Detection and Estimation of Helicopters Vibrations by

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Detection and Estimation of Helicopters Vibrations byAdaptive Notch Filters

Antoine Monneau, Nacer M’Sirdi, Sébastien Mavromatis, Guillaume Varra,Marc Salesse, Jean Sequeira

To cite this version:Antoine Monneau, Nacer M’Sirdi, Sébastien Mavromatis, Guillaume Varra, Marc Salesse, et al.. De-tection and Estimation of Helicopters Vibrations by Adaptive Notch Filters. 17th International Con-ference on Informatics in Control, Automation and Robotics, Jul 2020, Online Streaming, France.�10.5220/0009910302010207�. �hal-02936935�

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Detection and Estimation of Helicopters Vibrations by Adaptive NotchFilters∗

Antoine Monneau1,2, Nacer K. M’Sirdi1 a, Sebastien Mavromatis1, Guillaume Varra2, Marc Salesse2

and Jean Sequeira1

1Aix Marseille Universite, Universite de Toulon, CNRS, LIS, LIS UMR CNRS 7020, Marseille, France2Airbus Helicopters, Marignane, France

Keywords: Adaptive Spectral Analysis, Vibration detection, Adaptive Notch Filters, Aerospace Systems.

Abstract: This paper addresses online vibration detection in helicopters using Adaptive Filters. Adaptive Notch Filters(ANF) are used to estimate and track the time varying frequencies of the vibrations. We estimate and track theamplitudes and phases of time varying frequencies of the vibrations. This allows the detection of abnormaloscillations in the helicopter flight to keep control of the aircraft. In the application presented, we show thedetection of severe vibrations that occurred during a helicopter flight test. This proves the effectiveness ofproposed ANF to track and reject narrow band perturbations.

1 INTRODUCTION

1.1 Mechanical Vibrations ofHelicopters

Within helicopters structure, the excitation’s stressesare relatively important considering the mass of thefuselage and its flexibility. The large number of ro-tating parts aboard a helicopter generates vibrationsthat can be fed back into the flight loop by the aircraftflight control system (AFCS) (see in (NTSB, 2018)).For example, the main rotor of an Airbus H125 rotatesat 390 rpm, which generates vibrations of frequency6.5Hz and 19.5Hz within the air-frame.

When excitation go near to natural frequencies ofthe helicopter, this may lead to constraints in the me-chanical parts and increase the vibrations. The in-creasing need of comfort on board helicopters leadmanufacturers to develop anti-vibration systems. Theconsequences of vibrations aboard a helicopter canrange from the discomfort of the crew to the completedestruction of the aircraft.

The last case occurred on July 6, 2016 with thefirst prototype of the Bell 525 Relentless. Accord-ing to the US National Transportation Safety Board(NTSB), the in-flight breakup of the aircraft was

a https://orcid.org/0000-0002-9485-6429∗This work is supported by the SASV of the LIS (UMR

CNRS 7020) and Airbus Helicopters.

18161412

Time [s]108642

14

12

10

Fre

quency [H

z]

8

6

4

2

0

0.02

0.04

0.06

0.08

0

Pow

er

Spectr

al D

ensity [abs]

Figure 1: Power Spectral Density of the helicopter lateralacceleration ΓY pilot .

caused by severe vibration during a flight test at 185 kt(342 km/h), (see figure 1 (NTSB, 2018)). The vibra-tion at 6Hz was so high in the cockpit (17mm of dis-placement with a vertical load factor up to 3g) that thepilots were probably unable to see the control panelnor exit the flight case. Moreover, vibrations causedunintentional vertical control inputs by the pilot thatfurther amplified the phenomenon. A secondary feed-back loop was set by the aircraft’s attitude and head-ing reference system (AHRS). This system attempted

Monneau, A., M’Sirdi, N., Mavromatis, S., Varra, G., Salesse, M. and Sequeira, J.Detection and Estimation of Helicopters Vibrations by Adaptive Notch Filters.DOI: 10.5220/0009910302010207In Proceedings of the 17th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2020), pages 201-207ISBN: 978-989-758-442-8Copyright c© 2020 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

201

to correct the airframe’s vertical vibration, however itresponded to the 6Hz vibration and exacerbated ro-tor blades modes. While the rotor was losing speed,the excessive flapping of the blades became uncon-trollable and caused the main rotor to strike the heli-copter’s tail boom. This led to the crash of the aircraft.

