It is easy for the dedendum to produce fatigue crack in the transmission process, resulting in the bending fatigue fracture. The dedendum bending stress of gears can be calculated as [20]

(1) σF=YFaYSaYeYβFtbmnKAKVKFβKFα

where Ft is the rated tangential tooth force at the transverse pitch, b is the active face width, mn is the normal module, YFa is the tooth form factor, YSa is the bending stress concentration coefficient, Ye is the contact ratio factor, Yβ is the helix angle coefficient, KA is the work condition coefficient, KV is the dynamic load coefficient, KFβ is the longitudinal load distribution coefficient, KFα is the load distribution coefficient.

Dedendum bending fatigue strength of gears can be calculated as

(2) σFS=σFlimYSTYNTYδrelTYRrelTYX

where σFlim is the experimental gear bending fatigue strength, YST is the experimental gear tooth stress concentration coefficient, YNT is the life coefficient, YδrelT is the relative sensitive coefficient, YRrelT is the relative surface condition coefficient, YX is the size coefficient.

According to the stress-strength interference theory, the performance function of dedendum bending fatigue can be expressed as

(3) g1(X1)=σFS−σF

If the value of the performance function is negative, the structure is a failure. On the contrary, if the value is positive, the structure is safe. The border between the negative and positive domains is called the limit state (g1(X1)=0). X1 contains all the random variables in Eqs. (1) and (2).

The tooth contact fatigue is the common cause of gear failure and is affected by design geometry, material, manufacturing, and other variables. The contact stress of gears can be calculated as [20]

(4) σH=ZHZEZeZβFtd1bu±1uKAKVKHαKHβ

where d1 is the pinion pitch diameter, u is the gear ratio, KHα is the longitudinal load distribution coefficient, KHβ is the transverse load distribution coefficient, ZH is the nodal field coefficient, ZE is the elastic coefficient, Ze is the contact ratio coefficient, Zβ is the spiral angle coefficient.

Allowable contact fatigue strength of gears can be calculated as

(5) σHS=σHlimZNTZLZVZRZWZX

where σHlim is the experimental flank contact fatigue strength, ZNT is the life coefficient, ZL is the lubricant coefficient, ZV is the velocity coefficient, ZR is the tooth fineness coefficient, ZW is the hardening coefficient, ZX is the size coefficient.

The performance function of tooth contact fatigue can be expressed as

(6) g2(X2)=σHS−σH

X2 contains all the random variables in Eqs. (4) and (5). It can be seen that X1 and X2 have the same random variables, so there is a certain degree of the dependence relationship between the two failure modes. Therefore, it is necessary to establish a joint distribution among failure modes to describe their correlation accurately.

Sklar’s theorem: Any multivariate joint distribution can be written in terms of univariate marginal distribution functions and a Copula function, which describes the dependence structure between the two variables. In other words, the Copula function connects the joint distribution of multivariate random variables with their respective marginal distribution. The Copula function has the following properties [16,21,22].

(1) C(u1,u2,⋯,un) is the n-dimensional distribution function defined on [0,1]n, C:In=[0,1]n→[0,1];

(2) For vector u=(u1,u2,⋯,un), if there is any vector ui=0,i=1,2,⋯,n, Cn(u)=0;

(3) When ui,i≠k,i=1,2,⋯,n=1, ∀0≤uk≤1, Cn(u)=Cn(1,⋯,1,uk,1,⋯,1)=uk;

(4) C(u1,u2,⋯,un) is monotonically increasing for any of its variables.

Supposing that x1,x2,⋯,xn are random variables, their marginal distributions are F1(X1),F2(X2),⋯,Fn(Xn), respectively, and their joint distribution is H(x1,x2,⋯,xn). Then, there is a Copula function C(⋅) that connects the marginal distribution and the joint distribution.

