# Research on the Fault Diagnosis Method for Rolling Bearings Based on Improved VMD and Automatic IMF Acquisition

Authors: Ying Zhang; Anchen Wang
Journal: Shock and Vibration
 (2020)

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## Abstract

This paper proposes a novel method to improve the variational mode decomposition (VMD) method and to automatically acquire the sensitive intrinsic mode function (IMF). First, since fault signals are impulsive and periodic, a weighted autocorrelative function maximum (AFM) indicator is constructed based on the Gini index and autocorrelation function to serve as the optimization objective function. The mode numberK and the penalty parameter α of VMD are automatically obtained through an optimal parameter searching process underpinned by the improved particle swarm optimization (PSO) algorithm with a variety of inertia weights. This improvement solves one of the major drawbacks of the conventional VMD method, that is, the need to manually set parameters. Then, an optimal IMF automatic selecting process is performed for single-failure faults and compound faults, according to the principles of the maximum weighted AFM indicator and maximum spectrum peak ratio (SPR), respectively. The sensitive IMFs are then subjected to an envelope demodulation analysis to obtain the fault characteristic frequency. The results of simulations and experiments show that the proposed method can effectively identify fault characteristics early, especially compound faults, demonstrating great potential for real-world applications.


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## Body

## 1. Introduction

Developing techniques for monitoring rolling bearing conditions and diagnosing faults are key to realizing the transition from traditional regular maintenance to condition-based maintenance. Owing to harsh working environments and changing conditions, vibrations of rolling bearings are typically characterized by strong background noise, unsteadiness, and coupling modulation, which make it difficult to extract and identify fault characteristics. Nonetheless, accurately extracting fault characteristic frequencies from a mix of complex, nonstationary, and nonlinear vibration signals is essential for early fault identification [1].Time-frequency analysis methods simultaneously provide local information about nonstationary signals in both the time domain and frequency domain and can be used to extract fault characteristics from weak signals. Some commonly used time-frequency analysis methods include empirical mode decomposition (EMD) [2], ensemble empirical mode decomposition (EEMD) [3], and local mean decomposition (LMD) [4]. All of these are based on recursive decomposition and, thus, have the same disadvantages, such as endpoint effects and modal aliasing, which can impact accuracy in early detection of weak fault characteristics [5].Dragomirestkiy and Zosso [6] recently proposed the VMD method, which is based on solid mathematical foundations and can be used for broad applications in the field of mechanical fault diagnosis. The VMD method introduces a secondary penalty factor and a Lagrange multiplier to transform the constrained variational problem into a nonconstrained variational problem. This nonrecursive decomposition approach avoids some of the problems of recursive decomposition, such as the endpoint effect and the need to determine conditions for terminating the decomposition, and, in addition, effectively remedies the deficiencies of conventional decomposition methods and achieves high computational efficiency and good noise robustness. However, the VMD method also has a number of drawbacks. Most notably, the mode number K and the penalty parameter α must be set before starting the decomposition. Moreover, problems like aliasing between mode components or excessive decomposition may occur if parameters are not set properly. Therefore, adaptive acquisition of VMD parameters is of crucial importance.To tackle this challenge, Shi and Yang [7] proposed a step-by-step parameter optimization strategy which achieved good results in validation experiments using a wind turbine. Wang et al. [8] used the minimum average envelope spectrum entropy as the objective function and PSO to optimize the VMD parameters. The method was shown to be superior to EMD, LMD, and other envelope and wavelet transform methods. Yan et al. [9] proposed a composite fault detection method for rotating machinery based on optimal variational mode decomposition (OVMD) and a 1.5-dimensional envelope spectrum. The envelope spectrum entropy value was used as the objective function, and the optimal VMD parameters were obtained using the genetic algorithm, thereby achieving adaptive acquisition of the mode number and penalty parameter. In another approach, Zhang et al. [10] proposed a grasshopper optimization algorithm- (GOA-) based adaptive VMD parameter analysis method that works in the following way: a weighted kurtosis indicator is constructed based on the kurtosis and correlation coefficient; VMD parameters are optimized using the GOA algorithm with the weighted kurtosis indicator as the objective function; and the sensitive IMF is selected following the principle of maximum weighted kurtosis for extracting the fault characteristics. Liu et al. [11] used the principle of maximum kurtosis to obtain the optimal values of parameter pair K and α in order to achieve an accurate VMD. Lian et al. [12] proposed an adaptive VMD method that uses the principle of maximum kurtosis as an evaluation criterion to automatically determine the mode number.Based on the above studies, it can be concluded that constructing a suitable objective function and selecting an appropriate optimization algorithm are two critical issues in the adaptive acquisition of VMD parameters. Some indicators such as kurtosis [10–14], correlation coefficient [15], autoregressive (AR) model parameters [16], and entropy [8] have proven extremely useful in vibration signal analysis and fault characteristics extraction. As a sparse indicator, kurtosis is widely used in the fault diagnosis of rotating machinery and can effectively extract impulsive components of signals. The principle of maximum kurtosis is useful for constructing the objective function for VMD parameter optimization. However, kurtosis is more suitable for measuring the most important pulses in a signal. In addition to not being robust enough, kurtosis-based techniques focus more on individual pulses than on the desired periodic pulses [17].The Gini index is widely used to measure income inequality in countries and has recently been used in fault diagnosis of rotating machinery [18]. The index is not only very effective in identifying fault pulses but offers superb performance in preventing random pulses and disturbances. Miao et al. [19] used the Gini index to improve the kurtosis guidance map and were able to identify the resonance band, which contains the richest information. More specifically, the resonance band contains the fault pulse rather than interferences caused by fault modulation and random pulses. Kurtosis and the Gini index are both used as indicators but share the same shortcoming; that is, both measures are aimed at detecting the impact of fault pulses and ignore recurring periodic pulses.The autocorrelation function retains periodic signal information; however, phase information of the original signal is not retained. For white noise signals, the autocorrelation function quickly decays to zero at the fastest decay rate. For an impulse signal, the autocorrelation function also acquires impulse characteristics while retaining periodic characteristics. The autocorrelation function of a noise-containing impulse signal decays rapidly, but its amplitude becomes large during the impulse. This suggests that the autocorrelation function of an impulse signal combines characteristics of the autocorrelation function of white noise and impulse signal. Chen et al. [20] proposed an AFM indicator based on this particular feature of the autocorrelation function, which was then used as a quantitative evaluation criterion of the IMF value after complementary EEMD processing and achieved good results.Up to now, most studies in the literature have focused on single-failure faults (e.g., inner ring failure, outer ring faults, and rolling element faults of rolling bearings). In practice, most faults are compound and composed of two or more different types of faults. At present, there are many research studies on the diagnosis of compound faults. Wang et al. [21] proposed a modified VMD (MVMD) based on noise-assisted data analysis (NADA), which can extract compound fault features of gearbox under a strong noise environment. Yang et al. [22] proposed a fault diagnosis framework based on a novel dual-extreme learning machine (ELM) network that can quickly detect single and simultaneous failures in a multistage gearbox. Zhong et al. [23] proposed a fault feature extraction framework based on feature extraction and support vector machine (SVM), and the experimental results show that the method can effectively diagnose single and simultaneous faults of the gas turbine generator system (GTGS). Wang et al. [24] proposed a gearbox integrated fault diagnosis method based on ensemble LMD, defined as CELMD (Complementary Ensemble Local Mean Decomposition) which overcomes the pattern mixing phenomenon and can be used to extract composite fault features. Wang [25] proposed a method based on Multipoint Optimal Minimum Entropy Deconvolution Adjusted (MOMEDA) and EEMD, which is suitable for searching complex fault pulse signals under strong noise environment. Wang et al. [26] proposed a maximum kurtosis spectrum deconvolution (MKSED) applied to the fault diagnosis of the gearbox, and the experimental results show that the compound faults of gearbox can be successfully diagnosed by using MKSED.In this paper, a method for improving the VMD is proposed; a new weighted AFM indicator is constructed to serve as the objective function for parameter optimization. The improved PSO with a variety of inertia weights is used for optimal parameter searching. Mutual information of the decomposed components and the original signal is used to detect overdecomposition, upon which the decision to perform the search again is made. The VMD is performed after the optimal decomposition parameters are obtained. For single-failure faults, the weighted AFM indicator is used to select the sensitive IMF and for compound faults, SPR is used. Finally, the selected IMF undergoes an envelope demodulation analysis to extract the fault characteristic frequency. The following sections are arranged as follows. The VMD method is introduced in Section2. The construction of weighted WAFM index, IPSO algorithm, mutual information, and the method proposed in this paper are introduced in Section 3. The simulation signal and test signal are analyzed using the proposed method, respectively, in Section 4 and Section 5. Finally, the conclusions of this paper are drawn in Section 6.


