Shock and Vibration
Volume 2018 (2018), Article ID 8385021, 8 pages
https://doi.org/10.1155/2018/8385021

Research Article

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

Sheng-wei Fei

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

Correspondence should be addressed to Sheng-wei Fei; fsw@dhu.edu.cn

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.

Figure 1: The process of the selection of (embedding space dimension) and (time delay) of PSR and (kernel parameter) of RVM by BGSA.

As shown in Table 1, the first part of binary code string represents the embedding space dimension of PSR and denotes the length of binary code string representing the embedding space dimension of PSR. The second part of binary code string represents the time delay of PSR, and denotes the length of binary code string representing the time delay of PSR. The third part of binary code string represents the kernel parameter of RVM, and denotes the length of the binary code string representing the kernel parameter of RVM.

Table 1: Encoding (embedding space dimension) and (time delay) of PSR and (kernel parameter) of RVM.

...

...

...

Step 2. Check the position of each agent, and ensure the position of each agent in the search space. Check the binary codes representing the embedding space dimension of PSR, the binary codes representing the time delay of PSR, and the binary codes representing the kernel parameter of RVM of each agent, and ensure that the decimal value of the binary codes representing the embedding space dimension of each agent is 2 at least, one of the binary codes representing the time delay of each agent is “1” at least, and one of the binary codes representing the kernel parameter of each agent is “1” at least.

Step 3. Evaluate the Fitness of Each Agent. The training samples are divided equally into subsets of the samples, among which subsets of the samples are employed to train the RVM model, and the remaining subset is used to test the RVM model. Each subset can be employed as the testing subset. Then, the total diagnosis accuracy of the subsets of the samples can be obtained as follows:

where denotes the total number of the subsets of the samples, denotes the total number of the subsets of the samples with correct diagnosis, and is set to 5 here.

Here, the fitness of the agent is defined as follows:

Step 4. The gravitational and inertial masses can be calculated as follows:

where ( denotes the total number of iterations), denotes the fitness value of the agent at iteration , and for a minimization problem, and are, respectively, defined as follows:

Step 5. The gravitational constant can be calculated as follows:

where and are the constants.

Step 6. Calculate the Acceleration in Gravitational Field. First, the force acting on mass “” from mass “” can be calculated as follows:

where denotes gravitational constant at iteration , denotes the passive gravitational mass related to the agent, denotes the active gravitational mass related to the agent, denotes a small constant, and denotes the Euclidian distance between two agents and :

is an iteration function, which is expressed as follows:

where the function is used to be rounded to the nearest integer number here and is the constant.

Then,

where is a random number in the range of 0∼1.

By the law of motion, which is the acceleration of the agent at iteration in the direction can be given as follows:

where is the inertial mass of the agent.

As , can be given as follows:

Step 7. Calculate the Next Velocities of the Agents. The next velocity of an agent can be calculated as follows:

where is a uniform random variable in the range of 0∼1, denotes the agent’s velocity at iteration in dimension, and the maximum value of is set to 6 here.

Step 8. Calculate the Next Positions of the Agents. The next position of an agent can be calculated as follows:

where denotes the agent’s position at iteration in dimension, denotes the complement of , and denotes a random value in the range of 0∼1.

Step 9. Repeat Step 2 to Step 8 until the stopping condition is reached.

Step 10. Decode the best solution, and the optimized (embedding space dimension) and (time delay) of PSR and (kernel parameter) of RVM can be obtained.

5. Experimental Analysis

In the experiment, the bearing vibration data are obtained from “bearings vibration data set” of Case Western Reserve University, among which the fault data are collected under the condition of single point faults with a fault diameter of 0.014 inches [24]. Three groups of samples are derived from bearing vibration signals acquired under three different loads. In group 1, the samples are obtained based on the bearing vibration signal acquired under 1 HP motor load. In group 2, the samples are obtained based on the bearing vibration signal acquired under 2 HP motor load. In group 3, the samples are obtained based on the bearing vibration signal acquired under 3 HP motor load. Each group includes 200 samples, among which 50 samples represent normal state, 50 samples represent inner race fault, 50 samples represent outer race fault, and 50 samples represent ball fault.

The first 40 samples of each state in each group are employed as the training samples, and the remaining 10 samples of each state in each group are employed as the testing samples. Thus, the training samples include 480 samples, and the testing samples include 120 samples.

Four LWCs’ reconstructed signals of the bearing vibration signal are obtained by performing the three-level decomposition for the bearing vibration signal based on LWT. Figure 2 gives the four LWCs’ reconstructed signals of one of the samples representing the normal state in the training samples.

(a)

(b)

(c)

(d)

Figure 2: The four LWCs’ reconstructed signals of one of the samples representing the normal state in the training samples. (a) Reconstructed signal of lifting wavelet coefficient 1, (b) reconstructed signal of lifting wavelet coefficient 2, (c) reconstructed signal of lifting wavelet coefficient 3, and (d) reconstructed signal of lifting wavelet coefficient 4.