(7) H(x1,x2,⋯,xn)=C(F1(X1),F2(X2),⋯,Fn(Xn))

According to the inverse transformation of cumulative distribution function for marginal distribution xi=F−1(ui),i=1,2,⋯,n, the Copula function of the n-dimensional random variables can be formulated as

(8) C(u1,u2,⋯,un)=H[F−1(u1),F−1(u2),⋯,F−1(un)]

When the random variables are two-dimensional, the Copula function is C(u1,u2)=H[F−1(u1),F−1(u2)]. Combined with the above two failure modes, the reliability of gear transmission system considering failure dependence can be expressed as

(9) R=RC=1−P(g1≤0)−P(g2≤0)+P(g1≤0,g2≤0) =1−Fg1(g1)−Fg2(g2)+C(Fg1(0),Fg2(0)) =C(1,1)−C(Fg1(0),1)−C(1,Fg2(0))+C(Fg1(0),Fg2(0))

The normal operation of a mechanical structure depends on the relationship between strength and stress. In the design service period, if the stress at any moment is greater than the structural strength, the structure will fail; when the stress is less than the strength, the structure will appear cumulative fatigue damage, reducing the structure strength until it fails [23–27]. The traditional structural reliability analysis based on the static stress-strength interference model does not consider the influence of time-variant strength on reliability. Due to the impact of random factors such as material oxidation, load fluctuation, and environmental corrosion, the structural strength will gradually decrease in engineering practice, showing a deterioration trend called strength degradation [28–30]. The dynamic stress-strength interference model is constructed, as shown in Fig. 1.

As time goes on, the mean value of structural strength and its probability density distribution function f(r) decreases gradually, while the failure probability of the structure increases. When t=0, there is a certain safety margin between strength and stress, and the structure will not fail. At tx moment, the f(r) decreases to f(r)′ and intersects with the probability density function of structural stress g(s), leading to structural failure. The structural function considering the strength degradation can be expressed as

(10) Z(t)=g[R(t),S(t)]=R(t)−S(t)

where Z(t) is the reliability over time, R(t) is the random process of structural strength, S(t) is the random process of structural stress, t is the service time.

The structural reliability considering the strength degradation during the service period T is

(11) PS(T)=P{Z(t)>0,t∈[0,T]}=P{R(t)>S(t),t∈[0,T]}

Strength degradation is an external characterization of structural performance degradation under the action of various stochastic factors, which will lead to the reduction of structural safety and reliability. From the macroscopic point of view, strength degradation is a continuous process with randomness and irreversibility, which is caused by the characteristics of the structure as well as stress effects. The effects caused by strength degradation should be fully considered when performing structural dynamic reliability analysis. Stochastic process theory is usually used to describe the general law of strength degradation. Since the Gamma process [31,32] is an independent and non-negative incremental process, it is considered a natural choice to describe the degradation process. The Gamma process can describe both small degradation fluctuations and drastic steps, and is well characterized for the degradation mechanism of various structures. The probability density function Q describing the structural strength degradation using the Gamma process can be formulated as

(12) Ga(q|v,u)=uvΓ(v)qv−1exp⁡(−uq)I(0,∞)(q)

where Γ(v)=∫t=0∞tv−1e−tdt is the Gamma function, IA(q) is an indicative function, if q∈A, then IA(q)=1, if q∉A, then IA(q)=0, u>0 is the dimension parameter, v(t)>0 is the shape parameter.

The Gamma process has the following characteristics.

(1) The probability of Q(0)=0 is 1;

(2) When ζ>t≥0, Q(ζ)−Q(t)∼Ga(v(ζ)−v(t),u);

(3) Q(t) has the independent increment.

The mathematical expectation E(Q(t)) and variance Var(Q(t)) of the Gamma process are

(13) {E(Q(t))=v(t)uVar(Q(t))=v(t)u2

Although there are some fluctuations in the strength degradation process, it is generally a stationary random process during engineering experience. Therefore, the shape parameter is a linear function v(t)=at, where a is a constant. So, Eqs. (12) and (13) can be written as

(14) Ga(q|at,u)=uatΓ(at)qat−1exp⁡(−uq)I(0,∞)(q)

(15) {E(Q(t))=atuVar(Q(t))=atu2

Considering the essential characteristics of structural strength degradation, the P−S−N curve is used to determine the values of a and u. The steps are as follows:

(1) The S−N curve is transformed into the S−t curve. The α and β are two constants.

