abstract

In view of the harsh working environment of strip mill bearing, easy damage of holder and rolling body, large signal noise, difficult identification and high diagnostic speed requirements under actual working conditions, first, Proposed particle swarm optimization variational mode decomposition (particle swarm optimization variational mode decomposition, For PSO VMD) and the multivariate multiscale permutation entropy (multivariate multiscale permutation entropy, Fault diagnosis method for MMPE), Combined with the particle swarm optimization of the support vector machine (particle swarm optimization support vector machine, For short, PSO SVM) to achieve fault classification; next, The bearing vibration signal is VMD treated into several modal components (intrinsic mode functions, the abbreviated form of a name IMF), The optimal component is selected for the envelope analysis; then, According to the large difference of vertical horizontal axial vibration of rolling mill bearing and the large radial force and axial force, The MMPE value and the time domain index of the MMPE value can form the eigenvector; last, Validation of the method based on the PSO SVM model. Calculation and experimental results and set empirical mode decomposition (ensemble empirical mode decomposition, EEMD) and local mean decomposition (local mean decomposition, LMD) method shows that VMD MMPE can optimize the input of the model, improve the diagnostic accuracy and speed of the model, realize the bearing holder and rolling parts and different degrees of fault diagnosis, has important engineering significance.

Key words: rolling mill bearing; variational mode decomposition; envelope spectrum; multivariscale alignment entropy; particle swarm optimization support vector machine; fault diagnosis

 foreword

 Extracting the fault information from the bearing vibration signal plays a very effective role in the fault diagnosis [1]. Fault diagnosis includes signal acquisition, feature extraction and diagnosis and identification of [2]. The key to fault diagnosis is how to extract the fault feature information [3] from the nonlinear and non-stationary vibration signal. Huang et al [4] proposed the EMD algorithm, where the signal is decomposed into multiple modal components and is adaptive. In order to overcome the endpoint effect and mode aliasing, Wu et al. [5] added white noise to the decomposition process to form the EEMD algorithm. Zhang Chen et al. [6] used EEMD algorithm combined with the singular value entropy value to realize the rolling bearing vibration letter with noise, number the feature extraction, and used the singular value entropy size to judge the bearing fault. Tian Jing et al [7] uses EEMD to combine with airspace noise reduction. The suppression effect of bearing fault signal noise is better than wavelet analysis, and the fault characteristics are more prominent. The above studies show that the short-time Fourier transform, wavelet analysis and EMD algorithm can achieve better results in bearing fault diagnosis.

 With the continuous development of adaptive algorithms, LMD has been proposed to suppress endpoint effects and mode aliasing phenomena. Liu et al. [8] performed LMD decomposition of the gearbox fault signal, and selected the better components for Fourier transformation, from which the pitting fault feature frequency was extracted. Cheng Junsheng et al. [9] used LMD and EMD algorithm to analyze the fault signal of the gearbox respectively, and the results showed that LMD algorithm performed better than EMD algorithm in fault diagnosis. Variational mode decomposition addresses the disadvantages of endpoint effects and mode aliasing. Yang et al. [10] used EMD and VMD algorithm to extract the features of the turbine vibration signal respectively, and concluded that the signal processing effect of VMD algorithm was better than that of EMD algorithm. Jiang et al [11] used VMD algorithm to solve the problem of early fault diagnosis, the extraction effect was significantly enhanced, and the performance was better than EMD algorithm, which verified the advantages of VMD in noise robustness. Mohanty et al. [1213] used VMD and EMD algorithms to process the vibration signal of rolling bearings respectively, which proved that VMD can effectively overcome the problem of mode aliasing in the time-frequency analysis algorithm.

