vibration analysis of rotating machinery using time frequency analysis and wavelet techniques

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Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing 25 (2011) 2083–2101

0888-32

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/jnlabr/ymssp

Vibration analysis of rotating machinery using time–frequencyanalysis and wavelet techniques

F. Al-Badour a, M. Sunar a,n, L. Cheded b

a Mechanical Department, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabiab Systems Engineering Department, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia

a r t i c l e i n f o

Article history:

Received 24 August 2008

Received in revised form

19 January 2011

Accepted 27 January 2011Available online 4 February 2011

Keywords:

Short-time Fourier transform

Wavelet transform

Wavelet packet

Windowing

Vibration signal

Rotating machinery

70/$ - see front matter & 2011 Elsevier Ltd. A

016/j.ymssp.2011.01.017

esponding author.

ail address: [email protected] (M. Suna

a b s t r a c t

Time–frequency analysis, including the wavelet transform, is one of the new and

powerful tools in the important field of structural health monitoring, using vibration

analysis. Commonly-used signal analysis techniques, based on spectral approaches such

as the fast Fourier transform, are powerful in diagnosing a variety of vibration-related

problems in rotating machinery. Although these techniques provide powerful diagnos-

tic tools in stationary conditions, they fail to do so in several practical cases involving

non-stationary data, which could result either from fast operational conditions, such as

the fast start-up of an electrical motor, or from the presence of a fault causing a

discontinuity in the vibration signal being monitored. Although the short-time Fourier

transform compensates well for the loss of time information incurred by the fast Fourier

transform, it fails to successfully resolve fast-changing signals (such as transient

signals) resulting from non-stationary environments. To mitigate this situation, wavelet

transform tools are considered in this paper as they are superior to both the fast and

short-time Fourier transforms in effectively analyzing non-stationary signals. These

wavelet tools are applied here, with a suitable choice of a mother wavelet function, to a

vibration monitoring system to accurately detect and localize faults occurring in this

system. Two cases producing non-stationary signals are considered: stator-to-blade

rubbing, and fast start-up and coast-down of a rotor. Two powerful wavelet techniques,

namely the continuous wavelet and wavelet packet transforms, are used for the

analysis of the monitored vibration signals. In addition, a novel algorithm is proposed

and implemented here, which combines these two techniques and the idea of window-

ing a signal into a number of shaft revolutions to localize faults.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Rotating machines cover a wide range of critical facilities and provide the backbone of numerous industries, from gasturbines used in the production of electricity to turbo-machinery utilized to generate power in the aerospace industry. It isvital that these machines run safely over time and under different operational conditions, to ensure continuousproductivity and prevent any catastrophic failure, which would lead to extremely expensive repairs and may alsoendanger lives of the operating personnel.

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dx.doi.org/10.1016/j.ymssp.2011.01.017

F. Al-Badour et al. / Mechanical Systems and Signal Processing 25 (2011) 2083–21012084

Generally, a simple condition monitoring system is approached from a pattern classification perspective. It can bedecomposed into three general tasks: (1) data acquisition, (2) feature extraction and (3) condition classification [1]. Severalparameters may be used for condition monitoring, including temperature, pressure, oil analysis, noise and vibration. Themost common method is based on nondestructive vibration measurements using transducers such as accelerometers,velocity pickups and displacement probes. Vibration measurement provides a very efficient way of monitoring thedynamic conditions of a machine such as unbalance, misalignment, mechanical looseness, structural resonance, softfoundation and shaft bow.

1.1. Traditional Fourier-Based approaches to vibration analysis

Traditional vibration signal analysis has generally relied upon the spectrum analysis via the Fourier Transform (FT).Fourier analysis transforms a signal f(t) from a time-based domain to a frequency-based one, thus generating the spectrumFðoÞ that includes all of the signal’s constituent frequencies (fundamental and its harmonics) and which is defined as [2]

FðoÞ ¼Z 1�1

f ðtÞe�iot dt ð1Þ

Fuelled by its huge success in processing stationary signals in a wealth of application areas, an FT technique has enjoyedother interesting extensions. One such extension is in the particular area of vibrations and machine-health monitoring,called the fast Fourier transform (FFT)-based order analysis (OA) technique, including its order-tracking capability [3]. TheOA technique transforms the revolution domain into an order spectrum and any signal that is periodic in the revolutiondomain will appear as a peak in the order spectrum. By doing so, the OA technique tries to overcome the effect offrequency change on the FFT and hence allows for a better tracking of speed-driven harmonics in rotating machinery.However, the assumption in this technique is that the frequency change within a single time interval is small, so that thenecessity of a stationary signal for frequency transformation is not largely violated. If the frequency changes significantlywithin this time interval, then the FFT will yield an error in the actual value of the signal [4].

An important deficiency of the FFT is its inability to provide any information about the time dependence of thespectrum of the signal analyzed, as results are averaged over the entire duration of the signal. This feature becomes aproblem when analyzing non-stationary signals. In such cases, it is often beneficial to acquire a correlation between thetime and frequency contents of the signal. Non-stationary signals could be classified into two groups:

�
Evolutionary harmonic or frequency-modulated signals: these signals are generated by some underlying periodic time-varying phenomenon like a change in rotational speed during ‘‘start-up or coast-down’’. � Transient signals: these signals have short durations and an unpredictable time behavior, and are therefore viewed as
being random in nature. Examples of such signals are impact loading and rubbing.

