System reliability analysis method based on failure mode mixed copula

Abstract

The application relates to the technical field of system reliability analysis, in particular to a mixed Copula system reliability analysis method based on failure modes, which comprises the following steps: acquiring multi-failure mode degradation data, combining Bayesian inference to estimate parameters of a basic Copula function, rotating and transforming the basic Copula function to construct a mixed static Copula model, judging and optimizing the mixed static Copula model based on root mean square errors, extracting a time-varying correlation parameter sequence by using a sliding window, smoothing and denoising the time-varying correlation parameter sequence by using Kalman filtering, and constructing a mixed dynamic Copula model, and calculating the reliability of the system at different operation time points in combination with a failure mode limit state function. The application can represent the non-symmetrical tail correlation between the multiple failure modes and the time evolution law thereof, and improve the accuracy, stability and engineering applicability of system reliability analysis.

Description

Technical Field

[0001] This invention relates to the field of system reliability analysis technology, and in particular to a hybrid Copula system reliability analysis method based on failure modes. Background Technology

[0002] Complex electromechanical systems, during long-term service, are typically subjected to multiple factors such as load, vibration, temperature, and environmental media. Their critical components may simultaneously exhibit multiple failure modes, including wear, cracking, corrosion, and fatigue. These different failure modes are often not independent but exhibit dynamic coupling and nonlinear correlation characteristics as the degradation process progresses. If this correlation is ignored in system reliability analysis, or if only a fixed correlation coefficient and a static Copula model are used for simplification, it will be difficult to accurately reflect the asymmetric tail correlation structure and its time-varying evolution between failure modes, leading to discrepancies between reliability assessment results and actual conditions.

[0003] Furthermore, in engineering monitoring and testing scenarios, the number of degraded samples is usually limited, and the observation data is often accompanied by random disturbances and noise. Traditional parameter estimation methods are prone to problems such as large fluctuations in time-varying parameters, insufficient stability, and inaccurate identification of local correlation structures. Therefore, it is necessary to propose a system reliability analysis method that can characterize the asymmetric correlation structure of multiple failure modes, track the time-varying evolution of correlation relationships, and take into account the stability of parameter estimation. Summary of the Invention

[0004] To overcome the problems raised in the background art, this invention provides a reliability analysis method for hybrid Copula systems based on failure modes, comprising the following steps:

[0005] S1. Collect degradation data of multiple failure modes of mechanical structures and construct the limit state function of failure modes;

[0006] S2. Based on the degradation data, determine the edge distribution function of each failure mode, select the basic Copula function and estimate the parameters of the basic Copula function through Bayesian inference to obtain the basic parameter set;

[0007] S3. Based on the basic parameter set, perform rotational transformation on the basic Copula function and construct a hybrid static Copula model;

[0008] S4. Perform accuracy determination and weight optimization on the hybrid static Copula model. If the accuracy does not meet the preset threshold, iteratively optimize the model weights.

[0009] S5. Use a sliding window to process degraded data and extract time-varying correlation parameter sequences;

[0010] S6. Use a filtering algorithm to smooth and denoise the time-varying correlation parameter sequence to obtain the optimal time-varying parameters;

[0011] S7. Construct a hybrid dynamic Copula model based on optimal time-varying parameters, and establish a joint distribution model of the system by combining the marginal distribution function;

[0012] S8. Combining the failure mode limit state function and the system joint distribution model, calculate the reliability of the mechanical structure at different operating times and output the reliability analysis results.

[0013] Preferably, in step S1, multiple failure modes can be multiple failure modes of the same component or a single failure mode of each of the multiple components.

[0014] The expression for the limit state function is: ;

[0015] in, For a moment No. The limit state function for each failure mode. For the first Safety threshold for each failure mode For a moment No. A stochastic process function for the performance degradation of each failure mode.

[0016] Preferably, in step S2, the degraded data is converted into pseudo-observation data in the interval [0,1] by probability integral transformation and used as input to the basic Copula function;

[0017] The basic Copula functions include Gaussian Copula, Clayton Copula, and Frank Copula;

[0018] Bayesian inference uses the Markov chain Monte Carlo method to generate parameters, and the standard deviation of the suggestion distribution is adaptively adjusted during the iterative process to ensure convergence and stability.

[0019] Preferably, in step S3, the rotation transformation includes 0°, 90°, and 180° rotations, which are used to characterize the upper tail, lower tail, and asymmetric correlation features of the failure mode;

[0020] The hybrid static Copula model is composed of a linearly weighted combination of the components of each rotated basic Copula function. The expression for the hybrid static Copula model is: ;

[0021] in, For the constructed hybrid static Copula model function, For the first The basic Copula function components For the first Initial weights for the components of the basic Copula function For the first The parameters corresponding to the components of the basic Copula function; , These are the marginal distribution variables in the basic Copula functions of each failure mode when two failure modes exist.

