on the statistical nature of real sinusoids associated with rotating machinery

Digital Signal Processing 12, 471–483 (2002)doi:10.1006/dspr.2001.0404

On the Statistical Nature of Real SinusoidsAssociated with Rotating Machinery

Peter Sherman

Iowa State University, Ames, Iowa 50011E-mail: [email protected]

Sherman, P., On the Statistical Nature of Real Sinusoids Associated withRotating Machinery, Digital Signal Processing 12 (2002) 471–483.

This paper represents the first phase of an ongoing investigation into thenature of sinusoidal types of random processes associated with real worldphenomena. While perfect sinusoids exist only in the mathematical sense,their use as an approximation model in relation to real world phenomenahas been and continues to be widespread, often with much value. Themotivation for this work is the belief that knowledge of their deviationfrom such a model can provide additional useful information. The focus ofthis paper is on sinusoids in relation to random processes associated withrotating machinery. The tools used include mathematical limit theoremresults, standard signal processing tools including spectral estimation andKalman filtering, and basic statistics. Some noteworthy results includethe normality of amplitude and frequency probability distributions, thecharacterization of the same as stationary random processes, and thepotential to improve condition monitoring of rotating machinery. 2002

Elsevier Science (USA)

1. INTRODUCTION

The concept of a sinusoid arises in the study of phenomena in practically everyarea of science and engineering. Examples include vibration of rotating machin-ery [1], species extinction rates [2], earthquake prediction [3], atmospheric windprofiles [4], sonar [5], heart rate variability [6], and music quality, to name justa very few. However, a true sinusoid only exists in the mathematical sense. Evena sine wave generator does not generate a perfect sinusoid. A perfect sinusoid ischaracterized by three constants, namely, amplitude, frequency, and phase. Be-cause the phase parameter reflects only the relation of the sinusoid to the timeat which the sinusoid is first observed, the value of this parameter is dictated bythe observer. Both amplitude and frequency, however, are subject to change over

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472 Digital Signal Processing Vol. 12, No. 4, October 2002

time. And so it is these parameters that will be the focus of this paper. It is worthpointing out that the frequency of a sinusoid is simply the inverse of its period,since in many applications it is period, and not frequency variability, that is ofinterest.

Amplitude and frequency (or period) variability can occur slowly over time(relative to the period of the sinusoid), as well as within a single period. Here,the period refers to the time period associated with 360 degrees of angle. Inmany rotating machinery applications the speed of the machine will vary tosome degree slowly over time. If the basic natures of the associated signals, suchas vibration, sound, or pressure, are not influenced by the slow-time variations,then it may be possible to recover truly periodic signals by performing a time-to-angle transformation [7]. Even then, however, there may well remain intracyclevariability of sinusoidal data. A simple example of this is shown in Fig. 1 (left),where a sinusoid has been warped so that the first half of the period is greaterthan the second half. Such warping could arise, or example, due to the increasingload that a compressing piston sees, followed by a more rapid speed followingcombustion. If, indeed, the waveform in Fig. 1 were one period of a trulyperiodic waveform, which would almost never be the case for uniform timessampled data from a machine operating at nominally constant speed due toslight speed variations, then it has the Fourier series decomposition, a portion ofwhich is shown in Fig. 1. However, even though this decomposition is in termsof true sinusoids, there are problems with using the information provided inthis way to identify the warped nature of the sinusoid. One is that, withoutphase information, which is much more sensitive to noise than is amplitudeinformation, it turns out that there are infinitely many periodic waveforms thatcould yield the shown Fourier series magnitudes. Even if one were to use therelative amplitudes of the Fourier series to flag the possibility of such behavior,even a small amount of noise could hinder identification of the higher harmonics(the one at 1 Hz is 10 dB below the fundamental). Finally, a Fourier series is notwhat is really being sought, namely, the identification of a real sinusoid whoseshape is slightly warped due to intracycle speed variation.

FIG. 1. A warped sinusoid (left), and the corresponding first 10 Fourier series coefficientmagnitudes (right).

Peter Sherman: Real Sinusoids Associated with Rotating Machinery 473

The remainder of this paper is organized as follows. First, we give a verybrief review of some of the tools that are used to conduct the analysis. This isfollowed by an equally brief description of the machinery data to be analyzed. Welimit ourselves to two real sinusoids. One is associated with a no-fault condition,and the second, which is at essentially the same frequency, is associated with aplanted fault condition. Our analysis proceeds primarily with the aid of figures,as opposed to mathematical equations. The reader should be cautioned notto expect any fundamental breakthroughs or new techniques. Our analysis isexploratory, and uses tools already well established. The goal here is simply tobegin to attempt to shed some light on the behavior of real world sinusoids, andto evaluate how a combination of existing mathematical, signal processing, andsimple statistical tools might be used for that purpose.

