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WAVELET TRANSFORMS AND ITS APPLICATION TO FAULT DETECTION ASHISH K DARPE DEPARTMENT OF MECHANICAL ENGINEERING, IIT DELHI 1. INTRODUCTION Machine vibration analysis is one of the important tools for faults identification. There are two types of conventional analysis methods: time domain and the frequency domain. The frequency domain analysis is more attractive one because it can give more detailed information about the signal and its component frequencies whereas; the time domain analysis can give overall qualitative information. Generally, the machine vibration signal is composed of three parts: stationary vibration, random vibration, and noise. Traditionally, Fourier transform (FT) was used to perform such analysis. It is well known, however, that the Fourier analysis has some inherent limitations in the analysis of the non-linear phenomena and some short duration transient signals. Over the past 10 years, the wavelet theory has become one of the emerging and fast-evolving mathematical and signal processing tools for its many distinct merits. Different from the fast-Fourier transform (FFT), the wavelet transform can be used for multi –scale analysis of the signal through dilation and translation, so it can extract the time-frequency features of the signals effectively. Wavelets are mathematical functions that cut up data into different frequency components, and then study each component with a resolution matched to its scale. They have advantages over traditional Fourier methods in analyzing physical situations where the signal contains discontinuities and sharp spikes. Wavelet functions are distinguished from other transformations in that they not only dissect signals into their component frequencies, they also vary the scale at which the component frequencies are analyzed. Therefore wavelets, as component pieces used to analyze a signal, are limited in space. In other words, they have definite stopping points along the axis of a graph--they do not repeat to infinity like a sine or cosine wave does. The ability to vary the scale of the function as it addresses different frequencies also makes wavelets better suited to signals with spikes or discontinuities than traditional transformations such as the FT. 1 2. APPLICATION OF WAVELET TRANSFORMATION. Wavelet transformation has been applied in wide variety of engineering analysis. It includes machinery fault diagnosis, in particular, time–frequency analysis of signals, fault feature extraction, singularity detection for signals, de-noising and extraction of the weak signals, compression of vibration signals and system identification. By combining the time domain and classical Fourier analysis, the wavelet transform provides simultaneously spectral representation and temporal order of the signal decomposition components, which is also applied to perform the analysis of NDE ultrasonic signal received during the inspection of reinforced composite materials. Wavelet transform can be used to process acoustic emission signals to find faults such as micro-failure modes and their interaction in composites. For rotating machines fault detection, wavelet transform has been used for isolation and identification of electrical faults in induction machine, bearing and gear related faults. Recently wavelet transform has also been applied to detect the stable and propagating cracks. 3. WAVELET THEORY In this section a brief theory of wavelet transformation has been explained. To develop a better insight a comparison between wavelet transform and fourier transform has been made. 3.1 Fourier Transforms The Fourier transforms utility lies in its ability to analyze a signal in the time domain for its frequency content. The transform works by first translating a function in the time domain into a function in the frequency domain. The signal can then be analyzed for its frequency content because the Fourier coefficients of the transformed function represent the contribution of each sine and cosine function at each frequency. An inverse Fourier transform does just what you'd expect; transform data from the frequency domain into the time domain. 2 F () = F {f (t)} = - e -it f (t) dt (3.1) 3.2 Discrete Fourier Transforms The discrete Fourier transform (DFT) estimates the Fourier transform of a function from a finite number of its sampled points. The sampled points are supposed to be typical of what the signal looks like at all other times. The DFT has symmetry properties almost exactly the same as the continuous Fourier transform. In addition, the formula for the inverse discrete Fourier transform is easily calculated using the one for the discrete Fourier transform because the two formulas are almost identical. 