## 2. VMD

The VMD is an adaptive signal processing technique first proposed in 2014 that can be used to decompose a signal into a series of finite-bandwidth IMFs in a nonrecursive manner, thereby creating the right conditions for effective signal separation [6]. Migrating the signal decomposition process into the variational framework, VMD accomplishes signal decomposition by seeking the optimal solution of the variational model.


### 2.1. Construction of the Variational Problem

With VMD, the signal is decomposed intoK mode functions, each with a center frequency of ωk. Therefore, the VMD can be expressed as the variational problem of solving K mode functions whose aggregated bandwidth is the smallest, constrained by the sum of all mode functions which must be equal to the original signal:(1)minuk,ωk∑k∂tδt+jπtukte−jωkt22s.t.∑luk=f,where ωk=ω1,ω2,…,ωk represents the center frequency of each mode.The VMD can be obtained using the following procedure.The Hilbert transform is performed on each mode functionuk to obtain the analytical signal:(2)δt+jπtukt.Then, a preliminary estimation is performed on the center frequencye−jωkt of the analytical mixed signal of each mode. The frequency of each mode component is modulated by its corresponding baseband frequency:(3)δt+jπtukte−jωkt.The square of the gradient 2-norm of the above demodulated signal is calculated and the bandwidth of each modal function is estimated.


### 2.2. Solution to the Variational Problem

To solve the variational problem presented in Section2.1, the secondary penalty factor α and Lagrange multiplier λt are introduced and used to transform the constrained variational problem into an unconstrained variational problem. The secondary penalty factor serves as a guarantee of the accuracy of signal reconstruction when Gaussian noise is present, while the Lagrange multiplier makes the constraint more stringent. The augmented Lagrange expression is as follows:(4)Luk,ωk,λ=α∑k∂tδt+jπtukte−jωkt22+ft−∑kukt+λt,ft−∑kukt.Both the IMF and its center frequency are continuously updated using the alternate direction method of multipliers (ADMM). Finally, the saddle point of the above formula is obtained, and this is the optimal solution. The IMF of all frequency domains along with their center frequencies can be obtained using the following formulae:(5)u^kn+1w=f^w−∑i≠ku^iw+λ^w/21+2αω−ωk,(6)ω^kn+1=∫0∞ωu^w2dω∫0∞u^w2dω,where u^kn+1w is the result of the current residual amount f^w−∑u^iw after Wiener filtering.The above is an adaptive decomposition procedure of VMD, which is in essence a Wiener filtering process. Signal decomposition is realized by iteratively searching for the optimal solution of the variational model in the variational problem framework. The approach achieves good noise robustness and effectively avoids the endpoint effect and modal aliasing problem.


## 2.1. Construction of the Variational Problem

With VMD, the signal is decomposed intoK mode functions, each with a center frequency of ωk. Therefore, the VMD can be expressed as the variational problem of solving K mode functions whose aggregated bandwidth is the smallest, constrained by the sum of all mode functions which must be equal to the original signal:(1)minuk,ωk∑k∂tδt+jπtukte−jωkt22s.t.∑luk=f,where ωk=ω1,ω2,…,ωk represents the center frequency of each mode.The VMD can be obtained using the following procedure.The Hilbert transform is performed on each mode functionuk to obtain the analytical signal:(2)δt+jπtukt.Then, a preliminary estimation is performed on the center frequencye−jωkt of the analytical mixed signal of each mode. The frequency of each mode component is modulated by its corresponding baseband frequency:(3)δt+jπtukte−jωkt.The square of the gradient 2-norm of the above demodulated signal is calculated and the bandwidth of each modal function is estimated.


## 2.2. Solution to the Variational Problem

To solve the variational problem presented in Section2.1, the secondary penalty factor α and Lagrange multiplier λt are introduced and used to transform the constrained variational problem into an unconstrained variational problem. The secondary penalty factor serves as a guarantee of the accuracy of signal reconstruction when Gaussian noise is present, while the Lagrange multiplier makes the constraint more stringent. The augmented Lagrange expression is as follows:(4)Luk,ωk,λ=α∑k∂tδt+jπtukte−jωkt22+ft−∑kukt+λt,ft−∑kukt.Both the IMF and its center frequency are continuously updated using the alternate direction method of multipliers (ADMM). Finally, the saddle point of the above formula is obtained, and this is the optimal solution. The IMF of all frequency domains along with their center frequencies can be obtained using the following formulae:(5)u^kn+1w=f^w−∑i≠ku^iw+λ^w/21+2αω−ωk,(6)ω^kn+1=∫0∞ωu^w2dω∫0∞u^w2dω,where u^kn+1w is the result of the current residual amount f^w−∑u^iw after Wiener filtering.The above is an adaptive decomposition procedure of VMD, which is in essence a Wiener filtering process. Signal decomposition is realized by iteratively searching for the optimal solution of the variational model in the variational problem framework. The approach achieves good noise robustness and effectively avoids the endpoint effect and modal aliasing problem.


## 3. WAFM-IPSO-VMD

To eliminate the need to set parameters in advance, which is a drawback of the VMD, this paper proposes a method for improving the VMD by enabling adaptive parameter acquisition. The method includes three aspects: first, the weighted AFM indicator is constructed as the objective function; second, the improved PSO is used to search for the optimal parameters of the VMD algorithm and mutual information of the decomposed components, and the original signal is used to detect overdecomposition, and thus to decide whether or not to perform the search again; third, an optimal IMF automatic selecting process is performed for the single-failure fault and compound faults following the principles of the maximum weighted AFM and maximum SPR, respectively.


### 3.1. WAFM

When searching for the optimal parameter combination using the improved PSO, construction of the objective function is a critical step. If a fault occurs, a periodic impulse signal will appear in the mixed signal. As described before, the Gini index aims to detect the impact of fault pulses and ignores periodic pulses that repeatedly occur when a fault is present, whereas AFM can effectively extract periodic characteristics of the signal. To enable more effective fault detection, this paper proposes constructing the weighted AFM indicator based on characteristics of the Gini index and the AFM.