[0022] Preferably, in step S4, the empirical Copula is used as a benchmark, and the accuracy is determined by the root mean square error between the hybrid static Copula model and the empirical Copula. Specifically, the difference between the theoretical value of the hybrid static Copula model and the actual value of the empirical Copula is calculated point by point, the square of the difference is taken as the mean, and then the square root of the mean is taken to obtain the root mean square error.

[0023] If the root mean square error does not meet the preset threshold, the component weights of the basic Copula function are iteratively updated using the simplex optimization algorithm until the root mean square error meets the preset threshold.

[0024] Preferably, in step S5, the original time-varying parameter estimates of the basic Copula function components are calculated using maximum likelihood estimation within a sliding window to form a time-varying correlation parameter sequence;

[0025] The original time-varying parameter estimate is expressed as follows: ;

[0026] in, At any moment The original time-varying parameter estimates, The width of the sliding window. The left boundary of the sliding window. The right boundary of the sliding window. , They are the i-th group , ; , These are the marginal distribution variables in the basic Copula functions of the two failure modes, respectively, when two failure modes exist. These are the parameters corresponding to the components of the basic Copula function.

[0027] Preferably, the roughness index of the time-varying correlation parameter sequence is calculated, and the expression is: ;

[0028] in, Roughness index The total time length of the parameter sequence. For parameters At any moment The estimated value, In parameters time The estimated value;

[0029] The width of the sliding window is adaptively adjusted according to the roughness index. If the window is too large, increase the width of the sliding window; when If the window is too small, reduce the width of the sliding window.

[0030] Preferably, in step S6, the filtering algorithm is Kalman filtering. A state-space model containing state equations and observation equations is constructed based on the time-varying correlation parameter sequence. Kalman filtering is performed based on the state-space model to perform state estimation and smoothing and denoising on the time-varying correlation parameter sequence to obtain the optimal time-varying parameters.

[0031] Preferably, in step S7, the expression for the hybrid dynamic Copula model is: ;

[0032] in, For hybrid dynamic Copula model functions, It is the total number of components of the basic Copula function. Let be the dynamic weight of the k-th fundamental Copula function component at time t. For the first The basic Copula function components For the first The fundamental Copula function components at time 1 The optimal time-varying parameters; , These are the marginal distribution variables in the basic Copula functions of the two failure modes, respectively, when two failure modes exist.

[0033] System joint distribution model The expression is: ;

[0034] in, , When two failure modes exist, the time of each failure mode is... The marginal distribution function, , These represent the degradation amount for each of the two failure modes; The dynamic weights of the basic Copula function components at time t. For at any time The actual value of the parameter state.

[0035] Preferably, in step S8, for a series system, the system reliability is the probability that all failure modes are simultaneously in a safe state, and the reliability expression is: ;

[0036] in, , When two failure modes exist, time... The limit state functions for each of the two failure modes , For a moment The performance degradation stochastic process functions for each of the two failure modes , The safety thresholds corresponding to the two failure modes, For a moment No. Dynamic weights of the components of a basic Copula function; , These are the edge distribution functions of the two failure modes at the threshold values, respectively. No. The fundamental Copula function components at time 1 The optimal time-varying parameters, For the first The basic Copula function components It is the number of basic Copula functions.

[0037] The beneficial effects of this invention are as follows:

[0038] This invention effectively characterizes the asymmetric tail correlation features between multiple failure modes by rotating, expanding and weighting the basic Copula function; by introducing a sliding window and Kalman filtering mechanism, it can track the time-varying evolution of failure mode-related structures and reduce noise interference; by mapping the failure mode correlation relationship to the system reliability calculation process, it can improve the accuracy and stability of reliability assessment of complex systems under related failure conditions. Attached Figure Description

[0039] Figure 1 This is an overall flowchart of an embodiment of the present invention;

[0040] Figure 2 This is a detailed flowchart illustrating an embodiment of the present invention;

[0041] Figure 3 This is a comparison chart of system failure probabilities for different methods in Example 1 of the present invention;

[0042] Figure 4 This is a schematic diagram of the structure of Embodiment 2 of the present invention;

[0043] Figure 5 This is a schematic diagram of the structure of Embodiment 3 of the present invention;

[0044] Figure 6 This is a comparison chart of the failure probability of the ten-bar truss system in Embodiment 3 of the present invention;

[0045] Figure 7 This is a diagram showing the object and failure characteristics of Embodiment 4 of the present invention;

[0046] Figure 8 This is a graph showing the variation of Kendall's rank correlation coefficient over time in the high-speed train gear transmission system according to Embodiment 4 of the present invention.

