Dynamic reliability evaluation method for coupling faults of middle trough of scraper conveyor

Abstract

The present invention discloses a dynamic reliability evaluation method for coupling faults of a middle trough of a scraper conveyor. Approximation precision of a moment-based saddlepoint approximation method and a function attribute of dynamic t-copula are fully utilized, and the dynamic reliability evaluation method for the coupling faults of the middle trough of the scraper conveyor is provided, so that dynamic correlation between failure modes of the middle trough of the scraper conveyor under a small sample condition is more accurately described, and accuracy of dynamic reliability evaluation of the coupling faults of the middle trough of the scraper conveyor is improved.

Claims

1. A dynamic reliability evaluation method for coupling faults of a middle trough of a scraper conveyor, comprising the following steps: step 1: defining a data collection time point, collecting data samples such as a crack length, a middle plate width, a fracture toughness, a yield strength and a fatigue load of the middle trough of the scraper conveyor at different operation time points, and counting first four moment probability information of each category of the data samples; step 2: respectively defining performance functions in two failure modes according to fracture failure criteria and static strength failure criteria of the middle trough of the scraper conveyor; step 3: on a basis of the first four moment probability information of each category of the data samples of the middle trough of the scraper conveyor, building a probability distribution function of each failure mode by using a moment-based saddlepoint approximation method, and calculating failure probabilities of fracture failure mode and static strength failure mode at different time points; step 4: building a dynamic probability correlation structure between the fracture strength mode and the static strength failure mode of the middle trough of the scraper conveyor by using a dynamic copula function, and further building a dynamic joint probability distribution function between the failure modes; and step 5: performing dynamic reliability evaluation on the coupling faults of the middle trough of the scraper conveyor in combination with the probability distribution function and the dynamic joint probability distribution function of each failure mode of the middle trough of the scraper conveyor and by using a system reliability theory.

2. The dynamic reliability evaluation method for a coupling faults of a middle trough of a scraper conveyor according to claim 1, wherein for details in the step 1: the first four moment probability information of all the data samples of the middle trough of the scraper conveyor refers to a mean, a variance, skewness and kurtosis.

3. The dynamic reliability evaluation method for coupling faults of a middle trough of a scraper conveyor according to claim 1, wherein for details in the step 2: the performance function of the fracture failure mode is defined according to whether a maximum stress strength factor of the middle trough of the scraper conveyor exceeds the fracture toughness or not, and the performance function of the static strength failure mode is defined according to whether a structure resistance of the middle trough of the scraper conveyor is greater than the fatigue load or not.

4. The dynamic reliability evaluation method for a coupling faults of a middle trough of a scraper conveyor according to claim 1, wherein for details in the step 3: a form of the probability distribution function built by using the moment-based saddlepoint approximation method is $F_Y\left(y\right)=\Phi \left[{\omega }_{\mathrm{ye}}+\frac{1}{{\omega }_{\mathrm{ye}}}\ln \left(\frac{{\upsilon }_{\mathrm{ye}}}{{\omega }_{\mathrm{ye}}}\right)\right],$ wherein ω.sub.ye and ν.sub.ye are parameters of the function, and may be calculated by the following formulas:
ω.sub.ye=sign(t.sub.e)√{square root over (2[y.sub.et.sub.e−K.sub.Ys(t.sub.e)])} and
ν.sub.ye=t.sub.e√{square root over (K.sub.Ys.sup.(2)(t.sub.e))} wherein y.sub.e represents a standardized variable of a performance function state variable, K.sub.Ys represents a cumulant generating function of the standardized variable, K.sub.Ys.sup.(2) represents a second derivative of the cumulant generating function, and t.sub.e represents a saddle point value, and may be calculated by the following formulas: $t_1=\frac{-\sqrt{\left(16a_2{a}_3+{\left(y-a_1\right)}^2\right)b^2-4a_2\left(y-a_1\right)b+4a_2^2}+\left(y-a_1\right)b+2a_2}{4{\mathrm{ba}}_2}$ and $t_2=\frac{-\sqrt{\left(16a_2{a}_3+{\left(y-a_1\right)}^2\right)b^2-4a_2\left(y-a_1\right)b+4a_2^2}+\left(y-a_1\right)b+2a_2}{4{\mathrm{ba}}_2},$ wherein y is a second-order reliability index of the performance function, t.sub.1 and t.sub.2 represent two solutions of the saddlepoint equations, and a value meeting calculation reasonability is taken as a saddle point value in practical calculation; and $a_1=-\frac{9{\theta }_{\mathrm{Ys}}^3}{2{\left({\eta }_{\mathrm{Ys}}-3\right)}^2},a_2=\frac{-3{\theta }_{\mathrm{Ys}}^3+2{\eta }_{\mathrm{Ys}}-6}{4\left({\eta }_{\mathrm{Ys}}-3\right)},a_3=\frac{27{\theta }_{\mathrm{Ys}}^4}{4{\left({\eta }_{\mathrm{Ys}}-3\right)}^3},b=\frac{{\eta }_{\mathrm{Ys}}-3}{3{\theta }_{\mathrm{Ys}}}$ wherein θ.sub.Ys and η.sub.Ys respectively represent skewness and kurtosis of the performance function.