This important limitation of the FFT has led to the introduction of time–frequency signal processing tools, such as theShort-Time Fourier Transform (STFT), the Wigner-Ville Distribution (WVD) and others. The STFT maps a signal into a two-dimensional (2D) function of time and frequency. The difficulty in using the STFT is that the accuracy of extractingfrequency information is limited by the length of the window relative to the duration of the signal. Once the windowfunction is defined, the area (time-bandwidth product) of the window function in the time–frequency plane remains fixed,which means that the time and frequency resolutions cannot be increased simultaneously. Consequently, for an STFT,there is a trade-off between time and frequency resolutions [5]. The WVD has a good energy concentration in the time–frequency plane; but suffers from interference terms which appear in the same plane and tend to mislead the signalanalysis [6].

1.2. Wavelet-Based Approaches to vibration analysis

The wavelet transform (WT) is a relatively new and powerful tool in the field of signal processing, which overcomesproblems that other techniques face, especially in the processing of non-stationary signals. It allows the use oflong time intervals, where more precise low-frequency information is desired and also permits the use of shortertime intervals where accurate high-frequency information is desired. It is also employed for the accurate extraction ofnarrow-band frequency signals. The main advantage gained by using wavelets is the ability to perform a local analysisof a signal, or to zoom on any interval of time without losing the spectral information contained therein. The waveletanalysis is thus capable of revealing some hidden aspects of the data that other signal analysis techniques fail todetect. This property is particularly important for damage (crack) or fault detection applications. One possible drawback ofthe WT is that the frequency resolution may be quite poor in the higher frequency region. Hence, the WT stillfaces difficulties when trying to discriminate signals containing high frequency components, such as impact faults(like rubbing).

F. Al-Badour et al. / Mechanical Systems and Signal Processing 25 (2011) 2083–2101 2085

A wavelet c(t) is a waveform of effectively limited duration that has an average value of zero over time, as described bythe following Eq. (2):

Z 1�1

cðtÞdt¼ 0 ð2Þ

To achieve wide ranges of analyses and applications, and a higher signal-resolving power, the wavelet schemes usebasis functions other than sines and cosines, which constitute bases of the Fourier analysis. The wavelet functions arecomposed of a family of basis functions that are capable of describing a signal in a localized time (or space) and frequency(or scale) domains. Choosing the type of the basis function depends on the application as well as the computation effortsrequired. The appropriate selection of mother wavelets is very important, as results are heavily dependent on the chosenwavelet shape. Wavelets of various shapes exist, including the first-ever wavelet (Haar wavelet) [7] and other types suchas Mexican hat-shaped and Gaussian-shaped [6]. However, for singularity analysis and detection, wavelets should have animpulse-like shape to capture the sudden change in the signal. Morlet and Gaussian wavelets are found to have anexcellent representation of a singularity (discontinuity) as the vibration signal has a harmonic oscillation plus a singularityrepresenting the fault. It is desirable to have a linear phase scaling function in order to maintain a constant group delayand the envelope of the original vibration signal, which will avoid problems during the signal reconstruction process.Having a wavelet shape resembling that of the targeted fault greatly helps in the wavelet selection decision.

The wavelet transform and its other three types are further discussed in Section 4, where the details of the proposedhybrid wavelet-based approach are clearly expounded.

The paper is organized as follows: Section 1 reviews the two general approaches to vibration analysis, namely thetraditional FFT-based and the modern wavelet-based ones. Section 2 describes the experimental setup used and the datacollection procedure, whereas Section 3 reports on, and discusses, the experimental results obtained using the FFTapproach. In the key Section 4, the 3 main types of wavelets, Continuous Wavelet Transform (CWT), discrete wavelettransformation (DWT) and wavelet packet transform (WPT) used in vibration analysis are first reviewed and then both theCWT-based maxima modulus technique and WPT are applied to the data from the blade-to-stator rubbing test, and theirperformances are compared. This section also introduces the proposed novel hybrid approach based on a judicious use ofthese 2 tested techniques, i.e. the CWT-based maximal modulus technique and WPT. Finally, some conclusions are givenin Section 5.

2. Experimental setup and data collection

The test rig was built for research purposes to study blade and shaft vibrations of turbo-machinery (Fig. 1). It isprimarily designed to study blade vibrations using a remote sensing system via a set of strain gages bonded to one of theblades. It is also possible to measure shaft vibrations directly by proximity probes or indirectly by accelerometers throughtheir placement near shaft supports. The test rig consists mainly of the following elements:

1)
A 3-phase induction motor of 220 Volt and 60 Hz with three Horse Power (HP) connected to a variable speed controller. 2) Two shafts, one of which holds three unbalanced disks coupled with another shaft carrying a disk with 12 blades. 3) A 2 kW generator and a bank of resistance elements used as a braking mechanism. The test rig mechanical components
(motor, shafts and generator) are coupled with three flexible couplings. An adjustable rubbing device is attached to thebase plate at the circumference of the blades to induce blade rubbing during the experiment.

Mainly, a proximity probe is used to read the lateral shaft vibration and the strain-gage system is used to collect theblade vibration data. An accelerometer is also used, but its results are kept for comparison only and are not reported here

Fig. 1. Schematic drawing of the rotor kit.


due to the lack of space. The 5-mm probe is of model 10001 from METRIX Instruments and has a sensitivity of 200 mV/mil.The strain gages made by MM micro measurements are of standard 120 Ohm type and are put together into a set of fourconnected in a full-bridge arrangement to produce an accurate and information-rich signal. The strain gage set is poweredby a 9 V DC battery from Binsfeld transmitter and its output is taken by a receiver unit. The strain gages attached on bladesare arranged to capture only the blade bending deflection and not the blade torsional vibration, which is found to occur athigh frequencies from a prior finite element analysis of the system. The transducer outputs are then sent to the dataacquisition interface unit (DAIU) 208 of the Bently-Nevada Inc. (BNC), which is equipped with eight channels for inputand 2 references (key phasors) for speed. The DAIU is connected to a computer, where the Automated Diagnostics forRotating Equipment (ADRE) software of the BNC is installed for processing the vibration data and producing various plots.The main plots are: (1) time waveform, (2) frequency spectrum and (3) cascade (a three-dimensional plot of the frequencyspectrum versus the speed). A Labview-based general-purpose interface package developed and supplied by Teclution, isalso used to acquire and process the vibration input. This software is useful to extract the time wave signal and save it inone vector rather than splitting it into patches of vectors as the ADRE does. This is needed to analyze the transientvibration signals during the start-up and coast-down stages of the rotor kit.

The location of a transducer is very important, because the location plays a big role in the ability of the transducer tosense a certain signal energy level generated by the vibrating body. Fig. 1 shows the location of transducers marked withnumbers.

3. Experimental results using an FFT

Three experiments are carried out: 1) free vibration of blades, 2) blade-to-stator rubbing at different speeds and3) fast start-up and coast-down. The 1st experiment is run to locate the natural frequencies of blades. The 2nd and 3rdexperiments are conducted to simulate the occurrence of non-stationary and transient faults for the comparison of the FFT,time–frequency and WT techniques. The expected faults from these experiments are unbalance and misalignment. Theunbalance in rotating machinery is one of the main causes of vibration [8]. A perfectly balanced rotor is hard to achieve; soeven under normal operating conditions, the frequency spectrum using the traditional FFT technique may show a spike atthe operating speed. The shaft misalignment is the deviation of the relative shaft position from a collinear axis of rotationmeasured at points of power transmission when the equipment is running at normal conditions [9]. The shaftmisalignment is considered to be another major vibration problem that occurs in rotating machines. Traditionally, avibration engineer diagnoses the shaft misalignment in the frequency spectrum from the harmonics of multiples ofrotational speed, so-called at 2X, 3X, y, etc.

3.1. Free vibration of blades

The test is carried out by applying a tip deflection to the blade holding the strain gages as an initial condition and thenletting the blade vibrate freely. As a result, the blade oscillates at its natural frequencies and so the signal from the straingages is expected to yield the blade’s natural frequencies in the FFT-based frequency spectrum of the signal from the blade.Results show that the first blade bending mode occurs at 42 Hz. This was predicted to be 49 Hz for the stationary rotorfrom the finite element analysis (FEA) results, using the ANSYS package. The section on the finite element modeling andanalysis is omitted here solely for lack of space. The difference between the experimental and finite element results is to beexpected and may possibly be due to the assumed boundary conditions and neglected details in the finite element model.The second bending mode of the blades is found from the test to be 264 Hz and by an FEA as 256 Hz.

3.2. Blade-to-Stator Rubbing

It is well-known that the blade failure due to rubbing is one of the main causes of turbine failures. The blade-to-statorrubbing usually occurs during the start-up as a result of vibration displacement, due to an unbalance becoming equal to, orlarger than, the clearance between the blades and the stator. The rubbing may suddenly break the blades or may developslowly, eventually leading to failure. The test is conducted such that when the rotating system rotates at a constant speed,a blade rubbing excitation device suddenly touches one blade and is released thereafter, thus simulating the occurrence ofa blade-to-stator rubbing. The test is carried out at the speeds of 300, 600 and 1000 rpm (5, 10 and 16.7 Hz, respectively).Higher rotor speeds are not tested due to potential danger to the operator and damage to the test equipment. The blade-to-stator rubbing is monitored by the strain gage system via the remote sensing capability. When the blade rubs the stator,the blade is excited with an impulse force, which causes the blade to vibrate at its natural frequencies. These naturalfrequencies, arising from the first and second bending modes of the blade, are clearly shown in Fig. 2(a–c) as having valuesof 41, 44, 46, 262 and 266 Hz. The running frequencies of the rotor at 5 10 and at about 16.7 Hz also appear in the spectradue to the unbalance. The magnitude increase due to rubbing is very clearly indicated in these figures both in time andfrequency domains.

The shaft displacement was used as an indirect measure of blade vibrations, for which the power spectral density (FFT)result is shown in Fig. 3. One of the expected frequencies is the first blade bending frequency around 46 Hz, as previouslyindicated via the direct strain gage measurement. The spectrum shows the dominant frequency component at the running

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-6-4-20246

Time [sec]

Am

pl [V

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50 100 150 200 250

0.51

1.52

Frequency [Hz]

Am

pl [V

olt] Normal

Rubbing5 Hz

41 Hz

31Hz 56 Hz 262 Hz

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-4-20246

Time [sec]

Am

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0.20.40.60.8

11.21.4

Frequency [Hz]

Am

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10 Hz

44 Hz

46 Hz

262 Hz

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-5

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Time [sec]

Am

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50 100 150 200 250 3000

0.5

1

1.5

Frequency [Hz]

Am

pl [V

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46 Hz

16.25 Hz266 Hz

NormalRubbing

NormalRubbing

Fig. 2. Strain gage results with and without blade rubbing at running speeds of (a) 5 Hz, (b) 10 Hz, and (c) 16.7 Hz.


speed of 16.7 Hz and its harmonics due to presence of the unbalance and/or shaft misalignment. It also seems to indicatethe first blade bending frequency with low amplitude. No significant change is observed in signal frequency content duringrubbing. This may possibly be due to combination of an insufficient strength of the impact force causing the rubbing andthe lack of resolving power of the FFT in analyzing the non-stationary signal emanating from the transient rubbing process.Principally, because of the non-stationary character of the rubbing signal under analysis, such a situation is expected to begreatly improved by the application of the wavelet approach advocated in this paper. The proximity probe results at therotor speeds of 5 and 10 Hz show almost the same trend and hence are not reported here.