[0047] Figure 9 This is a diagram showing the system reliability analysis results in the high-speed train gear transmission system according to Embodiment 4 of the present invention;

[0048] Figure 10 This is a comparison diagram of relative errors in the gear transmission system of a high-speed train according to Embodiment 4 of the present invention. Detailed Implementation

[0049] Example 1: An embodiment of the present invention provides a reliability analysis method for hybrid Copula systems based on failure modes, such as... Figure 1 , 2 As shown, it includes the following steps:

[0050] S1. Collect degradation data of the system under different failure modes (at least two) within a preset operating cycle, including historical degradation data and online measured data; if no measured data is available, pseudo-observation data can be generated through simulation. Simultaneously, determine the performance degradation variables and safety thresholds corresponding to each failure mode, constructing a basic dataset for subsequent mechanical mechanism analysis and reliability calculations. For each failure mode, establish a failure mode limit state function based on equipment mechanics, degradation laws, and safety thresholds.

[0051] The system includes mechanical structures and engineering systems (such as crane main beams, trusses, and train gear transmission systems), and consists of multiple components or functional units with related failure modes. Different failure modes can be multiple failure modes of the same component within the system, or multiple components within the system, each with one failure mode.

[0052] The corresponding limit state function expressions for each failure mode are: ;

[0053] in, For a moment No. The limit state function for each failure mode is used. In this embodiment, two failure modes are collected, and the limit state function is selected. , For the first The safety threshold corresponding to each failure mode For a moment No. The performance degradation stochastic process function corresponding to each failure mode.

[0054] S2. Determine the edge distribution function corresponding to each failure mode based on the basic dataset. Historical degraded data is converted into pseudo-observation data within the interval [0,1]. The conversion method involves combining historical degraded data with a marginal distribution function to perform a probability integral transformation, which is used to eliminate the influence of dimensional differences on the identification of related structures. In this embodiment, Gaussian Copula, Clayton Copula, and Frank Copula are selected as basic Copula functions. Pseudo-observation data is used as input samples for the basic Copula functions, and Bayesian inference is used to estimate the initial parameters of each basic Copula function, obtaining a set of basic parameters used to describe the failure mode-related structures. During the Bayesian inference process, the Markov chain Monte Carlo method is used to generate candidate Copula parameters, and the perturbation amplitude of the candidate Copula parameters during the iteration process is controlled by the standard deviation of the proposed distribution.

[0055] The suggested expression for the standard deviation of the distribution is: ;

[0056] in, For the first The suggested distribution standard deviation at the next iteration For the first The proposed standard deviation of the distribution is updated in the next iteration. For the first Instantaneous acceptance rate of the next iteration To achieve the preset target acceptance rate, Let be the number of iterations. The standard deviation of the proposal distribution is adaptively updated based on the instantaneous acceptance rate, ensuring that the Markov chain maintains good search efficiency and convergence stability in the parameter posterior space.

[0057] S3. Based on the fundamental parameter set, perform 0-degree, 90-degree, and 180-degree rotation transformations on each fundamental Copula function. These rotation transformations only change the variable form of the Copula to adapt to asymmetric correlation structures. Building upon this, a hybrid static Copula model containing multiple candidate components is constructed using a linear weighting method, and the initial weights of each candidate component are determined. This step expands the model's morphology to complete the asymmetric correlation structure, characterizing the upper-tail correlation, lower-tail correlation, and asymmetric correlation structures that may occur between multiple failure modes. Finally, a hybrid static Copula model capable of characterizing a complete and complex correlation structure is obtained.

[0058] The expressions for performing rotation transformations of 0 degrees, 90 degrees, and 180 degrees are as follows: ;

[0059] In this embodiment, two failure modes were collected. , These are the marginal distribution variables in the basic Copula functions for each of the two failure modes. , They are respectively , After rotation, the three or more failure modes expand sequentially as follows: , , ...

[0060] The expression for the hybrid static Copula model is: ;

[0061] in, For the constructed hybrid static Copula model function, For the first The basic Copula function components For the first Each of the basic Copula function components has an initial weight, and the sum of these initial weights is 1. For the first The parameters corresponding to the components of the basic Copula functions; the upper limit of the summation sign is written as 9 because this implementation selects 3 basic Copula functions, each of which undergoes 3 rotation transformations, for a total of 100 components. .

[0062] S4. Compare the fitting accuracy of the hybrid static Copula model with that of the empirical Copula (a non-parametric baseline Copula directly calculated from pseudo-observation data, representing the true relevant structure of the failure mode). The fitting accuracy index is the root mean square error (RMSE) between the hybrid static Copula model and the empirical Copula. Specifically, calculate the difference between the theoretical value of the hybrid static Copula model and the actual value of the empirical Copula point by point, square the difference, take the mean, and then take the square root of the mean to obtain the RMSE. If the RMSE meets the preset threshold, proceed to step S5; otherwise, use the simplex optimization method to iteratively optimize the initial weights until the preset threshold is met. This step completes the weight optimization by comparing the deviation between the fitted model (hybrid static Copula) and the data baseline (empirical Copula).

