Transmission shaft time-varying reliability analysis method based on hybrid agent model

Abstract

The invention relates to the technical field of transmission shaft time-varying reliability analysis and design, in particular to a transmission shaft time-varying reliability analysis method based on a hybrid agent model. Comprising the following steps: analyzing a transmission shaft, converting an input transmission shaft random process into a combination conforming to independent normal distribution input variables, discretizing time, and extracting principal components meeting requirements by using a principal component analysis method; establishing a multiple Kriging model between the training sample and the principal component; and adaptively updating the training sample, and calculating the time-varying reliability. According to the method, the time-varying reliability of the transmission shaft is analyzed through the hybrid agent model, and the prediction precision and reliability of the model are effectively improved. Compared with the traditional method for directly calculating the failure probability by using input variables, the method has the advantages that the input dimension is reduced, the regression trend is prevented from being manually selected, and the subjective error is reduced; compared with a Monte Carlo method, the method has the advantages that the number of model calling times is reduced, and the calculation efficiency is improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of time-varying reliability analysis and design of transmission shafts, and in particular relates to a time-varying reliability analysis method of a transmission shaft based on a hybrid agent model. Background Art

[0002] The drive shaft is a crucial mechanical transmission component, a key power transmission element in vehicles like automobiles and aircraft. Its reliability directly impacts the safety and performance of the entire system. Over time, drive shafts can experience performance degradation or even failure due to various factors, including material fatigue, changes in external loads, and environmental factors. Therefore, conducting time-varying reliability analysis on drive shafts to predict their failure probability over their intended lifespan is crucial for ensuring safe operation and optimizing design.

[0003] Traditional drive shaft reliability analysis methods mainly rely on computationally intensive technologies such as complex finite element analysis and Monte Carlo simulation. Although these methods can provide relatively accurate results, they are computationally expensive, time-consuming, and difficult to adapt to rapidly changing design and analysis requirements. In addition, these methods usually require a large amount of experimental data and statistical information, and their application is limited in situations where data acquisition is difficult or costly. Based on the above problems, researchers have begun to explore reliability analysis methods based on proxy models. Proxy models (such as response surface models, Kriging models, etc.) can quickly approximate the responses of complex models by training on limited sample points, thereby greatly improving computational efficiency. For example, Shi Chenpeng et al. proposed a bridge digital twin construction method based on the Kriging proxy model in the invention patent with authorization announcement number CN117540464B, constructed a real bridge digital twin model based on the proxy model, and realized the digital twin phase response simulation based on the monitoring system. In his invention patent application, published as CN118364711A, Ning Weiwei proposed a bandgap optimization method for elastic wave materials based on the Kriging surrogate model and the Q-learning algorithm. The Q-learning algorithm combined with the Kriging surrogate model improves the efficiency of searching for the optimal solution in the optimization problem and reduces computational costs. However, a single surrogate model often struggles to accurately capture the multi-dimensional input and dynamic response characteristics of complex systems, especially in the case of high-dimensional input variables and nonlinear responses. Furthermore, traditional surrogate model methods often require separate model training for each time point when dealing with dynamically changing operating conditions. This not only increases the amount of computation but also limits the model's generalization capabilities.

[0004] In the time-varying reliability analysis of structures such as drive shafts, the input is often not a simple input variable, but a complex process containing random processes. These random processes usually originate from complex actual working conditions, such as load fluctuations, temperature changes, and the randomness of material properties. Due to the highly nonlinear characteristics of high-dimensional random processes, traditional analysis methods face the challenge of exploding computational complexity when directly processing them, resulting in limited accuracy and reliability of the analysis results. In order to accurately simulate and analyze input random processes, some studies have begun to adopt random process modeling techniques, such as the power spectral density method (PSD) and random process simulation technology. For example, Wu Yongxin et al. proposed an efficient simulation method for one-dimensional multivariate random processes based on frequency domain equal energy interpolation in the invention patent application publication number CN107657127A. By setting frequency interpolation points, calculating the Cholesky decomposition matrix of the power spectral density function, and using cubic spline interpolation and spectral representation, they achieved high-precision random process simulation. In their invention patent application, CN117075596A, He Zhou et al. proposed a method and system for robot path planning for complex tasks under environmental and motion uncertainty. These methods achieve this by decomposing the environment, building a stochastic process model, and constructing a finite Markov decision process. While these methods can better capture the statistical characteristics of random processes, they typically require significant computing resources and their computational complexity increases dramatically when dealing with multidimensional random processes.

[0005] Based on this, the present invention proposes a time-varying reliability analysis method for a transmission shaft based on a hybrid agent model, which reduces the complexity of the model by extending the optimal linear estimation technology, and adopts a hybrid weight strategy to improve the prediction accuracy, thereby solving the problems existing in the above-mentioned existing technologies and improving the prediction accuracy and reliability of the model. Summary of the Invention

[0006] In order to solve the above technical problems, the present invention proposes a time-varying reliability analysis method for a transmission shaft based on a hybrid proxy model. This method uses the extended optimal linear estimation (EOLE) technology as a linear combination of independent standard normal variables, uses the principal component analysis method to reduce the dimension of the input variables, uses the Kriging model to construct multiple proxy models between the input variables and the principal components, and uses hybrid weights to calculate the reliability, thereby effectively improving the prediction accuracy and reliability of the model. Compared with the traditional method of directly calculating the failure probability using input variables, this method reduces the input dimension, avoids the artificial selection of regression trends, and reduces subjective errors. Compared with the Monte Carlo method, this method reduces the number of model calls and improves computational efficiency.