 Entropy algorithms can measure the complexity of nonlinear time series of mechanical dynamics, Are widely used in the field of fault diagnosis, Such as the approximate entropy (approximate entropy, AE for short), sample entropy (sample entropy, SE) and permutation entropy (permutation entropy, PE) and other [14], Where the permutation entropy theory is simple, Strong noise resistance ability, Comparcomparative analysis only for adjacent sample points, Can obtain the corresponding feature information, Ability to capture weak changes in the sequence. The above algorithm only considers the complexity and dynamic mutation of a single scale time series, and ignores the useful information of other scales. Therefore, the scholars made the multi-scale coarse-graining of entropy, and proposed the multi-scale entropy algorithm. Literature [15] applies multiscale sample entropy (multiscale sample entro py, MSE) to rotational mechanical feature extraction. The time of sample entropy processing SMS number is not stable enough, and the actual application speed is slow, while the theory of permutation entropy is simple, which can ignore the amplitude factor, and the calculation speed is fast. After the introduction of multiple scales, the advantages can be continued. Pan Zhen et al. [16] introduced the multi-scale arrangement entropy (multiscale permutation entropy, MPE) into the one-way valve fault diagnosis, extracted the MPE feature of the VMD component signal, and realized the one-way valve fault diagnosis. Zhang ancai et al. [17] combined VMD with MPE and input the MPE value of the extracted component into the PSO PNN model to realize the fault diagnosis of rolling bearings. The author introduces the idea of multivariate signal processing into the MPE calculation, considers the vertical and axial vibration signals of rolling mill bearing, calculates the MPE value as the feature vector, proposes MMPE, and applies it to the fault diagnosis of rolling mill rolling bearing.

 Deep networks require a large amount of data for training, with high input vector dimension and long training time. Wang Yujing et al. [18] input spectrum signals into deep confidence network (deep belief network, DBN), and the training time is up to 6 114s. In the condition of small samples, the training data is insufficient, and the deep network training is less effective and slow. In the case of small sample data, SVM makes a compromise on the fitting accuracy of sample data and the complexity of learning, so as to achieve the best generalization ability and avoid the defect problem of neural network falling into the local extreme value. Yang et al. [19] extracted multi-domain features of rolling bearing fault data and formed feature vector, and realized fault diagnosis through SVM. Diego [20] uses the SVM diagnostic model to detect the rolling bearing fault data, distinguish the rolling bearing faults from the normal state, and realize the identification of bearing faults. Yuan Xianfeng et al. [21] used the grey Wolf algorithm to optimize the SVM, and combined with the autoencoder to realize the fault diagnosis of the rolling bearing.

Particle swarm optimization algorithm (particle swarm optimization, PSO) [22] is inspired by the foraging behavior of birds or fish, swarm intelligence optimization method, has good global optimization ability, can be used for hyperparameter optimization of SVM and VMD. The author for the rolling mill bearing vibration signal interference noise, rolling body and retention easy to damage, each direction vibration signal difference is bigger, factory production actual fault data is insufficient, training data sample and slow diagnosis speed, using VMD decomposition combined with envelope spectrum, broken bearing fault, and through the VMD MMPE fault feature extraction and characterization, improve the diagnosis accuracy and speed. Due to the lack of fault data and small data sample, it is difficult to accurately identify the fault site and damage degree of the mill bearing. Therefore, PSO SVM method is adopted to realize the fault diagnosis of different injuries in the same part of the mill bearing.

1 Mathematical model

1.1PSOVMD algorithm

1.1.1VMD algorithm

VMD is a kind of time-frequency signal processing method that can change the scale, which can select the number of decomposition mode components by itself, overcoming the disadvantages of mode mixing and endpoint effect of previous adaptive decomposition algorithm (EMD, EEMD and LMD). Moreover, THE essence of VMD is noise reduction by Wiener filter, which has good noise reduction effect.

The signal is decomposed into several submodes uk, each mode bandwidth is compact distributed in the frequency center, and the bandwidth is estimated through the L2 norm of the gradient. The VMD algorithm procedure is

min { uk }，{ ωk } { ∑k        ∂  t [( δ ( t )+ j πt ) uk ( t ) ] e -jωk t 2 2 } s.t.∑k = 1 K uk = f （1）

Where: uk is each mode, ω k is the center frequency of each mode, K is the number of decomposition layers, and δ (t) is the pulse function.

The Lagrange multiplication operator λ and the quadratic penalty factor α are introduced to solve the constrained variational problem

 L( { uk }，{ ωk }，λ )= α∑k        ∂  t [( δ ( t )+ j πt ) uk ( t ) ] e -jωk t 2 2 +        f ( t )-∑  k uk ( t ) 2 2 + λ( t )，f ( t )-∑k uk ( t ) （2）

 First, the number of decomposition modes is determined, initializing the submode u 1 k, the center frequency corresponding to the submode u 1 k is ω 1 k, the multiplication operator is λ 1, and the initial cycle parameter is n. The decomposition of the original signal into K IMF components is performed as follows.

