A time-varying reliability optimization design method and system of helical cylindrical gears

Abstract

The application discloses a kind of oblique-tooth cylindrical gear time-varying reliability optimization design method and system, the method includes: the gear reliability optimization design model of minimum volume of gear body is established as optimization goal, and time-varying reliability in whole life cycle is constraint;Coupling time-varying load spectrum and material strength degradation model is constructed gear contact fatigue time-varying limit state function;Time-varying reliability analysis module based on proxy model and active learning strategy is constructed to evaluate time-varying limit state function;Optimization algorithm is used to solve the gear reliability optimization design model, the time-varying failure probability of candidate scheme is evaluated by calling the time-varying reliability analysis module and accordingly the time-varying reliability constraint is processed, and the optimal gear design parameter meeting the constraint is output.The application realizes the efficient cooperation of high-precision evaluation and lightweight optimization design of time-varying reliability, solves the problem that traditional method cannot consider time-varying factor in optimization due to high analysis cost.

Description

Technical Field

[0001] This invention belongs to the field of reliability design technology for gear transmission systems, specifically relating to a time-varying reliability optimization design method and system for helical cylindrical gears. Background Technology

[0002] Gear transmission systems are core components in mechanical equipment, and their lightweight design is a major trend in green manufacturing and improving equipment performance. How to achieve lightweighting while ensuring the reliability of gears throughout their entire lifespan is a prominent challenge currently facing engineering design.

[0003] Existing gear reliability optimization design methods have the following three main shortcomings in addressing this challenge:

[0004] (1) Static model, distorted evaluation: Traditional design methods are usually based on static safety factor or instantaneous reliability model, which do not fully consider the dynamic fluctuation of load and the time-varying degradation of material strength, resulting in reliability evaluation deviating from actual working conditions, and the design results are either conservative or have potential risks.

[0005] (2) Low analysis efficiency and difficulty in iteration: When using traditional reliability analysis methods such as Monte Carlo simulation, each evaluation requires tens of thousands or even millions of high-cost simulations, which takes too long to calculate and cannot support the optimization design process that requires repeated iterations, thus becoming a bottleneck in engineering applications.

[0006] (3) Analysis and optimization are disconnected and difficult to coordinate: accurate time-varying reliability models are computationally expensive and difficult to embed directly into optimization loops; if the model is simplified for efficiency, accuracy is sacrificed. This disconnect between "accurate analysis" and "efficient optimization" makes it difficult to achieve coordinated design of lightweight and high reliability.

[0007] Therefore, there is an urgent need for a new gear design method that can simultaneously characterize time-varying properties, improve computational efficiency, and achieve synergistic optimization of reliability and lightweight design. Summary of the Invention

[0008] To address the aforementioned shortcomings, this invention aims to solve three key problems in traditional gear design: evaluation distortion due to neglecting time-varying factors, low efficiency of reliability analysis that cannot support optimization iteration, and difficulty in coordinating reliability and lightweight design.

[0009] To address this, the present invention provides a time-varying reliability optimization design method for helical cylindrical gears. The method constructs a time-varying reliability analysis module based on the PC-Kriging surrogate model, combined with a spatiotemporal collaborative active learning (LIF) strategy, to achieve efficient and accurate assessment of the time-varying failure probability of the gear. Based on this, with minimizing the overall volume of the gear pair as the optimization objective and the full-life-cycle contact fatigue reliability as the constraint, the optimization algorithm is driven to perform collaborative optimization, thereby achieving lightweight design while ensuring reliability.

[0010] The primary objective of this invention is to provide a time-varying reliability optimization design method for helical cylindrical gears, comprising the following steps:

[0011] Determine the initial design parameters of the gear;

[0012] 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.

[0013] Construct a time-varying limit state function for gear contact fatigue that couples a time-varying load spectrum with a material strength degradation model;

[0014] 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;

[0015] 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.

[0016] Preferably, the establishment of a gear reliability optimization design model with the goal of minimizing gear volume and constrained by time-varying reliability throughout the entire life cycle further includes:

[0017] The optimization objective function is set as minimizing the overall volume of the gear pair.