3.3. Start-up and Coast-down Test

Fig. 4 shows the signal acquired from the vertical proximity probe 1 depicted in Fig. 1. Note in this figure the burst-likenature of the vibration signal as the rotor approaches the target speed. The maximum level of vibration reached is about7 mils, which is enough to classify this vibration roughly. The Fourier spectrum of this signal, obtained with the traditionalFFT and shown in Fig. 5 is full of spikes with not much useful information to be extracted from it for the purpose of faultdetection. This clearly shows the lack of power of the FFT in revealing fault-related information from the spectrum of thetransient vibration signal.

Prior to the advent of time–frequency techniques, the above problem of the FFT technique was overcome by so-calledwaterfall or cascade plots. Also called speed spectral maps in the order analysis [3] terms, a cascade plot is a three-dimensional plot, where the signal is divided into patches and each patch is FFT-analyzed over a certain duration of time.As such, the time-varying nature of the spectrum is mitigated, but at a cost of irretrievably loosing part of the signal during

Fig. 3. Shaft FFT results with and without blade rubbing at 16.7 Hz.

Fig. 4. Vertical proximity probe 1 time-wave signal during start-up and coast-down.

Fig. 5. Vertical proximity probe 1 Fourier spectrum during start-up and coast-down.


Fig. 6. Vertical proximity probe 1 cascade plot during start-up and coast-down.


the patching process. Fig. 6 shows the cascade plot of the vertical proximity probe 1 indicating the distribution of vibra-tion frequency in Hz (on the horizontal axis) as a function of the rotor speed in RPM (on the vertical axis). The thirddirection out of the paper in the plot depicts the vibration amplitude. In Fig. 6, the motor acceleration was lowered toincrease the start-up time, allowing suffusion time for the data acquisition system to capture a large number of spectrumsduring the start-up. The result shows frequencies of the unbalance (1x), and those of the misalignment (1x and 2x).Despite a large number of spectral lines, some transition speeds could not be captured as a result of the non-continuousdata recording.

4. Proposed Hybrid Wavelet-Based Approach and experimental results using Time-Frequency and wavelet techniques

4.1. Review of Three Wavelet Transforms: CWT, DWT and WPT

In this section, the three major wavelet transforms are briefly reviewed: namely the CWT, DWT and WPT, with anemphasis on the CWT and WPT.

The WT is classified into three types: the CWT, DWT and WPT. J. Morlet along with an A. Grossmann formulated theCWT and defined it as the sum over all time of the signal multiplied by scaled and shifted versions of the mother waveletfunction [6]. Hence, the modulus of the CWT for the signal f(t) is given as

Cf ða,bÞ ¼

Z 1�1

xðtÞcða,b,tÞdt ð3Þ

where t is the time, a is the scale and b is the location or space. The square of the modulus of the CWT is often called thescalogram (SG) and is defined as

SGf ða,bÞ ¼ 9Cf ða,bÞ92ð4Þ

The CWT is one of the best transforms for singularity detection. Impact faults could be detected by finding thesingularity in the signal. The singularity is detected using the local maxima lines by finding the abscissa, where the waveletmodulus maxima converge at fine scales. The term ‘‘modulus maxima’’ describes all points on the scale-space distributionWf(a,bo) such that 9Wf ða,boÞ9 is locally maximum at a=ao, which implies that

@9Wf ða,boÞ9@a

��a ¼ ao

¼ 0 ð5Þ

Daubechies and Mallat are credited with the development of the wavelet from continuous to discrete signal analysis bydiscretizing the modulus of the CWT as [6]

Cf ða,kÞ ¼X

n

xðnÞcða,k,nÞ ð6Þ

with t¼ nDT and b¼ kDT, where DT is the sampling interval, n and k are integers, i.e. n, k=1, 2, y, N, where N is thenumber of samples.

Coifman, Meyer and Wickerhauser developed the wavelet packets, which are the bases formed by taking linearcombinations of the usual wavelet functions [6]. The WPT continuously decomposes both the ‘‘approximate’’ and ‘‘detail’’components of the signal, so as to, respectively, utilize low- and high-frequency components of the signal at various scales.This feature allows the extraction of signal features that combine both non-stationary and stationary characteristics [5].

The WPT is the latest technique in the family of wavelet transforms and is found to be one of the best analyzing tools ofvibration signals and fault detection. It is a generalization of the DWT and, as such, gives the user a much richercharacterization of the signal being analyzed. The WPT is faster than the CWT as it uses orthogonal and bi-orthogonal


bases with a better resolution in the high-frequency region. The detection method used is similar to that in spectralanalysis, where each fault has a characteristic frequency to be detected. So, faults could be monitored by monitoringpackets that have central frequencies equal or close to the fault frequency of interest. The power of WPT lies in itsfrequency resolution of signals with a short or small number of samples as compared to the FFT approach.

