1 Introduction

At present, the reliability analysis of structural system is mainly classified into two categories, i.e., the Monte Carlo simulation (MCS) method and the numerical approximation method. The approximation method often includes the moment method (MM)^[1-6], the perturbation method (PM)^[7-11], the response surface method (RSM)^[12-20], and the stochastic finite element method (SFEM)^[21-25]. Most of these methods involve the calculation of statistical moments. In particular, for multi-dimensional basic random variables, a complex relation exists between the numerical characteristics of the state functions and those of the random variables. However, the expression of these statistical moments lacks a complete and unified matrix form, intuitive and concise matrix description, and simple but efficient matrix algorithm.

The reliability sensitivity analysis is to conduct differential operation on the basis of a reliability analysis. Through the differential operation of reliability, it is possible to acquire the quantitative relation between the reliability and the fluctuations in the distribution parameters of the basic random variables and obtain an assessment on the impact of the changes in the design parameters on the structural system reliability, thus fully reflecting the extent of such impact, i.e., the sensitivity^[26-28]. Currently, the differential operation method of reliability has developed considerably. It can be roughly divided into two categories, i.e., approximate analysis method^[29-31] and numerical method^[32-34]. The commonly used methods include the global finite difference method (GFDM), the direct differentiation method (DDM), and the automatic differentiation method (ADM). The differential operation for studying structural system reliability is especially important to reliability sensitivity analyses, and the differential of the numerical characteristics of the state functions to those of the basic random variables is the key to studying the reliability sensitivity. Similarly, it is possible to reveal the nature of the complex relation by establishing a matrix expression between the numerical characteristics of the state functions and those of the basic random variables.

The moment method has been made a deep research. Alibrandi and Koh^[35] proposed a procedure based on the classical form in the presence of the random parameters and interval uncertain parameters. Du and Hu^[36] modified the form so that the truncated random variables could be transformed into truncated standard normal variables. Yao et al.^[37] used the general optimization solvers to search the mesh portal point (MPP) in the outer loop, and then reformulated the double-loop optimization problem into an equivalent single-level optimization (SLO) problem. Kang et al.^[38] developed a new matrix-based procedure to compute the sensitivity of the system failure probability with respect to the parameters that affected the correlation coefficients between the components. Guo and Lu^[39] presented a generalized non-probabilistic reliability methodology for the analysis and design of the structures with bounded uncertainties.

In reliability calculation, due to the different arrangement patterns for the random variable matrix, the matrix relation and analytical expression of the created state function and those of the random variable are also different. At present, in the reliability and its sensitivity analysis, the calculation for the numerical characteristics of the state function generally uses the methods in the literatures with the representative of Ref.^[40], and the numerical characteristics of the random variable are expressed in the form of the column vector. Besides, the straightening Kronecker technology is required to convert, respectively, the covariance matrix, the third moment matrix, and the fourth moment matrix responding to the column vector, and the created equation is the relation of the column vector for the numerical characteristics between the state function and the random variable.

The matrix theory and the Kronecker technology have been used in this paper to describe the numerical characteristics of the random variable by matrices (square matrix or rectangular array). The matrix operation expression of the first fourth moments for the state function has been completely deduced. Besides, the created equation is the operation relation between the first fourth moments for the state function and the matrix (square matrix or rectangular array) with the relevant numerical characteristics for the random variable. The matrix relation expression about the sensibility of the numerical characteristics for the state function to the mean value and the standard deviation for the random variable has been further deduced, fully reflects the general rules of the numerical characteristics for the state function and the basic random variable, and indicates other calculation methods of the state function to be special cases for such methods. Therefore, the equation deduced in this paper is a common equation with a matrix pattern, and the numerical characteristics for the state function, since the method is in a concise expression form, have avoided high-order and Kronecker product operations, reduced the calculation difficulty featured with a simple calculation method, and provided the convenience for the programming calculation.

A new derived method is used in this paper to analyze the fatigue reliability and reliability sensitivity of the neck part of the hydraulic pump piston, of which the accuracy is verified through Monte Carlo analog and comparison.

2 Matrix method of reliability analysis

As we all know, to calculate the reliability (or failure probability) for component and system, the probability density function or joint probability density function for the basic random variable is required. However, due to the complicated engineering practice and relatively insufficient statistic data, we cannot easily and precisely determine the distribution rule of the design parameters. If the distribution scheme cannot be determined, but the first fourth moments of the design parameters are known (i.e., mean value, mean square error, third moment, and fourth moment), the fourth moment technology can be used to determine the reliability of the structural system.