[0018] Geometric parameters that affect gear volume and contact fatigue performance are selected from the initial gear design parameters as design variables.

[0019] 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.

[0020] Preferably, the gear contact fatigue time-varying limit state function for constructing the coupled time-varying load spectrum and the material strength degradation model further includes:

[0021] Time-varying contact stress load spectrum generated based on gear system dynamics model;

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

[0023] Establish a residual strength model for gears based on the theory of cumulative damage;

[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] Preferably, the proxy model is the PC-Kriging hybrid proxy model.

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

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

[0028] The second objective of this invention is to provide a time-varying reliability optimization design system for helical cylindrical gears, the system comprising:

[0029] 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.

[0030] 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.

[0031] 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.

[0032] The beneficial effects of this invention are:

[0033] 1. By establishing a time-varying limit state equation that couples the load spectrum and the strength degradation model, the service state of the gear throughout its entire service life can be accurately reflected, significantly improving the accuracy of reliability assessment.

[0034] 2. A time-varying reliability analysis method based on the PC-Kriging surrogate model is proposed. Combined with a spatiotemporal collaborative active learning strategy, the failure probability can be efficiently and accurately approximated with a small number of simulations.

[0035] 3. By adopting a dynamic penalty function mechanism and an improved differential evolution algorithm, time-varying reliability constraints are integrated into the optimization process, which effectively achieves lightweight gear design while ensuring reliability throughout the entire life cycle. Attached Figure Description

[0036] Figure 1 This is a flowchart illustrating an embodiment of the time-varying reliability optimization design method for helical cylindrical gears according to the present invention.

[0037] Figure 2 This is the gear transmission dynamics model for this embodiment.

[0038] Figure 3 The corrected contact fatigue SN curve for 20CrMnTi in this embodiment;

[0039] Figure 4 This is a flowchart of the time-varying reliability analysis based on PC-Kriging in this embodiment;

[0040] Figure 5 This is a flowchart of the improved differential evolution algorithm in this embodiment;

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

[0042] Figure 7 This provides an interface for constructing the proxy model in the gear reliability optimization system of this embodiment;

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

[0044] Figure 9 This is the result visualization and analysis interface of the gear reliability optimization system in this embodiment. Detailed Implementation

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

[0046] To enable those skilled in the art to accurately understand this invention, the key terms involved in the specification will first be explained.

[0047] Terminology Explanation

[0048] Surrogate models, also known as meta-models or response surface models, are simplified mathematical models used to replace complex, high-precision numerical models (such as finite element analysis). They establish an approximate mapping between input parameters and output response using a finite number of sample points, significantly improving computational efficiency and finding wide application in optimization design and reliability analysis.

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

[0050] Kriging model: A geostatistical interpolation model based on statistics, often used as a surrogate model in engineering. It can not only provide predicted values ​​for unknown points, but also provide the mean squared error (uncertainty estimate) of the prediction, a characteristic that forms the basis for active learning.

[0051] PC-Kriging model: A hybrid surrogate model that combines the advantages of polynomial chaotic expansion (PCE) and the Kriging model. It uses PCE to capture global trends and then uses the Kriging model to model the residuals (local fluctuations), thereby improving the prediction accuracy and robustness for complex nonlinear responses.

[0052] Active learning, also known as adaptive sampling, is an efficient sample selection strategy. When building a surrogate model, it does not use all samples at once, but intelligently selects the most valuable "information points" for simulation based on information such as the uncertainty of the current model, aiming to obtain a high-precision model with minimal computational cost.

[0053] Spatiotemporal Co-learning Function (LIF): This invention proposes an improved active learning function. It simultaneously considers the prediction uncertainty (such as the U function) in the spatial dimension (design variables and random variables) and the failure risk in the temporal dimension (such as the expected improvement function EI). By introducing a time weight coefficient, it guides the algorithm to pay more attention to high-risk periods such as the later stages of the lifespan, thereby achieving efficient and accurate time-varying reliability analysis.

