bearing envelope analysis window selection using spectral kurtosis techniques

8/6/2019 Bearing Envelope Analysis Window Selection Using Spectral Kurtosis Techniques

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Bearing Envelope Analysis Window SelectionUsing Spectral Kurtosis Techniques

Eric Bechhoefer

Lead System Engineer

NRG SystemsHinesburg, VT 05461

Michael Kingsley

Mechanical Diagnostics andHealth Management Lead

Sikorsky Aircraft Corporation, Stratford, CT 06614

Praneet Menon

Mechanical Diagnostics & Tools Team Manager

Goodrich SIS

Vergennes, VT 05491

Abstract —Envelope Analysis is a well-known signal processing

technique for bearing fault detection. However, improper

window selection can result in poor fault detection performance.

Using a known fault data set, we quantify the performance of

spectral kurtosis (SK) and envelope kurtosis (EK) as a technique

for setting an optimal frequency and bandwidth window for the

envelope analysis. We establish a measure of effectiveness

(MOE): the correlation of fault energy with total spall length.

With this MOE, we evaluate the ability of SK/EK to predict the

optimal envelope analysis window.

Bearing Envelope Analysis; Spectral Kurtosis;

Freiquency/Bandwidth Selection; Inner Race Energy

I. I NTRODUCTION

Many condition based maintenance (CBM) systems employ bearing envelope analysis for the detection of bearing fault.This technique is preferable to simple order tracking or spectralanalysis in complex gearboxes, as the shaft rates and gear mesh

tones associated with the gearbox are orders of magnitudeslarger than the bearing frequency tones. Essentially, at the pointwhere a bearing frequency tone can be detected in spectralanalysis, the bearing is heavily damaged and it is likelycollateral damage has occurred to other components in thegearbox itself. A recent review of wind turbine gearbox failuresindicated that 20% of planetary gear failures were a result of oilcontamination from bearing failures [1].

Bearing envelope analysis (BEA) is an amplitudedemodulation technique, based on the high frequencyexcitation from quasi cyclo-stationary derived impulses due to bearing component fault. Industrial monitoring systems haveemployed these techniques since 1977 [2], with a number of

system improvements being implemented since then [3].The successful implementation of BEA is dependent on

proper selection of the frequency/bandwidth window. Anexample of sub-optimal window selection was reported in [4].After a teardown analysis of an oil cooler fan, the bearing wasfound to be damaged by corrosion. The installed health andusage monitoring system (HUMS) did not indicate any bearingfaults. Reprocessing of the time domain data recorded by theHUMS (moving the frequency/bandwidth window from 13-18

KHz to 20-25 KHz) resulted in fault detection and indicated aslow progression of fault over a period of months [5].

While post analysis is helpful, standardization of a methodor practice to optimally select a window is desirable. From a

certification perspective, the certifying body requires as aminimum, indirect evidence that the system will detect thefault. From an operations and maintenance perspective,logistics planning and profitability is directly related to areduction in unscheduled maintenance. Given the need to setwindow frequency/bandwidth without the experience of measuring from a damaged components, is there a way toselect better windows?

Boskoski and Urevc [6] used spectral kurtosis (SK) in order to determine the best window in order to determine thecondition of a test gearbox with seeded faults. This method of SK was first proposed by Dwyer [7] as a method to distinguish between transient and stationary sinusoidal signal with

Gaussian noise.We wish to quantify the performance of SK to find an

appropriate window by using a nominal bearing. Using SK inthe selection the frequency/bandwidth window, we apply thiswindow to a set of 9 bearings with measured damage. First, wedescribe the BEA algorithm. We then establish a measure of effectiveness (MOE), which allows one to quantify the performance of a given window against another window. TheMOE is evaluated using classical bearing defect calculationsfrom the BEA algorithm as well as Sikorsky AircraftCorporation’s average energy algorithm. Finally a comparisonof SK to envelope kurtosis (the kurtosis of the windowsenvelope prior to estimating the spectrum) is made.

II. THE E NVELOPE ALGORITHM

Bearing Envelope Analysis is based on demodulation of high frequency resonance associated with bearing elementimpacts. For rolling element bearings, when the rollingelements strike a local fault on the inner or outer race, or a faulton a rolling element strikes the inner or outer race, an impact is produced. These impacts modulate a signal at the associated bearing pass frequencies, viz. Cage Pass Frequency (CPF), BallPass Frequency Outer Race (BPFO), Ball Pass Frequency Inner Race (BPFI), and Ball Fault Frequency (BFF). This modulation



occurs through the addition of energy (from the impact) to thesignal, which also increases the measured RMS.

