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Gearbox fault diagnosis using adaptive redundant Lifting Scheme
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Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 20 (2006) 1992–2006 www.elsevier.com/locate/jnlabr/ymssp
Gearbox fault diagnosis using adaptive redundant Lifting Scheme Jiang Hongkaia,, He Zhengjiab, Duan Chendonga, Chen Pengc a School of Mechanical Engineering, Xi’an Jiaotong University, 710049, Xi’an, China State Key Lab for Manufacturing Systems Engineering, Xi’an Jiaotong University, 710049, Xi’an, China c Department of Environmental Science and Engineering, Faculty of Bioresources, Mie University, 1515 Kamihama-cho, Tsu, Mie 514-8507, Japan b
Received 30 January 2005; received in revised form 9 June 2005; accepted 18 June 2005 Available online 10 August 2005
Abstract Vibration signals acquired from a gearbox usually are complex, and it is difficult to detect the symptoms of an inherent fault in a gearbox. In this paper, an adaptive redundant lifting scheme for the fault diagnosis of gearboxes is developed. It adopts data-based optimisation algorithm to lock on to the dominant structure of the signal, and well reveal the transient components of the vibration signal in time domain. Both lifting scheme and adaptive redundant lifting scheme are applied to analyse the experimental signal from a gearbox with wear fault and the practical vibration signal from a large air compressor. The results confirm that adaptive redundant lifting scheme is quite effective in extracting impulse and modulation feature components from the complex background. r 2005 Elsevier Ltd. All rights reserved. Keywords: Adaptive redundant lifting scheme; Optimisation algorithm; Vibration signal; Fault diagnosis
1. Introduction Gearboxes are key parts in a wide range of mechanical systems. It is very important to detect incipient fault symptoms from gearboxes. Usually, vibration signals are acquired from Corresponding author
E-mail address: [email protected] (J. Hongkai). 0888-3270/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2005.06.001
ARTICLE IN PRESS J. Hongkai et al. / Mechanical Systems and Signal Processing 20 (2006) 1992–2006
1993
accelerometers mounted on the outer surface of a bearing housing. The signals consist of vibrations from the meshing gears, shafts, bearings, and other components. The useful information is corrupted and it is difficult to diagnose a gearbox from such vibration signals. Wavelet theory is a powerful tool for non-stationary signal analysis, and it has been successfully used in gearbox diagnosis. Boulahbal et al. [1] used the amplitude and phase maps of continuous wavelet transform together to extract fault features of a gearbox and obtained a more positive assessment of a tooth condition. Lin et al. [2,3] utilised the similarity between Morlet wavelet and an impulse shape, and detected tooth crack symptoms of a gearbox immersed in the noise. Loutridis et al. [4] used continuous wavelet transform and Hodlder exponents to classify gear faults and got better performance. In all the wavelet techniques mentioned above, researchers usually selected an appropriate wavelet function from a library of previously designed wavelet functions to match a specific fault symptom. Different types of mechanical faults have different waveform characteristics, even one wavelet function is selected, and it is not always the best wavelet function to detect a specific fault symptom in a gearbox. New wavelet method is needed to overcome the drawback. Lifting scheme is a spatial domain construction of biorthogonal wavelets developed by Sweldens [5–7]. It abandons the Fourier transform as design tool for wavelets, wavelets are no longer defined as translates and dilates of one fixed function. Compared with classical wavelet transform, Lifting scheme possesses several advantages, e.g. possibility of adaptive design, inplace calculations, irregular samples and integers-to-integers wavelet transforms. Lifting scheme provides a great deal of flexibility, it can be designed according to the properties of the given signal, and it ensures that the resulting transform is invertible. In this paper, we develop a new wavelet method called adaptive redundant lifting scheme. It is based on lifting scheme and designed to capture the transient components of the gearbox vibration by adaptive decomposition. In Section 2, the theory of lifting scheme is reviewed briefly. The data-based optimisation algorithm is described in Section 3. The method of adaptive redundant lifting scheme is presented in Section 4. In Section 5, adaptive redundant lifting scheme is applied to analyse the vibration signals of an experimental gearbox and a large compressor gearbox. Comparison with lifting scheme is also shown. Conclusions are given in Section 6.
2. Lifting scheme principle Lifting scheme is an entirely space domain wavelet construction. A typical lifting scheme procedure consists of three basic steps: split, predict and update [5–7]. Split: Let x(n) be an original data set. In this step, x(n) is divided into two disjoint even subset xe(n) and odd subset x0(n), where xe ðnÞ ¼ xð2nÞ and x0 ðnÞ ¼ xð2n þ 1Þ. Predict: If the original signal has a local correlation structure, then the even and odd subsets are highly correlated. We predict the odd coefficients x0(n) from the neighbouring even coefficients xe(n), and the prediction differences d(n) are defined as detail signal dðnÞ ¼ x0 ðnÞ Pðxe ðnÞÞ, T
Where P ¼ ½pð1Þ; . . . ; pðNÞ is the prediction operator.
(1)
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Update: This step forms a coarse approximation to the original signal. We combine the even coefficients and the linear combination of the prediction differences, and then the approximation signal c(n) are obtained cðnÞ ¼ xe ðnÞ þ UðdðnÞÞ,
(2)
~ is the update operator. Where U ¼ ½uð1Þ; . . . ; uðNÞ Iteration of the three steps on the output c(n), and then the detail signal and approximation signal at different levels are generated. Scaling function f(x) and wavelet function c(x) of lifting scheme can be derived from P and U by iteration algorithm. The program flow chart is illustrated in Fig. 1. The scaling function and wavelet function with N ¼ 6 and N~ ¼ 6 are shown in Fig. 2. The scaling function and wavelet function are symmetrical and compactly supported. The shape of the wavelet function is very similar to an impulse, and it is desirable to detect the transient components in the vibration signal. We can choose P and U according to the properties of the given signal. Since the design of lifting scheme without reference to Fourier techniques, each lifting step is always invertible, and the inverse transform of lifting scheme is the inversion of the lifting steps.