[0063] S5. The pseudo-observation data obtained in step S2 is segmented and scanned using a sliding window. Within each window, the original time-varying parameter estimates of the basic Copula function components are calculated using maximum likelihood estimation, forming a time-varying correlation parameter sequence.

[0064] The original time-varying parameter estimate is expressed as follows: ;

[0065] in, At any moment The original time-varying parameter estimates, The width of the sliding window. The left boundary of the sliding window. The right boundary of the sliding window. , Group i is respectively , ; Given the parameters corresponding to the components of the basic Copula function, this formula separately solves for the original time-varying parameter estimates of various basic Copula functions such as Gaussian, Clayton, and Frank.

[0066] To balance parameters Sensitivity and estimated stability, further parameter calculations The roughness index of the evolution curve is expressed as: ;

[0067] in, Roughness index The total time length of the parameter sequence. For parameters At any moment The estimated value, In parameters time The estimated value. When When the value is too large, it indicates that the parameter fluctuates too much and is easily affected by noise. Increasing the sliding window width can improve smoothness. If the sliding window is too small, it indicates that the model is not sensitive enough to local time-varying features. Reducing the sliding window width can improve tracking ability.

[0068] S6. Construct a state-space model based on the time-varying correlation parameter sequence, perform Kalman filtering based on the state-space model, perform state estimation and smoothing and denoising on the time-varying correlation parameter sequence, and obtain the smoothed time-varying parameters. The smoothed time-varying parameters are the optimal time-varying parameters.

[0069] The state-space model constructed based on time-varying correlation parameter sequences is as follows:

[0070] Using the time-varying correlation parameter sequence as the observation value, assuming that the parameter series evolves continuously over time, we construct the state equation and observation equation; combining the statistical characteristics of the time-varying correlation parameter sequence, we determine the noise variance and complete the state-space model construction.

[0071] The expressions for the state equation and the observation equation are as follows: ; ;

[0072] in, For at any time The actual value of the parameter state. For the previous moment The actual value of the parameter state. For at any time The original time-varying parameter estimates; It is process noise (system noise), which follows a mean of 0 and a covariance of . Gaussian distribution; The observed noise follows a Gaussian distribution with a mean of 0 and a covariance of R.

[0073] This step uses noisy observations (time-varying correlation parameter sequences) to back-calculate and approximate the true values ​​(the true state of Copula correlation parameters at each time point), reducing the high-frequency random fluctuations caused by engineering monitoring data and finite sample estimation.

[0074] S7. Update the initial values ​​with the initial weights as the weights, update the time-varying dynamic weights of each candidate component at different times according to the smoothed time-varying parameters, construct a hybrid dynamic Copula model, and establish a joint distribution model of the system by combining the edge distribution functions corresponding to each failure mode.

[0075] The expression for the hybrid dynamic Copula model is: ;

[0076] in, For hybrid dynamic Copula model functions, It is the total number of components of the basic Copula function. For the first The fundamental Copula function components at time 1 The optimal time-varying parameters, Let be the dynamic weight of the k-th fundamental Copula function component at time t. For the first The basic Copula function components.

[0077] System joint distribution model The expression is: ;

[0078] In this embodiment, two failure modes were collected. , These are the two failure modes at time [time]. The marginal distribution function shows that three or more failure modes expand sequentially. , These represent the degradation amount for each of the two failure modes; For hybrid dynamic Copula model functions, The dynamic weights of the basic Copula function components at time t. For at any time The actual value of the parameter state.

[0079] S8. Based on the limit state functions of each failure mode and the joint distribution model of the system, substitute the joint distribution model of the system into the failure threshold, calculate the reliability of the system at different operating times, and output the time-varying reliability analysis results of the system, which can be used to guide the design optimization and performance verification of mechanical mechanisms, components or engineering systems within the system.

[0080] For a series system, the system reliability is defined as the probability that all failure modes are in a safe state, and the reliability expression is: ;

[0081] in, , They are time points The limit state functions for each of the two failure modes , For a moment The performance degradation stochastic process functions for each of the two failure modes , The safety thresholds corresponding to the two failure modes, For a moment No. Dynamic weights of the components of a basic Copula function; , These are the function values ​​of the two marginal distribution functions at the threshold, i.e., at time [time]. The marginal probability of not failing; No. The fundamental Copula function components at time 1 The optimal time-varying parameters, For the first The basic Copula function components It is the number of basic Copula functions.

[0082] By following the above steps, the time-varying reliability curves and failure probability curves of the system under analysis at different operating times can be obtained. Based on the curves, the reliability decay rate, lifetime inflection point, reliability value at any time, failure time when reliability is below the threshold, etc., can be obtained for analysis to obtain the time-varying reliability analysis results of the system.