5. The dynamic reliability evaluation method for a coupling faults of a middle trough of a scraper conveyor according to claim 1, wherein for details in the step 4: the dynamic copula function uses a dynamic t-copula function in a form: $C_t\left(u_1,u_2.Math.\rho ,k\right)={\int }_{-\infty }^{t_k^{-1}\left(u_1\right)}{\int }_{-\infty }^{t_k^{-1}\left(u_2\right)}{\frac{1}{2\pi \sqrt{1-{\rho }^2}}\left[1+\frac{s^2-2\mathrm{\rho st}+t^2}{k\left(1-{\rho }^2\right)}\right]}^{-\left(k+2\right)/2}\mathrm{dsdt},$ wherein k and ρ are parameters of the dynamic t-copula function, k is an invariable parameter, and ρ is a time-varying parameter; a value of the time-varying parameter ρ at different time points is obtained in combination with the data samples collected at different time points through a maximum likelihood estimation method; and u.sub.1 and u.sub.2 represent obtained failure probabilities in the fracture failure mode and the static strength failure mode through calculation in the step 3.

6. The dynamic reliability evaluation method for a coupling faults of a middle trough of a scraper conveyor according to claim 1, wherein for details in the step 5: according to an obtained failure probabilities u.sub.1 and u.sub.2 in the fracture failure mode and the static strength failure mode of the middle trough of the scraper conveyor in the step 3, and the obtained dynamic joint probability distribution function between the two failure modes in the step 4, dynamic reliability of the coupling faults of the middle trough of the scraper conveyor is calculated by the following formula, specifically:
P.sub.f=u.sub.1+u.sub.2−P(u.sub.1,u.sub.2|ρ), wherein P(u.sub.1,u.sub.2|ρ) is a joint failure probability obtained through calculation according to the dynamic joint probability distribution function in the step 4; and through calculation on reliability at different time points, the dynamic reliability evaluation on the coupling faults of the middle trough of the scraper conveyor is obtained.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is a flow chart of the present invention.

(2) FIG. 2 is a structural diagram of a middle trough of a scraper conveyor according to an embodiment of the present invention.

(3) FIG. 3 is a probability density diagram of a dynamic t-copula function.

(4) FIG. 4 is a scatter diagram of the dynamic t-copula function.

(5) FIG. 5 is a dynamic reliability curve diagram of a coupling faults of a middle trough of a scraper conveyor.

DETAILED DESCRIPTION OF THE INVENTION

(6) The present invention is further described below with reference to the accompanying drawings and embodiments.

