CC BY
4ВЕСТНИК ТСГУ. 2023. № 3 (70)
ПРИБОРЫ И СИСТЕМЫ
YflK 621.822.6
Zou Yingyong, Zhao Wenzhuo, Liu Yupeng, Zhang Ziyang
RESEARCH ON ROLLING BEARING FAULT DIAGNOSIS METHOD BASED ON THE COMBINATION OF OPTIMIZED VMD AND BP NEUTAL NETWORK
Zou Yingyong - College of Machinery and Vehicle Engineering, Changchun University, Changchun, email: [email protected]; Liu Yupeng - College of Machinery and Vehicle Engineering, Changchun University, email: [email protected]; Zhao Wenzhuo - College of Machinery and Vehicle Engineering, Changchun University, email: [email protected]; Zhang Ziyang - College of Machinery and Vehicle Engineering, Changchun University, email: [email protected].
In order to solve the problem that the weak fault signal of the rolling bearing is not obvious, the accuracy of identifying the fault type is not high and the parameters of the variational modal decomposition (VMD) mainly rely on artificial settings, a fault diagnosis method based on sparrow search algorithm (SSA) optimized VMD parameters combined with BP neural network is proposed. First, use the sparrow search algorithm to optimize the number of modal decompositions and penalty factors of the VMD decomposition, and search the global to obtain the optimal parameter combination. the fault signal is decomposed by VMD using the optimized parameters, and the decomposed eigenmode components are imported into the BP neural network for pattern recognition. The results show that compared with EMD and unoptimized VMD, VMD with optimized parameters has a higher fault diagnosis rate of 99.53%, which further improves the fault diagnosis effect.
Keywords: rolling bearing, Sparrow optimization algorithm, variational modal decomposition, BP neural network, fault diagnosis.
1. Introduction
With the rapid development of technology, rolling bearings are widely used in machinery and equipment. At the initial stage of rolling bearing failure, timely identification and diagnosis of rolling bearing faults and replacement can effectively avoid large safety accidents or other economic losses. However, the rolling bearing
© Zou Yingyong, Zhao Wenzhuo, Liu Yupeng, Zhang Ziyang, 2023
BECTHHK TOry. 2023. № 3 (70)
fault vibration signal is often non-linear and non-smooth, so it is difficult to extract the characteristic information of the fault signal.
In 1998, Norden E Huang proposed Empirical Mode Decomposition (EMD) [1], which decomposes the signal into multiple Intrinsic Mode Function (IMF) components and then analyses the different components in the time-frequency domain, compared to wavelet decomposition. Compared with wavelet decomposition, the EMD algorithm does not need to set a basis function and is adaptive. However, the EMD algorithm suffers from endpoint effects and modal aliasing. Later, some scholars have improved it by proposing EEMD [2] and CEEMD [3], which can suppress endpoint effects and modal aliasing but are too computationally intensive.
In 2014, Dragomiretskiy K. proposed Variational Model Decomposition (VMD) [4], which decomposes the signal into a set number of IMF components by introducing a variational model that iterates over a set frequency domain bandwidth. the VMD algorithm greatly improves endpoint effects and is suitable for non- the VMD algorithm greatly improves the endpoint effect and is suitable for non-stationary sequences. The number of decompositions K and the penalty parameter a of the VMD algorithm were selected by Tang Guiji et al. using the particle swarm algorithm, and the improved VMD algorithm has a higher diagnostic accuracy.
2. Introduction to the relevant algorithms 2.1. Variational Modal Decomposition
The VMD algorithm is a fully non-recursive adaptive signal processing tool based on Wiener filtering. VMD can iteratively calculate the bandwidth and centre frequency of multiple components of a complex vibration signal through a varia-tional model, and further calculate the eigenmode component (IMF).