#### 3.1.1. Gini Index

The Gini index is an inequality index that is often used in economics to reflect the overall income gap [27]. In mechanical fault diagnosis, the sequence elements must be positive in order to calculate the Gini index; therefore, the envelope spectrum Gini index has been used to extract the bearing fault signal [19] and achieved good results. The envelope spectrum Gini index can be defined as follows:(7)Gini=1−2∑n−1Nxnx⟶N−n+1/2N,where the vector x⟶ is the envelope spectrum of the signal; x⟶=x1,x2,…,xN is a sequence of points in the envelope spectrum arranged in ascending order of amplitude; and 1,2,3,…,N are the new index numbers after sorting.0≤Gini≤1. In economics, the closer the Gini index is to zero, the more equal the income distribution is; the closer the Gini index is to 1, the more uneven the income distribution is. Thus, extracting a bearing fault signal from the vibration signal of a faulty machine by applying the principle of the maximum Gini index of the envelope spectrum is feasible.


#### 3.1.2. AFM

The autocorrelation function is extremely effective in ascertaining the periodicity of a signal and can determine whether the signal is noisy [20].The autocorrelation function describes how similar a signal is to itself at different times. For a random processxt, a signal that occurs at a certain moment in time is denoted xt and the same signal becomes xt+k after a certain period k. Self-covariance of the signal can be defined as(8)yt=covxt,xt+k=Ext−x¯xt+k−x¯.The autocorrelation coefficient ofxt and xt+k is(9)ρk=Ext−x¯xt+k−x¯VarxtVarxt+k.For a stationary process,Varxt=Varxt+k=σx2. So, the autocorrelation coefficient of xt and xt+k can be defined as(10)ρk=Ext−x¯xt+k−x¯σx2.The autocorrelation coefficient sequenceρk,k=0,1,…,k with the time interval of k serving as a variable is called the autocorrelation function and can be defined as(11)Rxxk=1Tσx2∑t=1T−kxt−x¯xt+k−x¯.The autocorrelation function is an even function. When performing the analysis, it is enough to analyze one-half of the autocorrelation function corresponding to the positive half of the axis. In signal processing, the autocorrelation function is defined as(12)Rxk=1T∑t=1T−kxt⋅xt+k.As a sequence, the autocorrelation function cannot describe the characteristics of the signal. To convert the autocorrelation function into a quantitative indicator, the root mean square value of the maximum value sequence (denoted as AFM [20]) of the autocorrelation function can be used as an index for evaluating the IMF. The AFM can be expressed by the following formula:(13)AFM=∑n=1NPn2N,where Pn is the maximum point sequence of the autocorrelation function of the signal envelope and N is the sequence length.


#### 3.1.3. WAFM

In view of the impulsive and periodic nature of fault signals, this paper uses the product of the Gini index and the AFM as the objective function for VMD parameter optimization. Since0≤Gini≤1 , multiplying the AFM by the Gini index is equivalent to multiplying the AFM by a weight. Hence, the new indicator is defined as the weighted AFM:(14)WAFM=Gini⋅AFM.


### 3.2. Improved PSO

The PSO algorithm is a population-based random search method proposed by Kennedy and Eberhart [28]. The algorithm can be detailed as follows:Suppose that the population space dimensions areN×D, where N represents the number of populations, D represents the dimensionality of particles; the D-dimensional vector Xi=xi1,xi2,…,xiD represents the position i=1,2,…,N of the ith particle and its velocity can be expressed as Vi=vi1,vi2,…,viD; pbest=pi1,pi2,…,piD represents the individual extremum; and gbest=g1,g2,…,gD represents the global extremum. After the individual and global extrema are obtained, the particle velocity and position are updated using the following formulae:(15)vijt+1=wvijt+c1r1pijt−xijt+c2r2pgit−xijt,(16)xijt+1=xijt+vijt+1,where c1 and c2 are learning factors, r1 and r2 are random numbers uniformly distributed within [0, 1], and w is the inertia weight.The inertia weight affects local convergence and global exploration ability of the algorithm. Global exploration is performed when the weight is large and a local search is carried out when the weight is small. The PSO algorithm with a nonlinear decrease of the inertia weight improves upon on the inertia weightw [29]. The improved PSO algorithm is more effective since the solution space is analyzed during the initial search period such that the algorithm quickly approaches the optimal solution. In the late stage of search, higher accuracy can be achieved through iterative fine-tuning. The formula for improving the weight is as follows:(17)w=itermax−iteritermaxnwmax−wmin+wmin,where wmax is the inertia weight after initialization, wmin is the final inertia weight, itermax is the total number of iterations, iter is the current number of iterations, and n is the nonlinear adjustment factor.


### 3.3. Automatic Acquisition of IMF

For single-failure faults, the sensitive IMF is automatically obtained using the principle of the maximum weighted AFM. For compound faults, the sensitive IMF is automatically obtained using the maximum SPR principle.The SPR can effectively extract the periodic impulse components of the signal and is immune to interference from abnormal spikes in time-domain signals [30]. A larger value of SPR indicates a larger periodic impulse component of the fault signal:(18)SPRI=k∑i=1HpIh∑i=1Ksk,(19)SPRO=k∑i=1HpOh∑k=1Ksk,(20)SPRB=k∑i=1HpBh∑k=1Ksk,(21)SPRC=k∑i=1HpCh∑k=1Ksk.Here,sk denotes the amplitude of each point on the envelope spectrum, k=1,2,…,K denotes the number of envelope lines; pIh,pOh,pBh,pCh denote the amplitude peaks corresponding to the fault characteristic frequencies of the estimated bearing inner ring’s frequencies—that is, the bearing outer ring, roller, and cage, respectively. Parameters h=1,2,…,H denote the number of harmonic waves (for the purpose of this paper, h = 3).


### 3.4. Mutual Information

Mutual information (MI) [31] can quantitatively represent the correlation between two random variables, and it is more accurate than the correlation coefficient method. Mutual information can effectively measure the coupling degree between IMF components. The stronger the coupling of variables is, the greater the mutual information is. It is defined as follows:(22)MIX,Y=∬sfx,yx,ylogfx,yx,yfxxfyydxdy,where S is the domain of X and Y, fx,yx,y is the joint distribution of X and Y, and fxx and fyy are the marginal distributions of X and Y, respectively.


### 3.5. Proposed Method

A flowchart of the method proposed in this paper is shown in Figure1. The procedure is as follows:(1)
Input vibration signalf; set the range of the VMD parameter γ=K,α to be optimized and initialize the IPSO parameters.(2)
Randomly initialize the particles’ positions, calculate the objective function of each particle, and save the position corresponding to the maximum WAFM. That is, the parameter combinationγ=K,α.(3)
Construct a weighted AFM indicator as the objective function for optimization. Use the IPSO algorithm to update the position of the particle swarm and calculate the objective function of the particle swarm. Determine whether overdecomposition occurs using the minimum value of mutual information from each decomposed component and the original signal. If the minimum value of mutual information is less than 0.1, the current optimal solution is not updated; otherwise, the optimal solution is updated.(4)
Judge whether the stop condition is met: if so, exit to with optimal solution; if not, repeat step 3.(5)
Perform VMD on the original signal using the determined optimal parameter setγ=K,α. For a single-failure fault, the sensitive IMF is automatically obtained using the principle of the maximum weighted AFM. For compound faults, the sensitive IMF is automatically obtained using the maximum SPR principle.(6)
Envelope demodulation analysis is performed on the IMF to obtain the fault characteristics.Figure 1
Flowchart of the parametric adaptive VMD method.


## 3.1. WAFM

When searching for the optimal parameter combination using the improved PSO, construction of the objective function is a critical step. If a fault occurs, a periodic impulse signal will appear in the mixed signal. As described before, the Gini index aims to detect the impact of fault pulses and ignores periodic pulses that repeatedly occur when a fault is present, whereas AFM can effectively extract periodic characteristics of the signal. To enable more effective fault detection, this paper proposes constructing the weighted AFM indicator based on characteristics of the Gini index and the AFM.


### 3.1.1. Gini Index

The Gini index is an inequality index that is often used in economics to reflect the overall income gap [27]. In mechanical fault diagnosis, the sequence elements must be positive in order to calculate the Gini index; therefore, the envelope spectrum Gini index has been used to extract the bearing fault signal [19] and achieved good results. The envelope spectrum Gini index can be defined as follows:(7)Gini=1−2∑n−1Nxnx⟶N−n+1/2N,where the vector x⟶ is the envelope spectrum of the signal; x⟶=x1,x2,…,xN is a sequence of points in the envelope spectrum arranged in ascending order of amplitude; and 1,2,3,…,N are the new index numbers after sorting.0≤Gini≤1. In economics, the closer the Gini index is to zero, the more equal the income distribution is; the closer the Gini index is to 1, the more uneven the income distribution is. Thus, extracting a bearing fault signal from the vibration signal of a faulty machine by applying the principle of the maximum Gini index of the envelope spectrum is feasible.


### 3.1.2. AFM

The autocorrelation function is extremely effective in ascertaining the periodicity of a signal and can determine whether the signal is noisy [20].The autocorrelation function describes how similar a signal is to itself at different times. For a random processxt, a signal that occurs at a certain moment in time is denoted xt and the same signal becomes xt+k after a certain period k. Self-covariance of the signal can be defined as(8)yt=covxt,xt+k=Ext−x¯xt+k−x¯.The autocorrelation coefficient ofxt and xt+k is(9)ρk=Ext−x¯xt+k−x¯VarxtVarxt+k.For a stationary process,Varxt=Varxt+k=σx2. So, the autocorrelation coefficient of xt and xt+k can be defined as(10)ρk=Ext−x¯xt+k−x¯σx2.The autocorrelation coefficient sequenceρk,k=0,1,…,k with the time interval of k serving as a variable is called the autocorrelation function and can be defined as(11)Rxxk=1Tσx2∑t=1T−kxt−x¯xt+k−x¯.The autocorrelation function is an even function. When performing the analysis, it is enough to analyze one-half of the autocorrelation function corresponding to the positive half of the axis. In signal processing, the autocorrelation function is defined as(12)Rxk=1T∑t=1T−kxt⋅xt+k.As a sequence, the autocorrelation function cannot describe the characteristics of the signal. To convert the autocorrelation function into a quantitative indicator, the root mean square value of the maximum value sequence (denoted as AFM [20]) of the autocorrelation function can be used as an index for evaluating the IMF. The AFM can be expressed by the following formula:(13)AFM=∑n=1NPn2N,where Pn is the maximum point sequence of the autocorrelation function of the signal envelope and N is the sequence length.


### 3.1.3. WAFM

In view of the impulsive and periodic nature of fault signals, this paper uses the product of the Gini index and the AFM as the objective function for VMD parameter optimization. Since0≤Gini≤1 , multiplying the AFM by the Gini index is equivalent to multiplying the AFM by a weight. Hence, the new indicator is defined as the weighted AFM:(14)WAFM=Gini⋅AFM.


## 3.1.1. Gini Index

The Gini index is an inequality index that is often used in economics to reflect the overall income gap [27]. In mechanical fault diagnosis, the sequence elements must be positive in order to calculate the Gini index; therefore, the envelope spectrum Gini index has been used to extract the bearing fault signal [19] and achieved good results. The envelope spectrum Gini index can be defined as follows:(7)Gini=1−2∑n−1Nxnx⟶N−n+1/2N,where the vector x⟶ is the envelope spectrum of the signal; x⟶=x1,x2,…,xN is a sequence of points in the envelope spectrum arranged in ascending order of amplitude; and 1,2,3,…,N are the new index numbers after sorting.0≤Gini≤1. In economics, the closer the Gini index is to zero, the more equal the income distribution is; the closer the Gini index is to 1, the more uneven the income distribution is. Thus, extracting a bearing fault signal from the vibration signal of a faulty machine by applying the principle of the maximum Gini index of the envelope spectrum is feasible.


## 3.1.2. AFM

The autocorrelation function is extremely effective in ascertaining the periodicity of a signal and can determine whether the signal is noisy [20].The autocorrelation function describes how similar a signal is to itself at different times. For a random processxt, a signal that occurs at a certain moment in time is denoted xt and the same signal becomes xt+k after a certain period k. Self-covariance of the signal can be defined as(8)yt=covxt,xt+k=Ext−x¯xt+k−x¯.The autocorrelation coefficient ofxt and xt+k is(9)ρk=Ext−x¯xt+k−x¯VarxtVarxt+k.For a stationary process,Varxt=Varxt+k=σx2. So, the autocorrelation coefficient of xt and xt+k can be defined as(10)ρk=Ext−x¯xt+k−x¯σx2.The autocorrelation coefficient sequenceρk,k=0,1,…,k with the time interval of k serving as a variable is called the autocorrelation function and can be defined as(11)Rxxk=1Tσx2∑t=1T−kxt−x¯xt+k−x¯.The autocorrelation function is an even function. When performing the analysis, it is enough to analyze one-half of the autocorrelation function corresponding to the positive half of the axis. In signal processing, the autocorrelation function is defined as(12)Rxk=1T∑t=1T−kxt⋅xt+k.As a sequence, the autocorrelation function cannot describe the characteristics of the signal. To convert the autocorrelation function into a quantitative indicator, the root mean square value of the maximum value sequence (denoted as AFM [20]) of the autocorrelation function can be used as an index for evaluating the IMF. The AFM can be expressed by the following formula:(13)AFM=∑n=1NPn2N,where Pn is the maximum point sequence of the autocorrelation function of the signal envelope and N is the sequence length.


## 3.1.3. WAFM

In view of the impulsive and periodic nature of fault signals, this paper uses the product of the Gini index and the AFM as the objective function for VMD parameter optimization. Since0≤Gini≤1 , multiplying the AFM by the Gini index is equivalent to multiplying the AFM by a weight. Hence, the new indicator is defined as the weighted AFM:(14)WAFM=Gini⋅AFM.


## 3.2. Improved PSO

The PSO algorithm is a population-based random search method proposed by Kennedy and Eberhart [28]. The algorithm can be detailed as follows:Suppose that the population space dimensions areN×D, where N represents the number of populations, D represents the dimensionality of particles; the D-dimensional vector Xi=xi1,xi2,…,xiD represents the position i=1,2,…,N of the ith particle and its velocity can be expressed as Vi=vi1,vi2,…,viD; pbest=pi1,pi2,…,piD represents the individual extremum; and gbest=g1,g2,…,gD represents the global extremum. After the individual and global extrema are obtained, the particle velocity and position are updated using the following formulae:(15)vijt+1=wvijt+c1r1pijt−xijt+c2r2pgit−xijt,(16)xijt+1=xijt+vijt+1,where c1 and c2 are learning factors, r1 and r2 are random numbers uniformly distributed within [0, 1], and w is the inertia weight.The inertia weight affects local convergence and global exploration ability of the algorithm. Global exploration is performed when the weight is large and a local search is carried out when the weight is small. The PSO algorithm with a nonlinear decrease of the inertia weight improves upon on the inertia weightw [29]. The improved PSO algorithm is more effective since the solution space is analyzed during the initial search period such that the algorithm quickly approaches the optimal solution. In the late stage of search, higher accuracy can be achieved through iterative fine-tuning. The formula for improving the weight is as follows:(17)w=itermax−iteritermaxnwmax−wmin+wmin,where wmax is the inertia weight after initialization, wmin is the final inertia weight, itermax is the total number of iterations, iter is the current number of iterations, and n is the nonlinear adjustment factor.


## 3.3. Automatic Acquisition of IMF

For single-failure faults, the sensitive IMF is automatically obtained using the principle of the maximum weighted AFM. For compound faults, the sensitive IMF is automatically obtained using the maximum SPR principle.The SPR can effectively extract the periodic impulse components of the signal and is immune to interference from abnormal spikes in time-domain signals [30]. A larger value of SPR indicates a larger periodic impulse component of the fault signal:(18)SPRI=k∑i=1HpIh∑i=1Ksk,(19)SPRO=k∑i=1HpOh∑k=1Ksk,(20)SPRB=k∑i=1HpBh∑k=1Ksk,(21)SPRC=k∑i=1HpCh∑k=1Ksk.Here,sk denotes the amplitude of each point on the envelope spectrum, k=1,2,…,K denotes the number of envelope lines; pIh,pOh,pBh,pCh denote the amplitude peaks corresponding to the fault characteristic frequencies of the estimated bearing inner ring’s frequencies—that is, the bearing outer ring, roller, and cage, respectively. Parameters h=1,2,…,H denote the number of harmonic waves (for the purpose of this paper, h = 3).


## 3.4. Mutual Information

Mutual information (MI) [31] can quantitatively represent the correlation between two random variables, and it is more accurate than the correlation coefficient method. Mutual information can effectively measure the coupling degree between IMF components. The stronger the coupling of variables is, the greater the mutual information is. It is defined as follows:(22)MIX,Y=∬sfx,yx,ylogfx,yx,yfxxfyydxdy,where S is the domain of X and Y, fx,yx,y is the joint distribution of X and Y, and fxx and fyy are the marginal distributions of X and Y, respectively.


## 3.5. Proposed Method

A flowchart of the method proposed in this paper is shown in Figure1. The procedure is as follows:(1)
Input vibration signalf; set the range of the VMD parameter γ=K,α to be optimized and initialize the IPSO parameters.(2)
Randomly initialize the particles’ positions, calculate the objective function of each particle, and save the position corresponding to the maximum WAFM. That is, the parameter combinationγ=K,α.(3)
Construct a weighted AFM indicator as the objective function for optimization. Use the IPSO algorithm to update the position of the particle swarm and calculate the objective function of the particle swarm. Determine whether overdecomposition occurs using the minimum value of mutual information from each decomposed component and the original signal. If the minimum value of mutual information is less than 0.1, the current optimal solution is not updated; otherwise, the optimal solution is updated.(4)
Judge whether the stop condition is met: if so, exit to with optimal solution; if not, repeat step 3.(5)
Perform VMD on the original signal using the determined optimal parameter setγ=K,α. For a single-failure fault, the sensitive IMF is automatically obtained using the principle of the maximum weighted AFM. For compound faults, the sensitive IMF is automatically obtained using the maximum SPR principle.(6)
Envelope demodulation analysis is performed on the IMF to obtain the fault characteristics.Figure 1
Flowchart of the parametric adaptive VMD method.


## 4. Simulation Signal

In order to verify the effectiveness of the method proposed in this paper, the following model is used to simulate the vibration signal of the failure of the bearing outer ring, and add random noise with an amplitude of 2. The simulation signal is as follows:(23)y1t=y0e−ξωntsinωn1−ξ2t,(24)yt=y1t+nt,where y1t denotes the impact signal of a bearing fault, an intrinsic frequency fn=3000Hz, ωn=2πfn; a displacement constant y0=1, a damping coefficient ξ=0.1, an impact fault frequency of 100 Hz, and a sampling frequency fs=20kHz, with a sampled point count of N=4096. The waveform of the simulation signal is shown in Figure 2.Figure 2
Time-domain waveform of simulation signal.It can be seen from Figure2 that, under the interference of random noise, the periodic pulse in the simulation signal has been buried, and it is difficult to find the periodic impulse signal in the time-domain waveform. The envelope demodulation analysis of the simulation signal is conducted, and the resulting envelope spectrum is shown in Figure 3, in which failure characteristic components cannot be observed.Figure 3
Hilbert envelope spectrum of simulation signal.


### 4.1. Optimization of the VMD Parameters

The simulation signal is being decomposed by the parametric adaptive VMD method proposed in this paper (search range of K: [2, 12]; search range ofα: [100, 5000]). The setting of PSO parameters in this paper is obtained by adjusting according to experience and previous studies on PSO. In this paper, two parameters need to be optimized, so the dimension of the particle is 2. According to the literature [32], PSO is not sensitive to population size. At present, the study of learning factors c1 and c2 is limited to the application of the problem. Considering the computational complexity and reliability, the population size was set as 10, c1 as 1.7, and c2 as 1.5 according to the experience after several tests and adjustments. The authors of literature [33] found that when the inertia weight belongs to [0.4, 0.95], the performance of particle swarm optimization algorithm will be greatly improved. So, this paper sets maximum inertia weight wmax=0.95 and minimum inertia weight wmin=0.4. Initialize the IPSO, and set the population size N = 10, particle dimension D = 2, maximum evolutionary generation number t = 20, acceleration coefficients c1=1.7, c2=1.5, maximum inertia weight wmax=0.95, minimum inertia weight wmin=0.4, and nonlinear modulation index n = 1.2 [29]. Figure 4 shows the iteration curve for the optimization search, from which approximate a global maximum max (WAFM) = 0.0782 and the optimal parameter combination of [10, 3146] can be obtained.Figure 4
IPSO convergence curve for VMD parameter optimization.


### 4.2. VMD and Sensitive IMF Selection

On this basis, the optimal parameter combination [10, 3146] is used to perform the VMD of the original signal. The waveform of each IMF component after decomposition is shown in Figure5(a). The histogram of IMF-WAFM is shown in Figure 5(b), in which it can be seen that the WAFM value of IMF4 is the highest, so it is selected for further envelope spectrum analysis, and the effect is shown in Figure 5(c).Figure 5
Analysis results of simulation signal. (a) The modes obtained by the improved VMD method. (b) WAFM indexes of IMF components. (c) Envelope spectrum of IMF4 component. (d) Envelope spectrum of IMF3 component. (e) Envelope spectrum of IMF5 component.
(a)(b)(c)(d)(e)Envelope spectrum analysis was also performed on the other IMF components. Due to limited space, only envelope spectrums of IMF3 and IMF5 were given, as shown in Figures5(d) and 5(e). In the envelope spectrum of IMF5 in Figure 5(e), failure characteristic components cannot be observed. Although prominent spectral line can be observed at the fault characteristic frequency fo in Figure 5(d), there is a lot of background noise. In contrast, in the envelope spectrum of sensitive IMF in Figure 5(c), the spectral lines at the fault characteristic frequency fo and double frequency 2fo are more prominent. In Figure 5(c), the rotation frequency fr can also be observed significantly. This suggests that the improved VMD method proposed in this paper can effectively decompose the signal component containing fault characteristic components from the original signal buried by noise, which proves the effectiveness of the method.


### 4.3. Comparison with Other Methods

In order to verify the advantages of WAFM index proposed in this paper, the same optimal combination [10, 3146] is used for VMD, and sensitive IMF is selected through the maximum kurtosis principle. It is found that IMF6 is a sensitive IMF and that no fault characteristic components can be observed from its envelope spectrum, as shown in Figure6.Figure 6
Envelope spectrum of sensitive IMF component.To verify the advantages of the optimization method introduced in this paper, the combinations of parameters [10, 400] and [5, 3000] are artificially selected for VMD. The sensitive IMF envelope spectrum selected with WAFM can be seen in Figure7. According to our findings, fault characteristic components cannot be observed in Figure 7(b). The fault characteristic frequency fo can be observed on the envelope spectrum of sensitive IMF in Figure 7(b), but it has been buried by noise. This reveals the fact that there is a gap between the method of artificial parameter selection and the method proposed in this paper.Figure 7
Analysis results of randomly selected parameter combinations. (a) Combination of parameters [10, 400]. (b) Combination of parameters [5, 3000].
(a)(b)For the purpose of further validating the advantage of the method proposed in this paper, EMD and EEMD algorithms were adopted to decompose the simulation signal. The two parameters of EEMD, ensemble number and amplitude coefficient, were set as 100 and 0.01 in [34]. After decomposition, maximum kurtosis principle is adopted to select the acquired IMFs. The time-domain waveforms of the sensitive IMFs are shown in Figures 8 and 9 [35, 36], in which fault characteristics cannot be observed. It is clear that the VMD method proposed in this paper is superior to both EMD and EMMD.Figure 8
Time-domain waveform of sensitive IMF component by the EMD method.Figure 9
Time-domain waveform of sensitive IMF component by the EEMD method.


## 4.1. Optimization of the VMD Parameters

The simulation signal is being decomposed by the parametric adaptive VMD method proposed in this paper (search range of K: [2, 12]; search range ofα: [100, 5000]). The setting of PSO parameters in this paper is obtained by adjusting according to experience and previous studies on PSO. In this paper, two parameters need to be optimized, so the dimension of the particle is 2. According to the literature [32], PSO is not sensitive to population size. At present, the study of learning factors c1 and c2 is limited to the application of the problem. Considering the computational complexity and reliability, the population size was set as 10, c1 as 1.7, and c2 as 1.5 according to the experience after several tests and adjustments. The authors of literature [33] found that when the inertia weight belongs to [0.4, 0.95], the performance of particle swarm optimization algorithm will be greatly improved. So, this paper sets maximum inertia weight wmax=0.95 and minimum inertia weight wmin=0.4. Initialize the IPSO, and set the population size N = 10, particle dimension D = 2, maximum evolutionary generation number t = 20, acceleration coefficients c1=1.7, c2=1.5, maximum inertia weight wmax=0.95, minimum inertia weight wmin=0.4, and nonlinear modulation index n = 1.2 [29]. Figure 4 shows the iteration curve for the optimization search, from which approximate a global maximum max (WAFM) = 0.0782 and the optimal parameter combination of [10, 3146] can be obtained.Figure 4
IPSO convergence curve for VMD parameter optimization.


## 4.2. VMD and Sensitive IMF Selection

On this basis, the optimal parameter combination [10, 3146] is used to perform the VMD of the original signal. The waveform of each IMF component after decomposition is shown in Figure5(a). The histogram of IMF-WAFM is shown in Figure 5(b), in which it can be seen that the WAFM value of IMF4 is the highest, so it is selected for further envelope spectrum analysis, and the effect is shown in Figure 5(c).Figure 5
Analysis results of simulation signal. (a) The modes obtained by the improved VMD method. (b) WAFM indexes of IMF components. (c) Envelope spectrum of IMF4 component. (d) Envelope spectrum of IMF3 component. (e) Envelope spectrum of IMF5 component.
(a)(b)(c)(d)(e)Envelope spectrum analysis was also performed on the other IMF components. Due to limited space, only envelope spectrums of IMF3 and IMF5 were given, as shown in Figures5(d) and 5(e). In the envelope spectrum of IMF5 in Figure 5(e), failure characteristic components cannot be observed. Although prominent spectral line can be observed at the fault characteristic frequency fo in Figure 5(d), there is a lot of background noise. In contrast, in the envelope spectrum of sensitive IMF in Figure 5(c), the spectral lines at the fault characteristic frequency fo and double frequency 2fo are more prominent. In Figure 5(c), the rotation frequency fr can also be observed significantly. This suggests that the improved VMD method proposed in this paper can effectively decompose the signal component containing fault characteristic components from the original signal buried by noise, which proves the effectiveness of the method.


## 4.3. Comparison with Other Methods

In order to verify the advantages of WAFM index proposed in this paper, the same optimal combination [10, 3146] is used for VMD, and sensitive IMF is selected through the maximum kurtosis principle. It is found that IMF6 is a sensitive IMF and that no fault characteristic components can be observed from its envelope spectrum, as shown in Figure6.Figure 6
Envelope spectrum of sensitive IMF component.To verify the advantages of the optimization method introduced in this paper, the combinations of parameters [10, 400] and [5, 3000] are artificially selected for VMD. The sensitive IMF envelope spectrum selected with WAFM can be seen in Figure7. According to our findings, fault characteristic components cannot be observed in Figure 7(b). The fault characteristic frequency fo can be observed on the envelope spectrum of sensitive IMF in Figure 7(b), but it has been buried by noise. This reveals the fact that there is a gap between the method of artificial parameter selection and the method proposed in this paper.Figure 7
Analysis results of randomly selected parameter combinations. (a) Combination of parameters [10, 400]. (b) Combination of parameters [5, 3000].
(a)(b)For the purpose of further validating the advantage of the method proposed in this paper, EMD and EEMD algorithms were adopted to decompose the simulation signal. The two parameters of EEMD, ensemble number and amplitude coefficient, were set as 100 and 0.01 in [34]. After decomposition, maximum kurtosis principle is adopted to select the acquired IMFs. The time-domain waveforms of the sensitive IMFs are shown in Figures 8 and 9 [35, 36], in which fault characteristics cannot be observed. It is clear that the VMD method proposed in this paper is superior to both EMD and EMMD.Figure 8
Time-domain waveform of sensitive IMF component by the EMD method.Figure 9
Time-domain waveform of sensitive IMF component by the EEMD method.


## 5. Experimental Signal Analysis

In order to verify the effectiveness of this method, it is applied to the actual data for analysis. The test rig is shown in Figure10. The bearing type is HRB6304, the sampling frequency is 10 kHz, and 4096 data points are sampled every 1 minute. Failure of bearing inner ring of bearing and compound failure of inner and outer ring were selected for analysis, respectively, and the failure diagrams are shown in Figure 11. Table 1 shows the multiple of fault frequency of each bearing component.Figure 10
Test rig.Figure 11
The tested bearing with fault. (a) The tested bearing with inner race fault. (b) The tested bearing with compound fault.
(a)(b)Table 1
Multiple of fault frequency of each bearing component.
TypeInner raceOuter raceCageBall63044.432.5660.3671.742


### 5.1. Single Fault Data Analysis

This section analyses the data of inner race fault, and the 111th minute signal of the experiment is extracted for decomposition. The velocity of the signal is at 3,000 rpm, and the bearing inner race fault characteristic frequencyfi=136.74Hz can be calculated according to Table 1. The time-domain waveform and its envelope spectrum of this signal are shown in Figure 12. It can be observed that no periodic impact is visible on the time-domain waveform, and there is no prominent spectral line near the fault characteristic frequency in Figure 12(b), so it is impossible to judge the state of bearing accurately.Figure 12
Waveform and envelope spectrum of inner race fault signal. (a) Waveform of fault signal. (b) Envelope spectrum of fault signal.
(a)(b)By using the method proposed in this paper, the signal is decomposed, and the approximate global maximum max (WAFM) = 0.1356 and the optimal parameter combination [K, α] = [10, 2416] can be obtained. On this basis, the optimal parameter combination [10, 2416] is used to perform the VMD of the original signal. The waveform of each IMF component after decomposition is shown in Figure 13(a). The histogram of IMF-WAFM is shown in Figure 13(b), in which it can be seen that the WAFM value of IMF8 is the highest, so it is selected for further envelope spectrum analysis, and the effect is shown in Figure 13(c).Figure 13
Analysis results of inner race fault signal. (a) The modes obtained by the improved VMD method. (b) WAFM indexes of IMF components. (c) Envelope spectrum of IMF8 component. (d) Envelope spectrum of IMF7 component. (e) Envelope spectrum of IMF9 component.
(a)(b)(c)(d)(e)Envelope spectrum analysis was also performed on the other IMF components. Due to limited space, only envelope spectrums of IMF7 and IMF9 were given, as shown in Figures13(d) and 13(e). In the envelope spectrum of IMF9 in Figure 13(e), failure characteristic components cannot be observed. The envelope spectrum of IMF9 component in Figure 13(d) shows that prominent spectral lines cannot be observed obviously at the inner race fault characteristic frequency fi. In contrast, in the envelope spectrum of sensitive IMF in Figure 13(c), the spectral lines at the fault characteristic frequency fi are more prominent. On this basis, the following conclusion can be drawn: the bearing inner ring has serious local defects.Similarly, in order to verify the advantages of WAFM index proposed in this paper, sensitive IMF is selected through the maximum kurtosis principle. It is found that IMF10 is a sensitive IMF, and its envelope spectrum is shown in Figure14. Although the envelope line amplitudes at the fault characteristic frequency fi and double frequency 2fi can be observed, there is a lot of background noise, so it shows that the effect is not as good as that of the method proposed in this paper.Figure 14
Envelope spectrum of sensitive IMF component.EMD and EEMD are used for signal decomposition, and kurtosis was used to select sensitive IMF. As shown by their time-domain waveform and envelope spectrum, although fault characteristic frequencyfi can be observed in Figure 15, there are more noise interference patterns in the envelope spectrum obtained in this way, so the effect is less obvious than that in Figure 13(c). Failure characteristic components cannot be observed in Figure 16. It is, thus, clear that the VMD-based decomposition method proposed in this paper is superior to both EMD and EEMD.Figure 15
Envelope spectrum of sensitive IMF component by EMD method.Figure 16
Time-domain waveform of sensitive IMF component by EEMD method.


### 5.2. Analysis of Compound Faults Data

A dataset of inner-outer ring coupling faults was used as a sample for analysis. The rotational speed indicated by the data was 2020 r/min. According to Table1, the characteristic frequency of the inner ring fault was fi=149.14Hz and the characteristic frequency of the outer ring fault was fo=86.39Hz. The time domain waveform diagram and envelope spectrum of the group of signals are shown in Figure 17. No salient periodic impact can be observed in Figure 17(a). In the envelope spectrum as shown in Figure 17(b), the characteristic of the outer ring fault can be clearly observed; however, the characteristic of the inner ring fault cannot, making it difficult to determine the bearing condition.Figure 17
Waveform and envelope spectrum of compound fault signal. (a) Waveform of fault signal. (b) Envelope spectrum of fault signal.
(a)(b)By using the method proposed in this paper, the signal is decomposed, and the approximate global maximum max (WAFM) = 0.2203 and the optimal parameter combination [K, α] = [6, 4605] can be obtained. On this basis, the optimal parameter combination is used to perform the VMD of the original signal. The waveform of each IMF component after decomposition is shown in Figure 18(a). The histograms of IMF-SPRI and IMF-SPRO are shown in Figures 18(b) and 18(c), in which it can be seen that the SPRI value of IMF1 is the highest and the SPRO value of IMF6 is the highest, so the IMFs are selected for further envelope spectrum analysis, and the effects are shown in Figures 18(d) and 18(e).Figure 18
Analysis results of compound fault signal. (a) The modes obtained by the improved VMD method. (b) SPRI indexes of IMF components. (c) SPRO indexes of IMF components. (d) Envelope spectrum of IMF1 component. (e) Envelope spectrum of IMF6 component.
(a)(b)(c)(d)(e)As shown in Figure18(d), the fault characteristic frequency fi of the inner ring can be observed in the envelope spectrum of the IMF1 component. Salient lines can be observed near the characteristic frequency fo of the outer ring fault and its double frequency 2fo in the envelope spectrum of the IMF6 component in Figure 18(e). This indicates serious defects on both the inner and outer rings.Signal decomposition was performed using EMD and EEMD, and the sensitive IMF was selected using SPR. After EMD processing, the sensitive IMF component obtained through SPRI-based selection or via SPRO-based selection were the same, as shown in Figure19. The fault characteristic frequency is clearly buried in noise to the extent that no prominent spectral lines can be observed at the location of the fault characteristic frequency.Figure 19
Envelope spectrum of sensitive IMF component by EMD method.After EEMD processing, the sensitive IMF of the inner ring fault obtained via SPRI-based selection is IMF3 and the sensitive IMF of the outer ring fault obtained through SPRO-based selection is IMF5. The envelope spectrum is shown in Figure20. No fault characteristic frequency can be observed in the sensitive IMFs of the inner ring and outer ring, indicating that the VMD-based decomposition method proposed in this paper is superior to EMD and EMMD.Figure 20
Envelope spectrum of sensitive IMF component by EEMD method. (a) Sensitive IMF component of inner race fault. (b) Sensitive IMF component of outer race fault.
(a)(b)In order to verify the advantage of SPR index, this paper uses WAFM maximum principle, AFM maximum principle, and kurtosis maximum principle to select sensitive IMFs.The two sensitive IMFs with the maximum WAFM value selected by WAFM maximum principle and AFM maximum principle are IMF2 (Figure21) and IMF6 (Figure 18(e)), respectively. IMF6 can observe the fault characteristic frequency of the outer ring, which is the same as the IMF selected by SPR, while IMF2 cannot observe the fault characteristic frequency.Figure 21
Envelope spectrum of IMF2.The two sensitive IMFs with kurtosis maximum principle are IMF2 (Figure21) and IMF4 (Figure 22), respectively. IMF2 cannot observe the fault characteristic frequency, and the fault frequency of inner and outer ring can be found in the envelope spectrum of IMF4, but there are many interferences, and the effect is not as good as that of the method in this paper.Figure 22
Envelope spectrum of IMF4.