2. TOOLS USED FOR ANALYSIS

The tools used for analysis of the data include (I) the minimum variance(MV) and associated autoregressive (AR) spectral families, along with theirtheoretical convergence properties [8], (ii) standard power spectral density(PSD) estimates, (iii) extended Kalman filtering , (iv) fixed order AR models, and(v) histogram, scatter plot, and correlation coefficient information. The familiesof MV and AR spectra are used to identify nominally sinusoidal components, andto provide input to the extended Kalman filter for tracking the time-varyingamplitude and frequency of a real sinusoid. The PSD is used primarily forcomparative purposes because it is the tool of choice for analysis of frequencyinformation contained in (assumedly stationary) random processes. Fixed orderAR models are used in an attempt to better characterize the time-varyingbehavior of amplitude and frequency estimates. Finally, basic statistical tools,including histogram and moment estimates, are used to get a better idea of thedistributional properties of frequency and amplitude information.

3. THE MACHINERY DATA TO BE ANALYZED

The data chosen for analysis is from the Westland Helicopter data base [9],This data base includes vibration data taken from a military helicopter underwell controlled test conditions, and for a variety of planted (i.e., intentionallyintroduced) fault conditions. Our analysis will focus on the vibration associatedwith accelerometer number 6 for the no-fault condition, and for a pinion-bearingfault near to the measurement location. The data were originally sampled at103,116.08 Hz. Because of the preponderance of energy below 20,000 Hz, thesedata were decimated by a factor of 5. The no-fault data were chosen because ofthe presence of a very strong sinusoidal component at 3150 Hz. This affords usthe opportunity to study a real world sinusoid without complications associatedwith a significant amount of noise corruption. The pinion-bearing data werechosen for two reasons. Since the strength of this strong sinusoid was notablyattenuated, this afforded the opportunity to study a potentially more complex


real world sinusoid. It also provided the opportunity to explore the potentialfor using only sparse information associated with a single sinusoid, as opposedto the totality of information contained in a PSD, to characterize the influenceof a mechanical fault. It should be noted that the above frequency of interestcorresponds to a number of component gear mesh frequencies, but not to thespiral bevel pinion gear mesh frequency, which is 1109 Hz. Moreover, the pinion-bearing theoretical defect frequency is at 311 Hz.

4. ANALYSIS

The top plots in Fig. 2 show the raw and band-pass-filtered time seriescorresponding to the no-fault and fault conditions. It is interesting to note thatthe influence of the fault is to enhance the modulation, while decreasing thepeak level of the sinusoid. Neither of these influences is to be expected, given thenature of the fault. In fact, the expected influence would be an increase of energyat the bearing defect frequency, which in this case is 311 Hz. However, the PSDplots in Fig. 2 reveal a decrease of energy at that frequency, and an increaseat a slightly lower frequency. Other than this very low-frequency region, theonly significant difference due to the fault is the peak values at approximately

FIG. 2. Plots of raw and filtered no-fault (left) and fault (right) data (top); PSD estimates of theabove data (bottom).


3150 Hz. The plots of the filtered data in this region show that there appear tobe two sinusoidal components in this region, regardless of the fault. The faultresults in a reduction of the stronger peak on the order of 15 dB.

Recall, however, that our goal is not to mathematically decompose a realsinusoid into true sinusoids, but rather to capture its time-varying amplitudeand frequency characteristics. To this end, we proceed now to investigate theconvergence properties of the MV(n) and AR(n) spectra as n goes to infinity. Forany fixed order, n, the nth order MV, or Capon spectrum, has been used (with asuitable scale factor) for estimating both line spectrum and continuous spectraldensity information. In [8] it was shown that the family of MV(n) spectraconverge monotonically to the line spectrum associated with sinusoids as theorder, n, goes to infinity. It was also shown that the corresponding AR(n) spectraconverge to infinity at the same frequencies, while converging to the continuousPSD at all other frequencies. The MV(n) and AR(n) estimated spectra for theno-fault and faulted raw data are shown in Fig. 3. For the no-fault data thesespectra clearly suggest the presence of a true sinusoid at approximately 3150 Hz.The same cannot be said of the spectra for the fault data. Even though theMV(n) spectra do not exhibit the asymptotic 3-dB drop between orders, aspredicted in [10], they also do not suggest convergence. The correspondingAR(n) spectra exhibit the same multiple-peak structure that the PSD in Fig. 2did. So it is possible that it is this increased modulation effect, relative to the no-fault data, that is responsible for the nonconvergence behavior of these spectra.