(7) As shown in FIG. 1, a dynamic reliability evaluation method for a coupling faults of a middle trough of a scraper conveyor provided by the present invention includes the following steps:

(8) step 1: collecting data samples such as a crack length, a middle plate width, fracture toughness, yield strength and a fatigue load of the middle trough at different operation time points in combination with practical work conditions of the middle trough of an underground scraper conveyor, and counting probability information such as a mean, a variance, skewness and kurtosis of all the data samples;

(9) step 2: defining a reliability performance function of fracture failure of the middle trough according to relationship between a maximum stress strength factor of the middle trough of the scraper conveyor and a fracture toughness value, and defining a reliability performance function of the static strength failure of the middle trough according to relationship between a structure resistance of the middle trough and a fatigue load;

(10) step 3: on the basis of first four moment probability information obtained in step 1, calculating first four moments of the middle trough in fracture failure and static strength failure modes, further obtaining probability distribution functions of each failure mode by using a saddlepoint approximation method based on four moments, and calculating failure probabilities in two failure modes at different time points;

(11) step 4: generating random response samples in each failure mode in combination with the obtained probability distribution functions of the middle trough of the scraper conveyor in different failure modes in step 3, performing normalization processing, performing calculation by a statistical method to obtain a rank correlation coefficient between different sample sequences, further estimating time-varying parameters of a dynamic t-copula function at different time points, and further building a dynamic joint probability distribution model based on the dynamic t-copula function; and

(12) step 5: calculating a coupling failure probability of the middle trough at different time points in combination with a marginal probability distribution function and the joint probability distribution model based on the dynamic t-copula function respectively obtained in step 3 and step 4 and by using a system reliability theory.

Embodiment

(13) In order to more sufficiently know characteristics and engineering applicability of the present invention, the present invention performs dynamic reliability evaluation on a coupling faults by aiming at a structure of a middle trough of a scraper conveyor as shown in FIG. 2.

(14) (1) Data samples (such as a crack length, a middle plate width, fracture toughness, yield strength and a fatigue load) of fracture failure and static strength failure of the middle trough of the scraper conveyor at different time points are collected to obtain first four moment probability information of each sample.

(15) (2) According to safety criteria of fracture failure and static strength failure of the middle trough of the scraper conveyor, reliability performance functions in fracture failure and static strength failure are respectively built, and are:
g.sub.1=K.sub.κ−K.sub.max and
g.sup.2=Q−S,

(16) wherein is fracture toughness of the middle trough of the scraper conveyor, K.sub.max, is a maximum stress strength factor of the middle trough of the scraper conveyor, K.sub.max is a function of random variables of a crack expansion length, a material attribute and the like, Q is structure resistance of the middle trough of the scraper conveyor, Q is a function of the crack expansion length, and S is a fatigue load.

(17) (3) On the basis of probability information of each random variable in the data samples, a statistical method is used for obtaining first four moments, i.e. the mean, the variance, the skewness and the kurtosis, of the performance functions in the fracture failure and static strength failure modes. A saddlepoint approximation method based on four moments is used for respectively obtaining the failure probabilities in the fracture failure and static strength failure at different time points through calculation. Specifically:

(18) a form of the probability distribution function built by using the moment-based saddlepoint approximation method is

(19) $F_Y\left(y\right)=\Phi \left[{\omega }_{\mathrm{ye}}+\frac{1}{{\omega }_{\mathrm{ye}}}\ln \left(\frac{{\upsilon }_{\mathrm{ye}}}{{\omega }_{\mathrm{ye}}}\right)\right],$

(20) wherein ω.sub.ye and ν.sub.ye are parameters of the function, and may be calculated by the following formulas:
ω.sub.ye=sign(t.sub.e)√{square root over (2[y.sub.et.sub.e−K.sub.Ys(t.sub.e)])} and
ν.sub.ye=t.sub.e√{square root over (K.sub.Ys.sup.(2)(t.sub.e))},

(21) wherein y.sub.e represents a standardized variable of a performance function state variable, K.sub.Ys represents a cumulant generating function of the standardized variable, K.sub.Ys.sup.(2) represents a second derivative of the cumulant generating function, and t.sub.e represents a saddle point value, and may be calculated by the following formulas:

(22) $t_1=\frac{-\sqrt{\left(16a_2{a}_3+{\left(y-a_1\right)}^2\right)b^2-4a_2\left(y-a_1\right)b+4a_2^2}+\left(y-a_1\right)b+2a_2}{4{\mathrm{ba}}_2}$
and