Additionally, the impact excites resonance at one or moreof the bearing fundamental frequency modes. This results inamplitude modulation of the resonant frequency in the timedomain.

Ideally, one wishes to filter around the bearing resonance.

Then the signal is enveloped by using the Hilbert transform(e.g. taking the spectrum of the amplitude of the analyticsignal). The analytic signal xa(t), is complex, composed of thereal part of the original signal, x(t), and the imaginary part,which is the Hilbert transform of x(t):

xa

t ( ) = x t ( )+ i H x t ( )( ) (1)

where

H x t ( )( )=1 2" x t ( ) t #$ d $ #%

%

& (2)

The Fourier transform of the analytic signal xa(t) is then:

X a f ( )= X f ( )+ i H x t ( )( )= {

2 X f

( ), f > 0

X f ( ), f = 0

0, f < 0

(3)

The imaginary part of the analytic signal is identical to the real

part, except for the phase being delayed by π/2. Note that the

spectrum of the analytic signal has spectrum only for the positive frequency of x(t). This definition allows the efficientcomputation of the Hilbert transform using the Fast Fourier Transform (FFT) to derive xa(t), then taking the inverse FFT or equation 3.

The envelope of analytic signal in (1) is calculated by:

a t ( ) = x t ( )2

+ H x t ( )( )2

(4)

The spectrum of the envelope is then the measure of bearingdefect at the appropriate passing frequency.

A. Implementation of the Envelope Algorithm

Window frequencies are obtained by band pass filtering. A band pass filter can be thought of the convolution of a low passfilter with a high pass filter. The length of the band pass filter,for a given low/high pass cut off is 2*n-1, or twice the length of the low pass filter by itself. This is important in filtering, whichin general is a n2 operation (even FFT filtering has an order of operation of n * log 2(n)).

Alternatively, consider the trigonometric identity:

cos a( )" cos b( ) = 12

cos a # b( )+ cos a+b( )[ ] (5)

By multiplying x(t) by an analytic signal with a frequency f ,which is the center of the BEA window and then low pass filter with a bandwidth equal to the bandwidth of the BEA, we canforego band pass filtering in lieu of the low pass filter. TheHilbert transformed signal is then:

cos ft ( ) x t ( )+ isin ft ( ) x t ( )" LPF = xa

t ( ) (6)

Multiplying x(t) by sin( ft ) changes the phase of x(t) by π/2, as

per (3). This also has advantages in that, after low passfiltering, one can decimate x(t) without aliasing and further reduce the order of operations.

B. Other Bearing Algorithms

While the envelope algorithm as described, measures theenergy associated with a given bearing component, one cansimilarly look at average energy as a general indicator of bearing health. For example, Sikorsky Aircraft Corporation hasimplemented an average of the spectral energy measurementthat is currently fielded in the S-92 HUMS system.Alternatively, one can measure the average energy of theenvelope. These two general bearing health algorithms, inaddition to the bearing fault energy (from the inner race), will be evaluated.

III. THE MEASURE OF EFFECTIVENESS

The data set consisted of ten bearings, the first being baseline/nominal, and the other nine having a notch punched

into the inner race to initiate a fault. Each of the nine bearingwas periodically run under load, then inspected, reassembledand run till the next inspection. A total of 5 inspections wereconducted on each bearing. Prior to each inspection, anacquisition was made (250,000 data points sampled at 100KHz). At each inspection, the total spall length was measured(figure 1).

Figure 1 Example of Spall Propagation

Figure 1 is an example of the propagation of the fault over

time. The red dot is the relative size of ball element.

For a prognostics health management system, not only mustthe system detect the fault, but it must measure some metricrelating to change of state, such as trend or condition indicator rate of change. In general, the measured energy at ball passfrequency inner race (BPFI) was 2 orders of magnitude larger for the faulted bearings then the baseline bearing. However, wewanted to reflect an MOE such that a large spall would berelated to large measured energies. It was observed that therewas a linear relationship between spall length and energy in anearlier study [8]. An ideal linear relationship would be:

BE = B0+ B

1*TSL (6)

where BE is the BPFI energy, B0 , is an offset value, B1 is ascale value, and TSL is the total spall length of the damage. Ameasure of this models fit is the correlation coefficient (r ).

A correlation coefficient (r ) of positive 1 would indicate alinear relationship between TSL and measured BPFI energy.For prognostics, this is advantageous as one could easilymeasure the rate of change of BPFI vs. time (or load) and



develop, given an appropriate fault model, an estimate of theremaining useful life. No correlation would be reflected by acorrelation coefficient of 0 and an inverse correlation would bereflected by a negative coefficient. As the correlationcoefficient is reduced from a positive 1 the prognostic model of remaining useful life becomes less reliable.