Since the spectra use lagged-product correlation estimates, it is also possiblethat the lack of exhibited convergence could be due, in part, to statisticalvariability associated with these estimates. While there has been some progress

FIG. 3. Estimates of the MV(n) (top) and corresponding AR(n) (bottom) spectra for the no-fault(left) and fault (right) raw data, using orders n = 20, 40, 80, and 160.


FIG. 4. MV(n) and AR(n) spectra for no-fault filtered data (top) and pinion-bearing fault data.(bottom); n = 20, 40, 80, and 160.

in obtaining the statistics of the AR(n) [11] and MV(n) [12] correlation-basedestimates for mixed spectrum random processes, our analysis here will notconsider this statistical influence due to the risks of distracting the reader frommore fundamental issues, and of a very lengthy and involved analysis. Recallthat the goal of this paper is to explore both the nature of real sinusoids andthe tools we have chosen to use for that purpose. We believe that the MV(n) andAR(n) spectral families have significant potential for that purpose. However,how they are used is equally important. For example, it is commonly held thatsuch spectra, for high enough orders, can capture all the important spectralinformation in a given bandwidth without the need for filtering. However,when attempting to take advantage of the convergence, as opposed to modelingcapabilities of the MV(n) and corresponding AR(n) spectra, there is a definiteadvantage in filtering. This is demonstrated in Fig. 4.

In particular, for the no-fault data the MV(n) spectra appear to havecompletely converged, as the n = 40, 80, and 160 spectra are identical. Thisstrongly suggests the existence of a true sinusoid. However, the correspondingAR(n) spectra do not exhibit the corresponding +3-dB asymptotic increase perorder doubling. Since the sinusoid at approximately 3150 Hz is not a truesinusoid, this contradictory behavior is not totally unexpected, especially inview of the PSD information in Fig. 2. There it is observed that while thereappear to be two closely spaced sinusoids for both the no-fault and fault data,the no-fault data have one very dominant peak. The fact that the MV spectrumhas resolving ability lower than that of the corresponding AR spectra wouldexplain this contradictory behavior. Specifically, at lower orders both spectrawould exhibit behavior consistent with a single sinusoid, while at higher orders


the AR spectra would actually reduce in magnitude with the emerging presenceof two peaks. This behavior of these spectra for the fault data is more obvious.Since both the PSD in Fig. 2 and the AR spectra in Fig. 4 suggest two sinusoidsof relatively equal power, it would appear to support our speculation regardingthe reason for the contradictory behavior of the MV and AR spectra. It is ourbelief that the convergence properties of the families of MV and AR spectra offersignificant potential for characterization of random processes involving realsinusoids. However, from the results in Fig. 4, as well as a similar contradictorybehavior noted in their use in the analysis of diesel vibration [1], it is apparentthat more research is needed to better understand their joint behavior forreal sinusoids. In order to attempt to capture the time-varying amplitude andfrequency behavior of the real sinusoid at approximately 3150 Hz, as opposedto modeling it as two true sinusoids, we used an extended Kalman filter (EKF),similar to that used in [1]. Specifically, the band-pass-filtered data were assumedto consist of a single sinusoid plus white noise. The sinusoid amplitude andfrequency were modeled as uncorrelated random walks. The choice of this modelis based, to a large extent, on our ignorance of their true behavior. As will beseen, however, it provides a sufficient characterization to allow more realisticmodels to be studied. A variety of model covariance values and band pass filterswas investigated. The results were extremely robust with respect to all of these.Furthermore, for both the no-fault and fault data the EKF model captured99.999% of the total energy in the data.

The time-varying amplitudes and frequencies for the no-fault and fault dataare illustrated in Fig. 5. There are clear differences in these time series for the

FIG. 5. EKF frequency (left) and amplitude (right) estimates for filtered no-fault (top) andpinion-bearing fault (bottom) filtered data.


