Bearing Fault Detection based on Stochastic Resonance Optimized by Levenberg-Marquardt Algorithm

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1 International Journal of Performability Engineering, Vol. 11, No. 1, January 2015, pp RAMS Consultants Printed in India Bearing Fault Detection based on Stochastic Resonance Optimized by Levenberg-Marquardt Algorithm XINGHUI ZHANG 1*, LEI XIAO 2, and JIANSHE KANG 1 1 Mechanical Engineering College, Shijiazhuang, CHINA 2 The State Key Lab of Mechanical Transmission, Chongqing University, Chongqing, CHINA (Received on April 14, 2014, revised on October 26, 2014) Abstract: Bearings are one of the most important components in rotating machineries because their failure could cause catastrophic disasters of whole system. Currently, one of the main problems when implementing bearing prognostics and health management is how to detect the incipient fault as soon as possible. This capability can enable the operators having sufficient time to implement preventive maintenance activities. For incipient fault, its vibration signal is relatively weak and always submerged in the noise, which makes the fault hard to be detected. Stochastic resonance is a good way to detect the weak signal in strong noise. However, the effect of the stochastic resonance depends on the adjustment of two parameters. Current parameter optimization methods are mainly depend on some random searching algorithms like particle swarm optimization, genetic algorithm etc. However, these methods may converge to local optima and need more searching time. So, the Levenberg-Marquardt algorithm is utilized to optimize the two parameters in this paper. The resonance effect is evaluated by signal-to-noise ratio. In order to validate the effectiveness of the stochastic resonance optimized by Levenberg- Marquardt, two bearing fault data sets were used. The analysis results state the proposed method could detect the fault earlier. Keywords: Bearings, stochastic resonance, fault diagnosis, Levenberg-Marquardt, parameter optimization 1. Introduction Bearings are one of the key components in some mechanical transmission systems like gearbox. Fault of these components will decline the transmission accuracy. Furthermore, some severe faults may cause system s catastrophic incidents. Therefore, detecting and diagnosing these faults as early as possible is a quite important work. Because sooner the fault is detected, more time maintainers have to take maintenance actions. This problem has drawn intensive attention during the past several decades, there are many theories and techniques are developed to detect and diagnose faults. These methods include multiwavelet denoising [1,2], empirical mode decomposition (EMD) [3,4], and Kurtogram method [5,6] etc. Although so many works have been presented, the research is still deficient especially when fault is at early stage. The fault signals are weak and often submerged in large amount of noise. So how to detect the incipient fault is a problem. As Benzi proposed the stochastic resonance (SR) for solving a climatic problem [7], it is widely used in many realms. It is also introduced to detect the weak signals from gear or bearing fault. Different from conventional methods, SR uses noise to amplify the weak signal rather than eliminate noise. The traditional de-noising or filtering method could weaken or filter the useless noise to some extent, but the usable signals could be weakened more or less simultaneously. Actually, the noise is a good and free energy to enhance the signals [8]. * Corresponding author s dynamicbnt@gmail.com 61

2 62 Xinghui Zhang, Lei Xiao, and Jianshe Kang Due to the apparent advantages of SR, it is intensively researched in mechanical fault detection and diagnosis. Due to the restriction of adiabatic approximation theory, initial works focus on small parameter signals which have low frequency and weak periodic force. The theory limits the values of the frequency and amplitude of the aperiodic signals and noise intensity are all smaller than one. But in real mechanical systems, the frequency, amplitude of the signals and the noise intensity are higher than one. Meanwhile the signals are aperiodic. These facts make the traditional SR is not suitable for detecting or diagnosing mechanical faults. Then scale normalized stochastic resonance [9], re-scaling frequency stochastic resonance [10,11] and frequency-shifted and re-scaling stochastic resonance [12] etc. are proposed to solve the problem. They make SR could detect any frequency signals possible. The effect of SR not only depends on the noise intensity but also the system parameters. The two parameters could determine the height of potential barrier. When the height of potential barrier is too high, the particle cannot jump into the other potential well, so the SR system cannot resonate. Otherwise, when the height of potential barrier is too small, the effect of resonance is not obvious. So, choose a pair of optimal system parameters is important. In current works, the two parameters are optimized by some kinds of random optimization algorithm like genetic algorithm [13,14], ant colony optimization [15] etc. These kinds of random searching algorithms make the search a long time due to a certain scale of population. Considering the numeric optimization algorithm could utilize the error information in each iteration, in this study, Levenberg-Marquardt (LM) algorithm is used to optimize the system parameters of SR. In present work, theory of classical SR is introduced in Section 2. The system parameters optimization by Levenberg-Marquardt algorithm is explained in Section 3. Validation by simulation signals and two bearing fault signals are given in Section 4 and the whole work is concluded in Section Brief Introduction of Classical SR Stochastic Resonance (SR) is a phenomenon where the weak signal could be enhanced because of the cooperative effect between the internal mechanism and the external periodic forcing. Taking the overdamped motion of Brownian particle in a bistable potential as an example, the SR model is as follows: dx ' U( x) st ( ) nt ( ) dt = + + (1) a 2 b 4 U( x) = x + x 2 4 (2) Where U(x) is the potential function for the system. x(t) is the output signal. s(t) is the input signal and n(t) is a Gaussian white noise. (t) 0 n(t) n(t + t) = 2Dδ t. D is n =, ( ) the noise intensity and means the statistical mean value calculation. The bistable system contains two stable points ± x= ± ab and an unstable point x=0. The height of potential barrier is determined by the system parameters, U = a 2 4b.

