# Fatigue Reliability Analysis of Motor Hanger for High-Speed Train Based on Bayesian Updating and Subset Simulation

Authors: Yonghua Li; Pengpeng Zhi; Yue Zhang; Bingzhi Chen; Yuedong Wang
Journal: Advances in Materials Science and Engineering
 (2020)

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## Abstract

In order to more accurately analyze the fatigue reliability of motor hanger for high-speed train and reduce the influence of uncertain factors, a Bayesian statistical method is introduced to propose a novel fatigue reliability analysis method based on Bayesian updating and subset simulation. First, considering the influence of various uncertain parameters on the first principal stress (FPS) of motor hanger, the ANSYS parametric design language (APDL) is used to establish the parametric model. The D-optimal design of experiment is carried out to calculate the FPS of the motor hanger. Second, the experimental data is fitted by the least square method to establish a polynomial response surface function which characterizes the FPS of the motor hanger, and analysis of variance (ANOVA) is carried out. On this basis, the variation trend of the FPS under parameter fluctuation is calculated, and its probability distribution characteristics are obtained. Based on the MATLAB platform, the Bayesian updating method is adopted to correct the probability and statistical characteristics of the FPS to improve the accuracy of prediction. Finally, the subset simulation (SS) method is used to calculate the fatigue failure probability of the motor hanger. The research results show that the proposed method helps to improve the accuracy and efficiency of fatigue reliability analysis.


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## Body

## 1. Introduction

In the design process of mechanical products, the analysis of structural static strength and fatigue strength is the key to ensure that the product meets the standard requirements. However, with the continuous improvement of the operation speed of high-speed trains, in order to ensure the safety and reliability of operation, the design of its key components must meet the requirements of antifatigue design and reliability [1]. Currently, the evaluation of the fatigue performance of key components for high-speed trains is mainly based on fatigue strength and cumulative fatigue damage. The evaluation index can be used to perform fatigue strength analysis and fatigue life prediction on the structure quickly and easily and is easy to compare with the experimental results. But, the disadvantages of this method are that the results of evaluation and analysis are too ideal to consider the design tolerance and manufacturing error, and the comparison with the experimental results is too single and not universal, which is not enough to reflect the fatigue reliability of all products. In response to the above problems, some scholars have combined the reliability theory and fatigue analysis theory to study the fatigue reliability of mechanical products and proposed some fatigue reliability analysis methods [2–12]. Chen et al. [13] proposed a fatigue reliability analysis method that considers the uncertainty of parameters, draws a fatigue limit diagram without safety factor, and gives a more reasonable fatigue reliability analysis result, which is helpful for lightweight design of products. Li et al. [14] established a robust optimization model based on Six Sigma to analyze the fatigue reliability of the pantograph collector head support. This method not only considers the influence of uncertain factors, but also improves the robustness of fatigue life prediction. Bayraktar et al. [15] considering the number of kilometers and load cycles, statistically evaluated the actual life value. And the theoretical and practical Wohler diagrams S-N is plotted to evaluate the fatigue reliability of the axle. Zhu et al. [16] presented a framework for fatigue reliability assessments and service life prediction based on the estimation of the evolution and probabilistic distribution of fatigue damage over time, and the effectiveness of the method is verified by taking the axle of train as an example. An accelerated life test (ALT) method is proposed by Lu et al. which is first employed to predict the fatigue life of a full-scale bogie frame [17]. This method provides a reference for reliability evaluation of large structures. Zuo et al. [18] studied the fatigue damage of the bogie frame and used the Bayesian updating method to analyze the fatigue reliability of the welded bogie frame. Hu et al. [19] combined finite element method with experiment to evaluate the fatigue reliability of bolster under the application of the design passenger number spectra based on stress-life interference model. The above studies have played an important role in fostering applications of reliability theory for fatigue problems in the structural safety field. For the design process, how to accurately evaluate the fatigue reliability of the structure without experimentation is the premise of predicting whether the design meets the requirements of the standard. Generally, reliability methods based on simulation or approximation have three types: (1) sampling-based methods, (2) moment matching methods, and (3) MPP-based methods. Sampling-based methods are easy to apply and can provide accurate probability estimations with sufficient simulations [20]. Thus, a Bayesian updating combined with subset simulation method is proposed in this article to address the problem.The contributions of this study are as follows: (1) a response surface function is presented to characterize the functional relationship between random variables and the FPS of the structure; (2) the probabilistic and statistical parameters of the FPS are updated by using Bayesian theory; (3) the limit state equation of fatigue reliability analysis is established; (4) the fatigue failure probability of the motor hanger is calculated by the subset simulation method.The reminder of the paper is organized as follows. Section2 reviews Bayesian updating and subset simulation method; we used both methods in fatigue reliability analysis. Section 3 provides details on the proposed approach. Section 4 uses the fatigue reliability analysis of the motor hanger as an example to illustrate its effectiveness and practicality. Finally, Section 5 summarizes and concludes.


