# Dynamic Fatigue Analysis of High-Speed Trains Gearbox Using Copula Function

Authors: Yumei Liu; Ting Hu; Yun Chen; Zixu Hao; Ming Liu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

As a key component of the transmission system of high-speed trains, the reliability of gearbox is crucial to the overall reliability and driving safety of high-speed trains. In this work, a comprehensive reliability model for the key parts of the gearbox including the driving and driven gears, bearings, and gearbox housing is developed, which combines the strength degradation parameters obtained by P-S-N curves of the corresponding material, and a stress-strength interference model based on the Poisson distribution of random stress and the Wiener strength degradation process. Further, a nested Copula function reliability model of the gearbox series system is developed using the binary Frank Copula function considering the correlation of failures of different parts and different failure modes. This model realizes the dynamic reliability analysis of the gearbox under different failure modes. The reliability analysis of the key parts of the gearbox and the gearbox series system of high-speed train achieved with this model are in line with the engineering practice. This method enables dynamic tracking and monitoring of gearbox reliability during the service life of high-speed train.


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## Body

## 1. Introduction

The gearbox of high-speed trains is a key component for power transmission, it is important to ensure driving safety [1], and its structure is shown in Figure 1, mainly consisting of large and small gears, bearings, input and output shafts, seals, gearbox housing, and hanging devices [2]. Through the fault statistics of high-speed trains, it can be seen that the driving small gear, driven large gear, bearings, and gearbox housing are the parts with high damage frequency and serious damage. When the train is working, the gearbox is subjected to wheel-rail excitation, internal gear meshing excitation, and vibration loads generated by the structure connected to the boom, and external excitation loads such as torque loads connected to the motor [3]. For a complex mechanical system such as a high-speed trains gearbox, there is direct contact or indirect contact between the parts [4], and there are various failure modes with correlation, and the strength of each part is generally considered to be independent of each other, but the mutual independence of the strength of the parts does not mean that the failure of each part is independent of each other [5], the gearbox failure is the result of the interaction of load and strength. It is more in line with the engineering practice to measure the reliability of series mechanical systems by considering the dynamic changes of gearbox’s load, the strength degradation phenomenon of parts, and the correlation of failure among parts.Figure 1
Structure of the high-speed trains gearbox.The stress-strength interference theory is the basis of reliability design of mechanical parts, and the reliability mainly depends on the level of stress and strength interference. The stress-strength interference model was proposed by Freudenthal; it was gradually and widely used in fatigue reliability design in the 1960s [6]. Traditionally, this model is used to deal with the reliability of parts and structures under static stress-strength interference. However, the stress and strength of mechanical parts are random during the operation of mechanical systems, and the strength of the parts will degrade due to corrosion, aging, and other factors, and failure will occur when the stress on the parts is greater than their strength.Haugen [7, 8] gradually refined the method of fatigue reliability design using stress-strength interference model, transformed the static strength probability distribution in the classical stress-strength interference model into the fatigue strength distribution at a specified lifetime, and proposed the well-known fatigue reliability stress-strength interference model in 1978. In recent years, stress-strength interference theory has been widely adopted in railway vehicle reliability research. Li [9] determined the equivalent stress based on the stress test data during the operation of the gearbox, established a reliability model of equivalent stress-fatigue strength, and analyzed the fatigue reliability of the gearbox housing under typical working conditions. Wang et al. [10] took a new high-speed trains gearbox as the research object, combined with line tests to analyze the effects of train operating speed, motor output torque and line conditions on the dynamic stress response and fatigue strength of the housing, established a gearbox equivalent force-fatigue strength interference reliability model, and analyzed the relationship between the fatigue reliability of the gearbox housing and the service mileage. In the research of the above scholars, the stress probability distribution of the research object is obtained by collecting the stress data of the real vehicle, but this collection method is not practical for some of devices in the gearbox system. In addition, the relationship of stress and strength, and the degradation of strength are not considered in the process of evaluating reliability. Wu et al. [11] used SIMPACK and MATLAB/Simulink to establish the electromechanical model of the gearbox housing and traction system of high-speed trains to analyze the dynamic stress of the gearbox housing, which provides a new idea of simulation analysis for establishing the stress-strength interference model to analyze the fatigue reliability of the gearbox system. When analyzing the reliability of gearbox system based on stress-strength interference theory, it is necessary to study the strength degradation of each part. The commonly used degradation models are based on the degradation models of stochastic processes. Wang et al. [12] proposed a residual life prediction method considering multivariate degradation correlations based on the degradation process of Wiener process modeling features with nonlinear drift parameters and multivariate normal distribution describing the correlation between multiple features. Dong et al. [13] used a binary Wiener process to model degradation processes with two performance characteristics and successfully applied it to the reliability assessment of railway tracks. Lv et al. [14] used a stress-strength interference model to study the reliability of gear systems considering the strength degradation.Since Sklar [15] proposed the Copula theory in 1959, Copula theory has been increasingly used in the reliability research of mechanical systems. In recent years, copula theory has been used more and more in reliability research of mechanical parts. The Copula function can connect the joint distribution of multidimensional random variables with marginal distributions, and accurately handle the failure modes of parts or nonlinear correlations between random variables. Yi-rui et al. [16] proposed a reliability evaluation model considering strength degradation, studied the correlation and degree of correlation between different failure modes with the help of Copula function, defined failure thresholds to express the marginal failure probability of correlation between different failure modes, and evaluated the reliability of mechanical structures. Li [17] used the Copula function and combined the stochasticness and fuzziness of the load to study the dynamic reliability design and calculation of the series and parallel systems. Gao et al. [18] introduced a hybrid Copula function into the correlated stress-strength reliability model and established a correlated model, and the validation results showed that the evaluation results of the model were more realistic. Peng et al. [19] proposed an inverse Gaussian (IG) process model to fit the binary Copula degradation model for the edge degradation process, and the accuracy of the model was verified based on heavy-duty machine tools.In summary, most of the current research is mainly aimed at the extension of the stress-strength interference theoretical model, combined with the Copula function to evaluate the reliability of a single part or a series and parallel system of the same type of parts. There are few reliability studies applied to complex series and parallel systems. Thus, a nested Copula function reliability model for different types of parts series gearbox systems is established in this study. Combining the theory of order statistics, the random stress of the Poisson process and the Wiener strength degradation process, the time-varying stress-strength interference model is determined, and the strength degradation parameters of the parts based on the Wiener process are determined by the random model, with the help of the P-S-N curve strength degradation of the materials, the model can provide support for the reliability correlation modeling of the gearbox system. In the process of building the model, the structural characteristics of the gearbox system of high-speed trains, the characteristics of the load acting on each key part, the strength degradation of each part, and the failure correlation of each key part, all those are fully considered to evaluate its dynamic reliability and make the reliability analysis results closer to the reality practice.The following work is mainly carried out in the following section. On the basis of the classical stress-strength interference model, the time-varying stress based on the Poisson distribution and the stress-strength interference model based on strength degradation of the Wiener process are established; the Copula function reliability model of the gearbox multi-part series system is established. Based on the previous model established, analyzing the dynamic reliability of three types of key parts of gearbox: large and small gears, bearings, and housing; establishing the nested Copula reliability model of the gearbox system and analyzing the dynamic reliability of the gearbox system under different failure modes are done. Finally, the study concludes with a conclusion and future research.