[0083] In this embodiment, to verify the accuracy of the method of the present invention, a classic time-varying reliability example is selected for analysis. The nonlinear limit state function of this example is: ;

[0084] in, and The variables are mutually independent normally distributed random variables with a mean of 3.5 and a standard deviation of 0.25. It is a time step that varies between 0 and 1. For a zero-mean stationary Gaussian process, its autocorrelation function is: ;

[0085] in, , It represents two different moments in time and is used to measure the relevance of a process over time.

[0086] This embodiment compares the method of the present invention with the Monte Carlo simulation method MCS, the time-varying reliability analysis method PHI2+, the time-varying reliability analysis method iTRPD based on multi-peak distribution, and the Kriging surrogate model based on the most probable failure point combined with the importance sampling method MPP-KIS. Figure 3As shown, the system failure probability increases monotonically with increasing service time. The failure probability curve obtained by the method of this invention is closest to the Monte Carlo simulation benchmark results, with the relative error remaining stable within the range of 0.54% to 0.74% at the selected discrete times. In contrast, the maximum relative errors of the time-varying reliability analysis method, the time-varying reliability analysis method based on multi-peak distribution, and the Kriging surrogate model based on the most probable failure point combined with the importance sampling method reach 29.44%, 5.46%, and 3.52%, respectively.

[0087] Meanwhile, a comparison of the dynamic location results of the most likely failure point shows that the prediction error of the method of the present invention for the key random variables is controlled within 0.5% throughout the time-varying process. This indicates that the hybrid dynamic Copula model can identify high-risk failure areas without relying on a large number of physical model calls and can stably output time-varying reliability analysis results.

[0088] Example 2: This example provides an application of a hybrid Copula system reliability analysis method based on failure modes in a corrosion-degraded simply supported beam structure, such as... Figure 4 As shown, it includes the following steps:

[0089] S1. A general-purpose bridge crane main beam with a rated lifting capacity of 100t is selected as the object of analysis. This crane is exposed to a humid and acidic corrosive environment for a long time. Corrosion will cause the cross-sectional dimensions of the box girder to gradually decrease, thereby degrading the structural stiffness and load-bearing capacity. To simplify the mechanical modeling, one main beam in the load-bearing structure is equivalent to bearing half of the lifting load. A simply supported beam with a span of .

[0090] S2. Using displacement failure and stress failure as two competing failure modes, establish limit state functions for each: ; ;

[0091] in, To allow for deflection, To allow stress, For elastic modulus, Let the moment of inertia of the cross section be... Here is the bending section modulus. The moment of inertia and bending section modulus of the box section are as follows: ; ;

[0092] in, and It refers to the cross-sectional geometric dimensions.

[0093] S3. Set random variables , , , and All follow a normal distribution. Among them, the load The mean is 500,000 and the standard deviation is 50,000; beam span The mean is 36400 and the standard deviation is 364; the elastic modulus is The mean is 210000 and the standard deviation is 21000; cross-sectional parameters The mean is 48 and the standard deviation is 3.6; cross-sectional parameters The mean is 180 and the standard deviation is 5.4. Since the cross-sectional loss caused by corrosion affects both the deflection response and the stress response, there is a nonlinear correlation between the two failure modes driven by a common degradation variable.

[0094] S4. Using the degradation response data obtained in steps S1 to S3, construct the independent hypothesis model, the hybrid static Copula model, and the hybrid dynamic Copula model of this invention, respectively, and use the Monte Carlo simulation results as a benchmark. To maintain consistency with the benchmark corrosion state, this embodiment selects the corrosion state at the final moment for reliability comparison.

[0095] S5. Calculation results show that the system failure probability obtained by Monte Carlo simulation is 0.3417; the system failure probability obtained by the independent assumption model is 0.4078, with a relative error of 19.34%; the system failure probability obtained by the hybrid static Copula model is 0.3452, with a relative error of 1.02%; and the system failure probability obtained by the method in this embodiment is 0.3427, with a relative error of only 0.29%. Therefore, the method of this invention can accurately describe the correlation between displacement failure mode and stress failure mode under corrosion degradation conditions and can significantly improve the accuracy of system reliability assessment.

[0096] Example 3: This example provides an application of a failure mode-based hybrid Copula system reliability analysis method in a ten-bar truss structure, such as... Figure 5 and Figure 6 As shown, it includes the following steps:

[0097] S1. A classic two-dimensional ten-bar truss system is selected as the object to be analyzed. The truss structure contains six nodes and ten members, of which nodes 5 and 6 are fixed nodes; the external loads include the vertical downward load acting on node 4, the vertical downward load acting on node 2, and the horizontal load acting on node 2.

[0098] S2. Consider two competing failure modes. The first failure mode is displacement-controlled failure, related to the vertical displacement of node 2; the second failure mode is stress-controlled failure, related to the maximum axial stress among all members. The two limit state functions are as follows: ; ;

[0099] in, For a moment Vertical displacement of node 2, To determine the allowable displacement threshold, this embodiment takes... =9.3mm; For the first root member at time Axial stress, This is the allowable stress threshold obtained through calibration using test samples.

