5.4 Gear Diagnostics
The vibrations generated by friction of gears are described in detail in Chapter 2, including vibrations generated under normal conditions and vibrations caused by various faults. It is important to point out that since the same tooth profile meshes in the same way every time, these vibrations tend to be deterministic. Randomness may intervene due to speed fluctuations, but they can be compensated for using the sequential tracking procedure in Chapter 3, Section 3.6.1, and the Time Synchronous Average (TSA) should always be used before the synchronous average. Note that this actually changes the "time" axis to the angle of rotation, but as long as the speed of the machine is nominally constant, the book will use the term "TSA". Randomness can also occur due to random loads, such as rock crushers or wind turbines, and then the situation becomes more complicated. When the load changes over time, but relatively slowly, it is usually possible to record the load between certain specified limits (best determined by trial and error) and then compare only the same load conditions. Data from several different load ranges can be used as a reference. The following discussion mainly covers situations where slight speed changes can be compensated for and the range of load changes is limited, but the cases regarding speed and load changes are briefly dealt with in Section 5.4.5.
5.4.1 TSA-based technology
Gear diagnostics have been studied over the years, and back in the 1970s, Stewart [14] came up with some powerful diagnostic tools that became the benchmark for gear diagnostics, especially for helicopter gearboxes, which represented a particularly complex situation. These tools are based on the analysis of the TSA and its spectrum performed on each gear, from which a number of "advantage parameters" are extracted. Probably the most powerful of these is FM4, defined as the kurtosis of the "residual signal" obtained after the removal of the normal gear meshing pattern, which helps to reveal local changes in the signal that the normal meshing pattern tends to mask these changes. Initially, the residual signal is obtained by subtracting the known gear meshing harmonics from the spectrum and returning to the time domain. It was later realized that this often leaves considerable modulation effects at the first and second harmonics of rotational speed, which are not related to local faults, so it may also be necessary to remove one or both pairs of sidelobes around each tooth meshing harmonic to obtain a residual signal.
Recently, Wang and Wong [15] of the Australian Defence Science and Technology Organisation (DSTO) suggested using a more flexible approach to linear prediction to remove the normal gear meshing pattern, as described in Chapter 3, Section 3.6. Figure 5.13 from [15] shows the improvement using linear prediction compared to traditional methods (note that the second-order sidelobes may not be removed due to the interference having two cycles in the recording).
Peter McFadden, a former DSTO member and later a faculty member at the University of Oxford, made several improvements to the TSA-based approach (as well as improvements to sequence tracking and TSA operations themselves, see Chapter 3, Sections 3.6.1 and 3.6.2). One of the important developments is a method for obtaining TSA for planetary and sun gears in planetary (planetary) gearboxes. In the case of sensors mounted on housings (usually mounted on or close to ring gears), the signal is greatly influenced by a specific planet near the measurement point. McFadden proposed to take advantage of this by windowing by corresponding to the short signal portion of the sensor that happens to have just one tooth passing through [16]. This is illustrated in Figure 5.14. The next time the same planet passes by the sensor, another tooth on the planetary gear (and the sun gear) will mesh, but depending on the kinematics, it is known exactly which tooth it is, so the window signal can be assigned to each tooth and a different "barrel" for each gear. After enough passes, not only will each tooth on each planetary gear and sun gear be encountered, but they will be encountered multiple times, so the average of the window portion of each tooth will eventually give the (evenly weighted) TSA signal for each tooth, and then by connecting them, the TSA for the entire gear can be obtained. This applies to each planetary gear as well as to the sun gear (whose signal meshing with all planetary gears can contribute to the TSA of each tooth). In the original paper [16], McFadden used a rectangular window that exactly corresponded to one tooth meshing period, but in later papers (e.g., [17]) he recommended the use of a Hanning window (called a tuki window in some papers) with a total length of twice the tooth meshing period. The difference is barely noticeable in the TSA waveform, but the spectrum of the latter has improved significantly. It should be noted that the tooth meshing frequency (and therefore the tooth engagement cycle) is the same for each pair of meshes in a planetary gearbox. For a fixed ring gear (most commonly), it is equal to the planetary bracket speed multiplied by the number of ring gear or ring teeth. Thus, it is a harmonic of the velocity of the planetary bracket but not the harmonic of the velocity of the sun gear.
