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Mechanical Systemsand

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Mechanical Systems and Signal Processing 21 (2007) 780–794

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Singularity analysis of the vibration signals by meansof wavelet modulus maximal method

Z.K. Penga,�, F.L. Chua, Peter W. Tseb

aDepartment of Precision Instruments, Tsinghua University, Beijing 100084, PR ChinabSmart Asset Management Laboratory, City University of Hong Kong, Tat Chee Ave, Hong Kong, PR China

Received 21 July 2005; received in revised form 7 November 2005; accepted 5 December 2005

Available online 30 January 2006

Abstract

Machine fault diagnosis is vital for safe services and non-interrupted production. The key issue in fault diagnosis is the

pattern recognition. A set of valid features will simplify the classifying operations and enhance the accuracy in diagnosis.

In this paper, a novel singularity based fault features is presented. Vibration signals collected under different machine

health conditions will show different patterns of singularities that can be measured quantitatively by the Lipschitz

exponents. The wavelet transforms modulus maximal (WTMM) method provides a simple but accurate method in

calculating the Lipschitz exponents. Therefore, the WTMM based Lipschitz exponent calculation as well as the method to

select the appropriate wavelet function for WTMM and its range of scale are introduced. Three parameters about the

singularity analysis are recommended. They are the number of Lipschitz exponents per rotation N, the mean value ma andthe relative standard deviation sa of the Lipschitz exponents that are obtained from the extracted features. To verify the

usefulness of the proposed methods, simulated signals and vibration signals generated by four types of faults commonly

occurred in a rotating machine, including the imbalance, the oil whirl, the coupling misalignment and the rub-impact, had

been used for testing purpose. The results show that the signal from the rub-impact possesses the highest singular value and

the widest range of singularity. The signal of the coupling misalignment ranked the second. Whilst, the signal of imbalance

is more regular or having the smallest singular value and the narrowest range of singularity. The results also prove that the

three parameters are excellent fault features for pattern recognition as they can well separate the four fault patterns.

r 2006 Elsevier Ltd. All rights reserved.

1. Introduction

Fault diagnosis is a technique that is often used to ensure the safe running of rotating machines. Hitherto,many kinds of fault diagnosis techniques have been developed, among which the vibration signal analysisbased methods have been the most widely used ones. Essentially, the fault diagnosis is a pattern recognitionproblem, for which the key step is to extract useful fault features from vibration signals through some suitablesignal processing methods. Now, various kinds of fault features are available. The well accepted featuresinclude frequency based [1], energy based [2] and wavelet coefficients based features [3], most of which usuallycan be obtained by the Fourier transform (FT) [1], wavelet transform (WT) [3] and some other time–frequency

e front matter r 2006 Elsevier Ltd. All rights reserved.

ssp.2005.12.005

ing author.

ess: [email protected] (Z.K. Peng).

www.elsevier.com/locate/jnlabr/ymssp

ARTICLE IN PRESSZ.K. Peng et al. / Mechanical Systems and Signal Processing 21 (2007) 780–794 781

analysis methods [4]. Besides these common fault features, there are still some other unfamiliar but ofteneffective fault features, such as the fractal dimension [5,6], which often involve the geometrical character ofsimilarity at different scales of the analysed signals. According to the study of Logan and Mathew [6], thefractal dimension can be used as indexes to tell apart the different conditions of working bearings, includingthe normal, outer race fault and inner race fault. Here, we intend to introduce a method for extracting anotherkind of geometrical character parameter, the Lipschitz exponent, from vibration signals. It is well known that,for a rotating machine, its vibration signals with different type of malfunctions often have differentsingularities. Intuitively speaking, the signals with strong singularity are often very disorder and will reversesitself more frequently while the weak singular signals are often very smoothing. For example, among the fourtypical rotating machine faults: rub-impact between stator and rotor, oil whirl, coupling misalignment andimbalance, the rub-impact fault will often generate the most irregular vibration signals, which means thesignals with a rub-impact fault are the most singular, while the vibration signals of an imbalance fault areusually the most smooth (hence regular) and so the least singular. The Lipschitz exponent is a good index forsingularity measure [7]. Usually, a large Lipschitz exponent indicates a regular point in the signal while a smallLipschitz exponent indicates a singular point. In this paper, the wavelet transform modulus maximal(WTMM) method will be used to calculate the Lipschitz exponents of the vibration signals with differentfaults. Further, some parameters of Lipschitz exponents will be extracted to give a comprehensive descriptionof the singularity characters of the vibration signals with different faults. This study will potentially lead to anew method for fault feature extraction.