[0100] S3. The cross-sectional area of ​​the member is divided into three groups, with the initial cross-sectional area of ​​each group following a normal distribution with a mean of 1000 mm² and a standard deviation of 50 mm². The three external loads vary within the ranges of 79.2 kN to 80.8 kN, 79.2 kN to 80.8 kN, and 9.9 kN to 10.1 kN, respectively. Furthermore, a common random degradation component and a local random degradation component are introduced during the degradation process of the member's cross-sectional area, so that the correlation between the displacement-controlled failure mode and the stress-controlled failure mode gradually increases over time.

[0101] S4. Five hundred thousand Monte Carlo simulation trajectories are used as the baseline results. The structural response is solved at each time step to obtain the vertical displacement of node 2 and the maximum absolute axial stress of each member, and the system failure probability is calculated based on the response. The comparison models include the independent assumption model, the hybrid static Copula model, and the hybrid dynamic Copula model of the present invention; wherein, the method of the present invention uses a sliding window with a window width of 7 to extract time-varying correlation parameters, and uses Kalman filtering to smooth the parameters and weight trajectories.

[0102] S5. The results show that the Kendall correlation coefficient gradually increases with service time, indicating that the correlation between displacement-driven failure mode and stress-driven failure mode is enhanced with the accumulation of structural degradation. Figure 5 The failure probability comparison results show that the method of the present invention is closest to the Monte Carlo simulation benchmark throughout the entire service life. For example, in At that time, the failure probability of the Monte Carlo baseline system was 0.0099, while the independent assumption model, the hybrid static Copula model, and the method of this invention yielded 0.0194, 0.0131, and 0.0110, respectively; At that time, the Monte Carlo baseline value was 0.2488, and the three comparison methods yielded values ​​of 0.3598, 0.2668, and 0.2537, respectively.

[0103] S6. Within the examined timeframe, the average relative error of the hybrid dynamic Copula model in this embodiment is 5.99%, while the average relative errors of the hybrid static Copula model and the independent assumption model are 17.26% and 67.58%, respectively. This result demonstrates that the method of this invention can effectively characterize the time-varying correlation between competing failure modes in a ten-bar truss system and provide more accurate system reliability prediction results during structural degradation.

[0104] Example 4: This example provides an application of a failure mode-based hybrid Copula system reliability analysis method in a high-speed train gear transmission system, such as... Figures 7 to 10 As shown, it includes the following steps:

[0105] S1. The power transmission gear in the gearbox of the high-speed train is selected as the object of analysis. In order to conduct system reliability analysis, the gear transmission system is equivalent to a series system. When either tooth surface contact fatigue failure or tooth root bending fatigue failure occurs, the system is judged to have failed.

[0106] S2. For the tooth surface contact fatigue failure mode, it is assumed that failure occurs when the tooth surface contact stress exceeds the material's allowable contact fatigue limit. Considering the effects of input torque, service factor, dynamic load factor, tooth width, module, and tooth surface wear on contact performance degradation, the limit state function for the contact fatigue failure mode is established: ;

[0107] in, This refers to the material's contact fatigue limit. For working contact stress, The elastic coefficient, For the use factor, This is the dynamic load factor. For input torque, For tooth width, For modulus, The transmission ratio is... For a moment The amount of tooth surface wear, This is the sensitivity coefficient to the effect of wear on contact fatigue load capacity.

[0108] S3. For the tooth root bending fatigue failure mode, it is assumed that fracture failure occurs when the actual tooth root bending stress exceeds the material's allowable bending fatigue limit. Considering the influence of crack propagation rate and initial crack size on bending load capacity, the limit state function for the tooth root bending fatigue failure mode is established: ;

[0109] in, This represents the bending fatigue limit of the material. This represents the actual bending stress at the tooth root. For the composite tooth profile coefficient, The initial crack length is... This represents the crack propagation rate. This is the sensitivity coefficient for the influence of crack size on load-bearing capacity. From the two limit state functions mentioned above, it can be seen that the tooth surface contact fatigue failure mode and the tooth root bending fatigue failure mode are not only affected by common uncertain variables such as input torque, service factor, dynamic load factor, tooth width, and module, but also by degradation variables such as tooth surface wear and crack propagation rate. Therefore, there is a nonlinear correlation between the two.

[0110] S4. Set that all uncertain variables follow a normal distribution, including the input torque. Usage coefficient Dynamic load factor Contact fatigue limit Bending fatigue limit Modulus Tooth width Tooth surface wear Crack propagation rate and initial crack length The mean and standard deviation of the above variables were determined based on actual operating condition measurement data and design specifications.