[0007] To achieve the above object, the present invention provides the following technical solutions:

[0008] The time-varying reliability analysis method of the transmission shaft based on the hybrid surrogate model includes:

[0009] Step 1: Analyze the transmission shaft, establish the sampling space R, and discretize the time interval;

[0010] Step 2: Extract N candidate sample points in the sampling space R to form a sample pool S, and extract N1 and N2 sample points from the sample pool S to form the training set E respectively. tra and validation set E m ;

[0011] Step 3: Calculate the validation set E m The distance matrix d(x) between the sample points and the spatial position measurement factor l k (x) and weight factor λ d (i);

[0012] Step 4: Calculate the training set E tra Each sample in N t The actual response value of the failure function g(x) at each time node constitutes the response matrix Y t ;

[0013] Step 5: Extract the response matrix Y using principal component analysis t The principal component h k ;

[0014] Step 6: According to the training samples and principal component h k , construct multiple Kriging mixed agent models;

[0015] Step 7: Determine the mixing weight coefficient w of multiple surrogate models in the multi-Kriging mixed surrogate model i ;

[0016] Step 8: According to the mixing weight coefficient w i Calculate the output mean μ and standard deviation σ of all sample points;

[0017] Step 9: Establish the convergence parameter ε m , evaluate the convergence parameter ε m Whether the convergence condition is met, if not, the learning function is used to adaptively select a new sample point x new , and the sample point x new Add training set E tra , go back to step 6; if satisfied, proceed to the next step;

[0018] Step 10: Calculate the time-varying reliability R based on the output values ​​of all sample points obtained in step 8.

[0019] According to the time-varying reliability analysis method of the transmission shaft based on the hybrid agent model provided by the present invention, the specific process of step 1 includes: analyzing the structural composition, function and working conditions of the transmission shaft, determining the failure mode of the transmission shaft structure and the corresponding failure function function g(x), converting the input random process of the transmission shaft into a combination of m input variables that conform to the independent normal distribution through the extended optimal linear estimation technology, obtaining the sampling space R, and obtaining the input variable x that affects the structural failure function function g(x), determining the number of all input variables x and the corresponding mean, standard deviation and distribution type, and discretizing the time interval into N according to actual needs. t Time nodes;

[0020] When analyzing the drive shaft, the internal torque on the drive shaft cross section is a random process that obeys a Gaussian distribution. The formula for converting the input drive shaft random process into a combination of m input variables that conform to an independent normal distribution using the extended optimal linear estimation technique is as follows:

[0021]

[0022] in, represents the approximate value of the response matrix Y(t), ζ i represents the input variable of the standard normal distribution, m represents the number of terms in the expansion, and λ i and φ i Represents the correlation matrix C Y The eigenvalues ​​and eigenvectors of μ Y (t) represents the average value of the internal torque on the transmission shaft section, σ Y (t) represents the standard deviation of the internal moment on the drive shaft cross section.

[0023] According to the time-varying reliability analysis method of the transmission shaft based on the hybrid agent model provided by the present invention, in step 3, the verification set E is calculated. m The distance matrix d(x) between the sample points, the spatial position measurement factor l k (x) and weight factor λ d The process of (i) includes:

[0024] For E m Any two sample points x i and x j , the calculation formula of the element d(i,j) of the distance matrix d(x) is as follows:

[0025]

[0026] in, represents the kth eigenvalue of the i-th sample, x j (k)represents the kth eigenvalue of the jth sample, and n represents the dimension of the sample feature;

[0027] Spatial position measurement factor l k The calculation formula for (x) is as follows:

[0028]

[0029] Weight factor λ d The calculation formula for (i) is as follows:

[0030]

[0031] Among them, d min (i,1) represents the sample point x i The distance between the nearest neighbor sample x(i,1), d min (i,2) represents the sample point x i The distance between it and the second nearest neighbor sample x(i,2).

[0032] According to the time-varying reliability analysis method of the transmission shaft based on the hybrid agent model provided by the present invention, the response matrix Y in step 4 is t for:

[0033]

[0034] Among them, y i (t j ) represents the input vector x i At time node t j The actual response value of the failure function g(x) at .

[0035] According to the time-varying reliability analysis method of the transmission shaft based on the hybrid surrogate model provided by the present invention, step 5 uses the principal component analysis method to extract the response matrix Y t The principal component h k The process includes:

[0036] Step 5.1: Response matrix Y t Normalize and obtain the normalized matrix Y C , extract the normalized matrix Y C The eigenvalues ​​and eigenvectors of

[0037] Step 5.2: Normalize the matrix Y C The eigenvalues ​​of are arranged in descending order, and the first n eigenvalues ​​λ that meet the accuracy requirements are extracted k and the corresponding eigenvector v k ;

[0038] Extract the first n eigenvalues ​​λ k The accuracy requirements are as follows:

[0039]

[0040] Among them, λ k Represents the normalized matrix Y C Eigenvalues ​​in descending order, C represents the percentage of total variance, N t Represents the number of time nodes;

[0041] Step 5.3: According to the normalized matrix Y C and the eigenvector v k Calculate the principal component h k ;

[0042] h k =Y C v k .

[0043] According to the time-varying reliability analysis method of the transmission shaft based on the hybrid agent model provided by the present invention, step 6 is based on the training samples and the principal component h k , the process of constructing a multiple Kriging mixed surrogate model includes:

[0044] Step 6.1: Based on the training set E tra The dimension dim and number N of training samples in , construct an initial proxy model set, and determine the number θ of proxy models included in the initial proxy model set;

[0045] Step 6.2: Use the training set E tra The training samples and the corresponding principal components are used to train the proxy model, and θ proxy models are constructed for each of the n principal components.

[0046] Step 6.3: Use the trained proxy model to test the validation set E m Make predictions and obtain the predicted mean and standard deviation of the principal components;

[0047] Step 6.4: Use the predicted values ​​of the principal components to inverse PCA to reconstruct the output and obtain the mean and standard deviation of the output.