The Lipschitz exponents have even been used for health monitoring. Hambaba and Huff used the Lipschitzexponents to detect the presence of damage in gears [8]. Robertson, Farrra and Sohn applied Lipschitzexponents to detect damage in structures. The results indicated that the Lipschitz exponent is very damage-sensitive [9]. Later the same authors [10] applied the Lipschitz exponent to various experimental signals toreveal underlying damage causing events and demonstrated that the Lipschitz exponent can be an effectivetool for identifying certain types of events that introduce discontinuities into the measured dynamic responsedata. Based on Lipschitz exponents, Peng et al. [11] had extracted four effective features to identify the shaftorbits of rotating machines. Sun and Tang proposed a singularity analysis based algorithm for bearing defectdiagnosis. They showed that the time location of impact in the bearing vibration signals can be capturedeffectively with their proposed algorithm [12]. Loutridis and Trochidis [13] employed the Lipschitz exponentto investigate two kinds of gear faults, cracked tooth and loss of tooth. And they found that Lipschitzexponent for each type of fault exhibits a constant value, which is not affected by loading conditions orrotational speed.

2. Wavelet modulus maximal [7]

Assume yðtÞ is a smooth function, whose integral is non-zero, that is,Z 1�1

yðtÞ dt ¼ 1 and yðtÞ ¼ Oð1=ð1þ t2ÞÞ.

A smooth function can be viewed as the impulse response of a low-pass filter. Let the wavelet function cðtÞbe the first derivative of the smooth function yðtÞ, that is, cðtÞ ¼ dyðtÞ=dt. The wavelet function cðtÞ mustsatisfy the admissibility condition

Cc ¼

ZjcðoÞj2

jojdoo1, (1)

where cðoÞ ¼RcðtÞe�iot dt. This condition requires that the waveform of the mother wavelet function must

be oscillating.Introduce the scale factor s to the function yðtÞ, and let ysðtÞ ¼ ð1=sÞyðt=sÞ. Let f ðtÞ be a finite-energy

function, that is, f ðtÞ 2 L2ðRÞ. The wavelet transform of f ðtÞ defined with cðtÞ is given by

W sf ðtÞ ¼ f ðtÞcsðtÞ ¼1

s

Z þ1�1

f ðtÞct� t

s

� �dt. (2)

ARTICLE IN PRESSZ.K. Peng et al. / Mechanical Systems and Signal Processing 21 (2007) 780–794782

Here the wavelet transform is defined by convolution, which is different from the standard definition of thewavelet transform [3], but they are the same in nature. By combining the definitions of cðtÞ, ysðtÞ and Eq. (2),we then obtain

W sf ðtÞ ¼ sd

dx½f ðtÞysðtÞ�. (3)

So the wavelet transforms W sf ðtÞ are the first derivatives of f ðtÞ smoothed by ysðtÞ at the scale s. It can beverified that the corresponding relations exist between the singularities of f ðtÞ and the local modulus maximalof W sf ðtÞ. The definitions of the local modulus maximal of the wavelet transform and the local modulusmaximal lines are given as follows.

At scale s0, we call the point ðs0; t0Þ as the local modulus maximal, if for all points belong to either the rightor the left neighbourhood of t0, jW s0f ðtÞjpjW s0f ðt0Þj. The modulus maximal line consists of the points thatare local maximal.

