Shock and Vibration

Volume 2019, Article ID 8213056, 20 pages

https://doi.org/10.1155/2019/8213056

## Fault Diagnosis of Variable Load Bearing Based on Quantum Chaotic Fruit Fly VMD and Variational RVM

^{1}Engineering Research Center for Metallurgical Automation and Measurement Technology of Ministry of Education, Wuhan University of Science and Technology, Wuhan 430081, China^{2}School of Electronic Information, Huang Gang Normal University, Huanggang 438000, China^{3}School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

Correspondence should be addressed to Huipeng Li; moc.361@piuhsey

Received 13 June 2018; Revised 11 November 2018; Accepted 29 November 2018; Published 3 January 2019

Academic Editor: Franck Poisson

Copyright © 2019 Bo Xu et al. 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

Under normal circumstances, bearings generally run under variable loading conditions. Under such conditions, the vibration signals of the bearing malfunctions are often nonstationary signals, which are difficult to process effectively. In order to accurately and effectively diagnose the failure types and damage degree of bearings under variable load conditions, an intelligent diagnostic model based on the variational mode decomposition (VMD) of quantum chaotic fruit fly optimization algorithm (QCFOA) and a multiclassification variational relevance vector machine (VRVM) is proposed. First, the key parameters of the VMD are selected using the QCFOA. Secondly, the known bearing fault signal is decomposed by the optimized VMD, and the center frequency and marginal spectral entropy (MSE) of each natural modal component are extracted to construct two-dimensional MSE. Then, the probit model is used to replace the logistic model, and a simpler and more practical multiclassification model is constructed. The two-dimensional MSE of each intrinsic modal component is used as a learning sample for VRVM. Finally, the bearing fault data under 1 hp load are taken as training samples, and the bearing fault data under two loads of 0 hp and 3 hp are used as test samples to verify the effectiveness of the intelligent diagnosis model. The experimental results show that the intelligent fault diagnosis method proposed in this paper can accurately diagnose the type of fault and the degree of damage and has high robustness.

#### 1. Introduction

Rolling bearings, which are the core components of rotating machinery, operate under harsh conditions and are easily damaged. It is necessary to monitor the fault. Once a fault occurs, it may cause huge economic losses and personal safety problems. In recent years, with the continuous update of knowledge in the field of digital signal processing and machine learning, intelligent fault diagnosis technology has become a major development trend [1]. Intelligent diagnosis is essentially a pattern recognition process, including two important steps of fault feature extraction and fault identification. In particular, how to effectively extract weak fault characteristics in the fault signal is key to fault diagnosis.

The vibration signal analysis is widely used in the fault diagnosis of bearing. In general, the vibration signals of the rolling bearing fault are mostly nonstationary signals, and the method suitable for processing nonstationary signals should be adopted. Empirical mode decomposition (EMD) [2] as a powerful tool of nonstationary signal processing has received extensive attention from researchers concerned with mechanical fault diagnosis. The method of bearing fault feature extraction based on EMD has been widely applied [3–5]. Inspired by the EMD method, Smith et al in [6] proposed another adaptive signal decomposition method named the “local mean decomposition (LMD)” in 2005, which has aroused the attention of a large number of scholars [7]. The EMD and LMD are excellent self-adaptive processing methods for nonstationary and nonlinear signals. However, they have unavoidable deficiencies, such as the end effect, overshoot, and mode mixing. Recently, many scholars proposed optimization and improvement methods for such deficiencies. However, both of them were regarded as recursive mode decomposition methods, and inherent defects are difficult to solve fundamentally. Thus, a new adaptive signal processing method named the “variation mode decomposition (VMD)” was proposed by Dragomiretskiy and Zosso [8]. The method abandons the recursive decomposition mode and effectively alleviates or avoids a series of deficiencies in the EMD and LMD methods.

The excellent features presented by VMD have been used by scholars in the field of fault diagnosis [9]. However, an important feature of VMD is that the number of intrinsic modal components and their penalty parameters must be set in advance; once unsuitable parameter values are selected, the decomposition results will be seriously affected. Therefore, against the selection of key parameter values of VMD, some scholars introduced the particle swarm optimization algorithm to select the key parameters of VMD optimally and achieved certain results [10]. However, the traditional heuristic optimization algorithms, such as particle swarm optimization (PSO) [11], genetic algorithm (GA) [12], ant colony optimization (ACO) [13], and cuckoo algorithm (CA) [14], have some problems such as parameter dependence, computational complexity, convergence speed, and optimization accuracy, which restrict their practical applications. To solve this problem, this paper introduces the fruit fly optimization algorithm (FOA) [15] to optimize the key parameters of VMD. This algorithm is a new swarm intelligence algorithm based on the bionics principle of fruit fly foraging behavior. It is applied in many fields [16–18]. However, its convergence accuracy is very sensitive to the initial value. Once the initial value is not selected properly, the search is likely to fall into a local optimum, and the convergence accuracy is low. This paper combines the quantum logistic chaotic mapping [19] and FOA and propose a quantum chaotic fruit fly algorithm in the three-dimensional space search, which strengthens the ergodicity, avoids the search process falling into the local optimal value, and improves the search efficiency. Then, using the local minimum value of the MSE as the fitness function, the quantum chaotic fruit fly optimization algorithm (QCFOA) is used to search two key parameters of VMD and obtain the optimal combination value of the key parameters. Furthermore, the optimized VMD is used to process the known fault signals, and the effective intrinsic mode function (IMF) component and its MSE are obtained.