1) Initialize the u 1 kω1 k λ1, n=0.

2) n=n + 1, performing the entire cycle. The value of k ranges from 1 to K, update uk u n + 1 k (ω) = f (ω) -i = kuk (ω) + λ (ω) 2 1 + α (ω - ω k) 2 (3) according to Equation (3)

Where: un + 1 k (ω), f (ω), λ (ω) and uk (ω) are Fourier transformed forms of un + 1 k (t), f (t), λ (t) and uk (t), respectively. According to update ωk ωkωn + 1 k (4) (0 ω | uk (ω) | 2 d ω 0 | uk (ω) | for all ω> 0 (4), update λ λ n + 1 (ω) = λ n (ω), + τ (f (ω) -k u n + 1 k (ω) (5) (5)

3) Repeat Step 2 until the iteration accuracy ε k  u  n + 1 k-u n k 2 2 /  u  n k <ε (6)

4) The cycle stops, and K IMF components are output.

1.1.2PSO algorithm

m particles are randomly initialized in solution space to form the initial population, and the current position of the i th particle is xi. The particle velocity is initialized as vi, and the velocity determines the motion of the particle. An adaptation value is determined by the objective function, during the iteration the particle will track itself and the current population to find the optimal solution. Let the extreme value found by each particle be Pi and the global pole found by the population Pg, search generation by generation until the optimal solution is obtained. The velocity and position update formulas of the particles are given as follows

vi + 1 = wvi + c 1 r1 ( Pi - xi )+ c2 r2 ( Pg - xi )（7） xi + 1 = xi + vi + 1 （8）

 Where: c1 is the local learning factor; c2 is the global learning factor; w is the inertia factor; r1; r2 is the random number evenly distributed between [0,1].

The PE mean of all components as fitness function optimizes K and α. PE can well reflect the complexity of time series. The smaller the PE value, the more regular the signal sequence and the more vibration shock features; the larger the PE value, the more noise and invalid features. The author selects the first 4 components as the optimal components. During the PSO optimization, the parameters are set as follows: the number of particles is 20; the number of iterations is 50; the local learning factor and the global learning factor are 1.8; and the inertia factor is 0.8.

1.2 Feature extraction based on multivariate multiscale permutation entropy

1.2.1 Multi-scale permutation entropy

 PE provides a quantitative description of the 1 D sequence with good noise resistance. Spatially reconstructed 1 set of time series to obtain Xi [xi, xi + τ,, xi + (m-1) τ] (9)

Where: m is the embedding dimension; and τ is the delay time.

For every Xi, there will be a m! For arrangement, the number of occurrence of any arrangement is Tr, and the corresponding probability of occurrence is Pr = TrN- (m-1) τ (r = 1,2,, R) (10)

The arrangement entropy HPE of different aligned sequences can be defined by the information entropy HPE = -r = 1 R Pr ln Pr (11)

PE = HPE ln (m!) （12）

According to equation (10) ~ (12), it is known that the embedding dimension m and the delay time τ will cause the change of the permutation entropy calculation results. According to experience, the embedding dimension taking 3~7 can achieve good results, and τ usually takes 1.

 The multiscale permutation entropy is essentially coarse-grained time series to obtain new time series. The process is: divide the time series X according to the length s elements; calculate the average of each time series according to Equation (13).y s j = 1 s ∑i =( j - 1 ) s + 1 js xi ( ) 1 ≤ j ≤ N s （13）

 Where: s is the scale factor; N is the original signal length.

Solving the permutation entropy for the new time series yields the multiscale alignment entropy.

1.2.2 Multivariate multi-scale permutation entropy

The MPE can achieve better results when processing the 1-dimensional vibration signal. In view of the particularity of the rolling mill working condition, the axis bears a large radial force and axial force in the rolling process, and there is a big difference between the vertical horizontal axial vibration signal, so the 3-dimensional vibration signal should be considered comprehensively, and MPE handles the signals in three directions in turn and cannot achieve the optimal feature extraction effect. Therefore, using the multivariate multiscale permutation entropy algorithm for multidimensional signal feature extraction can achieve better results.