Passive systems such as suspensions or active con-trol systems help to decrease significantly the level ofvibration. The design of such anti-vibration systemsis based on mechanical models of the structure fordamping oscillations near to the natural frequencies.The perturbation can be filtered based on the mechan-ical features (see (Krysinski and Malburet, 2007)).Passive damping and compensations can also be con-sidered.Sensors like accelerometers capture these vibrationsand using feedback stabilization or classical con-trollers generates controls which contain these oscil-lations. As a consequence, vibrations can be ampli-fied by the closed loop. The excitation must also bemodeled, which is not an easy task for aerodynamicprocesses, but it can help to design active vibrationcompensation approaches.

In this paper we focus on fast detection and robustadaptive estimation of the perturbation, for its com-pensation. After a spectral analysis of the vibration(measured by accelerometers) on board helicopters,the first conclusion was that vibration can appear sud-denly (see figure 1) with a narrow spectrum with andchanging frequency.

1.2 Objective and Contribution

We propose the use of a narrow band signal modelto describe the perturbation like in (Nehorai, 1985)or in (M’Sirdi and Landau, 1987). A notch filtercan be considered for perturbation detection and fil-tering (see figure 2). We propose the use of a band-width monitored in an Adaptive Notch Filter. Thenwe propose an Adaptive Narrow Band Signal predic-tion method to perform online fast detection of thevibration frequency. This will generate an alarm andwill track the time varying frequency.

This allow us to develop an efficient and very fastconverging frequency estimation algorithm and adap-tive filter (see figure 3).

It is worthwhile to note that the notch bandwidthdepends on the parameter 0 < r < 1. When r goesnear to one the notch becomes very narrow. for es-timation, we will start with r small (wide notch) andmake it go to one for a narrow notch.

The third and last contribution, of this paper, isthe estimation of the amplitudes and phase of the vi-bration in order to allow its compensation. The dis-

Normalized frequency f/fs

0 0.1 0.2 0.3 0.4 0.5

Am

plit

ude [dB

]

-50

-40

-30

-20

-10

0

10

r=0.5r=0.7r=0.8r=0.9r=0.95

Figure 2: Frequency response of notch filter Hi centeredon the normalized frequency f

fs= 0.2 with different notch

bandwidth parameter r.

crete transfer function representation do not catch theinformation of the signal power or amplitude. So itis necessary for detection to estimate either the signalpower (output of the ANF) or the amplitude and phaseof each frequency component.

This is developed assuming not only one but sev-eral frequencies in the vibrations. Note that the pro-posed Adaptive Notch Filters may be cascaded to de-tect and estimate several vibration components.

This paper is organized as follows. After this in-troduction, section 2 is devoted to some related previ-ous works in literature and background definition.

In section 3, we present the use of Adaptive NotchFilters (ANF) for online frequency estimation using arecursive maximum likelihood (RML) adaptation al-gorithm. This is followed by a recursive least squares(RLS) estimation of the amplitudes and phases of thevibration components. Section 4 presents an applica-tion of the proposed ANF and amplitude and phasesestimation (APE) method on real data acquired froman helicopter flying in critical vibration conditions.This study emphasizes the interest of using the pro-posed method based on ANF for online spectral es-timation of vibrations and shows the effectiveness ofthe proposed methods.

Figure 3: Frequency estimation using ANF.

ICINCO 2020 - 17th International Conference on Informatics in Control, Automation and Robotics

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2 PREVIOUS WORKS

Vibration spectral analysis is a commonly used toolin the field of industrial rotating machinery. In fact,with online processing of vibration signals, it is pos-sible to extract the current status of the machine. Incase of fault, the machine produces distinctive vibra-tion patterns that can be compared with those of ref-erence, thus enabling the fault detection, see (Bettaet al., 2002).

The characteristic patterns of vibratory signals areoften represented in the frequency domain. Mostspectral signal analyzer uses the fast Fourier trans-form (FFT) to extract the frequency characteristics ofthe vibratory signal. An important limitation to theuse of the FFT is its application to non stationary sig-nals. The windowing of the signal over a fixed periodmakes it impossible to obtain the correlation betweentime and frequency components.

If the frequencies vary significantly during this pe-riod, the FFT will generate an error for building thesignal spectrum. Varying frequencies are frequent ona helicopter, for example when rotors are speeding up.

