Time-varying reliability optimization design method and system for helical gear

Abstract

The invention discloses a time-varying reliability optimization design method and system for a helical gear. The method comprises the following steps: establishing a gear reliability optimization design model which takes gear volume minimization as an optimization target and takes full-life-cycle time-varying reliability as a constraint; constructing a gear contact fatigue time-varying limit state function coupling the time-varying load spectrum and the material strength degradation model; constructing a time-varying reliability analysis module based on an agent model and an active learning strategy so as to evaluate the time-varying limit state function; and an optimization algorithm is adopted to solve the gear reliability optimization design model, the time-varying failure probability of the candidate scheme is evaluated by calling the time-varying reliability analysis module, the time-varying reliability constraint is processed according to the time-varying failure probability, and optimal gear design parameters meeting the constraint are output. According to the method, efficient cooperation of time-varying reliability high-precision evaluation and lightweight optimization design is realized, and the problem that a time-varying factor cannot be considered in optimization due to too high analysis cost in a traditional method is solved.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of reliability design of gear transmission system, and particularly relates to a time-varying reliability optimization design method and system of helical cylindrical gears. BACKGROUND

[0002] Gear transmission system is a core basic component in mechanical equipment, and its lightweight design is a main trend of green manufacturing and performance improvement of equipment. How to ensure the service reliability of gears in the whole life cycle while realizing lightweight is a prominent challenge faced by current engineering design.

[0003] The existing gear reliability optimization design method mainly has the following three deficiencies when dealing with this challenge:

[0004] (1) Model static, evaluation distortion: traditional design methods are usually based on static safety factor or instantaneous reliability model, without fully considering the dynamic fluctuation of load and the time-varying degradation of material strength, which leads to the deviation of reliability evaluation from the actual working condition, and the design result is conservative or there is potential risk.

[0005] (2) Low analysis efficiency, difficult to iterate: when using traditional reliability analysis methods such as Monte Carlo simulation, tens of thousands or even millions of high-cost simulations need to be called each time for evaluation, which is time-consuming for calculation, and cannot support the optimization design process which needs to be iterated repeatedly, becoming a bottleneck for engineering application.

[0006] (3) Analysis and optimization are disconnected, and it is difficult to cooperate: the accurate time-varying reliability model is high in calculation cost and difficult to be directly embedded in the optimization cycle; if the model is simplified for efficiency, the accuracy is sacrificed. The disconnection between "accurate analysis" and "efficient optimization" makes it difficult to realize the collaborative design of lightweight and high reliability.

[0007] Therefore, a new gear design method is needed, which can represent time-varying characteristics, improve calculation efficiency, and realize the collaborative optimization of reliability and lightweight. SUMMARY

[0008] In view of the above deficiencies, the application aims to solve the three key problems that the traditional gear design ignores time-varying factors, leading to evaluation distortion, low reliability analysis efficiency, and difficulty in collaborative design of reliability and lightweight.

[0009] To this end, the application provides a time-varying reliability optimization design method for helical cylindrical gears. The method realizes efficient and accurate evaluation of the time-varying failure probability of the gears by constructing a time-varying reliability analysis module based on a PC-Kriging surrogate model and combining a spatiotemporal collaborative active learning strategy (LIF). On this basis, the optimization algorithm is driven to perform collaborative optimization with the minimization of the total volume of the gear pair as the optimization objective and the life cycle contact fatigue reliability as the constraint, so as to realize lightweight design while ensuring reliability.

[0010] The first object of the application is to provide a time-varying reliability optimization design method for helical cylindrical gears, comprising the following steps:

[0011] determining initial design parameters of the gears;

[0012] establishing a gear reliability optimization design model with the minimization of the gear volume as the optimization objective and the life cycle time-varying reliability as the constraint;

[0013] constructing a gear contact fatigue time-varying limit state function coupled with a time-varying load spectrum and a material strength degradation model;

[0014] constructing a time-varying reliability analysis module based on a surrogate model and an active learning strategy to evaluate the time-varying limit state function;

[0015] solving the gear reliability optimization design model by using an optimization algorithm, evaluating the time-varying failure probability of the candidate scheme by calling the time-varying reliability analysis module, and processing the time-varying reliability constraint accordingly to output the optimal gear design parameters that meet the constraint.

[0016] As a preferred embodiment, the establishment of the gear reliability optimization design model with the minimization of the gear volume as the optimization objective and the life cycle time-varying reliability as the constraint further comprises:

[0017] establishing the minimization of the total volume of the gear pair as the optimization objective function;

[0018] selecting geometric parameters that have an impact on the gear volume and contact fatigue performance from the initial design parameters of the gears as design variables;

[0019] defining the time-varying reliability constraint, which requires that the time-varying failure probability of the contact fatigue of the gear pair in the life cycle does not exceed a target value.

[0020] As a preferred embodiment, the construction of the gear contact fatigue time-varying limit state function coupled with the time-varying load spectrum and the material strength degradation model further comprises:

[0021] generating a time-varying contact stress load spectrum based on a gear system dynamics model;

[0022] The material contact stress-life curve is corrected to obtain a gear contact stress-life curve;

[0023] A gear residual strength model based on a cumulative damage theory is established.

[0024] Based on the load spectrum, the corrected contact stress-life curve and the residual strength model, the time-varying limit state function is constructed.

[0025] As preferred, the surrogate model is a PC-Kriging hybrid surrogate model.

[0026] As preferred, the active learning is a spatiotemporal collaborative active learning strategy.

[0027] As preferred, the optimization algorithm is an improved differential evolution algorithm, and the time-varying reliability constraint is processed through a dynamic penalty function mechanism.

[0028] The second object of the application is a time-varying reliability optimization design system for helical cylindrical gears, which comprises:

[0029] A design modeling and initialization module is configured to determine initial design parameters of the gear and establish a gear reliability optimization design model with a gear volume minimization as an optimization objective and containing a time-varying reliability constraint.