3.3 Windowed Fourier Transforms If f(t) is a non periodic signal, the summation of the periodic functions, sine and cosine, does not accurately represent the signal. You could artificially extend the signal to make it periodic but it would require additional continuity at the endpoints. The windowed Fourier transform (WFT) is one solution to the problem of better representing the non periodic signal. The WFT can be used to give information about signals simultaneously in the time domain and in the frequency domain. With the WFT, the input signal f(t) is chopped up into sections, and each section is analyzed for its frequency content separately. If the signal has sharp transitions, one can window the input data so that the sections converge to zero at the endpoints. This windowing is accomplished via a weight function that places less emphasis near the interval's endpoints than in the middle. The effect of the window is to localize the signal in time. 3.4 Fast Fourier Transform To approximate a function by samples, and to approximate the Fourier integral by the discrete Fourier transform, requires applying a matrix whose order is the number sample points n. Since multiplying an n X n matrix by a vector costs on the order of 2n arithmetic 3 operations, the problem gets quickly worse as the number of sample points increases. However, if the samples are uniformly spaced, then the Fourier matrix can be factored into a product of just a few sparse matrices, and the resulting factors can be applied to a vector in a total of order nlogn arithmetic operations. This is the so-called fast Fourier transform or FFT. Example: Figure 3.2 shows the FFT of the rub signal shown in Fig 3.1. Figure 3.1. Simulation signals of the rub-impact fault (ordinates are in number of rotations)(a) signal of the slight rub fault;(b)signal of the severe rub fault. (Peng et al., 2003). Figure 3.2. FFT spectrum of the rub-impact signal: (a) spectrum of the slight rub- impact fault (b) spectrum of the severe rub-impact fault. (Peng et al., 2003). 3.5 Similarities between Fourier and Wavelet Transforms The fast Fourier transform (FFT) and the discrete wavelet transform (DWT) are both linear operations that generate a data structure that contains log2 n segments of various lengths, usually filling and transforming it into a different data vector of length 2n. The mathematical properties of the matrices involved in the transforms are similar as well. The inverse transform matrix for both the FFT and the DWT is the transpose of the original. As a result, both transforms can be viewed as a rotation in function space to a 4 different domain. For the FFT, this new domain contains basis functions that are sines and cosines. For the wavelet transform, this new domain contains more complicated basis functions called wavelets, mother wavelets, or analyzing wavelets. Both transforms have another similarity. The basis functions are localized in frequency, making mathematical tools such as power spectra (how much power is contained in a frequency interval) and scalograms useful at picking out frequencies and calculating power distributions. 3.6 Dissimilarities between Fourier and Wavelet Transforms The most interesting dissimilarity between these two kinds of transforms is that individual wavelet functions are localized in space. Fourier sine and cosine functions are not. This localization feature, along with wavelets' localization of frequency, makes many functions and operators using wavelets "sparse" when transformed into the wavelet domain. One way to see the time-frequency resolution differences between the Fourier transform and the wavelet transform is to look at the basis function coverage of the time- frequency plane. Figure 3.3 shows a windowed Fourier transform, where the window is simply a square wave. The square wave window truncates the sine or cosine function to fit a window of a particular width. Because a single window is used for all frequencies in the WFT, the resolution of the analysis is the same at all locations in the time-frequency plane. Figure. 