[0054] Limit state equations: In structural reliability analysis, these are mathematical expressions used to define the "safe" and "failure" states of a system, usually denoted as... ,in These are basic variables. When... When the system is in a safe state; when At that time, the system is in a state of failure.

[0055] Load spectrum: A statistical representation of the changes in the magnitude, frequency, and sequence of loads borne by a mechanical structure during service over time. It 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 for this step is to select a set of parameters from the gear's geometric parameters that have a significant impact on the optimization objective (such as volume) and key performance (such as contact fatigue strength) and allow for adjustment in the design, as design variables for the optimization search, and to set a reasonable range of variation for each variable.

[0077] In this embodiment, five geometric parameters that significantly affect gear volume and contact fatigue performance are selected as design variables, i.e., a design variable vector. , representing the number of teeth on the driving gear, the number of teeth on the driven gear, the normal module, the tooth width, and the helix angle, respectively. The permissible ranges for these variables are determined based on engineering experience, manufacturing processes, and installation space, as shown in Table 2 below:

[0078] Table 2 Example Value Ranges of Design Variables

[0079]

[0080] S23: Define constraints based on gear design and reliability requirements.

[0081] This step aims to impose a series of constraints on the optimization model to ensure that the optimization results are both engineering feasible and reliable enough to meet requirements. These constraints mainly include three categories: geometric constraints, performance constraints, and the core time-varying reliability constraints.

[0082] To implement the above solution, this embodiment sets the following exemplary constraints:

[0083] 1. Geometric and performance constraints: These constraints ensure that gears are manufacturable, mesh correctly, and have good transmission performance. For example:

[0084] 1) Transmission ratio constraint: the optimized actual transmission ratio It must fall within the allowed range (e.g.) ).

[0085] 2) Overlap Constraint: In order to ensure the length of the double tooth meshing area and thus reduce transmission noise and impact, the total overlap constituting the end face overlap and axial overlap must meet a specific performance range, such as [6.1, 6.9].

[0086] 3) Tooth count constraint: In order to avoid undercutting and ensure smooth transmission, the minimum number of teeth required for helical gears to avoid undercutting is set, such as in this embodiment, the constraint range can be set to ≥17.

[0087] 4) Boundary constraints for module, tooth width, and helix angle: Their values ​​must 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 into account the fatigue notch factor. Size factor Surface machining coefficient Loading method coefficient Comprehensive influence coefficient, etc.

[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] In the formula, This represents the stress in the standard specimen's SN curve. Through this correction, an SN curve suitable for gear design is obtained, for example... This provides a basis for subsequent calculations of strength degradation and damage accumulation.

[0110] In this embodiment, the gear material is 20CrMnTi. By consulting literature and engineering handbooks, the fatigue constants of this material at a survival rate P of 99% (SN curve) are obtained as m = 28.5714 and C = 1.0822 × 10⁻⁶. 102 The SN curve of the standard specimen is modified by introducing the comprehensive influence.

[0111] Referring to the "Mechanical Design Handbook", select based on the gear's structure and surface finish. It is 1.36. It is 0.68. It is 1.21. The value is 0.85. The corrected expression for the tooth surface contact stress-life curve is obtained. The curve is as follows Figure 3 .

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

[0113] This sub-step aims to abandon the traditional assumption of constant strength and can employ a nonlinear degradation model based on cumulative damage theory to describe the gradual decrease in contact fatigue strength of gear materials as load accumulates and service time increases. This requires establishing residual strength. With initial strength Load history and time The functional relationship between them.

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

[0115] ;

[0116] In the formula, R(0) is the initial contact fatigue strength. This represents the maximum contact stress. The design life is given by t, where t is the usage time. The parameters are related to load and material. This model dynamically describes the process by which the strength R(t) of a gear gradually decreases due to the accumulation of internal material damage with each meshing impact during its service life.

[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 At that time, 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 surrogate model and active learning to efficiently evaluate time-varying limit state functions.