[0048] According to the hybrid surrogate model-based time-varying reliability analysis method for a transmission shaft provided by the present invention, in step 6.1, during the process of constructing an initial surrogate model set and determining the number θ of surrogate models included in the initial surrogate model set, the surrogate model construction method is determined by the following method:

[0049] When 1<N1≤dim+2, a Kriging model with a constant regression trend is constructed;

[0050] when When , two Kriging models with constant regression and linear regression are constructed;

[0051] when When , three Kriging models with constant regression, linear regression and quadratic regression are constructed;

[0052] Among them, N1 represents the number of training samples and dim represents the dimension of the input variable.

[0053] According to the time-varying reliability analysis method of the transmission shaft based on the hybrid surrogate model provided by the present invention, step 7 determines the hybrid weight coefficients w of multiple surrogate models in the multiple Kriging hybrid surrogate model. i The process includes:

[0054] Step 7.1: Calculate the global weight coefficient using mean square error weighting

[0055] Step 7.2: Use the spatial position measure factor l k (x) and the output standard deviation to calculate the local weight coefficient

[0056] Step 7.3: Combine global weight coefficients through agent ensemble method and the local weight coefficient Calculate the mixing weight coefficient w i .

[0057] According to the time-varying reliability analysis method of the transmission shaft based on the hybrid surrogate model provided by the present invention, the convergence parameter ε in step 9 is m for:

[0058]

[0059] Among them, y vi and Represent the validation set E m The actual function response value and the predicted value after the mixed surrogate model, E k represents the E value in the kth iteration, E k-1 Represents the root mean square error E value in the k-1th iteration;

[0060] The convergence criterion is: ε m ≤L, L is the preset threshold;

[0061] If the convergence criterion is not met, the sample point x with the minimum value of the U learning function is selected from the entire sample pool according to the U learning function. new Add training set E tra , retrain the model.

[0062] According to the time-varying reliability analysis method of the transmission shaft based on the hybrid surrogate model provided by the present invention, the reliability R in step 10 is:

[0063]

[0064] Among them, N represents the total number of samples, N y≤0 Represents the number of hybrid agent model output values ​​less than or equal to 0, N y>0 Represents the number of hybrid agent model output values ​​greater than 0, P f represents the time-varying failure probability.

[0065] The present invention has at least the following beneficial effects:

[0066] The present invention provides a time-varying reliability analysis method for a transmission shaft based on a hybrid surrogate model. The method converts a random process into a combination of normally distributed input variables, reduces the dimensionality of the input variables using principal component analysis, constructs multiple surrogate models between the input variables and the principal components using the Kriging model, and calculates the reliability using hybrid weights. Compared with the traditional method of directly calculating the failure probability using input variables, the method reduces the input dimensionality; at the same time, it avoids the artificial selection of regression trends and reduces subjective errors; compared with the Monte Carlo method, the method reduces the number of model calls and improves computational efficiency.

[0067] When this method uses principal component analysis to expand the optimal linear estimation technique, it characterizes the random process by representing the spectrum of the equivalent transformation of the random process as a combination of a series of relatively simple orthogonal functions and input variables, thereby reducing the complexity of the model and reducing the dimension of the input variables through the principal component analysis process.

[0068] When this method uses the Kriging model to construct multiple surrogate models between input variables and principal components, it uses the Kriging model to construct multiple surrogate models and adopts a mixed weight strategy to improve the prediction accuracy and reliability of the model. BRIEF DESCRIPTION OF THE DRAWINGS

[0069] The accompanying drawings, which constitute part of this application, are intended to provide a further understanding of this application. The exemplary embodiments and descriptions of this application are intended to explain this application and do not constitute an improper limitation on this application. In the accompanying drawings:

[0070] Figure 1 This is a flow chart of a method for analyzing time-varying reliability of a transmission shaft based on a hybrid agent model according to the present invention;

[0071] Figure 2 This is a simplified schematic cross-sectional view of a hollow circular cross-section transmission shaft in Example 2 of the present invention;

[0072] Figure 3 A convergence process curve diagram of the convergence parameters of the method proposed in Example 1 in Example 2 of the present invention;

[0073] Figure 4This is a comparison diagram of the convergence curves of the failure probability calculation results of the method proposed in Example 1 and the MCS method in Example 2 of the present invention;

[0074] Figure 5 This is a comparison chart of the convergence curves of the reliability calculation results of the method proposed in Example 1 and the MCS method in Example 2 of the present invention. DETAILED DESCRIPTION

[0075] It should be noted that, in the absence of conflict, the embodiments and features of the embodiments in this application can be combined with each other. The present application will be described in detail below with reference to the accompanying drawings and in combination with the embodiments.

[0076] Example 1:

[0077] See also Figure 1 This embodiment provides a time-varying reliability analysis method for a transmission shaft based on a hybrid surrogate model, comprising:

[0078] Step 1: Analyze the structural composition, function and working conditions of the transmission shaft, determine the failure mode of the transmission shaft structure and the corresponding failure function function g(x), convert the input transmission shaft random process into a combination of m input variables that conform to the independent normal distribution through the extended optimal linear estimation technology, obtain the sampling space R, and obtain the input variables x that affect the structural failure function function g(x), determine the number of all input variables x and the corresponding mean, standard deviation and distribution type, and discretize the time interval into N according to actual needs t Time nodes;

[0079] Step 2: Use the hypercube sampling method or other low-discrepancy sequence sampling method to extract N candidate sample points from the sampling space R of the variable obtained in step 1 to form a sample pool S. Then extract N1 and N2 sample points from the sample pool S to form the training set E. tra and validation set E m , and let m = 1 to record the number of iterations;

[0080] Step 3: Calculate the validation set E m The distance matrix d(x) between the sample points in the matrix is ​​used to calculate the spatial position measurement factor l. k (x) and weight factor λ d (i) for the subsequent determination of local weights and hybrid weights;