3.3. Fourier basis functions, time-frequency tiles, and coverage of the time- frequency plane An advantage of wavelet transforms is that the windows vary. In order to isolate signal discontinuities, one would like to have some very short basis functions. At the same time, 5 in order to obtain detailed frequency analysis, one would like to have some very long basis functions. A way to achieve this is to have short high-frequency basis functions and long low-frequency ones. This happy medium is exactly what one gets with wavelet transforms. Figure 3.4 shows the coverage in the time-frequency plane with one wavelet function, the Daubechies wavelet. Figure 3.4. Daubechies wavelet basis functions, time-frequency tiles, and coverage of the time-frequency plane. One thing to remember is that wavelet transforms do not have a single set of basis functions like the Fourier transform, which utilizes just the sine and cosine functions. Instead, wavelet transforms have an infinite set of possible basis functions. Thus wavelet analysis provides immediate access to information that can be obscured by other time- frequency methods such as Fourier analysis. For signal shown in Figure 3.1, we have the following scalograms. Figure 3.5. Scalogram of the rub-impact signals (a) scalogram of the slight rub (b) scalogram of the severe rub-impact fault (Peng et al., 2003). 6 3.7 The Wavelet Transform The driving force behind wavelet transforms (WTs) is to overcome the disadvantages embedded in short time Fourier transform (STFT), which provides constant resolution for all frequencies since it uses the same window for the analysis of the inspected signal x (t). On the contrary, WTs use multi-resolution, that is, they use different window functions to analyse different frequency bands of the signal x (t). Different window functions ψ(s,b,t);which are also called son wavelets, can be generated by dilation or compression of a mother wavelet ψ(t) , at different time frame. A scale is the inverse of its corresponding frequency. WTs can be categorised as discrete WTs or continuous WTs. For vibration-based fault diagnosis, usually continuous WTs are employed. A continuous type of wavelet transform (CWT) that is applied to the signal x(t) can be defined as, 1 t b w(a, b) a f (t ) dt a (3.2) Where • a is the dilation factor, • b is the translation factor and • ψ(t) is the mother wavelet. • 1/a is an energy normalization term that makes wavelets of different scale has the same amount of energy. 7 3.8 Wavelet Definition A wavelet is a waveform of effectively limited duration that has an average value of zero. Figure 3.7 (a) Morlet wavelet of arbitrary width and amplitude, with time along the x-axis. (b) Construction of the Morlet wavelet (blue dashed) as a Sine curve (green) modulated by a Gaussian (red). (www.paos.colorad.edu) 3.9 Wavelet Properties 1.The definition domain is compact support ,which ensures that the function is fast decaying, and so time localization can be obtained. 2.The admissibility condition C = (( () 2)/ ) d < (3.3) Where () = (t) e-it dt . This condition means that the waveform of the mother wavelet function must be oscillating i.e. the average value of the wavelet in the time domain must be zero. 3.10 Scaling of Wavelet • Scaling a wavelet simply means stretching (or compressing) it. • Low frequencies (high scales) correspond to a global information of a signal (that usually spans the entire signal), whereas high frequencies (low scales) correspond To detailed information of hidden pattern in the signal (that usually last for relatively short time.) 8 In terms of mathematical functions, if f(t) is a given function f(st) corresponds to a contracted (compressed) version of f(t) if s > 1 and to an expanded (dilated) version of f(t) if s < 1 Figure 3.8 shows scaling of wavelet at four different scales. Figure 3.8 Scaling of Wavelet (www.users.rowan.edu) 3.11 COMPUTATION OF THE CWT Interpretation of the equation (3.2) will be explained in this section. Let x(t) is the signal to be analyzed. The mother wavelet is chosen to serve as a prototype for all windows in the process. All the windows that are used are the dilated (or compressed) and shifted versions of the mother wavelet. There are a number of functions that are used for this purpose. The Morlet wavelet is mostly used for vibration based fault diagnosis, and it is used for the wavelet analysis of the vibration signals due to rubbing which are presented in the later chapters. Once the mother wavelet is chosen the computation starts with s=1(scale) and the continuous wavelet transform is computed for all values of s, smaller and larger than ``1''.The wavelet is placed at the beginning of the signal at the point 9 which corresponds to time=0. The wavelet function at scale ``1'' is multiplied by the signal and