## 2. Bayesian Updating Theory and Subset Simulation Method

### 2.1. Bayesian Updating Theory

In engineering practice, it is often necessary to make decisions and processes on the data that has already been obtained. These data may include observational data, experimental data, and simulation data. When acquiring new data, it is necessary to update the acquired data to ensure the reliability of the data. In view of the above problems, this paper uses Bayesian method to update and process the uncertainty and difference of simulation data [21, 22].The basic principle of Bayesian theory can be roughly expressed as using prior distribution and sample information to solve its posterior distribution; that is, the posterior distribution is obtained based on prior information and experimental data [23]. The specific steps for solving it are as follows:(1)
Determine the parameters that need to be solved and set them as random variables(2)
According to the sample data, the corresponding likelihood function and prior distribution function are constructed(3)
The posterior distribution function is obtained by Bayes equation, and the posterior distribution function is used to solve various reliability indexesBecause Bayes method makes full use of prior information and experimental data, the posterior distribution function is more consistent with the probability and statistical characteristics of random variables. The traditional Bayesian equation is given by(1)πθx=fxθπθpx,where πθx is the posterior distribution function; πθ is the prior distribution function; and px is the edge distribution function, and the two types are px=∑θ∈Θfxθπθ and px=∫θ∈Θfxθπθdθ, respectively.The distribution types of design parameters of key components for high-speed railway are mainly lognormal distribution and normal distribution, and the two probability distributions can be converted to each other. However, the FPS used for fatigue reliability analysis mainly obeys logarithmic distribution. Therefore, the prior distribution of lognormal distribution should be transformed into the prior distribution of normal distribution. The transformation equation can be expressed as(2)μ1=lnμ0−σ122,(3)σ1=ln1+σ0μ02,where μ0 and σ0 represent the mean and standard deviation of prior lognormal distribution, respectively, and μ1 and σ1 represent the mean and standard deviation of prior normal distribution, respectively.According to Bayesian theory [24], the posterior distribution function of normal distribution can be obtained from equations (2) and (3), which is written as(4)μ2=μ′σ12+μ1σ′2/n′+σθ2σ12+σ′2/n′+σθ2,(5)σ2=σ1σ′2/n′+σθ2σ12+σ′2/n′+σθ2,where μ2 and σ2 represent the mean and standard deviation of posterior normal distribution, respectively; μ′ and σ′ represent the mean and standard deviation of the normal distribution actually obtained from the evaluated parameter, respectively; n′ represents the number of statistical samples; and σθ represents the standard deviation caused by uncertainties.Combining equations (2)–(5), the statistical characteristic parameters expression of the evaluated parameters is obtained by Bayesian updating, which is given by(6)μ3=expμ2+σ222,(7)σ3=μ3expσ22−1,where μ3 and σ3 represent the mean and standard deviation of lognormal distribution after Bayesian updating, respectively.


### 2.2. Subset Simulation Method

The subset simulation method was first proposed by Au and Beck [25], and it is a method that can efficiently deal with high-dimensional and small probability problems. The method can represent the failure probability with a small value as the product of a series of large condition failure probabilities; that is, the simulation process of the small failure probability event is transformed into a series of large event simulation processes. If the intermediate failure event satisfies the nested relationship (F1⊃F2⊃⋯⊃Fm=F), the target failure probability can be rewritten as(8)PF=PF=PFm=PFmFm−1PFm−1=⋯=PF1∏i=2mPFiFi−1,where F is the target failure event, F=gX≤b, and b is the critical value of the structural response.In the reliability analysis, subset simulation starts with direct Monte Carlo in the first step. The probabilityP1 is estimated as(9)PF1=P˜1=1N∑k=1N′IF1xk,(10)IF1=1,xk∈F1,0,xk∉F1,where xkk=1,2,…,N′ is an independent and identically distributed sample generated by simulation of joint probability density function qx with input random vectors; N is the number of sample points per layer; and IF1 is indicator function.Similarly, the other conditional probabilities are calculated by the idea of equation (9). In the modified Metropolis algorithm, a group of one-dimensional proposal PDFs are used, instead of an n-dimensional proposal PDF which is used in Metropolis algorithm. Thus, the acceptance ratio of individual sample can remain nonvanishing in spite of the increasing of dimension [26, 27]. Thus, in order to efficiently generate conditional samples, this paper implements MCMC simulation based on improved Metropolis–Hastings algorithm.


## 2.1. Bayesian Updating Theory

In engineering practice, it is often necessary to make decisions and processes on the data that has already been obtained. These data may include observational data, experimental data, and simulation data. When acquiring new data, it is necessary to update the acquired data to ensure the reliability of the data. In view of the above problems, this paper uses Bayesian method to update and process the uncertainty and difference of simulation data [21, 22].The basic principle of Bayesian theory can be roughly expressed as using prior distribution and sample information to solve its posterior distribution; that is, the posterior distribution is obtained based on prior information and experimental data [23]. The specific steps for solving it are as follows:(1)
Determine the parameters that need to be solved and set them as random variables(2)
According to the sample data, the corresponding likelihood function and prior distribution function are constructed(3)
The posterior distribution function is obtained by Bayes equation, and the posterior distribution function is used to solve various reliability indexesBecause Bayes method makes full use of prior information and experimental data, the posterior distribution function is more consistent with the probability and statistical characteristics of random variables. The traditional Bayesian equation is given by(1)πθx=fxθπθpx,where πθx is the posterior distribution function; πθ is the prior distribution function; and px is the edge distribution function, and the two types are px=∑θ∈Θfxθπθ and px=∫θ∈Θfxθπθdθ, respectively.The distribution types of design parameters of key components for high-speed railway are mainly lognormal distribution and normal distribution, and the two probability distributions can be converted to each other. However, the FPS used for fatigue reliability analysis mainly obeys logarithmic distribution. Therefore, the prior distribution of lognormal distribution should be transformed into the prior distribution of normal distribution. The transformation equation can be expressed as(2)μ1=lnμ0−σ122,(3)σ1=ln1+σ0μ02,where μ0 and σ0 represent the mean and standard deviation of prior lognormal distribution, respectively, and μ1 and σ1 represent the mean and standard deviation of prior normal distribution, respectively.According to Bayesian theory [24], the posterior distribution function of normal distribution can be obtained from equations (2) and (3), which is written as(4)μ2=μ′σ12+μ1σ′2/n′+σθ2σ12+σ′2/n′+σθ2,(5)σ2=σ1σ′2/n′+σθ2σ12+σ′2/n′+σθ2,where μ2 and σ2 represent the mean and standard deviation of posterior normal distribution, respectively; μ′ and σ′ represent the mean and standard deviation of the normal distribution actually obtained from the evaluated parameter, respectively; n′ represents the number of statistical samples; and σθ represents the standard deviation caused by uncertainties.Combining equations (2)–(5), the statistical characteristic parameters expression of the evaluated parameters is obtained by Bayesian updating, which is given by(6)μ3=expμ2+σ222,(7)σ3=μ3expσ22−1,where μ3 and σ3 represent the mean and standard deviation of lognormal distribution after Bayesian updating, respectively.


