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<—Introduction—>
The diagnostic effect of the rolling bearing fault diagnosis model is affected by two key factors: the completeness of the feature information of the fault feature extraction part and the accuracy and effectiveness of the fault identification part classifier.
In order to improve the diagnostic effect, it is necessary to optimize the algorithm of these two parts, and use the simulation signal to conduct a preliminary verification of their effect to evaluate the performance and determine the algorithm parameters, and finally complete the design of the method.
<—Rolling bearing fault feature extraction method—>
The problem of multi-scale analysis is that it can only capture the low-frequency information of the measured sequence, and there is a loss of some high-frequency information, which will also lead to the loss of amplitude information, which will have a certain impact on the entropy value. To overcome these problems, this paper introduces hierarchy theory into the RCMFDE method. By constructing high-frequency and low-frequency operators, the high-frequency and low-frequency characteristics of the measured sequence are quantified at the same time, and the hierarchical decomposition principle is used to achieve this goal. Figure 3-1 below shows a schematic diagram of the hierarchical decomposition principle.
The HRCMFDE method proposed in this paper can evaluate the hidden dynamic characteristics in the low- and high-frequency components of vibration signals. With this approach, we are able to overcome the loss of amplitude information in coarse-grained processes to analyze the characteristics in the vibration signal more comprehensively.
The calculation steps of HRCMFDE are as follows:
Define the symbols Q0 and Q1, which represent the average operator and the differential operator of the vibration signal, respectively.
The time series {x₁, x₂, ..., x_N} of the vibration signal with length 2n is divided into the high-frequency component Q0(x) and the low-frequency component Q1(x):
The operator Qj can be expressed as a matrix form:
Formula: j ∈ {0,1}^j represents the node vector of level j, and when the number of layers is n, the construction vector ε=[ε₁, ε₂, ..., ε_n] ∈ {0,1}^n. The number of hierarchical nodes e can be calculated by the following formula:
According to the vector ε, obtain the hierarchical component xn,e corresponding to the th node of the nth layer:
The HRCMFDE of the original vibration signal x can be obtained by equation (3-6):
The performance of HRCMFDE is determined by five parameters: embedding dimension m, class number c, time delay d, maximum scale factor max τ, and number of layers n. The choice of parameters directly affects the effectiveness of HRCMFDE in measuring time series.
(1) Embedding dimension m: If m is too small, HRCMFDE cannot accurately observe the dynamic changes of nonlinear time series. Conversely, if m is too large, HRCMFDE cannot detect small changes, and the value range of m is 2~4.
(2) Class number c: When c is too small, two very different amplitude values are assigned for the same class. When c is too large, HRCMFDE is sensitive to noise, and the value range of c is an integer of 3~9.
(3) Time delay d:d has little effect on HRCMFDE, when d > 1, some frequency information may be lost, usually the smallest positive integer 1.
(4) Maximum scale factor max τ: If max τ is too small, HRCMFDE cannot fully extract the features of nonlinear time series. If max τ is too large, it is easy to produce unstable and unreliable entropy values. In addition, a larger max τ may reduce the computational efficiency of HRCMFDE. Therefore, for reliable results, a maximum scale factor of 20 max τ is usually chosen, which is sufficient to efficiently analyze time series.
(5) Number of layers n: For the number of layers n, the larger the n, the shorter the time series of the corresponding layer, which will lead to insufficient extraction of fault information. However, if n is too small, the decomposition of the original time series is incomplete, and the extracted feature dimension is low and insufficient, and n = 3 is generally set.
To investigate the sensitivity of HRCMFDE to the embedding dimension m and class number c, we perform parametric analysis using white noise (WGN) and 1/f noise as simulated signals. Each simulated signal is randomly generated and consists of 20 sets of noise signals with a mean of 0, an variance of 0, and a data length of 3000.
(a) and (b) in the figure show the time domain waveform and spectrum of one of the random WGN signals, while (c) and (d) show the time domain waveform and spectrum of one of the random 1/f noise signals. From the plot, it can be observed that the 1/f noise has poor stationarity, while WGN has higher uncertainty.