Another important contribution is the demodulation of TSA signals using the Hilbert transform technique [18] (harmonics of gear meshing frequencies as carriers) to visualize local changes that are not visible to the naked eye (especially phase modulation that is not sensitive to the eyes). Figure 5.15 (from [18]) shows a typical example in which the TSA signal of a cracked gear is demodulated. In fact, this gear failed within 103 service hours after the signal was recorded, and the failure was even more pronounced before the final failure, using signal analysis techniques that had not yet been developed at the time. In this case, the second harmonic of the tooth meshing frequency is demodulated because it is much stronger than the first harmonic from the point of view of acceleration and therefore has a better signal-to-noise ratio.
It is important to note that this technique is limited by the bandwidth of the modulated signal, since the maximum bandwidth that can be demodulated corresponds to the plus or minus half of the tooth meshing frequency, regardless of which harmonic is demodulated. This is illustrated in Section 5.4.2 below, where not only the acceleration signal but also the Transmission Error (TE) signal is demodulated in order to observe the interaction of amplitude and phase modulation in both cases. In the case of a fault excitation of very high-frequency resonances, the conditions for tooth meshing signal demodulation may not be satisfied. An example of this from a wind turbine gearbox is provided in Section 5.4.5. The example also shows that synchronous averaging cannot be used even in all cases, depending on the frequency range in which the fault manifests, as discussed in Section 5.4.5.
5.4.2 Transmission error as a diagnostic tool
Gear diagnostics are affected in part by the modification of the signal from the source to the measurement point, so it may be valuable to take measurements directly at the gear meshing point. If the encoder on the shaft can be precisely mounted, especially at the free (non-driven) end, it is possible to measure the transmission error (TE) even in the most accessible places in any case. If there is no drive torque on the shaft connecting the gear and the encoder, it will accurately follow the torsional vibration of the gear up to a very high frequency.
As described in Section 2.2.2, the transmission error represents the difference between the angular motion of the driven gear and the angular motion it would have if the transmission was fully conjugated, i.e., the output speed is constant for a constant input velocity. Although it is usually measured in terms of the angular deviation of one gear or the other, it only measures the linear deviation of the tangent of the base circle of the gears corresponding to the two gears, so it must be scaled by the gear proportions to achieve a common basis for both gears. Therefore, it is possible to measure the torsional vibration of each gear, scale one of them in proportion to the gears so that it is equivalent to the other in terms of linear motion, and then subtract it from the other. This is usually interpreted as output minus input, but can also be expressed in terms of the angular motion of either gear or the linear motion on the base circle. Scaling is simply the scaling of the amplitude, because the timeline is not related to the gear being considered.
Figure 5.16 shows the process of generating a TE signal from the torsional vibration of two axes in a gearbox with a simulated root crack from one gear. The ratio is 1:1 (32 teeth per gear), so the torsional vibration signal does not have to be scaled before subtracting. Torsional vibration is measured by phase demodulation of the encoder signal on each axis, using the procedure shown in Figure 3.30(b). Note that the phase of each gear exhibits some random velocity variation (the local slope of the phase curve), but TE as a difference is very regular and periodic with rotational speed. Also note the scale difference of the TE compared to the original torsional vibration signal. The speed is 3 Hz, so there are three rotations in the 1-second record.
Note further that the gear engagement signal is very clear (32 cycles per revolution) and the local deviation is noticeable (around 0.11 seconds). In this case, the local fault is directly apparent in the TE signal (which is not the case at all speeds and loads), but is significantly enhanced by three stages of signal processing, as shown in Figure 5.17.
This method of measuring gear TE was first published in [19] and was published by P.J. Part of Sweeney's doctoral dissertation work. The phase demodulation of the encoder signal (and the frequency analysis of its results) is done using the scaling processor method described in Section 3.3.2.2, using Bru ̈el & Kjær Analyser Type 2035. In Figure 5.18, the results are compared to the "pulse timing" method developed by Sweeney, which uses a 100 MHz clock to time the time interval between encoder pulses. This is based on a very simple principle that the time interval between encoder pulses gives a measure of the constant increment of the phase angle (and thus can be used for angle resampling as an alternative to the sequential tracking method described in Section 3.6.1). The reciprocal of the time interval can be calibrated based on instantaneous angular velocity and frequency modulation, and has the advantage of a fixed number of samples per revolution, so sequential tracking is not required. This method of measuring torsional vibration is also valuable for reciprocating machines and is therefore mentioned again in Section 5.6.2.2. At Sweeney's doctoral dissertation work, it was difficult to use such a high-frequency clock, and phase demodulation was more practical, but now that commercial data acquisition systems have 80 MHz clocks, pulse timing techniques have become more feasible.
It is worth noting that the results obtained by both methods are almost identical. The accuracy of the measurement is impressive, as it is primarily based on the accuracy of the sampling and timing clocks involved, independent of the accuracy of the analog instrument (previously used for gear TE measurements). The encoder used here, the Heidenhain model ROD260, has a high degree and recorded accuracy with an error of about 3-5 arc seconds. However, judging by the results, even if these small errors are concentrated in the low harmonics of the rotation (which may be dominated by the tiny eccentricity of the inscribed disc), they are even lower at the higher harmonics corresponding to the gear meshing frequency. In this particular case, TE's measured values at the low harmonic and gear meshing frequencies are about 50 arc seconds, and each arc second corresponds to about 0.025 microns of linear motion on the base circle.
Shaft encoders with recording errors are expensive, so Du and Randall [20] investigated ways to measure and correct encoder errors, especially for cheaper encoders. It is explained in [20] that by taking two measurements and swapping two encoders between them, a composite measure of the encoder error can be obtained by taking the differences, and the mean of the two results gives an estimate of the best true result. This simple procedure works with gears with a 1:1 ratio, so that the results of both measurements are approximately the same and the error is given directly.
Figure 5.19 shows typical results for a 1:1 ratio spur and helical gear pair (32 and 29 teeth, respectively) on the same test setup. Make sure that the alignment of the initial positions of both encoders is the same. Two cycles are shown. Note that the scaling of the TE is 10 times wrong.
It can be seen that although the TE at the tooth meshing frequency of the helical gear is much smaller than that of the spur gear (although the low-frequency component due to pitch error, etc. is basically the same), the encoder error is almost the same in both measurements, which helps to verify the method. This method is further validated by the results shown in Figure 5.20, which shows the measurement of a spur gear pair (32:49) with a non-1:1 ratio. In this case, the two encoders rotate at different speeds in two measurements, and the difference in the two measurements contains two versions of the combined encoder error, one version for each gear's rotation cycle. However, it can be extracted by synchronous averaging relative to the period of one gear, as shown in Figure 5.20. Note that the base cycle of TE now corresponds to 32 rotations of 49 gears (only two are shown). The encoder error extracted by synchronous averaging with respect to the period of the 32 gears is similar to that shown in Figure 5.19, albeit subjected to the smoothing effect of the average. Its spectral display error is concentrated in the low harmonics of rotational speed and is not much larger than that of more expensive encoders.
The error measured with a lower-cost encoder (Heidenhain ROD426) was small enough that Du did not find it necessary to compensate for it in the rendering in his dissertation work, some of which was reported in [21].
It is interesting to compare the results of the simultaneous measurement of TE demodulation with the results of the acceleration signal. Figure 5.21 shows the TE results for the same gear as Figure 5.16. It can be seen that the amplitude after demodulation clearly shows the fault, while the phase does not.
Figure 5.22 shows the equivalent results for the demodulation of the acceleration signal (but includes the amplitude of the demodulation of TE for comparison). Obviously, for acceleration, the phase-modulated signal contains more information about the crack. It can be speculated that this can be explained in part by the fact that the amplitude of the TE gives the change in torque directly, which may explain the change in frequency and phase modulation, but it should be remembered that the acceleration response will depend on the transfer function between the force at the meshing point and the acceleration at the measurement point.
Figure 5.23 shows a simple example where the pure amplitude modulation at the source can be converted into a mixture of amplitude and phase modulation by a transfer function. This is based on the fact that, as shown in Figure 3.29, the main difference between amplitude and phase/frequency modulation is the phase relationship of the side lobes on either side of the carrier. The phase change in the transfer function (not shown in Figure 5.23) will further increase this change between amplitude and phase modulation.
