International Journal of Rotating Machinery

Volume 2017 (2017), Article ID 3595871, 12 pages

https://doi.org/10.1155/2017/3595871

## Rolling Bearing Fault Signal Extraction Based on Stochastic Resonance-Based Denoising and VMD

^{1}School of Mechanical Engineering, Shenyang University of Technology, Shenyang 110870, China^{2}Liaoning Engineering Center for Vibration and Noise Control, Shenyang 110870, China

Correspondence should be addressed to Changzheng Chen

Received 6 April 2017; Revised 10 August 2017; Accepted 27 August 2017; Published 1 November 2017

Academic Editor: Hyeong Joon Ahn

Copyright © 2017 Xiaojiao Gu and Changzheng Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Aiming at the difficulty of early fault vibration signal extraction of rolling bearing, a method of fault weak signal extraction based on variational mode decomposition (VMD) and quantum particle swarm optimization adaptive stochastic resonance (QPSO-SR) for denoising is proposed. Firstly, stochastic resonance parameters are optimized adaptively by using quantum particle swarm optimization algorithm according to the characteristics of the original fault vibration signal. The best stochastic resonance system parameters are output when the signal to noise ratio reaches the maximum value. Secondly, the original signal is processed by optimal stochastic resonance system for denoising. The influence of the noise interference and the impact component on the results is weakened. The amplitude of the fault signal is enhanced. Then the VMD method is used to decompose the denoised signal to realize the extraction of fault weak signals. The proposed method was applied in simulated fault signals and actual fault signals. The results show that the proposed method can reduce the effect of noise and improve the computational accuracy of VMD in noise background. It makes VMD more effective in the field of fault diagnosis. The proposed method is helpful to realize the accurate diagnosis of rolling bearing early fault.

#### 1. Introduction

Rolling bearing is one of the most critical and easily damaged components in rotating machinery. Its running state is directly related to the performance of the whole mechanical system [1–3]. Therefore, it is significant to implement fault diagnosis for rolling bearings so as to prevent fatal malfunction of rotating machinery [4]. Vibration analysis is one of the most frequently used methods for health monitoring and fault diagnosis of rotating machinery. The vibration signal of mechanical equipment is rich in status information [5, 6]. When a rotating machine has incipient faults or works in a wicked environment, the useful fault signals are often submerged in strong background noise. It will seriously affect the accuracy of fault diagnosis. Therefore, the extraction of weak fault signal is a research hotspot in the field of signal processing [7].

In order to extract useful signals from complex original signals, many scholars have proposed a lot of effective weak signal extraction methods. One category is to extract weak signal from the perspective of denoising, such as wavelet transform, chaos theory, empirical mode decomposition (EMD), local mean decomposition (LMD), ensemble empirical mode decomposition (EEMD), and singular value decomposition (SVD). Under the influence of sampling frequency, the EEMD decomposition error is large. Meng and Xiang [8] proposed an improved EEMD which extracts the real intrinsic mode functions (IMF) by using the correlation coefficients between the original signal and IMFs. With this method, the pseudo low-frequency IMFs can be eliminated. Yi et al. [9] proposed the augmented quaternion singular spectrum analysis multichannel denoising method. This method has a better ability than multivariate EMD method in multisignal processing. Xie et al. [10] proposed an improved LMD based on extension characteristic wave method to eliminate the end effect. Yi et al. [11] proposed a convex optimization algorithm using nonconvex penalty functions based on SVD for extracting weak fault characteristics. Variational mode decomposition (VMD) is a new adaptive signal processing method proposed by Dragomiretskiy and Zosso in 2014 [12]. In this method, the frequency center and bandwidth of each component are determined by iteration searching for the optimal solution of the variational mode during the process of obtaining decomposition components. Therefore, the effective separation of the frequency domain can be realized adaptively. The existing study shows that in complicated signal decomposition the VMD has more advantage than EMD [13]. Compared with the recursive filtering mode of EEMD and LMD, VMD is a nonrecursive and variational mode decomposition. It has a solid theoretical basis. Its essence is a number of adaptive Wiener filter groups, which show better noise robustness. Through the reasonable control for the convergence condition, the influence of sampling on VMD is much smaller than EEMD and LMD. In terms of mode separation, the VMD can successfully separate two pure harmonic signals with similar frequencies. At present, the research results of the variational mode decomposition in the field of mechanical fault diagnosis show that VMD has better performance than LMD and EEMD in many aspects [14]. The fault feature extraction based on variational mode decomposition has important application value.