[0127] This step aims to build an efficient, autonomous, and complete time-varying failure probability assessment module to overcome the bottleneck of excessive computational cost in solving high-dimensional time-varying reliability problems using Monte Carlo simulation methods. For example... Figure 4 As shown, its core is to construct a PC-Kriging surrogate model that integrates the advantages of polynomial chaotic expansion (PCE) and the Kriging model, and to introduce an improved spatiotemporal collaborative learning function (LIF). Through an active learning strategy, it achieves efficient and high-precision estimation of the time-varying failure probability of gears with the fewest expensive simulations. This module is key to achieving efficient coupling between reliability analysis and optimization design.

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

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

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

[0131] To implement the above scheme, it is necessary to determine the main random variables affecting the contact fatigue performance of gears. This embodiment exemplarily selects five key parameters, assuming they follow a normal distribution, to form a random variable vector. , representing the normal module, tooth width, elastic modulus, Poisson's ratio, and input torque, respectively. Examples of their uncertainty variables are shown in Table 3 below:

[0132] Table 3. Probability distribution parameters of uncertain variables

[0133]

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

[0135] The technical solution for this sub-step is as follows: initial training samples are generated using Latin hypercube sampling, and a PC-Kriging surrogate model is constructed to approximate the limiting state function; an improved learning function that comprehensively considers spatial prediction uncertainty and time failure risk is used to intelligently guide the addition of points to the samples; the model is iteratively updated until convergence, and finally, the time-varying failure probability is efficiently predicted based on the converged surrogate model.

[0136] To implement the above solution, the process in this embodiment is as follows: Figure 4 As shown, the main steps include:

[0137] S421: Generate a candidate sample pool based on the probability distribution of the uncertain variables. Based on the probability distribution described in Table 3, use Monte Carlo sampling to generate the pool. The candidate samples constitute the sample pool. Furthermore, t is generated within a given observation time range. MCS The time sample pool consists of 1 time sample. Together, they form the initial sample pool, which is used for subsequent iterative updates of the proxy model and time-varying reliability solutions.

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

[0139] S423: Based on the time-varying limit state function, calculate the true performance response of the initial training sample set to form the initial training dataset. Then, for each initial training sample... Substitute the time-varying limit state function established by S3 The mechanical model is called to perform calculations and obtain the true response value. This forms the initial training dataset.

[0140] S424: Construct a PC-Kriging proxy model based on the initial training dataset. Utilize the initial training dataset to construct a PC-Kriging hybrid proxy model. This model combines the advantages of PCE in capturing global trends with Kriging in interpolating local fluctuations, enabling it to more accurately predict the performance response and its uncertainties at unknown points.

[0141] S425: Optimal addition point selection based on an improved learning function. To efficiently improve the model's accuracy near the failure boundary, a spatiotemporal collaborative integrated learning function (LIF) is defined to select the new sample points with the highest "learning value" from the candidate sample pool. This function also considers the prediction uncertainty of the spatial dimension (through the improved U-learning function). Failure risk in the time dimension (through an improved expectation boosting function) and time dimension (through an improved expectation boosting function) An exemplary spatiotemporal collaborative synthesis learning function (LIF) proposed in this embodiment is as follows:

[0142] ;

[0143] In the formula, Let LIF(X,t) be a minimal normal quantity. Select the sample that maximizes the LIF(X,t) value as the optimal addition point.

[0144] in For the improved U learning function, For the improved EI learning function.

[0145] Those skilled in the art will understand that the specific form of the learning function can be appropriately adjusted, as long as it comprehensively considers spatial uncertainty and the risk of temporal failure.

[0146] S426: Update the training set and surrogate model using the optimal addition. Calculate the optimal addition. The actual response value at the location Add it to the training set and retrain the updated PC-Kriging model.

[0147] S427: Determine if the model has converged. Repeat steps S425 to S426 until the convergence criterion is met. For example, the relative error between the failure probability estimated in the current iteration and the previous iteration can be set to be less than a specific threshold (e.g., ...). Stop when ).

[0148] S428: Calculate the time-varying failure probability of the convergent model. Based on a high-precision PC-Kriging model that ultimately converges. The system performs rapid prediction on all candidate sample pools, counts the number of failed samples, and estimates the time-varying failure probability. This can be represented as follows:

[0149] ;

[0150] In the formula, It is an indicator function, when If the value is 1, then the value is 0; otherwise, the value is 0.

