BWD型擺線針輪減速器設(shè)計(jì)及虛擬裝配研究【說(shuō)明書+CAD+PROE】,說(shuō)明書+CAD+PROE,BWD型擺線針輪減速器設(shè)計(jì)及虛擬裝配研究【說(shuō)明書+CAD+PROE】,bwd,擺線,減速器,設(shè)計(jì),虛擬,裝配,研究,鉆研,說(shuō)明書,仿單,cad,proe Mechanical Systems and Signal Processing 25 (2011) 485–520 Contents lists available at ScienceDirect Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp Review Rolling element bearing diagnostics—A tutorial$ Robert B. Randall a,n, Je′ ro? me Antoni b a School of Mechanical and Manufacturing Engineering, University of New South Wales, Sydney, NSW 2052, Australia b Laboratory Roberval of Mechanics, University of Technology of Compie′gne, 60205 Compie′gne, Cedex, France a r t i c l e i n f o Article history: Received 23 July 2010 Accepted 29 July 2010 Keywords: Rolling element bearings Diagnostics Cyclostationarity Spectral kurtosis Minimum entropy deconvolution Envelope analysis a b s t r a c t 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 speci?c 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 ef?cient. This sometimes requires the effects of small speed ?uctuations to be removed in advance, which can be achieved by order tracking, and so this topic is also ampli?ed 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 ?nal diagnosis is based on ‘‘envelope analysis’’ of the optimally ?ltered 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. & 2010 Elsevier Ltd. All rights reserved. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 1.1. Short history of bearing diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 2. Bearing fault models and cyclostationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 2.1. Localised faults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 $ Some of the material in this tutorial is adapted from related sections in the book Vibration-based Condition Monitoring: Industrial, Automotive and Aerospace Applications, by R.B. Randall, to be published by John Wiley and Sons. n Corresponding author. Tel.: + 61 2 9958 3591; fax: + 61 2 9663 1222. E-mail addresses: b.randall@unsw.edu.au (R.B. Randall), jerome.antoni@utc.fr (J. Antoni). 0888-3270/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2010.07.017 2.2. Extended spalls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 3. Separation of bearing signals from discrete frequency noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 3.1. Linear prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 3.2. Adaptive noise cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 3.3. Self-adaptive noise cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 3.4. Discrete/random separation (DRS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 3.5. Time synchronous averaging (TSA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 4. Enhancement of the bearing signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 4.1. Minimum entropy deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 4.2. Spectral kurtosis and the kurtogram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 4.2.1. Spectral kurtosis—de?nition and calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 4.2.2. Use of SK as a ?lter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 4.2.3. The kurtogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 4.2.4. The fast kurtogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 4.2.5. Wavelet denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 5. Envelope analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506 6. A semi-automated bearing diagnostic procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 6.1. Case history 1—helicopter gearbox. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 6.2. Case history 2—high speed bearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 6.3. Case history 3—radar tower bearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 Appendix A Cyclostationarity and spectral correlation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 A.1. Spectral correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 A.2. Spectral correlation and envelope spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 A.3. Wigner–Ville spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 Appendix B Order tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 1. 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 ?xed 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: nfr d BPFO ? 1 cosf 2 D e1T Ballpass frequency, inner race: nfr d t BPFI ? 1 cos f 2 D e2T Fundamental train frequency (cage speed): fr d FTF ? 2 1 D cos f e3T R.B. Randall, J. Antoni / Mechanical Systems and Signal Processing 25 (2011) 485–520 519 Fig. 1. Typical signals and envelope signals from local faults in rolling element bearings. Ball (roller) spin frequency: D ( d  2 ) BSFeRSFT? 2d 1 cos f D e4T where fr is the shaft speed, n is the number of rolling elements, and f 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 f 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 in?uenced 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 o2 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 signi?cantly 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 ?nd wide frequency ranges dominated by the bearing signal in a higher frequency range. The advantage of ?nding 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 ?ltered between the ?rst 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 ?rst to use the cepstrum to diagnose bearing faults, this relying on being able to ?nd 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 ?rst 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 ?rst having separated the bearing signal from strong background signals which generally mask it), a brief history will ?rst be given here on the development of bearing diagnostics, and a justi?cation for the choice of the proposed method. 1.1. Short history of bearing diagnostics 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 Fig. 2. Bearing fault pulses with and without random ?uctuations: (a), (d) time signals; (b), (e) raw spectra; and (c), (f) envelope spectra. 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 ?tting, and thus natural frequencies were not greatly modi?ed 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 (recti?ed) 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 ?rst 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 i