Fig. 1. The program flow diagram of scaling and wavelet functions.
Fig. 2. Scaling function and wavelet function with N ¼ 6 and N~ ¼ 6.
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Assuming the same P and U are chosen for the forward and inverse transform, the lifting construction guarantees perfect reconstruction for any P and U [8].
3. Initial prediction operator and update operator design Lifting scheme increases the flexibility to extract the features of a given data set. We can optimise the prediction operator and update operator to well reveal the feature components of the data set. In this section, we adopt Claypoole’s optimisation algorithm [8,9] to design the initial prediction operator P and update operator U, and make them adapt to the dominating structures of the data set at the corresponding level. The design of initial prediction operator and update operator based on the data characteristics is accomplished by the following procedures. A N-point initial prediction operator P is designed to suppress polynomial components upto order MoN, and the remaining N–M degrees of freedom are used to match the given signal. The initial prediction operator is as below P ¼ ½ pð1Þ; . . . ; pðNÞT .
(3)
Construct a M N matrix V, its element is ½V i;j ¼ ½2j N 1i1 ,
(4)
where i ¼ 1; 2; . . . ; M; j ¼ 1; 2; . . . ; N. We require that VP ¼ ½1; 0; 0; . . . 0T ,
(5)
The vector of prediction differences can be expressed as follows: e ¼ X 0 X e P.
(6)
The goal is to obtain the prediction coefficients that minimise the sum of squared prediction differences, namely min kX 0 X e Pk2 .
(7)
P
We solve Eqs. (5) and (7), then the initial prediction operator P that locks on to the dominant structure of a signal at the corresponding level is obtained. ~ ~ A N-point initial update operator U is designed by the following procedures, where NpN. Construct a 2N–1 -dimensional vector q, its element is defined as i ¼ 1; . . . N :
qð2i 1Þ ¼ pðiÞ;
i ¼ 1; . . . N 1 :
qð2iÞ ¼ 0;
qðNÞ ¼ 1.
Assume that K ¼ 4N 3, construct a K N~ matrix H, its element is 0, except Hðð2i 1Þ : ð2i þ 2N 3Þ; iÞ ¼ q, ~ where i ¼ 1; . . . ; N.
(8)
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And then, a N~ K matrix W is constructed, in which element is ½W m;n ¼ ½n N N~ þ 1m1 ,
(9)
~ n ¼ 1; 2; . . . ; K. where m ¼ 1; 2; . . . ; N, We calculate U from the following expression: WHU ¼ ½1; 0; 0; . . . 0T .
(10)
Since matrix H contains the initial prediction operator P, the initial update operator U also adapts to the dominant signal structure at the corresponding level. 4. Adaptive redundant lifting scheme design algorithm Now, we explain how to design adaptive redundant lifting scheme. The design includes two parts, the design of redundant prediction operator and update operator, and the construction of adaptive redundant lifting scheme. 4.1. Redundant prediction operator and update operator design The design of redundant prediction operator P[l] and redundant update operator U[l] (l is the decomposition level) is the key step in adaptive redundant lifting scheme. P[l] and U[l] are obtained by padding the initial prediction operator P and update operator U with zeros at the corresponding level l. Their design algorithm is as follows. Assuming the initial prediction operator P and update operator U at level l are designed with ~ N and N~ the algorithm in Section 3, where P ¼ fpm g, m ¼ 1; 2; . . . ; N; U ¼ fun g, n ¼ 1; 2; . . . ; N, are the coefficient number of P and U, respectively. Then the redundant prediction operator P[l] and redundant update operator U[l] at level l are designed by padding the prediction coefficient pm and update coefficient un with zeros. The redundant prediction coefficient p½l j is expressed as below ( l pm ; j ¼ 2 m ; j ¼ 1; . . . 2l N. p½l (11) j ¼ 0; ja2l m The redundant update coefficient u½l i is expressed as below ( un ; i ¼ 2l n ~ u½l ¼ ; i ¼ 1; . . . 2l N: i 0; ia2l n
(12)
The expressions of redundant prediction operator P[l] and update operator U[l] at level l are as ½l l l ~ ½l below P½l ¼ fp½l j g, j ¼ 1; . . . ; 2 N; U ¼ fui g, i ¼ 1; . . . ; 2 N. 4.2. Adaptive redundant lifting scheme construction We convert lifting scheme into adaptive redundant lifting scheme by getting rid of the splitting step. In adaptive redundant lifting scheme, the signal is predicted and updated with redundant prediction operator and update operator directly, and the length of approximation signal and detail signal for all levels is the same as the original signal. Adaptive redundant lifting scheme
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possesses time invariant property, which well keeps the information of approximation signal and detail signal, and is particularly important for fault feature extraction. An approximation signal cl at level l decomposed with adaptive redundant lifting scheme is presented by following equations: d lþ1 ¼ cl P½l cl , clþ1 ¼ cl U ½l d lþ1 ,
ð13Þ
where cl+1 and dl+1 are approximation signal and detail signal at level l+1. The decomposition procedures of adaptive redundant lifting scheme are shown in Fig. 3. Adaptive redundant lifting scheme is easily invertible, and the reconstruction procedure is directly achieved from the inverse transform of adaptive redundant lifting scheme decomposition.