In the solution process, the original signal is assumed to be decomposed into K IMF components. To ensure that the decomposed IMF components are modal components with a central frequency and finite bandwidth, the sum of the decomposed modes is constrained to be equal to the original signal. The constraint expression is as follows.
where, K is the number of modes to be decomposed; uk is the k-th mode component after decomposition;^ is the central freque-ncy of the mode component; f is the initial vibration signal.
To find the optimal solution to equation the Lagrange factor is introduced, the Lagrange augmentation function is obtained by replacing the constrained variational problem with the unconstrained variational problem. As follows.
K
L(uk,vk,A) = a ^ \\d[(s(t) * u^e-^ k=1
2
2
+\\rn - iKk=1uk(t^ + (m,rn - iKk=lUk(t)) (2)
where, A is the Lagrange factor; a is the quadratic penalty factor.
Using the alternating directional multiplier (ADMM) iterative algorithm, optimisation results are obtained for each modal component and centre frequency, and the saddle point of the Lagrange function is found. Multiple iterations of the optimization search yield uk, wk and A.
-jn+l fiM)—T,№kUj{to) + (A(M)/2) , ,
Uk 1+2a(u-uky , (3)
n+1 _ ¡"affi+Harfda
= OK+1(»)\2^, (4)
An+1(^) = An(u) + Y(f((o) - luK+Ku)), (5)
where y is noise tolerance; A is the Fourier transform of the corresponding function.
The iterative process of the VMD algorithm is as follows.
(a) Initialize Uk, Wk, A and the number of iterations N.
(b) Updating the parameters Uk, Wk by substituting the initial values into equations (3) and (4).
(c) Updating X " by equation (5).
(d) Convergence accuracy is determined by the following equation to determine the accuracy.
(6)
If the convergence condition is not satisfied and the maximum number of iterations has not been reached, return to step 2; otherwise complete the iteration and output the final results ilk, ^k.
2.2. Sparrow Search Algorithm
The Sparrow Search Algorithm (SSA) [5], proposed by Jiankai Xue in 2020, is an intelligent search algorithm that simulates the foraging and anti-predatory behaviour of sparrows to build a mathematical model. The algorithm was constructed as follows.
Construct a population of n sparrows.
2
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X =
X11 X21
X12 X22
Xld X2d
(7)
Xn1 xn2 ••• Xnd_
where, d is the dimension of the problem to be optimized. Construct a population fitness function. As follows.
Fx =
fiXii f[X21
X12 X22
X1d] X2d]
Xnd\
(8)
yt+1 _
XiJ =
(9)
.f[xn1 xn2
where f is the moderate value function.
In the sparrow population, the sparrows are divided into two main categories, discoverers and joiners. In the iterative process, discoverers are calculated iteratively according to the following equation.
{ Xi'j ^ eXP i-^—)' tf R2 < ST' [XiJ + Q • L, if R2> ST'
where, t is the number of current iterations; Xij is the position of the i-th sparrow population in the j-th dimension; itermax is the maximum number of iterations; a is a random number from 0 to 1; L is a matrix of 1*d with all elements 1; Q is a random number obeying a normal distribution; R2 is the early warning value of the sparrow population (R2 E [0,1]); ST is the safety value of the sparrow population (ST E [0.5,1]).
When the warning value is less than the safety value, R2<ST, there is no predator in the foraging environment and the finder searches for the optimal solution in this area; when the warning value is greater than the safety value, R2>ST, some sparrows in the environment find the predator and all sparrows need to fly to a safe location to continue foraging.
During the foraging process, some of the joiners will always monitor the discoverer, and when a better solution appears, compare the previous optimal solution and immediately replace the optimal solution if the result is better than the previous one. The iterative function of joiner positions is as follows.
Y i+1 — Xi, j =
\Q •exp
-xf
hj
,if i>n
2
XtP+1 + \XtJ - XtP+1\ •A++L, otherwise
(10)
where, XP is the current discoverer optimal position; XworSf is the global current worst position; A is a 1 xd matrix, with the elements of the matrix all randomized to 1 or -1, A+ = AT(AAT)-1.
x
\2
When i > n/2, the i-th joiner does not get food, is hungry and needs to go to other locations to search for higher energy.