## 2.2. Subset Simulation Method

The subset simulation method was first proposed by Au and Beck [25], and it is a method that can efficiently deal with high-dimensional and small probability problems. The method can represent the failure probability with a small value as the product of a series of large condition failure probabilities; that is, the simulation process of the small failure probability event is transformed into a series of large event simulation processes. If the intermediate failure event satisfies the nested relationship (F1⊃F2⊃⋯⊃Fm=F), the target failure probability can be rewritten as(8)PF=PF=PFm=PFmFm−1PFm−1=⋯=PF1∏i=2mPFiFi−1,where F is the target failure event, F=gX≤b, and b is the critical value of the structural response.In the reliability analysis, subset simulation starts with direct Monte Carlo in the first step. The probabilityP1 is estimated as(9)PF1=P˜1=1N∑k=1N′IF1xk,(10)IF1=1,xk∈F1,0,xk∉F1,where xkk=1,2,…,N′ is an independent and identically distributed sample generated by simulation of joint probability density function qx with input random vectors; N is the number of sample points per layer; and IF1 is indicator function.Similarly, the other conditional probabilities are calculated by the idea of equation (9). In the modified Metropolis algorithm, a group of one-dimensional proposal PDFs are used, instead of an n-dimensional proposal PDF which is used in Metropolis algorithm. Thus, the acceptance ratio of individual sample can remain nonvanishing in spite of the increasing of dimension [26, 27]. Thus, in order to efficiently generate conditional samples, this paper implements MCMC simulation based on improved Metropolis–Hastings algorithm.


## 3. The Analysis Method of Fatigue Reliability

### 3.1. Fatigue Life Prediction

Fatigue, as the main reason for failure of engineering structures and components, is the main factor affecting the safety of high-speed train operation. Therefore, more and more attention has been paid to antifatigue design in the design process. At present, the prediction of structural fatigue life is mainly based on theS-N curve or the P-S-N curve. The S-N curve represents the stress-life curve, and the P-S-N curve is the stress-life curve of the material under different survival conditions. The most commonly used form of S-N curve describing materials is the power function form, as follows:(11)S−m×N=C.Equation (11) can be rewritten as(12)logC−mlogS=logN,where m and C are parameters related to material, stress ratio, and loading mode; S represents stress; and N represents lifetime.According to the design requirements, the material suitable for the motor hanger of high-speed railway trains is S355J2G. However, the mechanical properties of S355J2G are difficult to obtain. For this reason, the material parameters of Q345 similar to its mechanical properties are selected for fatigue reliability analysis [28]. Table 1 gives the main parameters of the material P-S-N curve, and the P-S-N curve of the material is plotted according to its parameters, as detailed in Figure 1.Table 1
P-S-N parameter estimation results of Q345 steel.
Survival rateP
0.50
0.90
0.95
0.99
C
1.84655
×
10
33
10
30.008
10
29.202
10
27.650
m
11.481
10.357
10.078
9.543Figure 1
P-S-N curve of Q345 steel.


### 3.2. Limit State Equation for Fatigue Reliability Analysis

In this paper, the fatigue lifetime model is selected for fatigue reliability analysis. The model uses fatigue lifetime as the parameter for reliability calculation. The limit state equation can be expressed as(13)Z=N−N0=>0,=0,<0,where N is the calculated fatigue lifetime and N0 is the design lifetime; when Z > 0, the structure is in a safe state; when Z = 0, the structure is in a limit state; when Z < 0, the structure is in a failure state.From the definition of reliability, the fatigue reliability of the structure is the probability of the functionZ > 0, and the expression of the reliability is(14)R=PNZ>0.The failure probability of a structure can be expressed as(15)Pf=1−R=1−PNZ<0.