5. Applications To diagnose the gearbox faults effectively, the most important thing is to isolate the transient components from original complex vibration signals. We present several application examples to demonstrate the performance of the adaptive redundant lifting scheme method. 5.1. Experiment signal analysis In this section, adaptive redundant lifting scheme is used to analyse the experimental signals of a rotating machine. Fig. 4 shows the experimental test rig. The specification of the gears is shown in Table 1. c1+l cl − P [l]
U [l] d1+l
Fig. 3. Adaptive redundant lifting scheme decomposition.
Fig. 4. Rotating machine for test.
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Table 1 Specification of gears Module Width of the tooth Pressure angle (a) Number of teeth (normal) Number of teeth (test) Backlash of the normal gear Load torque
2 20 (mm) 201 55 75 0.5 (mm) 1.5 (Nm)
Fig. 5. Vibration signal of the test gear.
For a gear transmission, the meshing frequency fz is calculated by the formula f z ¼ nz=60,
(14)
where, n is the rotating speed of the test gear, z is the number of the test teeth. In this experiment, n ¼ 120 rpm, z ¼ 75. It follows from formula (14) that the meshing frequency of the test gear is 150 Hz. The vibration signal was acquired by an accelerometer mounted on the outer case of the gearbox. The sampling frequency was 25.6 kHz. The signal was low-pass filtered at 6 kHz. Fig. 5 shows the vibration signal of the test gear, in which the test gear was in wear state, and the data number is 1024. The vibration signal is complex, and transient component is hidden among many irrelevant components. The FFT spectrum of the vibration signal is illustrated in Fig. 6. The frequency components is abundant, the main frequency components are 567, 690 and 3410 Hz. We do not find the feature frequency components, namely, the meshing frequency 150 Hz and its harmonics. The vibration signal is analysed by using lifting scheme. The prediction operator and update operator are [0.0625, 0.5625, 0.5625, 0.0625] and [0.0313, 0.2813, 0.2813, 0.0313], respectively. Fig. 7 presents the approximation signals (a1–a2) and detail signals (d1–d2). No obvious feature components appear. We analyse the same vibration signal by adopting adaptive redundant lifting scheme. The prediction operator with 4 prediction coefficients and 3 vanishing moments is chosen, and the number of update coefficients is 4. Using the optimisation algorithm in Section 3, the initial
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Fig. 6. Spectrum of the vibration signal.
Fig. 7. Lifting scheme result for the vibration signal.
prediction operator and update operator which adapted to the signal characteristics at the corresponding level were calculated. The initial prediction operator and update operator at the first level are [0.1650, 0.1200, 1.2450, 0.2900] and [0.0881, 0.4519, 0.1106, 0.0256], and the initial prediction operator and update operator at the second level are [0.0701, 0.5854, 0.5396, 0.0549] and [0.0293, 0.2755, 0.2870, 0.0332]. Fig. 8 shows the decomposition result using adaptive redundant lifting scheme. The detail signal at level 2 distinctly discloses periodic impulses. The period is just about 0.0067 s, and the frequency is 150 Hz, which is equal to the meshing frequency of the test gear.
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Fig. 8. Adaptive redundant lifting scheme result for the vibration signal.
Fig. 9. Envelope spectrum of the detail signal at level 2.
The envelope spectrum of the detail signal at level 2 is illustrated in Fig. 9. It mainly consists of the meshing frequency 150 Hz and the second harmonic 300 Hz, which coincide with the characteristic frequency of the gear wear fault.
5.2. Practical signal analysis A large air compressor in a refinery consists of motor, gearbox, and compressor. Its structure sketch is shown in Fig. 10. The rotating speed of the motor is 2985 rpm, the rotating frequency of the high gear is 213 Hz and the transmitting ratio of the gearbox is 4.28125. Accelerometers were mounted to acquire
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Fig. 10. The structure sketch of air compressor.
Fig. 11. The vibration signal acquired from bearing bridge 5].
vibration signals at the corresponding measurement points on bearing cases 3#, 4#, 5# and 6#. The sampling frequency was 15 kHz. After overhaul the air compressor was running again. It was found that the vibration of gearbox was intense with ear-piercing noise. The vibration signal acquired from bearing case 5] is shown in Fig. 11, the data number is 512. The transient components are hidden in the signal. We can hardly find any useful diagnosis information from the vibration signal. The FFT spectrum of vibration signal acquired from bearing case 5] is shown in Fig. 12. It is distinct that there exist three frequencies of 1480, 2960 and 4231 Hz, and 213 Hz sidebands around them. While these three frequencies cannot be found in gearbox or compressor based on mechanical rotating condition. The 213 Hz is just equal to the rotating frequency of high gear and the compressor. It indicates that there existed modulation fault corresponding to the 213 Hz rotating frequency. What property does the modulation fault possess? It is difficult to give an answer according to the FFT spectrum shown in Fig. 12. The vibration signal acquired from bearing case 5] is analysed via lifting scheme. The prediction operator and update operator are [0.0117, 0.0977, 0.5859, 0.5859, 0.0977, 0.0117] and [0.0059, 0.0488, 0.2930, 0.2930, 0.0488, 0.0059], respectively. The lifting scheme approximations (a1a2) and details (d1d2) are shown in Fig. 13, no obvious amplitude
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Fig. 12. The FFT spectrum of vibration signal acquired from bearing bridge 5].