Therefore, one possible advantage of using TE as a diagnostic parameter is that it will allow more inference of the difference between amplitude and phase modulation.
Before leaving this topic, it will be demonstrated how to demodulate the spectrum around the gear meshing harmonics of the data shown in Figures 5.21 and 5.22.
Figure 5.24 shows the spectrum of the TE and acceleration signals. For the TE signal in Figure 5.24(a), the sidelobes have some degree of overlap outside the 0.5-1.5x gear meshing (TM) frequency range, which will result in slight distortion. For the acceleration signal in Figure 5.24(b), the same applies to the sidelobes (in the range between 1.5-2.5 times TM) that mesh the second harmonic around the gear, in which case demodulation is performed. Since the acceleration signal has been downsampled to the same sampling frequency as the TE signal, the TM frequency corresponds to line 526 in both cases.
5.4.3 Cepstrum analysis
In gear diagnostics, there are three main areas of application for cepstrum analysis:
- Collect harmonics and side lobes across uniform intervals.
- Helps to distinguish between the driver function (at TM) and the transfer function to the measurement point.
- Identify echoes, especially inverted echo pairs, as well as echo delay times.
- These three areas will be discussed separately below.
5.4.3.1 Collection of harmonic and sidelobe series
In Section 2.2.2.1 it is stated that uniform wear tends to increase the harmonics of the TM frequency, initially the second harmonic and later all harmonics, which can be seen directly in the spectrum. It is also noted in Section 2.2.2.2 that a change in both local and distributed changes results in a change in all other harmonics of the rotational frequency of the affected gear compared to the mean (e.g., Figure 2.12). Because these harmonics are distributed across the spectrum, and the independent series from each gear are mixed, and the transfer function is modified over a wide range of frequencies, it becomes difficult to "see the trees" when looking at the spectrum over a wide frequency range, or when scaling on a narrow band. The cepstran has the ability to collect all the members of each series into a set of spectral series that are easier to interpret, the first of which is the most important because it tells "how much each series is on average prominent above the spectral noise level". The higher spectral series are affected by artifacts such as windows used in frequency analysis.
Figure 5.25 is very effective in illustrating the different information provided by the spectrum and the cepstrum spectrum for the wear and refurbishment of a cement mill gearbox. After eight years of operation, the gearbox was badly worn and needed to be repaired. Repairs mainly consist of reversing worn gears to take advantage of unworn sides, as well as replacing worn bearings and modifying support structures. Since both sets of sides were cut at the same time on the same machine, it can be assumed that the repaired spectrum is very similar to the original spectrum when the gearbox was put into service, although unfortunately there was no recording at that time.
If this assumption holds, we can see that the higher harmonics of the TM frequency all increase during wear (although some are lower than the adjacent side lobes). The fact that some of the sidelobes are higher than the carrier component (TM harmonics) does indicate that there is a large amount of frequency modulation, as this is almost impossible to achieve with pure amplitude modulation. Even if some of the sidelobes are very prominent in the spectrum, it is difficult to compare them with other possible spectral series.
However, the cepstrum immediately shows that the main sidelobes after the repair are spaced at 8.3 Hz, i.e. the rotational speed of the input pinion, while these were also present before the repair (at a higher level), but there is also a very strong 25 Hz series (this is the main sidelobe around the TM harmonics seen in the spectrum). The wear of the pinion was measured and found to be in a "triangular" pattern, modulated every three revolutions. The manufacturer suspects that this is due to the long grinding of the gears in the workshop before. This most likely excites structural resonances close to 25 hertz and initiates a triangular wear pattern that becomes more severe in operation over time. During the repair process, the unworn flanks were not ground and the structural modifications most likely changed the resonance situation.
In any case, Figure 5.26 made a similar spectrum and cepstrum comparison four years after the repair. It is immediately clear from the cepstrum that the sidelobe pattern has changed very little and the triangular wear pattern has not been redeveloped.
On the other hand, it can be seen from the spectrum that some uniform wear has occurred as the second harmonic of the gear mesh frequency increases. Also note that the secondary ghost component (the same interval as the first ghost component from the TM harmonic of the first harmonic) is also reduced, even though the primary ghost component does not change.