For each potential window, one can generate the correlationcoefficient observed over the five acquisitions/inspections (see

figure 2). In figure 2, the measured energies ( BE ) have beennormalized such the maximum value is 1. In general, BPFIvalues of 10

-5g/Hz were measured for the baseline bearing and

10-3 g/Hz values were measured for the damaged bearings(energy vs. power).

As an example, consider a window from 4 to 45 KHz.(figure 2). The correlation coefficient is r = -0.36 . After thesecond acquisition, the BE actually decreased with time. Thiswould result in a poor prognostic. Conversely, the 20.5 to 45KHz window resulted in a correlation coefficient of r = 0.99,almost a perfect linear relationship between damage and BE . Inan operational system, the assurance of a linear relationship between damage and measured condition indicator will give a

good prognostic.For the Sikorsky Aircraft Corporation’s average spectral

energy algorithm (ASE) the measure of effectiveness (MOE)was the correlation coefficient of the average energy vs. TSL.Similarly, the average envelope energy vs. TSL was evaluated.

Figure 2 Example of correlation as a function of BEA

Window

IV. THE K URTOGRAM

Kurtosis is a non-dimensional quantity that measures therelative “peakedness” of a distribution relative to the Gaussiandistribution. Spectral kurtosis (SK) is a statistical parameter indicating how the impulsiveness of a signal varies withfrequency. As noted, faults associated with rolling element bearings give rise to modulated impulses. The SK will belarge in frequency windows where the fault signal is dominantand small where the spectrum is dominated by stationarysignals. Antoni [9] developed the kurtogram, which is a mapindicating the optimum center frequency and bandwidthcombination.

The kurtogram of a nominal bearing was generated for bothspectral and envelope kurtosis. The filtering was limited between 4 KHz and 45 KHz (Nyquest was 50 KHz). The lower limit was chosen because it is our experience that below 4KHz, there is significant gear mesh tones which would bemeasured in the SK. We partitioned the frequency space intofive octaves, such that each frequency band was halved witheach increase in octave:

TABLE I. K URTOGRAM FREQUENCY MAP

Octave Bandwidth Number of Bands

PSDWindow

1 41 KHz 1 37768

2 20.5 KHz 2 16384

3 10.25 KHz 4 8192

4 5.125 KHz 8 4096

5 2.5625 KHz 16 2048

6 1.2812 KHz 32 1024

One additional analysis issue is the selection of power spectral density (PSD) window and overlap length. As the bandwidth degreased, the PSD window length was decreased proportionally. This was done in order to have a frequency binresolution that was constant: approximately 1 Hz. This alsoresulted in 12 averages for the PSD for each octave/filter. Seefigure 3 for SK kurtogram and figure 4 for the envelopekurtosis (EK) kurtogram. Each bearing resulted in64experiments.

Figure 3 Spectral Kurtosis kurtogram of the Nominal

Bearing

Figure 3 indicated that the lower frequency windows of 4 to45 KHz (SK1) and 24.5 to 45 KHz (SK2) have the highestkurtosis, and would be the best windows for BEA. The EK kurtogram interestingly showed a different behavior (figure 4).

SK1

SK2



Figure 4 Envelope Kurtosis kurtogram of the Nominal

Bearing

It is interesting that the best spectral bands SK are sodifferent than that for EK. The two best bands for SK are the

base band (e.g. 4 to 45 KHz, centered on 24.5 KHz, and thesecond octave (24.5 to 45 KHz, centered on 34.75 KHz). The best bands for the EK where: 14.25 to 24.5 KHz (EK1), 24.5 to34.75 KHz (EK2), and 30.9 to 33.46 KHz (EK3). This wasthen compared to the measured correlation coefficient of inner race envelope energy to TSL. Figure 5 is an example of thecorrelation coefficient for Bearing 1. Pictorially it can be seenthat the SK kurtogram did not match the correlation coefficientas well as the EK kurtogram. For example, the 4-45 Khz(SK1) window showed disappointing results and wouldindicate the SK method did not produce a subset of reliablewindows to utilize. Based on this it’s clear the SK ismeasuring a different feature than EK in the baseline bearing.

Figure 5 Correlation of Inner Race Envelope Energy to

Total Spall Length

For each bearing, the correlation coefficient for each bearing was evaluated at each of these five windows,calculating the mean and median response for the inner race

envelope energy (table II), and the average envelope energyand ASE (table III).

