Rolling element bearing diagnostics—A tutorial

Abstract

This tutorial is intended to guide the reader in the diagnostic analysis of acceleration signals from rolling element bearings, in particular in the presence of strong masking signals from other machine components such as gears. Rather than being a review of all the current literature on bearing diagnostics, its purpose is to explain the background for a very powerful procedure which is successful in the majority of cases. The latter contention is illustrated by the application to a number of very different case histories, from very low speed to very high speed machines. The specific characteristics of rolling element bearing signals are explained in great detail, in particular the fact that they are not periodic, but stochastic, a fact which allows them to be separated from deterministic signals such as from gears. They can be modelled as cyclostationary for some purposes, but are in fact not strictly cyclostationary (at least for localised defects) so the term pseudo-cyclostationary has been coined. An appendix on cyclostationarity is included. A number of techniques are described for the separation, of which the discrete/random separation (DRS) method is usually most efficient. This sometimes requires the effects of small speed fluctuations to be removed in advance, which can be achieved by order tracking, and so this topic is also amplified in an appendix. Signals from localised faults in bearings are impulsive, at least at the source, so techniques are described to identify the frequency bands in which this impulsivity is most marked, using spectral kurtosis. For very high speed bearings, the impulse responses elicited by the sharp impacts in the bearings may have a comparable length to their separation, and the minimum entropy deconvolution technique may be found useful to remove the smearing effects of the (unknown) transmission path. The final diagnosis is based on “envelope analysis” of the optimally filtered signal, but despite the fact that this technique has been used for 40 years in analogue form, the advantages of more recent digital implementations are explained.

Introduction

Rolling element bearings are one of the most widely used elements in machines and their failure one of the most frequent reasons for machine breakdown. However, the vibration signals generated by faults in them have been widely studied, and very powerful diagnostic techniques are now available as discussed below.

Fig. 1 shows typical acceleration signals produced by localised faults in the various components of a rolling element bearing, and the corresponding envelope signals produced by amplitude demodulation. It will be shown that analysis of the envelope signals gives more diagnostic information than analysis of the raw signals. The diagram illustrates that as the rolling elements strike a local fault on the outer or inner race a shock is introduced that excites high frequency resonances of the whole structure between the bearing and the response transducer. The same happens when a fault on a rolling element strikes either the inner or outer race. As explained in [1], the series of broadband bursts excited by the shocks is further modulated in amplitude by two factors:

• The strength of the bursts depends on the load borne by the rolling element(s), and this is normally modulated by the rate at which the fault is passing through the load zone.

• Where the fault is moving, the transfer function of the transmission path varies with respect to the fixed positions of response transducers.

Fig. 1 illustrates typical modulation patterns for unidirectional (vertical) load on the bearing, at shaft speed for inner race faults, and cage speed for rolling element faults. The formulae for the various frequencies shown in Fig. 1 are as follows:

Ballpass frequency, outer race:

$$\text{BPFO}=\frac{n{f}_r}{2}\left\{1-\frac{d}{D}\cos \varphi \right\}$$

Ballpass frequency, inner race:

$$BPFI=\frac{n{f}_r}{2}\left\{1+\frac{d}{D}\cos \varphi \right\}$$

Fundamental train frequency (cage speed):

$$FTF=\frac{f_r}{2}\left\{1-\frac{d}{D}\cos \varphi \right\}$$

Ball (roller) spin frequency:

$$BSF(RSF)=\frac{D}{2d}\left\{1-{\left(\frac{d}{D}\cos \varphi \right)}^2\right\}$$

where f_r is the shaft speed, n is the number of rolling elements, and ϕ is the angle of the load from the radial plane. Note that the ballspin frequency (BSF) is the frequency with which the fault strikes the same race (inner or outer), so that in general there are two shocks per basic period. Thus the even harmonics of BSF are often dominant, in particular in envelope spectra.

These are however the kinematic frequencies assuming no slip, and in actual fact there must virtually always be some slip because the angle ϕ varies with the position of each rolling element in the bearing, as the ratio of local radial to axial load changes. Thus, each rolling element has a different effective rolling diameter and is trying to roll at a different speed, but the cage limits the deviation of the rolling elements from their mean position, thus causing some random slip. The resulting change in bearing frequencies is typically of the order of 1–2%, both as a deviation from the calculated value and also as a random variation around the mean frequency. This random slip, while small, does give a fundamental change in the character of the signal, and is the reason why envelope analysis often extracts diagnostic information not available from frequency analyses of the raw signal. It means that bearing signals can be considered as cyclostationary (see Appendix A). This also allows bearing signals to be separated from gear signals with which they are often mixed, as discussed below.