[0030] A time-varying failure probability analysis engine module is configured to construct a gear contact fatigue time-varying limit state function coupling load and strength degradation, and construct a time-varying reliability analysis module based on a surrogate model and an active learning strategy, for evaluating the time-varying limit state function and calculating a time-varying failure probability.

[0031] A reliability constraint driven optimization module is configured to solve the gear reliability optimization design model by using an optimization algorithm, evaluate the time-varying failure probability of a candidate scheme by calling the analysis module constructed by the time-varying reliability analysis engine module and process the time-varying reliability constraint accordingly, and finally output optimal gear design parameters meeting the constraint.

[0032] The application has the following beneficial effects:

[0033] 1. By establishing a time-varying limit state equation coupling a load spectrum and a strength degradation model, the full-life service state of the gear is truly reflected, and the reliability evaluation accuracy is significantly improved.

[0034] 2. A time-varying reliability analysis method based on a PC-Kriging surrogate model is proposed, and a spatiotemporal collaborative active learning strategy is combined to realize efficient and high-precision approximation of the failure probability with a small amount of simulation.

[0035] 3. Dynamic penalty function mechanism and improved differential evolution algorithm are adopted to integrate time-varying reliability constraint into the optimization process, so that the lightweight design of the gear is effectively realized while the life cycle reliability is ensured. BRIEF DESCRIPTION OF DRAWINGS

[0036] Figure 1 A total flowchart of the embodiment of the time-varying reliability optimization design method of the helical cylindrical gear of the application;

[0037] Figure 2 The gear transmission dynamics model of the embodiment;

[0038] Figure 3 The modified contact fatigue S-N curve of 20CrMnTi of the embodiment;

[0039] Figure 4 The flowchart of the time-varying reliability analysis based on PC-Kriging of the embodiment;

[0040] Figure 5 The flowchart of the improved differential evolution algorithm of the embodiment;

[0041] Figure 6 The parameter acquisition module interface in the gear reliability optimization system of the embodiment;

[0042] Figure 7 The proxy model construction interface in the gear reliability optimization system of the embodiment;

[0043] Figure 8 The optimization solution calculation interface in the gear reliability optimization system of the embodiment;

[0044] Figure 9 The result visualization analysis interface in the gear reliability optimization system of the embodiment. DETAILED DESCRIPTION

[0045] The preferred embodiments of the application will be described in more detail below with reference to the accompanying drawings. It should be understood that the following description is intended to illustrate the general principles of the application by way of example, and is not intended to limit the scope of protection defined by the appended claims.

[0046] In order for those skilled in the art to accurately understand the application, the key terms involved in the specification are first described.

[0047] Term explanation

[0048] Proxy model: also known as metamodel or response surface model, is a simplified mathematical model used to replace complex high-precision numerical model (such as finite element analysis). It establishes an approximate mapping relationship between input parameters and output responses through limited sample points, can significantly improve the calculation efficiency, and is widely used in optimization design and reliability analysis.

[0049] Polynomial Chaos Expansion (PCE): A surrogate modeling method based on orthogonal polynomial basis functions, used to quantify uncertainty. It represents the output response of a stochastic system as a weighted sum of a series of orthogonal polynomials, which can efficiently characterize the nonlinear relationship between random inputs and outputs.

[0050] Kriging Model: A geostatistical interpolation model based on statistics, commonly used as a surrogate model in engineering. It not only provides the predicted value of an unknown point, but also provides the mean square error (uncertainty estimate) of the prediction, which is the basis for implementing active learning.

[0051] PC-Kriging Model: A hybrid surrogate model that combines the advantages of Polynomial Chaos Expansion (PCE) and Kriging Model. It uses PCE to capture global trends and then uses the Kriging Model to model the residual (local fluctuations), thereby improving the prediction accuracy and robustness of complex nonlinear responses.

[0052] Active Learning: Also known as adaptive sampling, it is an efficient sample selection strategy. When building a surrogate model, it does not use all samples at once, but intelligently selects "information points" that are most valuable for improving model accuracy based on the current model's uncertainty and other information, aiming to obtain a high-precision model with the least computational cost.

[0053] LIF (Learning Function in Space-Time Coordination): An improved active learning function proposed by the invention. It considers both the prediction uncertainty (such as U-function) in the spatial (design variables and random variables) dimension and the failure risk (such as expected improvement function EI) in the temporal dimension, and by introducing a time weight coefficient, it guides the algorithm to pay more attention to high-risk periods in the later life, thereby achieving efficient and accurate time-varying reliability analysis.

[0054] Limit State Equation: In structural reliability analysis, a mathematical expression used to define the "safe" and "failure" states of a system, usually denoted as where is the basic variable. When , the system is in a safe state; when , the system is in a failure state.

[0055] Load Spectrum: A statistical representation of the size, frequency, and sequence of loads borne by mechanical structures over time, which is an important input for fatigue life prediction and reliability assessment.

[0056] Dynamic penalty function mechanism: This invention employs a constraint handling method in the optimization algorithm. The penalty function value dynamically increases with the number of optimization iterations. Initially, it allows the algorithm to explore near the infeasible region to maintain population diversity, while later it forces the search to converge to the feasible region that satisfies the constraints, effectively balancing global exploration and local optimization.

[0057] Latin hypercube sampling (LHS): A stratified sampling technique that can generate a uniformly distributed sample set with good spatial filling in a multidimensional parameter space. It is often used for initial sampling in surrogate model construction.

[0058] Monte Carlo simulation (MCS) is a numerical method based on random sampling that estimates mathematical expectations, integral values, or failure probabilities through a large number of repeated random trials. In reliability engineering, it is often considered a benchmark method for assessing failure probabilities, but it is computationally expensive.