(16) N=f(t),S=β/f(t)α

(2) Let SPi(tj)−SPi(tj+1)=ΔQ^ij, i=1,2,⋯,m, j=1,2,⋯,n denote the strength degradation of the S−t curve with survival rate Pi in the time [tj,tj+1]. Then the estimated values of the mean value and variance of the strength degradation are as follows.

(17) {μ^j=1m∑i=1mΔQ^ijσ^j2=1m−1∑i=1m(ΔQ^ij−1m∑i=1mΔQ^ij)2

(3) The estimated values of characteristic parameters a and u of the Gamma process are calculated. Finally, the final parameters are estimated by averaging a and u obtained from all-time intervals.

(18) {a^=1n∑j=1nuj2(μ^j+σ^j2)(tj+1−tj)(uj+1)u^=1n∑j=1nμ^jσ^j2

Combined with the above introduction, the flow chart of this paper is shown in Fig. 2.

Taking a certain type of EMU gear transmission system as the research object, the dynamic model is constructed according to actual geometric parameters and constraints. According to the 615 kW input power and 4,200 r/min driving gear speed, constant traction torque is applied to the model. The geometric parameters of the gear pair are shown in Table 1. The values of each random variable in X1 and X2 are obtained by referring to the standards and manuals, as shown in Table 2.

The load-time history of the gear transmission system is obtained by dynamic analysis. The stress-time history is obtained by substituting the load-time history into Eqs. (1) and (4), as shown in Fig. 3. It can be seen that the fluctuation of dedendum bending stress and tooth contact stress fluctuate randomly with time, respectively. Therefore, when the strength degenerates to a certain extent, the gear transmission system may fail, even if the average strength is not less than the stress.

The strength degradation is described by Gamma random process. The P−S−N curve of the gear material can be obtained by querying the corresponding mechanical material manual, as shown in Fig. 4, and the specific parameters are shown in Table 3. Since the P−S−N curves of gear and base materials are similar, this paper directly uses the P−S−N curves of the base material to characterize the gear performance.

Define the unit of operating time of the gearing system as h, the N=f(t)=4200×60t=2.52×105t. Substituting the data in Table 3 into Eq. (16) to obtain the S--t curve data, as shown in Eq. (19).

(19) {P=0.50,S=1024.49/2.52×105t−7.39P=0.90,S=1020.88/2.52×105t−6.06P=0.95,S=1020.38/2.52×105t−5.30P=0.99,S=1018.83/2.52×105t−4.77

The value of strength degradation D^ij in any time interval can be calculated from the function of the S--t curve. The parameters uj and aj of the Gamma process are obtained by substituting the D^ij in different time intervals into Eqs. (17) and (18), as shown in Table 4. Then, u = 29.6223, a = 19.2490.

Through the above analysis, the strength degradation of the gear pair can be described as a Gamma process with a shape parameter of 19.2490t and a dimension parameter of 29.6223. By substituting the dynamic stress-time history and Gamma process into Eq. (3) and Eq. (6), the dynamic fatigue reliability model of the gear is constructed. The performance function values of tooth contact and dedendum bending are obtained by numerical calculation.

Nonparametric kernel distribution estimation is used to select the optimal Copula function. The core idea of this method is to use nonparametric kernel distribution to estimate the marginal distribution of F(g1) and F(g2), and preliminarily screen out the Copula functions that can describe the correlation of failure modes. At the same time, the relevant parameters of each Copula function are calculated. Finally, the optimal Copula function is selected by comparing the Square Euclidean distance between each Copula function and the empirical Copula function.