It should be noted that the argument about variation of rolling diameter with load angle applies equally to taper roller and spherical roller bearings, since by virtue of their kinematics, the ratio of roller diameter to race diameter varies with the axial position, and so there is only one position where there is no slip. The slip on either side of this position is in opposite directions, and generates opposing friction forces which balance, but the location of the no-slip diameter is strongly influenced by the point of maximum pressure between the rollers and races, and is thus dependent on the ratio of axial to radial load, which varies with the rotational position of the roller in the bearing. The same argument cannot be made for parallel roller bearings, which are unable to sustain an axial load, but on the other hand, they would rarely have negative clearance, and the rollers are only compelled to roll in the load zone. Thus, when they enter the load zone, they will tend to have a random position in the clearance of the cage, and the repetition frequency would have a stochastic variation as for other bearing types, even if the deviation of the mean value from the kinematic frequency is less.

Fig. 2 shows the basic reason why there is often no diagnostic information in the raw spectrum. This shows acceleration signals from a simulated outer race fault, with and without random slip. Spectra are shown for both the raw signal and the envelope. The individual bursts are simulated as the impulse response (IR) of a single degree of freedom (SDOF) system with just one resonance, but this could be the lowest of a series. As is quite common, the assumed resonance frequency is two orders of magnitude higher than the repetition frequency of the impacts. The Fourier series for the periodically repeated IRs are samples of the frequency response function (FRF) of one IR. Because the FRF is measured in terms of acceleration, the spring line at low frequencies is a ω² parabola, with zero value and zero slope at zero frequency. Thus, the low harmonics of the repetition frequency have very low magnitude and are easily masked by other components in the spectrum. If the signal were perfectly periodic, the repetition frequency could be measured as the spacing of the harmonic series in the vicinity of the resonance frequency, but as illustrated in Fig. 2(e), the higher harmonics smear over one another with even a small amount of slip (here 0.75%). However, the envelope spectra (Fig. 2(c), (f)) show the repetition frequency even with the small amount of slip, even though the higher harmonics in the latter case are slightly smeared.

As mentioned, the lowest resonance frequencies significantly excited are often, but not universally, very high with respect to the bearing characteristic frequencies. It would for example not be the case for gas turbine engines, where the fault frequencies are often in the kHz range. Even so, the low harmonics of the bearing characteristic frequencies are almost invariably strongly masked by other vibration components, and it is generally easier to find wide frequency ranges dominated by the bearing signal in a higher frequency range. The advantage of finding an uncontaminated frequency band encompassing several harmonics of the characteristic frequency is that bearing fault signals are generally impulsive, but cannot be recognised as such unless the frequency range includes at least ten or so harmonics. If a pulse train is lowpass filtered between the first and second harmonics of the repetition frequency, the result is a sinewave, with no impulsivity at all. The most powerful bearing diagnostic techniques depend on detecting and enhancing the impulsiveness of the signals, and so the fact that low harmonics of the bearing characteristic frequencies can sometimes be found in raw spectra is basically ignored in the rest of this paper. This is because the authors believe that the purpose of a tutorial is to give details of the most widely applicable method to solve the problem at hand, rather than a catalogue of all publications on the subject, which is more the function of a review. As a counter example, a paper by one of the authors [2] was the first to use the cepstrum to diagnose bearing faults, this relying on being able to find separated harmonics of the bearing frequency over a reasonably wide frequency range. It was a high speed machine (an auxiliary gearbox running at 3000 rpm), and a reasonable number of the first 20 or 30 harmonics were separated and gave a component in the cepstrum. On the other hand, the primary method recommended in this paper, envelope analysis, performed equally well if not better in that case, and does not require the harmonics to be separated, as illustrated in Fig. 2, so the cepstrum method has little application.

Even though this tutorial concentrates primarily on the method of envelope analysis (after first having separated the bearing signal from strong background signals which generally mask it), a brief history will first be given here on the development of bearing diagnostics, and a justification for the choice of the proposed method.