[0059] Example 1

[0060] The following example uses a pair of helical cylindrical gears in a car transmission, along with the attached... Figure 1 The overall flowchart shown illustrates in detail the implementation steps of the method of the present invention. The objective of this embodiment is to design a set of methods that meet contact fatigue reliability requirements (e.g., target failure probability) throughout the entire life cycle, while fully considering the randomness of the load and the degradation of material strength. A value of 2.28% is acceptable, corresponding to a reliability level of... The gear parameters with the smallest overall volume.

[0061] S1: Determine the initial design parameters of the gear.

[0062] Based on the application conditions and installation space requirements, select the gear and extract its geometric parameters as the initial design parameters for the gear.

[0063] This step aims to select a set of gear pairs as the design starting point based on the application conditions of the gear transmission system (such as the transmitted power and speed) and installation space constraints, and to extract its complete geometric parameters, thereby establishing accurate initial benchmarks and parameter boundaries for subsequent optimization modeling and reliability assessment.

[0064] This embodiment selects a pair of gears from a helical cylindrical gear reducer in an automobile as the optimization design object, and extracts key parameters of their geometric dimensions. These parameters form the basis for defining design variables and applying engineering constraints in subsequent steps. The main extracted parameters include: the number of teeth on the driving and driven gears. , Tooth width Normal module Pressure angle α, helix angle Addendum coefficient h *， Pitch coefficient c* And gear materials.

[0065] For clarity, a set of exemplary initial parameters used in the gear pair in this embodiment are shown in Table 1.

[0066] Table 1 Gear Parameters

[0067]

[0068] S2: Establish a gear reliability optimization design model with the goal of minimizing gear volume and constrained by time-varying reliability throughout the entire life cycle.

[0069] This step aims to transform the engineering design task into a computable mathematical optimization problem. Its core is to construct a mathematical model that aims to minimize the overall volume of the gear pair while strictly satisfying a series of geometric, performance, and probabilistic reliability constraints. This model serves as the basis for the subsequent optimization algorithm, and its construction mainly includes three parts: defining the objective function, selecting design variables, and determining the constraints. The specific method for this step is as follows:

[0070] S21: Based on the geometric characteristics of gears, establish an optimization objective function with volume minimization as the core.

[0071] The optimization objective of this embodiment is to achieve lightweight gear transmission, specifically by minimizing the overall volume of the gear pair. Specifically, this embodiment establishes minimizing the total volume of the driving and driven gears as the objective function for optimization design, in order to achieve lightweight gear transmission.

[0072] The volume of a gear is a function of its basic geometric parameters, and those skilled in the art can estimate it using different formulas based on the gear's geometric characteristics. For example, one formula is based on the tip circle diameter. The following is an example of calculating the tooth width b:

[0073] ;

[0074] In the formula, and These are the tip circle diameters of the driving and driven gears, respectively, and their values ​​can be determined by the number of teeth. Normal module helix angle The basic geometric parameters are calculated.

[0075] S22: Select design variables from the initial design parameters of the gear and determine their value range.

[0076] The technical solution of the step is: from the geometric parameters of the gear, a group of parameters which have significant influence on the optimization target (such as volume) and key performance (such as contact fatigue strength) and are allowed to be adjusted in the design are selected as design variables of the optimization search, and a reasonable variable range is set for each variable.

[0077] In the embodiment, five geometric parameters which have significant influence on the volume and contact fatigue performance of the gear are selected as design variables, that is, a design variable vector , representing the number of driving gear teeth, the number of driven gear teeth, the normal modulus, the tooth width and the helix angle. The allowed value range of these variables is determined based on engineering experience, manufacturing process and installation space, and examples are shown in Table 2 as follows:

[0078] Table 2: Example value range of design variables

[0079]

[0080] S23: Based on the gear design requirements and reliability requirements, define the constraint conditions.

[0081] This step aims to impose a series of constraints on the optimization model to ensure that the optimization result is both engineering feasible and meets the required reliability. These constraints mainly include three categories: geometric constraints, performance constraints and core time-varying reliability constraints.

[0082] To implement the above scheme, the embodiment sets the following example constraint conditions:

[0083] 1. Geometric and performance constraints: This type of constraint ensures that the gear is manufacturable, correctly meshed and has good transmission performance. For example:

[0084] 1) Transmission ratio constraint: the actual transmission ratio after optimization needs to fall within the allowed interval (such as ).

[0085] 2) Coincidence degree constraint: To ensure the length of the double-tooth meshing area, thereby reducing transmission noise and impact, the total coincidence degree composed of the face coincidence degree and the axial coincidence degree needs to meet a specific performance interval, such as the specific range set to [6.1, 6.9].

[0086] 3) Tooth number constraint: To avoid undercutting and ensure smooth transmission, according to the minimum number of helical gears that do not undercut, such as the constraint range ≥ 17 set in this embodiment.

[0087] 4) Modulus, tooth width, helix angle boundary constraint: the values need to be within the standard sequence or a reasonable range (see Table 2)

[0088] 2. Time-varying reliability constraint (core constraint): This embodiment takes into account the degradation of material properties over time and the dynamic fluctuation of load, and introduces time-varying reliability as the core constraint, requiring the gear pair to maintain reliability throughout the entire design life cycle. Contact fatigue time-varying failure probability Not exceeding the allowable target value The general mathematical expression of this constraint can be:

[0089] ;

[0090] in, The limit state function considering the time-varying characteristics of load and strength (will be established in step S3) is then developed. This represents a failure event. In this example, the target failure probability is set to . (corresponding reliability) ).

[0091] S24: Construct an optimized design model.

[0092] This sub-step aims to formally establish a reliability optimization design model for helical cylindrical gears based on the optimization objective (minimizing gear volume), selected design variables, and set geometric and reliability constraints defined above.

[0093] The reliability optimization design model for helical cylindrical gears constructed in this embodiment can be expressed as a minimization problem:

[0094] ;

[0095] In the formula, V is the objective function for minimizing the gear volume; Representing the One constraint condition.

[0096] S3: Construct the time-varying limit state function for gear contact fatigue that couples the time-varying load spectrum with the material strength degradation model.