The wavelet transform is a powerful alternative tothe Fourier transform, which is adapted to the studyof non-stationary signals. In particular, it is able toperform local spectral analysis over any time inter-val (zoom) without losing its frequency information.An application using continuous wavelets transform(CWT) to the study of vibrations of rotating machinesis presented in (Al-Badour et al., 2011) and (Qin et al.,2011).

An auto-regressive model can be used to charac-terize the vibrations of a mechanical assembly. Forexample in (Wang and Wong, 1986), an AR filter isset to output a low residual signal during nominal op-eration of a helicopter transmission box. When a faultoccurs, for example when a tooth fatigue crack devel-ops, the residual signal increases because of the vari-ation of the vibration spectrum. In fact, the AR filteris no longer centered on the nominal frequencies.

3 REAL TIME SPECTRALANALYSIS

Mechanical vibrations in a helicopter may resultsfrom numerous rotating parts that generate oscillatorymotions. This appear when a transmission shaft isunbalanced or the pressure on the blades of the heli-copter varies alternately.

3.1 Frequency Components Estimationand Tracking

The signals describing the vibrations can be seen as asum of sinusoidal components with time varying fre-quencies f i(k), amplitudes Ci(k) and phases βi(k) :

yk =

n

∑i=1

Cik sin(2π f i

kTsk+βik) (1)

Counteracting these oscillatory signals in noise relieson accurate online detection by means of estimationof their time varying parameters. The recently pro-posed adaptive identification algorithm deduces thefrequency estimation of narrow band signals basedon Adaptive Notch Filters (ANF), see (M’Sirdi andal., 2018). Online amplitudes and phases estimationare made using Weighted Recursive Least-Squares(WRLS) algorithm on a Fourier series decompositionof the signal.

3.2 Frequencies Estimation

Adaptive Notch Filters are well known to be very ef-ficient for extracting frequencies of signals composedof sinusoidal components, see (M’Sirdi and Landau,1987). For example, the following second order ANFHi(z−1) is proposed to catch the ith sinusoidal compo-nent (frequency f i) of a given signal:

Hi(z−1) =1+aiz−1 + z−2

1+ raiz−1 + r2z−2 (2)

• ai = −2cos(2π f iTs) is the notch filter parameterwith Ts as the signal sampling period

• 0 < r < 1 the notch bandwidth parameterFigure 2 shows the amplitude of the frequency re-sponse of Hi with different r. A small value of thisparameter is associated with a large filter bandwidthwhereas r close to 1 is associated with thin filteringfor an accurate frequency detection.

The ANFs are cascaded when there are severalsinusoidal components to be removed. The transferfunction can be written ∏

pi=1 Hi(z−1) with i ∈ [1; p],

for p components to estimate. When the ANFs haveconverged, each cell will remove one component.When using the ANFs cascaded of equation:∏

pi=1,i 6= j Hi(z−1) with i ∈ [1; p] and i 6= j, all sinu-

soidal components are removed except the one of thefrequency f j.

Consequently, the remaining signal noted y jk is

written:

y jk =

p

∏i = 1i 6= j

1+aiz−1 + z−2

1+ raiz−1 + r2z−2 · yk (3)

Detection and Estimation of Helicopters Vibrations by Adaptive Notch Filters

203

Filtering of the remaining signal y jk with the last notch

filter H j(z−1) will give us the prediction error for theestimation of the frequency component f j:

εjk = H j(z−1)y j

k =1+a jz−1 + z−2

1+ ra jz−1 + r2z−2 · yjk (4)

Minimization of this prediction error εjk will lead to

estimate the error gradient:

ψjk−1 =−

dεjk

da j=

(1− r)(1− rz−2)

(1+ ra jz−1 + r2z−2)2 · yjk−1 (5)

Real time implementation of the frequency estimationleads to use the following Recursive Maximum Like-hood (RML) algorithm for an output Prediction ErrorMethod (PEM):

f or j = 1, ..., p do1) compute the prediction ε

jk

2) a jk = a j

k−1 +F jk−1ψ

jk−1ε

jk

3) F jk =

F jk−1

(λ+ψjk−1F j

k−1ψjk−1)

(6)

for each time instant k, where:

• a jk =−2cos(2π f j

k Ts)

• F jk is the adaptation gain

• 0 < λ < 1 is the forgetting factor

Several second order notch filter cells are applied in acascaded way to removed the frequency componentssuccessively from the prediction equation error. Theimplementation structure is shown on the figure 4.The cascaded cells are adapted in a recursive manner.In fact, the current cell input is the output predictionerror of the previous ones.