As a result, spectrum appears to be best suited for detecting uniform faults in gears (such as uniform wear), while cepstrum provides more information about non-uniform faults while indicating which gear they are located on.
Figure 5.27 illustrates another aspect of using a cepstrum to collect all members of a given series. The cepstrum frequency value corresponding to a particular series represents the average interval of all members of the series and is therefore more accurate than the measurement of a single interval. Of course, roughly the same accuracy can be achieved by using harmonics or sidelobe cursors (the former is more accurate), but the cepstrum immediately provides accurate values without the need to find a single member of the series. In this case, the gearbox was tested at the end of the production line. One of them clearly shows faults in the spectrum and champs. During the test, first gear was activated, with an input speed of 35.6 Hz and an output speed of 5.4 Hz. The cepstrum indicates that the fault produced harmonics spaced 10.4 Hz, a value accurate enough to rule out the possibility that it was a second harmonic (10.8 Hz) at a speed of first gear. In fact, it does correspond to second gear, albeit turning without load. A local "notch" on the second-gear gear still produces a pulse, even if it is not loaded.
It is important to note that the amplitude version of the "Analytical Chambled" should be used when performing such applications, as described in Section 3.4.3, as this always shows the correct position of the cepstrum peak, regardless of whether the evenly spaced series pass through zero frequency, or from the scaled spectrum. As such, it can be used in the edited spectrum to exclude components that are not relevant to the diagnostic problem at hand, or simply to reduce the size of the transformations required to calculate the cepstrufre.
Figure 5.28 illustrates how editing the spectrum before calculating the cepstrum can aid in the diagnostic process. Figure 5.28(a) shows the spectrum of a single-stage gearbox, from zero frequency to about 1.5 times the gear meshing frequency, and the corresponding cepstip. The latter appears to contain harmonics corresponding to the speeds of the two axes. However, in (b), the spectrum is edited to remove components below the meshing frequency of about half of the gears, and then it can be seen that the cepstran component corresponding to the 121 Hz axis is significantly reduced. This may be because they are low harmonics of that shaft speed rather than modulated sidelobes around the gear meshing frequency and therefore have nothing to do with the condition of the gear. Figure 5.28(c) shows a similarly edited spectrum, measured a month later, when the positioning of the 121 Hz axis changed. The corresponding cepstrum harmonics reappear, indicating that the gear is now modulating the gear meshing frequency.
However, if you want to edit the cepstruf, for example to remove a series of sidelobes or harmonics, you should edit the complex values of the analyzed chamstrand so that you can make a forward transformation to a logarithmic spectrum (one-sided). As shown in Figure 5.29, the 50 Hz harmonics have been removed from the entire spectrum by disabling the corresponding components in the cepstrum (and a small number of lines around the actual peak). Note that even though the cepstrum does not provide any information about the sidelobe distribution, the edited spectrum may be useful for judging the distribution without the shielding influence of other series. The same removal can also be achieved by synchronous averaging, but this will require a rpm signal.
5.4.3.2 Separate the excitation function from the transfer function
The simplest example of the benefits of cepstrum in this regard is that the useful part of the cepsant, which represents the excitation function at the gear mesh, is rarely affected by the measurement point and therefore by the different signal transmission paths transmitted to the measurement point.
Figure 5.30 shows an example of spectrum and cepstrum for two measurement points in the same gearbox. Although the spectrum on the logarithmic amplitude scale looks very similar at first glance, it differs in detail. For example, there is a peak close to 2.7 kilohertz near measurement point 1, but there is a valley floor at the same frequency at measurement point 2. It can be shown that [22] the low cepstrum region will be contributed by both the excitation and the transfer function, but the high cepstran region is almost entirely dominated by the excitation function, and the two measurement points appear to be the same.
Based on the theory of [22], Figure 5.31 (from [23]) shows an example in which the excitation is removed from the low cepstrum portion of the cepstrum and the gears are compared with those with and without root cracking. The results show that for both cases, the transfer function changes very little. In particular, the resonant frequency of the transfer path does not appear to change, thus confirming that the difference is due to a change in the excitation function (the crack teeth mainly change the force at the mesh). Part of the excitation function located at the low cepstrum frequency comes from the harmonics of the gear meshing frequency in the spectrum, and in this case, the corresponding harmonics in the cepstrum can be removed by using a "comb lifter" given by the |sin x/x|function, where the zero point is adjusted to the interval of the harmonics to be removed.