[0097] This step aims to reveal the physical nature of the performance degradation of gear pairs throughout their entire life cycle and provide accurate criteria for quantitative reliability assessment. Its core is the construction of a time-varying limit state equation coupling "dynamic contact stress" and "nonlinear degradation intensity" to dynamically describe the "stress-intensity" interference process, thereby accurately determining whether the gear experiences contact fatigue failure at any given service time. The establishment of this model forms the basis for the subsequent efficient time-varying reliability analysis in step S4.

[0098] S31: Generate time-varying contact stress load spectrum based on gear dynamics model.

[0099] This sub-step aims to obtain dynamic load inputs for fatigue reliability assessment by establishing a dynamic model of the gear system.

[0100] Specifically, firstly, a dynamic model is established based on gear design parameters (such as the number of teeth and the module). Figure 2 The translational-torsional coupling model is shown. By solving the dynamic response of this model under a given working condition, the dynamic meshing force of the gear pair can be obtained. Subsequently, based on contact mechanics theories (such as Hertzian theory) and relevant standards (such as ISO standards), this dynamic meshing force was used... As a key input, the time-varying contact stress on the tooth surface is calculated. This results in a contact stress load spectrum that characterizes the randomness and time-varying nature of the load. An example of a general formula for calculating contact stress is shown below:

[0101] ;

[0102] In the formula, , , , The correlation coefficient is d1, where d1 is the pitch circle diameter of the pinion. For tooth width, The gear ratio, This is the load factor.

[0103] By analyzing the entire gear and its life cycle By performing simulations, the contact stress-load spectrum describing the randomness and time-varying nature of the load can be obtained. .

[0104] S32: Correct the material contact stress-life curve to obtain the gear contact stress-life curve.

[0105] To obtain a stress-life curve that better reflects the actual fatigue performance of gears, the contact stress-life curve (i.e., contact fatigue SN curve) of standard material specimens needs to be modified based on the actual structure, size, surface condition, and load characteristics of the gears.

[0106] The correction usually takes fatigue notch factor into account Size factor Surface machining coefficient Loading method coefficient Comprehensive influence coefficient.

[0107] In this embodiment, the SN curve of the standard material sample is typically represented as follows: Corrected stress It can be represented as:

[0108] ;

[0109] wherein, is the stress in the standard specimen S-N curve. By this modification, the S-N curve applicable to gear design is obtained, for example provides the basis for subsequent strength degradation and damage accumulation calculation.

[0110] In this embodiment, the gear material is 20CrMnTi, by consulting literature and engineering manual, the fatigue constant m = 28.5714, C = 1.0822 x 10 102 , the S-N curve of the standard specimen is modified by introducing the comprehensive influence.

[0111] According to the structure and surface processing of the gear, select 1.36, 0.68, 1.21, 0.85 according to the Mechanical Design Manual. The modified gear surface contact stress-life curve expression is Figure 3 .

[0112] S33: Construct the gear residual strength model.

[0113] This sub-step aims to abandon the traditional assumption of constant strength, and a nonlinear degradation model based on cumulative damage theory can be used to describe the process of gradually reducing the contact fatigue strength of gear material with the increase of load accumulation and service time. This requires the establishment of a residual strength function relationship between the initial strength , load history and time .

[0114] An exemplary gear residual strength model used in this embodiment is as follows:

[0115] ;

[0116] wherein, R(0) is the initial contact fatigue strength, is the maximum contact stress, is the design life, t is the service time, is a parameter related to load and material. This model dynamically describes the process of gradually reducing the strength R(t) of the gear material due to internal damage accumulation with each meshing impact during service.

[0117] It is understood that the above model is one specific way of implementing the present invention. Those skilled in the art can also derive or use other forms of nonlinear degradation functions to describe the time-varying characteristics of intensity based on different cumulative damage theories (such as the Miner criterion, the Corten-Dolan model, etc.), all of which fall within the scope of the present invention.

[0118] S34: Construct the time-varying limit state function for gear contact fatigue.

[0119] This sub-step aims to construct a time-varying limit state function for determining gear contact fatigue failure based on dynamic load spectrum, contact stress-life curve, and gear strength degradation model.

[0120] To implement the above scheme, this embodiment constructs the limit state function according to the following steps:

[0121] S341: Load spectrum processing. The time-history load spectrum of tooth surface contact stress generated by S31. The rainflow counting method is used for statistical processing to transform the complex time-varying stress waveform into a series of cyclic load blocks with different stress amplitudes and mean values, providing input for subsequent fatigue damage accumulation calculation.

[0122] S342: Fatigue life calculation. Based on the contact fatigue SN curve of the gear material modified by S32, the number of fatigue failure cycles (i.e., ultimate life) corresponding to each stress level obtained in S341 is calculated.

[0123] S343: Construct the time-varying limit state function. Combining the strength degradation model described in S33 and considering the cumulative damage caused by load cycling in S341, establish a function based on design variables. Gear contact fatigue failure limit state function as a function of time t This function characterizes the instantaneous safety margin as a function of time t under a specific design x. Its core definition is the instantaneous residual strength of the gear. The difference between the stress and the current equivalent working stress. An example of a common function form is shown below:

[0124] ;

[0125] in, The equivalent contact stress is calculated to account for the cumulative effect of the load history. When When, it means at The gear is always in a safe state; when If the value is zero, it indicates that contact fatigue failure occurred at that moment. Life-cycle reliability requirements are in place throughout the entire design life. Inside, there is no such thing as making At that moment.

[0126] S4: Construct a time-varying reliability analysis module based on a surrogate model and active learning to efficiently evaluate the time-varying limit state function.

[0127] This step aims to build an efficient and autonomous complete time-varying failure probability evaluation module to overcome the bottleneck of high computational cost of Monte Carlo simulation method in solving high-dimensional time-varying reliability problems. As shown in Figure 4 , the core is to build a PC-Kriging surrogate model that combines the advantages of polynomial chaos expansion (PCE) and Kriging model, and introduce an improved spatio-temporal collaborative learning function (LIF) to achieve efficient and high-precision estimation of gear time-varying failure probability with the least number of expensive simulations through active learning strategy. This module is the key to efficient coupling of reliability analysis and optimization design.