[0081] Step 4: Calculate the training set E tra Each sample in N t The actual response value of the failure function g(x) at each time node constitutes the response matrix Y t , get the response matrix Y t The form is:

[0082]

[0083] Among them, y i (t j ) represents the input vector x i At time node t j The actual response value of the failure function g(x) at ;

[0084] Step 5: Extract the response matrix Y using principal component analysis (PCA) t The principal component h k ;

[0085] Step 6: According to the training samples and principal component h k , construct multiple Kriging mixed agent models;

[0086] Step 7: Determine the mixing weight coefficient w of multiple surrogate models in the multi-Kriging mixed surrogate model i ;

[0087] Step 8: According to the mixing weight coefficient w i Calculate the output mean μ and standard deviation σ of all sample points;

[0088] Among them, according to the mixing weight coefficient w i When calculating the output mean and standard deviation of all sample points, the formula for calculating the weighted output y is as follows:

[0089]

[0090] Its mean μ and standard deviation σ are:

[0091]

[0092] Step 9: Establish the convergence parameter ε m , evaluate the convergence parameter ε m Whether the convergence condition is met, if not, the learning function is used to adaptively select a new sample point x new , and the sample point x new Add training set E tra , go back to step 6; if satisfied, proceed to the next step;

[0093] Step 10: Calculate the time-varying reliability R based on the output values ​​of all sample points obtained in step 8;

[0094] Among them, the time-varying failure probability P is calculated using the output values ​​of all sample points f The formula for reliability R is as follows:

[0095]

[0096] Among them, N represents the total number of samples, N y≤0 Represents the number of hybrid agent model output values ​​less than or equal to 0, N y>0 Represents the number of hybrid agent model output values ​​greater than 0, P f represents the time-varying failure probability.

[0097] Specifically, in the process of analyzing the structural composition, function, and operating conditions of the transmission shaft in step 1, the internal torque on the transmission shaft cross section is a random process that obeys a Gaussian distribution and can be characterized by its mean, standard deviation, and autocorrelation coefficient. The extended optimal linear estimation technique can be used to transform this random process into a combination of independent normally distributed input variables. The formula is as follows:

[0098]

[0099] in, represents the approximate value of the response matrix Y(t), ζ i represents the input variable of the standard normal distribution, m represents the number of terms in the expansion, and λ i and φ i Represents the correlation matrix C Y The eigenvalues ​​and eigenvectors of μ Y (t) represents the average value of the internal torque on the transmission shaft section, σ Y (t) represents the standard deviation of the internal moment on the drive shaft section. Correlation matrix C Y for:

[0100]

[0101] Among them, ρ Y (t i ,t j )(i,j=1,2,...,N t ) represents the autocorrelation coefficient.

[0102] Specifically, in step 3, calculate the validation set E m The distance matrix d(x) between the sample points, the spatial position measurement factor l k (x) and weight factor λ d The process of (i) includes:

[0103] Step 3.1: The distance matrix d(x) is used to measure the distance between sample points, usually using Euclidean distance. For any two sample points x in the distance matrix d(x) i and x j , the calculation formula of the element d(i,j) of the distance matrix is ​​as follows:

[0104]

[0105] in, represents the kth eigenvalue of the i-th sample, x j (k) represents the kth eigenvalue of the jth sample, and n represents the dimension of the sample feature. This formula calculates the distance between sample points i and j in multidimensional space.

[0106] Step 3.2: Spatial position measurement factor l k The calculation formula for (x) is as follows:

[0107]

[0108] Step 3.3: Weight factor λ d (i) It is used to reflect the relative distance between a sample point and its adjacent sample points. The calculation formula is as follows:

[0109]

[0110] Among them, d min (i,1) represents the sample point x i The distance between the nearest neighbor sample x(i,1), d min (i,2) represents the sample point x i The distance between the sample point x(i,2) and the second nearest neighbor. The sin function in this formula ensures that the measurement coefficient ranges between 0 and 1. This coefficient reflects the distance between the sample point x(i,2). i The proportional relationship between the distance to its nearest neighbor samples.

[0111] Specifically, step 5 uses principal component analysis (PCA) to extract the response matrix Y t The principal component process includes:

[0112] Step 5.1: Response matrix Y t Normalize and obtain the normalized matrix Y C , extract the normalized matrix Y C The eigenvalues ​​and eigenvectors of

[0113] Step 5.2: Normalize the matrix Y C The eigenvalues ​​of are arranged in descending order, and the first n eigenvalues ​​λ that meet the accuracy requirements are extracted k (k=1,2,...,n) and the corresponding eigenvector v k (k=1,2,...,n);

[0114] It should be noted that the accuracy requirements of the first n eigenvalues ​​extracted in step 5.2 must meet the following requirements:

[0115]

[0116] Among them, λ k (k=1,2,...,N t ) represents the normalized matrix Y C The eigenvalues ​​are arranged in descending order, C represents the percentage of the total variance, usually C>95%, N t Represents the number of time nodes.

[0117] Step 5.3: According to the normalized matrix Y C and the eigenvector v k Calculate the principal component h k (k=1,2,...,n);

[0118] It should be noted that the principal component h k The calculation formula is as follows:

[0119] h k =Y C v k

[0120] Among them, Y C According to step 5.1, the response matrix Y t The normalized matrix obtained by normalization, v k Y is extracted according to step 5.2 C The eigenvectors of in descending order.