The selection of the mother wavelet function plays a big role in the efficiency of the WPT. From a simulation point ofview, it is found that Daubechies and discrete Meyer are the best wavelets to be used in vibration signal analysis. Thisjudgment was reached as a result of conducting some simulation-based experiments to select the best wavelets forvibration signals, which are impulsive in nature. It is usually preferable to rearrange the wavelet packets based on theirfrequency order, rather than on their natural one, as the packet index (j,p) labeled as (j,k) in equations below. A waveletpacket function W is a function with three integer indices of (j,k,n) defined as

Wnj,kðtÞ ¼ 2j=2Wnð2jt�kÞ ð7Þ

where integers j and k indicate the index scale and translation operations, respectively, and the index n is called themodulation or oscillation parameter. The first two-wavelet packet functions are the usual scaling and mother waveletfunctions of f(t) and c(t), respectively, given by

W00,0ðtÞ ¼fðtÞ and W1

0,0ðtÞ ¼cðtÞ ð8Þ

The wavelet packet coefficients oj,n,k of the signal f(t) can be computed via the following equation [5]:

oj,n,k ¼

Zf ðtÞWn

j,kðtÞdt ð9Þ

The WT, briefly described above, was introduced to overcome the drawbacks of the STFT, WVD and other related time–frequency transforms. Ping et al. [1] used the first-order wavelet gray moment to indicate quantitatively faults in rotatingmachines and compared results with those obtained with the FFT. In order to detect the low frequency signals, Li et al. [7]used Haar wavelets, which under different and particularly large scales, have good low-pass filter characteristics in thefrequency domain, so as to allow the use of Haar continuous wavelet transform for this purpose. This transform wasapplied to the diagnosis of various types of machine faults. Miao and Makis [10] used the modulus of maxima lines andproposed an on-line fault classification system with an adaptive model re-estimation algorithm. The machine conditionwas then identified by selecting the Hidden Markov Model (HMM), which maximized the probability of a givenobservation sequence. The authors validated their HMM model on three sets of real gearbox vibration data to classifythe normal and failure conditions. For a ball-bearing fault detection by the HMM classifier, Purushotham et al. [11] usedthe vibration signals from ball-bearings having single and multiple point defects on inner race, outer race, ball fault andcombination of these faults. Bearing race faults were detected via the DWT. Then, the HMM model was employed as aclassifier, and the vibration signals were decomposed up to four levels, using the Daubechies 2 wavelet.

Adewusi and Al-Bedoor [12] analyzed the start-up and steady-state vibration signals of a rotor with a propagatingtransverse crack by scalograms and space-scale energy distribution graphs. The start-up results showed that the crackreduced the critical speed of the rotor system. The steady-state results indicated that the propagating crack causedchanges in vibration amplitudes with the frequencies corresponding to 1X (rotor speed), 2X and 4X harmonics. Zouet al. [13] analyzed the torsional vibration signal of a cracked rotor by using the CWT using Daubechies 10 mother waveletand Hilbert transform. Their analysis led to the detection of a deep crack by two peaks in a three-dimensional waveletspectrum and that of a shallow crack by six peaks. Junsheng et al. [14] utilized the CWT in fault diagnosis for rollerbearings, where an impulse response wavelet was used to target characteristics of faulty roller bearings’ vibration signals.They concluded that the impulse response wavelet was superior to the Morlet wavelet in the scale-wavelet powerspectrum comparison method.

Zanardelli et al. [15] developed three wavelet-based methods to predict the failure of electric motors. A total of 15mother wavelets were tried with the expectation that one or two of them would be finally used. Antonino-Daviu et al. [16]diagnosed the rotor bar failures in induction machines based on the analysis of the stator current during the start-up, usingthe DWT. The method yielded a better diagnosis when compared with the Fourier analysis. Figarella and Jansen [17]addressed the singularity detection of an electrical motor current signal using the CWT as related to the brush condition.

The literature survey revealed that applying the WT as a filtering operation was found to be very efficient in thevibration signal analysis, especially for non-stationary signals. Many fault classification or detection algorithms proposedby researchers were based on pattern recognition and soft computing (learning) techniques (e.g. Fuzzy, Neural Network,Neuro-Fuzzy, etc.). But these techniques are not well-suited for real-time applications and analysis of non-stationaryvibration signals, because they are used to model situations and/or processes under certain stationary conditions with theprocessing done offline in some cases (as with neural nets and fuzzy sets, etc.). Having recognized the strong resolvingpower of the WT and its successes reported by researchers in various areas, we carried out some simulation work in thearea of fault detection in rotating machinery and the results were found to be encouraging [18–20]. In this paper and asexplained below in sub-Section 4.2, our first main contribution is to show that despite the fact that the WPT surpasses theCWT in terms of speed and spectral characterization of the vibration signal, the latter (CWT), when supplemented with themaxima modulus technique, can outperform the former (WPT) in detecting impulsive faults. Our second maincontribution, as explained in sub-Section 4.3, is to propose a novel and powerful wavelet-based signal processing tool


that judiciously combines the strengths of both transforms, i.e. the rich spectral characterization and speed of the WPT andthe ability of detecting impulsive faults, of the CWT used in conjunction with the maxima modulus technique. This newscheme was tested on real experimental data obtained from a custom-designed rotating machine (test rig).

4.2. Blade-to-Stator Rubbing Analysis using CWT-Based Maxima Modulus Technique and WPT

In the blade-to-stator rubbing test, two different signals are collected: the first one is from the strain gage systemattached to one blade and the second one is via the proximity probe 2. The test is carried out at the speeds of 5, 10 and16.7 Hz.

The CWT was applied to the strain gage signals and the results are shown in Figs. 7 and 8 where the intensity scale is linear.As seen in this figure, the CWT results clearly distinguish between normal and rubbing conditions by detecting the suddenchange in blade vibration amplitude as a discontinuity in the scale vs. space domain, which is shown in the figure as a linecrossing all scales (frequencies) at the instant of rubbing. The CWT is implemented using Gaussian wavelet and the localizationof the discontinuity or singularity in the time wave signal is achieved using the modulus maxima lines. The modulus maxima

Fig. 7. CWT (upper part) and modulus maxima lines using Gaussian wavelet (lower part) for strain gage signal with rotor operating at 5 Hz

(normal operation).


lines are expected to converge at the instant of rubbing. The results are shown in Figs. 7–11 for the rotor speeds of 5, 10 and16.7 Hz. Fig. 7 shows the test results at 5 Hz with normal operation, that is the condition with no rubbing, and it is clear that nosign of discontinuity or singularity can be seen. On the other hand, Fig. 8 shows a clear convergence at the time of rubbing, i.e.at a space around 370. This is the location where the blade starts to vibrate rapidly due to the rubbing impact. Fig. 10 showsthree convergence points at 10 Hz: one just before, one during and one just after the rubbing, as indicated by arrows in thisfigure. A possible explanation of the existence of the pre- and post-rubbing convergence points is that, in addition to sensingthe rubbing on the blade to which it is bonded, the strain gage is also picking up the vibrational impact of the rubbing onadjacent blades. Similar plausible explanations can be made for the modulus maxima lines shown in Fig. 11 at 16.7 Hz rotorspeed, although, in this figure, only the rubbing instant is indicated by an arrow. These results together show that the methodof local maxima is able to indicate and localize the rubbing instant.

The singularity detection scheme through the CWT is outlined below:

1)

Fig(wi

The signal is chosen to have a length of 2n and a zero mean.
2) An appropriate mother wavelet is selected. In our case Gaussian wavelet was used.
. 8. CWT (upper part) and modulus maxima lines using Gaussian wavelet (lower part) for strain gage signal with rotor operating at 5 Hz

th rubbing).

Fig. 9. Strain-gage time-wave signal (upper part) and modulus maxima lines using Gaussian wavelet (lower part) with rotor operating at 10 Hz

(normal operation).

Fig. 10. Strain-gage time-wave signal (upper part) and modulus maxima lines using Gaussian wavelet (lower part) with rotor operating at 10 Hz

(with rubbing).


Fig. 11. Strain gage time-wave signal (upper part) and modulus maxima lines using Gaussian wavelet (lower part) with rotor operating at 16.7 Hz

(with rubbing).


3)
The CWT is then implemented with the selected mother wavelet and the transformation is carried out up to scale 64(selection of the scale interval is heavily dependent upon the chosen sampling frequency and wavelet function).
4)
The modulus maxima are evaluated from the CWT coefficients. 5) The local maxima lines are plotted. 6) The singularity, if any, is then presented as a convergence of maxima lines down to the finest scale used.
The WPT was applied to the same strain gage signal for the purpose of comparing its performance with that of the CWTtransform. In the implementation of the WPT, two mother wavelets were used: the first one is an anti-symmetric wavelet(Daubechies 8�db8) and second is a symmetric one (Discrete Meyer�dmey). The WPT results for the dmey wavelet are,respectively, shown in Figs. 12–14 for the rotor speeds of 5, 10 and 16.7 Hz, where the WPT is decomposed up to level 6. Itis clearly presented in these figures that, due to blade rubbing, the energy level of packet 6 is increased. Furthermore, byusing the (dmey) wavelet, the discontinuity is captured after finding the best tree based on Shannon entropy(Figs. 12(a), 13(a) and 14(a)), but the results are not as clear as those obtained with the CWT. It can be seen that theCWT, when supplemented with the modulus maxima technique, identifies the blade rubbing more clearly than the WPTalone. The modulus maxima technique uses in effect some derivative properties of the CWT, which seem to have a closersimilarity (in terms of signal sharpness and continuity, etc.) with the detected singularity than the properties of theWPT alone.

4.3. Novel hybrid approach for Blade-to-Stator Rubbing

The novel approach proposed here consists of a judicious combination of two wavelet-based processing tools, i.e. WPTand CWT supplemented with the modulus maxima technique. The shaft vibration signal is analyzed using this approach. Itwas shown before that the FFT was not clearly detecting the blade-to-stator rubbing on the proximity probe signal (Fig. 3).The proposed hybrid algorithm yields a powerful way to detect the impact fault from proximity probe signals. The WPTdecomposes the proximity probe signal into detailed and approximate coefficients, which are then reconstructedseparately. After that, the CWT is applied to evaluate the modulus maxima. The algorithm steps are listed in the followingchart (Fig. 15) [21].

Note that various types of faults commonly encountered in the vibration analysis are well-documented, together withtheir frequency ranges. Hence, such a-priori knowledge can always be exploited to guide one’s investigation. Lack of thisknowledge would not be a limitation, but would certainly entail an exhaustive search for the actual fault as variouspackets would have to be tested. But once a particular packet (and hence a scale) is selected based on the system’s naturalfrequency band and a full scan of all other packets, then the corresponding frequency can be readily obtained based on the

Space

Inde

x

100 200 300 400 500 600 700 800 900 1000

4

8

33706965666463

Normal conditionSpace

Inde

x

100 200 300 400 500 600 700 800 900 1000

19

20

10

8

6768706965666463

Blade 2nd Bendingfrequency

Singularity

Blade to stator rubbing

0 2 4 6 8 100

20

40

60

80

100

Packet no.