## 2. Stress-Strength Interference Model

### 2.1. Introduction to the Stress-Strength Interference Model

Assuming that the probability density functions of stressS and fatigue strength Z are fs⋅ and fσ⋅, the interference between S and Z is shown in Figure 2, and the shaded part indicates the “interference zone of stress-strength”.Figure 2
Stress-strength interference.The failure of a part usually occurs in the interference area and increases with the increase of the area of the interference area. The reliabilityRs for a given stress s is defined as(1)Rs=∫s+∞fσzdz.Both stress and strength are continuous random variables defined on−∞,+∞. Rs is a function of the variable. According to the full probability formula, it can be known that the probability the strength is greater than the stress, the formula of the reliability R is(2)R=PZ>S=∫−∞+∞fssRsds=∫−∞+∞fss∫s+∞fσzdzds=∫−∞+∞fss1−Fσsds,where Fσs is the distribution function of part strength.


### 2.2. Stress-Strength Reliability under Random Loading Conditions Based on the Poisson Process

In the actual situation of a mechanical system, the load on each part or component becomes a random variable, due to the external environment, or its own micro-stress or its own wear. If the classical stress-strength interference model is used to analyze the reliability of parts and components, then it will produce a large error. Therefore, the stress-strength interference model considering time-varying load is established.Assuming that the strength degradation of components is not considered, the components will not fail when a random loads is applied n times, the probability is(3)Pz>smax=Pz>s1,z>s2,…,z>sn,where z is the strength of the components, s1,s2,…,sn are n samples drawn from a random load sample, and smax is the maximum load, where smax=maxs1,s2,…,sn.LetFss denote the distribution function of the random load s, the probability density function is fss. When the load is applied on the parts n times, the distribution function of the maximum load X is(4)FXx=Fsxn.The probability density function is(5)fXs=nFssn−1fss.Combining formulas (2) and (5), when the load is applied n times on the parts, the reliability is(6)Rn=∫−∞+∞fXs∫x+∞fσzdzds=∫−∞+∞nFssn−1fss∫s+∞fσzdzds.The Poisson process can describe events that occur independently in mutually exclusive times and intervals, in the process of reliability analysis of mechanical parts, the cumulative action times of random loads on the parts are in line with the Poisson process [20]. Therefore, the Poisson process is used to describe the process of random load action. Let the total number of load actions is Nt, and the process can be described by the Poisson process with parameter λ. The probability of random load at any moment t is(7)PNt−N0=n=λtnn!e−λt.Combined with formulas (6) and (7), the dynamic reliability of the parts at any moment t can be obtained as(8)Rt=∑n=0∞RnPNt−N0=n=∑n=0∞λtnn!e−λt∫−∞∞fσz∫s+∞nFssn−1fssdsdz.A Taylor expansion of equation (8), it can be obtained.(9)Rt=∫−∞∞fσze−λt−λt∑n=0∞λtnn!Fszndz=∫−∞∞fσzeFsz−1λtdz.


### 2.3. Stress-Strength Reliability Based on Strength Degradation of the Wiener Processes

When the mechanical parts are working in the actual situation, the strength will gradually decrease with the increase of the load action times, and the reliability model need to consider the strength degradation, which is more in line with the engineering practice. Based on formula (9), the reliability of the parts considering strength degradation can be obtained as(10)Rt=∫−∞∞fσze∫0tFsz,τ−1λdτdz,where Dt represents the random process of strength degradation, then at any time t, the strength of the part is(11)Zt=Z−Dt.When the stressAt occurs at any moment t, the failure probability of the mechanical structure is(12)PFt=PZ−Dt<S|At=1−FsZ−Dt.The reliability of the part can be obtained as(13)Rt=∫−∞∞fσze∫0tFsZ−Dτ−1λdτdz.The strength degradation of mechanical parts, such as fatigue, corrosion, cracking, and wear degradation, will show a smooth increasing trend with the increase of the fatigue load action times, and there will not be large fluctuations or jumps [21]. The Wiener process has the property of describing smooth and small increments, so the Wiener process can be used in this study to describe the random law of strength degradation [22]. When the degradation X follows the Wiener process, then the degradation at each moment t can be expressed as(14)Xt=at+bBt,where a is the drift coefficient, b is the diffusion coefficient, Bt,t≥0 is the standard Brownian motion, Xt adds a trend term at to describe the monotonic process of product performance degradation, so that if the amount of strength degradation at time t is denoted by Xt, the mean is EXt=at, and the variance is VarXt=b2t.The probability density function of the increment is(15)ftx=12πb2texp−x22b2t.The Werner process degradation modeling method is as follows: assuming that there aren samples. The performance degradation is zero at the initial time, and the sample performance degradation is made from time t1 to tm, which are measured for m times, and the following values are obtained:(16)Xt11Xt12⋯Xt1mXt21Xt22⋯Xt2m⋮⋮⋱⋮Xtn1Xtn2⋯Xtnm.Denote△xij as the amount of performance degradation of sample i, which between moments of tij−1 and tij, then △xij=Xtij−Xtij−1, i=1,2,…,n;j=1,2,…,m. From the Wiener process property: △xij∼Na△tij,b2△tij, where △tij=tij−tij−1.The likelihood function can be obtained as(17)La,b2=∏i=1n∏j=1m12πb2△tijexp−△xij−a△tij2b2△tij.The maximum likelihood estimates of the parametera and b2 can be described as(18)a^=∑i=1nXim∑i=1ntim,b2^=1∑i=1nm∑i=1n∑j=1m△xij2△tij−∑i=1nXim2∑i=1ntim.Assuming that the stress and strength of the part obey the normal distribution, respectively:S∼Nμ1,σ12, Z∼Nμ2,σ22. Let Y=Z−S, the reliability of the part considering strength degradation can be expressed as(19)Rt=∫−∞∞12πσ12exp−12z−μ1σ122⋅exp−λ⋅∫0t1−∫0+∞12πσ12+σ22exp−12y−μ1−μ2σ12+σ222⋅∫0y12πb2texp−x22b2tdxdydτdz.


## 2.1. Introduction to the Stress-Strength Interference Model

Assuming that the probability density functions of stressS and fatigue strength Z are fs⋅ and fσ⋅, the interference between S and Z is shown in Figure 2, and the shaded part indicates the “interference zone of stress-strength”.Figure 2
Stress-strength interference.The failure of a part usually occurs in the interference area and increases with the increase of the area of the interference area. The reliabilityRs for a given stress s is defined as(1)Rs=∫s+∞fσzdz.Both stress and strength are continuous random variables defined on−∞,+∞. Rs is a function of the variable. According to the full probability formula, it can be known that the probability the strength is greater than the stress, the formula of the reliability R is(2)R=PZ>S=∫−∞+∞fssRsds=∫−∞+∞fss∫s+∞fσzdzds=∫−∞+∞fss1−Fσsds,where Fσs is the distribution function of part strength.


