A semi-analytical time-varying reliability analysis method
Patent Information
- Application Number
- CN202611299239.2
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2026-08-26
- Publication Date
- 2026-09-29
AI Technical Summary
[0003]然而，谐波减速器可靠性分析涉及高维、强耦合的复杂随机过程，传统时变可靠性分析方法存在计算效率低下、数值求解精度不足、算法鲁棒性较差等问题，无法精准预测结构的时变可靠性，还极易在功能函数非线性程度较高、随机变量时变特征显著的工况下出现计算失效的问题，极大限制了该类方法在复杂装备结构时变可靠性定量评估的工程适用性
[0015]（2）提出了一种基于误差驱动的子区间自适应泰勒展开技术，在随机变量全域精度不满足要求时对随机变量的参数取值区间自适应剖分，有效提高了时变可靠性的求解精度。
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Abstract
Description
Technical Field
[0001] This invention belongs to the field of data analysis technology, specifically relating to a semi-analytical time-varying reliability analysis method. Background Technology
[0002] With the continuous upgrading of modern engineering technology, the transmission systems of high-end equipment are becoming increasingly sophisticated and complex. As a core precision transmission component, the service conditions and performance degradation patterns of harmonic reducers are becoming increasingly complex. Traditional static reliability analysis methods for assessing the reliability of harmonic reducers often neglect the inherent time-dependent nature of various uncertainties. For key time-varying parameters such as alternating transmission loads that fluctuate continuously with service time, material stiffness and fatigue strength that decay over time, and transmission clearances that evolve with operating conditions, it is impossible to accurately reconstruct the actual transmission service state of the harmonic reducer throughout its entire lifecycle, resulting in poor consistency between reliability assessment results and actual operating conditions. Therefore, research on time-varying reliability analysis methods for harmonic reducer systems is urgently needed.
[0003] However, the reliability analysis of harmonic reducers involves complex stochastic processes with high dimensions and strong coupling. Traditional time-varying reliability analysis methods suffer from problems such as low computational efficiency, insufficient numerical solution accuracy, and poor algorithm robustness. They cannot accurately predict the time-varying reliability of the structure and are prone to computational failures under operating conditions with high degree of nonlinearity of the function and significant time-varying characteristics of random variables. This greatly limits the engineering applicability of such methods for quantitative assessment of the time-varying reliability of complex equipment structures. Summary of the Invention
[0004] The purpose of this invention is to overcome the shortcomings of existing technologies and provide a semi-analytical time-varying reliability analysis method. This method is designed for time-varying reliability analysis of harmonic reducers in design verification, life assessment, and in-service maintenance. Inputs include flexspline geometry, material properties, load spectrum, degradation patterns, and target service time. Outputs include failure probabilities at various times, reliability curves, and a judgment result indicating whether design reliability requirements are met. This significantly improves the efficiency of time-varying reliability solution while ensuring computational accuracy.
[0005] To achieve the above objectives, the present invention adopts the following technical solution: a semi-analytical time-varying reliability analysis method, comprising the following steps: Step S1: Establish a time-varying function based on the structural parameters, material parameters, transmission load, and failure mode of the harmonic reducer; considering the influence of random variables and random processes, the time-varying function is expressed as: (1) Where t is time, Given n random variables, There are m random processes; when When the harmonic reducer is considered to have failed; when At that time, it was considered that the harmonic reducer was safe.
[0006] Step S2: Determine the failure criterion based on the time-varying function established in Step S1, and calculate the cumulative time-varying failure probability within the target service time interval: (2) in, express The probability of; Step S3: Discretize the time-dependent stochastic processes; use the optimal linear estimation expansion EOLE to convert each stochastic process into several time-independent standard random variables: (3) Where, μ Y (t) is the mean of the random process, σ Y (t) is the standard deviation of the random variable, p is the autocorrelation function ρ Y The number of cutoff terms in (t1,t2), C Y λ(t) is the covariance matrix between the target time t and all discrete nodes, where λ is the covariance matrix between the target time t and all discrete nodes. h and φ h These are the h-th eigenvalue and eigenvector of the covariance matrix, respectively, Z. h There are h time-independent random variables; based on equation (3), the time-varying function of the harmonic reducer can be further expressed as: (4) in, N represents discrete time points within the time interval [0, T]. t The number of discrete time points in the time interval; The time-dependent random process is the transmission torque.