Most important information of a signal is carried by the position and the value of the local modulusmaximal of the wavelet transform. In the field of signal processing, the information carried by the localmodulus maximal of the wavelet transform is used to detect singularities, to eliminate noise and to reconstructsignals. In other words, because of the relation between the modulus maximal of the wavelet and the localsingularities, the signal can be represented and analysed through the local modulus maximal of the wavelettransform.

In mathematics, the local regularity of a function can be measured with Lipschitz exponent a. In the nextsection, the relations between Lipschitz a and the modulus maximal of the wavelet will be briefly introduced.

3. Lipschitz exponent [7]

Many words, such as discontinuity, disorder, smoothness, etc., are often used to describe the geometricalcharacteristics of signals, but these items can only give qualitative descriptions. On the other hand, theLipschitz exponent a can give a quantitative description for the geometrical characteristics of signals, and itcan represent the regularity of functions, i.e. continuity and differentiability.

Let f ðtÞ be a finite-energy function, that is, f ðtÞ 2 L2ðRÞ. We call function f ðtÞ be Lipschitz a(noapnþ 1),at t0, if and only if there exist two constants K and h040, and a polynomial PnðhÞ of order n, such that forhph0,

f ðt0 þ hÞ � PnðhÞjpK jhja. (4)

A higher Lipschitz exponent a implies better regularity of the function f ðtÞ, that is, a more smooth functionf ðtÞ. The classical tool for measuring the Lipschitz exponent a of the function f ðtÞ is to study the asymptoticdecay of its Fourier transform f ðoÞ, but this can only give a global regularity condition because the Fouriertransform cannot localise the information along the spatial variable t. The Fourier transform is therefore notwell adapted to measure the local Lipschitz regularity of functions. On the contrary, the wavelet transform canmeasure the local Lipschitz regularity of functions since the coefficients of the wavelet transform are onlydetermined by the characteristics of the neighbourhood of t0 and the scale s.

Let the wavelet transform of the function f ðtÞ be defined over (a, b), that is, x0 2 ða; bÞ. We assume that thereexist a scale s040 and a constant C, such that for 8t 2 ða; bÞ and sos0, all the modulus maximal of W sf ðtÞ

belong to a cone defined by jt� t0jpCs. Then the function f ðtÞ is Lipschitz a at t0 (a is smaller than theexponent number n of the vanishing moment of the function f ðtÞ), if and only if there exists a constant A suchthat for all modulus maximal in the cone

log2 jW sf ðtÞjplog2 Aþ a log2 s. (5)

Formula (5) shows that the relation between log2jW sf ðtÞj and the scale s is determined by the Lipschitzexponent a, and the relation is expressed especially clear through the wavelet modulus maximal. It shows thatwhen a40 the wavelet modulus maximal increases with the scale s, and when ao0, the wavelet modulusmaximal decreases with the scale s. Formula (5) also offers a simple method to calculate the Lipschitzexponent a of the singularity point, that is, the Lipschitz regularity at t0 is the slope of the straight lines that

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Z.K. Peng et al. / Mechanical Systems and Signal Processing 21 (2007) 780–794 783

remain above log2jW sf ðtÞj, on a logarithmic scale, as follows:

a ¼ log2 jW sf ðtÞj=log2 s. (6)

Now, let us give a little more explanations about the Lipschitz exponent, which is also known as the Holderexponent, is a tool that provides information about the regularity of a signal. In essence, the regularityidentifies to what order a function is differentiable. For instance, if a signal f ðtÞ is only once differentiable atthe t0 point, it has a Lipschitz exponent a ¼ 1, its representative sample is the ramp function shown in Fig. 1.The estimated Lipschitz exponent at Point 2 by Eq. (6) is about 1.0599. If the signal is discontinuous butbounded in the neighbourhood of t0, such as a step function, then the Lipschitz exponent at the t0 point is 0.The Dirac Delta function then has a Lipschitz exponent a ¼ �1 since it is unbounded at the impulse point, asFig. 2 shows, the estimated Lipschitz exponent is about �0.9286. From these examples, one can see that thereis a relationship between the Lipschitz exponent of a function and its derivatives and primitives. Taking thederivative of a function decreases its regularity by 1 and integrating increases it by 1.

