Shock and Vibration

Volume 2019, Article ID 4954920, 7 pages

https://doi.org/10.1155/2019/4954920

## The Hybrid Method of VMD-PSR-SVD and Improved Binary PSO-KNN for Fault Diagnosis of Bearing

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

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

Received 5 September 2018; Revised 1 November 2018; Accepted 15 November 2018; Published 2 January 2019

Academic Editor: Adam Glowacz

Copyright © 2019 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

Fault diagnosis of bearing based on variational mode decomposition (VMD)-phase space reconstruction (PSR)-singular value decomposition (SVD) and improved binary particle swarm optimization (IBPSO)-K-nearest neighbor (KNN) which is abbreviated as VPS-IBPSOKNN is presented in this study, among which VMD-PSR-SVD (VPS) is presented to obtain the features of the bearing vibration signal (BVS), and IBPSO is presented to select the parameter *K* of KNN. In IBPSO, the calculation of the next position of each particle is improved to fit the evolution of the particles. The traditional KNN with different parameter *K* and trained by the training samples with the features based on VMD-SVD (VS-KNN) can be used to compare with the proposed VPS-IBPSOKNN method. The experimental result demonstrates that fault diagnosis ability of bearing of VPS-IBPSOKNN is better than that of VS-KNN, and it can be concluded that fault diagnosis of bearing based on VPS-IBPSOKNN is effective.

#### 1. Introduction

Defects of bearing can lead to serious damage for the entire mechanical system [1–4], so it is very important to study the reliable fault diagnosis method to prevent the bearing from malfunction [5–7]. The features of the bearing vibration signal (BVS) are key to the fault diagnosis results of bearing. Thus, in this study, variational mode decomposition (VMD)-phase space reconstruction (PSR)-singular value decomposition (SVD) which is abbreviated as VPS is presented to obtain the features of the BVS. VMD [8–10] can decompose the signal into a set of band-limited intrinsic mode functions (BLIMFs) with certain sparsity properties. In this study, the BVS can be decomposed into several BLIMFs by VMD. By PSR for BLIMFs of the BVS, the dynamic characteristics of BLIMFs of the BVS can be reflected.

K-nearest neighbor (KNN) classifier is a simple and reliable classification method [11]. KNN classifier is a multiclassification method, which can recognize the several states of bearing simultaneously. However, the selection of the parameter *K* of KNN has a certain influence on the classification performance of KNN. The improved binary particle swarm optimization (IBPSO) is presented to select the parameter *K* of KNN. In IBPSO, the calculation of the next position of each particle is improved to fit the evolution of the particles.

In this study, the hybrid method of VMD-PSR-SVD and IBPSO-KNN (VPS-IBPSOKNN) is presented for fault diagnosis of bearing. The traditional KNN with different parameter *K* and trained by the training samples with the features based on VMD-SVD can be used to compare with the proposed VPS-IBPSOKNN method. The experimental result demonstrates that fault diagnosis ability of bearing of VPS-IBPSOKNN is better than that of VS-KNN.

#### 2. K-Nearest Neighbor Classifier

In the KNN classifier, for a new sample to be classified, its distance to each sample in the sampling set must be computed, and the new sample is classified to the class that contains the most samples from this set of closest *K* instances [12, 13].

The Euclidean distance approach is employed in the KNN model in this study, and the Euclidean distance between two samples and is described as follows:

KNN classifier is a multiclassification method, which can recognize the several states of bearing simultaneously.

#### 3. Variational Mode Decomposition

The signal can be decomposed into a set of band-limited intrinsic mode functions (BLIMFs) with certain sparsity properties by VMD [8,14–16], and VMD can be used to decompose the signal into a set of BLIMFs around the center frequencies according to the following constrained optimization formula:where denotes the time series signal, denotes the decomposed BLIMF, denotes the center frequency of BLIMF, denotes the number of BLIMFs, denotes the Dirac distribution, and denotes the convolution operator.

The minimization problem in equation (2) is transformed into the unconstrained optimization problem:where denotes the data fidelity constraint factor, and denotes the Lagrangian multiplier.

#### 4. Feature Extraction of BVS Based on VPS

In this study, the BVS is decomposed into four BLIMFs by VMD. Assuming the data set of the BLIMF is described as and defining as embedding space dimension and as time delay, the PSR signal of the BLIMF is given as follows:

SVD for matrix which is the PSR signal of the BLIMF can be performed, and define as the singular values of the matrix .

The singular values of the PSR signals of the four BLIMFs of the BVS constitute a vector as . By calculating the relative values of the elements in the vector as follows: , the features of the BVS based on VPS are described as . When is less than or equal to , the features of the BVS based on VPS can be described as .

#### 5. Parameter Optimization of KNN Based on IBPSO

In binary particle swarm optimization (BPSO), solutions are encoded as binary vectors, the position of the *i*th particle is defined by the following vector:where denotes the position of particle in the dimension and denotes the number of the particles.

In traditional BPSO [17, 18], the next position of each particle is calculated according to the following formula:where denotes the iteration counter, denotes the particle’s velocity at the iteration in the dimension, denotes the *i*th particle’s position at the iteration in the dimension, and denotes the random value in the range of 0∼1.

In this study, the parameter *K* of the KNN model is selected by IBPSO, and is employed as transform function to fit the evolution of the particles instead of in IBPSO.

Thus, in IBPSO, the next position of each particle is calculated as follows:where denotes the complement of , denotes the random value in the range of 0∼1, and the maximum value of is set to 6 here.

Figure 1 shows the process of the selection of the parameter *K* of KNN by IBPSO, which can be described in detail as follows: