Shock and Vibration

Volume 2018, Article ID 8385021, 8 pages

https://doi.org/10.1155/2018/8385021

## Fault Diagnosis of Bearing by Utilizing LWT-SPSR-SVD-Based RVM with Binary Gravitational Search Algorithm

College of Mechanical Engineering, Donghua University, Shanghai 201620, China

Correspondence should be addressed to Sheng-wei Fei; nc.ude.uhd@wsf

Received 20 April 2018; Accepted 3 July 2018; Published 2 September 2018

Academic Editor: Adam Glowacz

Copyright © 2018 Sheng-wei Fei. 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

The fault diagnosis method of bearing based on lifting wavelet transform (LWT)-self-adaptive phase space reconstruction (SPSR)-singular value decomposition (SVD)-based relevance vector machine (RVM) with binary gravitational search algorithm (BGSA) is presented in this study, among which LWT-SPSR-SVD (LSS) is presented for feature extraction of the bearing vibration signal, the dynamic characteristics of lifting wavelet coefficients' (LWCs') reconstructed signals of the bearing vibration signal can be reflected by SPSR for LWCs' reconstructed signals of the bearing vibration signal, and BGSA is used to select the embedding space dimension and time delay of phase space reconstruction (PSR) and kernel parameter of RVM. In order to show the superiority of LWT-SPSR-SVD-based RVM with BGSA (LSS-BGSA-RVM), the traditional RVM trained by the training samples with the features based on LWT-SVD (LS-RVM) is used to compare with the proposed LSS-BGSA-RVM method. The experimental result demonstrates that compared with LS-RVM, LSS-BGSA-RVM can achieve the higher diagnosis accuracy for bearing.

#### 1. Introduction

Bearing is the important component of mechanical equipment, and the reliable fault diagnosis method of bearing is key to ensuring its safe operation, which is helpful to safe operation of mechanical equipment [1–9]. Support vector machine (SVM) classifier [10–12] has a good ability to solve the classification problems, which has been applied in fault diagnosis of bearing [13]. Relevance vector machine (RVM) based on sparse Bayesian framework has a sparser representation than SVM, which has a better application prospect in fault diagnosis of bearing. However, the selection of the kernel parameter of RVM has a certain influence on its classification performance. Furthermore, the dynamic characteristics of the decomposed signals of the bearing vibration signal should be considered, which can be helpful to obtain the excellent features.

Therefore, in this study, lifting wavelet transform (LWT)-self-adaptive phase space reconstruction (SPSR)-singular value decomposition (SVD)-based RVM with binary gravitational search algorithm (BGSA) is presented and applied for fault diagnosis of bearing, among which LWT-SPSR-SVD (LSS) is presented for feature extraction of the bearing vibration signal. By SPSR for lifting wavelet coefficients’ (LWCs’) reconstructed signals of the bearing vibration signal, the dynamic characteristics of LWCs’ reconstructed signals of the bearing vibration signal can be reflected. SPSR for LWCs’ reconstructed signals of the bearing vibration signal can be helpful to obtain the excellent features.

The different embedding space dimension and time delay of phase space reconstruction (PSR) can obtain different PSR signals, which has an influence on the performance of the diagnosis model. Furthermore, the selection of the kernel parameter of RVM has a certain influence on its classification performance. Gravitational search algorithm (GSA) is an intelligent optimization algorithm based on the law of gravity [14–19], and BGSA can be used to solve the optimization problems in the binary space. In BGSA, the heavy masses corresponding to good solutions move more slowly than lighter ones, which can guarantee the algorithm’s exploitation step. Thus, BGSA is employed to select the embedding space dimension and time delay of PSR and kernel parameter of RVM. In order to show the superiority of LWT-SPSR-SVD-based RVM with BGSA (LSS-BGSA-RVM), the traditional RVM trained by the training samples with the features based on LWT-SVD (LS-RVM) is used to compare with the proposed LSS-BGSA-RVM method.

#### 2. Feature Extraction Method of Bearing Vibration Signal Based on LSS

##### 2.1. SPSR for LWCs’ Reconstructed Signals of Bearing Vibration Signal

In this study, the bearing vibration signal is decomposed into four LWCs’ reconstructed signals with different frequency ranges by performing the three-level decomposition for the bearing vibration signal based on LWT. The different embedding space dimension and time delay of PSR can obtain different PSR signals, which has an influence on the performance of the diagnosis model, so SPSR is used instead of PSR here. By SPSR for LWCs’ reconstructed signals of the bearing vibration signal, the dynamic characteristics of LWCs’ reconstructed signals of the bearing vibration signal can be reflected. Thus, SPSR for lifting LWCs’ reconstructed signals of the bearing vibration signal can be helpful to obtain the excellent features.

Assume that the data set of the LWC’s reconstructed signal is described as and define as embedding space dimension and as time delay, the SPSR signal of the LWC’s reconstructed signal is given as follows:

##### 2.2. SVD for SPSR Signals

SVD [20, 21] for matrix which is the SPSR signal of the LWC’s reconstructed signal can be performed as follows:where is a matrix with ; is an orthogonal matrix with ; is an orthogonal matrix with ; and , or its transposition, where is the zero matrix, , and are the singular values of the matrix , , .

##### 2.3. Obtaining the Features of Bearing Vibration Signal Based on LSS

The singular values of the SPSR signals of the four LWCs’ reconstructed signals of the bearing vibration signal constitute a vector as . Calculate the relative values of the elements in the vector as follows:

Thus, the features of the bearing vibration signal based on LSS are described as .

When is less than or equal to , the features of the bearing vibration signal based on LSS can be described as .

#### 3. RVM Classifier

Given a set of training samples , the likelihood function obeys the Bernoulli distribution [22]:where , denotes the input vector; , denotes the corresponding output target, , “0” and “1” denote two classes which the training samples belong to; and denotes the number of training samples; is a predefined logistic sigmoid function, ; and is the weight vector.

Here, radial basis function (RBF) kernel can be described as the following equation, which can be used to construct the RVM,where denotes the RBF kernel parameter.

#### 4. Optimizing the Embedding Space Dimension and Time Delay of PSR and Kernel Parameter of RVM Based on BGSA

In this study, BGSA is used to select (embedding space dimension) and (time delay) of PSR and (kernel parameter) of RVM. In BGSA, solutions are encoded as binary vectors; agents can be considered as objects, and their performance can be measured by their masses [23]. Figure 1 shows the process of the selection of (embedding space dimension) and (time delay) of PSR and (kernel parameter) of RVM by BGSA, which can be described in detail as follows:

*Step 1*. Encode (embedding space dimension) and (time delay) of PSR and (kernel parameter) of RVM, and randomly initialize the positions of agents in the search space. The position of the agent is defined by the following vector:where denotes the position of the agent in the dimension and denotes the number of the agents.