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

Transcription

1 Mechanical Systems and Signal Processing () 4 Mechanical Systems and Signal Processing Singularity analysis of the vibration signals by means of wavelet modulus maximal method Z.K. Peng a,, F.L. Chu a, Peter W. Tse b a Department of Precision Instruments, Tsinghua University, Beijing 4, PR China b Smart Asset Management Laboratory, City University of Hong Kong, Tat Chee Ave, Hong Kong, PR China Received July ; received in revised form November ; accepted December Available online 3 January 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 m a and the relative standard deviation s a 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 Elsevier Ltd. All rights reserved.. 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 analysis based methods have been the most widely used ones. Essentially, the fault diagnosis is a pattern recognition problem, for which the key step is to extract useful fault features from vibration signals through some suitable signal processing methods. Now, various kinds of fault features are available. The well accepted features include frequency based [], energy based [] and wavelet coefficients based features [3], most of which usually can be obtained by the Fourier transform (FT) [], wavelet transform (WT) [3] and some other time frequency Corresponding author. address: pengzhike@tsinghua.org.cn (Z.K. Peng). -3/$ - see front matter r Elsevier Ltd. All rights reserved. doi:./j.ymssp...

2 analysis methods [4]. Besides these common fault features, there are still some other unfamiliar but often effective fault features, such as the fractal dimension [,], which often involve the geometrical character of similarity at different scales of the analysed signals. According to the study of Logan and Mathew [], the fractal dimension can be used as indexes to tell apart the different conditions of working bearings, including the normal, outer race fault and inner race fault. Here, we intend to introduce a method for extracting another kind 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 different singularities. Intuitively speaking, the signals with strong singularity are often very disorder and will reverses itself more frequently while the weak singular signals are often very smoothing. For example, among the four typical rotating machine faults: rub-impact between stator and rotor, oil whirl, coupling misalignment and imbalance, the rub-impact fault will often generate the most irregular vibration signals, which means the signals with a rub-impact fault are the most singular, while the vibration signals of an imbalance fault are usually the most smooth (hence regular) and so the least singular. The Lipschitz exponent is a good index for singularity measure []. Usually, a large Lipschitz exponent indicates a regular point in the signal while a small Lipschitz 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 different faults. Further, some parameters of Lipschitz exponents will be extracted to give a comprehensive description of the singularity characters of the vibration signals with different faults. This study will potentially lead to a new method for fault feature extraction. The Lipschitz exponents have even been used for health monitoring. Hambaba and Huff used the Lipschitz exponents to detect the presence of damage in gears []. Robertson, Farrra and Sohn applied Lipschitz exponents to detect damage in structures. The results indicated that the Lipschitz exponent is very damagesensitive []. Later the same authors [] applied the Lipschitz exponent to various experimental signals to reveal underlying damage causing events and demonstrated that the Lipschitz exponent can be an effective tool for identifying certain types of events that introduce discontinuities into the measured dynamic response data. Based on Lipschitz exponents, Peng et al. [] had extracted four effective features to identify the shaft orbits of rotating machines. Sun and Tang proposed a singularity analysis based algorithm for bearing defect diagnosis. They showed that the time location of impact in the bearing vibration signals can be captured effectively with their proposed algorithm []. Loutridis and Trochidis [3] employed the Lipschitz exponent to investigate two kinds of gear faults, cracked tooth and loss of tooth. And they found that Lipschitz exponent for each type of fault exhibits a constant value, which is not affected by loading conditions or rotational speed.. Wavelet modulus maximal [] Assume yðtþ is a smooth function, whose integral is non-zero, that is, Z yðtþ dt ¼ and yðtþ ¼Oð=ð þ t ÞÞ. A smooth function can be viewed as the impulse response of a low-pass filter. Let the wavelet function cðtþ be the first derivative of the smooth function yðtþ, that is, cðtþ¼dyðtþ=dt. The wavelet function cðtþ must satisfy the admissibility condition Z j ^cðoþj C c ¼ d o o, () joj where ^cðoþ ¼ R cðtþe iot d t. 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 y s ðtþ ¼ð=sÞyðt=sÞ. Let f ðtþ be a finite-energy function, that is, f ðtþ L ðrþ. The wavelet transform of f ðtþ defined with cðtþ is given by W s f ðtþ ¼f ðtþc s ðtþ ¼ Z þ f ðtþc t t dt. () s s Z.K. Peng et al. / Mechanical Systems and Signal Processing () 4