3.3 Tracking Capabilities

The notch filter bandwidth parameter rk is time vary-ing: starting from a value r0 to r f according to thefollowing expression:

rk = rdrk−1 +(1− rd)r f (7)

0 < rk < 1 defines the position of the filter poles alongfrequency radials in the z plan. rk ≈ 0 means thatpoles are close to the origin whereas rk ≈ 1 meansthat poles are close to the unit circle (narrow band-width). rd is the exponential coefficient of the filterbandwidth evolution, it is typically 0.99. The conver-gence and performance of frequency estimation us-ing ANF are developed in (M’Sirdi and al., 2018) and(M’Sirdi et al., 1988).

Figure 4: Frequencies estimation stage of ANF algorithm.

3.4 Amplitudes and Phases Estimation

Once the frequencies of the signal components f i areknown or estimated, we can use a Weighted Recur-sive Least Squares (WRLS) to estimate amplitude andphase of each component.

The signal defined in equation 1 can be decom-posed in a Fourier basis as follows:

yk =

p

∑i=1

[gik cos(2π f i

kTsk)+hik sin(2π f i

kTsk)]+vk (8)

where Cik =

√gi

k2+hi

k2 is the amplitude of the ith

frequency component and βik its phase, at time k

(tan(βik) = gi

k/hik).

The parameter vector θk and regression vector Φk aredefined as follows:

θk =[g1

k g2k ... gp

k h1k h2

k ... hpk

]T and Φk = [Ck,Sk]T

Ck =[cos(2π f 1

k Tsk) · · · cos(2π f pk Tsk)

]Sk =

[sin(2π f 1

k Tsk) · · · sin(2π f pk Tsk)

](9)

The Fourier parameters gi and hi are estimated usingthe WRLS:

ε0k = yk− θT

k−1Φk

Gk =1

λ0

(Gk−1−

Gk−1ΦTk ΦkGk−1

λ0+ΦTk Gk−1Φk

)θk = θk−1 +GkΦkε0

k

(10)

where ε0k is the a priori prediction error, Gk the adapta-

tion gain and λ0 the exponential forgetting factor typ-ically chosen between 0.98 and 0.995.

The exponential convergence of this algorithm hasbeen proved, provided that the data are informative(persistent excitation condition). The proof uses theresults in (Johnson and col., 1982) and (M’Sirdi andal., 2018). This means the existence narrow bandcomponents in the signal.


204

4 APPLICATION

In order to test the performance of the online spectralestimation algorithm, we apply the signals measuredon board of a prototype helicopter during a flight test.The objective is to determine if the algorithm is ableto detect the predominant frequency (6Hz) and its as-sociated amplitude.

4.1 Helicopter Lateral Vibration Case

We focus here on the lateral acceleration signal mea-sured by the accelerometer placed under the pilot’sseat (ΓY pilot ). This location is one of the most sensi-tive to vibrations, it is far from the center of gravityof the helicopter and subjected to the lateral bendingmode excitations. This mode of the fuselage includesleft/right yawing motion of the main rotor mast andlateral motion of the forward cabin. The motions ofthe airframe are illustrated in figure 5, that shows anoverlay of the fuselage at its minimum and maximumdisplacement for the first lateral bending mode. Theforward part of the cabin (which includes the pilotseat and the attitude heading reference system) seesmainly 6 Hz right-left lateral motion.

Figure 5: Helicopter first lateral bending mode (exaggeratedfor comprehension).

When looking at the evolution of the spectrum of lat-eral acceleration over time on figure 1, no significantvibration is present on the first seconds of the flighttest record. From 6s, we observe a small oscilla-tion varying around 8Hz. At 13s a 6Hz vibration isperceived and its amplitude increases strongly untilreaching 0.5g at 16s. Then the vibration decreasesbecause the test pilot changed the flight case to avoiddamaging the structure of the helicopter.