[0121] Specifically, step 6 is based on the training samples and the principal component h k , the process of constructing a multiple Kriging mixed surrogate model includes:

[0122] Step 6.1: Based on the training set E tra The dimension dim and number N of training samples in , construct the initial proxy model set, and determine the number θ of proxy models included in the initial proxy model set, that is, the number of regression trends;

[0123] It should be noted that in step 6.1, when constructing the initial proxy model set and determining the number θ of proxy models included in the initial proxy model set, the proxy model set is constructed using the following method:

[0124] When 1<N1≤dim+2, a Kriging model with a constant regression trend is constructed, that is, θ=1;

[0125] when When , two Kriging models with constant regression and linear regression are constructed, that is, θ = 2;

[0126] when When , three Kriging models with constant regression, linear regression and quadratic regression are constructed, that is, θ = 3;

[0127] Among them, N1 represents the number of training samples and dim represents the dimension of the input variable.

[0128] Step 6.2: Use the training set E tra The training samples and the corresponding principal components are used to train the proxy model, and θ proxy models are constructed for each of the n principal components.

[0129] It should be noted that in step 6.2, θ proxy models are constructed for each of the n principal components, and the total number of proxy models is θ×n.

[0130] Step 6.3: Use the trained proxy model to test the validation set E m Make predictions and obtain the predicted mean and standard deviation of the principal components;

[0131] Step 6.4: Use the predicted values ​​of the principal components to perform inverse PCA decomposition to reconstruct the output and obtain the mean and standard deviation of the output;

[0132] It should be noted that step 6.4 uses the predicted value of the principal component to inverse PCA decomposition and reconstruct the output, and the calculation formula is:

[0133]

[0134] in, represents the dynamic output reconstructed by the ith proxy model at time t, Represents the average value of output at each moment; The mean and standard deviation of are obtained from the mean and standard deviation of the principal component predictions, and the calculation formula is:

[0135]

[0136] in, They represent the mean and standard deviation of the prediction of the kth principal component by the ith surrogate model.

[0137] Specifically, step 7 determines the mixing weight coefficient w of multiple proxy models in the multiple Kriging mixed proxy model i The process includes:

[0138] Step 7.1: Calculate the global weight coefficient using mean square error weighting

[0139] It should be noted that step 7.1 uses mean square error weighting to calculate the global weight coefficient The formula for calculating the mean square error is as follows:

[0140]

[0141] Among them, N2 represents the validation set E m The number of samples in x k Represents the validation set E m The kth sample point in y i (x k ) represents the i-th agent model at x k The actual response value of the functional function at .

[0142] Global weight coefficient The calculation formula is as follows:

[0143]

[0144] Wherein, α (α<1) and β (β<0) control the average value and the weight of a single agent model respectively. In this embodiment, α=0.05 and β=-1.

[0145] Step 7.2: Use the spatial position measure factor l from step 3 k (x) and the output standard deviation in step 6.4 to calculate the local weight coefficient

[0146] It should be noted that in step 7.2, the spatial position measurement factor l is used k (x) and the standard deviation of the output to calculate the local weight coefficient The formula is as follows:

[0147]

[0148] Among them, w ik Represents the point-by-point weight factor of the i-th proxy model at the k-th sample point, using a 0 / 1 weighting strategy: it is equal to 1 for the proxy model with the smallest output standard deviation and equal to 0 for other proxy models.

[0149] Step 7.3: Combine global weight coefficients through agent ensemble method and the local weight coefficient Calculate the mixing weight coefficient w i .

[0150] It should be noted that step 7.3 calculates the mixing weight coefficient w i The calculation formula is as follows:

[0151]

[0152] in, and Represent the global weight and local weight of the i-th agent model, λ d (i) is the weighting factor.

[0153] Specifically, step 9 defines the convergence criteria including when the mean square error between two iterations is small enough, the convergence parameter ε m for:

[0154]

[0155] Among them, y vi and Represent the validation set E m The actual function response value and the predicted value after the mixed surrogate model, E k represents the E value in the kth iteration, E k-1 represents the E value in the k-1th iteration.

[0156] The convergence criterion is: ε m ≤L;

[0157] Wherein, L represents a preset threshold, which is set to 0.0005 in this embodiment.

[0158] If the convergence criterion is not met, the sample point x with the minimum value of the U learning function is selected from the entire sample pool according to the U learning function. new Add training set E tra , retrain the model.

[0159] Example 2:

[0160] like Figure 2 、 Figure 3 、 Figure 4 and Figure 5 As shown, the difference from the above embodiment 1 is that, in order to verify the feasibility and credibility of the method described in the above embodiment 1, this embodiment uses a hollow circular cross-section transmission shaft as an example to verify the method described in the above embodiment 1. Figure 2 This is a simplified schematic diagram of a hollow circular section, and a rectangular coordinate system is established in the figure.

[0161] Figure 2 Where R represents the outer radius of the transmission shaft, r represents the inner radius of the transmission shaft, ρ represents the distance from the center O when calculating the shear stress of the infinitesimal element dA, and α represents the angle between the infinitesimal element and the y-axis.

[0162] Step 1: Consider Figure 2 For the hollow circular section transmission shaft shown in the figure, the shear stress of the infinitesimal element dA is:

[0163]

[0164] Where T represents the resultant moment on the section, I pRepresents the second polar moment of the cross section about point O. Since the farther away from the axis, the greater the shear stress, the maximum shear stress is at the outer radius, and the maximum shear stress is:

[0165]

[0166] Considering the maximum shear stress τ0 that the material used for the transmission shaft can withstand, the shear failure function g(x) of the hollow circular section transmission shaft in this example is established as:

[0167]

[0168] consider Figure 2 The outer radius R, inner radius r and the maximum shear stress τ0 that the material can withstand of the hollow circular section transmission shaft are input variables, and the resultant moment T on the section is 5M. 2 (t)C m , C m =0.4422 represents the moment coefficient, M(t) is a random process, and its average value is μ T (t), standard deviation is σ T (t), autocorrelation coefficient ρ T (t1, t2) is given by:

[0169]

[0170] Among them, the constants a, b, and c are:

[0171]

[0172] The distribution information of the input variables is shown in Table 1.