[0111] S5. Based on the competing failure physical model constructed in steps S2 to S4, N=2000 sets of degradation sample data are generated in MATLAB, and the original degradation data are transformed into pseudo-observation samples in the interval [0,1] through probability integral transformation. The Bayesian inference and hybrid dynamic Copula modeling method of this invention are used for parameter identification and model selection.

[0112] S6. First, basic Copula static parameter estimation was performed on the data section during the mid-degradation period. Gaussian Copula, Clayton Copula, and Frank Copula were selected as the models to be fitted, and the MCMC chain length was set to 5000 iterations, with the first 1000 iterations considered as the burning period. The parameter estimation results show that the posterior mean of the Gaussian Copula parameters is 0.5074 and the standard deviation is 0.0143, the posterior mean of the Clayton Copula parameters is 0.7469 and the standard deviation is 0.0409, and the posterior mean of the Frank Copula parameters is 3.4564 and the standard deviation is 0.1483. The posterior distributions of each parameter are concentrated in a relatively narrow confidence interval, indicating that the Bayesian MCMC estimation process has good convergence and stability.

[0113] S7. Further, the single static Copula model, the hybrid static Copula model, and the hybrid dynamic Copula model of this invention are compared using the AIC and BIC criteria. The results show that the hybrid dynamic Copula model of this embodiment has the lowest AIC and BIC values ​​within 100 observation windows throughout the entire degradation process, which are -66493.08 and -66319.58, respectively, significantly better than the single Copula model and the hybrid static Copula model. This indicates that introducing time-varying parameters and time-varying weights can more accurately fit the non-stationary correlation structure in gear competitive failure data.

[0114] S8, such as Figure 8 As shown, the vertical axis represents the Kendall's rank correlation coefficient (τ), and the horizontal axis represents service time. This indicates that the Kendall's rank correlation coefficient between tooth surface wear depth and tooth root crack length in the gear transmission system exhibits significant time-varying characteristics with service time. In the initial stage (t < 4), the correlation is weak and fluctuates significantly. In the intermediate stage, the mutual promotion effect between failure modes begins to strengthen, and the correlation coefficient gradually increases. In the final service stage (t > 7), wear-induced dynamic loads accelerate crack propagation, causing the correlation to rapidly increase and eventually reach 0.4427. The method of this invention can continuously track this nonlinear growth trend while maintaining smooth parameter trajectories through a sliding window and Kalman smoothing.

[0115] S9, such as Figure 9 As shown, throughout the entire service life, the reliability curve of the gear transmission system obtained by the hybrid dynamic Copula model of this invention maintains good consistency with the Monte Carlo simulation benchmark results, especially in the mid-to-late degradation stage, where it shows a high degree of agreement. In contrast, the independent assumption model ignores the positive correlation and coupling effect between the two competing failure modes, resulting in a larger reliability prediction deviation; although the hybrid static Copula model can consider the correlation, its parameters are fixed and cannot reflect the tail-related structures that are enhanced in the later stages. Figure 10Further relative error results show that the method of the present invention has minimal error in most service periods, and can improve the accuracy and stability of reliability assessment of EMU gear transmission system under relevant failure conditions.

[0116] This invention has been described through embodiments. Those skilled in the art will understand that various changes or equivalent substitutions can be made to these features and embodiments without departing from the spirit and scope of the invention. Furthermore, under the teachings of this invention, these features and embodiments can be modified to adapt to specific situations and materials without departing from the spirit and scope of the invention. Therefore, this invention is not limited to the specific embodiments disclosed herein, and all embodiments falling within the scope of the claims of this application are within the protection scope of this invention.

Claims

1. A reliability analysis method for hybrid Copula systems based on failure modes, characterized in that, Includes the following steps: S1. Collect degradation data of multiple failure modes of mechanical structures and construct the failure mode limit state function; Degradation data includes historical degradation data and online measured data. When there is no online measured data, pseudo-observation data is generated through simulation. S2. Based on the degradation data, determine the edge distribution function of each failure mode, select the basic Copula function and estimate the parameters of the basic Copula function through Bayesian inference to obtain the basic parameter set; S3. Based on the basic parameter set, perform rotational transformation on the basic Copula function and construct a hybrid static Copula model; The rotation transformation includes 0°, 90°, and 180° rotations, which are used to characterize the upper tail, lower tail, and asymmetric correlation features of the failure modes. The hybrid static Copula model is composed of a linearly weighted combination of the rotated basic Copula function components. The expression of the hybrid static Copula model is: ； in, For the constructed hybrid static Copula model function, For the first The basic Copula function components For the first Initial weights for the components of the basic Copula function For the first The parameters corresponding to the components of the basic Copula function; , These are the marginal distribution variables in the basic Copula functions of each failure mode when two failure modes exist; S4. Perform accuracy determination and weight optimization on the hybrid static Copula model. If the accuracy does not meet the preset threshold, iteratively optimize the model weights. S5. Use a sliding window to process degraded data and extract time-varying correlation parameter sequences; S6. Use a filtering algorithm to smooth and denoise the time-varying correlation parameter sequence to obtain the optimal time-varying parameters; S7. Construct a hybrid dynamic Copula model based on optimal time-varying parameters, and establish a joint distribution model of the system by combining the marginal distribution function; S8. Combining the failure mode limit state function and the system joint distribution model, calculate the reliability of the mechanical structure at different operating times and output the reliability analysis results.