4. Practical considerations in computations

For practical computation of the Lipschitz exponents, some problems must be treated carefully.First, we should choose a wavelet function with suitable vanishing moments n. This implies that the wavelet

function satisfiesZ þ1�1

tkcðtÞ dt ¼ 0 8k; 0pkonþ 1: (7)

Theoretically, each singular point of a function has at least one corresponding modulus maximal line, andsometimes may have more than one [7]. The number of the modulus maximal lines relates to the number ofvanishing moments of the wavelet. This often increases linearly with the vanishing moments of the waveletfunction. More modulus maximal lines will make singular points confused and increase the computation cost,

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Z.K. Peng et al. / Mechanical Systems and Signal Processing 21 (2007) 780–794784

therefore, we must choose a wavelet function with as few vanishing moments as possible, but with enoughmoments to detect the Lipschitz exponents of the highest order that we are interested in. For example, whenthe upper bound of the Lipschitz exponent a is up close to the integral number n, it is better to choose awavelet function with at least n vanishing moments. In this paper, the Lipschitz exponents that we areinterested in are within 2, so it is sufficient to use a wavelet function with only two vanishing moment. Here,the Sombrero wavelet function is used, whose Fourier transform is shown as follows:

c_ðoÞ ¼ o2 expð�o2=2Þ. (8)

Second, we should choose a suitable scale range, between which the wavelet coefficients along maximal moduluslines will be used to estimate the Lipschitz exponents by Eq. (6). Notice, not all wavelet coefficients along maximalmodulus lines will obey the rule described as the formula (5), this can be seen clearly in Fig. 3, which will be used toestimate the Lipschitz exponents of a set of vibration signals with an imbalance fault in the following section.Usually, only the wavelet coefficients at high frequency region will satisfy formula (5) and therefore can be used toestimate the Lipschitz exponents, but the coefficients at low frequency region will not. This is because the wavelettransforms in the low frequency range have only poor time resolution, and so the coefficients on the maximalmodulus lines will contain the information of not only their corresponding points but also their neighbouringpoints. On the contrary, the coefficients in the high frequency range will reflect the information of theircorresponding points with relative better precision. After determining the scale range, we can use lines toapproximate the maximal modulus lines within the selected range with the least-square method and take the sloperatios of these obtained lines as the Lipschitz exponents. Each maximal modulus line will give a Lipschitzexponent. Fig. 3 contains 44 maximal modulus lines, and so with them, 44 Lipschitz exponents will be obtained.

The final consideration is that, for singularity detection, it is better to use continuous wavelet transforms,with which a more precise localisation of the singular points can be found. As for the discrete wavelettransforms, even though they can provide an efficient method for computing the wavelet transforms, theircompact forms might allow singularities to go undetected unless they align themselves with precise points inthe time-scale plane.

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5

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log2 (a)

log2

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Fig. 3. The wavelet coefficients along maximal modulus lines (44 lines in total).


A common problem with wavelet transform is end-effects [8], which are the errors in the wavelet transformresulting from performing a convolution on a finite-length signal and are unavoidable for any convolutionoperation. The end-effects will generate some pseudo-maximal modulus lines, for example, the Points 1 and 3in Fig. 1. There are a number of ways to deal with this problem including zero-padding and mirroring.Furthermore, noise components will generate some maximal modulus lines as well, but which often are of veryshort lengths, therefore we can discard them easily just to delete these lines whose lengths are smaller than aconstant threshold.