[0151] S429: Evaluate the computational stability of time-varying failure probabilities based on convergent models. Calculate the coefficient of variation (COV) of the failure probability estimates. If it meets preset accuracy requirements (e.g., ...), ... Then the output will be... Otherwise, the candidate sample pool needs to be expanded and the analysis re-analyzed 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: The gear reliability optimization design model is solved by using an optimization algorithm. The time-varying reliability analysis module is called to evaluate the time-varying failure probability of the candidate scheme and process the time-varying reliability constraints accordingly. The optimal gear design parameters that satisfy the constraints are then output.

[0155] This step aims to efficiently solve the reliability optimization design model established in S2 using a global optimization algorithm, ultimately outputting a lightweight gear design scheme that meets the time-varying reliability constraints throughout the entire lifecycle. Its core is an improved differential evolution (DE) algorithm, which initializes the population through chaotic mapping, adaptively adjusts control parameters, and combines a dynamic penalty function mechanism to handle reliability constraints, thereby robustly and efficiently searching for the optimal solution within a complex design space.

[0156] The process is as follows Figure 5 As shown. The specific steps are as follows:

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

[0158] The technical solution for this sub-step is to configure the parameters of the improved differential evolution algorithm and adopt a strategy based on chaotic mapping to generate the initial population, so as 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. Maximum number of iterations wait.

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

[0161] S512: Generate a chaotic sequence. Iteratively generate a chaotic variable sequence using the Logistic mapping formula. This sequence exhibits ergodicity and pseudo-randomness. The mapping formula can be exemplified as follows:

[0162] ;

[0163] in, To control the parameters, a value of 4 is typically used to ensure a chaotic state. The initial value is a non-specific value within the interval (0,1).

[0164] S513: Map the chaotic sequence to the design space to generate the initial population. (The chaotic sequence is then mapped to the design space.) Each value is linearly mapped to the design variables defined in S2. upper and lower bound intervals This process generates each individual in the initial population. Quantity:

[0165] .

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

[0167] The technical solution for this sub-step is as follows: In each generation of evolution, adaptive mutation and crossover operations are performed to generate new individuals (experimental vectors), and their fitness is calculated using the time-varying reliability analysis module established by S4; a dynamic penalty function mechanism is used to integrate reliability constraints into fitness evaluation, guiding the population to converge toward the feasible region.

[0168] To implement the above plan, each individual in the population... Repeat the following operations until the termination condition is met:

[0169] S521: Based on the initial population, adaptive mutation is performed to generate a mutation vector. To increase search capability, the mutation factor... No longer fixed, but from a... The parameters are randomly generated from a Cauchy distribution of location parameters. This adaptive... Perform a difference operation on three randomly selected distinct individuals to generate a mutation vector. ;

[0170] ;

[0171] In the formula, It is a variable factor; , , These are three distinct individuals randomly selected from the current population.

[0172] S522: Cross the mutation vector with the original individual to generate the experimental vector. The mutation vector... Compared with the original individual (target vector) According to the adaptively generated crossover probability Perform cross operations to generate test vectors .

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

[0174] 1) Perform time-varying reliability assessment on the test vector using step S4: [The test vector is then...] (Representing a set of gear design parameters) are input into the PC-Kriging time-varying reliability analysis module established in step S4 to calculate the design scheme. Time-varying failure probability .

[0175] 2) Constraint violation calculation: Based on the S2 time-varying reliability constraint (i.e. ), calculate the constraint violation degree of the current scheme. :

[0176] .

[0177] 3) Based on time-varying reliability assessment and violation rate, construct a dynamic penalty fitness function and calculate the fitness of the test vector: The objective function (total volume) is used to... The degree of constraint violation is determined by a series of iterations. The final fitness value is formed by combining dynamically increasing penalty functions. This function can be defined as:

[0178] ;

[0179] In the formula, This represents the total volume of the gear. This is a dynamic penalty factor. The design results in lighter penalties in the early stages of optimization and increased penalties in later stages. An example is as follows:

[0180] .