Throughout the population, sparrows that are aware of the danger make up between 10 % and 20 % of the population, the location is randomly generated, and the equation for updating the location after a sparrow engages in anti-predatory behaviour is as follows.
iXbest + P\XiJ — Xbest\, if fi > fg,
xh + ^W^)' 'f f' = f- (11)
where, Xbest is the global currently optimal position; K is a step control parameter obeying a normal distribution; ft is a step control parameter with a random number ft E [-1,1]; fi is the current individual fitness value; fg is the global best fitness value; fw is the global worst fitness value; £ is a constant to avoid the denominator being zero.
When fi> fg, the sparrow is at the edge of the population and vulnerable to predators; when fi > fg, the sparrow in the middle of the population is also at risk of predation and needs to move closer to other sparrows to reduce the risk of predation.
2.3. SSA optimization parameter method
It has been demonstrated in the literature [6,7] using simulated signals that the choice of parameters [K, a] has a significant effect on the VMD decomposition results, and there is no clear rule.
The artificial selection of decomposition parameters can only obtain relatively optimal decomposition results, but not optimal decomposition results. In response to this result, SSA optimization of VMD parameters is proposed to globally search for optimal decomposition results.
Table 1
VMD parameters
Name Symbol Name Symbol
Number of decompositions K Time delay factor T
Penalty factor a Central frequency update parameter DC
Accuracy parameter £ Central frequency initial parameter Init
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3.1 SSA-VMD optimization parameters
To be able to find the best combination of [K, a] accurately, Choose Permutation Entropy (PE) as the optimization function. PE can effectively express the complexity of time series [8, 9]. The calculation procedure is as follows.
(a) Assume that there is a time series of length n as X = {x(1), x(2), •••, x(n)}.
(b) Given the embedding dimension m and a time delay t, take m consecutive samples for each sample point and the signal is reconstructed to obtain the reconstructed vector in m-dimensional space as follows. Xt = [x(i),x(i + 1), — ,x(i + (m-1)* I)}
(c) After arranging the reconstructed vectors in ascending order, we get {jl,)2—jm}.
(d) The order of occurrence after ranking is counted and the probability P^ (k < ml) of occurrence of each case is calculated and normalised to obtain the entropy of ranking, with the following formula.
rr _ (-ZhPiMPd) nr.
H =-- (12)
PE is normalised to better reflect the degree of regularity of the time series. The smaller PE indicates the more regular the time series is; conversely, the more random the time series is [10].
PE was calculated for rolling bearings in four different states, choosing the dimension m and time delay, as shown below.
1
0.95
0.9
S: 0.85 2
W 0.8
c o
3 0.75 S. 0.7 0.65 0.6 0.55
Normal condition Inner ring failure Rolling element failure Outer ring failure
2.5 3 3.5 4 4.5
Dimension Fig.1. Permutation entropy
5.5
start
Initialization parameters Generate a random population of sparrows
Calculating individual sparrow optimisation function values
Output the optimal result
Update individual sparrow positions and calculate the optimization function again
Record the current position of the sparrow and update the optimal result
I
Fig.2. SSA-VMD algorithm flow chart
It can be seen from Figure 1 that PE of the rolling bearing for the four states are most easily distinguished when the dimension is chosen to be 5 and the time delay is chosen to be 1.
The SSA algorithm uses PE as the optimization function, decomposing the initial signal into K eigenmodal components by VMD, and performing the calculation of PE for the different components. The minimum entropy is searched globally throughout the interval, at which point the decomposition parameter is the optimal solution. The SSA-VMD flow chart is shown in Fig. 2.
The specific steps of SSA-VMD are as follows.
(a) Initialize the SSA algorithm and set the parameters of the algorithm.
(b) Introduce the rolling bearing vibration signal, decompose the signal into K eigenmodal components by VMD, each component is a sparrow individual, use the alignment entropy as the optimization function, search for the minimum alignment entropy value, and output the current optimal solution.