When the distribution of the basic random variable vector X=(X₁, X₂, …, X_n)^T cannot be sure and the first four moments of the basic random variable vector are known, the reliability index^[40] is

(1)

where μ_g, σ_g, θ_g, and η_g are the first four moments of the state function.

With the fourth moment method, we can obtain the reliability estimator as follows:

(2)

where Φ(·) is the standard normal distribution.

Theoretically, the probability density function of the state function and the integral of the probability density function are difficult to be deduced in mathematics. Therefore, the fourth moment method, which is a numerical method, is a practical and effective way for the reliability analysis. According to the first four moments of the basic random variable vector X=(X₁, X₂, …, X_n)^T, the first four moments of the state function g(X) can be obtained, and then the reliability index β and the reliability R can be obtain.

3 New matrix description of numerical characteristics of state function

According to the known first four moments of the basic random variable vector X=(X₁, X₂, …, X_n)^T, the matrix description of the first four moments of the state function can be obtained by the Taylor series expansion approximation method. According to the first-order approximate Taylor series expansion of the state function, we have

(3)

where is the derivative with the state function to the random variable in the mean value μ.

With the first-order approximate Taylor series expansion, we can obtain the mean of the state function as follows:

(4)

The variance of the state function g(X) is

(5)

where C₂(X) is the covariance matrix^[41] of the random variable X expressed by

and the matrix expression of the variance of the state function is

(6)

According to the Taylor series expansion expression, with the first-order approximate of the state function g(X), we can obtain the third moment of the state function θ_g as follows:

(7)

According to the property of the Kronecker product, we have

(8)

Then, Eq. (7) can be expressed as follows:

(9)

where C₃(X) is the third moment matrix^[41] of the random variable X expressed by

According to the Taylor series expansion expression, g(X) is the first-order approximate of the state function, and the fourth moment of the state function η_g is

(10)

With Eq. (8), we can obtain the fourth moment of the state function η_g as follows:

(11)

where C₄(X) is the fourth moment matrix^[41] of the random variable X expressed by

Therefore, the second moment expression and the fourth moment expression for the reliability analysis of the structural system have been deduced by the matrix theory and described the matrix form of the fourth-order statistical moment completely.

4 Partial derivatives of first four moments of state functions with respect to mean value of random variable μ_X 4.1 Partial derivative of state function mean value with respect to mean μ_g value of basic random variable μ_X

(12)

where , , …, and are the partial derivatives of the state function mean value with respect to the mean value variable of each random variable.

4.2 Partial derivative of standard deviation of state function with respect to mean vector of basic random variable vector

According to the matrix derivative equation

(13)

we can obtain the partial derivative of σ_g² with respect to μ_X as follows:

(14)

(15)

4.3 Partial derivative of third moment of state function with respect to mean vector of basic random variable vector

According to Eq. (13), the partial derivative of θ_g with respect to μ_X is

(16)

4.4 Partial derivative of fourth moment of state function with respect to mean vector of basic random variable vector

With Eq. (13), we can obtain the partial derivative of η_g with respect to μ_X as follows:

That is

(17)

5 Partial derivative of numerical characteristics of state functions with respect to standard deviation vector σ_X of random variables 5.1 Partial derivative with mean value μ_g of state function with respect to standard deviation vector σ_X of basic random variable vector

With the first-order approximate of the Taylor series expansion of the state function g(X), i.e., μ_g=g(μ_X), where μ_g is the function of μ_X and is independent of σ_X^T, we have

(18)

5.2 Partial derivative of variance of state function with respect to standard deviation vector of basic random variable vector

According to the matrix theory, we can transform C₂(X) to

(19)

where (ρ) is the correlation coefficient matrix, and diag (·) is the diagonal matrix. Substituting Eq. (19) into Eq. (6), we can obtain the variance of the state function as follows:

(20)

where

(21)

(22)

Set

Then, we can obtain the variance of the state function as follows:

(23)

According to the matrix differential operation equation (13), we can obtain the partial derivative of σ_g² with respect to σ_X^T by

where C_2μ(X) is the symmetric matrix, C_2μ(X)=(C_2μ(X))^T, and =1. Therefore,

(24)

The partial derivative of the state function standard deviation with respect to the standard deviation vector is

(25)

5.3 Partial derivative of third moment of state function with respect to standard deviation vector of basic random variable vector

Transform the third moment matrix C₃(X) by

(26)

where (ρ₃) is the n × n² -order matrix (the correlation coefficient matrix of the third moment of the random variable). Substituting Eq. (26) into Eq. (9) yields the third moment of the state function as follows:

(27)

(28)

Set C_3μ(X)=, where C_3μ(X) is independent of σ_X. Then, we have

(29)

With Eq. (13), we can obtain the partial derivative with the third moment of the state function with respect to the standard deviation vector by

(30)

5.4 Partial derivative of fourth moment of state function with respect to standard deviation vector of basic random variable vector

Transform the fourth moment matrix of the random variable as follows:

(31)

where ρ₄ is the n² × n²-order matrix (the correlation coefficient matrix of the fourth moment of the random variable).