[0007] Step S4: Generate random samples based on the probability distribution of each random variable, and call the harmonic reducer mechanical model or finite element model at discrete time points to obtain the function response; by determining whether the function response of each sample is not greater than 0 within the target time interval, obtain the reference failure probability: (5) in, This is the failure assessment index for sample points, and its expression is as follows: (6).
[0008] Step S5: To reduce the number of times the harmonic reducer mechanical model or finite element model is repeatedly called, a Taylor expansion is performed on the time-varying function with the mean of each random variable as the expansion center at the initial time, establishing a semi-analytical surrogate model that can directly calculate the function response. (7) Among them, X u and Z u These are the means of random variables X and Z, respectively, and g(X) = Z. u Z u , t k ) is the true value of the time-varying function of the harmonic reducer at the mean point of the random variable at time t0. It is the Taylor expansion approximation of the time-varying function of the harmonic reducer at time t0. and These are the gradients of the time-varying function of the harmonic reducer with respect to the random variables X and Z at the mean points of X and Z, respectively.
[0009] Step S6: Adaptively determine the Taylor expansion order using an error-driven approach; considering the nonlinearity of the function, perform a higher-order Taylor expansion at the mean of the random variable, and compare the true model response with the semi-analytical prediction using validation samples. (8) Where α and β are the differential order indices of the time-varying function of the harmonic reducer with respect to random variables X and Z, respectively, and R n It is the remainder of a higher-order Taylor expansion.
[0010] Step S7: Set verification sampling points within the engineering analysis interval; if the global maximum relative error exceeds the allowable error threshold, gradually increase the Taylor expansion order until the error meets the requirements or reaches the preset global maximum relative error; define the range of the mean of each random variable plus or minus 6 times the standard deviation as the engineering analysis interval; the preset global maximum relative error is: (9) Wherein, ε is the preset global maximum relative error; if the preset global maximum relative error ε is greater than the preset allowable error threshold, the Taylor expansion order is increased step by step, the higher-order Taylor approximation expression of the time-varying function of the harmonic reducer is iteratively updated, and the error verification process is repeated until the preset global maximum relative error ε is less than the preset allowable error threshold.
[0011] Step S8: When the error still does not meet the requirements after reaching the highest expansion order, adaptively partition the random variable interval where the error exceeds the limit, and construct a Taylor expansion model at the mean of each sub-interval: (10) Among them, X u,ξ and Zu,η These are the mean values of the ξ-th and η-th subintervals of random variables X and Z, respectively. After dividing the range of parameter values of random variables into multiple subintervals, the global maximum relative error of all verification sampling points is recalculated according to equation (8). When calculating the Taylor approximate prediction value of any sample point, the range of the subinterval to which the sample point belongs is first determined, and the Taylor expansion function of the corresponding subinterval is selected to calculate the approximate prediction value of the time-varying function of the harmonic reducer. If the preset global maximum relative error ε obtained after subdividing the range of parameters of the random variable into a single sub-interval is still greater than the preset allowable error threshold, then the sub-interval that does not meet the accuracy requirements will continue to be subdivided into a second sub-interval. The iterative process of sub-interval adaptive Taylor expansion, sampling verification, and error determination will be repeated until the preset global maximum relative error ε is less than the preset allowable error threshold.
[0012] Step S9: After the maximum relative error of each sub-interval meets the requirements, the corresponding semi-analytical model is called according to the sub-interval to which the sample belongs, and the function value, failure judgment index and cumulative time-varying failure probability of all samples at each discrete time are calculated; the reliability curve is further obtained and compared with the design reliability threshold, so as to provide a basis for the optimization of flexible wheel size, material selection, determination of rated load and maintenance cycle.
[0013] In engineering applications, the time-varying failure probability of the harmonic reducer can be quickly updated simply by updating the material degradation parameters, load process parameters, and condition monitoring data over time. When the predicted reliability is lower than the specified threshold, the reducer will be prompted to reduce load, adjust structural parameters, or perform early maintenance.
[0014] This invention proposes a semi-analytical time-varying reliability analysis method. The main innovative points of this invention are as follows: (1) A semi-analytical time-varying reliability calculation method is proposed, which approximates the function by Taylor expansion at the mean of the random variable, effectively improving the solution efficiency of time-varying reliability.