Although not further developed in this book, cepstrum can be used to blindly determine the dynamic properties of a structure, at least if the response is dominated by a single excitation function [24].
5.4.3.3 Identify echo and inverted echo pairs
In Section 3.4, it is stated that the echo presents a series of harmonics in the cepstrum spectrum, with an interval equal to the echo delay time. For forward echoes, harmonics are in alternating series, but as shown in Figure 5.32, when the echo is negative or inverted, all harmonics are negative. Therefore, when the signal portion of the analyzed part contains an inverted echo, the sum of all spectral components should become more negative.
In [25] El Badaoui et al. defined the Moving Cepstral Integral (MCI) to take advantage of this fact to detect the effect of tooth spalling in the gearbox vibration signal. The idea is that the signal produced when the mating tooth exits from the spalling will be an inverted copy of the signal at the time of entry and will produce an inverted echo. As shown in Figure 5.34(2a–2c), this is emphasized in the case of acceleration. If one time window is moved along the record, the length of which is between one and two times the TM period, and then the cepstrum is calculated for each position and "integral" is taken on all the cepstrums, then the MCI should become negative when any inverted echoes corresponding to the spalling are found within the window. Figure 5.33 from [25] illustrates that this is in fact true for the actual measured spalling. In [26], the same authors, in collaboration with researchers at INSA in Lyon, France, demonstrated that the same results were obtained when spalling was simulated on gears. In other words, the rather simple assumption made in the paper that the second derivative of the linear displacement of the mating tooth at the time of entry and exit spalling will correspond to the acceleration response actually has some basis.
In [27], Endo et al. showed that when there is root cracking on the gear, the MCI also becomes negative, as this tends to produce two inverted echo pairs. This will be discussed further in Section 5.4.4 on separating spalling and cracking.
5.4.4 Separation of spalling and cracking
The method in Section 5.4.1 allows for the separation of local and distributed faults, but does not distinguish between spalling and root cracking. This distinction is important because cracking often has a very different prognosis than flaking, with a tendency to fail more quickly. This issue has not been addressed at this time, but some recent studies have used finite element analysis to simulate to understand the difference between spalling and cracking. Both studies only involved spur gears.
In [27], Endo et al. modeled root cracking as a thin slit located at the base of the tooth, which mainly led to a change in stiffness of the affected tooth upon meshing. As shown in Figure 5.34(1a–1c), this results in a two-stage deviation between the meshing mode and the normal meshing mode of the healthy tooth, the first in which the load is shared by the healthy tooth pair and the tooth pair involving the cracked tooth, and then a stage where only the cracked tooth is in the engaged state. When the results are doubly differentiated to get an approximation of the acceleration response, two pairs of reversed echoes can be seen, as described in Section 5.4.3.3 above. Endo et al. also simulated spalling at different widths and under-edge lengths on the tooth surface and concluded that TE is subject to geometric errors (almost independent of load) and is affected by tooth making, meaning that mating teeth enter further into the tooth as the spalling expands on the tooth. The resulting simulated TE (and its double differential) is shown in Figure 5.34(2a–2c).
From the simulations, Endo et al. found that not only cracking would lead to a negative bias in the MCI, but also that two pairs of inverted echoes would be evident in the windowed cepstral. Since the timeline depends only on how long the cracking teeth are in mesh, the echo delay time is independent of the cracking depth, as shown in Figure 5.35. On the other hand, the change in TE is found to be directly proportional to the depth of the crack as the stiffness of the tooth decreases.
The equivalent case for spalling is shown in Figure 5.36. In this case, the echo delay time of the cepstrum is proportional to the size of the spalling.
In [28], Endo et al. presented experimental confirmation of their results and confirmed that the effect of spalling is independent of load, while the effect of cracking depends on load. It is worth mentioning that the cracks simulated in this experiment are formed by EDM and therefore do not occur naturally. Mark and Reagor [29] have recently shown that for naturally occurring root cracking, plastic deformation of the crack tip tends to result in a fixed geometric deviation of the tooth, which will provide a load-independent component in TE in addition to the load-dependent component.