TABLE II. SK AND EK R ESPONSE FOR 5 FREQ. BANDS: I NNER R ACE

E NVELOP E NERY

Band 4-45 24.5-45 14.2-24 24-35 31-33

Kurtosis SK1 SK2 EK1 EK2 EK3

Brng1 0.08 0.93 0.96 0.97 0.92 Brng2 0.29 0.87 0.58 0.00 0.28 Brng3 -0.36 0.99 0.91 0.94 0.75 Brng4 0.61 0.85 0.81 0.87 0.77 Brng5 -0.20 0.97 0.78 0.93 0.32 Brng6 -0.55 0.75 0.42 0.48 0.51 Brng7 -0.56 0.69 -0.83 -0.92 0.87 Brng8 -0.37 0.12 0.48 -0.32 0.38 Brng9 0.91 0.82 0.28 0.75 0.72 Mean -0.02 0.78 0.49 0.41 0.61 Median -0.20 0.85 0.58 0.75 0.72 Table II suggests that the BE measurement based on inner race

envelope energy is a stochastic process. But where there is

some between bearing measurement, there are definite trends.For example, the 24.5-45 KHz (EK2) window resulted in a

mean correlation coefficient of 0.78. The high frequency

window (31-33 KHz) showed similar performance to the 24.5

to 45 KHz window, which is consistent with a bearing

resonance in the range of 31 to 33 KHz.

TABLE III. SK AND EK R ESPONSE FOR 5 FREQ. BANDS: AVERAGE

E NVELOPE E NERGY AND AVERAGE SPECRAL ENERGY (SAC ALGORITHM)

Band 4-45 24-45 14 -24 24-35 31-33

Kurtosis SK1 SK2 EK1 EK2 EK3

Avg

Eng

Brng1 0.97 0.97 0.97 0.99 0.93 0.96

Brng2 0.99 0.99 0.81 0.95 0.86 0.87

Brng3 0.89 0.98 0.93 0.98 0.93 0.95Brng4 0.77 0.87 0.83 0.92 0.86 0.84

Brng5 0.94 0.97 0.85 0.95 0.91 0.86

Brng6 -0.46 0.97 0.31 0.85 0.74 0.18

Brng7 -0.04 0.95 0.58 0.97 0.99 0.75

Brng8 0.85 0.97 0.92 0.93 0.88 0.51

Brng9 0.86 0.84 0.90 0.97 0.93 0.71

Mean 0.64 0.95 0.79 0.95 0.89 0.73

Median 0.86 0.97 0.85 0.95 0.91 0.84

V. DISCUSION OF R ESULTS

The results highlight the variability between bearings. For

example, bearing 7 has both negative and positive correlationcoefficients, depending on the frequency band. As noted, theSK gave different results than the EK, suggesting that theanalysis techniques are measuring different features. For example, the difference in SK value for the best and second to best band is less than 1%. However, the MOE results for SK changed from mean r of -0.2 to r = 0.78. The differences between the MOE results of the EK appear more consistent andappear to be a better tool for guiding the choice of thewindows.

SK1

EK1 EK2

EK3

SK2

EK1 EK2

EK3



The SK and EK gave similar results for the averageenvelope energy. The ASE algorithm performed exceedinglywell with the exception of the 4-45 Khz (SK1) window. Thevariability seen with the BE based on inner race envelopeenergy were significantly reduced. It is tempting to use theaverage energy algorithms exclusively. However, for complexgearboxes, gear mesh or gear clash can excite the higher frequencies spectral, which could potentially cause elevatedaverage envelope or ASE energies. Unless the energy windowcan be modified accordingly this would cause a false alarm.Another item to consider is if isolation of which bearing isdegraded is required the ASE algorithm does not providesufficient information to accomplish isolation for complexgearboxes with multiple bearings. A hybrid of the twotechniques should be evaluated. For example a technique thatutilized ASE algorithm and BPFI for modeling remaininguseful life could be developed to highlight strengths of twotechniques and compensate for each of the algorithmsindividual inherent weaknesses.

The results suggest that EK gives more consistent performance than SK. Some considerations for this suggestionare:

• The EK frequency bands were smaller, allowing theuse of a shorter FFT length for analysis. For anembedded application, this is a significant advantage.

• While the mean correlation coefficient was higher for the SK, the variance in the results for EK was lower,representing a reduction in the probability of misseddetection.

• The first window (SK1) recommended by SK technique missed the better windows that existed in thehigher octave bands of kutogram.