[0173] Table 1: Distribution information of input variables

[0174]

[0175] The extended optimal linear estimation technique is used to transform the random process M(t) into a combination of five independent input variables that obey the standard normal distribution. The calculation formula is as follows:

[0176]

[0177] in, represents the approximate value of M(t), ζ i represents the input variable of the standard normal distribution, λ i and φ i Represents the correlation matrix C Y Arrange the five largest eigenvalues ​​and eigenvectors in descending order. Correlation matrix C Y for:

[0178]

[0179] Among them, ρ T (t i ,t j ) represents the autocorrelation coefficient.

[0180] After the random process M(t) is transformed into a combination of five independent standard normally distributed input variables using the extended optimal linear estimation technique, the input dimension is transformed into eight dimensions. The distribution information of all input variables is shown in Table 2.

[0181] Table 2: Distribution information of input variables

[0182]

[0183] Discretize the time interval [0.12,12] considered in the case into N t =100 time nodes.

[0184] Step 2: Latin hypercube sampling can ensure both randomness and relative uniformity of samples. Sobol sequence can achieve the same high-quality distribution of samples as Latin hypercube sampling. At the same time, it does not require the number of samples to be determined in advance or to be stored, and can generate unlimited samples as needed. Therefore, the case uses Sobol sequence to extract N = 100,000 input sample points described in step 1 to form the sample pool S, and then extract N1 = 30 and N2 = 200 sample points again to form the training set E respectively. tra and validation set E m , and let m = 1 to record the number of iterations.

[0185] Step 3: Calculate the validation set E m The distance matrix d(x) between the sample points in the matrix is ​​used to calculate the spatial position measurement factor l. k (x) and weight factor λ d (i), used for the subsequent determination of local weights and mixing weights.

[0186] For any two sample points x i and x j , the calculation formula of the element d(i,j) of the distance matrix is ​​as follows:

[0187]

[0188] in, represents the kth eigenvalue of the i-th sample, and n represents the dimension of the sample feature. This formula calculates the distance between two sample points i and j in multidimensional space.

[0189] Spatial position measurement factor l kThe calculation formula for (x) is as follows:

[0190]

[0191] Weight factor λ d (i) It is used to reflect the relative distance between a sample point and its adjacent sample points. The calculation formula is as follows:

[0192]

[0193] Among them, d min (i,1) represents the sample point x i The distance between the nearest neighbor sample x(i,1), d min (i,2) represents the sample point x i The distance between the sample point x(i,2) and the second nearest neighbor. The sin function in this formula ensures that the measurement coefficient ranges between 0 and 1. This coefficient reflects the distance between the sample point x(i,2). i The proportional relationship between the distance to its nearest neighbor samples.

[0194] Step 4: Calculate the training set E tra Each sample in N t =The actual response values ​​of the failure function g(x) at 100 time nodes, forming the response matrix Y t :

[0195]

[0196] Among them, y i (t j ) represents the input vector x i At time node t j The actual response value of the failure function at .

[0197] Step 5: Extract the response matrix Y using principal component analysis (PCA) t The principal component h k

[0198] Step 5.1: Response matrix Y t Normalize to get Y C , extract the normalized matrix Y C The eigenvalues ​​and eigenvectors of .

[0199] Step 5.2: Y C The eigenvalues ​​of are arranged in descending order, and the first three eigenvalues ​​λ that meet the accuracy requirements are extracted. k (k=1,2,3) and the corresponding eigenvector v k (k=1,2,3). The accuracy requirements of the first three eigenvalues ​​extracted must meet the following requirements:

[0200]

[0201] Among them, λ k (k=1,2,...,N t ) represents the normalized matrix Y C The eigenvalues ​​are arranged in descending order, C represents the percentage of the total variance, usually C>95%, N t Represents the number of time nodes.

[0202] Step 5.3: According to the normalized matrix Y C and the eigenvector v k Calculate the principal component h k (k=1,2,3).

[0203] Principal component h k The calculation formula is as follows:

[0204] h k =Y C v k

[0205] Among them, Y C According to step 5.1, the response matrix Y t The normalized matrix obtained by normalization, v k Y is extracted according to step 5.2 C The eigenvectors of in descending order.

[0206] Step 6: Based on the input variables and principal component h determined in step 1 k , construct multiple Kriging mixed surrogate models

[0207] Step 6.1: Based on the training set E tra The dimension dim and number N of training samples in determine the number θ of proxy models included in the initial proxy model set:

[0208] When 1<N1≤dim+2, a Kriging model with a constant regression trend is constructed, that is, θ=1;

[0209] when When , two Kriging models with constant regression and linear regression are constructed, that is, θ = 2;

[0210] when When θ=3, three Kriging models with constant regression, linear regression and quadratic regression are constructed.

[0211] Since the initial number of training samples N1 = 30 and the dimension of the input variable dim = 8, it satisfies:

[0212]

[0213] Therefore, the number of initial proxy models is θ = 2, and two Kriging models with constant regression and linear regression are constructed.

[0214] Step 6.2: Use the training set E tra The training samples and the corresponding principal components are used to train the proxy model, and θ proxy models are constructed for each of the n principal components. The total number of proxy models is θ×n.

[0215] Step 6.3: Use the trained proxy model to test the validation set E m Make predictions and get the predicted mean and standard deviation of the principal components.

[0216] Step 6.4: Use the predicted values ​​of the principal components to perform inverse PCA decomposition to reconstruct the output and obtain the mean and standard deviation of the output. The calculation formula is:

[0217]

[0218] in, represents the dynamic output reconstructed by the ith proxy model at time t, Represents the average output at each moment. The mean and standard deviation of are obtained from the mean and standard deviation of the principal component predictions, and the calculation formula is:

[0219]

[0220] Among them, μ ki , σ ki represents the mean and standard deviation of the prediction of the kth principal component by the i-th surrogate model.