Fig. 13. Lifting scheme result for the vibration signal.
modulation signals and periodic impulses are found. As a result, it is difficult to draw any conclusive result from Fig. 13. We use adaptive redundant lifting scheme to analyse the same vibration signal. The prediction operator with 6 prediction coefficients and five vanishing moments is adopted, and the number of update coefficients is 6. Using the algorithm in Section 3, we calculated the initial prediction operator and update operator that adapted to the signal characteristics at the corresponding level. The initial prediction operator and update operator at the first level are [0.0059, 0.0684, 0.5274, 0.6445, 0.1269, 0.0176] and [0.0073, 0.0561, 0.3076, 0.2783, 0.0415, 0.0044], respectively; and the initial prediction operator and update operator at the second level are [0.0449, 0.2636, 0.9179, 0.2540, 0.0683, 0.0215] and [0.0024, 0.0073, 0.2100, 0.37600, 0.0903, 0.0142],
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respectively. The approximation signal at level 2 is shown in Fig. 14. From Fig. 14, the amplitude modulation in the gearbox vibration is clearly revealed, and there are seven amplitude modulation signals. The seven amplitude modulation signals spaced regularly at the time intervals correspond to the seven rotating period of the high gear. Fig. 15 illustrates the detail signal at level 2. From Fig. 15, distinct evenly spaced impulses can be observed from the detail signal, and the frequency of the impulses is identified as 213 Hz, which exactly is in accordance with the rotating frequency of high gear. Adopting adaptive redundant lifting scheme, we are able to identify the amplitude modulation signals and the time locations of the leading impulses in the gearbox vibration and determine that the fault occurred in the gearbox. In order to further find the gearbox fault, we use Hilbert envelope analysis to demodulate the approximation signal at level 2. The envelope of the approximation signal, as shown in Fig. 16, highlights periodicity in the amplitude modulation signals. The envelope spectrum of the approximation signal at level 2 is shown in Fig. 17. It is dominated by the impulse frequency 213 Hz, and its second and fifth harmonic. From Fig. 17, we can conclude that the impulse frequency 213 Hz, which corresponds to the rotating frequency of the high gear, is the modulation frequency of the gearbox fault. What resulted in the occurrence of an impact in every rotating period of high gear? We study the structure of the gearbox in Fig. 10. Helical gears were adopted to transmit force in the air compressor. The thrust plate was mounted on the high gear, which transmits axial force to the low
Fig. 14. Adaptive redundant lifting scheme decomposition approximation signal at level 2.
Fig. 15. Adaptive redundant lifting scheme decomposition detail signal at level 2.
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Fig. 16. The envelope of the approximation signal at level 2.
Fig. 17. The envelope spectrum of the approximation signal at level 2.
Fig. 18. Gears assembled improperly.
gear when the helical gears mesh. If the gears are assembled improperly, or there exists misalignment between the axles of high gear and the compressor, the plane of the thrust plate does not parallel the plane of low gear, as shown in Fig. 18. Then, rub-impact between the thrust plate and the low gear occurs once in every rotating period of the high gear, and it has excited the three distinct frequencies of 1480 Hz, 2960 Hz and 4231 Hz in Fig. 12, which are natural frequencies of the gearbox. It is obvious that rub-impact causes the intense vibration with ear-piercing noise in the gearbox [10]. After stoppage, the parallel condition of the gears and the alignment of axles of high gear and compressor were inspected and assembled. The machine set was adjusted well. When it was
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running again, the vibration became normal and there was no ear-piercing noise. A successful mechanical fault diagnosis using adaptive redundant lifting scheme has been achieved.
6. Conclusions In this paper, we have proposed an adaptive redundant lifting scheme for gearbox fault diagnosis. Firstly, the initial prediction operator and update operator are designed by adopting data-based optimisation algorithm, they are padded with zeros, and the redundant prediction operator and update operator are obtained, which adapt to the characteristics of the signal at the corresponding level. Then, the adaptive redundant lifting scheme algorithm is constructed by using the redundant prediction operator and update operator, in which the splitting step is omitted, and the signal at each level is predicted and updated directly, the data number for all levels remains the same. The proposed method is tested with the analysis of the experimental signal of a gearbox and the practical signal of a large air compressor. The results show that it performs better than lifting scheme that used the fixed prediction operator and update operator. Adaptive redundant lifting scheme clearly highlights the periodic impulses caused by wear fault in the gearbox vibration, and the amplitude modulation signal and impulses caused by rub-impact in the compressor by adaptive decomposition. It is obvious that adaptive redundant lifting scheme is an effective wavelet method to reveal the transient components hidden in the vibration signals.
Acknowledgements This work was supported by the key project of National Natural Science Foundation of China (No. 50335030), Doctor Program Foundation of University of China (No. 20040698026), National Basic Research Program of China (2005CB724106) and Natural Science Foundation of Xi’an Jiaotong university.
Appendix A. Nomenclature x xe xo c d P U f C P[l] U[l]
original data set even subset odd subset approximation signal detail signal prediction operator update operator scaling function wavelet function redundant prediction operator at level l redundant update operator at level l
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References [1] D. Boulahbal, M. Frid Golnaraghi, F. Ismail, Amplitude and phase wavelet maps for the detection of cracks in geared systems, Mechanical Systems and Signal Processing 13 (3) (1999) 423–426. [2] J. Lin, L. Qu, Feature extraction based on Morlet wavelet and its application for mechanical fault diagnosis, Journal of Sound and Vibration 234 (1) (2000) 135–148. [3] J. Lin, M.J. Zuo, Gearbox fault diagnosis using adaptive wavelet filter, Mechanical System and Signal Processing 17 (6) (2003) 1259–1269. [4] S. Loutridis, A. Trochidis, Classification of gear faults using Hoelder exponents, Mechanical System and Signal Processing 18 (5) (2004) 1009–1030. [5] W. Sweldens, The lifting scheme: a custom-design construction of biorthogonal wavelets, Applied Computer Harmonic Analysis 3 (2) (1996) 186–200. [6] W. Sweldens, The lifting scheme: A construction of second generation wavelets, SIAM Journal of Mathematical Analysis 29 (2) (1997) 511–546. [7] I. Daubechies, W. Sweldens, Factoring wavelet transform into lifting steps, Journal of Fourier Analysis Applications 4 (3) (1998) 245–267. [8] R. Claypoole, G. Davis, W. Sweldens, Nonlinear wavelet transforms for image coding via lifting, IEEE Transactions on Image Processing 12 (12) (2003) 1449–1459. [9] R.L. Claypoole Jr., Adaptive Wavelet Transforms via lifting, Thesis: Doctor of Philosophy, Electrical and Computer Engineering, Rice University, Houston, Texas, October 1999. [10] Z.J. He, Y.Y. Zi, Q.F. Meng, et al., Fault Diagnosis Principle of Non-stationary Signal and Applications to Mechanical Equipment, Higher Education Press, Beijing, 2001, pp. 133–146.
Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 20 (2006) 1992–2006 www.elsevier.com/locate/jnlabr/ymssp
Gearbox fault diagnosis using adaptive redundant Lifting Scheme Jiang Hongkaia,, He Zhengjiab, Duan Chendonga, Chen Pengc a School of Mechanical Engineering, Xi’an Jiaotong University, 710049, Xi’an, China State Key Lab for Manufacturing Systems Engineering, Xi’an Jiaotong University, 710049, Xi’an, China c Department of Environmental Science and Engineering, Faculty of Bioresources, Mie University, 1515 Kamihama-cho, Tsu, Mie 514-8507, Japan b
Received 30 January 2005; received in revised form 9 June 2005; accepted 18 June 2005 Available online 10 August 2005
Abstract Vibration signals acquired from a gearbox usually are complex, and it is difficult to detect the symptoms of an inherent fault in a gearbox. In this paper, an adaptive redundant lifting scheme for the fault diagnosis of gearboxes is developed. It adopts data-based optimisation algorithm to lock on to the dominant structure of the signal, and well reveal the transient components of the vibration signal in time domain. Both lifting scheme and adaptive redundant lifting scheme are applied to analyse the experimental signal from a gearbox with wear fault and the practical vibration signal from a large air compressor. The results confirm that adaptive redundant lifting scheme is quite effective in extracting impulse and modulation feature components from the complex background. r 2005 Elsevier Ltd. All rights reserved. Keywords: Adaptive redundant lifting scheme; Optimisation algorithm; Vibration signal; Fault diagnosis
1. Introduction Gearboxes are key parts in a wide range of mechanical systems. It is very important to detect incipient fault symptoms from gearboxes. Usually, vibration signals are acquired from Corresponding author
E-mail address: [email protected] (J. Hongkai). 0888-3270/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2005.06.001
ARTICLE IN PRESS J. Hongkai et al. / Mechanical Systems and Signal Processing 20 (2006) 1992–2006
1993
accelerometers mounted on the outer surface of a bearing housing. The signals consist of vibrations from the meshing gears, shafts, bearings, and other components. The useful information is corrupted and it is difficult to diagnose a gearbox from such vibration signals. Wavelet theory is a powerful tool for non-stationary signal analysis, and it has been successfully used in gearbox diagnosis. Boulahbal et al. [1] used the amplitude and phase maps of continuous wavelet transform together to extract fault features of a gearbox and obtained a more positive assessment of a tooth condition. Lin et al. [2,3] utilised the similarity between Morlet wavelet and an impulse shape, and detected tooth crack symptoms of a gearbox immersed in the noise. Loutridis et al. [4] used continuous wavelet transform and Hodlder exponents to classify gear faults and got better performance. In all the wavelet techniques mentioned above, researchers usually selected an appropriate wavelet function from a library of previously designed wavelet functions to match a specific fault symptom. Different types of mechanical faults have different waveform characteristics, even one wavelet function is selected, and it is not always the best wavelet function to detect a specific fault symptom in a gearbox. New wavelet method is needed to overcome the drawback. Lifting scheme is a spatial domain construction of biorthogonal wavelets developed by Sweldens [5–7]. It abandons the Fourier transform as design tool for wavelets, wavelets are no longer defined as translates and dilates of one fixed function. Compared with classical wavelet transform, Lifting scheme possesses several advantages, e.g. possibility of adaptive design, inplace calculations, irregular samples and integers-to-integers wavelet transforms. Lifting scheme provides a great deal of flexibility, it can be designed according to the properties of the given signal, and it ensures that the resulting transform is invertible. In this paper, we develop a new wavelet method called adaptive redundant lifting scheme. It is based on lifting scheme and designed to capture the transient components of the gearbox vibration by adaptive decomposition. In Section 2, the theory of lifting scheme is reviewed briefly. The data-based optimisation algorithm is described in Section 3. The method of adaptive redundant lifting scheme is presented in Section 4. In Section 5, adaptive redundant lifting scheme is applied to analyse the vibration signals of an experimental gearbox and a large compressor gearbox. Comparison with lifting scheme is also shown. Conclusions are given in Section 6.
2. Lifting scheme principle Lifting scheme is an entirely space domain wavelet construction. A typical lifting scheme procedure consists of three basic steps: split, predict and update [5–7]. Split: Let x(n) be an original data set. In this step, x(n) is divided into two disjoint even subset xe(n) and odd subset x0(n), where xe ðnÞ ¼ xð2nÞ and x0 ðnÞ ¼ xð2n þ 1Þ. Predict: If the original signal has a local correlation structure, then the even and odd subsets are highly correlated. We predict the odd coefficients x0(n) from the neighbouring even coefficients xe(n), and the prediction differences d(n) are defined as detail signal dðnÞ ¼ x0 ðnÞ Pðxe ðnÞÞ, T
Where P ¼ ½pð1Þ; . . . ; pðNÞ is the prediction operator.