## 2.2. Stress-Strength Reliability under Random Loading Conditions Based on the Poisson Process

In the actual situation of a mechanical system, the load on each part or component becomes a random variable, due to the external environment, or its own micro-stress or its own wear. If the classical stress-strength interference model is used to analyze the reliability of parts and components, then it will produce a large error. Therefore, the stress-strength interference model considering time-varying load is established.Assuming that the strength degradation of components is not considered, the components will not fail when a random loads is applied n times, the probability is(3)Pz>smax=Pz>s1,z>s2,…,z>sn,where z is the strength of the components, s1,s2,…,sn are n samples drawn from a random load sample, and smax is the maximum load, where smax=maxs1,s2,…,sn.LetFss denote the distribution function of the random load s, the probability density function is fss. When the load is applied on the parts n times, the distribution function of the maximum load X is(4)FXx=Fsxn.The probability density function is(5)fXs=nFssn−1fss.Combining formulas (2) and (5), when the load is applied n times on the parts, the reliability is(6)Rn=∫−∞+∞fXs∫x+∞fσzdzds=∫−∞+∞nFssn−1fss∫s+∞fσzdzds.The Poisson process can describe events that occur independently in mutually exclusive times and intervals, in the process of reliability analysis of mechanical parts, the cumulative action times of random loads on the parts are in line with the Poisson process [20]. Therefore, the Poisson process is used to describe the process of random load action. Let the total number of load actions is Nt, and the process can be described by the Poisson process with parameter λ. The probability of random load at any moment t is(7)PNt−N0=n=λtnn!e−λt.Combined with formulas (6) and (7), the dynamic reliability of the parts at any moment t can be obtained as(8)Rt=∑n=0∞RnPNt−N0=n=∑n=0∞λtnn!e−λt∫−∞∞fσz∫s+∞nFssn−1fssdsdz.A Taylor expansion of equation (8), it can be obtained.(9)Rt=∫−∞∞fσze−λt−λt∑n=0∞λtnn!Fszndz=∫−∞∞fσzeFsz−1λtdz.


## 2.3. Stress-Strength Reliability Based on Strength Degradation of the Wiener Processes

When the mechanical parts are working in the actual situation, the strength will gradually decrease with the increase of the load action times, and the reliability model need to consider the strength degradation, which is more in line with the engineering practice. Based on formula (9), the reliability of the parts considering strength degradation can be obtained as(10)Rt=∫−∞∞fσze∫0tFsz,τ−1λdτdz,where Dt represents the random process of strength degradation, then at any time t, the strength of the part is(11)Zt=Z−Dt.When the stressAt occurs at any moment t, the failure probability of the mechanical structure is(12)PFt=PZ−Dt<S|At=1−FsZ−Dt.The reliability of the part can be obtained as(13)Rt=∫−∞∞fσze∫0tFsZ−Dτ−1λdτdz.The strength degradation of mechanical parts, such as fatigue, corrosion, cracking, and wear degradation, will show a smooth increasing trend with the increase of the fatigue load action times, and there will not be large fluctuations or jumps [21]. The Wiener process has the property of describing smooth and small increments, so the Wiener process can be used in this study to describe the random law of strength degradation [22]. When the degradation X follows the Wiener process, then the degradation at each moment t can be expressed as(14)Xt=at+bBt,where a is the drift coefficient, b is the diffusion coefficient, Bt,t≥0 is the standard Brownian motion, Xt adds a trend term at to describe the monotonic process of product performance degradation, so that if the amount of strength degradation at time t is denoted by Xt, the mean is EXt=at, and the variance is VarXt=b2t.The probability density function of the increment is(15)ftx=12πb2texp−x22b2t.The Werner process degradation modeling method is as follows: assuming that there aren samples. The performance degradation is zero at the initial time, and the sample performance degradation is made from time t1 to tm, which are measured for m times, and the following values are obtained:(16)Xt11Xt12⋯Xt1mXt21Xt22⋯Xt2m⋮⋮⋱⋮Xtn1Xtn2⋯Xtnm.Denote△xij as the amount of performance degradation of sample i, which between moments of tij−1 and tij, then △xij=Xtij−Xtij−1, i=1,2,…,n;j=1,2,…,m. From the Wiener process property: △xij∼Na△tij,b2△tij, where △tij=tij−tij−1.The likelihood function can be obtained as(17)La,b2=∏i=1n∏j=1m12πb2△tijexp−△xij−a△tij2b2△tij.The maximum likelihood estimates of the parametera and b2 can be described as(18)a^=∑i=1nXim∑i=1ntim,b2^=1∑i=1nm∑i=1n∑j=1m△xij2△tij−∑i=1nXim2∑i=1ntim.Assuming that the stress and strength of the part obey the normal distribution, respectively:S∼Nμ1,σ12, Z∼Nμ2,σ22. Let Y=Z−S, the reliability of the part considering strength degradation can be expressed as(19)Rt=∫−∞∞12πσ12exp−12z−μ1σ122⋅exp−λ⋅∫0t1−∫0+∞12πσ12+σ22exp−12y−μ1−μ2σ12+σ222⋅∫0y12πb2texp−x22b2tdxdydτdz.


## 3. Reliability Model of the Series System Based on Copula Function

### 3.1. Copula Function Theory

Copula function is a function that connects the joint distribution and marginal distribution of multiple variables [15], so it is also called “connection function”, which can describe the correlation of multiple random variables. X1,X2,…,Xn are n-dimensional random variables with marginal distribution function: F1x1,F2x2,…,Fnxn. If the marginal distribution function is continuous, then there exists an unique n-dimensional Copula function Cu1,u2,…,un, for x1,x2,…,xn∈R:(20)Fx1,x2,…,xn=CF1x1,F2x2,…,Fnxn.From the definition of Copula function, it is known that when studying the correlation between multidimensional random variables, the marginal distribution functions of the variables and their correlations can be split up and studied separately. Citing the most widely used binary Copula function, assuming thatCu,v is a binary Copula function, the joint distribution function of the two-dimensional random variables X,Y can be expressed as(21)Hx,y=Cu,v;θ,where u=Fx,v=Gy is the marginal distribution function of the variables X and Y, and θ is a self-contained parameter of the Copula function. It follows that its joint probability density function hx,y can be expressed as(22)hx,y=∂2H∂x∂y=fXxgYycu,v;θ,where fXx and gYy are the marginal probability density functions of X and Y, and cu,v;θ are the density function of the Copula function, which can be expressed as(23)cu,v;θ=∂Cu,v;θ∂u∂v.When building a Copula function model, it is necessary to determine the marginal distribution of the random variables, choose an appropriate Copula function to describe the correlation structure among the random variables, and estimate the parameters in the Copula model. In this study, the nonparametric method is used to determine the distribution of random variables, and the most commonly used method is the kernel density estimation method, which means that the overall distribution is determined by the sample observation data using the kernel density estimation method. The marginal distribution may contain unknown parameters and the copula function selected also contains unknown parameters, so parameter estimation is necessary.For the unknown parametersθ in the binary copula function, the great likelihood estimation method can be used for calculation. The likelihood function of the sample with respect to the parameters θ is(24)Lθ=∏i=1nhxi,yi|θ=∏i=1ncui,vi|θfXxigYyi.The log-likelihood function can be obtained as(25)lnLθ=∑i=1nlncui,vi;θ+∑i=1nlnfXxi+∑i=1nlngYyi.Solving the maximum point of the formula (25), the maximum likelihood estimation value θ^ of the unknown parameters θ in the copula function can be obtained as(26)θ^=argmaxθlnLθ=∑i=1nlncui,vi|θ.