FIG. 6. Histograms of frequency (left) and amplitude (right) estimates for no-fault (top) andfault (bottom) data.

no-fault versus the fault data. For example, the heavier frequency modulationbehavior associated with the fault data results in temporal regions whereinthe filtered data is close to 0. These regions cause the EKF to lose trackingability, and yield frequency estimates that are well outside of the actual rangeof activity, hence the very large excursions in estimated frequency. Theseexcursions are periodic, with a period corresponding to the modulation period ofapproximately 0.05 s. The frequency and amplitude histograms for the no-faultand fault data are shown in Fig 6. The frequency histogram for the fault datawas truncated in order to alleviate the frequency estimates related to poor EKFtracking. Both frequency and amplitude information is dramatically influencedby the presence of the fault. Recall that the frequency range of interest here hasno known relation to the characteristic frequency region associated with such afault. Nonetheless, such strong differences suggest that there may well be otherfrequency regions of equal, if not greater, ability to capture the presence of afault.

Scatterplots of frequency versus amplitude for the no-fault and fault dataare shown in Fig. 7. Again, there is a distinct difference between the no-fault and fault conditions. The no-fault condition reveals a mild negativecorrelation (−0.41) between amplitude and frequency. For the fault conditionthe correlation is almost zero (0.07). The very narrow range of frequenciesfor the real sinusoid shown in Fig. 6 results in the very peaked nature of thefault histogram. Even though the overall correlation between frequency andamplitude information is only modest for the no-fault data, and is essentiallyzero for the fault data, the coherence plots in Fig. 8 suggest that there are indeedcorrelations in specific frequency ranges.


FIG. 7. Amplitude versus frequency scatterplots for the filtered no-fault (left) and fault(right) data.

To further explore the time series structure of the frequency and amplitudeestimates provided by the EKF then MV(n) and AR(n) tools used to analyzethe measurement data were applied. The results are shown in Fig. 9. Onenotable result indicated in the MV spectra is the presence of a strong sinusoidalcomponent (0 dB) in the no-fault frequency time series at twice the frequencybeing studied. This component is essentially absent (−90 dB) for the faultcondition. In this same time series the AR spectra indicate a very clear differencein the continuous spectral structure between the no-fault and fault conditions.In particular, the no-fault frequency time series exhibits an AR(2) shape, witha spectral peak around 2000 Hz. For the fault condition it becomes an AR(1)

shape, and increases by 20 dB at lower frequencies. A closer look at thelow-frequency behavior of the spectra of Fig. 9 requires that the frequencyand amplitude time series be decimated in order to take advantage of theconvergence properties of the MV and AR spectra (c.f. [10] for a detaileddiscussion). These data were decimated by a factor of 10. Application of theMV and AR tools to this data resulted in the plots in Fig. 10. For the no-

FIG. 8. Magnitude-squared coherence estimates between frequency and amplitude estimatesfor the no-fault (left) and fault (right) data.


FIG. 9. MV and AR spectra for no-fault (top 4) and fault (bottom 4) EKF estimates.

fault condition the 20-Hz modulation behavior discussed above is revealed asstrong sinusoidal components at this same frequency in both the frequencyand amplitude MV (and AR) spectra. There is no evidence whatsoever of sucha periodicity in the frequency time series for the fault data, even though theamplitude time series for both the no-fault and fault conditions retains a 20-Hzperiodicity.


FIG. 10. MV and AR spectra for fault (top 4) and fault (bottom 4) EKF estimates.

5. SUMMARY AND CONCLUSIONS

The purpose of this effort was to investigate the potential of a combination ofsignal processing and basic statistical tools for characterization of real sinusoids.This was done in the context of sinusoids associated with no-fault and faultvibration data from a military helicopter power system. The fault addressed


was a pinion-bearing fault having a characteristic fault frequency of 311 Hz.Rather than investigating this frequency region it was decided to investigatethe region around 3150 Hz. In that region it was observed that not only werestrong sinusoids present, but that the bearing fault had a significant influenceof the data structure. For both the no-fault and fault conditions it was notedthat two sinusoidal components spaced 20 Hz apart were present, but thatthe fault resulted in a significant attenuation of one of the two. The 3150-Hzfrequency happens to be a gear mesh frequency. Because it is also the 10thharmonic of the fault frequency, it is possible that it is the 10th harmonicthat is responsible for the change in the spectral structure in this region.However, since it is a relatively high harmonic, and since the change was sosignificant, one might posit that there are other influences of the fault on thepower system vibration characteristics. While there is no direct support herefor such speculation, the investigation resulted in a number of very significantfindings. First, it was found that a model for a single real sinusoid, having time-varying amplitude and frequency, was able to completely characterize the band-pass data including the two theoretical sinusoidal components. Analysis of theamplitude and frequency time series provided some novel insight into the realsinusoid that goes well beyond a simplistic two-sinusoid model suggested bytraditional spectral analysis. For example, it was noted that even though theamplitude and frequency of the real sinusoid associated with the fault datawas close to 0, there was a strong correlation in specific frequency regions.It was also found that the frequency and amplitude data both had a strongperiodic component at approximately two times the frequency of interest,regardless of machine condition. While the same was true for the amplitudetime series in the range corresponding to the fault frequency, for the frequencytime series the presence of the fault all but eliminated any periodic behavior.Finally, both histogram and scatter plot information revealed such distinctdifferences between the no-fault and fault conditions that discernment of thesetwo conditions based on this information would be trivial. It is not the intent tosuggest that information related to a single real sinusoid should be used insteadof classic fault frequency information. However, our analysis suggests that afault may influence data in a far greater way than has been considered to date.