## 3.1. Fatigue Life Prediction

Fatigue, as the main reason for failure of engineering structures and components, is the main factor affecting the safety of high-speed train operation. Therefore, more and more attention has been paid to antifatigue design in the design process. At present, the prediction of structural fatigue life is mainly based on theS-N curve or the P-S-N curve. The S-N curve represents the stress-life curve, and the P-S-N curve is the stress-life curve of the material under different survival conditions. The most commonly used form of S-N curve describing materials is the power function form, as follows:(11)S−m×N=C.Equation (11) can be rewritten as(12)logC−mlogS=logN,where m and C are parameters related to material, stress ratio, and loading mode; S represents stress; and N represents lifetime.According to the design requirements, the material suitable for the motor hanger of high-speed railway trains is S355J2G. However, the mechanical properties of S355J2G are difficult to obtain. For this reason, the material parameters of Q345 similar to its mechanical properties are selected for fatigue reliability analysis [28]. Table 1 gives the main parameters of the material P-S-N curve, and the P-S-N curve of the material is plotted according to its parameters, as detailed in Figure 1.Table 1
P-S-N parameter estimation results of Q345 steel.
Survival rateP
0.50
0.90
0.95
0.99
C
1.84655
×
10
33
10
30.008
10
29.202
10
27.650
m
11.481
10.357
10.078
9.543Figure 1
P-S-N curve of Q345 steel.


## 3.2. Limit State Equation for Fatigue Reliability Analysis

In this paper, the fatigue lifetime model is selected for fatigue reliability analysis. The model uses fatigue lifetime as the parameter for reliability calculation. The limit state equation can be expressed as(13)Z=N−N0=>0,=0,<0,where N is the calculated fatigue lifetime and N0 is the design lifetime; when Z > 0, the structure is in a safe state; when Z = 0, the structure is in a limit state; when Z < 0, the structure is in a failure state.From the definition of reliability, the fatigue reliability of the structure is the probability of the functionZ > 0, and the expression of the reliability is(14)R=PNZ>0.The failure probability of a structure can be expressed as(15)Pf=1−R=1−PNZ<0.


## 4. Fatigue Reliability Analysis of Motor Hanger for High-Speed Train

Fatigue reliability analysis is an analytical method to improve product quality by combining fatigue lifetime analysis and reliability design. It is essential to ensure that the structural design meets the standard requirements. As the load-bearing structure between the motor and the bogie frame, whether the performance and service time of the structure can meet the requirements is directly related to the normal operation of the train. To this end, this paper takes the motor hanger as the research object and uses Bayesian updating and subset simulation to explore its fatigue reliability, which provides a new idea and method for the fatigue life prediction in the design stage. The specific process of fatigue reliability analysis is provided in Figure2.Figure 2
The analysis process of fatigue reliability.


### 4.1. Finite Element Analysis of Motor Hanger

The premise of fatigue reliability analysis for motor hanger is to calculate its FPS under fatigue conditions. The research object of this paper is the motor hanger of high-speed train, which is bolted to the bogie frame to fix the motor. The motor hanger is mainly plate welded, and the partial opening is used to reduce the quality. Figure3 shows the specific installation position and structure type. The numerical analysis by the finite element method is performed to obtain the FPS of the motor hanger. It is modeled by shell and solid mode, given in Figure 4. The motor hanger is modeled using beam188, shell181, solid 185, rigid, and rbe3 elements. It is meshed to have 50,879 elements, including 21,766 triangular and quadrilateral shell elements and 28,973 tetrahedral and hexahedral solid elements.Figure 3
Geometric model of motor hanger.Figure 4
Finite element model of motor hanger.In the finite element model, the reasonable application of load cases is the key to ensure the accuracy of the analysis results. Generally, the boundary conditions of the test components under normal operating conditions should be consistent with those under test assembly conditions. The elastic connector on the bogie can be replaced by a rigid member, but the direction and magnitude of the load acting on the test piece cannot be changed. Figure5 displays the load direction of motor hanger, which is positive upward. The loads are calculated according to EN13749 standard and analyzed by finite element method. Figure 6 plots the result of the finite element analysis. It can be clearly seen from the figure that the maximum FPS is 119.23 MPa, which occurs on the motor boom.Figure 5
Load direction of motor hanger.Figure 6
FPS plot of motor hanger.