(23) $t_2=\frac{-\sqrt{\left(16a_2{a}_3+{\left(y-a_1\right)}^2\right)b^2-4a_2\left(y-a_1\right)b+4a_2^2}+\left(y-a_1\right)b+2a_2}{4{\mathrm{ba}}_2},$

(24) wherein y is a second-order reliability index of the functional function, t.sub.1 and t.sub.2 represent two solutions of a saddlepoint equation, and a value meeting calculation reasonability is taken as a saddle point value in practical calculation; and

(25) $a_1=-\frac{9{\theta }_{\mathrm{Ys}}^3}{2{\left({\eta }_{\mathrm{Ys}}-3\right)}^2},a_2=\frac{-3{\theta }_{\mathrm{Ys}}^3+2{\eta }_{\mathrm{Ys}}-6}{4\left({\eta }_{\mathrm{Ys}}-3\right)},a_3=\frac{27{\theta }_{\mathrm{Ys}}^4}{4{\left({\eta }_{\mathrm{Ys}}-3\right)}^3},b=\frac{{\eta }_{\mathrm{Ys}}-3}{3{\theta }_{\mathrm{Ys}}}$

(26) wherein θ.sub.Ys and η.sub.Ys respectively represent skewness and kurtosis of the functional function.

(27) (4) By considering dynamic correlation of fracture failure and static strength failure, a dynamic t-copula function is used for building a dynamic correlation model between two failure modes. A probability density diagram of a dynamic t-copula function is shown in FIG. 3, and a scatter diagram of the dynamic t-copula function is shown in FIG. 4.

(28) On the basis of reliability performance functions of the middle trough of the scraper conveyor in fracture failure and static strength failure, a rank correlation coefficient between the two failure modes is obtained by a sampling method through calculation, and variation rules of the time-varying parameters of the dynamic t-copula function are further obtained. For details in step 4:

(29) the dynamic copula function uses a dynamic t-copula function in a form:

(30) 0$C_t\left(u_1,u_2.Math.\rho ,k\right)={\int }_{-\infty }^{t_k^{-1}\left(u_1\right)}{\int }_{-\infty }^{t_k^{-1}\left(u_2\right)}{\frac{1}{2\pi \sqrt{1-{\rho }^2}}\left[1+\frac{s^2-2\mathrm{\rho st}+t^2}{k\left(1-{\rho }^2\right)}\right]}^{-\left(k+2\right)/2}\mathrm{dsdt},$

(31) wherein k and p are parameters of the dynamic t-copula function, k is an invariable parameter, and p is a time-varying parameter; a value of the time-varying parameter p at different time points may be obtained in combination with the data samples collected at different time points through a maximum likelihood estimation method; and u.sub.1 and u.sub.2 represent obtained failure probabilities in the fracture failure mode and the static strength failure mode through calculation in step 3.

(32) (5) On the basis of the failure probabilities of the fracture failure and static strength failure of the middle trough of the scraper conveyor and a joint probability failure probability of the dynamic t-copula function, dynamic reliability of the middle trough of the scraper conveyor under a coupling faults condition at different time points (load action times) is calculated, and a reliability curve is shown in FIG. 5.

(33) For details in step 5:

(34) according to the obtained failure probabilities u.sub.1 and u.sub.2 in the fracture failure mode and the static strength failure mode of the middle trough of the scraper conveyor in step 3, and the obtained dynamic joint probability distribution function between the two failure modes in step 4, dynamic reliability of the coupling faults of the middle trough of the scraper conveyor is calculated by the following formula, specifically:
P.sub.f=u.sub.1+u.sub.2−P(u.sub.1,u.sub.2|ρ),

(35) wherein P(u.sub.1,u.sub.2|ρ) is a joint failure probability obtained through calculation according to the dynamic joint probability distribution function in step 4; and

(36) through calculation on reliability at different time points, the dynamic reliability evaluation on the coupling faults of the middle trough of the scraper conveyor is obtained.