A second reason for the investigation of the real sinusoid work in this paperwas to begin to gain some appreciation for the effect of real sinusoids on signal-processing tools designed for idealized sinusoids. In particular, it was observedthat the convergence properties of the estimated MV(n) and correspondingAR(n) spectra can exhibit contradictory behavior, even for sinusoids with verylittle period variation. For example, the MV(n) plots for the no-fault conditionin Fig. 4 suggest, without question, convergence. But if this is the case, then thecorresponding AR(n) plots should be growing at the asymptotic 3 dB per orderdoubling. In Fig. 4 they exhibit no such behavior. This points to the need for astatistical analysis of the influence of small period variability on the convergenceproperties of the estimated AR(n) and MV(n) spectra.


ACKNOWLEDGEMENTS

This work was supported, in part, by the U.S. Air Force Office of Scientific Research, GrantF49620-98-1-0252.

REFERENCES

1. Sherman, P. J. and White, L. B., Periodic spectral analysis of diesel vibration data. J. AcousticSoc. Am. 98 (1995), 3285–3301.

2. Raup, D. M. and Sepkoski, J. J., Periodicity of extinctions in the geologic past. Proc. Natl. Acad.Sci. U.S.A. 81 (1984), 801–805.

3. Safek, E., Analytical approach to calculation of response spectra from seismologic models ofground motion. Earthquake Eng. Struct. Dyn. 16 (1988), 121–134.

4. Wikle, C. and Sherman, P. J., Using the family of MV spectra to determine periodicities inatmospheric data. J. Climate 8 (1995), 2352–2363.

5. Johnson, D., The application of spectral estimation methods to bearing estimation problems.Proc. IEEE 70 (1982), 1018–1028.

6. Myers, G. A., et al., Power spectral analysis of heart rate variability in sudden cardia death:comparison of other methods. IEEE Trans. Biomed. Eng. 33 (1986), 1149–1156.

7. Lau, S. S. and Sherman, P. J., The influence of period variation on time/frequency analysis ofthe Westland helicopter data. In Proc. 9th IEEE SP Workshop on Statistical Signal and ArrayProcessing, Portland, OR, September 14–18, 1998, pp. 180–183.

8. Foias, C., Frazho, A. E., and Sherman, P. J., A geometric approach to the maximum likelihoodspectral estimator for sinusoids in noise. IEEE Trans. Inform. Theory 34 (1988), 1066–1070.

9. Available at http://wisdom.arl.psu.edu/Westland/data/.10. Sherman, P. J. and Lou, K. N., On the family of ML spectral estimates for mixed spectrum

identification. IEEE Trans. Acoustics Speech Signal Process. 39 (1991), 644–655.11. Lau, S. S. and Sherman, P. J., Asymptotic statistical properties of the AR model for mixed

spectrum estimation. In Proc. IEEE ICASSP ’98, Seattle, WA, May 12–15, 1998, pp. 2289–2292.12. Liu, Xiao Hu and Sherman, P. J., Asymptotic statistical properties of the Capon MV spectral

estimator for mixed spectrum processes. In Proc. 9th IEEE SP Workshop on Statistical Signaland Array Processing, Portland, OR, September 14–18, 1998, pp. 328–331.

PETER J. SHERMAN received his Ph.D. in mechanical engineering from the University ofWisconsin—Madison in 1984. He was a faculty member in the area of systems and control in MEat Purdue University until 1991. Since 1991 he has held a joint appointment in the Departmentof Statistics and the Department of Aerospace Engineering and Engineering Mechanics at IowaState University. His research interests span te theoretical areas of systems, control, statistics andstochastic processes, signal processessing, and vibrations and acoustics. His applications interestsinclude machinery and human condition monitoring and, more generally, statistical dynamics inphysics, engineering, and human relationships.

on the statistical nature of real sinusoids associated with rotating machinery

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