[0128] The specific method of this step is as follows:

[0129] S41: Identify and quantify uncertainty parameters to construct a random input vector.

[0130] The technical solution of this sub-step is: analyze the uncertainty factors in the design, manufacturing and service process of the gear, select the key variables, and define their probability distribution characteristics to construct a random input vector for reliability analysis.

[0131] To implement the above scheme, the main random variables that affect the contact fatigue performance of the gear need to be determined. This embodiment exemplarily selects five key parameters, assuming that they follow normal distribution, to form a random variable vector , which represents the normal modulus, tooth width, elastic modulus, Poisson's ratio and input torque, respectively. The uncertainty variable examples are shown in Table 3 as follows:

[0132] Table 3 Probability distribution parameters of uncertainty variables

[0133]

[0134] S42: Perform time-varying reliability analysis based on PC-Kriging surrogate model and spatio-temporal collaborative active learning.

[0135] The technical solution of this sub-step is: use Latin hypercube sampling to generate initial training samples, construct a PC-Kriging surrogate model to approximate the limit state function, and use an improved learning function that considers spatial prediction uncertainty and temporal failure risk to intelligently guide sample point addition. Update the model iteratively until convergence, and finally efficiently predict the time-varying failure probability based on the converged surrogate model.

[0136] To implement the above scheme, the flow of this embodiment is shown in Figure 4 , which mainly includes the following steps:

[0137] S421: Generating candidate sample pool according to uncertainty variable probability distribution. According to the probability distribution described in Table 3, a candidate sample pool is generated by Monte Carlo sampling . In addition, t MCS time samples are generated within the given observation time range to form a time sample pool , which together form an initial sample pool for subsequent iterative updating of the surrogate model and time-varying reliability solution.

[0138] S422: Generating initial training sample set based on candidate sample pool. In the random variable space, a small number (such as m) of sample points are selected by Latin hypercube sampling (LHS), and paired with randomly selected time points to form an initial training sample set .

[0139] S423: Based on the time-varying limit state function, the true performance response of the initial training sample set is calculated to form an initial training data set. Each initial training sample is substituted into the time-varying limit state function established in S3 , and the mechanical model is called to calculate the true response value , forming an initial training data set.

[0140] S424: Building PC-Kriging surrogate model based on initial training data set. Using the initial training data set, a PC-Kriging hybrid surrogate model is constructed . This model combines the advantages of PCE in capturing global trends and Kriging in interpolating local fluctuations, and can more accurately predict the performance response and uncertainty of unknown points.

[0141] S425: Selecting optimal add points based on improved learning function. To efficiently improve the accuracy of the model near the failure boundary, a spatiotemporal collaborative comprehensive learning function (LIF) is defined to select the most "learning value" new sample points from the candidate sample pool . This function considers both the prediction uncertainty in the spatial dimension (through the improved U learning function ) and the failure risk in the time dimension (through the improved expected improvement function ). An exemplary spatiotemporal collaborative comprehensive learning function (LIF) proposed in this embodiment is as follows:

[0142] ;

[0143] wherein is a very small normal quantity. The sample with the maximum value of LIF(X, t) is selected as the optimal add point.

[0144] wherein is an improved U learning function, is an improved EI learning function.

[0145] It is understood by those skilled in the art that the specific form of the learning function can be adjusted as appropriate, as long as it can comprehensively consider the spatial uncertainty and the time failure risk.

[0146] S426: Update the training set and the surrogate model using the optimal add-on point. Calculate the true response value at the optimal add-on point , add it to the training set, and retrain and update the PC-Kriging model.

[0147] S427: Determine whether the model converges. Repeat steps S425 to S426 until the convergence criterion is met. For example, it can be set to stop when the relative error of the failure probability estimated by the current iteration and the previous iteration is less than a certain threshold (e.g. ).

[0148] S428: Calculate the time-varying failure probability of the converged model. Based on the final converged high-precision PC-Kriging model , perform fast prediction on all candidate sample pools, count the number of failure samples, and estimate the time-varying failure probability , which can be exemplarily represented as follows:

[0149] ;

[0150] wherein, is an indicator function that takes 1 when , and 0 otherwise.

[0151] S429: Based on the time-varying failure probability of the converged model, evaluate its calculation stability. Calculate the coefficient of variation (COV) of the failure probability estimate, if it meets the preset accuracy requirement (e.g. ), output ; otherwise, the size of the candidate sample pool needs to be expanded and reanalyzed to ensure the statistical stability of the results.

[0152] In this embodiment, the preset accuracy calculation can be performed using the following formula:

[0153] ;

[0154] S5: Solve the gear reliability optimization design model using an optimization algorithm, evaluate the time-varying failure probability of the candidate scheme by calling the time-varying reliability analysis module, and process the time-varying reliability constraint accordingly, and output the optimal gear design parameters that meet the constraint.

[0155] ​This step aims to use a global optimization algorithm to efficiently solve the reliability optimization design model established in S2, and finally output a lightweight gear design scheme that meets the time-varying reliability constraints throughout the life cycle. The core is to use an improved differential evolution (DE) algorithm. This algorithm initializes the population through chaotic mapping, adaptively adjusts the control parameters, and combines a dynamic penalty function mechanism to handle reliability constraints, thereby robustly and efficiently searching for optimal solutions in complex design spaces.

[0156] The flow is shown in Figure 5 . The specific steps are as follows:

[0157] S51: Optimization algorithm setting and population initialization.

[0158] The technical solution of this sub-step is to configure the parameters of the improved differential evolution algorithm, and use a strategy based on chaotic mapping to generate the initial population, in order to enhance the diversity and uniform distribution of the population in the feasible region, and avoid premature convergence.