It is certain that the pilot felt this vibration fromthe 13th second, given its amplitude and frequency.At this moment, the problem of the crew was nolonger the detection of the anomaly but to leave the

Figure 6: Helicopter first vertical bending mode (exagger-ated for comprehension).

current flight case. Given the low amplitude of thevibrations (3mm of amplitude at 6Hz), the pilot wasable to recover. But in some extreme case, the severityof vibrations prevents the pilot to act on the controls.Only automatic system could recover the helicopter tosafe flight conditions.

Two automatic actions could be envisaged:

• Exit the flight case after the detection of severevibrations (for example: automatic action on thecontrols to increase the rotational speed of themain rotor)

• Filter the AFCS output controls to eliminate theabnormal oscillations amplified by the feedbackloops.

We present in the results the online spectral analy-sis of the lateral acceleration ΓY pilot performed withadaptive notch filters. Severe lateral vibrations can bedetected when the amplitude of the ΓY pilot frequencycomponents exceeds a given threshold. In parallel allcontrols sent to actuators can be analysed, in partic-ular the yaw control δped (tail rotor collective pitchcontrol). If a strong vibration is detected on ΓY pilotand its frequency is present in the controls spectrum,it could be filtered using the associated notch filter.

The detection and tracking of significant oscilla-tions and the estimation of their frequencies must befast enough to be able to filter them efficiently. Theadaptive notch filter method is particularly suited tothe filtering of unexpected vibration since it directlygives the notch filter to be used and thus eliminate thepropagation of corresponding oscillations in the con-trol loop.

4.2 Methods Setting

ANF. The parameter estimates a j are set to zero ini-tially. For the ANF bandwidths, r0 is 0.5, and r f ischosen such that the poles of Hi are as close as possi-


205

Time [s]

0 2 4 6 8 10 12 14 16 18 20

Lateral A

ccele

ratio

n [g]

-0.8-0.6-0.4-0.2

00.20.40.60.8

ΓY pilot

Amplitude estimate a1Amplitude estimate a2

Time [s]0 2 4 6 8 10 12 14 16 18 20

Frequencie

s [H

z]

0

2

4

6

8

10

12

14 Frequency estimate f1

Frequency estimate f2

Time [s]0 2 4 6 8 10 12 14 16 18 20

AN

F O

utput [g]

-0.1

-0.05

0

0.05

0.1ANF Output ǫ

Figure 7: Online spectral estimation of helicopter lateralacceleration ΓY pilot .

ble to the unit circle. We use r f = 0.85. We choosep = 2 to study the frequency content of ΓY pilot andp = 3 for δped .WRLS. The initial value of the adaptation gain is setto a large value, typically G0 = 100. The forgettingfactor λ0 is set to 0.99, and the parameters vector θ(0)is initially set to 0.

4.3 Results

We present on figure 7 the online spectral analysisof the lateral acceleration signal ΓY pilot . The secondcurve of this figure depicts the frequencies estimate ofthe two main oscillations using ANF algorithm. We

Time [s]

0 2 4 6 8 10 12 14 16 18 20

Yaw

contr

ol [%

]

2

4

6

8

10

12

14

Yaw control δped measured

Yaw control δped filtered by ANF

Time [s]

0 2 4 6 8 10 12 14 16 18 20

Frequencie

s [H

z]

0

2

4

6

8

10

12

14Frequency estimate f1



Figure 8: Online spectral estimation of helicopter yaw con-trol δped .

observe that the first frequency estimate f1 is vary-ing until 6s where it converges to 8.2Hz. In parallel,the weighted recursive least squares algorithm esti-mates the amplitude of this first oscillation using thecurrent frequency estimate. Amplitude estimate a1 isshowed in green on the first curve, it never exceeds0.1g. A second frequency band is tracked by f2 andconverges to 6Hz from the 13th second. The ampli-tude estimate a2 of this frequency band shows a rapidincrease from 14s to 16s when it reaches 0.28g. Thethird curve represents the output of the filter, we ob-serve that it gradually increases from 11s to 16s, thisis due to the presence of others frequency componentsthan 8.2Hz and 6Hz. In fact when looking at the spec-trum evolution, other frequency components appearfrom 11s at higher frequencies. This is due to mainrotor modes that develops during this high dynamicalphase. These frequencies are not estimated by the twonotch filters, they appear as error. This spectral analy-sis can be used to detect exaggerate vibration and exitthe current flight conditions. For example, at 15s, theinformation of exceedance of 0.2g at 6Hz can be sendto the autopilot computer which would act on the ac-tuators to change the main rotor states and thereforedecrease the level of vibration.