A. Artificiality of the Testing or Biasing of Results

In general, real application of condition monitoring will nothave intermediate inspection and teardowns of the gearbox. Itis understood that no matter how careful the bearing inspectionwere, there will be difference in alignment/buildup of the bearing after each inspection. These cannot but effectsubsequent acquisition values (although, significant time wasallowed between each acquisition to mitigate these effects).Additionally, the general level of damage is high. Dependingon the definition of failed (we take this to be a spall the lengthof the roller element), the bearings were in a failed state after the 3

rdinspection. It would be preferable to do similar testing

on bearings with less damage, as the results may be less linear.

The condition monitoring for these bearing was almostideal. The accelerometer was mounted directly on the bearing

cap, and there was no other rotating equipment other the shaftdriving the bearing. In a complex gearbox, the interaction of multiple gears, shafts, bearing etc, could generate differentresults.

B. Application of Envelop Kurtosis

In general the best technique for selecting bearing windowsfor analysis is through utilization of seeded fault data. Asdemonstrated in this paper one needs to evaluate the correlationcoefficient of the degraded condition to the selected bearing

window. Additional analysis to consider is the magnitude of separation from healthy to faulted data, as well as, magnitudeof separation of selected window is from the noise floor of thesensor. Both of these will drive the probability of detection and probability of false alarms. However, it is common to not havesufficient seeded fault data to analytically evaluate bearingwindows for all dynamic components. Seeded fault data can be both costly and time consuming to capture. With the lack of relevant seeded fault data the picking of the appropriatewindow for bearing analysis becomes a challenge. While nota perfect methodology, the use of Envelope Kurtosis to selectwindows is a tool that should be considered when seeded faultdata is not available to help guide the selection of windows for bearing analysis.

R EFERENCES

[1] Maintenace and Management wind power. (references)

[2] Randall, R., B., “Frequency Analysis” Bruel & Kjaer, 1977.

[3] Carter, D., “Rolling Elemetn Bearing Condition Testing Method andApparatus”, U.S. Patent 5,477,730, Dec 24, 1995.

[4] McCain,B.,“TearDownAnalysis(TDA)for the UH-60 Axial Fan,”

CSTE-DTC-RT- E-CS, 4 Aug 2008.[5] Bechhoefer E., Menon, P., “Bearing Envelope Analysis Window

Selection”. Annual Conference of the Prognositics and HealthManagement Society, 2009.

[6] Boskoski, P., Urevc, A., “Bearing fault detection with application to the

PHM Data challenge”, Annual Conference of the Prognositics andHealth Management Society, 2009.

[7] Dwyer, R. “Detection of non-gaussian signals by frequency domainkurtosis estimation”. Acoustics, Speech, and Signal Processing, IEEEInternational Conference on ICASSP, 8:607–610, 1983.

[8] Bechhoefer, E., He, D., “Bearing Damage Condition Indicator Correlation,” Center for Rotorcraft Innovation Project: 07-B- 6-59-S2.1

[9] Antoni. J., Randall, R., “The Spectral Kurtosis: Application to theSurveillance and Diagnostics of Rotating Machines”, Mechanical Systems and Signal Processing submitted for publication, 2004.

[10]

Sawalhi, N., Randall, R., “The Application of Spectral Kurtosis toBearing Diagmostics” Proceeding of Acoutics 2004, 3-5 November 2004, Could Coast, Austrlia.

Eric Bechhoefer is the lead system engineer at NRGSystems. A former naval aviator, Dr. Bechhoefer recently joined NRG from the aerospace industry. Dr. Bechhoefer has16 patents and over 50 papers related to condition monitoringof rotating equipment.

Michael Kingsley has 14 years of experience with UTCdeveloping diagnostic and prognostic systems. His backgroundincludes 10 years at Pratt & Whitney where he was mostrecently the Vibrations IPT Lead for the Joint Strike Fighter program. Previous to becoming the Vibrations IPT Lead,

Michael’s career spanned roles of leadership in the test,operability, and software groups. He is currently theMechanical Diagnostics and Health Management Lead for Sikorsky. Mike has achieved both a Bachelor of Sciencedegree and a Master of Science degree in ElectricalEngineering. His undergraduate work was performed at theUniversity of Michigan. He received his graduate degree fromRensselaer Polytechnic Institute.



Praneet Menon works at Goodrich, SIS as an EngineeringTeam Manager and Mechanical Diagnostics Lead. His teamfocuses on Mechanical Diagnostics R&D efforts as well asdevelops analytical tools. Currently his research focuses onseeded fault testing for various gear and bearing faults and will progress towards gear CI-computation verification and

generation of new CIs using advanced signal processingtechniques. He has his BS in Aeronautical & MechanicalEngineering from Rensselaer Polytechnic Institute and iscurrently pursuing his Masters in Mechanical Engineering atthe University of Vermont.

bearing envelope analysis window selection using spectral kurtosis techniques

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