In the following section, the wavelet transform maximal modulus method will be used to estimate theLipschitz exponents to analyse the singularities of some vibration signals with different types of fault,including the imbalance, oil whirl, coupling misalignment and rub-impact between stator and rotor.

5. Singularity analysis of vibration signals

Since the vibration signals of different faults will have different singularities, and hence different regularities,and the Lipschitz exponents can measure the regularity and singularity of a signal, it is feasible to use theLipschitz exponents to extract the singularity features of the vibration signals, further for the fault diagnostics.To describe the singularities of vibration signals more comprehensively, some parameters will be extractedfrom Lipschitz exponents.

First, a strong singular signal will often contain more singular points, so the number of singular pointsshould be regarded as an index for the singularity measure. The number of Lipschitz exponents is relative tothe number of the singular points in the analysed signal, and therefore the number of Lipschitz exponents canbe used as a substitute for the singularity measure. For rotating machines’ vibration signal analysis, thenumber of Lipschitz exponents per rotation (denoted as N) will be appropriate for the singularity measure,

N ¼Total Number of Lipschitz Exponents

Number of Rotations. (9)

Second, a Lipschitz exponent a can only measure the singularity of its corresponding point. To describe theglobal singularity of a signal, the mean value of all Lipschitz exponents can be used, which is defined asfollows:

ma ¼1

N

XN

i¼1

ai, (10)

where N is the total number of the Lipschitz exponents, and ai is the value of the ith Lipschitz exponent.

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Bearing and supporter

Motor

Stator Rotor

Sensor Coupling

Fig. 4. Experimental test rig.


Finally, vibration signals with different faults often cover a different singularity range that will be reflectedby the variety ranges of their Lipschitz exponents. Here, the standard deviation of Lipschitz exponents will beused to measure their variability. For a set of Lipschitz exponents with a small mean, even if their standarddeviation is small, their relative deviation maybe large and we can still think the amplitude of their variety islarge. But for a set of Lipschitz exponents with a large mean, the condition is the contrary. Therefore, here therelative standard deviation is used, as defined in the following equation:

sa ¼sa

ma, (11)

where sa is the standard deviation of Lipschitz exponents a, defined as

sa ¼1

N

XN

i¼1

ðai � maÞ2

" #1=2. (12)

With these parameters defined by Eqs. (9)–(11), we can characterise the vibration signals’ singularities verywell.

All experimental vibration signals analysed in this paper were sampled from an experimental test rig, shownin Fig. 4, which is composed of the rotor, a driving motor, journal bearings and couplings, with a samplingspeed of 1.6 kHz by non-contact eddy current transducers. The rotating speed was 3000 rpm. In the followingsections, all sets of the vibration signals contain 512 samples, that is, 16 rotations.

5.1. Imbalance

Imbalance [14], the most common problem for rotating machines, will always exist in all rotating machinesin a slight or serious form. Usually, the imbalance will not cause much damage to machines, but when theimbalance becomes serious, it will lead other destructive faults, for example rub-impact between the rotor andstator. Therefore, it would be necessary and useful to detect the imbalance as early as possible and to takesuitable steps to prevent the development of the imbalance.

Generally, the vibration signals with imbalance are very approximate to the sinusoidal signals and so arevery smooth. This means that the imbalance vibration signals are often of strong regularities or weaksingularities. Figs. 5 and 6 are two sets of vibration signals with imbalance and their respective analysis results.

For Fig. 5, this set of vibration signal contains 32 modulus maximal lines that, actually, direct to themaximal and minimal points of the analysed signal with exactness. For their corresponding Lipschitzexponents, the least amin is 1.8017 and the biggest amax is 1.9564 and their mean value ma is 1.8783, whichimplies that this signal has strong regularity and weak singularity. Additionally, the sa of 0.0252 indicates avery small singularity variety in this signal, and it can be seen that, in this case, the maximal points often havebigger Lipschitz exponents than the minimum points, that is, the maximal points are often more regular thanthe minimum points.