Then, substituting Eq. (31) into Eq. (11) yields

(32)

where

(33)

Substituting Eqs. (28) and (33) into Eq. (32) yields

(34)

Set Then, we have

(35)

With Eq. (13), we can obtain the partial derivative of the fourth moment of the state function with respect to the standard deviation vector by

(36)

6 Partial derivative of reliability index β_FM with respect to numerical characteristic of state function

The partial derivative of the reliability index β_FM with respect to the mean value μ_g, the standard deviation σ_g, the third moment θ_g, and the fourth moment η_g are

(37)

(38)

(39)

(40)

7 Matrix method of reliability sensitivity analysis

Since the physical, geometrical, and load parameters in the complex structural system are numerous, the parameter, which has a great effect on the structure system, must be concerned when the structural system is designed and modified, the sensitivity degree of the parameter effect on the product characteristic is researched, and the effect of the parameter change is estimated. The change in the structural system must be pointed out when the parameters change. Then, the sensitivity, reanalysis, and redesign must be analyzed so as to make the structural characteristic optimum. This section provides the matrix formula for the derivative computation of the first four moment functions, and expresses the vector and matrix forms of the derivative computation expressions by the matrix theory, the Kronecker product, and the probability and statistics theory. The matrix relation expression about the derivative of the first four moments of the state function and the random variable is established, and the theory and technology support for the reliability analysis and assess is provided.

The derivative with the reliability of system to the mean value and the mean squared deviation of the random parameter vector X=(X₁, X₂, …, X_n)^T are

(41)

(42)

(43)

where φ(·) is the standard normal probability density function.

Substitute all known conditions and Eqs. (12), (15), (16), (17), (18), (25), (30), (36), (37), (38), (39), (40), and (43) into the reliability sensitivity equations (41) and (42). Then, we can obtain the reliability sensitivity information and . The differential relation equation with the reliability of the structural system to the basic random variable distribution parameters can be deduced by the matrix differential theory, which is clearer mathematically and more convenient for the calculation.

The gradient of the reliability grad R(μ_Xi, σ_Xi) is approximated by the following function:

(44)

8 Examples

The key component of the hydraulic pump is always under pulsating the alternating loading. The alternative stress level is higher and higher when the key components provide a direction of high pressure, high speed, and extramalization. Therefore, the antifatigue reliability design and the finite life analysis of the key component of the hydraulic pump must be paid attention to. Because of the complexity of the hydraulic piston pump, the traditional strength design method cannot reflect the pulsating stress and fatigue strength condition exactly. Therefore, the strength design method based on the mechanical reliability design theory must be researched.

In the hydraulic system, the minimum length of the plunger inside the cylinder is L_0s=24×10^-3 m, the rotation rate of the cylinder is n=1 500 r·min^-1, the effective stress concentration factor is K_σ=1.55, the size factor is ε_σ=0.87, the superficial mass factor is β_σ=0.95, the sensitivity coefficient is ψ_σ=0.2, and the endurance limit of the symmetric cycle is s_-1=4.30 × 10⁸ MPa. Table 1 lists the statistical properties of the random variables of the plunger pump and the obtained antifatigue reliability and sensitivity, where γ is the swashplate angle of the hydraulic pump, m_z is the mass of the plunger, d is the diameter of the plunger, p_d is the work stress of the plunger, R is the radius of the distribution circle of the plunger, f is the friction coefficient of the plunger and cylinder, d_C is the diameter of the plunger bulb, D_r is the diameter of the plunger neck, d_r is the diameter of the oil pipe inside the plunger, and L_C is the length of the plunger neck.