3Tl
(c) compare whether the current optimal solution is smaller than the previous optimal solution, if it is not smaller than the previous optimal solution, then search again; conversely, replace the previous sparrow position with the latest position and update the optimal solution.
(d) judge whether the maximum number of iterations is reached, if not, return to step b and loop again; if the maximum number of iterations is met, stop the loop and output the optimal solution.
3.2. Diagnostic process based on the combination of SSA-VMD and BP neural network
The specific steps are as follows.
(a) Acquisition of bearing vibration signals for bearings in non-faulty states, respectively.
(b) optimise the parameters of the VMD algorithm (number of decompositions and penalty factors) using SSA.
(c) VMD decomposition of the signal using the optimized parameters to obtain multiple IMF components.
(d) select the IMF component with the lowest correlation and find its spectrum to form a dataset.
(e) Import the dataset into a BP neural network [11, 12] for training and output the diagnosis results.
The flow chart is shown in Fig. 3.
Rolling bearing failure signals
SSA selects the best k, a VMD decomposition
parameters for VMD
Calculate IMF frequencies and compose them into a data set
BP neural network for fault type identification
Fig.3. Bearing fault diagnosis flowchart
Zou Yingyong, Zhao Wenzhuo,
HECnMC TCry. 2m № 3 (70) Liu Yupeng, Zhang Ziyang
4. Validation of SSA optimization results
4.1 Data analysis
The experimental data in this thesis uses the rolling bearing vibration data collected by Case Western Reserve University. The vibration signals at the drive end are selected, and the vibration signals are divided into four categories: normal signals, inner ring fault signals, rolling element fault signals and outer ring fault signals. Each of the four types of data was taken in 171 groups, with 1000 sampling points in each group.
The following diagram shows the time and frequency domain images of the four types of vibration signals in order.
Normal condition
0.3 t/s
Inner ring failure
I 11nLk^.iLii.ùu, iL^Lti. it.tiiL.ii. i. lIiiLLiLU Li^Lit[..ii.itjti.i>Liiiiiiiii.iui JIiiIi^Il.II
S. 0 1-1
< -0.5
0.2 0.3 0.4
t/s
Rolling element failure
1.1j
0.3 t/s
Outer ring failure
-a 4 -:§ 2 '
I"2
..........................I............................
0.3 t/s
Fig.4. Time domain diagram of rolling bearing signals in four states
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Normal condition
3 0.02 'ä.o.oi
"JI-'-l
3000 Frequency/Hz Inner ring failure
ilJlllilil. lniii.
lill. ubk. .
) 3000 A
Frequency/Hz Rolling element failure
j-i-liLhl.. - -.1 ■
3000 Frequency/Hz Outer ring failure
ä 0.1 's. 0.05 < 0
iL
Frequency/Hz
Fig.5. Frequency domain diagram of rolling bearing signal in four states
Fig. 4 shows the signal time domain diagram, from the diagram can be seen in the normal state of the bearing amplitude is the smallest, the smoothest, while the fault state bearing signal amplitude is much greater than the normal state, there is a periodic shock signal, resulting in a more complex signal. And the noise interference is too large, the fault signal is submerged in the noise signal, making it difficult to directly distinguish the type of fault and the degree of damage. Fig. 5 shows the signal frequency domain diagram, although the normal state signal is clearly distinguished from the fault signal, but there is no obvious distinction between the fault signal, and the noise interference is too large, it is difficult to directly find the fault frequency, so the combination of SSA-VMD and BP is used to diagnose the rolling bearing fault.
4.2. SSA-VMD optimization and comparative validation
Taking the inner circle fault as an example, the grey wolf optimization algorithm GWO and the sparrow optimization algorithm SSA were used to optimize the VMD parameters respectively. The number of populations was set to 50, the number of iterations was 30, the average test was 20, the range of K values was set to [3, 8], the range of a values was set to [500, 3000], and the optimization functions all used the ranking entropy.