then integrated over all times. The result of the integration is then multiplied by the constant number 1/sqrt{s}. This multiplication is for energy normalization purposes so that the transformed signal will have the same energy at every scale. The final result is the value of the transformation, i.e., the value of the continuous wavelet transform at time zero and scale s=1. In other words, it is the value that corresponds to the point t =0, s=1 in the time-scale plane. The wavelet at scale s=1 is then shifted towards the right by “b” amount to the location t=b , and the above equation is computed to get the transform value at t=b , s=1 in the time-frequency plane. This procedure is repeated until the wavelet reaches the end of the signal. One row of points on the time- scale plane for the scale s=1 is now completed. Then, “s” is increased by a small value. Note that, this is a continuous transform, and therefore, both time (b) and “s” must be incremented continuously. However, if this transform needs to be computed by a computer, then both parameters are increased by a sufficiently small step size. This corresponds to sampling the time-scale plane. The above procedure is repeated for every value of “s”. Every computation for a given value of “s” fills the corresponding single row of the time-scale plane. When the process is completed for all desired values of s, the CWT of the signal has been calculated. The Figures 3.9 below illustrate the entire process step by step. 10 (a) (b) 11 (c) Figure 3.9 Computation of the CWT(a) scale s=1(b) scale s=5 (c) scale s=20(www.users.rowan.edu) 3.12 Different Shapes of Wavelet Wavelet transforms comprise an infinite set. The different wavelet families make different trade-offs between how compactly the basis functions are localized in space and how smooth they are. In Figure 3.10, several different wavelet families are illustrated. Figure. 3.10. Several different families of wavelets. 12 3.13 Scalogram For all (t), x (t) L2 (R), the inverse wavelet transform of x (t) is defined as x(t) = (1/ C) a-2 wx (a,b;) ab (t) da db (3.4) Since the wavelet transform does not lose any information, and the energy is preservative for the transform. So the following equation is tenable: < x(t) , x(t) >= x(t)2 dt = (1/ C) a-2 da wx (a,b;)2 db (3.5) wx (a,b;)2 a b/ C a2 represents the total energy of a domain centered at(a ,b )with scale interval a and time interval b. and wx (a,b;)2 is defined as the wavelet scalogram. It shows how the energy of the signal varies with time and frequency. The wavelet scalogram has been widely used for the analysis of non-stationary signal. In order to illustrate the characteristics of the scalogram , a signal X (t) is considered . Figure3.11 shows the signal X (t)in the time domain and it can be expressed by X= 6 sin 2(10t)+ 6 sin 2(20t) 0 t < 0.5 (3.8) X= 6 sin 2(20t)+ 6 sin 2(30t) 0.5 t < 1 Figure 3.11 Test signal X(t) 13 Figure 3.12 FFT of numerical test signal X(t) Figure 3.13 Scalogram for signal X (t) Although we know that the signal X (t) contains three components whose frequencies are 10, 20 and 30 Hz, respectively, from the spectrum (Figure 3.12) we cannot determine whether the component exists during all the signal lifespan or only part of the lifespan. 14 However, through the signal scalogram, using Morlet Wavelet, we are not only able to recognize, the three components but also able to know the time at which they exist. It is to be noted that for Scalogram scale a is inversely proportional to the frequency, this means that low scale value shows high frequency and vice-versa. In figure (4) vertical- axis is represented by scale factor “a” which is inversely proportional to frequency. Horizontal-axis as time. Bright region in scalogram shows the presence of that particular frequency at the particular time. a=8 represents frequency =10 Hz which is present in the time span 0 < t< 0.5. a= 4 represents frequency =20 Hz which is present in the time span 0 < t< 1 a= 3 represents frequency =30 Hz which is present in the time span 0.5 < t < 1 The above analysis of simulated test signals X(t) with the help of FFT and CWT (Continuous Wavelet Transform) using Morlet Wavelet helps to understand the applicability of Wavelet transformation for rubbing. Such representation is very helpful to reveal the time intervals as well as the frequency range of the occurrence of some transient disturbances due to typical faults like rub in rotors and structures. 