### 3.2. Commonly Used Copula Functions

After a lot of research by many scholars, a variety of specific forms of Copula functions have been summarized [23]. According to the different generating elements of Copula functions, the commonly used Copula functions are Gaussian Copula functions, t-Copula functions, and Archimedean Copula functions. Archimedes Copula function has the advantages of simple form and combinability that other Copula functions do not have. Among the Archimedean Copula functions, Frank Copula function, Clayton Copula function, and Frank Copula function are the most commonly used binary Copula functions in reliability research. The description of these five Copula functions is shown in Table 1.Table 1
Commonly used five copula functions.
Copula functionCDF functionCu,v;θPDF functionCu,v;θGaussian copula1/1−θ2exp−ξ12θ2−2ξ1ξ2θ+ξ22/21−θ2,ξ1=Φ−1u,ξ2=Φ−1vΦθΦ−1u,Φ−1vt-Copulatθ,ktk−1u,tk−1v1/2π1−θ21+u2−2uvθ+v2/k1−θ2−k+2/2Frank copula−1/θlne−θu−1e−θv−1/e−θ−1−θe−θ−1e−θu+v/e−θ−1+e−θu−1e−θv−12Gumbel copulaexp−−lnuθ+−lnvθ1/θCu,v;θlnulnvθ−1/uv−lnuθ+−lnvθ2−1/θ⋅−lnuθ+−lnvθ1/θ+θ−1Clayton copulamaxu−θ+v−θ−11/θ,01+θuv−θ−1u−θ+v−θ−1−2−1/θThere are obvious differences in the Copula functions. It can be seen from the probability density plots of the five Copula functions in Figure3: Gaussian copula, t-copula function, and Frank copula function have symmetric tails and can capture the symmetric tail correlations between random variables; t-copula function has thicker tails, it is more sensitive to tail correlation changes between random variables, and can capture symmetric tail correlations between random variables better; Frank Copula has a simple structure and strong adaptability, and can meet the needs of most fields of application; Gumbel Copula is suitable for two-dimensional random vectors with asymmetric tails, and upper tail correlation and lower tail asymptotic independence. Clayton Copula is suitable for applications with asymmetric tails, lower tail correlation, and upper tail asymptotic independent two-dimensional random vectors.Figure 3
Distribution of five types of copulas: (a) PDF of Gaussian Copula; (b) PDF oft-Copula; (c) PDF of Frank Copula; (d) PDF of Clayton Copula; (e) PDF of Gumbel Copula.
(a)(b)(c)(d)(e)


### 3.3. Reliability Model of the Series Mechanical System

Suppose that the series system is composed ofn units, and its joint distribution function is Fyy1,y2,…yn, Yit=σi−si, Yi represents the functional function of the ith part at time t, σi represents the strength of the part, and si represents the stress of the part, then the dynamic reliability of the series system [24] is(27)Rst=PY1t>0,Y2t>0,…,Ynt>0=Δ1−R1t1Δ1−R2t1⋯Δ1−Rnt1Cw1,w2,…wn,where Δ represents the difference symbol, Δw1w2Cw,γ0=Cw2,γ0−Cw1,γ0 represents the one-fold difference, Δγ1γ2Δw1w2Cw,γ=Cw2,γ2−Cw2,γ1−Cw1,γ2+Cw1,γ1 represents the double integral, and so on, Rjt is the dynamic reliability of the part, and wj=Fjt=1−Rjt,j=1,2,…,n.When calculating the reliability of a specific mechanical system, the multivariate Gaussian Copula function [25] and the Archimedes Copula function are usually selected to deal with the correlation problem.


### 3.4. Dynamic Reliability of Driving and Driven Gears

During the process of power transmission, the driving and driven gears in the high-speed trains gearbox are subjected to external excitation from other parts as well as internal meshing impact excitation. The main failure models are fatigue failure, including root fracture, pitting, wear, flaking, galling, root fracture, and tooth surface damage [26]. Based on their failure forms, when calculating contact stress and bending stress of gears, the action of random time-varying load and strength degradation need to be considered, then the stress-strength interference models of tooth face contact strength and tooth root bending strength of driving and driven gears are established, so as to derive the dynamic reliability of contact fatigue and bending fatigue of driving and driven gears.According to GB/T3480-1997, the calculated contact stress and calculated bending stress of gears are(28)St=Kt⋅CNm=ZNZVZLZWZRZX⋅Cftm,where ZN is the contact strength life factor, taken ZN=1.1; ZV is the speed factor, taken ZV=1.006; ZL is the lubricant factor, taken ZL=1.01; ZW is the work hardening factor, taken ZW=1; ZR is the roughness factor, taken ZR=1.016; and ZX is the dimension factor for contact strength calculation, taken ZX=1.(29)Sb=Kb⋅CNm=YSTYNTYδrelTYRrelTYX⋅Cftm,where YST is the tress correction factor, taken YST=2; YNT is the life factor of bending strength, taken YNT=1.08; YδrelT is the relative tooth root fillet sensitivity factor, taken YδrelT=1.01; YRrelT is the relative tooth root surface condition factor, taken YRrelT=1.12; and YX is the bending strength calculation factor, taken YX=1.The probability distribution of the contact stress on the tooth surface of the driving gear and the driven gear obeyssH∼N458,35.512. The probability distribution of the bending stress at the root of the driving gear obeys sF1∼N95.74,1.322, and the probability distribution of the bending stress at the root of the driven gear obeys sF2∼N93.17,1.292. The tooth surface contact fatigue strength and tooth root bending fatigue strength of the driving and driven gears are, respectively, obeyed:(30)σH∼N1031.25,61.882,σF∼N575.79,46.062.The amount of strength degradation is a random variable, and this study uses the Wiener process to describe the random strength degradation law of each part in the gearbox. The degradation law of gear strength is related to the nature of the material and the working condition of the work condition, so the strength degradation law of driving and driven gears will be different. The P-S-N curve of the material can reflect a more comprehensive stress-life relationship, and the P-S-N curves can reflect the probabilistic information of three aspects, as shown in Figure4.Figure 4
P-S-N curve of the material.According to data collection, the fatigue test data and P-S-N curve of gear material 17CrNiMo6 were obtained [27]. The characteristic parameters in the Wiener process degradation model are estimated, which combines the P-S-N curve of the gear material. The specific calculation process is shown in Figure 5.Figure 5
Parameter estimation of the Wiener process.Based on the P-S-N curve of the gear material and the Wiener parameter estimation process, the Wiener process parameters of the driving gear and the driven gear can be calculated as shown in Table2.Table 2
Strength degradation parameters of the driving and driven gears.
Driving gearDriven gearContact fatiguea51.610049.1722b29.143826.6691Bending fatiguea116.6235105.8101b62.712457.3872Combining the stress distribution, strength distribution, and strength degradation process parameters of gears determined above, the dynamic reliability of the driving and driven gears can be obtained, as shown in Figure6.Figure 6
(a) Fatigue reliability of the four-point angular contact ball bearing. (b) Fatigue reliability of the outer cylindrical roller bearing. (c) Fatigue reliability of the inner cylindrical roller bearing. (d) Fatigue reliability of the outer tapered roller bearing. (e) Fatigue reliability of the inner tapered roller bearing.
(a)(b)(c)(d)(e)It can be seen from Figures6(a) and 6(b), the tooth face contact fatigue reliability of gears decreases relatively faster than the tooth root bending fatigue reliability, which is in line with the conclusion that the main form of gear failure is contact fatigue failure. In addition, when the working time of high-speed trains is within 2000 hours, the contact fatigue reliability of gears in the gearbox is around 0.97, which is in line with the secondary maintenance standard of the corresponding vehicle type, and the gearbox should be maintained by oil change. This shows that the calculation method of this study is in line with the practical application of engineering.Figure 7
(a) Fatigue reliability of driving gear. (b) Fatigue reliability of driven gear.
(a)(b)


### 3.5. Dynamic Reliability of Different Types of Bearings

The bearings arrangement in the gearbox is shown in Figure1. A four-point angular contact ball bearing (FPACBB) is mounted on the side of the input shaft, it is near the motor to carry the axial load when the motor drives the input shaft to rotate. The mounting cylindrical roller bearings at both the ends of the driving gear are used to bear the radial load when the input shaft rotates, and the mounting tapered roller bearings (TRB) at both ends of the output shaft bear both the radial and axial loads of the output shaft. The distribution of contact stress and strength of rolling bearings are considered as normal distribution [28]. The contact stress calculation of rolling bearings in gearbox is calculated by referring to the algorithm in Harris and Kotzalas [29]. After calculation and analysis, it can be obtained: the surface contact stress distribution of the four-point contact ball bearing obeys sb1∼N1066.5,106.652, the surface contact stress distribution of the outer cylindrical roller bearing of the driving gear obeys sb2∼N671.29,67.122, and the surface contact stress distribution of the inner cylindrical roller bearing of the driving gear obeys, the surface contact stress distribution of the outer tapered roller bearing of the driven gear obeys sb4∼N852.34,85.232, and the surface contact stress of the inner tapered roller bearing of the driven gear obeys sb5∼N696.64,69.662, the fatigue strength of the bearing material obeys σb∼N1718.18,171.82.Based on the algorithm in Figure7 and the P-S-N curve of bearing steel GCr15A, the Wiener strength degradation parameters of each bearing are determined as shown in Table 3.Table 3
Strength degradation parameters of the bearings.
Strength degradation parameters of bearingsabFour-point contact ball bearing14.784411.9669Outer cylindrical roller bearing14.784411.9669Inner cylindrical roller bearing14.784411.9669Outer tapered roller bearing16.211813.4591Inner tapered roller bearing16.211813.4591Based on the above stress distribution, strength distribution, and strength degradation process parameters of each bearing, the dynamic reliability of each bearing can be obtained, as shown in Figure6.It can be seen from Figure6, among the various types of bearings, the reliability of four-point contact ball bearings decreases faster compared to other bearings, which is due to the more complex load with four-point ball bearings. In addition, the reliability of four-point ball bearings can be maintained above 0.96, when the high-speed trains run within 2000 hours, and the reliability of the other two types of bearings are above 0.99, which is in line with the secondary maintenance standard. It shows that the reliability calculation method for bearings is in line with the engineering practice.


### 3.6. Dynamic Reliability of the Gearbox Housing

The gearbox housing is not only subjected to external excitation from the wheel track caused by track unevenness, but also the internal excitation caused by the time-varying meshing stiffness and transmission error of gear transmission and other excitation such as motor torque load, resulting in fatigue cracks in the housing [9]. The gearbox housing is distinguished from other standard parts by stress analysis due to its complex shape. The equivalent stress at the weak point of the housing is obtained by finite element analysis or line stress spectrum calculation [30], and the probability distribution of the equivalent stress is determined, according to the damage consistency principle as the stress distribution of the gearbox housing. The distribution of equivalent stress calculated under specified operating conditions obeys s∼N24.91,0.832.The fatigue strength of the gearbox housing depends on material properties, and it can be considered to be normal distribution. The material of the gearbox housing is AlSi7Mg0.3, according to the fatigue strength standard of cast aluminum alloy in BS EN 1999-1-3-2007, the fatigue strength variation coefficient is taken to be 0.1, and the fatigue strength of the housing obeysσ∼N29.94,2.992.Based on the P-S-N curve of the gearbox housing material [14], the Wiener strength degradation parameters of gearbox housing are determined as shown in Table 4.Table 4
Strength degradation parameters of the gearbox housing.
Degradation of contact fatigue strength of the gearbox housingabGearbox housing77.811450.2314Based on the parameters of the stress distribution, strength distribution, and strength degradation of the gearbox housing, the dynamic reliability of the gearbox housing can be obtained as shown in Figure8.Figure 8
Fatigue reliability of the gearbox housing.It can be seen from Figure8, under the premise of satisfying the casting quality and no excessive external impact, the fatigue reliability of the gearbox housing keeps decreasing, when the train’s running time is increasing, which indicates that the wear state of the transmission system parts will affect the reliability of the housing. When the train’s running time is within 4000 hours, the reliability of the housing can be maintained above 0.98, which meets the mileage requirements for level III and level IV maintenance of the vehicle. It shows that the above calculation method for gearbox housing is in line with the actual application of the engineering.


## 3.1. Copula Function Theory

Copula function is a function that connects the joint distribution and marginal distribution of multiple variables [15], so it is also called “connection function”, which can describe the correlation of multiple random variables. X1,X2,…,Xn are n-dimensional random variables with marginal distribution function: F1x1,F2x2,…,Fnxn. If the marginal distribution function is continuous, then there exists an unique n-dimensional Copula function Cu1,u2,…,un, for x1,x2,…,xn∈R:(20)Fx1,x2,…,xn=CF1x1,F2x2,…,Fnxn.From the definition of Copula function, it is known that when studying the correlation between multidimensional random variables, the marginal distribution functions of the variables and their correlations can be split up and studied separately. Citing the most widely used binary Copula function, assuming thatCu,v is a binary Copula function, the joint distribution function of the two-dimensional random variables X,Y can be expressed as(21)Hx,y=Cu,v;θ,where u=Fx,v=Gy is the marginal distribution function of the variables X and Y, and θ is a self-contained parameter of the Copula function. It follows that its joint probability density function hx,y can be expressed as(22)hx,y=∂2H∂x∂y=fXxgYycu,v;θ,where fXx and gYy are the marginal probability density functions of X and Y, and cu,v;θ are the density function of the Copula function, which can be expressed as(23)cu,v;θ=∂Cu,v;θ∂u∂v.When building a Copula function model, it is necessary to determine the marginal distribution of the random variables, choose an appropriate Copula function to describe the correlation structure among the random variables, and estimate the parameters in the Copula model. In this study, the nonparametric method is used to determine the distribution of random variables, and the most commonly used method is the kernel density estimation method, which means that the overall distribution is determined by the sample observation data using the kernel density estimation method. The marginal distribution may contain unknown parameters and the copula function selected also contains unknown parameters, so parameter estimation is necessary.For the unknown parametersθ in the binary copula function, the great likelihood estimation method can be used for calculation. The likelihood function of the sample with respect to the parameters θ is(24)Lθ=∏i=1nhxi,yi|θ=∏i=1ncui,vi|θfXxigYyi.The log-likelihood function can be obtained as(25)lnLθ=∑i=1nlncui,vi;θ+∑i=1nlnfXxi+∑i=1nlngYyi.Solving the maximum point of the formula (25), the maximum likelihood estimation value θ^ of the unknown parameters θ in the copula function can be obtained as(26)θ^=argmaxθlnLθ=∑i=1nlncui,vi|θ.


## 3.2. Commonly Used Copula Functions

After a lot of research by many scholars, a variety of specific forms of Copula functions have been summarized [23]. According to the different generating elements of Copula functions, the commonly used Copula functions are Gaussian Copula functions, t-Copula functions, and Archimedean Copula functions. Archimedes Copula function has the advantages of simple form and combinability that other Copula functions do not have. Among the Archimedean Copula functions, Frank Copula function, Clayton Copula function, and Frank Copula function are the most commonly used binary Copula functions in reliability research. The description of these five Copula functions is shown in Table 1.Table 1
Commonly used five copula functions.
Copula functionCDF functionCu,v;θPDF functionCu,v;θGaussian copula1/1−θ2exp−ξ12θ2−2ξ1ξ2θ+ξ22/21−θ2,ξ1=Φ−1u,ξ2=Φ−1vΦθΦ−1u,Φ−1vt-Copulatθ,ktk−1u,tk−1v1/2π1−θ21+u2−2uvθ+v2/k1−θ2−k+2/2Frank copula−1/θlne−θu−1e−θv−1/e−θ−1−θe−θ−1e−θu+v/e−θ−1+e−θu−1e−θv−12Gumbel copulaexp−−lnuθ+−lnvθ1/θCu,v;θlnulnvθ−1/uv−lnuθ+−lnvθ2−1/θ⋅−lnuθ+−lnvθ1/θ+θ−1Clayton copulamaxu−θ+v−θ−11/θ,01+θuv−θ−1u−θ+v−θ−1−2−1/θThere are obvious differences in the Copula functions. It can be seen from the probability density plots of the five Copula functions in Figure3: Gaussian copula, t-copula function, and Frank copula function have symmetric tails and can capture the symmetric tail correlations between random variables; t-copula function has thicker tails, it is more sensitive to tail correlation changes between random variables, and can capture symmetric tail correlations between random variables better; Frank Copula has a simple structure and strong adaptability, and can meet the needs of most fields of application; Gumbel Copula is suitable for two-dimensional random vectors with asymmetric tails, and upper tail correlation and lower tail asymptotic independence. Clayton Copula is suitable for applications with asymmetric tails, lower tail correlation, and upper tail asymptotic independent two-dimensional random vectors.Figure 3
Distribution of five types of copulas: (a) PDF of Gaussian Copula; (b) PDF oft-Copula; (c) PDF of Frank Copula; (d) PDF of Clayton Copula; (e) PDF of Gumbel Copula.
(a)(b)(c)(d)(e)


## 3.3. Reliability Model of the Series Mechanical System

Suppose that the series system is composed ofn units, and its joint distribution function is Fyy1,y2,…yn, Yit=σi−si, Yi represents the functional function of the ith part at time t, σi represents the strength of the part, and si represents the stress of the part, then the dynamic reliability of the series system [24] is(27)Rst=PY1t>0,Y2t>0,…,Ynt>0=Δ1−R1t1Δ1−R2t1⋯Δ1−Rnt1Cw1,w2,…wn,where Δ represents the difference symbol, Δw1w2Cw,γ0=Cw2,γ0−Cw1,γ0 represents the one-fold difference, Δγ1γ2Δw1w2Cw,γ=Cw2,γ2−Cw2,γ1−Cw1,γ2+Cw1,γ1 represents the double integral, and so on, Rjt is the dynamic reliability of the part, and wj=Fjt=1−Rjt,j=1,2,…,n.When calculating the reliability of a specific mechanical system, the multivariate Gaussian Copula function [25] and the Archimedes Copula function are usually selected to deal with the correlation problem.


## 3.4. Dynamic Reliability of Driving and Driven Gears

During the process of power transmission, the driving and driven gears in the high-speed trains gearbox are subjected to external excitation from other parts as well as internal meshing impact excitation. The main failure models are fatigue failure, including root fracture, pitting, wear, flaking, galling, root fracture, and tooth surface damage [26]. Based on their failure forms, when calculating contact stress and bending stress of gears, the action of random time-varying load and strength degradation need to be considered, then the stress-strength interference models of tooth face contact strength and tooth root bending strength of driving and driven gears are established, so as to derive the dynamic reliability of contact fatigue and bending fatigue of driving and driven gears.According to GB/T3480-1997, the calculated contact stress and calculated bending stress of gears are(28)St=Kt⋅CNm=ZNZVZLZWZRZX⋅Cftm,where ZN is the contact strength life factor, taken ZN=1.1; ZV is the speed factor, taken ZV=1.006; ZL is the lubricant factor, taken ZL=1.01; ZW is the work hardening factor, taken ZW=1; ZR is the roughness factor, taken ZR=1.016; and ZX is the dimension factor for contact strength calculation, taken ZX=1.(29)Sb=Kb⋅CNm=YSTYNTYδrelTYRrelTYX⋅Cftm,where YST is the tress correction factor, taken YST=2; YNT is the life factor of bending strength, taken YNT=1.08; YδrelT is the relative tooth root fillet sensitivity factor, taken YδrelT=1.01; YRrelT is the relative tooth root surface condition factor, taken YRrelT=1.12; and YX is the bending strength calculation factor, taken YX=1.The probability distribution of the contact stress on the tooth surface of the driving gear and the driven gear obeyssH∼N458,35.512. The probability distribution of the bending stress at the root of the driving gear obeys sF1∼N95.74,1.322, and the probability distribution of the bending stress at the root of the driven gear obeys sF2∼N93.17,1.292. The tooth surface contact fatigue strength and tooth root bending fatigue strength of the driving and driven gears are, respectively, obeyed:(30)σH∼N1031.25,61.882,σF∼N575.79,46.062.The amount of strength degradation is a random variable, and this study uses the Wiener process to describe the random strength degradation law of each part in the gearbox. The degradation law of gear strength is related to the nature of the material and the working condition of the work condition, so the strength degradation law of driving and driven gears will be different. The P-S-N curve of the material can reflect a more comprehensive stress-life relationship, and the P-S-N curves can reflect the probabilistic information of three aspects, as shown in Figure4.Figure 4
P-S-N curve of the material.According to data collection, the fatigue test data and P-S-N curve of gear material 17CrNiMo6 were obtained [27]. The characteristic parameters in the Wiener process degradation model are estimated, which combines the P-S-N curve of the gear material. The specific calculation process is shown in Figure 5.Figure 5
Parameter estimation of the Wiener process.Based on the P-S-N curve of the gear material and the Wiener parameter estimation process, the Wiener process parameters of the driving gear and the driven gear can be calculated as shown in Table2.Table 2
Strength degradation parameters of the driving and driven gears.
Driving gearDriven gearContact fatiguea51.610049.1722b29.143826.6691Bending fatiguea116.6235105.8101b62.712457.3872Combining the stress distribution, strength distribution, and strength degradation process parameters of gears determined above, the dynamic reliability of the driving and driven gears can be obtained, as shown in Figure6.Figure 6
(a) Fatigue reliability of the four-point angular contact ball bearing. (b) Fatigue reliability of the outer cylindrical roller bearing. (c) Fatigue reliability of the inner cylindrical roller bearing. (d) Fatigue reliability of the outer tapered roller bearing. (e) Fatigue reliability of the inner tapered roller bearing.
(a)(b)(c)(d)(e)It can be seen from Figures6(a) and 6(b), the tooth face contact fatigue reliability of gears decreases relatively faster than the tooth root bending fatigue reliability, which is in line with the conclusion that the main form of gear failure is contact fatigue failure. In addition, when the working time of high-speed trains is within 2000 hours, the contact fatigue reliability of gears in the gearbox is around 0.97, which is in line with the secondary maintenance standard of the corresponding vehicle type, and the gearbox should be maintained by oil change. This shows that the calculation method of this study is in line with the practical application of engineering.Figure 7
(a) Fatigue reliability of driving gear. (b) Fatigue reliability of driven gear.
(a)(b)