The GWO optimization is shown in the following figure.
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Iteration
a)
Convergence urve
fe 0-6111 8
-g 0.611
§ 0 6109
CD
o
$ 0.6108
CO
0.6107
\ ' [-GWO |
15 Iteration
b)
Fig.6. GWO optimized VMD curve
The GWO algorithm is to record the position of the relative optimal solution of the function in the interval and update the interval around the relative optimal solution as the next optimization search interval, so the method leads to the appearance of a local optimal solution. It can be seen from Figure 6 that the first few optimizations of the VMD parameters appear to fall into a local optimum, as shown in Fig. 6, a, with an alignment entropy value of 0.61109, which GWO identifies as reaching the optimal solution, thus only repeatedly confirming in the smaller interval around it, causing it to fall into an optimal solution.
At the 10th iteration of the GWO optimization, its alignment entropy value reaches an optimal solution of 0.61061, the smallest of the 20 test results and the same value as the SSA optimization below, which should be the optimal solution. In the 20 tests, the GWO convergence position is in iterations 7 to 12, and the convergence accuracy is not high, so the SSA optimization parameters are proposed.
The SSA optimization algorithm is not fixed to a certain interval, which greatly avoids the generation of local optimal solutions. In the 20 tests, the SSA algorithm is very stable, and all of them keep converging after 5 iterations, and directly reach the minimum value of the alignment entropy 0.61061, the convergence accuracy is
BECTHHK TOry. 2023. № 3 (70)
obviously higher than that of the GWO algorithm, the following figure shows the optimization curve of the SSA algorithm.
Fig. 7. SSA optimized VMD curve
As with the normal signals, the parameters of the remaining three types of bearing condition signals are also optimized. The following table shows the optimised [K, a] parameters of the four types of bearing condition signals.
Table2
_VMD parameters after optimization_
Type of fault
[K, a]
Normal condition Inner ring failure Rolling element failure Outer ring failure
[8, 2503.7]
[6, 2547.81]
[6, 2647.97]
[8, 520.29]
5. Result identification verification
The signal is decomposed by VMD with optimized parameters and the frequency signal of the IMF component is calculated. The frequency signals are fed directly into a BP neural network for fault diagnosis. Of the input signal spectrum, 60 % is used as the training signal, 20 % as the validation signal and 20 % as the test signal for fault identification of the four types of signals.
The results of EMD decomposition, VMD decomposition without optimization (K=6, a=2000) and the optimized results were used as a control group and the results are shown in the following figures.
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2.5
a
£ 1.5 □
(0 1L
0.5
0 0 0
CD
00 0 (3D
3.5
O.
1.5
-e-1
-e-1
250 300
50 100 150 200 Sample
a) EMD decomposition diagnosis result
Test result
-1-#-i-#-
50
150
100 Sample
b) Unoptimized VMD diagnosis results
200
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Test result
1.5 -
1 ................................—'-1-1-
0 50 100 150 200
Sample
c) Optimize VMD diagnosis results Fig.8. Fault diagnosis results under different decompositions
As can be seen from Figure 8, the EMD diagnostic accuracy is 95.66 %, the unoptimized VMD decomposition diagnostic accuracy is 97.00 % and the optimized diagnostic accuracy is 99.53 %. In comparison, it is clear that the SSA-VMD optimizes the fault diagnosis recognition rate, identifies the type of fault more effectively and improves the accuracy of fault diagnosis for rolling bearings.
Conclusion
In this thesis, SSA-optimized VMD parameters were proposed and combined with BP neural network to achieve fault diagnosis for rolling bearings; the rolling bearing vibration signal disclosed by Case Western Reserve University was used as the data, than it was simulated and verified.
The results show that the SSA-optimized VMD is more effective in identifying rolling bearing faults. Compared with GWO optimization, SSA converges faster, has higher accuracy and is easier to achieve the global optimal solution; compared with EMD and unoptimized VMD, the optimizd VMD has a higher identification rate.