[0181] This design allows for lighter penalties in the early stages of optimization, enabling the algorithm to explore regions near the feasible region and even infeasible regions, maintaining global search capabilities; while in the later stages of optimization, the penalties increase sharply, strongly driving the population to converge to the feasible region that satisfies the reliability constraints.

[0182] S524: Compare the fitness values ​​of the experimental vector and the original individual, and select the individual with the lower fitness value. (Compare experimental vectors) fitness value With the original individual fitness value Since the optimization objective is to minimize the size, individuals with smaller fitness values ​​(i.e., better) are retained for the next generation of the population.

[0183] S525: Determine whether the algorithm terminates based on the convergence condition. Repeat steps S521 to S524 until the preset maximum number of iterations is reached. If the fitness value of the best individual in the population improves by less than the given tolerance over several consecutive generations, the algorithm is considered 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 best fitness, that is, the optimal gear design parameter combination that minimizes the total volume of the driving and driven gears under the premise of strictly satisfying the time-varying reliability constraints throughout the entire life cycle.

[0185] Following the optimization process, the final output is the optimal combination of gear design parameters that achieves lightweight design while satisfying time-varying reliability constraints. The comparison of key design parameters and volume before and after optimization is shown in Tables 4 and 5 below.

[0186] Table 4. Examples of comparisons before and after optimization of key gear design parameters.

[0187]

[0188] Table 5. Example of comparison of objective functions before and after optimization

[0189]

[0190] Using the aforementioned reliability optimization design method, and under the premise of fully considering the uncertainties of operating conditions and material parameters and satisfying the reliability constraints of gear contact stress, the gear volume V can be calculated to be 631827.70 mm². 3 Up to the optimized 564530.36mm 3 The volume is reduced by approximately 10.56%.

[0191] The above embodiments are only used to illustrate the technical solutions and implementation effects of the present invention, and are not intended to limit the present invention. 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, surrogate model construction methods, active learning function forms, optimization algorithms and their parameters, etc., without departing from the principles of the present invention. All such changes fall within the scope of protection defined by the claims of the present invention.

[0192] Example 2

[0193] This embodiment provides a time-varying reliability optimization design system for helical cylindrical gears, used to execute the time-varying reliability optimization design method as described in Embodiment 1. The system is deployed on a computing device, such as... Figure 1 The overall process shown is executed automatically by the system, and it includes the following modules that are connected in sequence via communication:

[0194] 1. The design modeling and initialization module is configured to perform the functions of steps S1 and S2 in Example 1, namely, to determine the initial design parameters of the gear and establish a gear reliability optimization design model with the gear volume minimization as the optimization objective and including time-varying reliability constraints, thus transforming the engineering design requirements into a mathematical model of an optimization problem that can be solved iteratively by a computer. Specifically, it consists of the following three collaboratively working units:

[0195] 1) Parameter input and design variable management unit, used to receive the initial geometric parameters and design requirements of the gear, and select and define parameters including the number of teeth based on these parameters. Normal module Tooth width helix angle The design variables and their value ranges correspond to steps S1 and S22 of Example 1.

[0196] 2) Constraint configuration unit, used to configure geometric performance constraints (such as transmission ratio and overlap ratio constraints) and core time-varying reliability constraints for the optimization model. This corresponds to step S23 in Embodiment 1.

[0197] 3) Optimize the model building unit, used to define the total volume of the gear pair. The optimization objective function is minimized, and the design variables and constraints are integrated to construct a complete reliability optimization design mathematical model, corresponding to steps S21 and S24 of Implementation Example 1.

[0198] 2. The time-varying failure probability analysis engine module is configured to perform the functions of steps S3 and S4 in Example 1, namely, to construct the time-varying limit state function of gear contact fatigue coupled with load and strength degradation, and to construct a time-varying reliability analysis module based on a surrogate model and active learning strategy to evaluate the limit state function and calculate the time-varying failure probability. It is the core analysis component of the system. Specifically, it includes:

[0199] 1) Physical model building unit, used to establish the gear system dynamic model as described in step S31, in order to generate the time-varying contact stress load spectrum. ; and for performing material SN 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) The surrogate model training and optimization unit, the core of the analysis engine, is used to perform the time-varying reliability analysis as described in step S4. It further includes:

[0201] Uncertainty quantization unit, used to identify and quantize random input vectors as described in step S41 and Table 3. and its probability distribution.