Comparing the signal in Fig. 6 with the signal in Fig. 5, it can be found that the signal in Fig. 6 has moremodulus maximal lines, about 2.75 in each rotation, and is less regular. Its minimal Lipschitz exponentamin ¼ 1:5777, maximal exponent amax ¼ 1:8653 and mean value ma ¼ 1:7430. All of them are smaller than

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Fig. 6. A set of imbalance vibration signal and its CWT, MML and Lipschitz exponents (N ¼ 2:75, amin ¼ 1:5777, amax ¼ 1:8653,ma ¼ 1:7430, sa ¼ 0:0734, sa ¼ 0:0421).



those in Fig. 5. Conversely, its relative standard deviation sa of Lipschitz exponents is larger than that of thesignal in Fig. 5. Actually, such a relative standard deviation sa ¼ 0:0421 is still very small, which implies thatthe singularity variety of this signal is very small as well. Obviously, among all the modulus maximal lines, 32long lines direct to the maximal and minimum points exactly just as in Fig. 5, and the other new added linesare often with only relative short length.

With the above analysis, a simple conclusion can be made, that is, for vibration signals with imbalance,usually, there are at least 2 and not more than 3 maximal modulus lines in every rotation, and these lines oftendirect some points with strong regularities and weak singularities since their corresponding Lipschitzexponents are somewhat large. Furthermore, the variability regularity of these are usually very small, whichcan be seen by their standard deviation sa.

5.2. Oil whirl

For a rotor-bearing-foundation system, the self-excited vibration of the oil film force between the bearingand the journal may cause the film to collapse. Under certain conditions, vibration will increase suddenly atthese positions and spread over the whole system in a short time, which will cause strong vibrations. Inaddition, the difference between the whirl frequency and the rotating frequency will cause alternating stressesin the rotor, which may cause more severe harm to the rotor system than the synchronous vibration caused byimbalance does. As an important engineering problem, the oil whirl often happens in practice.

The oil whirl [15] in a rotor-bearing system will often cause some low frequency components in vibrationsignals, such as the 1/2X component, and the signals are usually smooth and hence of weak singularities. Intotal six different sets of oil whirl vibration signals have been analysed in our study. Figs. 7 and 8 show two ofthem and their respective analysis results. It can be seen that the signals with oil whirl have more modulusmaximal lines than those of with imbalance fault. Often more than 2.5 but less than 4 in per rotation.Additionally, their mean values ma are often between 1.5 and 1.8 (the ma is 1.7126 in Fig. 7 and 1.5704 in Fig. 8)and smaller than those of imbalance signals. All these indicate that the signals of oil whirl faults are often more

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Fig. 7. A set of oil whirl vibration signal and its CWT, MML and Lipschitz exponents (N ¼ 2:8125, amin ¼ 1:4312, amax ¼ 1:8903,ma ¼ 1:7126, sa ¼ 0:0853, sa ¼ 0:0498).

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Fig. 8. A set of oil whirl vibration signal and its CWT, MML and Lipschitz exponents (N ¼ 4, amin ¼ 1:0319, amax ¼ 1:7474, ma ¼ 1:5704,sa ¼ 0:1828, sa ¼ 0:1164).


singular or less regular than those of imbalance faults. Even being more singular, in fact, the signals of oilwhirl faults can be still regarded as very regular for their mean values ma are large, often bigger than 1.5.Singularity ranges that the oil whirl vibration signals covered are small as well, but somewhat larger than thoseof the imbalance fault. Through comparing their respective relative standard deviations sa, we can see this.

In conclusion, the oil whirl vibration signals are often very regular, but a little more singular than theimbalance vibration signals. In every rotation, more than 2 but at most 4 maximal modulus lines would becontained, the mean values of their Lipschitz exponents are often bigger than 1.5 but smaller than 1.8.Additionally, their singularity ranges are usually small but bigger than those of the imbalance.