[0159] S511: To implement the above scheme, first set the basic parameters of the algorithm, such as the population size , the maximum number of iterations , etc.

[0160] Unlike traditional random initialization, this embodiment introduces Logistic chaotic mapping to generate the initial population. The specific steps are as follows:

[0161] S512: Generate a chaotic sequence. Use the Logistic mapping formula to generate a chaotic variable sequence , which has ergodicity and pseudo-randomness. The mapping formula can be shown as follows:

[0162] ;

[0163] Where, is the control parameter, usually taking 4 to ensure the chaotic state; is a non-specific value initial value in the interval (0, 1).

[0164] S513: Map the chaotic sequence to the design space to generate the initial population. Map each value of the chaotic sequence to the upper and lower bound intervals of each design variable defined in S2 , thereby generating the components of each individual in the initial population:

[0165] .

[0166] S52: Adaptive evolution optimization and fitness evaluation of the initial population.

[0167] The technical solution of the sub-step is: in each generation evolution, adaptive mutation and cross operation are performed to generate new individuals (trial vectors), and the time-varying reliability analysis module established by S4 is used to calculate the fitness thereof; a dynamic penalty function mechanism is used to integrate the reliability constraint into the fitness evaluation, so as to guide the population to converge to the feasible region.

[0168] To implement the above scheme, the following operations are cyclically performed on each individual in the population until the termination condition is met:

[0169] S521: adaptive mutation is performed according to the initial population individual to generate a mutation vector. To increase the search ability, the mutation factor is no longer fixed, but is randomly generated from a Cauchy distribution with as a position parameter. The adaptive difference operation is performed on three mutually different individuals selected at random to generate a mutation vector ;

[0170] ;

[0171] In the formula, is the mutation factor; , , are three individuals selected at random from the current population and different from each other.

[0172] S522: the mutation vector is crossed with the original individual to generate a trial vector. The mutation vector is crossed with the original individual (target vector) according to the adaptive generated cross probability to generate a trial vector .

[0173] S523: establish the fitness evaluation based on the dynamic penalty function. This step is the core of connecting the optimization algorithm and the reliability analysis, and specifically includes:

[0174] 1) time-varying reliability evaluation of the trial vector is performed using step S4: the trial vector (representing a group of gear design parameters) is input into the PC-Kriging time-varying reliability analysis module established in step S4, and the time-varying failure probability of the design scheme is calculated.

[0175] 2) constraint violation degree calculation: according to the S2 time-varying reliability constraint (i.e. ), the constraint violation degree of the current scheme is calculated: ​​

[0176] .

[0177] 3) Based on the time-varying reliability assessment and the violation degree, a dynamic penalty fitness function is constructed to calculate the fitness of the trial vector: the objective function (total volume of the gear ) is combined with the constraint violation degree through a penalty function that grows dynamically with the iteration number to form the final fitness value . The function can be defined as:

[0178] ;

[0179] wherein is the total volume of the gear, is the dynamic penalty factor. The design of the dynamic penalty factor makes the penalty lighter in the early stage of optimization and increases in the later stage. An example is as follows:

[0180] .

[0181] The design makes the penalty lighter in the early stage of optimization, allowing the algorithm to explore the areas near the feasible region or even the infeasible region, and maintains the global search ability; in the later stage of optimization, the penalty increases sharply, and the population is strongly driven to converge to the feasible region that satisfies the reliability constraint.

[0182] S524: Compare the fitness value of the trial vector with the fitness value of the original individual, and select the individual with the smaller fitness value. The fitness value of the trial vector is compared with the fitness value of the original individual . Since the optimization goal is to minimize the volume, the individual with the smaller fitness value (i.e., better) is retained to enter the next generation population.

[0183] S525: Determine whether the algorithm is terminated according to the convergence condition. Repeat steps S521 to S524 until the maximum number of iterations is reached, or the fitness value of the optimal individual in the population improves by less than a given tolerance in consecutive generations, at which point the algorithm is determined to have converged and the optimization algorithm is terminated.

[0184] Through the above steps, the improved differential evolution algorithm finally outputs the individual with the optimal fitness, i.e., the optimal gear design parameter combination that minimizes the total volume of the driving and driven gears while strictly satisfying the time-varying reliability constraint throughout the life cycle.

[0185] After the optimization process, the optimal gear design parameter combination that realizes lightweight design is finally output under the premise of meeting the time-varying reliability constraint. The comparison of key design parameters and volume before and after optimization is shown in Tables 4 and 5.​

[0186] Table 4 Comparison example of key design parameters of gear before and after optimization

[0187]

[0188] Table 5 Comparison example of objective function before and after optimization

[0189]

[0190] By the reliability optimization design method, under the premise of fully considering the working condition and material parameter uncertainty and meeting the gear contact stress reliability constraint condition, the gear volume V can be obtained by calculation from the initial design of 631827.70 mm 3 to the optimized 564530.36 mm 3 , the volume is reduced by about 10.56%.

[0191] The above examples are only used to specifically illustrate the technical solutions and implementation effects of the present application, and are not a limitation on the present application. Those skilled in the art should understand that various substitutions, modifications and improvements can be made to the gear type, specific design variables and constraints, reliability model parameters, proxy model construction method, active learning function form, optimization algorithm and its parameters, etc. without departing from the principles of the present application. These changes all fall within the protection scope defined by the claims of the present application.

[0192] Example 2

[0193] The present embodiment provides a time-varying reliability optimization design system for helical cylindrical gears, which is used to execute the time-varying reliability optimization design method as described in Example 1. The system is deployed on a computing device, such as a computer, as shown in the overall flowchart Figure 1 The system automatically executes the overall flowchart, which includes the following modules connected in sequence:

[0194] 1. Design modeling and initialization module, configured to perform the functions of steps S1 and S2 in Example 1, i.e. configured to determine the initial design parameters of the gear, and to establish a gear reliability optimization design model with the minimization of gear volume as the optimization objective and containing time-varying reliability constraints, to convert engineering design requirements into an optimization problem mathematical model that can be iteratively solved by a computer. It is specifically composed of the following three units working cooperatively:

[0195] 1) Parameter input and design variable management unit, for receiving initial geometric parameters and design requirements of the gear, and based on this, selecting and defining design variables including number of teeth , normal modulus , tooth width , helix angle and their value ranges, corresponding to steps S1 and S22 of Example 1.