[0221] Step 7: Determine the mixing weight coefficient w of multiple surrogate models in the multi-Kriging mixed surrogate model i

[0222] Step 7.1: Calculate the global weight coefficient using mean square error weighting

[0223] The formula for calculating the mean square error is as follows:

[0224]

[0225] Among them, N2=200 is the number of samples in the validation set, x k is the kth sample point in the validation set, y i (x k ) is the i-th agent model in x k The actual response value of the functional function at .

[0226] Global weight coefficient The calculation formula is as follows:

[0227]

[0228] Wherein, α (α<1) and β (β<0) control the average value and the weight of a single agent model respectively. In this embodiment, α=0.05 and β=-1.

[0229] Step 7.2: Use the spatial position measure factor l k (x) and the standard deviation of the output to calculate the local weight coefficient The formula is as follows:

[0230]

[0231] Among them, w ik Represents the point-by-point weight factor of the i-th proxy model at the k-th sample point, using a 0 / 1 weighting strategy: it is equal to 1 for the proxy model with the smallest output standard deviation and equal to 0 for other proxy models.

[0232] Step 7.3: Calculate the mixing weight coefficient w through the agent integration method i The calculation formula is as follows:

[0233]

[0234] in, and Represent the global weight and local weight of the i-th agent model, λ d (i) represents the weight factor.

[0235] Step 8: According to the mixing weight coefficient w i When calculating the output mean and standard deviation of all sample points, the formula for calculating the weighted output y is as follows:

[0236]

[0237] Its mean μ and standard deviation σ are:

[0238]

[0239] Step 9: Establish the convergence parameter ε m , evaluate the convergence parameter ε m Whether the convergence conditions are met, the convergence parameters are:

[0240]

[0241] Among them, y vi and Represent the actual functional response value of the validation set and the predicted value after the hybrid surrogate model, Ek represents the E value in the kth iteration.

[0242] Define the convergence threshold L = 0.0005, if ε m > L uses the learning function to adaptively select new sample points x new , and x new Add training set E tra , go back to step 6; if ε m ≤L, then proceed to the next step. The convergence process curve is as follows Figure 3 shown.

[0243] Step 10: Based on the output values ​​of all sample points obtained in step 8, first calculate the failure probability P f , and then calculate the time-varying reliability R:

[0244]

[0245] Among them, N represents the total number of samples, N y≤0 Represents the number of hybrid agent model output values ​​less than or equal to 0.

[0246] The failure probability convergence curve and reliability convergence curve are obtained as follows: Figure 4 、 Figure 5 shown.

[0247] Step 11: Under the premise of the same external conditions and the same specifications of the transmission shaft, the comparative effect of the method described in Example 1 of the present invention and the typical method in the example results is shown in Table 3.

[0248] Table 3: Comparison results of the method of Example 1 of the present invention and the typical method

[0249] Analytical methods Required sample size Failure probability Coefficient of variation MCS <![CDATA[10 5 ]]> 0.00323 0.0056 Typical methods 30+102 0.00323 0.0556 Method of the present invention 30+70 0.00323 0.0556

[0250] The comparison results shown in Table 3 demonstrate that, compared to traditional MCS analysis methods and typical analysis methods, the hybrid surrogate model-based transmission shaft time-varying reliability analysis method described in Example 1 of the present invention effectively maintains computational accuracy while requiring fewer samples and shortening runtime. This demonstrates that the hybrid surrogate model-based transmission shaft time-varying reliability analysis method described in Example 1 of the present invention can more accurately and reliably perform transmission shaft time-varying reliability analysis.

[0251] While embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions, and variations may be made to these embodiments without departing from the principles and spirit of the invention, and that the scope of the invention is defined by the appended claims and their equivalents.

Claims

1. A time-varying reliability analysis method for a transmission shaft based on a hybrid surrogate model, characterized by: include: Step 1: Analyze the transmission shaft, establish the sampling space R, and discretize the time interval; Step 2: Extract N candidate sample points in the sampling space R to form a sample pool S, and extract N1 and N2 sample points from the sample pool S to form the training set E respectively. tra and validation set E m ; Step 3: Calculate the validation set E m The distance matrix d(x) between the sample points and the spatial position measurement factor l k (x) and weight factor λ d (i); Step 4: Calculate the training set E tra Each sample in N t The actual response value of the failure function g(x) at each time node constitutes the response matrix Y t ; Step 5: Extract the response matrix Y using principal component analysis t The principal component h k ; Step 6: According to the training samples and principal component h k , construct multiple Kriging mixed agent models; Step 7: Determine the mixing weight coefficient w of multiple surrogate models in the multi-Kriging mixed surrogate model i ; Step 8: According to the mixing weight coefficient w i Calculate the output mean μ and standard deviation σ of all sample points; Step 9: Establish the convergence parameter ε m , evaluate the convergence parameter ε m Whether the convergence condition is met, if not, the learning function is used to adaptively select a new sample point x new , and the sample point x new Add training set E tra , go back to step 6; if satisfied, proceed to the next step; Step 10: Calculate the time-varying reliability R based on the output values ​​of all sample points obtained in step 8.