3.14 Frequency –Scale relation for Scalogram As explained above scale is inversely related the frequency. Though there is no exact mathematical relation for this, with approximation it can de stated as Fa= x Fc/ s (3.9) ( Matlab 6.5 Help) Fa = pseudo frequency ( for the scale value s ) = sampling frequency s = Scale Fc = central frequency of mother wavelet in Hz. Centralfrequency of the Morlet wavelet used in this project is .0.8125 Hz ( Matlab 6.5 ) 15 4. SOME CASE STUDIES To elucidate the effectiveness of wavelet Scalogram, a few case studies are now discussed. a) Rotor-Stator Rub Detection (Experimental Study) b) Crack detection in a rotating shaft (Analytical Study) 4.1 Rub Detection in a rotor bearing system. In the first case experimental results related to rotor-stator rub are discussed in which wavelet transform tool available in Matlab software is used and results presented. Figure 4.1 shows the experimental set-up of the real rotor system. Figure 4.2 shows vibratory signal flow diagram for identifying Rub. It is composed of a rotor and a stator, a driving motor, bearings and couplings. Vibration signals were collected, for both horizontal and vertical direction, from the rotor system using non-contact eddy-current transducers at a sampling rate of 2 kHz. The experiment was done at three different rotational speed. 4.1.2 Experimental Setup Casing Rotor Disc Figure 4.1 Rotor-casing setup 16 Rotor-stator arrangement Motor Figure 4.2(a) Experimental Setup Rotor Disc Casing (Stator) Figure 4.2( b) Experimental Setup 17 4.1.3 Experimental Results Experiment for rub between rotor and stator was conducted at different rotational speeds. Rub between rotor and casing was introduced by lowing down the upper part of the casing till it touched the rotor disc and then tightened it up at its new position with help of nut arrangement. The shaft vibrations in both horizontal and vertical direction, was measured with the help proximity probe fixed at the probe holder. The vibration signals was then transferred to oscilloscope through proximitor and vibration monitor and the data was recorded in the oscilloscope. The following figures shows the time-domain waveform , FFT spectrum and the wavelet scalogram (using Morlet wavelet) of the vibration signal at different rotational speeds with both rubbing and without rubbing . Figure 4.3 (a)Time domain signal at 588 rpm in vertical direction(a) without-rub (b) with-rub To get a better information of high frequencies and their occurring time scalogram with low scale values is shown in the following figures and indicate 18 (a) High Frequencies present Figure 4.4 (b) Scalogram with lower range of scale “a”, at 588 rpm (a) without rub (b)with-rub absence of higher frequencies. But 4.4 (b) shows higher frequencies at the scale values of a=9-13 , and these are occurring after every 200 time sample, i.e after one cycle . Thus the rubbing is occurring once every cycle. Their approximate frequency can be calculated by using equation 3.9. 19 Figure 4.5 is the 3D plot of scalogram showing the values of wavelet coefficient on the z- axis. In the Figure the main peak indicate the rotational frequency, and the lower Peaks shows presence of higher frequencies. (a) (b) Figure 4.5 Scalogram 3-D at lower scale, at 588 rpm (a) without rub (b) with rub. Figure 4.5 (b) shows distinct periodic peaks at higher frequencies(lower scale) due to rubbing which is not visible in Figure4.5 (a). Thus scalogram provides a better information in both time and frequency domain. Three set vibrations signals along the horizontal direction were recorded separately at three different rpm i.e. 588 rpm, 912 rpm and 960 rpm. The following figures present horizontal vibration due to rubbing at 588 rpm using time domain waveform along with their FFT spectra and scalogram. 20 (a) No rubbing at the end of the signal (b) Figure 4.6 Time domain signal at 588 rpm in horizontal direction (a) without-rub (b) with-rub Figure 4.7 Scalogram 3-D for signal shown in Figure 4.6b. In Figure 4.7 that represents 3D scalogram for signal shown in Figure 4.6b, a periodic bright peaks can be seen which indicate high frequency impacts due to rubbing. In the 21 same scalogram we can observe the absence of these high frequencies at latter stage of the signal. Thus this provides the confirmation that the rubbing is not present for a very small time span at end of the recorded signal. Thus the wavelet scalogram is able to pick up information about the frequency content as well as its temporal representation with better clarity and detail. Experimental signals had been collected at different rotational