3 Z.K. Peng et al. / Mechanical Systems and Signal Processing () 4 Here the wavelet transform is defined by convolution, which is different from the standard definition of the wavelet transform [3], but they are the same in nature. By combining the definitions of cðtþ, y s ðtþ and Eq. (), we then obtain W s f ðtþ ¼s d dx ½f ðtþy sðtþš. (3) So the wavelet transforms W s f ðtþ are the first derivatives of f ðtþ smoothed by y s ðtþ at the scale s. It can be verified that the corresponding relations exist between the singularities of f ðtþ and the local modulus maximal of W s f ðtþ. The definitions of the local modulus maximal of the wavelet transform and the local modulus maximal lines are given as follows. At scale s, we call the point ðs ; t Þ as the local modulus maximal, if for all points belong to either the right or the left neighbourhood of t, jw s f ðtþjpjw s f ðt Þj. The modulus maximal line consists of the points that are local maximal. Most important information of a signal is carried by the position and the value of the local modulus maximal of the wavelet transform. In the field of signal processing, the information carried by the local modulus maximal of the wavelet transform is used to detect singularities, to eliminate noise and to reconstruct signals. In other words, because of the relation between the modulus maximal of the wavelet and the local singularities, the signal can be represented and analysed through the local modulus maximal of the wavelet transform. In mathematics, the local regularity of a function can be measured with Lipschitz exponent a. In the next section, the relations between Lipschitz a and the modulus maximal of the wavelet will be briefly introduced. 3. Lipschitz exponent [] Many words, such as discontinuity, disorder, smoothness, etc., are often used to describe the geometrical characteristics of signals, but these items can only give qualitative descriptions. On the other hand, the Lipschitz exponent a can give a quantitative description for the geometrical characteristics of signals, and it can represent the regularity of functions, i.e. continuity and differentiability. Let f ðtþ be a finite-energy function, that is, f ðtþ L ðrþ. We call function f ðtþ be Lipschitz a(noapn þ ), at t, if and only if there exist two constants K and h 4, and a polynomial P n ðhþ of order n, such that for hph, f ðt þ hþ P n ðhþjpkjhj a. (4) A higher Lipschitz exponent a implies better regularity of the function f ðtþ, that is, a more smooth function f ðtþ. The classical tool for measuring the Lipschitz exponent a of the function f ðtþ is to study the asymptotic decay of its Fourier transform f ^ ðoþ, but this can only give a global regularity condition because the Fourier transform cannot localise the information along the spatial variable t. The Fourier transform is therefore not well adapted to measure the local Lipschitz regularity of functions. On the contrary, the wavelet transform can measure the local Lipschitz regularity of functions since the coefficients of the wavelet transform are only determined by the characteristics of the neighbourhood of t and the scale s. Let the wavelet transform of the function f ðtþ be defined over (a, b), that is, x ða; bþ. We assume that there exist a scale s 4 and a constant C, such that for t ða; bþ and sos, all the modulus maximal of W s f ðtþ belong to a cone defined by jt t jpcs. Then the function f ðtþ is Lipschitz a at t (a is smaller than the exponent number n of the vanishing moment of the function f ðtþ), if and only if there exists a constant A such that for all modulus maximal in the cone log jw s f ðtþjplog A þ a log s. () Formula () shows that the relation between log jw s f ðtþj and the scale s is determined by the Lipschitz exponent a, and the relation is expressed especially clear through the wavelet modulus maximal. It shows that when a4 the wavelet modulus maximal increases with the scale s, and when ao, the wavelet modulus maximal decreases with the scale s. Formula () also offers a simple method to calculate the Lipschitz exponent a of the singularity point, that is, the Lipschitz regularity at t is the slope of the straight lines that