The traditional diagnosis method usually trains the fault diagnosis model under a certain load and has great limitations in diagnosing the state of equipment under a certain load. In practical applications, many mechanical devices work under variable load conditions. Both the load and the damage degree will affect the amplitude of the fault characteristic frequency of the bearing. Therefore, the single MSE cannot effectively characterize the damage degree of the fault in the variable load condition (the radial load is mainly discussed in this paper). Because the bearing is running under different load conditions, the normal contact load between the rolling body and the raceway will change, which leads to the change in the natural vibration frequency of the bearing. To address these challenges, the center frequency is introduced in this paper, and the one-dimensional MSE is extended into two-dimensional MSE as the learning sample of the variational correlation vector machine. Then, the method is used to identify the various types of faults and the damage degree of the bearing under variable load.

The technology of intelligent diagnosis for mechanical faults is changing fast. Artificial neural network (ANN) and support vector machine (SVM), as intelligent recognizers, have received the most extensive attention, and a multitude of recent research efforts have been made to explore the mechanical fault diagnosis. However, the ANN algorithm requires a large number of training samples and has some inherent problems, such as black box operation, low generalization ability, and overlearning [20–24]. Similarly, the SVM also has some unavoidable deficiencies. The method cannot get the probabilistic prediction and the uncertainty in prediction [25–29].

RVM is a new machine learning algorithm based on support vector machine (SVM) and Bayesian theory framework [30]. Compared with the SVM method, it can directly give the uncertainty of the result while giving the diagnosis results. The RVM training process needs fewer parameters, and its solution is more sparse [31]. So, the probability output of RVM accords with the actual mechanical fault diagnosis process and has high applied research value. However, when the standard RVM is very limited in the size of the data sample, the computational cost of the training is very high. For this problem, Bishop [32] proposed a method of computing and solving RVM by means of the variational method, named the “VRVM.” In the case of very limited data samples, this method is better than the type II maximum edge likelihood estimation and can give the posterior distribution of parameters and hyperparameters. Compared with the standard RVM, the practicability and performance of the VRVM are better. In addition, VRVM is the same as the standard RVM, and its classification and regression are all mapped by logistic function. Therefore, when the regression problem is converted into a classification problem, the noise variable must be ignored, and the true value of the model cannot be accurately estimated. Therefore, the probit model is used instead of the logistic model in this paper, which makes the classification problem and the regression problem organically combined to avoid the approximate derivation of the logistic model from the continuous output to the discrete output mapping and to reduce the amount of computation.

Finally, the proposed method is used to diagnose the variable load fault data collected from the failure platform to verify the effectiveness and robustness of the method.

#### 2. The Principle of VMD

In the VMD algorithm, the IMF is redefined as an AM-FM signal, which removes the loop iteration method used by the EMD algorithm. Instead, the signal decomposition process is transferred to the variational structure. By constructing and solving the constrained variational problem and decomposing the original signal into a specified number of IMF components, the construction process of the corresponding variational problem is summarized as follows.

For each IMF component , the following analytic signal is obtained through the Hilbert transform:

For each analytic signal, a central frequency is estimated, and the frequency of each analytical signal is transformed to the baseband by shifting frequency:

The Gaussian smoothing index of the frequency shift signal is used to estimate the bandwidth of each IMF component, and then the corresponding constraint variational model is expressed aswhere represents the decomposition of the IMF component, is the number, represents the frequency center of each component, is the input signal, and is the estimated center frequency. In order to obtain the abovementioned constraint variation problem, the augmented Lagrange function is introduced as follows:where is the quadratic penalty parameter and is the Lagrange multiplier. The saddle point of the augmented Lagrange function is obtained by using the alternating direction multiplier algorithm, which is the optimal solution of equation (2) constraint variational model, and the original signal is decomposed into narrowband IMF components. The solution procedure for the variational model is as follows (Algorithm 1).