5.3. Coupling misalignment

Coupling misalignment [16], one of the most familiar faults, often denotes the slant or misalignmentbetween the axes of two nearby rotors. When a misalignment fault occurs, a series of undesired dynamicresponses will appear in the rotor system, such as coupling deflection, bearing abrasion and oil collapsing.Therefore, it is very important to find misalignment as early as possible for ensuring the safe running of themachines.

Coupling misalignment will often cause some high frequency components in vibration signals, in which thetypical component is the 2X component. In this study, two sets of misalignment signals have been studied. Thesignal in Fig. 9 was sampled under a slight fault condition and the other in Fig. 10 was sampled under a seriousfault condition. Obviously, the singularities have been enhanced with the fault severity increasing; both thenumber of Lipschitz exponents and some other parameters extracted from Lipschitz exponents have shownthis. For the slight fault condition, the vibration signal contains only 4 modulus maximal lines per rotation,but the signal for the serious fault condition has more than 7 lines, which means the signal in the serious faultcondition has more singular points. In addition, with the fault severity increasing, the signal’s Lipschitzexponents decrease. It can be seen that the minimal and maximal Lipschitz exponents of the signal with slight

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Fig. 9. A set of misalignment vibration signal and its CWT, MML and Lipschitz exponents (N ¼ 4, amin ¼ 1:0982, amax ¼ 1:7616,ma ¼ 1:5496, sa ¼ 0:1566, sa ¼ 0:1010).

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Fig. 10. A set of misalignment vibration signal and its CWT, MML and Lipschitz exponents (N ¼ 7:5625, amin ¼ 0:3546, amax ¼ 1:3285,ma ¼ 0:9118, sa ¼ 0:2020, sa ¼ 0:2215).



fault are all bigger than those of with the serious fault, as well as the mean values ma. In the serious condition,the minimal Lipschitz exponent amin is even below 0.5, and the mean value ma is below 1. These are impossiblefor the oil whirl and imbalance. In addition, the signal with serious misalignment will cover a wider singularityrange than that with slight fault as well, as the relative standard deviations sa shows.

In conclusion, for coupling misalignment fault, the vibration signals are often more singular than those withimbalance or oil whirl, and with the fault severity increasing, the signal’s singularities will increase as well. Thesignal will often contain more than 4 but less than 8 singular points at each rotation. The relative standarddeviations sa are always bigger than 0.1 and usually smaller than 0.25.

5.4. Rub-impact

For rotating machines, rub-impact [17,18] between rotor and stator is a kind of serious malfunction, whichoften happens at the positions with small clearances. This fault will present a serious hazard to machines. Forexample, the rub-impact between blades and seals could make the blade break down. The factors that influencerub-impact between rotor and stator are complicated, and the vibration phenomenon of a rub-impact rotorsystem is also complicated. A rub fault will cause not only periodic motions but also quasi-periodic motions.

The rubbing caused impacts occur between a rotor and a stator can be regarded as multiple impulsive forces actingon the rotor and the stator. For rotating machines, these short and sudden impulsive forces will lead to vibrationsignals containing many impulse like components that usually are very singular. Therefore, we can image that thesignals with rub-impact faults will be more singular than the signals with others fault, and in fact is just the case.

Figs. 11–13 give three sets of vibration signals with rub-impact faults, among which the signal in Fig. 11 wassampled in slight rub-impact condition and the other two were sampled in serious condition. It can be seenthat, just like the coupling misalignment, the rub-impact vibration signals’ singularities will enhance with thefault severity increasing, which is reflected by the increasing of the number of Lipschitz exponents in perrotation N and the decreasing of the mean values ma. For the slight rub signal in Fig. 11, the N is 8.0625 andthe ma is 0.9353, but for the serious rub signals in Figs. 12 and 13, the N are 11.5 and 12.125 and the ma are

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Fig. 11. A set of rub-impact vibration signal and its CWT, MML and Lipschitz exponents (N ¼ 8:0625, amin ¼ �0:0775, amax ¼ 1:3124,ma ¼ 0:9353, sa ¼ 0:1871, sa ¼ 0:2001).