2. The reliability analysis method for hybrid Copula systems based on failure modes according to claim 1, characterized in that, In step S1, the multiple failure modes can be multiple failure modes of the same component or a single failure mode of each of the multiple components. The expression for the limit state function is: ； in, For a moment No. The limit state function for each failure mode. For the first Safety threshold for each failure mode For a moment No. A stochastic process function for the performance degradation of each failure mode.

3. The reliability analysis method for hybrid Copula systems based on failure modes according to claim 1, characterized in that, In step S2, the degraded data is transformed into pseudo-observation data in the interval [0,1] by probability integral transformation and used as input to the basic Copula function; The basic Copula functions include Gaussian Copula, Clayton Copula, and Frank Copula; The Bayesian inference uses the Markov chain Monte Carlo method to generate parameters, and the iterative process adaptively adjusts the standard deviation of the suggestion distribution to ensure convergence and stability.

4. The reliability analysis method for hybrid Copula systems based on failure modes according to claim 1, characterized in that, In step S4, the empirical Copula is used as a benchmark. The accuracy is determined by the root mean square error between the hybrid static Copula model and the empirical Copula. The difference between the theoretical value of the hybrid static Copula model and the actual value of the empirical Copula is calculated point by point. The squared difference is then averaged, and the square root of the average is taken to obtain the root mean square error. If the root mean square error does not meet the preset threshold, the component weights of the basic Copula function are iteratively updated using the simplex optimization algorithm until the root mean square error meets the preset threshold.

5. The reliability analysis method for hybrid Copula systems based on failure modes according to claim 1, characterized in that, In step S5, the original time-varying parameter estimates of the basic Copula function components are calculated using maximum likelihood estimation within the sliding window, forming a time-varying correlation parameter sequence; The original time-varying parameter estimate is expressed as follows: ； in, At any moment The original time-varying parameter estimates, The width of the sliding window. The left boundary of the sliding window. The right boundary of the sliding window. , Group i is respectively , ; , These are the marginal distribution variables in the basic Copula functions of the two failure modes, respectively, when two failure modes exist. These are the parameters corresponding to the components of the basic Copula function.

6. The reliability analysis method for hybrid Copula systems based on failure modes according to claim 5, characterized in that, The roughness index of the time-varying correlation parameter sequence is calculated using the following expression: ； in, Roughness index The total time length of the parameter sequence. For parameters At any moment The estimated value, In parameters time The estimated value; The width of the sliding window is adaptively adjusted according to the roughness index. If the window is too large, increase the width of the sliding window; when If the window is too small, reduce the width of the sliding window.

7. The reliability analysis method for hybrid Copula systems based on failure modes according to claim 1, characterized in that, In step S6, the filtering algorithm is Kalman filtering. A state-space model containing state equations and observation equations is constructed based on the time-varying correlation parameter sequence. Kalman filtering is performed on the state-space model to estimate the state and smooth and denoise the time-varying correlation parameter sequence to obtain the optimal time-varying parameters.

8. The reliability analysis method for hybrid Copula systems based on failure modes according to claim 1, characterized in that, In step S7, the expression for the hybrid dynamic Copula model is: ； in, For hybrid dynamic Copula model functions, It is the total number of components of the basic Copula function. Let be the dynamic weight of the k-th fundamental Copula function component at time t. For the first The basic Copula function components For the first The fundamental Copula function components at time 1 The optimal time-varying parameters; , These are the marginal distribution variables in the basic Copula functions of the two failure modes, respectively, when two failure modes exist. The system joint distribution model The expression is: ； in, , When two failure modes exist, the time of each failure mode is... The marginal distribution function, , These represent the degradation amount for each of the two failure modes; The dynamic weights of the basic Copula function components at time t. For at any time The actual value of the parameter state.

9. The reliability analysis method for hybrid Copula systems based on failure modes according to claim 1, characterized in that, In step S8, for a series system, the system reliability is the probability that all failure modes are simultaneously in a safe state, and the reliability expression is: ； in, , When two failure modes exist, time... The limit state functions for each of the two failure modes , For a moment The performance degradation stochastic process functions for each of the two failure modes , The safety thresholds corresponding to the two failure modes, For a moment No. Dynamic weights of the components of a basic Copula function; , These are the edge distribution functions of the two failure modes at the threshold values, respectively. No. The fundamental Copula function components at time 1 The optimal time-varying parameters, For the first The basic Copula function components It is the number of basic Copula functions.

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