### 4.2. Establishment of Response Surface Function

The purpose of establishing the response surface function is to obtain the functional relationship between the random variables and the FPS and then to observe the influence of the uncertainties on it. As one of the surrogate models, the response surface can more accurately represent the functional relationship between input and output. Compared to other surrogate models, the response surface belongs to the display function that can be more easily used for reliability calculations [29]. Due to the poor approximation ability of the linear response surface, the quadratic response surface with cross terms is adopted. The basic equation is given by(16)y=∑i=1nciixi2+∑i>jncijxixj+∑i=1ncixi+c0,where n is the number of uncertain variables, c0 is the constant, and ci, cii, and cij are the polynomial coefficients, respectively.According to the stress plot in Figure6, the random variables which have great influence on the stress plot are selected as random variables. Considering the uncertainty of design parameters, the upper and lower limits of size parameters are set according to tolerance, and the upper and lower limits of load are obtained according to test conditions. Table 2 displays the specific parameters for random variables.Table 2
Statistical characteristic of random variables.
Design parameters
Sign
Unit
Lower limit
Mean value
Upper limit
Steel plate thickness 1
TH1
mm
7.7
8.0
8.3
Steel plate thickness 2
TH2
mm
9.7
10.0
10.3
Vertical load 1
F
1
N
23040.0
25600.0
28160.0
Vertical load 2
F
2
N
23040.0
25600.0
28160.0The premise of establishing response surface function is to do the design of experiment for random variables. According to the design of experiment scheme, the finite element model is invoked to solve the problem, and the corresponding response of each scheme is obtained. In this paper, the D-optimal design of experiment method is used to carry out the experimental design; see Table3.Table 3
Statistical characteristic of random variables.
Experiment number
Factors
Response
TH1 (mm)
TH2 (mm)
F
1 (N)
F
2 (N)
S1 (MPa)
1
7.967
10.300
25625.600
26880.000
143.48
2
7.916
9.700
23040.000
25472.000
144.88
3
7.700
9.700
28160.000
23040.000
149.18
4
7.925
10.300
25676.800
23040.000
147.27
5
8.300
9.700
23961.600
23040.000
137.75
6
7.979
9.967
23756.800
28160.000
143.39
7
7.700
10.300
23040.000
28160.000
148.68
8
8.300
10.300
23040.000
23040.000
137.42
9
8.060
9.700
26368.000
24925.400
142.13
10
8.300
10.300
28160.000
28160.000
137.42
…
…
…
…
…
…
18
7.916
9.988
28160.000
28160.000
144.68
19
7.937
10.021
28160.000
24003.000
144.25
20
8.300
9.700
23040.000
28160.000
137.75
21
8.300
10.186
28160.000
23040.000
137.49
22
8.300
10.063
25011.200
26137.600
137.56
23
7.700
10.300
28160.000
25651.200
148.68
24
7.700
10.300
23040.000
28160.000
148.68
25
8.300
10.300
28160.000
28160.000
137.42According to (16), the experimental data in Table 3 is fitted by least squares method to obtain a response surface function, which characterizes the function relationship between the random variable and the FPS.(17)S1=+426.60+43.658×TH1−73.68×TH2+3.457×10−3×F1−4.444×10−3×F2−2.50×TH1×TH2+1.6×10−4×TH1×F1+3.99×10−4×TH1×F2+4.929×10−5×TH2×F1−4.103×10−4×TH2×F2−2.909×10−9×F1×F2−3.265×TH12+5.156×TH22−10−7×F12+1.036×10−7×F22.The fitting accuracy of the response surface function is the basis and premise to ensure the accuracy of the subsequent reliability analysis. According to the fitting resultR2=0.9964, indicating that the fitting accuracy is high. To further verify the accuracy of the response surface function, the ANOVA is adopted to analyze the function in this paper. Some results are provided in Table 4.Table 4
ANOVA for response surfaceS1 quadratic model.
Source model
Sum of squares
Df
Mean square
F value
P-value
Model
567.09
14
40.51
199.31
<0.0001
Significant
TH1
391.63
1
391.63
1927.02
<0.0001
TH2
0.065
1
0.065
0.32
0.5833
F
1
0.10
1
0.10
0.50
0.4948
F
2
1.36
1
1.36
6.69
0.0271
…
…
…
…
…
…
TH
1
2
0.26
1
0.26
1.30
0.2805
TH
2
2
0.76
1
0.76
3.74
0.0820
F
1
2
1.28
1
1.28
6.31
0.0308
F
2
2
1.29
1
1.29
6.34
0.0305
Residual
2.03
10
0.20
Lack of fit
2.03
5
0.41
Pure error
0.000
5
0.000
Cor total
569.12
25In Table4, the F and P values are the main indicators used to evaluate the significance of the response surface function and its random variables. The larger the F value, the more significant it is; on the contrary, the smaller the P value, the more significant it is. As can be seen from Table 4, the function generally presents a very significant level, P<0.0001, indicating that the fitting accuracy of response surface is high. Among the four random variables, random variable TH1 is the most significant, which indicates that the small fluctuation of the random variable TH1 has a greater impact on the FPS. The other three random variables did not exhibit a very significant level, only that their small fluctuations had little effect on the FPS. Because of space, the interaction among random variables cannot be given. Through analysis, it is found that the interaction among random variables in response surface function is relatively low, which indicates that the interaction between random variables has less influence on response. By comparing the results of the first and second significance of random variables, it can be seen that the second significance of TH1 decreases and the second significance of TH2, F1, and F2 increases, indicating that the selection of random variables is reasonable. In summary, the response surface function can be used for fatigue reliability analysis of subsequent motor hanger.


### 4.3. Determination of the FPS Distribution Type and Bayesian Updating

For equation (13), the uncertainty of the calculated lifetime N results in three different states of the structure. As the main factor affecting the calculated lifetime N, accurate probability distribution of the FPS is the key to improve the accuracy of reliability analysis. For this reason, the FPS under random parameter fluctuation is obtained by using Monte Carlo method for 10000 calculations based on equation (17). The FPS is fitted by MATLAB 2015b to determine its probability distribution characteristics, as shown in Figure 7. From Figure 7, it can be seen that the FPS obeys lognormal distribution, and the fitting curve has high accuracy, R2=0.99349. In order to accurately estimate the fatigue reliability of the structure, simulation data are used as real measurement data for Bayesian updating because there is no experimental data in the design process. The prior distribution data of the FPS are obtained from the experimental data of similar products. The data of prior distribution obey the lognormal distribution of N4.945,0.041, and the real measured data obey the lognormal distribution of N4.962,0.0425. Two kinds of distribution data are substituted into equations (2)–(7) to obtain the probability and statistical parameters of the FPS after Bayesian updating. Figure 8 plots the probability distribution of the FPS before and after Bayesian updating. It can be seen from Figure 8 that the mean of FPS value after the Bayesian updating is slightly smaller than that of the FPS obtained by the actual engineering, and the corresponding variance is also reduced. This proves that the Bayesian updating method not only reduces the uncertainty of FPS, but also reduces the distortion and uncertainty of the detected data. In the design sense, Bayesian updating reduces the uncertainty of the simulation results.Figure 7
Lognormal distribution fitting of FPS.Figure 8
Comparison of probability and statistics characteristics of FPS.