speeds. It can be concluded that the Scalogram (CWT) have distinct advantage for the rub-impact signal analysis and they can characterize the dynamical behaviour of the rub-impact rotor better and can extract the rub impact feature well. 4.2 Case Study II: Crack Detection Fatigue cracks are a potential source of catastrophic failures in rotors. Researchers have put in considerable effort to develop a foolproof and reliable strategy to detect cracks in rotors. Figure 4.8. Breathing behaviour of crack This case study presents a novel way to detect fatigue transverse cracks in rotating shafts. The proposed detection methodology exploits both the typical nonlinear breathing phenomenon of the crack and the coupling of bending-torsional vibrations due to the 22 presence of crack for its diagnosis. A transient torsional excitation is applied for a very short duration at specific angular orientation of the rotor and its effect in the lateral vibrations is investigated. Wavelet transforms is used in revealing the transient features of the resonant bending vibrations, which are set up for a short duration of time upon transient torsional excitation. Variation of peak absolute value of wavelet coefficient (of the transient lateral vibration response) with angle at which torsional excitation is applied is investigated. The correlation of this variation with the breathing pattern of the crack is studied. The sensitivity of the proposed methodology to the depth of crack is investigated. The detection methodology gives a vibration response signature that closely correlates with and is very specific to the behaviour of the transverse surface crack in a horizontal rotor. The response features are not likely to be exhibited by other common rotor faults under similar excitation conditions making the proposed detection methodology unambiguous. The fact that the rotor is not required to stop and that the detection process is applied for a rotating shaft with only a short period transient external excitation makes the methodology more convenient. Figure 4.9 Finite element model of the rotor bearing system. Consider a rotor segment containing a single transverse surface crack. A beam finite element containing a transverse surface crack of depth a as shown in Figure 4.9. To study the coupled bending-torsional vibrations of a cracked rotor, a simply supported rotor with 23 a single centrally situated disc of mass 1kg is considered. A single transverse surface crack is assumed just adjacent to the central disc. The total rotor span is divided into 14 elements of equal length (Figure 4.9). A crack finite element is used to represent the crack. Rest of the rotor is modeled with Timoshenko beam elements with six degrees of freedom per node. The torsional excitation is applied to the cracked rotor with depth ratio of a 0.2 rotating at 22rad/sec. The torsional excitation Te= Mtsin(221t) at frequency of 221rad/sec is applied for a very short duration of 0.02857sec (equal to period of one cycle). The frequency 221rad/sec is the natural frequency of bending vibrations of the rotor. The exact timing of the application of transient torsional excitation during rotation of the cracked rotor can be varied. Figure 4.10 shows the response of the cracked rotor ( a 0.2 ) when the transient torsional excitation is applied when rotor is oriented at =00. Figure 4.10. Unbalance response of a cracked rotor ( a 0.2 ) with transient torsional excitation applied at =00 during 5th revolution =22rad/sec. 24 The transient torsional excitation is evident from the torsional vibration signal in Figure 4.10 at the start of 5th rotation (c=14400). The torsional excitation causes a broadband excitation as seen in the spectrum of the torsional vibration signal (Figure 4.10h). Although a transient torsional excitation was applied, the time domain signal in bending and longitudinal vibrations do not show any sign of transient disturbance to the rotor after c=14400. It may be mentioned that the excitation is applied when =00 (angle turned during the 5th cycle of rotation) and when cumulative angle turned by the rotor from the start is c=14400. Figure 4.11 shows response of the cracked rotor when the same transient torsional excitation is applied at =1800 during the 5th rotation (i.e., c=16200). In this case, Figures 4.11a and 4.11b show some noticeable changes in the response for a very short duration for a couple of cycles or rotation (c=16200 - 25200) after the application of excitation. The changes in the form of high frequency ripples are more