## 3.5. Dynamic Reliability of Different Types of Bearings

The bearings arrangement in the gearbox is shown in Figure1. A four-point angular contact ball bearing (FPACBB) is mounted on the side of the input shaft, it is near the motor to carry the axial load when the motor drives the input shaft to rotate. The mounting cylindrical roller bearings at both the ends of the driving gear are used to bear the radial load when the input shaft rotates, and the mounting tapered roller bearings (TRB) at both ends of the output shaft bear both the radial and axial loads of the output shaft. The distribution of contact stress and strength of rolling bearings are considered as normal distribution [28]. The contact stress calculation of rolling bearings in gearbox is calculated by referring to the algorithm in Harris and Kotzalas [29]. After calculation and analysis, it can be obtained: the surface contact stress distribution of the four-point contact ball bearing obeys sb1∼N1066.5,106.652, the surface contact stress distribution of the outer cylindrical roller bearing of the driving gear obeys sb2∼N671.29,67.122, and the surface contact stress distribution of the inner cylindrical roller bearing of the driving gear obeys, the surface contact stress distribution of the outer tapered roller bearing of the driven gear obeys sb4∼N852.34,85.232, and the surface contact stress of the inner tapered roller bearing of the driven gear obeys sb5∼N696.64,69.662, the fatigue strength of the bearing material obeys σb∼N1718.18,171.82.Based on the algorithm in Figure7 and the P-S-N curve of bearing steel GCr15A, the Wiener strength degradation parameters of each bearing are determined as shown in Table 3.Table 3
Strength degradation parameters of the bearings.
Strength degradation parameters of bearingsabFour-point contact ball bearing14.784411.9669Outer cylindrical roller bearing14.784411.9669Inner cylindrical roller bearing14.784411.9669Outer tapered roller bearing16.211813.4591Inner tapered roller bearing16.211813.4591Based on the above stress distribution, strength distribution, and strength degradation process parameters of each bearing, the dynamic reliability of each bearing can be obtained, as shown in Figure6.It can be seen from Figure6, among the various types of bearings, the reliability of four-point contact ball bearings decreases faster compared to other bearings, which is due to the more complex load with four-point ball bearings. In addition, the reliability of four-point ball bearings can be maintained above 0.96, when the high-speed trains run within 2000 hours, and the reliability of the other two types of bearings are above 0.99, which is in line with the secondary maintenance standard. It shows that the reliability calculation method for bearings is in line with the engineering practice.


## 3.6. Dynamic Reliability of the Gearbox Housing

The gearbox housing is not only subjected to external excitation from the wheel track caused by track unevenness, but also the internal excitation caused by the time-varying meshing stiffness and transmission error of gear transmission and other excitation such as motor torque load, resulting in fatigue cracks in the housing [9]. The gearbox housing is distinguished from other standard parts by stress analysis due to its complex shape. The equivalent stress at the weak point of the housing is obtained by finite element analysis or line stress spectrum calculation [30], and the probability distribution of the equivalent stress is determined, according to the damage consistency principle as the stress distribution of the gearbox housing. The distribution of equivalent stress calculated under specified operating conditions obeys s∼N24.91,0.832.The fatigue strength of the gearbox housing depends on material properties, and it can be considered to be normal distribution. The material of the gearbox housing is AlSi7Mg0.3, according to the fatigue strength standard of cast aluminum alloy in BS EN 1999-1-3-2007, the fatigue strength variation coefficient is taken to be 0.1, and the fatigue strength of the housing obeysσ∼N29.94,2.992.Based on the P-S-N curve of the gearbox housing material [14], the Wiener strength degradation parameters of gearbox housing are determined as shown in Table 4.Table 4
Strength degradation parameters of the gearbox housing.
Degradation of contact fatigue strength of the gearbox housingabGearbox housing77.811450.2314Based on the parameters of the stress distribution, strength distribution, and strength degradation of the gearbox housing, the dynamic reliability of the gearbox housing can be obtained as shown in Figure8.Figure 8
Fatigue reliability of the gearbox housing.It can be seen from Figure8, under the premise of satisfying the casting quality and no excessive external impact, the fatigue reliability of the gearbox housing keeps decreasing, when the train’s running time is increasing, which indicates that the wear state of the transmission system parts will affect the reliability of the housing. When the train’s running time is within 4000 hours, the reliability of the housing can be maintained above 0.98, which meets the mileage requirements for level III and level IV maintenance of the vehicle. It shows that the above calculation method for gearbox housing is in line with the actual application of the engineering.


## 4. Dynamic Reliability Analysis of the Gearbox Based on Copula Function

### 4.1. Copula Function Model of the Gearbox

High-speed trains gearbox system is a series mechanical system consisting of driving and driven gear pairs, bearings and gearbox housing. An important condition for the gearbox system not to fail is that none of the above key components fails. When the number of parts of the series mechanical system is large, it is difficult to solve through the integration of the joint probability density function of parts. If the gearbox system in this study is expressed through a Copula function for the failure correlation of all key parts, the calculation will be complicated due to high dimension. Therefore, based on the system failure hierarchy correlation, a fully nested Copula function is considered to realize the modeling of the high-dimensional Copula function. In modeling the Copula function of gearbox system, two different levels of failure correlations are considered between different failure modes of the same part and between different parts of the system.Any N-ary Archimedes Copula function can be equivalent to a binary Archimedes Copula function, and conversely, it can also be constructed and calculated by a binary Archimedes Copula function, this property provides the possibility to construct the nested Copula function models. The binary Frank Copula function is selected to obtain a sketch of the failure-related nested Copula model structure of the high-speed trains gearbox system, as shown in Figure9.Figure 9
The nested Copula function models of the gearbox system.In Figure9, Miji=1,2,…8,j=1,2 is the performance function of the kth part in the ith failure mode, i=1,2,…8 represents the driving gear, driven gear, outer cylindrical roller bearing, inner cylindrical roller bearing, four-point contact ball bearing, outer tapered roller bearing, inner tapered roller bearing, and gearbox housing. j=1,2 means contact fatigue failure and bending fatigue failure, respectively, it should be noted that M81 represents the fatigue failure function of the gearbox housing. C1⋅,C2⋅ is the Copula function describing the two failure modes of the driving and driven gears, and C3⋅,C4⋅…C9⋅ is a Copula function that describes the failure correlation between parts.In this study, the nonparametric kernel density estimation method and the great likelihood estimation method are combined to solve the marginal distribution of each random variable and the parameters of the Copula function. Before calculating the parameters of the Copula function by the great likelihood estimation method, the failure samples of different parts or different failure modes are determined, so that the marginal distribution functions of the part failure samples can be determined. Based on the distribution functions of each part and each failure mode determined in the previous section, the Monte Carlo method is used for random sampling, and the part with negative difference between strength and stress of each part or each failure mode is recorded as the failure sample, and its marginal distribution function is obtained by nonparametric kernel density estimation, and the parameters of the Frank Copula function at each level are determined according to the great likelihood estimation process in the previous section, as shown in Table5.Table 5
Parameter values of Frank Copula function.
Copula functionParameter values for the frank copula functionθ13.9735θ25.3857θ322.3798θ422.0796θ518.6860θ67.3525θ746.5217θ811.6168θ933.0648