ARTICLE IN PRESS

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0.6112 and 0.5074, respectively. Furthermore, the relative standard deviations sa increase with the faultseverity increasing, which indicates that a signal in a serious fault conditions will often cover a widersingularity range than that in slight fault condition. In all three sets of rub-impact vibration signals, there aresome points whose Lipschitz exponents are smaller than 0, which means that the signals are very singular atthese points, and without exception, their maximal Lipschitz exponents amax are all not more than 1.5 andtheir mean values ma are all not more than 1. All these indicate that the signals with rub-impact faults areusually very singular. In fact, the rub-impact vibration signals are the most singular among the four kinds offault vibration signals analysed here. The slight rub-impact vibration signal will have similar singularities tothe signal with serious coupling misalignment that is the most singular among those signals analysedpreviously, and the rub-impact signal has a large sa, often bigger than 0.2 and even bigger than 0.5.

With above analysis, we can know that, among the four kinds of fault including the imbalance, oil whirl,coupling misalignment and rub-impact, the rub-impact vibration signals are often the most singular. Their maare always smaller than 1 and their sa are often bigger than 0.2. At every rotation, they usually contain at least8 singular points on average and even more. Additionally, they may contain some very singular points whoseLipschitz exponents are smaller than 0. With the rub-impact degree becoming serious, the vibration signalswill become more singular as well.

6. Conclusions and prospects

In this paper, the WTMM method has been used to analyse the singularity characteristics of the rotatingmachine’s vibration signals with different faults, including imbalance, oil whirl, coupling misalignment andrub-impact. This study is mainly based on the fact that the Lipschitz exponent, which can be calculated withthe WTMM method easily, can give a quantitative analysis for the signal’s singularity. For analysis, someparameters are extracted, such as the number of Lipschitz exponents per rotation, denoted as N, the meanvalue of the Lipschitz exponents ma and the relative standard deviation of Lipschitz exponents sa. With theseparameters, we can describe the signal singularity easily and comprehensively.

With the previous analysis results, it can be seen that, among the four kinds of fault including theimbalance, oil whirl, coupling misalignment and rub-impact, the rub-impact vibration signals have the largestN, hence the most of maximal modulus lines in per rotation, and the smallest ma, and therefore the rub-impactvibration signals are the most singular. The coupling misalignment’s are the next. On the other hand, theimbalance vibration signals have the smallest N and the largest ma, so they are the most regular. The oil whirlvibration signals are the second most regular. Additionally, as the sa shows, the rub-impact vibration signalscover widest singularity ranges, the coupling misalignment vibration signals cover the second widestsingularity ranges while the narrowest are the imbalance vibration signals. Actually, for these kinds of fault, itcan be concluded that the more singular the vibration signals are, the more wide singularity ranges they willcover. As the aforementioned analysis, we can also see that for a special fault, the more serious the faultseverity is, the more singular its vibration signal will be and the more wide singularity range its signal willcover, especially for the rub-impact and the coupling misalignment fault.

Actually, we can find that these singularity based parameters, defined in Eqs. (9)–(11), are a set of excellentfault features, which have separated the four kinds of fault very well. For fault diagnosis, a set of effective faultfeatures is very important, with which it only need a simple classifier in the classifying operation to give anaccurate diagnosis result, and therefore it can be expected that these singularity based features will play animportant role in the fault diagnostics of rotating machines in the future.

Furthermore, the results show that, with the fault severity increasing, the vibration signals’ singularities andsingularity ranges will increase as well, and therefore one can evaluate the fault severity through measuring thevibration signals’ singularities and singularity ranges.

Acknowledgements

The work described in this paper was supported partially by the Trans-Century Training ProgrammeFoundation for the Talents by the Ministry of Education, China and partially by a grant from the ResearchGrants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 1100/02E).


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