perceptible in horizontal vibrations. However, being for a very short duration, these vibrations are not properly represented in the frequency domain signal of the vibrations shown in Figures 4.11e and 4.11f. The spectra in Figures 4.11e and 4.11f are almost identical to those in Figures 4.10e and 4.10f although the time domain signals are slightly different as the disturbance is noticed in Figures 4.11a and 4.11b. When the torsional excitation is applied at the two different angular orientation of the rotor, Figures 4.10 and 4.11 show that there is small and barely noticeable change in time domain signal. There is no change in the longitudinal vibration response in either case. When the excitation is applied at =1800, the crack is on the underside (in the tensile region) and hence is fully open. A fully open crack allows cross coupling between torsional and bending vibration and torsional excitation generates resonant bending vibrations that dies out quickly and are not sustained beyond couple of cycles of rotation. In contrast, when the torsional excitation applied at =00, the crack is at the top side (in the compressive region) and hence is fully closed. In this case the closed crack prevents cross coupling of torsional and bending vibrations; the transient torsional excitation therefore fails to generate any additional response in the bending vibrations. It is also noticed from the bending stiffness variation of the cracked rotor (Figure 4.11i) that the application of excitation does not alter the periodic variation of the stiffness indicating an 25 unchanged opening and closing (breathing) of the crack. The observation is important as it ensures that a different quality of rotor response is generated depending upon whether the excitation is applied at =00 (crack closed) or at =1800 (crack closed). Figure 4.11. Unbalance response of a cracked rotor ( a 0.2 ) with transient torsional excitation applied at =1800 during 5th revolution =22rad/sec. Although a qualitatively different lateral vibration response is observed from Figures 4.10 and 4.11, the response dies down quickly causing detection of such transient phenomenon a difficult task using traditional signal analysis techniques such as frequency domain analysis. This is evident from Figures 4.11e and 4.11f. Wavelet transforms are better suited for such situations as they have time/space localization feature. Wavelet analysis provides an excellent access to information that may be obscured by the other techniques such as Fourier analysis. 26 Over the past 15 years, wavelet transforms (WT) have become one of the fast- emerging and effective mathematical and signal processing tools for its distinct advantage in analyzing signals, particularly the transient ones. Using different window functions through dilation and translation of a prototype function called mother wavelet, the wavelet transform can provide multi-resolution (multi-scale) analysis of a signal. Hence it can extract time-frequency features of a signal effectively, which are obscured in the traditional Fourier Transform analysis. The limitation of Fourier Transform stem from the fact that the integral transform is applied on the signal globally and hence in the process the time localization of the spectral component is not highlighted and is lost if the transient decays quickly. To reveal the subtle but important information hidden in the transient vibration signal generated due to transient torsional excitation, the response estimated using a nonlinear rotor dynamic analysis of the cracked rotor is analyzed using continuous wavelet transforms. Matlab’s (version 7) standard functions are used in the analysis with Morlet mother wavelet. Figure 4.12 shows the 3D and 2D wavelet scalogram of torsional vibration signal applied at =1800 (c=16200). It may be observed that the application of excitation is observed in the form of large wavelet coefficient value after c=16200 for a short duration thereafter. The large coefficients are observed for a wide scale range (from scale value of 4 to 100 and beyond) indicating that wide band of frequencies are excited at c=16200. 