8.1 Mechanical analyses of plunger of hydraulic plunger pump 8.1.1 Force analysis of plunger

Figure 1 shows the force diagram of the plunger blub, where F_A is the axial force, F_C is the shear force, and M is the flexural moment, and their values are

(45)

(46)

(47)

In Fig. 1, γ is the swashplate angle, d_C is the diameter of the plunger bulb, L_C is the length of the plunger neck, and F_N is the force sliding shoes exerted on the plunger^[41]. The value of F_N is

(48)

where F_d is the value of the resultant force of the work stress on the plunger, F_r is the value of the relative inertia force, F_ey is the projection of the convected inertial force on the y-axis, f is the friction coefficient of the plunger and cylinder, and K_p is the structural parameter. They are expressed as follows:

where l₀ is the length of the plunger at the optional position inside the cylinder, l is the plunger length, and l₂=(6l₀l -4l₀ ² -3fdl₀)/(12_l -6fd -6l₀).

8.1.2 Stress analysis of plunger

According to the force analysis of the plunger, the resultant forces of the axial stress and the bending stress work on the cross-section I-I. Then, we can obtain the stress caused by the axial component of F_N as follows:

(49)

where A' is the cross-sectional area of the cross-section I-I defined by .D_r and d_r are the diameters of the cross-section I-I and the plunger internal oil hole, respectively.

Bending the stress caused by the axial component of F_N yields the maximum crushing stress at the cross-section I-I as follows:

(50)

The upper extreme point of the cross-section I-I is a dangerous point, where the maximum combined stress is

(51)

8.1.3 Endurance limit of component

The relation of the endurance limit of the pulsation cycle s_r0 under the asymmetry pulsation cycle stress and the endurance limit of the symmetric cycle of the material s_-1 is

(52)

where s_r0 is the fatigue limit of the component, K_σ is the effective stress concentration factor, β is the superficial mass factor, ε_σ is the size factor, and ψ_σ is the sensitivity coefficient.

8.2 Antifatigue reliability design and analysis of plunger in hydraulic plunger pump

According to interference theory, we can obtain the state function as follows:

(53)

where the basic random parameter vector is X=(γ, m_z, d, p_d, R, f, d_C, D_r, d_r, L_C)^T, the mean value is E(X), the covariance matrix is C₂(X), the third moment matrix is C₃(X), the fourth moment matrix is C₄(X), and the basic random parameter vector is

However, the distribution of them cannot be confirmed.

The partial derivative with the state function g(X) to the basic random parameter vector X is

(54)

Substituting all the known conditions and related data into the mean values and the variance formula of the state function of the plunger, then substituting the reliability index and reliability formula, we can obtain the antifatigue reliability index and reliability as follows: β_FM=2.616 3, and R_FM=0.995 6.

With the MCS method, we have R_MC=0.995 8.

Figure 2 shows the change curves of the antifatigue reliability of the plunger R_FM over the angular displacement of the plunger φ (0≤φ≤π). From the figure, we can see that the reliability will increase when the angular displacement increases. The imaginary line represents the results obtained by the MCS method and the provided method in this paper. The two results agree each other very well.

9 Antifatigue reliability sensitivity analysis of plunger in hydraulic plunger pump

Table 2 is the reliability sensitivity of the reliability R_FM to the mean vector μ_X and the standard deviation vector σ_X of the Basic random parameter vector

According to the reliability sensitivity matrix , we can see that the reliability of the hydraulic plunger pump increases with the increase in the mean value of the diameter of the plunger neck D_r, and the incidence of failures increases with the increases in the mean values of the swashplate angle γ, the mass of the plunger m_z, the diameter of the plunger d, the work stress of the plunger p_d, the radius of the distribution circle of the plunger R, the friction coefficient of the plunger and cylinder f, the diameter of the plunger bulb d_C, the diameter of the oil pipe inside the plunger d_r, and the length of the plunger neck L_C.

According to the reliability sensitivity matrix , the incidence of failures would increase with the increases in the standard deviations of the basic random parameters. The standard deviation is used as a normalization factor to make the sensitivity dimensionless^[42]. The changed parameters are the diameter of the plunger neck D_r, the work stress of the plunger p_d, the friction coefficient of the plunger and cylinder f, the diameter of the plunger d, the swashplate angle of the hydraulic pump γ, the diameter of the plunger bulb d_C, the length of the plunger neck L_C, the diameter of the oil pipe inside plunger d_r, the mass of the plunger m_z, and the radius of the distribution circle of the plunger R.

9.1 Change rule of reliability sensitivity of plunger with respect to its rotation angle φ

φ is the angular coordinate of the plunger at work, the reliability sensitivity of the plunger changes with the change of the rotation angle φ. Figure 3 shows the change rule of the mean sensitivity of the plunger with respect to its rotation angle φ, and Fig. 4 shows the change rule of the standard-deviation sensitivity of the plunger with respect to its rotation angle φ.