[0015] (2) An error-driven sub-interval adaptive Taylor expansion technique is proposed. When the accuracy of the random variable in the whole domain does not meet the requirements, the parameter value interval of the random variable is adaptively divided, which effectively improves the solution accuracy of time-varying reliability. Attached Figure Description
[0016] Figure 1 This is the overall flowchart of the present invention.
[0017] Figure 2 A comparison of the calculation results of time-varying failure probability of harmonic reducers using different methods. Detailed Implementation
[0018] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.
[0019] Example 1
[0020] like Figure 1 The semi-analytical time-varying reliability analysis method shown includes the following steps: Step S1: Establish a time-varying function based on the structural parameters, material parameters, transmission load, and failure mode of the harmonic reducer; considering the influence of random variables and random processes, the time-varying function is expressed as: (1) Where t is time, Given n random variables, There are m random processes; when When the harmonic reducer is considered to have failed; when At that time, it was considered that the harmonic reducer was safe.
[0021] Step S2: Determine the failure criterion based on the time-varying function established in Step S1, and calculate the cumulative time-varying failure probability within the target service time interval: (2) in, express The probability of; Step S3: Discretize the time-dependent stochastic processes; use the optimal linear estimation expansion EOLE to convert each stochastic process into several time-independent standard random variables: (3) Where, μ Y (t) is the mean of the random process, σ Y (t) is the standard deviation of the random variable, p is the autocorrelation function ρ Y The number of cutoff terms in (t1,t2), C Y λ(t) is the covariance matrix between the target time t and all discrete nodes, where λ is the covariance matrix between the target time t and all discrete nodes. h and φ h These are the h-th eigenvalue and eigenvector of the covariance matrix, respectively, Z. h There are h time-independent random variables; based on equation (3), the time-varying function of the harmonic reducer can be further expressed as: (4) in, N represents discrete time points within the time interval [0, T]. t The number of discrete time points in the time interval; The time-dependent random process is the transmission torque.
[0022] Step S4: Generate random samples based on the probability distribution of each random variable, and call the harmonic reducer mechanical model or finite element model at discrete time points to obtain the function response; by determining whether the function response of each sample is not greater than 0 within the target time interval, obtain the reference failure probability: (5) in, This is the failure assessment index for sample points, and its expression is as follows: (6).
[0023] Step S5: To reduce the number of times the harmonic reducer mechanical model or finite element model is repeatedly called, a Taylor expansion is performed on the time-varying function with the mean of each random variable as the expansion center at the initial time, establishing a semi-analytical surrogate model that can directly calculate the function response. (7) Among them, X u and Z u These are the means of random variables X and Z, respectively, and g(X) = Z. u Z u , t k ) is the true value of the time-varying function of the harmonic reducer at the mean point of the random variable at time t0. It is the Taylor expansion approximation of the time-varying function of the harmonic reducer at time t0. and These are the gradients of the time-varying function of the harmonic reducer with respect to the random variables X and Z at the mean points of X and Z, respectively.
[0024] Step S6: Adaptively determine the Taylor expansion order using an error-driven approach; considering the nonlinearity of the function, perform a higher-order Taylor expansion at the mean of the random variable, and compare the true model response with the semi-analytical prediction using validation samples. (8) Where α and β are the differential order indices of the time-varying function of the harmonic reducer with respect to random variables X and Z, respectively, and R n It is the remainder of a higher-order Taylor expansion.
[0025] Step S7: Set verification sampling points within the engineering analysis interval; if the global maximum relative error exceeds the allowable error threshold, gradually increase the Taylor expansion order until the error meets the requirements or reaches the preset global maximum relative error; define the range of the mean of each random variable plus or minus 6 times the standard deviation as the engineering analysis interval; the preset global maximum relative error is: (9) Wherein, ε is the preset global maximum relative error; if the preset global maximum relative error ε is greater than the preset allowable error threshold, the Taylor expansion order is increased step by step, the higher-order Taylor approximation expression of the time-varying function of the harmonic reducer is iteratively updated, and the error verification process is repeated until the preset global maximum relative error ε is less than the preset allowable error threshold.