[0196] 2) Constraint configuration unit, for configuring geometric performance constraints (such as gear ratio, overlap constraints) and core time-varying reliability constraints for the optimization model , corresponding to step S23 of embodiment 1.

[0197] 3) Optimization model construction unit, for defining the total volume of the gear pair Minimize the optimization objective function, and integrate the design variables and constraints to construct a complete reliability optimization design mathematical model, corresponding to steps S21 and S24 of embodiment 1.

[0198] 2. Time-varying failure probability analysis engine module, configured to perform the functions of steps S3 and S4 in embodiment 1, i.e. configured to build a gear contact fatigue time-varying limit state function coupling load and strength degradation, and to build a time-varying reliability analysis module based on a proxy model and an active learning strategy, for evaluating the limit state function and calculating the time-varying failure probability. It is the core analysis component of the system. Specifically includes:

[0199] 1) Physical model construction unit, for establishing the gear system dynamics model as described in step S31 to generate the time-varying contact stress load spectrum ; and for performing material S-N curve correction and residual strength degradation modeling as described in steps S32 and S33, thereby constructing the time-varying limit state function as described in step S34 .

[0200] 2) Proxy model training and optimization unit, which is the core of the analysis engine, for performing time-varying reliability analysis as described in step S4. It further includes:

[0201] Uncertainty quantification unit, for identifying and quantifying the random input vector and its probability distribution as described in step S41 and table 3.

[0202] Sample management unit, for performing the generation of the Monte Carlo candidate sample pool and the Latin hypercube initial training sample set as described in steps S421 and S422.

[0203] PC-Kriging model construction unit, for performing the initial construction and iterative updating of the hybrid proxy model as described in step S424.

[0204] Active learning controller, integrated with a spatiotemporal collaborative comprehensive learning function , for performing the decision logic of screening the optimal addition point as described in step S425.

[0205] Convergence judgment and probability calculation unit, configured to perform the iteration convergence judgment as described in steps S427 and S428, and to calculate the time-varying failure probability based on the final agent model , whose calculation logic and formula are completely the same as those of Embodiment 1.

[0206] 3. Reliability constraint driven optimization module, configured to perform the function of step S5 in Embodiment 1, i.e. configured to solve the gear reliability optimization design model by using an optimization algorithm; in the solving process, the time-varying reliability analysis module is called to evaluate the time-varying failure probability of the candidate scheme and to process the time-varying reliability constraint accordingly, and finally the optimal gear design parameters meeting the constraint are output. Specifically, it includes:

[0207] 1) Improved differential evolution algorithm engine, configured to perform the optimization search. It includes:

[0208] Chaos initialization unit, configured to perform the population initialization based on Logistic chaos mapping as described in steps S511 and S512.

[0209] Adaptive evolution operation unit, configured to perform the adaptive mutation and crossover operation as described in steps S521 and S522 to generate the trial vector .

[0210] 2) Dynamic penalty function fitness evaluator, a core component for connecting optimization and reliability analysis, configured to perform the fitness evaluation as described in step S523. Its working process is as follows: the time-varying failure probability analysis engine module is called to calculate the failure probability corresponding to the current trial vector ; the constraint violation degree is calculated; and the penalty factor defined as in Embodiment 1 is dynamically adjusted with the iteration number , and the final fitness function is constructed to perform the evaluation.

[0211] 3) Optimization iteration controller, configured to control the selection operation and termination judgment as described in steps S524 and S525, until the optimal design parameter combination is output.

[0212] 4. Result output and visualization module, configured to receive and display the optimal result output by the reliability constraint driven optimization module. Its display content strictly corresponds to the optimization result of Embodiment 1, including: the design variable comparison table before and after optimization (such as Table 4), the target function volume comparison value (such as Table 5), the full life cycle reliability curve, and the calculation efficiency index.

[0213] The modules of the system work in cooperation in the above-mentioned order and data interface: the output of the design modeling and initialization module provides the deterministic parameters and model framework for the time-varying failure probability analysis engine module; the analysis engine module trains the converged PC-Kriging surrogate model as a fast predictor, which is provided to the reliability constraint driven optimization module for calling at each fitness evaluation; the optimization module transmits the optimal parameter set obtained by optimization to the result output and visualization module for final presentation. Through the above process, the system completely and automatically implements all the method steps described in embodiment 1.

[0214] In summary, through the cooperative operation of the above-mentioned four modules, the system realizes the encapsulation of the high-cost analysis process of time-varying reliability into an autonomous and complete analysis service that can be efficiently and repeatedly called by optimization algorithms. The system fundamentally changes the traditional serial process of “analysis” and “optimization” in reliability optimization design, forming an integrated and automated design closed loop with high-precision surrogate model as the bridge and dynamic constraint processing as the driver, thereby efficiently realizing the lightweight design goal while ensuring the reliability of the gear throughout its life cycle.

[0215] Embodiment 3

[0216] The embodiment provides an electronic device for implementing the time-varying reliability optimization design method of the helical cylindrical gear described above. The electronic device can be a workstation, a server, a high-performance computing cluster node, or a cloud virtual machine, etc.

[0217] The electronic device of the embodiment includes a processor, a memory, a communication interface, and a system bus. The memory can include volatile memory (such as RAM) and non-volatile memory (such as ROM, flash memory, or hard drive). The system bus couples the system components including the processor and the memory together.

[0218] The memory stores a computer program (i.e., software system) that can be executed by the processor. When the computer program is executed by the processor, the electronic device as a whole implements the steps of the method described in embodiment 1, or implements all or part of the functions of the system as described in embodiment 2.