4 Z.K. Peng et al. / Mechanical Systems and Signal Processing () 4 3 Point Point Point log (wt) 4 - Point Point -4 Point log (a) Fig.. The ramp function and its analysis results (a ¼ :344, a ¼ :, a 3 ¼ :4). remain above log jw s f ðtþj, on a logarithmic scale, as follows: a ¼ log jw s f ðtþj=log s. () Now, let us give a little more explanations about the Lipschitz exponent, which is also known as the Ho lder exponent, is a tool that provides information about the regularity of a signal. In essence, the regularity identifies to what order a function is differentiable. For instance, if a signal f ðtþ is only once differentiable at the t point, it has a Lipschitz exponent a ¼, its representative sample is the ramp function shown in Fig.. The estimated Lipschitz exponent at Point by Eq. () is about.. If the signal is discontinuous but bounded in the neighbourhood of t, such as a step function, then the Lipschitz exponent at the t point is. The Dirac Delta function then has a Lipschitz exponent a ¼ since it is unbounded at the impulse point, as Fig. shows, the estimated Lipschitz exponent is about.. From these examples, one can see that there is a relationship between the Lipschitz exponent of a function and its derivatives and primitives. Taking the derivative of a function decreases its regularity by and integrating increases it by. 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 satisfies Z þ t k cðtþ dt ¼ k; pkon þ : () Theoretically, each singular point of a function has at least one corresponding modulus maximal line, and sometimes may have more than one []. The number of the modulus maximal lines relates to the number of vanishing moments of the wavelet. This often increases linearly with the vanishing moments of the wavelet function. More modulus maximal lines will make singular points confused and increase the computation cost,

5 4 Z.K. Peng et al. / Mechanical Systems and Signal Processing () log (wt) log (a) Fig.. The Dirac Delta function and its analysis results (a ¼ :). therefore, we must choose a wavelet function with as few vanishing moments as possible, but with enough moments to detect the Lipschitz exponents of the highest order that we are interested in. For example, when the upper bound of the Lipschitz exponent a is up close to the integral number n, it is better to choose a wavelet function with at least n vanishing moments. In this paper, the Lipschitz exponents that we are interested in are within, 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 ¼o expð o =Þ. () Second, we should choose a suitable scale range, between which the wavelet coefficients along maximal modulus lines will be used to estimate the Lipschitz exponents by Eq.().Notice,notallwavelet coefficients along maximal modulus lines will obey the rule described as the formula (), this can be seen clearly in Fig. 3, which will be used to estimate 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 () and therefore can be used to estimate the Lipschitz exponents, but the coefficients at low frequency region will not. This is because the wavelet transforms in the low frequency range have only poor time resolution, and so the coefficients on the maximal modulus lines will contain the information of not only their corresponding points but also their neighbouring points. On the contrary, the coefficients in the high frequency range will reflect the information of their corresponding points with relative better precision. After determining the scale range, we can use lines to approximate the maximal modulus lines within the selected range with the least-square method and take the slope ratios of these obtained lines as the Lipschitz exponents. Each maximal modulus line will give a Lipschitz exponent. 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 wavelet transforms, even though they can provide an efficient method for computing the wavelet transforms, their compact forms might allow singularities to go undetected unless they align themselves with precise points in the time-scale plane.

6 Z.K. Peng et al. / Mechanical Systems and Signal Processing () 4 log (wt) - I High Frequency Region II Low Frequency Region log (a) Fig. 3. The wavelet coefficients along maximal modulus lines (44 lines in total). A common problem with wavelet transform is end-effects [], which are the errors in the wavelet transform resulting from performing a convolution on a finite-length signal and are unavoidable for any convolution operation. The end-effects will generate some pseudo-maximal modulus lines, for example, the Points and 3 in Fig.. 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 very short lengths, therefore we can discard them easily just to delete these lines whose lengths are smaller than a constant threshold. In the following section, the wavelet transform maximal modulus method will be used to estimate the Lipschitz 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.. 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 the Lipschitz 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 extracted from Lipschitz exponents. First, a strong singular signal will often contain more singular points, so the number of singular points should be regarded as an index for the singularity measure. The number of Lipschitz exponents is relative to the number of the singular points in the analysed signal, and therefore the number of Lipschitz exponents can be used as a substitute for the singularity measure. For rotating machines vibration signal analysis, the number of Lipschitz exponents per rotation (denoted as N) will be appropriate for the singularity measure, Total Number of Lipschitz Exponents N ¼. () Number of Rotations Second, a Lipschitz exponent a can only measure the singularity of its corresponding point. To describe the global singularity of a signal, the mean value of all Lipschitz exponents can be used, which is defined as follows: m a ¼ N X N i¼ a i, () where N is the total number of the Lipschitz exponents, and a i is the value of the ith Lipschitz exponent.