In order to further investigate the effects of the embedding dimension m and class number c on HRCMFDE, we discuss two kinds of noise under different values. Take m and c as univariate, and calculate the mean and standard deviation of each level node. Since the value range of m is 2~4, we select M = 2, M = 3 and M = 4; The value range of c is 3~9, and we choose C = 3, C = 6 and C = 9.
The value effect of the embedded dimension m is shown in (a) and (c) in the figure, which show the mean and standard deviation of entropy under WGN and 1/f noise, respectively; (b) and (d) in the figure show the mean and standard deviation of entropy at WGN and 1/f noise. At m = 2, both at WGN and at 1/f noise, the mean curve is smooth and the standard deviation value is minimal.
Considering the effect of the degree of dispersion of the noise data on the results, we will discuss the coefficient of variation CV values for the three m-values. By calculating the ratio of the standard deviation to the mean, the CV value is obtained, and the magnitude of the CV value is inversely proportional to the calculation stability. The results in the table more objectively show the superiority of m = 2 in HRCMFDE.
As shown in (a) and (c) in the figure, the mean and standard deviation of entropy at WGN are shown; Figures (b) and (d) show the mean and standard deviation of entropy at 1/f noise. It can be seen intuitively that at both WGN and 1/f noise, the average value fluctuates the least when c = 6. The standard deviation curve is the most cluttered when c = 3, while the standard deviation curves are closer when c = 6 and c = 9.
The CV values that are further calculated are shown in the results in the table. When c takes 6 and 9, it is not obvious which value is superior, and it can be seen that it has little effect on the stability of entropy. In order to ensure consistency and rigor, we uniformly take c = 6 in subsequent experiments in this article.
To verify the superiority of the proposed HRCMFDE algorithm in terms of stability, we compare four different FDE-derived algorithms. The parameter selection in the experiment is shown in Table 3-5.
Due to the different principles and feature extraction methods of different algorithms, the extracted feature dimensions are also different. Therefore, in the experiment, we conducted two comparisons. One group is HFDE and HRCMFDE, as shown in (a) and (b) in the figure, where the abscissa represents the number of hierarchical nodes as the feature dimension; The other group is MFDE and RCMFDE, as shown in (c) and (d) in the figure, where the abscissa represents the magnitude of the scale factor as the feature dimension.
As shown in 3-6 in the figure, in general, the curves of four different entropy curves are roughly similar under WGN and 1/f noise signals, indicating that the noise signal type has little effect on the stability of the entropy calculation. By comparing subplots (a) and (b), we can see that HFDE is not as stable as HRCMFDE under any noise condition.
This is because HRCMFDE introduces a process of sliding coarse-graining, which makes the information considered more complete, thereby improving the stability of the calculation. Looking at the subplots (c) and (d), we can see that both curves are trending downward. This is because a single coarse-grained process in MFDE and RCMFDE results in incomplete information, resulting in different entropy at each scale.
In addition, comparing subgraphs (a) and (c), and subgraphs (b) and (d), we find that the curve tilt of RCMFDE is significantly improved, indicating that the stability of entropy is improved after the fine recombination algorithm of MFDE. However, compared with HRCMFDE, RCMFDE still has some shortcomings, which verifies the contribution of the introduction of hierarchy theory to solving the problem of information loss caused by coarse-grainedness.
When measuring nonlinear dynamic changes, HRCMFDE has the lowest error rate and performs better than MFDE, HFDE, and RCMFDE.
<—Rolling bearing fault identification method—>
The PSO algorithm was first proposed by Eberhart et al. in 1995 and was inspired by the information-sharing behavior of social animals such as birds. It finds the best solution to a problem by sharing information about individuals in the group. The flow of the PSO algorithm is shown in Figure 3-7.
Suppose there are n particles in space, each with D dimensions. They form a group. Each particle contains two basic properties, position and velocity, where position represents a possible solution to a problem. In the tth iteration, the position and velocity of the i-th particle are expressed as:
You can assess whether a particle has reached a good position by calculating the fitness value of the particle's current position. In each iteration of particle swarm optimization, the particles update their velocity according to the formula (3-9):
where j ∈ [1, D], ω represents the inertia weight, (g_t) _ij represents the global optimal position of the population at the tth iteration, and (p_t) _ij represents the individual optimal position of the ith particle. C1 and C2 represent the acceleration constant, respectively, and their range is between 0~2. (i_1) _r and (i_2) _r represent random numbers in the range [0,1].
According to the formula
to adjust its speed
To avoid blind searching for particles, the position and velocity of particles are limited to [min[X,X]] and [max[V,V]]. It is found that by adjusting the inertia weight ω, the global search and the local search can be balanced. When the value of ω is large, it can help the particle avoid falling into the local minimum. When the ω value is small, it can promote the convergence of the algorithm. Therefore, the concept of adaptive inertia weights is introduced, i.e. as the iteration progresses, ω decreases linearly according to the formula (3-11).
where maxω represents the initial inertia weight, minω represents the final inertia weight, t represents the current number of iterations, and maxt represents the maximum value of the iteration.
For further analysis of the optimization ability of the PSO algorithm, 4 classical test functions are used in this section to evaluate its performance. In the experiment, the parameter selection of the PSO algorithm is shown in Table 3-7. where pN represents the population size, maxT represents the maximum number of iterations, c1 and c2 are the acceleration coefficients, maxV and minV are the initial velocity range, and ω is the inertia coefficient.
The four test functions introduced include the Sphere, Rosenbrock, Ackley, and Rastrigin functions. Their function expressions and related parameter settings are shown in Table 3-8. The dimension d of these test functions is set to 30, and the search space and target optimal value are set according to the information in the table. When the target optimal value is reached, it is considered successful convergence.
According to Table 3-7, the parameters were set, and the PSO optimization algorithm was used to perform 30 experiments on the above four test functions. The average optimal fitness values for the convergence process of each test function are calculated and the convergence curves are plotted logarithmic in base 10, as shown in Figures 3-8 a), b), c) and d).
From the convergence curve of each test function, it can be observed that the convergence speed of the PSO algorithm is quite fast. In order to further study its convergence performance, the average number of convergences of the algorithm successfully convergence (that is, reaching the target optimal value) is recorded to reflect the convergence speed of the algorithm. The results are shown in Table 3-9.
It can be seen that these four test functions have completed convergence before they are halfway through. In addition, the average optimal fitness value and standard deviation after 1000 iterations were calculated. These data are used to illustrate the convergence stability of the PSO algorithm regardless of the target optimal value.
The PSO algorithm not only shows excellent ability in convergence speed, but also shows excellent performance in convergence stability. Therefore, it can be used as an efficient optimization tool to improve the fault identification ability of classifiers.
<—Epilogue—>
In the simulation experiment, we randomly generate two noise signals, discuss the parameter settings of the HRCMFDE algorithm, and compare them with other FDE-based feature extraction methods. The results show that the HRCMFDE algorithm shows considerable results.
We also analyze the performance of the PSO optimization algorithm in the PSO-ELM classifier, using four classical test functions to evaluate its performance. In terms of convergence speed and convergence stability, PSO algorithm shows significant advantages, which provides a guarantee for the effect of fault identification.
Finally, the HRCMFDE feature extraction method and the PSO-ELM fault identification method are combined to construct a rolling bearing fault diagnosis model.
<—References—>
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ZHANG Long, CHENG Junliang, LI Xinglin, et al. Quantitative evaluation of rolling bearing faults based on adaptive band impact strength[J]. Journal of Vibration and Shock, 2018, 37(19): 30-38.
[3] Wen Xin. Research on early fault diagnosis technology of rolling bearings based on wavelet packet decomposition and graph theory[D].Shandong University, 2020.)
[4] LIU Misty Moon. Research on motor bearing fault diagnosis method based on vibration signal[D].Harbin Institute of Technology, 2016.)
WU Haibin, CHEN Yinsheng, ZHANG Tinghao, et al. Improved fault diagnosis of rolling bearings combining multi-scale amplitude-sensing permutation entropy with random forest[J]. Optics and Precision Engineering, 2020, 28(03): 621-631.
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