### 4.4. Fatigue Reliability Analysis Based on Subset Simulation

The FPS of the motor hanger after Bayesian updating obeys the lognormal distribution ofN4.9451,0.0409. The symmetrical cyclic load spectrum specified in the EN13749 standard is selected, and the stress ratio is R = -1. The P-S-N curves with survival rates of 95% and 99% are selected, respectively, and the design lifetime is 10 million cycles. According to equations (11)–(13), the fatigue failure probability of the motor hanger is calculated by SS. Figures 9 and 10 display the CDF curves for fatigue failure probabilities at different survival rates, respectively. The estimated fatigue failure probabilities and the number of required samples are listed in Table 5.Figure 9
CDF of survival rate at 95%.Figure 10
CDF of survival rate at 99%.Table 5
Estimated failure probabilities and the number of samples with differentp0.
Survival rateP (%)
Conditional failure probabilityp0
0.10
0.15
0.20
0.25
0.30
95
Failure probabilityPf
0.005
0.0055
0.005
0.005
0.0086
Number of samples
560
540
680
650
620
99
Failure probabilityPf
0.1
0.15
0.1950
0.1950
0.2340
Number of samples
380
370
360
350
340It can be seen from Figures9 and 10 that the overall fatigue failure probability with survival rate of 99% is higher than the survival rate of 95%, indicating that the selection of the material P-S-N curve has a direct impact on the fatigue reliability analysis results of the structure, and should be paid attention to during the design process. Then, Figures 9 and 10 plot the CDF for P0 = 0.15, 0.20, 0.25, and 0.30. The purpose of this comparison is to see how SS behaves differently by adopting different values of P0 for the fatigue reliability analysis of motor hanger; P0 is the conditional failure probability of subset simulation. In Figure 9, the CDF plot with P0 = 0.3 significantly deviates from the rest. However, the CDF plot with P0 = 0.1 significantly deviates from the rest. It can be seen that the selection of P0 is very important for estimating the fatigue failure probability. Although Zuev et al. [30] proposed the values of P0 selected from the interval [0.1, 0.3] would produce similar efficiency with similar accuracy, the accuracy is still different for different subjects. For the fatigue reliability analysis of the motor hanger, when the survival rate is 95%, the failure probability obtained by Monte Carlo simulation is 0.0015; P0 = 0.10, 0.15, 0.20, and 0.25 are suitable for reliability analysis, while P0 = 0.30 is not proper. Similarly, when the survival rate is 99%, the failure probability obtained by Monte Carlo simulation is 0.1880; P0 = 0.15, 0.20, and 0.25 are suitable for reliability analysis, while P0 = 0.1 and 0.30 are not proper. For the calculation of Monte Carlo failure probability, 10 000 sample points are used, while the SS method is only 4%–7% of Monte Carlo, which greatly improves the calculation efficiency.


## 4.1. Finite Element Analysis of Motor Hanger

The premise of fatigue reliability analysis for motor hanger is to calculate its FPS under fatigue conditions. The research object of this paper is the motor hanger of high-speed train, which is bolted to the bogie frame to fix the motor. The motor hanger is mainly plate welded, and the partial opening is used to reduce the quality. Figure3 shows the specific installation position and structure type. The numerical analysis by the finite element method is performed to obtain the FPS of the motor hanger. It is modeled by shell and solid mode, given in Figure 4. The motor hanger is modeled using beam188, shell181, solid 185, rigid, and rbe3 elements. It is meshed to have 50,879 elements, including 21,766 triangular and quadrilateral shell elements and 28,973 tetrahedral and hexahedral solid elements.Figure 3
Geometric model of motor hanger.Figure 4
Finite element model of motor hanger.In the finite element model, the reasonable application of load cases is the key to ensure the accuracy of the analysis results. Generally, the boundary conditions of the test components under normal operating conditions should be consistent with those under test assembly conditions. The elastic connector on the bogie can be replaced by a rigid member, but the direction and magnitude of the load acting on the test piece cannot be changed. Figure5 displays the load direction of motor hanger, which is positive upward. The loads are calculated according to EN13749 standard and analyzed by finite element method. Figure 6 plots the result of the finite element analysis. It can be clearly seen from the figure that the maximum FPS is 119.23 MPa, which occurs on the motor boom.Figure 5
Load direction of motor hanger.Figure 6
FPS plot of motor hanger.


