2043 BWD型擺線針輪減速器設(shè)計(jì)及虛擬裝配研究

2043 BWD型擺線針輪減速器設(shè)計(jì)及虛擬裝配研究,bwd,擺線,減速器,設(shè)計(jì),虛擬,裝配,研究,鉆研 Mechanical Systems and Signal Processing 25 (2011) 48 5–520Contents lists available at ScienceDirectMechan ical Syste ms and Sign al Processingjournal homepage: www.elsevier.com/locate/jnlabr/ymsspReviewRolling element bearing diagnosti cs—A tutoria l$Robert B. Randall a ,n , Je′ ro? me Antoni ba School of Mechanical and Manufacturing Engineering, University of New South Wales, Sydney, NSW 2052, Australiab Laboratory Roberval of Mechanics, University of Technology of Compi e′ gne, 60205 Compi e′ gne, Cedex, Francea r t i c l e i n f oArticle history:Received 23 July 2010Accepted 29 July 2010Keywords:Rolling element bearings Diagnostics CyclostationaritySpectral kurtosisMinimum entropy deconvolutionEnvelope analysisa b s t r a c t This tutori al is inten ded to guide the reader in the diagnos tic anal ysis of acce leration sign als from rolling el ement bearin gs, in partic ular in the pres ence of strong m asking sign als from ot her mach ine compo nents such as gears. Rather than being a review of all the curr ent lite rature on bearing diagnos tics, its purpose is to expl ain the bac kgrou nd for a very powerf ul proced ure whi ch is succes sful in the majori ty of cases. The lat ter con tention is ill ustrated by the ap plication to a number of very diffe rent case his tories, from very low speed to very high speed mach ines. The speci?c characte ristics of rolling ele m ent beari ng si gnals are expl ained in great detail, in particular the fact that they are not per iodic, but stochas tic, a fact which allows them to be separated from dete rmin istic sign als such as from gears. They can be modell ed as cyclosta tionary for so m e purpo ses, but are in fact not strictly cyclos tationary (at le ast for local ised defects) so the term pseudo -cyclosta tionary has been co ined. An appe ndix on cyclo stationari ty is inclu ded. A number of techniq ues are described for the sepa ration, of which the dis crete/ra ndom separation (DRS) m ethod is usual ly most ef?ci ent. This sometimes requ ires the effects of small speed ?uctuatio ns to be remov ed in ad vance, whi ch can be ach ieved by order track ing, and so this topic is also am pli?ed in an ap pendix. Signals from local ised faul ts in bearin gs are impuls ive, at least at the sour ce, so tech niques are descr ibed to ide ntify the frequency ban ds in whi ch this im pulsivity is m ost m arked, using spectral kur tosis. For very high speed be arin gs, the im pulse responses elicited by the sharp im pacts in the bearin gs may have a compar able length to their separa tion, and the min imum entro py de convolution tech nique may be fou nd useful to remove the sme aring eff ects of the (un kno w n) tra nsmission path. The ?nal diagnos is is based on‘‘env elope analy sis’’ of the optim ally ?ltered sign al, but des pite the fact that this tech nique has be en us ed for 40 years in analo gue form, the adva ntages of more recent dig ital im plem entatio ns are explai ned.& 2010 Elsevier Ltd. All rig hts reserve d.Cont ents1. Intro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4861.1. Sho rt history of bearing diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4882. Bear ing fault m odels and cy clostation arity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4892.1. Localised faults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490$ Some of the material in this tutorial is adapted from related sections in the book Vibration-based Condition Monitoring: Industrial, Automotive andAerospace 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.2010.07.0172 R.B. Randall, J. Antoni / Mechanical Systems and Signal Processing 25 (2011) 48 5–520t2.2. Extend ed spalls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4913. Sep aration of be aring signals from discr ete frequency noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4933.1. Linear pred iction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4933.2. Adaptive noise canc ellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4943.3. Self-a daptive noise canc ellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4953.4. Discrete /random separation (DRS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4963.5. Time synchro nous averagi ng (TSA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4974. Enh ancement of the be aring sign als . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4984.1. Minim um entr opy deconv olution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4994.2. Spectral kur tosis and the kur togra m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5014.2.1. Spectr al kur tosi s— de?nition and calc ulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5014.2.2. Use of SK as a ?lter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5024.2.3. The kur togram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5034.2.4. The fast ku rtogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5044.2.5. Wavelet denoi sing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5055. En velope an alysi s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5066. A semi-a utom ated bearing diagnos tic proced ure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5086.1. Case history 1— he licopter gearbo x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5096.2. Case history 2— high speed bearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5116.3. Case history 3— radar tower bearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512Ap pendix A Cyc lostat ionarity and spectr al co rrelatio n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514A.1. Spectral cor relati on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515A.2. Spectral cor relati on and envelo pe sp ectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516A.3. Wigner –V ille spec trum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516Ap pendix B Ord er track ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516Ref erences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5191. Introdu ctionRolling 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 dBPFO ? 1 cosf2 D e1TBallpass frequency, inner race:nfr dBPFI ? 1 cos f2 D e2TFundamental train frequency (cage speed):fr dFTF ? 2 1 D cos f e3TR.B. Randall, J. Antoni / Mechanical Systems and Signal Processing 25 (2011) 48 5–520 487Fig. 1. Typical signals and envelope signals from local faults in rolling element bearings.Ball (roller) spin frequency:D ( d 2 )BSFeRSFT? 2d 1 cos fD e4Twhere 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 is488 R.B. Randall, J. Antoni / Mechanical Systems and Signal Processing 25 (2011) 48 5–520two 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 o 2 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 componentsin 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 at3000 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 diagnosticsOne 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