[0202] The sample management unit is used to perform 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] The PC-Kriging model building unit is used to execute the hybrid agent model as described in step S424. The initial construction and iterative updates.

[0204] Active learning controller, integrating spatiotemporal collaborative comprehensive learning function Used to perform the selection of optimal points as described in step S425. The decision-making logic.

[0205] The convergence judgment and probability calculation unit is used to perform the iterative convergence judgment as described in steps S427 and S428, and calculate the time-varying failure probability based on the final surrogate model. Its calculation logic and formula are exactly the same as those in Example 1.

[0206] 3. A reliability constraint-driven optimization module, configured to perform the function of step S5 in embodiment 1, that is, configured to solve the gear reliability optimization design model using an optimization algorithm; during the solution process, the time-varying reliability analysis module is called to evaluate the time-varying failure probability of candidate schemes and process the time-varying reliability constraints accordingly, finally outputting the optimal gear design parameters that satisfy the constraints. Specifically, this includes:

[0207] 1) An improved differential evolution algorithm engine for performing the optimization search. It includes:

[0208] The chaos initialization unit is used to perform population initialization based on Logistic chaotic mapping as described in steps S511 and S512.

[0209] The adaptive evolutionary operation unit is used to perform adaptive mutation and crossover operations as described in steps S521 and S522 to generate trial vectors. .

[0210] 2) The dynamic penalty function fitness evaluator, a core component connecting optimization and reliability analysis, is used to perform the fitness evaluation as described in step S523. Its workflow is as follows: it calls the time-varying failure probability analysis engine module to calculate the current test vector. Corresponding failure probability ; Calculate the degree of constraint violation ; and according to the method defined in Example 1, with the number of iterations Dynamically adjusted penalty factor Formula, construct the final fitness function To conduct an evaluation.

[0211] 3) Optimize the iterative controller to control the selection operation and termination judgment as described in steps S524 and S525 until the optimal combination of design parameters is output.

[0212] 4. The results output and visualization module is configured to receive and display the optimal results output by the reliability constraint-driven optimization module. Its displayed content strictly corresponds to the optimization results of Example 1, including: a comparison table of design variables before and after optimization (as shown in Table 4), a comparison value of the objective function volume (as shown in Table 5), the whole life cycle reliability curve, and computational efficiency indicators.

[0213] The system's modules work collaboratively in the aforementioned order and through their data interfaces: the output of the design modeling and initialization module provides deterministic parameters and a model framework to the time-varying failure probability analysis engine module; the analysis engine module trains a converged PC-Kriging surrogate model as a fast predictor, which is then used by the reliability constraint-driven optimization module during each fitness evaluation; the optimization module transmits the optimal parameter set obtained through optimization to the result output and visualization module for final presentation. Through this process, the system fully and automatically implements all the method steps described in Example 1.

[0214] In summary, this embodiment of the system, through the coordinated operation of the four modules mentioned above, encapsulates the high-cost analysis process of time-varying reliability into a fully autonomous analysis service that can be efficiently and repeatedly invoked by optimization algorithms. This system fundamentally changes the traditional sequential process of separating "analysis" and "optimization" in reliability optimization design, forming an integrated and automated design closed loop driven by a high-precision surrogate model and dynamic constraint processing. This allows for the efficient achievement of lightweight design goals while ensuring the reliability of gears throughout their entire lifecycle.

[0215] Example 3

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

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

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

[0219] Specifically, when the processor executes the computer program, it is able to:

[0220] 1. Call parameters and modeling logic to determine the initial design parameters of the gear, and establish an optimization design model with the goal of minimizing volume and including time-varying reliability constraints.