In the proposed LSS-BGSA-RVM method, the features of the bearing vibration signal are obtained by using the feature extraction method of the bearing vibration signal based on LSS, and BGSA is used to select (embedding space dimension) and (time delay) of PSR and (kernel parameter) of RVM. The value range of the embedding space dimension of PSR is [2, 2³−1], and the intervals of the adjacent values of the embedding space dimension are 1; thus, the length of binary code string representing the embedding space dimension is 3. The value range of the time delay of PSR is [1, 2³−1], and the intervals of the adjacent values of the time delay are 1; thus, the length of binary code string representing the time delay is 3. The value range of the kernel parameter is [1, 2¹⁰−1], and the intervals of the adjacent values of the kernel parameter are 1; thus, the length of binary code string representing the kernel parameter is 10. Obviously, in this case, , so the features of the bearing vibration signal based on LSS can be described as here. Then, the diagnosis model of bearing is established by three LSS-BGSA-RVMs with the form of binary tree, among which LSS-BGSA-RVM1 is employed to separate the normal state from the fault state, LSS-BGSA-RVM2 is employed to separate inner race fault from other faults (outer race fault and ball fault), and LSS-BGSA-RVM3 is employed to separate outer race fault from ball fault. The values of the respective embedding space dimensions, time delays, and kernel parameters of LSS-BGSA-RVM1, LSS-BGSA-RVM2, and LSS-BGSA-RVM3 can be obtained by BGSA.

In order to show the superiority of the proposed LSS-BGSA-RVM method, the LS-RVM method is used to compare with the proposed LSS-BGSA-RVM method. In the LS-RVM method, the features of the bearing vibration signal are obtained by using the feature extraction method of the bearing vibration signal based on LWT-SVD. Define as the singular values of the matrix which is composed of the four LWCs’ reconstructed signals of the bearing vibration signal,

, and the singular values of the matrix composed 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 LWT-SVD are described as . Obviously, in this case, , so the features of the bearing vibration signal based on LWT-SVD can be described as here. Moreover, in the LS-RVM method, the grid method is used to select the kernel parameter of RVM; here, the value range of the kernel parameter is [1, 2¹⁰−1], and the intervals of the adjacent values of the kernel parameter are 1. Then, the diagnosis model of bearing is established by three LS-RVMs with the form of binary tree, among which LS-RVM1 is employed to separate the normal state from the fault state, LS-RVM2 is employed to separate inner race fault from other faults (outer race fault and ball fault), and LS-RVM3 is employed to separate outer race fault from ball fault. The values of the respective kernel parameters of LS-RVM1, LS-RVM2, and LS-RVM3 can be obtained by the grid method.

The diagnosis accuracy (DA) described in the following equation is used to evaluate the performance of the diagnosis models,

where is the number of testing samples and is the number of testing samples with correct diagnosis in the case.

The features of two samples representing outer race fault and ball fault, respectively, in a set of samples based on LWT-SVD can be shown in Figure 3, and for the same samples as above, their features based on LSS in LSS-BGSA-RVM3 can be shown in Figure 4. As in shown in Table 2, only one testing sample is incorrectly diagnosed by using LSS-BGSA-RVM, and 119 testing samples are correctly diagnosed by using LSS-BGSA-RVM; then, the diagnosis accuracy of bearing by using LSS-BGSA-RVM is 99.17%. However, 12 testing samples are incorrectly diagnosed by using LS-RVM, and 108 testing samples are correctly diagnosed by using LS-RVM; then, the diagnosis accuracy of bearing by using LS-RVM is 90%. The experimental result demonstrates that LSS-BGSA-RVM can achieve the higher diagnosis accuracy of bearing than LS-RVM, and the proposed LSS-BGSA-RVM method for fault diagnosis of bearing is feasible.

(a)

(b)

Figure 3: The features of two samples representing outer race fault and ball fault, respectively, in a set of samples based on LWT-SVD. (a) The features of the sample representing outer race fault, and (b) the features of the sample representing ball fault.

(a)

(b)

Figure 4: The features of two samples representing outer race fault and ball fault, respectively, in a set of samples based on LSS in LSS-BGSA-RVM3. (a) The features of the sample representing outer race fault, and (b) the features of the sample representing ball fault.

Table 2: The comparison of the diagnosis accuracy of bearing between LSS-BGSA-RVM and LS-RVM.

The number of the testing samples
Classifier
The number of the testing samples with correct diagnosis
Diagnosis accuracy (%)

120
LSS-BGSA-RVM
119
99.17

LS-RVM
108
90

6. Conclusion

In this study, the fault diagnosis method of bearing based on LSS-based RVM with BGSA is presented, among which LSS is presented for feature extraction of the bearing vibration signal, and BGSA is used to select the embedding space dimension and time delay of PSR and kernel parameter of RVM. 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. 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. SPSR for LWCs’ reconstructed signals of the bearing vibration signal can be helpful to obtain the excellent features. The experimental result demonstrates that compared with LS-RVM, LSS-BGSA-RVM can achieve the higher diagnosis accuracy for bearing.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The author confirms that there are no conflicts of interest.

Acknowledgments

This project is supported by the “Fundamental Research Funds for the Central Universities (no. 2232017D-14).”

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