In Fig. 3, Sensitivity 1 shows the change rule of the mean sensitivity of the swashplate angle of the hydraulic pump with respect to its rotation angle φ, Sensitivity 2 shows the change rule of the mean sensitivity of the mass of the plunger with respect to its rotation angle φ, Sensitivity 3 shows the change rule of the mean sensitivity of the diameter of the plunger with respect to its rotation angle φ, Sensitivity 4 shows the change rule of the mean sensitivity of the work stress of the plunger with respect to its rotation angle φ, Sensitivity 5 shows the change rule of the mean sensitivity of the radius of the distribution circle of the plunger with respect to its rotation angle φ, Sensitivity 6 shows the change rule of the mean sensitivity of the friction coefficient of the plunger and the cylinder with respect to its rotation angle φ, Sensitivity 7 shows the change rule of the mean sensitivity of the diameter of the plunger bulb with respect to its rotation angle φ, Sensitivity 8 shows the change rule of the mean sensitivity of the diameter of the plunger neck with respect to its rotation angle φ, Sensitivity 9 shows the change rule of the mean sensitivity of the diameter of the oil pipe inside the plunger with respect to its rotation angle φ, Sensitivity 10 shows the change rule of the mean sensitivity of the length of the plunger neck with respect to its rotation angle φ. From Fig. 3, we can see that the range of the rotation angle φ is 0≤φ≤π, and the absolute value of each mean sensitivity is the maximum at the dangerous point φ=0 (the top dead center). The results also show that the influence degree of the plunger mean value with respect to the reliability is the most at this time.

In Fig. 4, Sensitivity 11 shows the change rule of the standard-deviation sensitivity of the swashplate angle of the hydraulic pump with respect to its rotation angle φ, Sensitivity 12 shows the change rule of the standard-deviation sensitivity of the mass of the plunger with respect to its rotation angle φ, Sensitivity 13 shows the change rule of the standard-deviation sensitivity of the diameter of the plunger with respect to its rotation angle φ, Sensitivity 14 shows the change rule of the standard-deviation sensitivity of the work stress of the plunger with respect to its rotation angle φ, Sensitivity 15 shows the change rule of the standard-deviation sensitivity of the radius of the distribution circle of the plunger with respect to its rotation angle φ, Sensitivity 16 shows the change rule of the standard-deviation sensitivity of the friction coefficient of the plunger and cylinder with respect to its rotation angle φ, Sensitivity 17 shows the change rule of the standard-deviation sensitivity of the diameter of the plunger bulb with respect to its rotation angle φ, Sensitivity 18 shows the change rule of the standard-deviation sensitivity of the diameter of the plunger neck with respect to its rotation angle φ, Sensitivity 19 shows the change rule of the standard-deviation sensitivity of the diameter of the oil pipe inside plunger with respect to its rotation angle φ, Sensitivity 20 shows the change rule of the standard-deviation sensitivity of the length of the plunger neck with respect to its rotation angle φ. From Fig. 4, we can see that the absolute value of each standard-deviation sensitivity is also the maximum in the dangerous point φ=0 (the top dead center). The results also show that the influence degree of the plunger mean with respect to the reliability is the most at this time. Therefore, in reliability sensitivity design, aiming at the dangerous point φ=0 (the top dead center) is correct.

The proportion of the parametric sensitivities is shown in Fig. 5, which can provide the basis for reliability design.

10 Conclusions

This paper establishes the matrix relation between the numerical characteristics of the state functions and those of the basic random variables, fully reveals the inherent necessary relation between them, and provides a complete and unified universal matrix expression for the relations of these statistical moments. It establishes the matrix relation for the differential of the numerical characteristics of the state function to those of the basic random variables, and provides a universal matrix expression for the differential relations of these statistical moments, which provides an essential matrix description for the structural system reliability and reliability sensitivity analysis, achieves a perfect matrix for the statistical moment functions, and facilities the implementation of the procedures and formats for the structural system reliability and reliability sensitivity analysis. In engineering examples, an antifatigue reliability analysis and a reliably sensitivity analysis are conducted for the plungers in the hydraulic pump system, and a comparison is made by use of the Monte Carlo method. The method is fully demonstrated for its correctness and practicality. Therefore, it lays a solid theoretical foundation for the reliability analysis and design of the structural system and its sensitivity study.