7 Z.K. Peng et al. / Mechanical Systems and Signal Processing () 4 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 reflected by the variety ranges of their Lipschitz exponents. Here, the standard deviation of Lipschitz exponents will be used to measure their variability. For a set of Lipschitz exponents with a small mean, even if their standard deviation is small, their relative deviation maybe large and we can still think the amplitude of their variety is large. But for a set of Lipschitz exponents with a large mean, the condition is the contrary. Therefore, here the relative standard deviation is used, as defined in the following equation: s a ¼ s a, () m a where s a is the standard deviation of Lipschitz exponents a, defined as " s a ¼ X # N = ða i m N a Þ. () i¼ With these parameters defined by Eqs. () (), we can characterise the vibration signals singularities very well. All experimental vibration signals analysed in this paper were sampled from an experimental test rig, shown in Fig. 4, which is composed of the rotor, a driving motor, journal bearings and couplings, with a sampling speed of. khz by non-contact eddy current transducers. The rotating speed was 3 rpm. In the following sections, all sets of the vibration signals contain samples, that is, rotations... Imbalance Imbalance [4], the most common problem for rotating machines, will always exist in all rotating machines in a slight or serious form. Usually, the imbalance will not cause much damage to machines, but when the imbalance becomes serious, it will lead other destructive faults, for example rub-impact between the rotor and stator. Therefore, it would be necessary and useful to detect the imbalance as early as possible and to take suitable steps to prevent the development of the imbalance. Generally, the vibration signals with imbalance are very approximate to the sinusoidal signals and so are very smooth. This means that the imbalance vibration signals are often of strong regularities or weak singularities. Figs. and are two sets of vibration signals with imbalance and their respective analysis results. For Fig., this set of vibration signal contains 3 modulus maximal lines that, actually, direct to the maximal and minimal points of the analysed signal with exactness. For their corresponding Lipschitz exponents, the least a min is. and the biggest a max is.4 and their mean value m a is.3, which implies that this signal has strong regularity and weak singularity. Additionally, the s a of. indicates a very small singularity variety in this signal, and it can be seen that, in this case, the maximal points often have bigger Lipschitz exponents than the minimum points, that is, the maximal points are often more regular than the minimum points. Comparing the signal in Fig. with the signal in Fig., it can be found that the signal in Fig. has more modulus maximal lines, about. in each rotation, and is less regular. Its minimal Lipschitz exponent a min ¼ :, maximal exponent a max ¼ :3 and mean value m a ¼ :43. All of them are smaller than

8 Z.K. Peng et al. / Mechanical Systems and Signal Processing () Fig.. A set of imbalance vibration signal and its CWT, MML and Lipschitz exponents ( N ¼, a min ¼ :, a max ¼ :4, m a ¼ :3, s a ¼ :43, s a ¼ :) Fig.. A set of imbalance vibration signal and its CWT, MML and Lipschitz exponents ( N ¼ :, a min ¼ :, a max ¼ :3, m a ¼ :43, s a ¼ :34, s a ¼ :4).