We present on figure 8 the online spectral analysisof the yaw control signal δped . The second graphicof this figure shows the 3 frequencies estimates ofthe signal. Note that the 6Hz oscillation (in red) is


206

present from 11s. The first graphic shows δped mea-sured (in blue) and its filtered counterpart (in red) us-ing the notch filter centered on the second frequencyestimate f2. In practice, we could trigger the filter-ing of 6Hz at 15s after severe vibration detection onΓY pilot .

5 CONCLUSION ANDPERSPECTIVES

Online estimation of the spectral component of vi-brations has been proposed based on use of AdaptiveNotch Filters. Frequency tracking is done using a Re-cursive Maximum Likelihood (RML) algorithm andAmplitudes and Phases Estimation (APE) adapted bya Weighed Recursive Least Square (WRLS) algo-rithm. This method allows on estimation and trackingof the narrow band frequencies and their amplitudesand phases.

The main interest of the ANF is the fast trackingof time-varying frequencies and allows to get directlyaccurate prediction of the vibration components to becompensated. Moreover, this algorithm, being veryfast and with reduced complexity can be easily imple-mented on computing resources of an aircraft thanksto its low number of coefficients to estimate (one coef-ficient for each frequency component). The proposedapproach has been applied to the case of an helicopterflying in flight test conditions and having a severe vi-bration.

The application results validate the good perfor-mance and efficiency of the proposed algorithms tocharacterize and track oscillations. Moreover, a strat-egy based on detection and filtering has been sug-gested to prevent propagation of abnormal oscilla-tions in the flight control loop. The proposed APE al-lows predictions of the vibrations in case where largeprediction horizon can be needed.

The experimental validations on helicopters andthe integration with active vibration control systemsis in progress. Extensions of this work will be done incase of longer prediction horizon is needed (e.g. forintegration of the compensation in the control loop)and for model prediction control of the helicopter.

The main concern of the future work will be in-tegration of the ANF in the helicopter autonomousflight robust control, to enhance the robustness versusperturbations.

REFERENCES

Al-Badour, F., Sunar, M., and Cheded, L. (2011). Vibrationanalysis of rotating machinery using time frequencyanalysis and wavelet techniques. In Mechanical Sys-tems and Signal Processing, volume 25, pages 2083–2101.

Betta, G., Liguori, C., and al. (2002). A dsp-based fft-analyzer for the fault diagnosis of rotating machinebased on vibration analysis. In IEEE Transactions onInstrumentation and Measurement, volume 51, pages1316–1322.

Johnson, R. M. and col. (1982). Exponential convergenceof rls with exponential forgetting factor. System andcontrol letters.

Krysinski, T. and Malburet, F. (2007). Mechanical Vibra-tions: Active and Passive Control. ISTE.

M’Sirdi, N. and al. (2018). Adaptative notch filters for pre-diction of narrow band signals. In 7th InternationalConference on Systems and Control, Valencia, Spain.

M’Sirdi, N. and Landau, I. (1987). Adaptive evolutionaryspectrum analysis for narrow band signals. In ICASSP,IEEE International Conference on Acoustics, Speechand Signal Processing.

M’Sirdi, N. K., Tjokronegoro, H. R., and Landau, I. D.(1988). An rml algorithm for retrieval of sinusoidswith cascaded notch filters. In ICASSP-88., Interna-tional Conference on Acoustics, Speech, and SignalProcessing, volume 4, pages 2484–2487.

Nehorai, A. (1985). A minimal parameter adaptive notchfilter with constrained poles and zeros. In Proceed-ings of ICASSP 85, IEEE International Conference onAcoustics, Speech, and Signal Processing, volume 10,pages 1185–1188.

NTSB (2018). Bell 525 crash report dca16fa199. Techni-cal report, The National Transportation Safety Board(NTSB), Washington, USA.

Qin, Z., Chen, L., and Bao, X. (2011). Continuous wavelettransform for non-stationary vibration detection withphase-otdr. In Optics express, volume 20, pages20459–20465.

Wang, W. and Wong, A. (1986). Autoregressive model-based gear fault diagnosis. Journal of Vibration andAcoustics, 1274:172–179.


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