet kurtosis plot and extracting the signal from the bearing fault. It was found to be a filter with a bandwidth of 517 Hz and a center frequency of 2755 Hz. The kurtosis has increased to a staggering value of 541. This clearly shows that kurtosis cannot be used directly as an indicator of fault severity; For example, if the machine were to run at twice the speed, the kurtosis might be halved. Therefore, the ratio of the fault repetition frequency to the typical resonant frequency of the excitation should be considered when evaluating the kurtosis corresponding to the machine failure. The damping ratio of the shock response also affects the kurtosis (hence the benefits of MED). This is also the case mentioned in Section 5.4.5, namely a gear failure in the low-speed section of a wind turbine gearbox [35], where a relatively high SK value is found.
The harmonic marker in Figure 5.53 is set at the spherpass frequency of 4.79 Hz, but you will see that there is also a multiples component of half the frequency. The reason for the appearance of the half-frequency is the peculiar structure of these bearings, in which the track is V-shaped, with 118 rollers (of the same length and diameter) mounted alternately, in the direction of ±45◦, so that only every other roller is in contact with the side of the V-shape. As a result, a fault on one side will produce a pulse at half the frequency. As mentioned above, there are sidebands of rotational velocity (0.082 Hz) around harmonics of half-spherical pass frequency, and low harmonics of 0.082 Hz.
Obviously, the semi-automated procedure used to extract the bearing signal in Figure 5.53(a) will detect the bearing fault at a very early stage before it becomes apparent above the background gear signal, as is the case in Figure 5.52(a).