9 Z.K. Peng et al. / Mechanical Systems and Signal Processing () 4 those in Fig.. Conversely, its relative standard deviation s a of Lipschitz exponents is larger than that of the signal in Fig.. Actually, such a relative standard deviation s a ¼ :4 is still very small, which implies that the singularity variety of this signal is very small as well. Obviously, among all the modulus maximal lines, 3 long lines direct to the maximal and minimum points exactly just as in Fig., and the other new added lines are 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 and not more than 3 maximal modulus lines in every rotation, and these lines often direct some points with strong regularities and weak singularities since their corresponding Lipschitz exponents are somewhat large. Furthermore, the variability regularity of these are usually very small, which can be seen by their standard deviation s a... Oil whirl For a rotor-bearing-foundation system, the self-excited vibration of the oil film force between the bearing and the journal may cause the film to collapse. Under certain conditions, vibration will increase suddenly at these positions and spread over the whole system in a short time, which will cause strong vibrations. In addition, the difference between the whirl frequency and the rotating frequency will cause alternating stresses in the rotor, which may cause more severe harm to the rotor system than the synchronous vibration caused by imbalance does. As an important engineering problem, the oil whirl often happens in practice. The oil whirl [] in a rotor-bearing system will often cause some low frequency components in vibration signals, such as the /X component, and the signals are usually smooth and hence of weak singularities. In total six different sets of oil whirl vibration signals have been analysed in our study. Figs. and show two of them and their respective analysis results. It can be seen that the signals with oil whirl have more modulus maximal lines than those of with imbalance fault. Often more than. but less than 4 in per rotation. Additionally, their mean values m a are often between. and. (the m a is. in Fig. and.4 in Fig. ) and smaller than those of imbalance signals. All these indicate that the signals of oil whirl faults are often more Samples Samples Fig.. A set of oil whirl vibration signal and its CWT, MML and Lipschitz exponents ( N ¼ :, a min ¼ :43, a max ¼ :3, m a ¼ :, s a ¼ :3, s a ¼ :4).

10 Z.K. Peng et al. / Mechanical Systems and Signal Processing () 4 Samples Samples Fig.. A set of oil whirl vibration signal and its CWT, MML and Lipschitz exponents ( N ¼ 4, a min ¼ :3, a max ¼ :44, m a ¼ :4, s a ¼ :, s a ¼ :4). singular or less regular than those of imbalance faults. Even being more singular, in fact, the signals of oil whirl faults can be still regarded as very regular for their mean values m a are large, often bigger than.. Singularity ranges that the oil whirl vibration signals covered are small as well, but somewhat larger than those of the imbalance fault. Through comparing their respective relative standard deviations s a, we can see this. In conclusion, the oil whirl vibration signals are often very regular, but a little more singular than the imbalance vibration signals. In every rotation, more than but at most 4 maximal modulus lines would be contained, the mean values of their Lipschitz exponents are often bigger than. but smaller than.. Additionally, their singularity ranges are usually small but bigger than those of the imbalance..3. Coupling misalignment Coupling misalignment [], one of the most familiar faults, often denotes the slant or misalignment between the axes of two nearby rotors. When a misalignment fault occurs, a series of undesired dynamic responses 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 the machines. Coupling misalignment will often cause some high frequency components in vibration signals, in which the typical component is the X component. In this study, two sets of misalignment signals have been studied. The signal in Fig. was sampled under a slight fault condition and the other in Fig. was sampled under a serious fault condition. Obviously, the singularities have been enhanced with the fault severity increasing; both the number of Lipschitz exponents and some other parameters extracted from Lipschitz exponents have shown this. 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 lines, which means the signal in the serious fault condition has more singular points. In addition, with the fault severity increasing, the signal s Lipschitz exponents decrease. It can be seen that the minimal and maximal Lipschitz exponents of the signal with slight

11 Z.K. Peng et al. / Mechanical Systems and Signal Processing () Fig.. A set of misalignment vibration signal and its CWT, MML and Lipschitz exponents ( N ¼ 4, a min ¼ :, a max ¼ :, m a ¼ :4, s a ¼ :, s a ¼ :) Fig.. A set of misalignment vibration signal and its CWT, MML and Lipschitz exponents ( N ¼ :, a min ¼ :34, a max ¼ :3, m a ¼ :, s a ¼ :, s a ¼ :).