27 (a) (b) Figure 4.12. Wavelet scalogram of the torsional vibration signal with excitation applied at c=16200 i.e., =1800 a) 3D b)2D plot 28 The broadband torsional excitation also excites the torsional resonance in the torsional vibration signal as observed in Figure 4.12b. The scale value of 8.2 maps to the frequency of 125Hz, which is the torsional natural frequency of the rotor. At the scale value of 8.2 and c=16200, a gradually decreasing coefficient value is observed from the fading white patch between c=16200 and c=19000. This indicates a decaying torsional vibration signal of torsional natural frequency. The 3D scalogram of lateral vibration signal is shown in Figure 4.13a. The x-axis represents the cumulative angle turned by the rotor (c), the y-axis represents the scale value and the z-axis represents the absolute values of wavelet coefficients. The figure represents the wavelet scalogram of the time domain signal of lateral vibration shown in Figure 4.11b; wherein the excitation is applied at =1800. The high energy concentration is observed at scale values of 293 and 146, which represents 1x and 2x harmonic frequency components for the rotational speed of 3.5Hz. The figure also shows a gradually decreasing ridge (shown encircled) from angle of rotation c=16200 at the scale value of 29 that corresponds to the bending natural frequency (35Hz) of the cracked rotor. The same 3D plot for the scale value from 1 to 100 is shown in Figure 4.13b, that clearly shows the bending natural frequency vibration initiated during the 5th cycle of rotation after the application of excitation at c=16200. 29 (a) (b) Figure 4.13. Wavelet scalogram of the bending vibration signal with excitation applied at c=16200 i.e., =1800 a) scale range 1-350 b) scale range 1-100 30 (c) (d) Peak Ca,b value at c=16930 Figure 4.13. Wavelet scalogram of the bending vibration signal with excitation applied at c=16200 i.e., =1800 c) 3D scalogram; scale range 1-40 d) 2D scalogram; scale range 1- 40 31 As the scale value of interest is 29, the 2D wavelet map with reduced scale range (Figure 4.13) shows the generation of bending resonance vibration initiated after c=16200 and gradually decays. It may also be observed that although the resonant bending vibrations are seen immediately following the application of torsional excitation at c=16200, the peak value of wavelet coefficient is observed at c=16930 (indicated by dark brown patch) as it takes finite time to build up resonant vibration. The vibrations decay after c=16930 as seen in Figure 4.13d. The excitation is applied in torsional mode and resonant vibrations are observed in the bending mode, indicating coupling of vibrations. Wavelet transform is now applied to the bending vibration signal shown in Figure 4.10b. All other conditions being the same, only timing of the excitation is different in this case. Instead of applying at =1800 (c=16200), the excitation is applied at =00 (c=14400) when the crack is closed as observed from the stiffness variation in Figure 4.11i. Compared to Figures 4.13b and 4.13d, Figures 4.14a and 4.14b shows a complete absence of the ridge representing transient resonant bending vibrations at scale value of 29 beyond c=14400. The wavelet map thus can pick up the transient part of the signal and locate the presence of these transients in a time-frequency representation shown in Figures 4.13 and 4.14. The exact timing of excitation and the frequency of the transient disturbance can be judged easily from a single graph. The simulation is repeated for different depths of crack and it has been found that the wavelet coefficients are highly sensitive to the depth of crack (Figure 4.15) and hence the strategy can be adopted to detect crack at its early stage of progress. Figure 4.16 shows that even for a very shallow depth crack (crack depth 5% of diameter of crack), the presence of high frequencies can be detected using wavelet scalogram and the frequencies would have been undetected using conventional signal analysis techniques. 32 (b) Figure 4.14. Wavelet scalogram of the bending vibration signal with excitation applied at c=14400 i.e., =00 a) 3D scalogram; scale range 1-100 b) 3D scalogram; scale range 1-40 Figure 4.15. Sensitivity of Cap,b to crack depth ratio a 33 Figure 4.16. Wavelet scalogram of the bending vibration signal with excitation applied at =1800 for shallow crack ( a =0.05) CONCLUDING REMARKS: Wavelet Transform is an excellent tool for detection of non-stationary vibration signals. Features that are obscured during Fourier Transformation are revealed with better clarity using wavelet transform. For short duration transient signals, as the wavelet transform preserves the time and frequency information, the cause of such signals can be easily correlated. The technique has been successfully applied for detection of rotor rub and also tried for unique identification of rotor crack. 34

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