[0026] Step S8: When the error still does not meet the requirements after reaching the highest expansion order, adaptively partition the random variable interval where the error exceeds the limit, and construct a Taylor expansion model at the mean of each sub-interval: 10) Among them, X u,ξ and Z u,η These are the mean values of the ξ-th and η-th subintervals of random variables X and Z, respectively. After dividing the range of parameter values of random variables into multiple subintervals, the global maximum relative error of all verification sampling points is recalculated according to equation (8). When calculating the Taylor approximate prediction value of any sample point, the range of the subinterval to which the sample point belongs is first determined, and the Taylor expansion function of the corresponding subinterval is selected to calculate the approximate prediction value of the time-varying function of the harmonic reducer. If the preset global maximum relative error ε obtained after subdividing the range of parameters of the random variable into a single sub-interval is still greater than the preset allowable error threshold, then the sub-interval that does not meet the accuracy requirements will continue to be subdivided into a second sub-interval. The iterative process of sub-interval adaptive Taylor expansion, sampling verification, and error determination will be repeated until the preset global maximum relative error ε is less than the preset allowable error threshold.
[0027] Step S9: After the maximum relative error of each sub-interval meets the requirements, the corresponding semi-analytical model is called according to the sub-interval to which the sample belongs, and the function value, failure judgment index and cumulative time-varying failure probability of all samples at each discrete time are calculated; the reliability curve is further obtained and compared with the design reliability threshold, so as to provide a basis for the optimization of flexible wheel size, material selection, determination of rated load and maintenance cycle.
[0028] In engineering applications, the time-varying failure probability of the harmonic reducer can be quickly updated simply by updating the material degradation parameters, load process parameters, and condition monitoring data over time. When the predicted reliability is lower than the specified threshold, the reducer will be prompted to reduce load, adjust structural parameters, or perform early maintenance.
[0029] Example 2
[0030] This embodiment uses the time-varying reliability assessment of fatigue fracture of the flexure in a harmonic reducer.
[0031] In the design of harmonic reducers, the function of fatigue fracture of the flexure in a harmonic reducer can be expressed as: (11) Among them, S σ S is the safety factor of the flexible wheel under normal stress alone. τ S is the safety factor of the flexible wheel under shear stress alone. f It is the safety factor threshold, γ z =0.7 is a coefficient representing the influence of shear stress on the fatigue fracture failure of the flexural gear. In the design of harmonic reducers, a safety factor greater than 1.5 is required to ensure the reliability and safety of the reducer. Uncertainty parameters affecting the reliability of harmonic reducers are shown in Table 1. Considering material aging, the degradation relationships of bending fatigue limit and shear fatigue limit over time are as follows: and In the operation of a harmonic reducer, the torque T t (t) uses a mean of 30 N·m, a standard deviation of 1.5 N·m, and an autocorrelation function of . The stochastic process is approximated. In the time-varying reliability assessment of harmonic reducers, the EOLE method is used to approximate the torque T. t (t) is transformed into a linear combination of three random variables, and the time interval is uniformly discretized into N. t =10 time points, Monte Carlo sample size set to N m =10 5 In the proposed adaptive sub-interval Taylor expansion method, the highest Taylor expansion order is set to 4, the maximum number of partitions of the random variable space is set to 4, and the number of validation sampling points N is [not specified]. s =10, preset global maximum relative error ε=0.001.
[0032] This embodiment is implemented according to the following engineering steps: Step C1: Determine the evaluation object and criteria. Taking the flexure of a harmonic reducer as the object, fatigue fracture of the flexure is taken as the target failure mode. Design parameters such as flexure length, wall thickness, pitch circle radius, stress coefficient, stress concentration coefficient, dynamic load coefficient, elastic modulus, and initial fatigue limit are read, and a safety factor of 1.5 is used as the reliability judgment threshold.
[0033] Step C2: Establish a time-varying input model. The manufacturing errors and material dispersion parameters listed in Table 1 are used as random variables; the operating torque is described as a stochastic process with a mean of 30, a standard deviation of 1.5, and a given autocorrelation function; the bending fatigue limit and shear fatigue limit are updated over service time according to a given degradation relationship.