12 fault are all bigger than those of with the serious fault, as well as the mean values m a. In the serious condition, the minimal Lipschitz exponent a min is even below., and the mean value m a is below. These are impossible for the oil whirl and imbalance. In addition, the signal with serious misalignment will cover a wider singularity range than that with slight fault as well, as the relative standard deviations s a shows. In conclusion, for coupling misalignment fault, the vibration signals are often more singular than those with imbalance or oil whirl, and with the fault severity increasing, the signal s singularities will increase as well. The signal will often contain more than 4 but less than singular points at each rotation. The relative standard deviations s a are always bigger than. and usually smaller than...4. Rub-impact Z.K. Peng et al. / Mechanical Systems and Signal Processing () 4 For rotating machines, rub-impact [,] between rotor and stator is a kind of serious malfunction, which often happens at the positions with small clearances. This fault will present a serious hazard to machines. For example, the rub-impact between blades and seals could make the blade break down. The factors that influence rub-impact between rotor and stator are complicated, and the vibration phenomenon of a rub-impact rotor system 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 acting on the rotor and the stator. For rotating machines, these short and sudden impulsive forces will lead to vibration signals containing many impulse like components that usually are very singular. Therefore, we can image that the signals with rub-impact faults will be more singular than the signals with others fault, and in fact is just the case. Figs. 3 give three sets of vibration signals with rub-impact faults, among which the signal in Fig. was sampled in slight rub-impact condition and the other two were sampled in serious condition. It can be seen that, just like the coupling misalignment, the rub-impact vibration signals singularities will enhance with the fault severity increasing, which is reflected by the increasing of the number of Lipschitz exponents in per rotation N and the decreasing of the mean values m a. For the slight rub signal in Fig., the N is. and the m a is.33, but for the serious rub signals in Figs. and 3, the N are. and. and the m a are Fig.. A set of rub-impact vibration signal and its CWT, MML and Lipschitz exponents ( N ¼ :, a min ¼ :, a max ¼ :34, m a ¼ :33, s a ¼ :, s a ¼ :).

13 Z.K. Peng et al. / Mechanical Systems and Signal Processing () Fig.. A set of rub-impact vibration signal and its CWT, MML and Lipschitz exponents ( N ¼ :, a min ¼ :3, a max ¼ :, m a ¼ :, s a ¼ :, s a ¼ :4) Fig. 3. A set of rub-impact vibration signal and its CWT, MML and Lipschitz exponents ( N ¼ :, a min ¼ :4, a max ¼ :, m a ¼ :4, s a ¼ :3, s a ¼ :44).

14 . and.4, respectively. Furthermore, the relative standard deviations s a increase with the fault severity increasing, which indicates that a signal in a serious fault conditions will often cover a wider singularity range than that in slight fault condition. In all three sets of rub-impact vibration signals, there are some points whose Lipschitz exponents are smaller than, which means that the signals are very singular at these points, and without exception, their maximal Lipschitz exponents a max are all not more than. and their mean values m a are all not more than. All these indicate that the signals with rub-impact faults are usually very singular. In fact, the rub-impact vibration signals are the most singular among the four kinds of fault vibration signals analysed here. The slight rub-impact vibration signal will have similar singularities to the signal with serious coupling misalignment that is the most singular among those signals analysed previously, and the rub-impact signal has a large s a, often bigger than. and even bigger than.. 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 m a are always smaller than and their s a are often bigger than.. At every rotation, they usually contain at least singular points on average and even more. Additionally, they may contain some very singular points whose Lipschitz exponents are smaller than. With the rub-impact degree becoming serious, the vibration signals will become more singular as well.. Conclusions and prospects In this paper, the WTMM method has been used to analyse the singularity characteristics of the rotating machine s vibration signals with different faults, including imbalance, oil whirl, coupling misalignment and rub-impact. This study is mainly based on the fact that the Lipschitz exponent, which can be calculated with the WTMM method easily, can give a quantitative analysis for the signal s singularity. For analysis, some parameters are extracted, such as the number of Lipschitz exponents per rotation, denoted as N, the mean value of the Lipschitz exponents m a and the relative standard deviation of Lipschitz exponents s a. With these parameters, 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 the imbalance, oil whirl, coupling misalignment and rub-impact, the rub-impact vibration signals have the largest N, hence the most of maximal modulus lines in per rotation, and the smallest m a, and therefore the rub-impact vibration signals are the most singular. The coupling misalignment s are the next. On the other hand, the imbalance vibration signals have the smallest N and the largest m a, so they are the most regular. The oil whirl vibration signals are the second most regular. Additionally, as the s a shows, the rub-impact vibration signals cover widest singularity ranges, the coupling misalignment vibration signals cover the second widest singularity ranges while the narrowest are the imbalance vibration signals. Actually, for these kinds of fault, it can be concluded that the more singular the vibration signals are, the more wide singularity ranges they will cover. As the aforementioned analysis, we can also see that for a special fault, the more serious the fault severity is, the more singular its vibration signal will be and the more wide singularity range its signal will cover, especially for the rub-impact and the coupling misalignment fault. Actually, we can find that these singularity based parameters, defined in Eqs. () (), are a set of excellent fault features, which have separated the four kinds of fault very well. For fault diagnosis, a set of effective fault features is very important, with which it only need a simple classifier in the classifying operation to give an accurate diagnosis result, and therefore it can be expected that these singularity based features will play an important 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 and singularity ranges will increase as well, and therefore one can evaluate the fault severity through measuring the vibration signals singularities and singularity ranges. Acknowledgements Z.K. Peng et al. / Mechanical Systems and Signal Processing () 4 3 The work described in this paper was supported partially by the Trans-Century Training Programme Foundation for the Talents by the Ministry of Education, China and partially by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU /E).