## 4.2. Establishment of Response Surface Function

The purpose of establishing the response surface function is to obtain the functional relationship between the random variables and the FPS and then to observe the influence of the uncertainties on it. As one of the surrogate models, the response surface can more accurately represent the functional relationship between input and output. Compared to other surrogate models, the response surface belongs to the display function that can be more easily used for reliability calculations [29]. Due to the poor approximation ability of the linear response surface, the quadratic response surface with cross terms is adopted. The basic equation is given by(16)y=∑i=1nciixi2+∑i>jncijxixj+∑i=1ncixi+c0,where n is the number of uncertain variables, c0 is the constant, and ci, cii, and cij are the polynomial coefficients, respectively.According to the stress plot in Figure6, the random variables which have great influence on the stress plot are selected as random variables. Considering the uncertainty of design parameters, the upper and lower limits of size parameters are set according to tolerance, and the upper and lower limits of load are obtained according to test conditions. Table 2 displays the specific parameters for random variables.Table 2
Statistical characteristic of random variables.
Design parameters
Sign
Unit
Lower limit
Mean value
Upper limit
Steel plate thickness 1
TH1
mm
7.7
8.0
8.3
Steel plate thickness 2
TH2
mm
9.7
10.0
10.3
Vertical load 1
F
1
N
23040.0
25600.0
28160.0
Vertical load 2
F
2
N
23040.0
25600.0
28160.0The premise of establishing response surface function is to do the design of experiment for random variables. According to the design of experiment scheme, the finite element model is invoked to solve the problem, and the corresponding response of each scheme is obtained. In this paper, the D-optimal design of experiment method is used to carry out the experimental design; see Table3.Table 3
Statistical characteristic of random variables.
Experiment number
Factors
Response
TH1 (mm)
TH2 (mm)
F
1 (N)
F
2 (N)
S1 (MPa)
1
7.967
10.300
25625.600
26880.000
143.48
2
7.916
9.700
23040.000
25472.000
144.88
3
7.700
9.700
28160.000
23040.000
149.18
4
7.925
10.300
25676.800
23040.000
147.27
5
8.300
9.700
23961.600
23040.000
137.75
6
7.979
9.967
23756.800
28160.000
143.39
7
7.700
10.300
23040.000
28160.000
148.68
8
8.300
10.300
23040.000
23040.000
137.42
9
8.060
9.700
26368.000
24925.400
142.13
10
8.300
10.300
28160.000
28160.000
137.42
…
…
…
…
…
…
18
7.916
9.988
28160.000
28160.000
144.68
19
7.937
10.021
28160.000
24003.000
144.25
20
8.300
9.700
23040.000
28160.000
137.75
21
8.300
10.186
28160.000
23040.000
137.49
22
8.300
10.063
25011.200
26137.600
137.56
23
7.700
10.300
28160.000
25651.200
148.68
24
7.700
10.300
23040.000
28160.000
148.68
25
8.300
10.300
28160.000
28160.000
137.42According to (16), the experimental data in Table 3 is fitted by least squares method to obtain a response surface function, which characterizes the function relationship between the random variable and the FPS.(17)S1=+426.60+43.658×TH1−73.68×TH2+3.457×10−3×F1−4.444×10−3×F2−2.50×TH1×TH2+1.6×10−4×TH1×F1+3.99×10−4×TH1×F2+4.929×10−5×TH2×F1−4.103×10−4×TH2×F2−2.909×10−9×F1×F2−3.265×TH12+5.156×TH22−10−7×F12+1.036×10−7×F22.The fitting accuracy of the response surface function is the basis and premise to ensure the accuracy of the subsequent reliability analysis. According to the fitting resultR2=0.9964, indicating that the fitting accuracy is high. To further verify the accuracy of the response surface function, the ANOVA is adopted to analyze the function in this paper. Some results are provided in Table 4.Table 4
ANOVA for response surfaceS1 quadratic model.
Source model
Sum of squares
Df
Mean square
F value
P-value
Model
567.09
14
40.51
199.31
<0.0001
Significant
TH1
391.63
1
391.63
1927.02
<0.0001
TH2
0.065
1
0.065
0.32
0.5833
F
1
0.10
1
0.10
0.50
0.4948
F
2
1.36
1
1.36
6.69
0.0271
…
…
…
…
…
…
TH
1
2
0.26
1
0.26
1.30
0.2805
TH
2
2
0.76
1
0.76
3.74
0.0820
F
1
2
1.28
1
1.28
6.31
0.0308
F
2
2
1.29
1
1.29
6.34
0.0305
Residual
2.03
10
0.20
Lack of fit
2.03
5
0.41
Pure error
0.000
5
0.000
Cor total
569.12
25In Table4, the F and P values are the main indicators used to evaluate the significance of the response surface function and its random variables. The larger the F value, the more significant it is; on the contrary, the smaller the P value, the more significant it is. As can be seen from Table 4, the function generally presents a very significant level, P<0.0001, indicating that the fitting accuracy of response surface is high. Among the four random variables, random variable TH1 is the most significant, which indicates that the small fluctuation of the random variable TH1 has a greater impact on the FPS. The other three random variables did not exhibit a very significant level, only that their small fluctuations had little effect on the FPS. Because of space, the interaction among random variables cannot be given. Through analysis, it is found that the interaction among random variables in response surface function is relatively low, which indicates that the interaction between random variables has less influence on response. By comparing the results of the first and second significance of random variables, it can be seen that the second significance of TH1 decreases and the second significance of TH2, F1, and F2 increases, indicating that the selection of random variables is reasonable. In summary, the response surface function can be used for fatigue reliability analysis of subsequent motor hanger.