[0034] Table 1 Distribution of parameters affecting the uncertainty of the safety factor of harmonic reducer
[0035] Step C3, Discrete Stochastic Process and Service Time. EOLE is used to transform the torque stochastic process into three independent, time-independent random variables, and the target service time interval is evenly divided into a preset number of time nodes, so that each sample can obtain the torque history and function response within the complete service time interval.
[0036] Step C4: Construct and validate the semi-analytical model. Use the mean of the random variable as the initial expansion center, set the highest Taylor expansion order to 4, and the maximum subinterval subdivision order to 4; arrange validation sampling points in each analysis interval. When the global maximum relative error between the true function value and the semi-analytical predicted value exceeds a given threshold, increase the expansion order or the subdivision error exceeding the limit interval until the accuracy requirements are met.
[0037] Step C5: Calculate time-varying reliability. Generate Monte Carlo samples and call the corresponding Taylor expansion model according to the sub-interval to which the sample belongs to calculate the flexural fatigue function of each sample at all time points; if the function is not greater than 0 at any time, the sample is recorded as a failure, and the cumulative failure probability and reliability at each time point are calculated accordingly.
[0038] Step C6: Forming an engineering judgment. The obtained reliability curve is compared with the design reliability requirements to identify the time point when reliability drops to the threshold. During the design phase, this can be used to adjust the flexible impeller wall thickness, material fatigue limit, or rated torque; during the service phase, it can be used to determine load reduction strategies and maintenance cycles. Furthermore, the calculation error and number of function calls of the semi-analytical method are verified using Monte Carlo results based on the extreme value method as a reference.
[0039] Figure 2 Table 2 shows the time-varying reliability analysis results of harmonic reducers using different reliability analysis methods. Table 3 compares the failure probability results of harmonic reducers calculated by different methods at t=5. Figure 2 The results in Table 2 show that the first-order Taylor expansion at the mean value results in a large deviation from the Monte Carlo reference solution, making it impossible to accurately predict the fatigue fracture behavior of the flexure in the harmonic reducer. However, the adaptive sub-interval Taylor expansion method can accurately obtain the failure probability of the harmonic reducer at different times, requiring only a small amount of computation of the function. Figure 2 The results in Table 2 verify the efficiency and accuracy of the proposed semi-analytical method AMSEVM in calculating the time-varying reliability of harmonic reducers.
[0040] Table 2 Comparison of failure probability calculation results for harmonic reducers using different methods at t=5 .
Claims
1. A semi-analytical time-varying reliability analysis method, characterized in that, Includes the following steps: Step S1: Establish a time-varying function based on the structural parameters, material parameters, transmission load, and failure mode of the harmonic reducer; Step S2: Determine the failure criterion based on the time-varying function established in Step S1, and calculate the cumulative time-varying failure probability within the target service time interval: （2） in, express The probability of; Step S3: Discretize the time-dependent stochastic process; Step S4: Generate random samples based on the probability distribution of each random variable, and call the harmonic reducer mechanical model or finite element model at discrete time points to obtain the function response; by determining whether the function is not greater than 0 in each sample within the target time interval, obtain the reference failure probability. Step S5: In order to reduce the number of times the harmonic reducer mechanical model or finite element model is repeatedly called, Taylor expansion is performed on the time-varying function with the mean of each random variable as the expansion center at the initial time to establish a semi-analytical surrogate model that can directly calculate the function response. Step S6: Adaptively determine the Taylor expansion order using an error-driven method; Step S7: Set up verification sampling points in the engineering analysis interval; If the global maximum relative error exceeds the allowable error threshold, the Taylor expansion order is gradually increased until the error meets the requirements or reaches the preset global maximum relative error. Step S8: When the error still does not meet the requirements after reaching the highest expansion order, adaptively divide the random variable interval where the error exceeds the limit, and construct Taylor expansion models at the mean of each sub-interval. Step S9: After the maximum relative error of each sub-interval meets the requirements, the corresponding semi-analytical model is called according to the sub-interval to which the sample belongs, and the function value, failure judgment index and cumulative time-varying failure probability of all samples at each discrete time are calculated; the reliability curve is further obtained and compared with the design reliability threshold, so as to provide a basis for the optimization of flexible wheel size, material selection, determination of rated load and maintenance cycle.