## 5.1. Single Fault Data Analysis

This section analyses the data of inner race fault, and the 111th minute signal of the experiment is extracted for decomposition. The velocity of the signal is at 3,000 rpm, and the bearing inner race fault characteristic frequencyfi=136.74Hz can be calculated according to Table 1. The time-domain waveform and its envelope spectrum of this signal are shown in Figure 12. It can be observed that no periodic impact is visible on the time-domain waveform, and there is no prominent spectral line near the fault characteristic frequency in Figure 12(b), so it is impossible to judge the state of bearing accurately.Figure 12
Waveform and envelope spectrum of inner race fault signal. (a) Waveform of fault signal. (b) Envelope spectrum of fault signal.
(a)(b)By using the method proposed in this paper, the signal is decomposed, and the approximate global maximum max (WAFM) = 0.1356 and the optimal parameter combination [K, α] = [10, 2416] can be obtained. On this basis, the optimal parameter combination [10, 2416] is used to perform the VMD of the original signal. The waveform of each IMF component after decomposition is shown in Figure 13(a). The histogram of IMF-WAFM is shown in Figure 13(b), in which it can be seen that the WAFM value of IMF8 is the highest, so it is selected for further envelope spectrum analysis, and the effect is shown in Figure 13(c).Figure 13
Analysis results of inner race fault signal. (a) The modes obtained by the improved VMD method. (b) WAFM indexes of IMF components. (c) Envelope spectrum of IMF8 component. (d) Envelope spectrum of IMF7 component. (e) Envelope spectrum of IMF9 component.
(a)(b)(c)(d)(e)Envelope spectrum analysis was also performed on the other IMF components. Due to limited space, only envelope spectrums of IMF7 and IMF9 were given, as shown in Figures13(d) and 13(e). In the envelope spectrum of IMF9 in Figure 13(e), failure characteristic components cannot be observed. The envelope spectrum of IMF9 component in Figure 13(d) shows that prominent spectral lines cannot be observed obviously at the inner race fault characteristic frequency fi. In contrast, in the envelope spectrum of sensitive IMF in Figure 13(c), the spectral lines at the fault characteristic frequency fi are more prominent. On this basis, the following conclusion can be drawn: the bearing inner ring has serious local defects.Similarly, in order to verify the advantages of WAFM index proposed in this paper, sensitive IMF is selected through the maximum kurtosis principle. It is found that IMF10 is a sensitive IMF, and its envelope spectrum is shown in Figure14. Although the envelope line amplitudes at the fault characteristic frequency fi and double frequency 2fi can be observed, there is a lot of background noise, so it shows that the effect is not as good as that of the method proposed in this paper.Figure 14
Envelope spectrum of sensitive IMF component.EMD and EEMD are used for signal decomposition, and kurtosis was used to select sensitive IMF. As shown by their time-domain waveform and envelope spectrum, although fault characteristic frequencyfi can be observed in Figure 15, there are more noise interference patterns in the envelope spectrum obtained in this way, so the effect is less obvious than that in Figure 13(c). Failure characteristic components cannot be observed in Figure 16. It is, thus, clear that the VMD-based decomposition method proposed in this paper is superior to both EMD and EEMD.Figure 15
Envelope spectrum of sensitive IMF component by EMD method.Figure 16
Time-domain waveform of sensitive IMF component by EEMD method.


## 5.2. Analysis of Compound Faults Data

A dataset of inner-outer ring coupling faults was used as a sample for analysis. The rotational speed indicated by the data was 2020 r/min. According to Table1, the characteristic frequency of the inner ring fault was fi=149.14Hz and the characteristic frequency of the outer ring fault was fo=86.39Hz. The time domain waveform diagram and envelope spectrum of the group of signals are shown in Figure 17. No salient periodic impact can be observed in Figure 17(a). In the envelope spectrum as shown in Figure 17(b), the characteristic of the outer ring fault can be clearly observed; however, the characteristic of the inner ring fault cannot, making it difficult to determine the bearing condition.Figure 17
Waveform and envelope spectrum of compound fault signal. (a) Waveform of fault signal. (b) Envelope spectrum of fault signal.
(a)(b)By using the method proposed in this paper, the signal is decomposed, and the approximate global maximum max (WAFM) = 0.2203 and the optimal parameter combination [K, α] = [6, 4605] can be obtained. On this basis, the optimal parameter combination is used to perform the VMD of the original signal. The waveform of each IMF component after decomposition is shown in Figure 18(a). The histograms of IMF-SPRI and IMF-SPRO are shown in Figures 18(b) and 18(c), in which it can be seen that the SPRI value of IMF1 is the highest and the SPRO value of IMF6 is the highest, so the IMFs are selected for further envelope spectrum analysis, and the effects are shown in Figures 18(d) and 18(e).Figure 18
Analysis results of compound fault signal. (a) The modes obtained by the improved VMD method. (b) SPRI indexes of IMF components. (c) SPRO indexes of IMF components. (d) Envelope spectrum of IMF1 component. (e) Envelope spectrum of IMF6 component.
(a)(b)(c)(d)(e)As shown in Figure18(d), the fault characteristic frequency fi of the inner ring can be observed in the envelope spectrum of the IMF1 component. Salient lines can be observed near the characteristic frequency fo of the outer ring fault and its double frequency 2fo in the envelope spectrum of the IMF6 component in Figure 18(e). This indicates serious defects on both the inner and outer rings.Signal decomposition was performed using EMD and EEMD, and the sensitive IMF was selected using SPR. After EMD processing, the sensitive IMF component obtained through SPRI-based selection or via SPRO-based selection were the same, as shown in Figure19. The fault characteristic frequency is clearly buried in noise to the extent that no prominent spectral lines can be observed at the location of the fault characteristic frequency.Figure 19
Envelope spectrum of sensitive IMF component by EMD method.After EEMD processing, the sensitive IMF of the inner ring fault obtained via SPRI-based selection is IMF3 and the sensitive IMF of the outer ring fault obtained through SPRO-based selection is IMF5. The envelope spectrum is shown in Figure20. No fault characteristic frequency can be observed in the sensitive IMFs of the inner ring and outer ring, indicating that the VMD-based decomposition method proposed in this paper is superior to EMD and EMMD.Figure 20
Envelope spectrum of sensitive IMF component by EEMD method. (a) Sensitive IMF component of inner race fault. (b) Sensitive IMF component of outer race fault.
(a)(b)In order to verify the advantage of SPR index, this paper uses WAFM maximum principle, AFM maximum principle, and kurtosis maximum principle to select sensitive IMFs.The two sensitive IMFs with the maximum WAFM value selected by WAFM maximum principle and AFM maximum principle are IMF2 (Figure21) and IMF6 (Figure 18(e)), respectively. IMF6 can observe the fault characteristic frequency of the outer ring, which is the same as the IMF selected by SPR, while IMF2 cannot observe the fault characteristic frequency.Figure 21
Envelope spectrum of IMF2.The two sensitive IMFs with kurtosis maximum principle are IMF2 (Figure21) and IMF4 (Figure 22), respectively. IMF2 cannot observe the fault characteristic frequency, and the fault frequency of inner and outer ring can be found in the envelope spectrum of IMF4, but there are many interferences, and the effect is not as good as that of the method in this paper.Figure 22
Envelope spectrum of IMF4.


## 6. Conclusions

(1)
This paper proposes a method for improving VMD and automatically acquiring the sensitive IMF. The proposed strategy solves the problem of setting parameter combination for the conventional VMD method.(2)
The weighted AFM indicator constructed in this paper takes into account the impulsive and periodic nature of fault signals and makes it possible to search for the optimal VMD parameter set using the IPSO algorithm. Moreover, the proposed method has the benefit of adaptive parameter acquisition and the decomposition result was found to be superior compared to that of other methods such as EMD and EEMD.(3)
In view of the differences between single-failure faults and compound faults, this paper proposes using two different principles (principle of maximum weighted AFM and principle of maximum SPR) for selecting the sensitive IMF, thereby solving the difficult problem of accurately detecting compound faults. The proposed method could potentially provide technical support for engineering applications.



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*Source: 6216903-2020-05-22.xml*