(1)
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Update: This step forms a coarse approximation to the original signal. We combine the even coefficients and the linear combination of the prediction differences, and then the approximation signal c(n) are obtained cðnÞ ¼ xe ðnÞ þ UðdðnÞÞ,
(2)
~ is the update operator. Where U ¼ ½uð1Þ; . . . ; uðNÞ Iteration of the three steps on the output c(n), and then the detail signal and approximation signal at different levels are generated. Scaling function f(x) and wavelet function c(x) of lifting scheme can be derived from P and U by iteration algorithm. The program flow chart is illustrated in Fig. 1. The scaling function and wavelet function with N ¼ 6 and N~ ¼ 6 are shown in Fig. 2. The scaling function and wavelet function are symmetrical and compactly supported. The shape of the wavelet function is very similar to an impulse, and it is desirable to detect the transient components in the vibration signal. We can choose P and U according to the properties of the given signal. Since the design of lifting scheme without reference to Fourier techniques, each lifting step is always invertible, and the inverse transform of lifting scheme is the inversion of the lifting steps.
Fig. 1. The program flow diagram of scaling and wavelet functions.
Fig. 2. Scaling function and wavelet function with N ¼ 6 and N~ ¼ 6.
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Assuming the same P and U are chosen for the forward and inverse transform, the lifting construction guarantees perfect reconstruction for any P and U [8].
3. Initial prediction operator and update operator design Lifting scheme increases the flexibility to extract the features of a given data set. We can optimise the prediction operator and update operator to well reveal the feature components of the data set. In this section, we adopt Claypoole’s optimisation algorithm [8,9] to design the initial prediction operator P and update operator U, and make them adapt to the dominating structures of the data set at the corresponding level. The design of initial prediction operator and update operator based on the data characteristics is accomplished by the following procedures. A N-point initial prediction operator P is designed to suppress polynomial components upto order MoN, and the remaining N–M degrees of freedom are used to match the given signal. The initial prediction operator is as below P ¼ ½ pð1Þ; . . . ; pðNÞT .
(3)
Construct a M N matrix V, its element is ½V i;j ¼ ½2j N 1i1 ,
(4)
where i ¼ 1; 2; . . . ; M; j ¼ 1; 2; . . . ; N. We require that VP ¼ ½1; 0; 0; . . . 0T ,
(5)
The vector of prediction differences can be expressed as follows: e ¼ X 0 X e P.
(6)
The goal is to obtain the prediction coefficients that minimise the sum of squared prediction differences, namely min kX 0 X e Pk2 .
(7)
P
We solve Eqs. (5) and (7), then the initial prediction operator P that locks on to the dominant structure of a signal at the corresponding level is obtained. ~ ~ A N-point initial update operator U is designed by the following procedures, where NpN. Construct a 2N–1 -dimensional vector q, its element is defined as i ¼ 1; . . . N :
qð2i 1Þ ¼ pðiÞ;
i ¼ 1; . . . N 1 :
qð2iÞ ¼ 0;
qðNÞ ¼ 1.
Assume that K ¼ 4N 3, construct a K N~ matrix H, its element is 0, except Hðð2i 1Þ : ð2i þ 2N 3Þ; iÞ ¼ q, ~ where i ¼ 1; . . . ; N.
(8)
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And then, a N~ K matrix W is constructed, in which element is ½W m;n ¼ ½n N N~ þ 1m1 ,
(9)
~ n ¼ 1; 2; . . . ; K. where m ¼ 1; 2; . . . ; N, We calculate U from the following expression: WHU ¼ ½1; 0; 0; . . . 0T .
(10)
Since matrix H contains the initial prediction operator P, the initial update operator U also adapts to the dominant signal structure at the corresponding level. 4. Adaptive redundant lifting scheme design algorithm Now, we explain how to design adaptive redundant lifting scheme. The design includes two parts, the design of redundant prediction operator and update operator, and the construction of adaptive redundant lifting scheme. 4.1. Redundant prediction operator and update operator design The design of redundant prediction operator P[l] and redundant update operator U[l] (l is the decomposition level) is the key step in adaptive redundant lifting scheme. P[l] and U[l] are obtained by padding the initial prediction operator P and update operator U with zeros at the corresponding level l. Their design algorithm is as follows. Assuming the initial prediction operator P and update operator U at level l are designed with ~ N and N~ the algorithm in Section 3, where P ¼ fpm g, m ¼ 1; 2; . . . ; N; U ¼ fun g, n ¼ 1; 2; . . . ; N, are the coefficient number of P and U, respectively. Then the redundant prediction operator P[l] and redundant update operator U[l] at level l are designed by padding the prediction coefficient pm and update coefficient un with zeros. The redundant prediction coefficient p½l j is expressed as below ( l pm ; j ¼ 2 m ; j ¼ 1; . . . 2l N. p½l (11) j ¼ 0; ja2l m The redundant update coefficient u½l i is expressed as below ( un ; i ¼ 2l n ~ u½l ¼ ; i ¼ 1; . . . 2l N: i 0; ia2l n
(12)
The expressions of redundant prediction operator P[l] and update operator U[l] at level l are as ½l l l ~ ½l below P½l ¼ fp½l j g, j ¼ 1; . . . ; 2 N; U ¼ fui g, i ¼ 1; . . . ; 2 N. 4.2. Adaptive redundant lifting scheme construction We convert lifting scheme into adaptive redundant lifting scheme by getting rid of the splitting step. In adaptive redundant lifting scheme, the signal is predicted and updated with redundant prediction operator and update operator directly, and the length of approximation signal and detail signal for all levels is the same as the original signal. Adaptive redundant lifting scheme
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possesses time invariant property, which well keeps the information of approximation signal and detail signal, and is particularly important for fault feature extraction. An approximation signal cl at level l decomposed with adaptive redundant lifting scheme is presented by following equations: d lþ1 ¼ cl P½l cl , clþ1 ¼ cl U ½l d lþ1 ,
ð13Þ
where cl+1 and dl+1 are approximation signal and detail signal at level l+1. The decomposition procedures of adaptive redundant lifting scheme are shown in Fig. 3. Adaptive redundant lifting scheme is easily invertible, and the reconstruction procedure is directly achieved from the inverse transform of adaptive redundant lifting scheme decomposition.
5. Applications To diagnose the gearbox faults effectively, the most important thing is to isolate the transient components from original complex vibration signals. We present several application examples to demonstrate the performance of the adaptive redundant lifting scheme method. 5.1. Experiment signal analysis In this section, adaptive redundant lifting scheme is used to analyse the experimental signals of a rotating machine. Fig. 4 shows the experimental test rig. The specification of the gears is shown in Table 1. c1+l cl − P [l]
U [l] d1+l
Fig. 3. Adaptive redundant lifting scheme decomposition.
Fig. 4. Rotating machine for test.
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Table 1 Specification of gears Module Width of the tooth Pressure angle (a) Number of teeth (normal) Number of teeth (test) Backlash of the normal gear Load torque
2 20 (mm) 201 55 75 0.5 (mm) 1.5 (Nm)
Fig. 5. Vibration signal of the test gear.
For a gear transmission, the meshing frequency fz is calculated by the formula f z ¼ nz=60,
(14)
where, n is the rotating speed of the test gear, z is the number of the test teeth. In this experiment, n ¼ 120 rpm, z ¼ 75. It follows from formula (14) that the meshing frequency of the test gear is 150 Hz. The vibration signal was acquired by an accelerometer mounted on the outer case of the gearbox. The sampling frequency was 25.6 kHz. The signal was low-pass filtered at 6 kHz. Fig. 5 shows the vibration signal of the test gear, in which the test gear was in wear state, and the data number is 1024. The vibration signal is complex, and transient component is hidden among many irrelevant components. The FFT spectrum of the vibration signal is illustrated in Fig. 6. The frequency components is abundant, the main frequency components are 567, 690 and 3410 Hz. We do not find the feature frequency components, namely, the meshing frequency 150 Hz and its harmonics. The vibration signal is analysed by using lifting scheme. The prediction operator and update operator are [0.0625, 0.5625, 0.5625, 0.0625] and [0.0313, 0.2813, 0.2813, 0.0313], respectively. Fig. 7 presents the approximation signals (a1–a2) and detail signals (d1–d2). No obvious feature components appear. We analyse the same vibration signal by adopting adaptive redundant lifting scheme. The prediction operator with 4 prediction coefficients and 3 vanishing moments is chosen, and the number of update coefficients is 4. Using the optimisation algorithm in Section 3, the initial
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Fig. 6. Spectrum of the vibration signal.
Fig. 7. Lifting scheme result for the vibration signal.
prediction operator and update operator which adapted to the signal characteristics at the corresponding level were calculated. The initial prediction operator and update operator at the first level are [0.1650, 0.1200, 1.2450, 0.2900] and [0.0881, 0.4519, 0.1106, 0.0256], and the initial prediction operator and update operator at the second level are [0.0701, 0.5854, 0.5396, 0.0549] and [0.0293, 0.2755, 0.2870, 0.0332]. Fig. 8 shows the decomposition result using adaptive redundant lifting scheme. The detail signal at level 2 distinctly discloses periodic impulses. The period is just about 0.0067 s, and the frequency is 150 Hz, which is equal to the meshing frequency of the test gear.
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Fig. 8. Adaptive redundant lifting scheme result for the vibration signal.
Fig. 9. Envelope spectrum of the detail signal at level 2.
The envelope spectrum of the detail signal at level 2 is illustrated in Fig. 9. It mainly consists of the meshing frequency 150 Hz and the second harmonic 300 Hz, which coincide with the characteristic frequency of the gear wear fault.
5.2. Practical signal analysis A large air compressor in a refinery consists of motor, gearbox, and compressor. Its structure sketch is shown in Fig. 10. The rotating speed of the motor is 2985 rpm, the rotating frequency of the high gear is 213 Hz and the transmitting ratio of the gearbox is 4.28125. Accelerometers were mounted to acquire
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Fig. 10. The structure sketch of air compressor.
Fig. 11. The vibration signal acquired from bearing bridge 5].
vibration signals at the corresponding measurement points on bearing cases 3#, 4#, 5# and 6#. The sampling frequency was 15 kHz. After overhaul the air compressor was running again. It was found that the vibration of gearbox was intense with ear-piercing noise. The vibration signal acquired from bearing case 5] is shown in Fig. 11, the data number is 512. The transient components are hidden in the signal. We can hardly find any useful diagnosis information from the vibration signal. The FFT spectrum of vibration signal acquired from bearing case 5] is shown in Fig. 12. It is distinct that there exist three frequencies of 1480, 2960 and 4231 Hz, and 213 Hz sidebands around them. While these three frequencies cannot be found in gearbox or compressor based on mechanical rotating condition. The 213 Hz is just equal to the rotating frequency of high gear and the compressor. It indicates that there existed modulation fault corresponding to the 213 Hz rotating frequency. What property does the modulation fault possess? It is difficult to give an answer according to the FFT spectrum shown in Fig. 12. The vibration signal acquired from bearing case 5] is analysed via lifting scheme. The prediction operator and update operator are [0.0117, 0.0977, 0.5859, 0.5859, 0.0977, 0.0117] and [0.0059, 0.0488, 0.2930, 0.2930, 0.0488, 0.0059], respectively. The lifting scheme approximations (a1a2) and details (d1d2) are shown in Fig. 13, no obvious amplitude
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Fig. 12. The FFT spectrum of vibration signal acquired from bearing bridge 5].