15 4 Z.K. Peng et al. / Mechanical Systems and Signal Processing () 4 References [] A. Muszynska, Rotor-to-stationary element rub-related vibration phenomena in rotating machinery literature survey, Shock and Vibration Digest () 3. [] R. Al-Balushi, B. Samanta, Gear fault diagnosis using energy-based features of acoustic emission signals, Journal of Systems and Control Engineering () 4 3. [3] W.J. Wang, P.D. McFadden, Application of wavelets to gearbox vibration signals for fault detection, Journal of Sound and Vibration () 3. [4] N. Baydar, A. Ball, A comparative study of acoustic and vibration signals in detection of gear failures using Wigner Ville distribution, Mechanical Systems and Signal Processing (). [] C. Craig, R.D. Neilson, J. Penman, The use of correlation dimension in condition monitoring of systems with clearance, Journal of Sound and Vibration 3 (). [] D. Logan, J. Mathew, Using the correlation dimension for vibration fault diagnosis of rolling element bearings.. Basic concepts, Mechanical Systems and Signal Processing () 4. [] S. Mallat, W.L. Hwang, Singularity detection and processing with wavelet, IEEE Transactions on Information Theory 3 () 43. [] A. Hambaba, A.E. Huff, Multiresolution error detection on early fatigue cracks in gears, IEEE Aerospace Conference Proceedings () 3 3. [] A.N. Robertson, C.R. Farrar, H. Sohn, Singularity detection for structural health monitoring using holder exponents, Mechanical Systems and Signal Processing (3) 3 4. [] H. Sohn, A.N. Robertson, C.R. Farrar, Holder exponent analysis for discontinuity detection, Structural Engineering and Mechanics (4) 4 4. [] Z. Peng, Y. He, Z. Chen, F. Chu, Identification of the shaft orbit for rotating machines using wavelet modulus maxima, Mechanical Systems and Signal Processing () 3 3. [] Q. Sun, Y. Tang, Singularity analysis using continuous wavelet transform for bearing fault diagnosis, Mechanical Systems and Signal Processing () 4. [3] S. Loutridis, A. Trochidis, Classification of gear faults using Hoelder exponents, Mechanical Systems and Signal Processing (4) 3. [4] P. Sundararajan, S.T. Noah, An algorithm for response and stability of large order non-linear systems application to rotor systems, Journal of Sound and Vibration 4 () 3. [] S. Theodossiades, S. Natsiavas, On geared rotordynamic systems with oil journal bearings, Journal of Sound and Vibration 43 () 4. [] Y.S. Lee, C.W. Lee, Modelling and vibration analysis of misaligned rotor-ball bearing systems, Journal of Sound and Vibration 4 () 3. [] Z. Peng, Y. He, Q. Lu, F. Chu, Feature extraction of the rub-impact rotor system by means of wavelet analysis, Journal of Sound and Vibration (3). [] F. Chu, Z. Zhang, Bifurcation and chaos in a rub-impact Jeffcott rotor system, Journal of Sound and Vibration ().