References
1. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis / Huang N.E., Shen Z., Long S.R. et al. // Proceedings Mathematical Physical & Engineering Sciences. 1998. 454 (1971): 903-995.
2. Wu Z., Huang N.E. Ensemble empirical mode decomposition: a noise-assisted data analysis method // Advances in Adaptive Data Analysis. 2011. 1 (01).
3. Yeh J.R., Shieh J.S., Huang N.E. Complementary Ensemble Empirical Mode Decomposition: a Novel Noise Enhanced Data Analysis Method // Advances in Adaptive Data Analysis. 2010. 2 (2).
4. Dragomiretskiy K., Zosso D. Variational Mode Decomposition // IEEE Transactions on Signal Processing. 2014. 62 (3): 531-544.
5. Xue Jiankai. Research and application of a novel swarm intelligence optimization technique. Donghua University, 2020.
6. Fault diagnosis of rolling bearings based on parameter optimization VMD and sample entropy / Liu Jianchang, Quan He Yu, Xia He Kan, Li Zhenhua // Journal of Automation. 2021. Vol. 11.
7. Wang J., Guo S.W. Application of adaptive VMD algorithm in rolling bearing fault diagnostics// Electromechanical Engineering Technology. 2020. 49 (11): 161-164.
8. Zhang Shuli, Wu Zijun. Mining gear fault identification based on improved wavelet thresholding and alignment entropy // Coal Technology. 2021. 40 (10): 196-199.
9. Tian Y., Liu Q.Y. VMD improved multiscale alignment entropy and LLTSA for train wheel damage diagnosis // Mechanical Science and Technology. 2021. 40 (10): 15301535.
10. Wu Zhe, Zhang Qiang, Huang Huameng. Research on fault diagnosis of composite planetary gears based on multi-scale alignment entropy // Mechanical Design and Manufacture. 2020. (09): 182-186.
11. Fault diagnosis of cut-off arm of roadheader based on optimized BP neural network / Liu Qiang, Zhang Chao, Wei Ming, Chen Qing, Li Nuowei // Coal Mining Machinery. 2020. 41 (12): 146-149.
12. Ma Ziyao, Cheng Linlin, Li Yue. A BP neural network-based localization study of RSSI and ultrasound // Computer Age. 2021. (11): 16-20.
Заглавие: Исследование метода диагностики неисправностей подшипников качения на основе сочетания оптимизированного алгоритма VMD и нейронной сети BP
Авторы:
Цзоу Инюн - Чанчуньский университет (КНР) Чжао Вэньчжо - Чанчуньский университет (КНР) Лю Юйпэн - Чанчуньский университет (КНР) Чжан Цзыян - Чанчуньский университет (КНР)
ЗГ|
ВЕСТНИК ТОГУ. 2023. № 3 (70)
Аннотация: При диагностировании подшипников качения приходится решать проблему низкого уровня вибросигнала, влияющего на точность определения типа неисправности невысока. Параметры декомпозиции вариационно-модальной декомпозиции (VMD) в основном зависят от настроек, метода диагностики неисправностей, основанного на оптимизированном алгоритме поиска sparrow (SSA). Предложены параметры VMD в сочетании с нейронной сетью BP. Сначала используется алгоритм поиска Sparrow для оптимизации количества модальных разложений и штрафных коэффициентов декомпозиции VMD и выполняется глобальный поиск для получения оптимальной комбинации параметров. Сигнал неисправности разлагается VMD с использованием оптимизированных параметров и разложенные компоненты собственной моды импортируются в нейронную сеть BP для распознавания образов. Результаты показывают, что по сравнению с EMD и неоптимизированным VMD, VMD с оптимизированными параметрами имеет более высокую точность диагностики неисправностей - 99,53%, что повышает эффективность диагностики неисправностей.
Ключевые слова: Подшипник качения, алгоритм оптимизации Sparrow, вариационно-модальная декомпозиция, нейронная сеть BP, диагностика неисправностей.