3 Bearing Fault Detection based on Stochastic Resonance Optimized by Levenberg-Marquardt Algorithm 63 Figure 1: Potential Function U(x) in a Bistable SR System The position of Brownian particle is determined by initial condition due to lack of periodic signal and noise, and it would not change. When there is a periodic signal, the Brownian particle just moves in one potential well. It cannot cross the barrier and jump into the other potential well. But the Brownian particle s energy will be accumulated due to the impact of noise. If the noise intensity is large enough, the Brownian particle will cross the barrier and jump between the two potential wells continuously. When the transition rate of the particle matches the periodic input signal, the periodic input signal will be enhanced with SR. 3. Parameters Optimized by Levenberg-Marquardt Algorithm To optimize the two parameters, local signal to noise ratio (LSNR) is introduced as the criterion. 2 AoF0± 1 maximize LSNR = 10 log10 (3) 2 2 AoF AoF0± 1 Where, AoF 0±1 is the amplitudes of the resonated frequency and the two closed frequencies. LM algorithm is used to optimize the system parameters to make the effect of SR could be more obvious. LM algorithm could be regarded as an adjuster between the Gauss Newton algorithm and the method of gradient descent. ( J T J + λ diag ( J T J )) d = J T y f ( β) (4) f ( β ) Where, J is the Jacobian matrix J =, λ is the damping factor, δ is the increment, y β is the empirical datum vector. In this study, y is a certain LSNR, f(β) is the model curve with parameter β. Therefore, a simple optimization flow chart is given as Figure 2. In Figure 2, f s, h and LSNR m are sampling frequency, time calculation step and given maximum LSNR. In this study, the output signals are calculated by fourth-order Runge-Kutta discretisation. Taking the partial derivative of LSNR with respect to system parameters a and b, three parts should be calculated. They are the derivative of LSNR with respect to AoF, the derivative of AoF with respect to the signal after Runge-Kutta discretisation (x) and the derivative of x with respect to a and b LSNR 10 AoF AoF 2 AoF ( AoF AoF ± 0± 1) = (5) AoF ln10 AoF0± 1 AoF AoF ( 0± 1)

4 64 Xinghui Zhang, Lei Xiao, and Jianshe Kang AoFk window 2p i = exp ( j 1) ( k 1) x N 2 N j = 1, 2,, N (6) Where, N is the length of the signal, window is the window function when Fast Fourier Transformation (FFT) is implemented. 2 x x 1 = 3 a 2 1+ ax + bx 4 x x 1 = 3 b 4 1+ ax + bx (7) Therefore, the Jacobian matrix can be calculated when equation (5) multiply equation (6) then multiply equation (7). Hessian matrix (H) can be calculated according to Jacobian matrix, then the hybrid Hessian matrix (H_lm) is given as follows. T H = J J (8) H _ lm = H + li (9) The increment of the parameters is calculated according to error information. E = LSNR LSNR (10) m 1 H _lm T J E δ = (11) The parameters a and b should be updated as follows. a = a+ δ a, (12) b = b+ δb where, δ a and δ b are the increments of a and b respectively. Initializing a, b, f s, h, LSNR m Calculating the output signal by fourth-order Runge-Kutta discretisation Calculating LSNR of the output signal Calculating Jacobian matrix, Hessian matrix and increment Updating a, b and λ N Satisfying the terminal condition? Outputting the best parameters a and b Figure 2: The Simple Flow Chart of SR Optimized by LM Algorithm Y