Fig. 13. Lifting scheme result for the vibration signal.
modulation signals and periodic impulses are found. As a result, it is difficult to draw any conclusive result from Fig. 13. We use adaptive redundant lifting scheme to analyse the same vibration signal. The prediction operator with 6 prediction coefficients and five vanishing moments is adopted, and the number of update coefficients is 6. Using the algorithm in Section 3, we calculated the initial prediction operator and update operator that adapted to the signal characteristics at the corresponding level. The initial prediction operator and update operator at the first level are [0.0059, 0.0684, 0.5274, 0.6445, 0.1269, 0.0176] and [0.0073, 0.0561, 0.3076, 0.2783, 0.0415, 0.0044], respectively; and the initial prediction operator and update operator at the second level are [0.0449, 0.2636, 0.9179, 0.2540, 0.0683, 0.0215] and [0.0024, 0.0073, 0.2100, 0.37600, 0.0903, 0.0142],
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respectively. The approximation signal at level 2 is shown in Fig. 14. From Fig. 14, the amplitude modulation in the gearbox vibration is clearly revealed, and there are seven amplitude modulation signals. The seven amplitude modulation signals spaced regularly at the time intervals correspond to the seven rotating period of the high gear. Fig. 15 illustrates the detail signal at level 2. From Fig. 15, distinct evenly spaced impulses can be observed from the detail signal, and the frequency of the impulses is identified as 213 Hz, which exactly is in accordance with the rotating frequency of high gear. Adopting adaptive redundant lifting scheme, we are able to identify the amplitude modulation signals and the time locations of the leading impulses in the gearbox vibration and determine that the fault occurred in the gearbox. In order to further find the gearbox fault, we use Hilbert envelope analysis to demodulate the approximation signal at level 2. The envelope of the approximation signal, as shown in Fig. 16, highlights periodicity in the amplitude modulation signals. The envelope spectrum of the approximation signal at level 2 is shown in Fig. 17. It is dominated by the impulse frequency 213 Hz, and its second and fifth harmonic. From Fig. 17, we can conclude that the impulse frequency 213 Hz, which corresponds to the rotating frequency of the high gear, is the modulation frequency of the gearbox fault. What resulted in the occurrence of an impact in every rotating period of high gear? We study the structure of the gearbox in Fig. 10. Helical gears were adopted to transmit force in the air compressor. The thrust plate was mounted on the high gear, which transmits axial force to the low
Fig. 14. Adaptive redundant lifting scheme decomposition approximation signal at level 2.
Fig. 15. Adaptive redundant lifting scheme decomposition detail signal at level 2.
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Fig. 16. The envelope of the approximation signal at level 2.
Fig. 17. The envelope spectrum of the approximation signal at level 2.
Fig. 18. Gears assembled improperly.
gear when the helical gears mesh. If the gears are assembled improperly, or there exists misalignment between the axles of high gear and the compressor, the plane of the thrust plate does not parallel the plane of low gear, as shown in Fig. 18. Then, rub-impact between the thrust plate and the low gear occurs once in every rotating period of the high gear, and it has excited the three distinct frequencies of 1480 Hz, 2960 Hz and 4231 Hz in Fig. 12, which are natural frequencies of the gearbox. It is obvious that rub-impact causes the intense vibration with ear-piercing noise in the gearbox [10]. After stoppage, the parallel condition of the gears and the alignment of axles of high gear and compressor were inspected and assembled. The machine set was adjusted well. When it was
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running again, the vibration became normal and there was no ear-piercing noise. A successful mechanical fault diagnosis using adaptive redundant lifting scheme has been achieved.
6. Conclusions In this paper, we have proposed an adaptive redundant lifting scheme for gearbox fault diagnosis. Firstly, the initial prediction operator and update operator are designed by adopting data-based optimisation algorithm, they are padded with zeros, and the redundant prediction operator and update operator are obtained, which adapt to the characteristics of the signal at the corresponding level. Then, the adaptive redundant lifting scheme algorithm is constructed by using the redundant prediction operator and update operator, in which the splitting step is omitted, and the signal at each level is predicted and updated directly, the data number for all levels remains the same. The proposed method is tested with the analysis of the experimental signal of a gearbox and the practical signal of a large air compressor. The results show that it performs better than lifting scheme that used the fixed prediction operator and update operator. Adaptive redundant lifting scheme clearly highlights the periodic impulses caused by wear fault in the gearbox vibration, and the amplitude modulation signal and impulses caused by rub-impact in the compressor by adaptive decomposition. It is obvious that adaptive redundant lifting scheme is an effective wavelet method to reveal the transient components hidden in the vibration signals.
Acknowledgements This work was supported by the key project of National Natural Science Foundation of China (No. 50335030), Doctor Program Foundation of University of China (No. 20040698026), National Basic Research Program of China (2005CB724106) and Natural Science Foundation of Xi’an Jiaotong university.
Appendix A. Nomenclature x xe xo c d P U f C P[l] U[l]
original data set even subset odd subset approximation signal detail signal prediction operator update operator scaling function wavelet function redundant prediction operator at level l redundant update operator at level l
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