One of the earliest papers on bearing diagnostics was by Balderston [3] of Boeing in 1969. He recognised that the signals generated by bearing faults were primarily to be found in the high frequency region of resonances excited by the internal impacts, and investigated the natural frequencies of bearing rings and rolling elements, which were often to be found in the response vibrations. He pointed out that at such high frequencies, in the tens of kHz, measurable acceleration levels corresponded to extremely small displacements, which could be accommodated in the clearance space between surface asperities of a bearing ring in its housing, even after fitting, and thus natural frequencies were not greatly modified by the mounting. Shortly after, in 1970, Weichbrodt and Smith [4] used synchronous averaging to expose local faults in both bearings and gears. In the former case they sometimes performed averaging on the (rectified) envelope signals. Braun [5] made a fundamental analysis of synchronous averaging in 1975, and the basic technique was also applied to bearing signals [6]. This appears to be one of the first references to the fact that bearing signals are not completely periodic, with a random variation in period. Braun made an analysis of the effects of jitter (of the synchronising signal) and likened this to the random spacing of bearing response impulses. This model, which is effectively Model 1 in the next section (Fig. 4), was much later shown to be incorrect, and so this approach has not been expanded on in this tutorial, even though it can give satisfactory results in some situations.

At around that time, the “high frequency resonance technique” (HFRT), later called “envelope analysis”, was developed (see [7] and the first 15 references of [1]). Even though this is described in the previous section as solving the problem of smearing of high harmonics (Fig. 2), this was not the main reason for its development, since it probably was not recognised at the time because of the limited resolution of FFT analysis. The main reason for its development was to shift the frequency analysis from the very high range of resonant carrier frequencies, to the much lower range of the fault frequencies, so that they could be analysed with good resolution. The frequency shifting was done using analogue rectifiers. Even in McFadden and Smith’s classic paper on the modelling of bearing fault signals [1], the fault pulses are treated as periodic.

This concept of demodulating high frequency resonant responses led to the development of a number of bearing diagnostic methods, where the demodulated frequency was the resonance of the transducer itself. This includes the “Shock Pulse Meter” (SPM), marketed for some time by the SKF bearing company, and the “Spike Energy” method marketed by IRD. These used the resonance of a conventional accelerometer as the main carrier, in the former case with bandpass filtering around a well-defined frequency of about 32 kHz, and in the latter case a highpass filtering at about 15 kHz, with more tolerance for the transducer resonance. Systems including acoustic emission (AE) transducers, with frequency ranges from 50 kHz to 1 MHz, were also introduced at that time. While often being effective in improving the signal/noise ratio of bearing signal to background noise, this was not universally the case. Ref. [8] describes situations where the transducer resonance happened to coincide with other excitations, such as turbulence and cavitation in pumps, and therefore gave false readings. The authors recommended choosing the appropriate resonance frequency for demodulation in each case.

There has long been a discussion on how to choose the optimum bandwidth for the demodulation associated with envelope analysis. Some recommended searching for a peak at high frequency in response spectra, on the assumption that it would be excited by bearing faults, while others suggested that a hammer tap test would be more likely to identify bearing resonances. In the authors’ opinion, prior to the development of the spectral kurtosis (SK) based methods in the current tutorial, the best approach was to demodulate the band with the biggest dB change from the original condition, although this does require having reference signals with the bearings in good condition.

Such methods are not discussed further in this tutorial because the authors believe that the methods proposed herein solve the problems in the vast majority of cases.

Another approach to bearing diagnostics that can be found in the literature, but is not discussed here, are statistical methods based on pattern recognition. These rely on training a pattern recognition system with typical signals representing the different classes to be distinguished. There are two main reasons why such methods are not treated here. One is that they require large amounts of data for the training, and it is very rare that sufficient data can be acquired by experiencing actual faults in practice, including all permutations and combinations of fault type, location, size, machine load and speed, etc., in particular for expensive critical machines. Most published results are not non-dimensionalised and would only apply to a particular bearing on a particular machine for which the system was trained. It is likely that some of these problems will be overcome by fault simulation in the future. The other reason is that the authors believe that the quasi-deterministic approach proposed in this tutorial covers the vast majority of situations, as exemplified by the wide range of different cases treated in Section 6, without requiring excessive amounts of data from failures. Even so, the reader is referred to the Tutorial on “Natural Computing” [9] for a detailed discussion of methods based on pattern recognition.