### 4.2. Dynamic Reliability of the Gearbox under Different Failure Modes

The reliability of each subsystem and the dynamic reliability of the gearbox can be obtained, which is based on the nested Copula function model of the gearbox system and the functional functions of each nested system. For comparison, the reliability of the gearbox system is calculated according to the above three cases: failure-related, mutually independent, and fully correlated; the dynamic reliability of the high-speed trains gearbox system is calculated and shown in Figure10.Figure 10
Dynamic reliability of gearbox system in three different cases.It can be seen from Figure10, the dynamic reliability of high-speed trains gearbox in these three cases gradually decreases with the increase of service time, and the dynamic reliability of gearbox in failure-related cases is between the theoretical calculated value of complete correlation of weak link theory and the theoretical calculated value of failure-independent assumption, which is in line with the engineering practice. The dynamic reliability analysis of high-speed trains gearbox considering random time-varying load, strength degradation and failure-related relationships can predict and evaluate the reliability of gearbox at the design stage, so as to optimize and improve the preliminary design, and can also realize dynamic tracking of gearbox reliability during the service life of gearbox, so as to ensure the operational reliability of gearbox and the timeliness of inspection and maintenance.


## 4.1. Copula Function Model of the Gearbox

High-speed trains gearbox system is a series mechanical system consisting of driving and driven gear pairs, bearings and gearbox housing. An important condition for the gearbox system not to fail is that none of the above key components fails. When the number of parts of the series mechanical system is large, it is difficult to solve through the integration of the joint probability density function of parts. If the gearbox system in this study is expressed through a Copula function for the failure correlation of all key parts, the calculation will be complicated due to high dimension. Therefore, based on the system failure hierarchy correlation, a fully nested Copula function is considered to realize the modeling of the high-dimensional Copula function. In modeling the Copula function of gearbox system, two different levels of failure correlations are considered between different failure modes of the same part and between different parts of the system.Any N-ary Archimedes Copula function can be equivalent to a binary Archimedes Copula function, and conversely, it can also be constructed and calculated by a binary Archimedes Copula function, this property provides the possibility to construct the nested Copula function models. The binary Frank Copula function is selected to obtain a sketch of the failure-related nested Copula model structure of the high-speed trains gearbox system, as shown in Figure9.Figure 9
The nested Copula function models of the gearbox system.In Figure9, Miji=1,2,…8,j=1,2 is the performance function of the kth part in the ith failure mode, i=1,2,…8 represents the driving gear, driven gear, outer cylindrical roller bearing, inner cylindrical roller bearing, four-point contact ball bearing, outer tapered roller bearing, inner tapered roller bearing, and gearbox housing. j=1,2 means contact fatigue failure and bending fatigue failure, respectively, it should be noted that M81 represents the fatigue failure function of the gearbox housing. C1⋅,C2⋅ is the Copula function describing the two failure modes of the driving and driven gears, and C3⋅,C4⋅…C9⋅ is a Copula function that describes the failure correlation between parts.In this study, the nonparametric kernel density estimation method and the great likelihood estimation method are combined to solve the marginal distribution of each random variable and the parameters of the Copula function. Before calculating the parameters of the Copula function by the great likelihood estimation method, the failure samples of different parts or different failure modes are determined, so that the marginal distribution functions of the part failure samples can be determined. Based on the distribution functions of each part and each failure mode determined in the previous section, the Monte Carlo method is used for random sampling, and the part with negative difference between strength and stress of each part or each failure mode is recorded as the failure sample, and its marginal distribution function is obtained by nonparametric kernel density estimation, and the parameters of the Frank Copula function at each level are determined according to the great likelihood estimation process in the previous section, as shown in Table5.Table 5
Parameter values of Frank Copula function.
Copula functionParameter values for the frank copula functionθ13.9735θ25.3857θ322.3798θ422.0796θ518.6860θ67.3525θ746.5217θ811.6168θ933.0648


## 4.2. Dynamic Reliability of the Gearbox under Different Failure Modes

The reliability of each subsystem and the dynamic reliability of the gearbox can be obtained, which is based on the nested Copula function model of the gearbox system and the functional functions of each nested system. For comparison, the reliability of the gearbox system is calculated according to the above three cases: failure-related, mutually independent, and fully correlated; the dynamic reliability of the high-speed trains gearbox system is calculated and shown in Figure10.Figure 10
Dynamic reliability of gearbox system in three different cases.It can be seen from Figure10, the dynamic reliability of high-speed trains gearbox in these three cases gradually decreases with the increase of service time, and the dynamic reliability of gearbox in failure-related cases is between the theoretical calculated value of complete correlation of weak link theory and the theoretical calculated value of failure-independent assumption, which is in line with the engineering practice. The dynamic reliability analysis of high-speed trains gearbox considering random time-varying load, strength degradation and failure-related relationships can predict and evaluate the reliability of gearbox at the design stage, so as to optimize and improve the preliminary design, and can also realize dynamic tracking of gearbox reliability during the service life of gearbox, so as to ensure the operational reliability of gearbox and the timeliness of inspection and maintenance.


## 5. Conclusion

In this study, the dynamic reliability of the high-speed train gearbox system is evaluated by considering the multiple random load effects on each key part, the strength degradation of the part, and the nonlinear failure-related relationships of the part failure. In the evaluation process, the Poisson process, the stress-strength interference theory considering time-varying load, the strength degradation theory based on the Wiener process, and the Copula function theory are combined to consider different hierarchical relationships of the gearbox system, so as to establish the Copula function nested model of the reliability of the gearbox system, and calculate the reliability of the gearbox system. This study establishes a Copula function nested reliability model for a series system of key components of a gearbox system. The model can comprehensively and flexibly describe various failure correlations in the gearbox system and avoid the problem of “dimensional explosion” in the calculation process. The time-varying stress-strength interference model of each key component is established, considering the strength degradation based on the Wiener process, and the degradation parameters of the components are determined by the strength degradation model of the material P-S-N curve, which provides the theoretical basis for the reliability model of the gearbox system. By solving the dynamic reliability of each part and the gearbox system, the reliability results of considering the failure-related reliability of the parts are consistent with the engineering reality, when compared with the two cases of completely related and independent parts. The reliability analysis method in this study can monitor the reliability dynamics of the gearbox during its actual operating life cycle.In the study, the Wiener process is used to describe the random pattern of strength degradation of parts, which is consistent with the general characteristics of strength degradation, but whether it is consistent with the actual phenomenon of strength degradation of gearbox systems, it needs to be further verified based on the subsequent engineering practice. Although the failure correlation model of gearbox system is constructed by Copula function, which has a rigorous mathematical basis, the nature of system failure for a complex nonlinear elastic system like gearbox needs further study. On the basis of the previous study, the fatigue damage characteristics of key parts during actual operation are collected, and the reliability analysis of gearboxes under the interaction of multiple failure modes needs to be further studied, and a multi-parameter composite Copula function is established for the accurate calculation of reliability. The service process of high-speed trains gearboxes is disturbed by many uncertain factors, considering theories and methods such as fuzzy theory and interval analysis to further study the performance failure degradation under uncertain conditions, and realize the real-time and accurate analysis of the dynamic reliability of high-speed trains gearboxes.



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*Source: 1088747-2022-07-30.xml*