2. The semi-analytical time-varying reliability analysis method according to claim 1, characterized in that, In step S1, considering the influence of random variables and random processes, the time-varying function is expressed as follows: （1） Where t is time, Given n random variables, There are m random processes; when When the harmonic reducer is considered to have failed; when At that time, it was considered that the harmonic reducer was safe.
3. The semi-analytical time-varying reliability analysis method according to claim 1, characterized in that, In step S3, the optimal linear estimation expansion EOLE is used to convert each stochastic process into several time-independent standard random variables: （3） Where, μ Y (t) is the mean of the random process, σ Y (t) is the standard deviation of the random variable, p is the autocorrelation function ρ Y The number of cutoff terms in (t1,t2), C Y λ(t) is the covariance matrix between the target time t and all discrete nodes, where λ is the covariance matrix between the target time t and all discrete nodes. h and φ h These are the h-th eigenvalue and eigenvector of the covariance matrix, respectively, Z. h There are h time-independent random variables; based on equation (3), the time-varying function of the harmonic reducer can be further expressed as: （4） in, N represents discrete time points within the time interval [0, T]. t The number of discrete time points in the time interval; The time-dependent random process is the transmission torque.
4. The semi-analytical time-varying reliability analysis method according to claim 1, characterized in that, In step S4, the reference failure probability is: （5） in, This is the failure assessment index for sample points, and its expression is as follows: （6）。 5. The semi-analytical time-varying reliability analysis method according to claim 1, characterized in that, In step S5, the semi-parse proxy model: Among them, X u and Z u These are the means of random variables X and Z, respectively, and g(X) = Z. u Z u , t k ) is the true value of the time-varying function of the harmonic reducer at the mean point of the random variable at time t0. It is the Taylor expansion approximation of the time-varying function of the harmonic reducer at time t0. and These are the gradients of the time-varying function of the harmonic reducer with respect to the random variables X and Z at the mean points of X and Z, respectively.
6. The semi-analytical time-varying reliability analysis method according to claim 1, characterized in that, In step S6, considering the nonlinearity of the function, a higher-order Taylor expansion is performed at the mean of the random variable, and the true model response is compared with the semi-analytical prediction using validation samples. （8） Where α and β are the differential order indices of the time-varying function of the harmonic reducer with respect to random variables X and Z, respectively, and R n It is the remainder of a higher-order Taylor expansion.
7. The semi-analytical time-varying reliability analysis method according to claim 1, characterized in that, In step S7, the range of the mean of each random variable plus or minus 6 times the standard deviation is determined as the engineering analysis interval; the preset global maximum relative error: （9） Wherein, ε is the preset global maximum relative error; if the preset global maximum relative error ε is greater than the preset allowable error threshold, the Taylor expansion order is increased step by step, the higher-order Taylor approximation expression of the time-varying function of the harmonic reducer is iteratively updated, and the error verification process is repeated until the preset global maximum relative error ε is less than the preset allowable error threshold.
8. The semi-analytical time-varying reliability analysis method according to claim 1, characterized in that, In step S8, the Taylor expansion model is: （10） Among them, X u,ξ and Z u,η These are the mean values of the ξ-th and η-th subintervals of random variables X and Z, respectively. After dividing the range of parameter values of random variables into multiple subintervals, the global maximum relative error of all verification sampling points is recalculated according to equation (8). When calculating the Taylor approximate prediction value of any sample point, the range of the subinterval to which the sample point belongs is first determined, and the Taylor expansion function of the corresponding subinterval is selected to calculate the approximate prediction value of the time-varying function of the harmonic reducer. If the preset global maximum relative error ε obtained after subdividing the range of parameters of the random variable into a single sub-interval is still greater than the preset allowable error threshold, then the sub-interval that does not meet the accuracy requirements will continue to be subdivided into a second sub-interval. The iterative process of sub-interval adaptive Taylor expansion, sampling verification, and error determination will be repeated until the preset global maximum relative error ε is less than the preset allowable error threshold.
9. A semi-analytical time-varying reliability analysis method according to any one of claims 1 to 8, characterized in that, In engineering applications, the time-varying failure probability of the harmonic reducer can be quickly updated simply by updating the material degradation parameters, load process parameters, and condition monitoring data over time. When the predicted reliability is lower than the specified threshold, the reducer will be prompted to reduce load, adjust structural parameters, or perform early maintenance.