The marginal distribution functions of F(g1) and F(g2) are determined by the kernel distribution estimation method, and the accuracy of the marginal distribution functions is verified by comparing with the empirical distribution function. The results are shown in Fig. 5. It can be seen that the kernel distribution estimation and the empirical distribution coincide, so the marginal distribution functions obtained by the kernel distribution estimation can be used to construct the Copula function.

The bivariate frequency histogram (shown in Fig. 6) is drawn as the joint density function for the two failure modes to choose the appropriate Copula function. Fig.6 shows the frequency histogram has tail symmetry, so the Gauss Copula function and t-Copula function are preliminarily selected to fit the dependence relationship of two failure modes. The expressions of the Gauss Copula function and t-Copula function are as follows:

(20) CGa(g1,g2,R)=∫−∞Φ−1(g1)∫−∞Φ−1(g2)12π1−R2exp⁡(2RX1X2−X12−X222(1−R2))dX1dX2

(21) Ct(g1,g2;R,m)=TR,m(tm−1(g1),tm−1(g2))

where Φ−1(⋅) is the inverse function of the standard normal distribution function. R is an n-dimensional coefficient matrix. TR,m(⋅) is the n dimension distribution function with the coefficient matrix R, and the degree of freedom m. tm−1 is the inverse function of the one-dimensional distribution function.

The parameters estimation results for the Gauss Copula and t-Copula functions are calculated, respectively. The coefficient matrix R of Gauss Copula is as follows:

(22) RGa=[1.00000.92690.92691.0000]

The coefficient matrix R and degree of freedom m of t-Copula are as follows:

(23) Rt=[1.00000.92760.92761.0000],m=7.6936

It is substituting the R and m into Eqs. (20) and (21) to calculate the density function and distribution function of the two Copula functions. The Square Euclid distance between the alternative Copula function and the empirical Copula function was then calculated through Eqs. (24) and (25). Eq. (24) is the expression of the empirical Copula function. Eq. (25) is the Square Euclid distance of the Gauss Copula function and t-Copula function, reflecting the fitting degree of the Copula function to the failure dependence. The smaller the distance, the better the fitting effect. The results are shown in Table 5.

(24) C^(u,v)=1n∑i=1nI[Fn(xi)≤u]I[Gn(yi)≤v],u,v∈[0,1]

(25) dgau2=∑i=1n|C^n(ui,vi)−Cgau(ui,vi)|2,dt2=∑i=1n|C^n(ui,vi)−Ct(ui,vi)|2

where I[⋅] is a characteristic function, when Fn(xi)≤u, the I[Fn(xi)≤u]=1, otherwise, I[Fn(xi)≤u]=0.

According to the results in Table 5, the Square Euclid distance between the t-Copula function and the empirical Copula function was the smallest. It indicates that the t-Copula function is more appropriate to describe the dependence relationship. The density function and distribution function of the t-Copula is shown in Fig. 7. It can be seen that the t-Copula function has good tail correlation characteristics, which can well fit the dependence relationship between the tooth contact fatigue and dedendum bending fatigue.

Substituting the above-calculated parameters into Eq. (9), the dynamic fatigue reliability of gear transmission system considering failure dependence is obtained, as shown in Fig. 8.

It can be seen from Fig.8 that the reliability gradually decreases due to the strength degradation with the increase of service time. The dynamic fatigue reliability of the gear transmission system has an inevitable fluctuation. The main reason for the reliability fluctuation is that both the dynamic contact stress and dedendum bending stress has randomness. Because of the dependence relationship between the two failure modes, the reliability is lower than that of the single failure mode, and the fluctuation is more obvious. The results show that the failure dependence greatly influences the reliability of the gear transmission system. In this paper, only infant mortality (assume 400 hours) is analyzed, and the reliability is following the description of the bathtub curve. The study provides a theoretical basis for considering the correlation in the design stage. Compared with the deterministic reliability analysis, the result is more suitable for engineering practice.