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

[0222] 3. Invoke the optimization solution logic to drive the optimization algorithm to perform iterative optimization. During the iteration, call the reliability analysis module to evaluate the time-varying failure probability to handle reliability constraints, and finally output the optimal gear design parameters.

[0223] This electronic device can receive gear design parameters, material properties, load spectrum and reliability targets input by the user through a communication interface. It can also transmit the final optimized design parameters, reliability curves and performance comparison results to a display device or computer-aided design (CAD) system.

[0224] Example 4

[0225] This embodiment provides a computer-readable storage medium for storing a computer program that implements the time-varying reliability optimization design method for the above-described helical cylindrical gear.

[0226] The computer-readable storage medium can be any tangible medium that contains or stores a program, such as, but not limited to: USB flash drive, portable hard drive, read-only memory (ROM), random access memory (RAM), disk (including hard disk and floppy disk), optical disk (including CD-ROM and DVD-ROM), or 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 (e.g., the processor of the electronic device in Embodiment 3), the processor is able to perform the steps of the method described in Embodiment 1, or control the corresponding device to perform the functions of the system described in Embodiment 2.

[0228] The computer program includes a series of instructions that, when executed, specifically instruct the processor to complete the full process as detailed in Example 1.

[0229] Through this storage medium, the time-varying reliability optimization design scheme of the helical cylindrical gear described in the embodiments of the present invention can be saved, distributed and deployed in the form of a software product, which facilitates its widespread application in industrial design environments.

[0230] The above description is merely a preferred embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

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 the PC-Kriging hybrid agent model and a spatiotemporal collaborative active learning strategy is constructed to evaluate the time-varying limit state function. The spatiotemporal collaborative active learning strategy uses a comprehensive learning function that simultaneously considers the prediction uncertainty in the spatial dimension and the failure risk in the time dimension to screen the addition samples used to update the PC-Kriging hybrid agent model. An optimization algorithm is used to solve the gear reliability optimization design model. The optimization algorithm handles time-varying reliability constraints through a dynamic penalty function mechanism. The penalty factor of the dynamic penalty function is set to be small in the early stage of optimization iteration to make the penalty for constraint violation lighter, and large in the later stage of optimization iteration to make the penalty for constraint violation increase sharply. The optimization algorithm evaluates the time-varying failure probability of candidate schemes by calling the time-varying reliability analysis module and processes the time-varying reliability constraints accordingly, outputting the optimal gear design parameters that satisfy the constraints.

2. The method according to claim 1, characterized in that, The establishment of a gear reliability optimization design model with the goal of minimizing gear volume and constrained by time-varying reliability throughout the entire life cycle 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 residual strength model for gears based on the theory of cumulative damage; 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 optimization algorithm is an improved differential evolution algorithm, and the time-varying reliability constraint is handled through a dynamic penalty function mechanism.

5. 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 PC-Kriging hybrid surrogate model and a spatiotemporal collaborative active learning strategy, for evaluating the time-varying limit state function and calculating the time-varying failure probability; wherein, the spatiotemporal collaborative active learning strategy uses a comprehensive learning function that simultaneously considers the prediction uncertainty of the spatial dimension and the failure risk of the time dimension to filter the addition samples used to update the PC-Kriging hybrid surrogate model; The reliability constraint-driven optimization module is configured to execute an optimization algorithm to solve the gear reliability optimization design model. The optimization algorithm processes the time-varying reliability constraints through a dynamic penalty function mechanism. The penalty factor of the dynamic penalty function is set to be smaller in the early stages of optimization iteration to reduce the penalty for constraint violations, and larger in the later stages of optimization iteration to drastically increase the penalty for constraint violations. The optimization algorithm evaluates the time-varying failure probability of candidate solutions by calling the analysis module constructed by the time-varying reliability analysis engine module and processes the time-varying reliability constraints accordingly, ultimately outputting the optimal gear design parameters that satisfy the constraints.

6. The system according to claim 5, characterized in that, The optimization algorithm used in the reliability constraint-driven optimization module is an improved differential evolution algorithm, and the time-varying reliability constraints are handled through a dynamic penalty function mechanism.

7. 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 4.

8. 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 4.

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