Ene

rgy

%

Normal condition

0 2 4 6 8 100

20

40

60

80

100

Packet no.

Ene

rgy

%


Fig. 12 (a). WPT of strain gage signal at 5 Hz using dmey at level 6. (b). WPT energy for strain gage signal at 5 Hz using dmey at level 6.

Normal condition

Normal condition

Space

Inde

x

100 200 300 400 500 600 700 800 900 1000

19858641

10

8

6768706965666463

Singularity

Blade 2nd bendingfrequency

Space

Inde

x

100 200 300 400 500 600 700 800 900 1000

4

8

6768706965666463

0 10 20 300

20

40

60

80

100

Packet no.

Ene

rgy

%

0 10 20 300

20

40

60

80

100

Packet no.

Ene

rgy

%



Fig. 13 (a). WPT of strain gage signal at 10 Hz using dmey at level 6. (b). WPT energy for strain gage signal at 10 Hz using dmey at level 6.


Normal condition

Normal condition Blade to stator rubbing

Singularity

Blade 2ndBending frequency

0 10 20 300

20

40

60

80

100

Packet no.

Ene

rgy

%

0 10 20 300

20

40

60

80

100

Packet no.

Ene

rgy

%


Fig. 14 (a). WPT of strain gage signal at 16.7 Hz using dmey at level 6. (b). WPT energy for strain gage signal at 16.7 Hz using dmey at level 6.


sampling frequency through the relation

Fa ¼Fc

aDð10Þ

where Fa is the frequency of the scale, Fc is the central frequency of the used wavelet, a is the scale and D is the samplingfrequency.

The proximity probe results of the proposed method are shown in Fig. 16(a) and (b), for the tests carried out at 10 and16.7 Hz, respectively. These figures show both the time waveforms reconstructed from the detail coefficients of theproximity probe signal as well as the local maxima results using the db10 wavelet. It is very clear from these results thatthe time waveform represents the shaft response as the blade impact occurs.

Another method for the detection and localization of impulsive defects proposed here involves a certain preprocessingof the probe signal before feeding it through the same algorithm used to detect the blade rubbing through shaft vibration.This alternative approach proposes to first divide the time wave signal into shaft revolutions using a window function andthen analyzing each single shaft revolution separately. The key idea behind slicing up the signal prior to processing it isdue to the fact that when analyzing short-time signals the frequency resolution obtained with the wavelet transform is farbetter than that with the FFT. This will lead to a better and more accurate localization of impulsive-type of faults such asthose resulting from a blade rubbing, and can be efficient even for low-frequency faults such as misalignment andunbalance.

Traditional methods for locating blade rubbing utilize a source of color on the tip of blades, so when a blade touches thestator casing it leaves a mark on the stator, hence the operator can detect and pinpoint the location of the rubbing duringmaintenance operation. With the above algorithm, the blade rubbing is located from the vibration signal, where the timespace is converted to the angular space. Figs. 17 and 18 show results using the proposed algorithm, applied on the shaftvibration for the blade rubbing at the speed of 16.7 Hz.

4.4. Start-up and Coast-Down STFT and WPT results

In the start-up and coast-down phases, the signal has two properties: it is Frequency-Modulated (FM) as well asAmplitude-Modulated (AM). During the start-up, the time-wave signal was collected from the vertical proximity probe 1as shown in Fig. 4, its frequency spectrum and cascade plots were given in Figs. 5 and 6 respectively. In order to run a

WPT up to level 3

Reconstruct the detailed component

of (3, 2)

CWT for the reconstructed signal

Evaluate the modulus maxima

lines

Modulus maxima lines indicate the

rubbing

Fig. 15. Algorithm of novel hybrid approach for the blade-to-stator rubbing.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-0.1

-0.05

0

0.05

0.1

Am

pl. [

mil]

Time wave signal

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.01

0.02

0.03

0.04

Time [sec]

Am

pl. [

mil]

maxima at scale 0.07

@ 0.336 sec.

transient envelope

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-0.2

-0.1

0

0.1

0.2

Am

pl. [

mil]

Time wave signal

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.02

0.04

0.06

Time [sec]

Am

pl. [

mil]

maxima at scale 0.07

@ 0.155 sec.

Transient envelopdue to blade rub

Fig. 16. Time wave and local maxima of reconstructed signal from detail coefficients using db10 of the proximity probe signal (a) at 10 Hz and

(b) at 16.7 Hz.


time–frequency technique on the test-rig during the transient phases of start-up and coast-down, and compare the resultswith those of the WPT, the STFT analysis was performed on the signal from this probe. The STFT distribution was based ona 512 point-Hanning window and was computed on a log scale. Fig. 19 shows the shaft’s first natural frequency (criticalspeed) and blade’s second bending mode, and also reveals the presence of shaft misalignment (2X and 3X).

The WPT is carried out using the discrete Meyer (dmey) wavelet distributed in the frequency order. The WPT resultsfrom the vertical probe are presented in Fig. 20, where the distribution shows the lines of 1X, 2X, 3X and the shaft’s firstcritical speed. As stated above, the WPT conserves all the coefficients, detailed and approximate ones, which provide theanalyst with a good tool for a thorough analysis. It is also possible to reconstruct each packet coefficient separately.

0 pi/3 2pi/3 pi 4pi/3 5pi/3 2pi

0 pi/3 2pi/3 pi 4pi/3 5pi/3 2pi

-1

0

1

Am

pl. [

mil]

Time wave signal

-1

-0.5

0

0.5

1A

mpl

. [m

il]maxima at scale 0.01 cycle no. 1

Fig. 17. Modulus of maxima lines at scale 0.01 for proximity probe signal for normal operation at 16.7 Hz.