2. The method for analyzing time-varying reliability of a transmission shaft based on a hybrid surrogate model according to claim 1, wherein: The specific process of step 1 includes: analyzing the structural composition, function and working conditions of the transmission shaft, determining the failure mode of the transmission shaft structure and the corresponding failure function function g(x), converting the input transmission shaft random process into a combination of m input variables that conform to the independent normal distribution through the extended optimal linear estimation technology, obtaining the sampling space R, and obtaining the input variables x that affect the structural failure function function g(x), determining the number of all input variables x and the corresponding mean, standard deviation and distribution type, and discretizing the time interval into N according to actual needs. t Time nodes; When analyzing the drive shaft, the internal torque on the drive shaft cross section is a random process that obeys a Gaussian distribution. The formula for converting the input drive shaft random process into a combination of m input variables that conform to an independent normal distribution using the extended optimal linear estimation technique is as follows: in, represents the approximate value of the response matrix Y(t), ζ i represents the input variable of the standard normal distribution, m represents the number of terms in the expansion, and λ i and φ i Represents the correlation matrix C Y The eigenvalues ​​and eigenvectors of μ Y (t) represents the average value of the internal torque on the transmission shaft section, σ Y (t) represents the standard deviation of the internal moment on the drive shaft cross section.

3. The method for analyzing time-varying reliability of a transmission shaft based on a hybrid surrogate model according to claim 1, wherein: In step 3, calculate the validation set E m The distance matrix d(x) between the sample points, the spatial position measurement factor l k (x) and weight factor λ d (i) Process include: For E m Any two sample points x i and x j , the calculation formula of the element d(i,j) of the distance matrix d(x) is as follows: in, represents the kth eigenvalue of the i-th sample, x j (k) represents the kth eigenvalue of the jth sample, and n represents the dimension of the sample feature; Spatial position measurement factor l k The calculation formula for (x) is as follows: Weight factor λ d The calculation formula for (i) is as follows: Among them, d min (i,1) represents the sample point x i The distance between the nearest neighbor sample x(i,1), d min (i,2) represents the sample point x i The distance between the second nearest neighbor sample x(i,2).

4. The method for analyzing time-varying reliability of a transmission shaft based on a hybrid surrogate model according to claim 1, wherein: The response matrix Y in step 4 t for: Among them, y i (t j ) represents the input vector x i At time node t j The actual response value of the failure function g(x) at .

5. The method for analyzing time-varying reliability of a transmission shaft based on a hybrid surrogate model according to claim 1, wherein: Step 5: Extract the response matrix Y using principal component analysis t The principal component h k The process includes: Step 5.1: Response matrix Y t Normalize and obtain the normalized matrix Y C , extract the normalized matrix Y C The eigenvalues ​​and eigenvectors of Step 5.2: Normalize the matrix Y C The eigenvalues ​​of are arranged in descending order, and the first n eigenvalues ​​λ that meet the accuracy requirements are extracted k and the corresponding eigenvector v k ; Extract the first n eigenvalues ​​λ k The accuracy requirements are as follows: Among them, λ k Represents the normalized matrix Y C Eigenvalues ​​in descending order, C represents the percentage of total variance, N t Represents the number of time nodes; Step 5.3: According to the normalized matrix Y C and the eigenvector v k Calculate the principal component h k ; h k =Y C v k 。 6. The method for analyzing time-varying reliability of a transmission shaft based on a hybrid surrogate model according to claim 1, wherein: Step 6: Based on the training samples and principal component h k , the process of constructing a multiple Kriging mixed surrogate model includes: Step 6.1: Based on the training set E tra The dimension dim and number N of training samples in , construct an initial proxy model set, and determine the number θ of proxy models included in the initial proxy model set; Step 6.2: Use the training set E tra The training samples and the corresponding principal components are used to train the proxy model, and θ proxy models are constructed for each of the n principal components. Step 6.3: Use the trained proxy model to test the validation set E m Make predictions and obtain the predicted mean and standard deviation of the principal components; Step 6.4: Use the predicted values ​​of the principal components to inverse PCA to reconstruct the output and obtain the mean and standard deviation of the output.

7. The method for analyzing time-varying reliability of a transmission shaft based on a hybrid surrogate model according to claim 5, wherein: In step 6.1, when constructing the initial proxy model set and determining the number θ of proxy models included in the initial proxy model set, the proxy model construction method is determined by the following method: When 1<N1≤dim+2, a Kriging model with a constant regression trend is constructed; when When , two Kriging models with constant regression and linear regression are constructed; when When , three Kriging models with constant regression, linear regression and quadratic regression are constructed; Among them, N1 represents the number of training samples, and dim represents the dimension of the input variable.

8. The method for analyzing time-varying reliability of a transmission shaft based on a hybrid surrogate model according to claim 5, wherein: Step 7: Determine the mixing weight coefficient w of multiple surrogate models in the multi-Kriging mixed surrogate model i The process includes: Step 7.1: Calculate the global weight coefficient using mean square error weighting Step 7.2: Use the spatial position measure factor l k (x) and the output standard deviation to calculate the local weight coefficient Step 7.3: Combine global weight coefficients through agent ensemble method and the local weight coefficient Calculate the mixing weight coefficient w i .

9. The method for analyzing time-varying reliability of a transmission shaft based on a hybrid surrogate model according to claim 1, wherein: Convergence parameter ε in step 9 m for: Among them, y vi and Represent the validation set E m The actual function response value and the predicted value after the mixed surrogate model, E k represents the E value in the kth iteration, E k-1 Represents the root mean square error E value in the k-1th iteration; The convergence criterion is: ε m ≤L, L is the preset threshold; If the convergence criterion is not met, the sample point x with the minimum value of the U learning function is selected from the entire sample pool according to the U learning function. new Add training set E tra , retrain the model.

10. The method for analyzing time-varying reliability of a transmission shaft based on a hybrid surrogate model according to claim 1, wherein: The reliability R of step 10 is: Among them, N represents the total number of samples, N y≤0 Represents the number of hybrid agent model output values ​​less than or equal to 0, N y>0 Represents the number of hybrid agent model output values ​​greater than 0, P f represents the time-varying failure probability.

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