For the n-dimensional time series Xk, i, the multivariate multiscale coarse-grained sequences are

y s k，j = 1 s ∑i =( j - 1 ) s + 1 js xk，i （k = 1，2，⋯，n）（14）

 The time series is Ym (i) = [y 1, i, y 1, i + τ 1,, y 1, i + (m1-1) τ 1, y 1, y 2, i, y 2, i + τ 2, y 2, i + (m2-1) τ 2,, yd, i, yd, yd, i + τ d,, yd, i + (md-1) τ d (] 15)

 Where: M = [m1, m2,, md] is the embedded dimension vector; Γ = [τ 1, τ 2,, τ d] is the delay time vector.

For the d-dimensional time series Ym (i), find the relative probability of each dimensional data and combine equations (10) and (11) to obtain the final multiscale permutation entropy H MMPE = -j = 1 J Pj ln Pj (16) where: J is the number of permutations common to the multivariate signal sequences, J = km!, And k is the input signal dimension.

The normalized form is MMPE = H MMPE (x, t, m, τ), t = 1s H MMPE (x, t, m, τ) (17)

1.3PSOSVM algorithm

The support vector machine used by the author is part of the LibSVM toolkit, and the appropriate penalty factor C and the kernel function parameter g are required to be determined when using it. Using PSO, the algorithm optimized C and g and takes the SVM recognition accuracy as its fitness function. PSO, the parameters are set as follows: the number of particles is 20; the number of iterations is 50; both local learning factor and global learning factor are 1.8; and inertia factor is 0.8.

1.4 Fault model based on VMD MMPE and PSO SVM

PSO VMD The algorithm combined with envelope analysis can effectively solve the problem of large noise of bearing vibration signal and difficult fault frequency extraction. The preliminary diagnosis of fault bearing can be realized by envelope spectrum, but the diagnosis depends on experience, and the feature extraction and characterization of bearing fault signals can be realized by MMPE algorithm. Therefore, the author combined PSO VMD and MMPE to achieve preliminary diagnosis through VMD and envelope spectrum, and then calculated the MMPE value of each component of the three direction signal, to achieve the characterization of bearing fault characteristics, and formed the feature vector input PSO SVM model for training, to achieve the diagnosis and classification of faults. VMD MMPE, the value optimizes the input of SVM, improves the calculation speed and accuracy, solves the problem of insufficient initial training samples and low diagnostic accuracy, finally establishes a fault diagnosis model combining PSO VMD envelope spectrum, MMPE and PSO optimization SVM, and finally realizes the fault diagnosis of different faults of rolling bearing and different loss degree of the same fault. The fault diagnosis process is shown in Figure 1.

2 Fault diagnosis experiment of rolling roller and holder

 The equipment in this experiment mainly includes experimental rolling mill, sensor and data acquisition equipment. The test platform of plate-strip rolling mill bearing is shown in Figure 2. The mill parameters are as follows: the roll diameter is 120 mm; the length is 90 mm; the main motor speed is 18 r/min; the maximum rolling force is 12 t. The vibration sensor is a YS8202 acceleration sensor, and the pressure sensor model is a HZC 01.

The working roller bearing of the experimental mill is a single cylindrical roller bearing, and the model is NU1012. 8 sets of normal bearings, 2 sets of rolling body wear bearings, 2 sets of damaged holder bearings and 2 sets of rolling body peeling bearings were collected. The 4 kinds of experimental bearings are shown in Figure 3.

During the rolling experiment, the rolling speed was set as 5.04 r/min, collecting the vibration signal of x, y and z axis, the rolling force signal of the rolling drive side and operating side and the torque signal of the upper and lower drive shaft, and the sampling frequency was 2 kHz. The signal time domain diagram is shown in Figure 4. The first 4s rolling mill is in no-load state with smallamplitude. After 51s, the amplitude of steel throwing phenomenon rises sharply, so the signals at both ends need to be discarded during analysis.

Source: Ji Jiang, Zhao Chen, Wang Yongqin