0 pi/3 2pi/3 pi 4pi/3 5pi/3 2pi

-1

0

1

Am

pl. [

mil]

Time wave signal

0 pi/3 2pi/3 pi 4pi/3 5pi/3 2pi0

0.5

1

Time [sec]

Am

pl. [

mil]

maxima at scale 0.01cycle no. 4

Fig. 18. Modulus of maxima lines at scale 0.01 for proximity probe signal with blade rubbing at 16.7 Hz.


The reconstruction process using the WPT is presented in Fig. 21 for various packet indices. In Fig. 21(b), the vibrationlevels indicate the presence of unbalance during the steady-state. From the reconstructed coefficient and before the rotorreaches the steady-state speed, the misalignment 2X coincides with the shaft critical speed (time=2 sec. in Fig. 21(c)). Incomparing the performance of the WPT with that of the STFT, it is clear that the former goes further than the latter, in thatit allows a more accurate and detailed characterization of the faults at various scales (frequencies).

5. Conclusion

The main theme of this work was to study the application of wavelets, in particular the WPT transform, to faultdetection in rotating machinery, an area of paramount importance to various industries. All of the signals used in thisstudy were experimentally obtained from a custom-built rotor kit to simulate on a laboratory scale the main operatingconditions of rotating machinery in a wide range of industries involving equipments such as turbines, compressors andfans. Although not reported here, a finite element analysis of the kit was carried out to identify certain dynamiccharacteristics of the system using the ANSYS package, so as to compare them with their experimentally-derivedcounterparts.

In this paper, the effective use of the traditional FFT approach in yielding good results in fault detection understationary operations was demonstrated and its major failing in detecting faults from non-stationary signals was clearlyexposed via the study of the case involving a blade-to-stator rubbing of the rotor kit, which produced a typical impulsive

Spectrogram

Time [ms]

Freq

uenc

y [k

Hz]

2000 4000 6000 8000 10000 120000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

shaftnatural frequency

1X, 2X & 3X

blade 2nd bending frequency

Fig. 19. Time–frequency distribution based on spectrogram for the vertical probe 1 during start-up to 42 Hz and coast-down to zero.

Frequency Order, for WP using dmey wavelet

space

Inde

x

0 2000 4000 6000 8000 10000 12000 14000 16000

178176175143144146145149150148147155156158157153154152151135136138137141142140139131132134133129130128127

1X,2X, &3Xshaft

1st critical speed

Fig. 20. Index-space distributions (frequency order) based on WPT dmey wavelet, for the vertical probe 1 during start-up to 42 Hz and coast-down

to zero.


and non-stationary test signal. In this test, it was shown that the FFT of the signals extracted from the proximity probes didnot reveal any change in amplitude due to rubbing.

In order to remedy this situation, a new combined approach was proposed that relies heavily on the well-acknowledgedstrength of the wavelet transforms of WPT and CWT in successfully and effectively analyzing non-stationary signals. Theproposed approach exploited the fact that the WPT, largely undiscovered and untested in vibration analysis, provides a

Fig. 21 (a). Reconstruction packet with index 140 holds the shaft free vibration by the critical speed. (b). Reconstruction packet with index 139 holds the

1X during the steady-state operation. (c). Reconstruction packet with index 152 holds the 2X during the steady-state operation. (d). Reconstruction

packet with index 150 holds the 3X during the steady-state operation.


much richer spectral characterization of a signal than the CWT by decomposing the detail signal along the approximateone at every scale. The high-frequency content of impulsive signals (such as rubbing faults), and in particular the timelocation of the singularity, was then shown to be effectively recovered from a signal reconstructed from the informationextracted from the detail components of the vibration signal at various scales. The proposed combined approach also owesits power to the use of the modulus maxima technique, which in a way sharpens its resolving power by exploiting gradientinformation of the CWT in the scale-space domain. In the following, a summary of the main wavelet-related findings ofthis paper is given to reiterate the main characteristics of the various techniques used successfully here with a view toencouraging their uses in other similar applied areas.

1.
The WPT is a powerful tool for detailed feature extraction. 2. For impulsive faults like rubbing, using a combination of the WPT and CWT gives an effective method for fault analysis
and detection.
3. The usage of modulus of local maxima lines is a powerful tool for the singularity or discontinuity detection. 4. The selection of a mother wavelet is very important, because it greatly affects fault detection results. Although there is
no optimal way of selecting the best mother wavelet for the application at hand, a sensible and practical way is tocompare the shape of the fault under consideration with the wavelet function to be used. Symmetric wavelets(e.g. Gaussian wavelet) were found to be more effective in singularity analysis and a very narrow pulse-like anti-symmetric wavelet, such as db10 or any higher-order of Daubechies family was also found to perform well.

5.
The idea of windowing the signal into a number of shaft revolutions and analyzing each window separately to localizethe defect was found to be very helpful in tests and could effectively be used in the detection of cracked gear teeth or inthe analysis of blade vibration problems. The two main advantages of this signal-splitting approach could besummarized as: 1) involving shorter signals and hence dealing with shorter computational time and smaller memoryrequirements, with a possible further enhancement through the use of a parallel computational structure, and 2) betterlocalization of defects in rotating machinery through utilization of a better frequency resolution of the wavelet duringthe processing of shorter signals.
Acknowledgements

The authors greatly acknowledge the support of King Fahd University of Petroleum & Minerals for this work.

References

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vibration analysis of rotating machinery using time frequency analysis and wavelet techniques

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