[0219] Specifically, when the processor executes the computer program, it can:

[0220] 1. Call the parameter and modeling logic to determine the initial design parameters of the gear and establish an optimization design model targeting volume minimization and containing time-varying reliability constraints.

[0221] 2. Call the time-varying failure analysis logic to build a time-varying limit state function coupling load and strength degradation, and train a reliability analysis module based on surrogate model and active learning strategy.

[0222] 3. Call the optimization solving logic to drive the optimization algorithm to iterate and optimize, in which the reliability analysis module is called to evaluate the time-varying failure probability to handle the reliability constraint, and finally output the optimal gear design parameters.

[0223] The electronic device can receive the user-input gear design parameters, material properties, load spectrum and reliability target through the communication interface, and can also transmit the final optimization design parameters, reliability curve and performance comparison result to a display device or a computer-aided design (CAD) system.

[0224] Embodiment 4

[0225] The embodiment provides a computer-readable storage medium for storing a computer program for implementing the time-varying reliability optimization design method of the helical cylindrical gear.

[0226] The computer-readable storage medium can be any tangible medium containing or storing programs, such as but not limited to: a U disk, a mobile hard disk, a read-only memory (ROM), a random access memory (RAM), a magnetic disk (including a hard disk and a floppy disk), an optical disk (including a CD-ROM and a DVD-ROM), or a cloud storage space, etc.

[0227] The storage medium stores a computer program (instructions). When the computer program is read and executed by one or more processors (for example, the processor of the electronic device in embodiment 3), the processor can execute the steps of the method as described in embodiment 1, or control the corresponding device to realize the functions of the system as described in embodiment 2.

[0228] The computer program includes a series of instructions, which when executed, are specifically used to guide the processor to complete the complete process as described in detail in embodiment 1.

[0229] Through the storage medium, the time-varying reliability optimization design scheme of the helical cylindrical gear described in the embodiment of the application can be saved, distributed and deployed in the form of a software product, facilitating wide application in an industrial design environment.

[0230] The above is only a preferred embodiment of the application, and is not used to limit the protection scope of the application. Any modification, equivalent replacement, improvement, etc. made within the spirit and principles of the application shall be included in the protection scope of the application.

Claims

1. A time-varying reliability optimization design method for helical cylindrical gears, characterized in that, Includes the following steps: Determine the initial design parameters of the gear; A gear reliability optimization design model is established with the goal of minimizing gear volume and constrained by time-varying reliability throughout the entire life cycle. Construct a time-varying limit state function for gear contact fatigue that couples a time-varying load spectrum with a material strength degradation model; A time-varying reliability analysis module based on an agent model and an active learning strategy is constructed to evaluate the time-varying limit state function; An optimization algorithm is used to solve the gear reliability optimization design model. The time-varying reliability analysis module is called to evaluate the time-varying failure probability of the candidate schemes and process the time-varying reliability constraints accordingly, and output the optimal gear design parameters that satisfy the constraints.

2. The method according to claim 1, characterized in that, The establishment of the gear reliability optimization design model, which takes minimizing gear volume as the optimization objective and time-varying reliability throughout the entire life cycle as the constraint, further includes: The optimization objective function is set as minimizing the overall volume of the gear pair. Geometric parameters that affect gear volume and contact fatigue performance are selected from the initial gear design parameters as design variables. The time-varying reliability constraint is defined such that the probability of contact fatigue time-varying failure of the gear pair over its entire life cycle does not exceed the target value.

3. The method according to claim 1, characterized in that, The gear contact fatigue time-varying limit state function, which constructs a coupled time-varying load spectrum and a material strength degradation model, further includes: Time-varying contact stress load spectrum generated based on gear system dynamics model; The material contact stress-life curve is corrected to obtain the gear contact stress-life curve; Establish a gear residual strength model based on cumulative damage theory; Based on the load spectrum, the corrected contact stress-life curve, and the residual strength model, the time-varying limit state function is constructed.

4. The method according to claim 1, characterized in that, The proxy model is a PC-Kriging hybrid proxy model.

5. The method according to claim 1 or 4, characterized in that, The active learning mentioned is a spatiotemporal collaborative active learning strategy.

6. The method according to claim 1, characterized in that, The optimization algorithm is an improved differential evolution algorithm, and the time-varying reliability constraint is handled through a dynamic penalty function mechanism.

7. A time-varying reliability optimization design system for helical cylindrical gears, characterized in that, include: The design modeling and initialization module is configured to determine the initial design parameters of the gear and establish a gear reliability optimization design model with the goal of minimizing gear volume and including time-varying reliability constraints. The time-varying failure probability analysis engine module is configured to construct a time-varying limit state function for gear contact fatigue coupled with load and strength degradation, and to construct a time-varying reliability analysis module based on a surrogate model and an active learning strategy to evaluate the time-varying limit state function and calculate the time-varying failure probability. The reliability constraint-driven optimization module is configured to execute an optimization algorithm to solve the gear reliability optimization design model. It evaluates the time-varying failure probability of candidate solutions by calling the analysis module built by the time-varying reliability analysis engine module and processes the time-varying reliability constraints accordingly. Finally, it outputs the optimal gear design parameters that satisfy the constraints.

8. The system according to claim 7, characterized in that, This includes satisfying at least one of the following conditions: The proxy model constructed in the analysis module is a PC-Kriging hybrid proxy model; The active learning strategy adopted is a spatiotemporal collaborative active learning strategy; The optimization algorithm used in the optimization solution module is an improved differential evolution algorithm, and the time-varying reliability constraints are handled through a dynamic penalty function mechanism.

9. An electronic device comprising a processor and a memory, the memory storing a computer program, characterized in that, When the processor executes the computer program, it implements the steps of the method according to any one of claims 1 to 6.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by a processor, it implements the steps of the method according to any one of claims 1 to 6.

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