5 Bearing Fault Detection based on Stochastic Resonance Optimized by Levenberg-Marquardt Algorithm 65 If the new a or b is smaller than 0, it should be regenerated due to the restriction of physical significance for SR system. The update of λ is determined by the new error (E new ) compared with the former. λ = λ 10 Enew < E (13) λ = λ 10 Enew E When λ becomes smaller, LM algorithm approaches Gauss Newton algorithm which performs well for searching local optima. Otherwise, when λ becomes bigger, LM algorithm approaches gradient descend algorithm which performs well for searching global optima. 4. Illustrative Example 4.1 Validation by Simulation Signal The proposed method is validated by a simulation signal. The amplitude A=0.1, the noise intensity D=0.5, the original signal frequency f o =0.02. So, the simulation signal x1 = Asin ( 2π fo ) + 2D rand. The data length DL=5000, the sampling frequency f s =20, the time calculation step h=1/f s. The window function is Hamming. Therefore, the waveform and frequency spectrum of x 1 are as follows. Figure 3: (a) The waveform of simulation signal; (b) Frequency spectrum of simulation original signal From Figure 3, the target frequency is submerged in the noise. To resonate the target frequency, the initial system parameters are random in the interval (0,10). The terminal condition is maximum iteration epoch is 200. By LM algorithm, a=0.9209, b= , the resonated signal and its frequency spectrum are given as follows. From Figure 4, the target frequency could be detected, and the system resonates. 4.2 Validation by Bearing Outer Fault Signal The test rig [16] was equipped with a NICE bearing with the following parameters: roller diameter inch, pitch diameter inch, number of elements is 8, and the contact angle equal to 0. The input shaft rotation rate is 25Hz. The sample frequency is 48,828 Hz and lasting 3 seconds. For outer race fault conditions, we acquire three data respectively under load 270 lbs. Through the geometric parameters of NICE bearing, the fault characteristic frequencies can be calculated as illustrated in Table 1. And we define

6 66 Xinghui Zhang, Lei Xiao, and Jianshe Kang the four fault characteristic frequencies such as Ball Pass Frequency Inner Race (BPFI), Ball Pass Frequency Outer Race (BPFO), Ball Spin Frequency (BSF), and Fundamental Train Frequency (FTF). Figure 4: (a) Result of stochastic resonance; (b) Frequency spectrum of stochastic resonance result; (c) Local amplify of (b) Table 1: Fault Characteristic Frequencies at the Input Frequency of 25 Hz Type of Faults Fault Characteristic Frequencies (Hz) Outer race fault Inner race fault Ball fault Cage fault The waveform and frequency spectrum of the signal is as follows.

7 Bearing Fault Detection based on Stochastic Resonance Optimized by Levenberg-Marquardt Algorithm 67 Figure 5: (a) Raw Time Signal of Bearing Outer Race Fault; (b) Frequency Spectrum of Raw Signal From Figure 5, the signals do not meet the small parameter s condition, the signals are preprocessed based on the method proposed by Tan et al. [12]. Then a series parameters are set, the resampling frequency is 400, the rescale ratio is 600, the carrier frequency is 78, the estimated frequency of periodic signal is 80 (approximately equal to the outer race fault frequency). The initial a and b are in the interval (0, 10). After optimized by LM algorithm, a=1.1826, b= , and the detected frequency is The output signal and its spectrum are given in Figure 6. Figure 6: (a) Result of StochasticResonance; (b) Frequency Spectrum of Stochastic Resonance Result; (c) Local amplify of (b)

8 68 Xinghui Zhang, Lei Xiao, and Jianshe Kang 4.3 Validation by Bearing Ball Fault Signal of Planetary Gearbox This is a bearing fault case of commercial wind turbine [16]. A bearing of intermediate shaft has ball fault. The fault frequency is 24.3 Hz. Sampling frequency is 48,828 Hz. After processed using stochastic resonance, the result and its frequency spectrum are illustrated in Figure 7. The resampling frequency is 400, rescale ratio is 600, carrier frequency is 20, estimated frequency of periodic signal is 25 (approximately equal to the ball fault frequency). The optimized parameters a and b are and respectively. Figure 7: (a ) Stochastic Resonance of Bearing Ball Fault Signal; (b) Frequency Spectrum 5. Conclusions Different from existing methods which search the optimal system parameters for SR model, in this paper, Levenberg-Marquardt which is a numeric optimization method is introduced to solve this problem. Compared with random searching methods, this method could avoid selecting optima from solutions generated by population. That could shorten searching time and decline calculated amount. The proposed method is validated by simulation signal and two bearing fault signals. The results illustrate this method could detect the target frequency and characteristic frequency of the bearing fault. Acknowledgement: The authors would like to thank anonymous referees for their remarkable comments and great support by Key Project supported by National Science Foundation of China( ); Natural Science Foundation project of Chongqing (CSTC, 2009BB3365), and the Fundamental Research Funds for the State Key Laboratory of Mechanical Transmission, Chongqing University(SKLMT-ZZKT-2012 MS 02). References [1] Chen, J., Y. Zi, Z. He and X. Wang. Adaptive Redundant Multiwavelet Denoising with Improved Neighboring Coefficients for Gearbox Fault Detection. Mechanical Systems and Signal Processing, 2013; 38:

9 Bearing Fault Detection based on Stochastic Resonance Optimized by Levenberg-Marquardt Algorithm 69 [2] Sun, H., Y. Zi and Z. He. Wind Turbine Fault Detection using Multiwavelet Denoising with the Data-driven Block Threshold. Applied Acoustics, 2014; 77: [3] Georgoulas, G., T. Loutas, C. D. Stylios and V. Kostopoulos. Bearing Fault Detection based on Hybrid Ensemble Detector and Empirical Mode Decomposition. Mechanical Systems and Signal Processing, 2013; 41: [4] Dybała, J. and R. Zimroz. Rolling Bearing Diagnosing Method based on Empirical Mode Decomposition of Machine Vibration Signal. Applied Acoustics 2014; 77: [5] Zhang, Y. and R. B. Randall. Rolling Element Bearing Fault Diagnosis based on the Combination of Genetic Algorithms and Fast Kurtogram. Mechanical Systems and Signal Processing, 2009; 23: [6] Wang, D., P. W. Tse and K. L. Tsui. An Enhanced Kurtogram Method for Fault Diagnosis of Rolling Element Bbearings. Mechanical Systems and Signal Processing, 2013; 35: [7] Benzi, R., G. Parisi, A. Sutera and A. Vulpiani. A Theory of Stochastic Resonance in Climatic Change. Siam Journal on Applied Mathematics 1983; 43: [8] Benzi, R., A. Sutera and A. Vulpiani. The Mechanism of Stochastic Resonance. Journal of Physics A: Mathematical and General 1981; 14: L453-L457. [9] Yang, D. X. and N. Q. Hu. Detection of Weak Aperiodic Signal based on Stochastic Resonance. Third International Symposium on Instrument Science and Technology, Xi an, China, August 18-22, 2004; [10] Leng, Y. S., Y. G. Leng, T. Y. Wang and Y. Guo. Numerical Analysis and Engineering Application of Large Parameter Stochastic Resonance. Journal of Sound and Vibration 2006; 292: [11] Leng, Y., Y. Guo and D. Zhou. Characteristic Signal Detection Based on Re- Scaling Frequency Stochastic Resonance and Its Application in Fault Diagnosis. ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Las Vegas, Nevada, September 4 7, 2007; [12] Tan, J., X. Chen, J. Wang, H. Chen, H. Cao, Y. Zi and Z. He. Study of Frequency- Shifted and Re-scaling Stochastic Resonance and its Application to Fault Diagnosis. Mechanical Systems and Signal Processing 2009; 23: [13] Hou, Z., J. Yang, Y. Wang and K. Wang. Weak Signal Detection based on Stochastic Resonance Combining with Genetic Algorithm. 11 th IEEE Singapore International Conference on Communication Systems 2008; Guangzhou: [14] He, Q., J. Wang, F. Hu and F. Kong. Wayside Acoustic Diagnosis of Defective Train Bearings based on Signal Resampling and Information Enhancement. Journal of Sound and Vibration 2013; 332: [15] Lei, Y., D. Han, J. Lin and Z. He. Planetary Gearbox Fault Diagnosis using an Adaptive Stochastic Resonance Method. Mechanical Systems and Signal Processing 2013; 38: [16] Fault Data Sets: last accessed 2014 Xinghui Zhang is a Ph.D. student of Mechanical Engineering College, Shijiazhuang, China. His research is focused on fault detection, diagnostics and prognostics, and performance-based contracting.

10 70 Xinghui Zhang, Lei Xiao, and Jianshe Kang Lei Xiao is a Ph.D. student of Chongqing University, Chongqing, China. Her main research direction is the remaining useful life prediction based on data-driven models. Jianshe Kang received the Ph.D. degree in mechatronical engineering from Beijing Institute Technology, Beijing, China. He is a professor in Mechanical Engineering College. His research interests include reliability analysis and optimization, and condition monitoring.