The other category is to use the noise to enhance the fault signal by means of stochastic resonance (SR). Because of its obvious advantages in weak signal detection, SR has become a hotspot in the field of signal processing in recent years [15, 16]. Han et al. [17] combined wavelet transform with stochastic resonance theory, which can detect several high frequency weak signals in strong noise background by adjusting the amplitude of wavelet multiscale decomposition. Rother et al. [18] combined stochastic resonance with EMD, which is applied to detect deviations in strip travel of hot strip mill. The results show that this method can reduce downtimes and extend the lifetime of components. Barbini et al. [19] focused on the use of SR in a two-dimensional system of gradient type for detection of weak signals submerged in Gaussian noise. The two-dimensional system is more accurate and more reliable than the traditional one-dimensional system. Castiglione et al. [20] used bistable stochastic resonance and Woods-Saxon stochastic resonance in bearing faults detection, respectively. Experimental data shows that each SR method has its own characteristic features. Gao et al. [21] constructed corresponding Duffing oscillator based on chaos theory to monitor weak signal, which reduces the SNR threshold compared with the traditional monitoring method. Lu et al. [22] enhanced stochastic resonance by full-wave signal construction, which surpass the traditional stochastic resonance in the field of fault signal processing and can be used in areas related to weak signal detection. Lin et al. [23] presented a new additive noise driving and aperiodic chirp signal stochastic dynamical system, which can improve the system performance of localization and moving target detection. Breen et al. [24] used the mechanical stochastic resonance to detect weak signals, which harvest weak periodic signals via the movement of an inverted pendulum. Shi et al. [25] combined analytical mode decomposition (AMD) and EEMD with adaptive stochastic resonance, which can improve the decomposition results of EEMD. This method has a good effect in the rolling bearing fault diagnosis. Xiang and Zhong [26] presented a two-step method of Duffing oscillator and stochastic resonance. And this method combined stochastic resonance with the artificial bee colony algorithm. It is favorable to detect weak signal from a strong noise environment. Liu et al. [27] combined wavelet reconstruction and stochastic resonance. The reconstruction scale was selected by comparing the wavelet entropy of each decomposition scale. This fault detection method is more accurate than the conventional wavelet reconstruction. The above studies which combine the stochastic resonance with the signal processing method or the optimization method show that the appropriate application of the stochastic resonance method can make the signal processing method have a better effect in fault diagnosis. Stochastic resonance has important application value in fault diagnosis under strong noise background.

Combining the respective advantages of VMD and SR, this study presents a weak signal extraction method of rolling bearing fault based on VMD and quantum particle swarm optimization (QPSO) adaptive stochastic resonance. First, the parameters of stochastic resonance system are optimized according to the original signal feature and quantum particle swarm optimization algorithm. Thereafter, the original signal is processed by the stochastic resonance so that the noise frequency component is weakened and the fault frequency component is enhanced. Lastly, the signal which is processed by stochastic resonance is decomposed by VMD method. The useful signal features are extracted. The rolling bearing measured signal analysis shows that the proposed method can discharge most of the interfering signal generated by the background noise so that the components obtained by the VMD can better reflect the fault signal information and enhance the useful signal amplitude. The detection results are more accurate and reliable.

The rest of this paper is organized as follows. Section 2 provides a brief description of the principal theory of VMD, SR, and QPSO. Section 3 introduces the proposed fault diagnosis process and discusses the simulation study to validate the performance of the proposed method. Section 4 discusses the case study to validate the practical application value of this method. Lastly, Section 5 presents the conclusion.

#### 2. Basic Theory

##### 2.1. Variational Mode Decomposition

VMD algorithm is a new nonrecursive variational mode signal decomposition method, which can decompose the complex input signal into a set of discrete mode components [28]. The implementation steps of the VMD algorithm are as follows.

*Step 1. *The Hilbert transform is performed for each mode function to obtain its analytical signal, as follows:

*Step 2. *The estimated center frequency of is mixed. The spectrum of each mode is modulated to the corresponding base band, as follows:

*Step 3. *Calculate the gradient square norm of the above demodulated signal and estimate the bandwidth of each mode signal. The variational problem is constructed as follows: where represents variational mode components. represents the center frequency of each component. is the number of variational mode components.* t* represents the time. represents the partial derivative of . represents the impulse function. is the square root of −1. represents the circular frequency. represents the input signal.

*Step 4. *In order to transform the constraint variational problem into the unconstrained problem, the quadratic penalty factor is introduced to ensure the accuracy of signal reconstruction in noise condition. The Lagrangian multiplier is introduced to ensure the stringency of the constraint condition. Through introducing the two parameters, the extended Lagrangian expression is obtained as follows:

*Step 5. *Solve the extended Lagrangian expression by using the alternate direction method of multipliers (ADMM). Specific steps are as follows:

() Initialize , , , .

() Repeat .

() Update , , and , and do

() Judge whether or not the following conditions are met. If met, the iteration is stopped; otherwise, return to execute Step .where is the convergence stopped condition. is the iteration number.

##### 2.2. Stochastic Resonance

Benzi et al. [29] first introduced the mechanism of stochastic resonance in 1981 and used SR in climatic change. In the field of signal processing, stochastic resonance can transfer the energy of some noise signals to useful signals, which is beneficial to the identification of weak signals submerged in noise [30–32]. The noise signal is input to the bistable or multistable nonlinear system. The system performance can be optimized by adjusting the parameters so that the effect of stochastic resonance achieves the best. The equation of the bistable Langevin system is as follows:where is the potential function.where and are system parameters. The potential function (8) has one unstable equilibrium point and two stable equilibrium points. The height of the potential barrier is . represents an input signal with amplitude and frequency . represents the Gaussian white noise signal with intensity , mean 0, and variance 1. The bistable Langevin system structure is shown in Figure 1.