## 4.3. Determination of the FPS Distribution Type and Bayesian Updating

For equation (13), the uncertainty of the calculated lifetime N results in three different states of the structure. As the main factor affecting the calculated lifetime N, accurate probability distribution of the FPS is the key to improve the accuracy of reliability analysis. For this reason, the FPS under random parameter fluctuation is obtained by using Monte Carlo method for 10000 calculations based on equation (17). The FPS is fitted by MATLAB 2015b to determine its probability distribution characteristics, as shown in Figure 7. From Figure 7, it can be seen that the FPS obeys lognormal distribution, and the fitting curve has high accuracy, R2=0.99349. In order to accurately estimate the fatigue reliability of the structure, simulation data are used as real measurement data for Bayesian updating because there is no experimental data in the design process. The prior distribution data of the FPS are obtained from the experimental data of similar products. The data of prior distribution obey the lognormal distribution of N4.945,0.041, and the real measured data obey the lognormal distribution of N4.962,0.0425. Two kinds of distribution data are substituted into equations (2)–(7) to obtain the probability and statistical parameters of the FPS after Bayesian updating. Figure 8 plots the probability distribution of the FPS before and after Bayesian updating. It can be seen from Figure 8 that the mean of FPS value after the Bayesian updating is slightly smaller than that of the FPS obtained by the actual engineering, and the corresponding variance is also reduced. This proves that the Bayesian updating method not only reduces the uncertainty of FPS, but also reduces the distortion and uncertainty of the detected data. In the design sense, Bayesian updating reduces the uncertainty of the simulation results.Figure 7
Lognormal distribution fitting of FPS.Figure 8
Comparison of probability and statistics characteristics of FPS.


## 4.4. Fatigue Reliability Analysis Based on Subset Simulation

The FPS of the motor hanger after Bayesian updating obeys the lognormal distribution ofN4.9451,0.0409. The symmetrical cyclic load spectrum specified in the EN13749 standard is selected, and the stress ratio is R = -1. The P-S-N curves with survival rates of 95% and 99% are selected, respectively, and the design lifetime is 10 million cycles. According to equations (11)–(13), the fatigue failure probability of the motor hanger is calculated by SS. Figures 9 and 10 display the CDF curves for fatigue failure probabilities at different survival rates, respectively. The estimated fatigue failure probabilities and the number of required samples are listed in Table 5.Figure 9
CDF of survival rate at 95%.Figure 10
CDF of survival rate at 99%.Table 5
Estimated failure probabilities and the number of samples with differentp0.
Survival rateP (%)
Conditional failure probabilityp0
0.10
0.15
0.20
0.25
0.30
95
Failure probabilityPf
0.005
0.0055
0.005
0.005
0.0086
Number of samples
560
540
680
650
620
99
Failure probabilityPf
0.1
0.15
0.1950
0.1950
0.2340
Number of samples
380
370
360
350
340It can be seen from Figures9 and 10 that the overall fatigue failure probability with survival rate of 99% is higher than the survival rate of 95%, indicating that the selection of the material P-S-N curve has a direct impact on the fatigue reliability analysis results of the structure, and should be paid attention to during the design process. Then, Figures 9 and 10 plot the CDF for P0 = 0.15, 0.20, 0.25, and 0.30. The purpose of this comparison is to see how SS behaves differently by adopting different values of P0 for the fatigue reliability analysis of motor hanger; P0 is the conditional failure probability of subset simulation. In Figure 9, the CDF plot with P0 = 0.3 significantly deviates from the rest. However, the CDF plot with P0 = 0.1 significantly deviates from the rest. It can be seen that the selection of P0 is very important for estimating the fatigue failure probability. Although Zuev et al. [30] proposed the values of P0 selected from the interval [0.1, 0.3] would produce similar efficiency with similar accuracy, the accuracy is still different for different subjects. For the fatigue reliability analysis of the motor hanger, when the survival rate is 95%, the failure probability obtained by Monte Carlo simulation is 0.0015; P0 = 0.10, 0.15, 0.20, and 0.25 are suitable for reliability analysis, while P0 = 0.30 is not proper. Similarly, when the survival rate is 99%, the failure probability obtained by Monte Carlo simulation is 0.1880; P0 = 0.15, 0.20, and 0.25 are suitable for reliability analysis, while P0 = 0.1 and 0.30 are not proper. For the calculation of Monte Carlo failure probability, 10 000 sample points are used, while the SS method is only 4%–7% of Monte Carlo, which greatly improves the calculation efficiency.


## 5. Conclusions

This study presents a novel fatigue reliability analysis method based on Bayesian updating and subset simulation to improve the accuracy of structural fatigue performance evaluation at the design stage. In order to reduce the uncertainty of fatigue lifetime calculated by simulation analysis, the Bayesian method is used to update the probability and statistical parameters of the FPS for the motor hanger. Compared with the original data, the mean and variance of the FPS after the update are reduced, which improves the robustness of the data. Meanwhile, the SS method is adopted to calculate the fatigue failure probability under different survival rates and different conditions failure probabilities, and the CDF curve of fatigue failure probability is given. The fatigue failure probability of motor hanger underP-S-N curve with 95% survival rate is P0 = 0.1 Pf = 0.005, P0 = 0.15 Pf = 0.0055, P0 = 0.20 Pf = 0.005, P0 = 0.25 Pf = 0.005. The fatigue failure probability of motor hanger under P-S-N curve with 99% survival rate is P0 = 0.15 Pf = 0.15, P0 = 0.20, Pf = 0.1950, P0 = 0.25 Pf = 0.1950. This not only proves the influence of the P-S-N curve with different survival rates on the fatigue failure probability, but also verifies that the reasonable selection of conditional failure probability is helpful to improve the accuracy of the SS. In a word, the proposed method helps the design stage to more accurately assess whether the fatigue lifetime of the structure meets the standard requirements. At the same time, it is also applicable to the fatigue reliability analysis of other large complex structures.



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*Source: 3012471-2020-03-27.xml*