Another recently published study on simulating root cracking and spalling was carried out by Jia and Howard [30], who performed the simulation in different ways, specifically spalling. The latter is modeled as a round hole on the tooth surface with almost no geometric deviation due to the absence of a tooth top arc. Therefore, the main effect of the spalling they simulated was the change in contact stiffness after the effective width of the tooth decreased. Even so, they confirmed the findings of Endo et al. that the effects of cracking are directly related to the time the cracked teeth are in mesh, while the effects of spalling are generally shorter. When TE and acceleration are demodulated around TM harmonics, they find other differences when plotted in polar plots. In other words, the real and imaginary parts of the analytic signal (represented by amplitude and phase in Figure 5.15(b) and (c)) are plotted against each other.
This simulation work has not yet been extended to helical and other more complex gears, such as spiral bevel gears and helical bevel gears, and it is clear that it will be more difficult to determine the length of engagement time of the cracked tooth in these gears, especially if the crack does not extend to the full width of the tooth.
5.4.5 Diagnosis of variable speed and variable load gears
Many of the above methods rely on being able to analyze signals recorded at steady speeds and loads. In cases where the velocity changes are relatively slow, this can usually be compensated for by order tracking (Section 3.6), as is the case in Figures 3.42 and 3.44. When the speed changes rapidly, this may be done over a smaller range and accompanied by a rapid load change. In some cases, such as in the mining industry and wind turbines, the load varies over a wide range, sometimes randomly and over a relatively short period of time. This section provides some literature references on this specialized topic.
In cases where the load is still cyclical, but it may be second-order cyclic rather than cyclical, as in some mining machinery, sometimes some form of time-frequency analysis can be used to obtain a measure of the approximate instantaneous load, which is then compensated. This is the method used in [31], but is based on the fact that the load is a case of deterministic change in laboratory tests. Another method based on the AR model was proposed in [32] and was again used for deterministic load changes, i.e., load abrupt changes and sinusoidal changes. In [33-34], Bartelmus and Zimroz describe the problems encountered with planetary gearboxes for bucket wheel excavators. Here, the load is cyclical as it varies with each bucket as it is mined, but with random variations that make the load signal second-order cycle. They found that in a worn gearbox (without serious local failures), planetary carrier motion was more sensitive to load, resulting in greater modulation, both in amplitude and frequency, which could be used as a status indicator or characteristic. Instantaneous motor speed or current can be used as a load indicator, while the regression slope, which associates the state characteristics with the load, reliably measures the state of wear.
For wind turbines, the load can vary over a wide range in less than a minute. For the high-speed portion of the gearbox, this may still allow for analysis using conventional techniques by selecting a signal section for a specific load range. However, a failure (crack teeth on the ring gear) in the low-speed part of the gearbox is described in [35], which is not detected by conventional monitoring techniques. Many parameters were monitored continuously, and the time signal was recorded relatively randomly. The kurtosis of the overall signal reacts to the imminent occurrence of the failure, but the other parameters do not. When analyzing the recorded signals, a quick kurtosis plot [12] found that the values of the SK occasionally showed higher values in the months leading up to the last failure, which can be used as a predictor of failures, especially if the recordings were performed periodically to obtain higher load conditions. The kurtosis found filter bandwidth is concentrated around 11 kHz, which explains why conventional diagnostic techniques cannot be used in this case. The sampling frequency used was 25 kHz, which was required to cover the resonant frequency caused by the fault, but since the repetition rate of the planetary carrier around the ring gear was about 0.3 Hz, the length of recording required to perform the TSA would have to contain 75000 samples. The frequency stability required for a successful TSA will be an order of magnitude of this order of magnitude (1 : 750 000), which is obviously impossible, even if the speedometer signal is located in the low-frequency part of the system. In fact, it is located on the generator shaft, and calculations show that the elastic twist of the input shaft relative to the output shaft will be 0.3°, even over the entire load range. The potential for demodulation using gear meshing frequencies is even less, since this frequency for planetary gearboxes is about 30 Hz, and the maximum permissible modulation frequency will be half of it. The paper concludes that SK provides the best early indication of such a failure and, most likely, envelope analysis by optimizing the filtered signal will point to the source of the failure.