Gear design optimization method based on Bayesian algorithm

Abstract

The invention discloses a gear design optimization method based on a Bayesian algorithm, and relates to the technical field of gear design, a tooth surface geometric model is established based on a pure rolling relation of a rack and a gear, a load rigidity coupling model is established in combination with a meshing partitioning and tooth direction slicing method, and contact and bending stress is obtained; driving and non-driving side pressure angles, tooth addendum coefficients, tooth root fillet radius coefficients, radial variables and tooth widths are used as design parameters, a multi-objective function with the maximum tooth root bending stress, the maximum contact stress being minimized and the average contact ratio being maximized is set, a Markov blanket structure is introduced to screen key variables, a proxy model is established by using a Gaussian process, and the maximum tooth root bending stress and the maximum contact stress are optimized. Iterative optimization is carried out on the basis of an expected hyper-volume improved acquisition function, the number of iterations, the lifting amplitude and the operation time limit are used as termination conditions, and optimal parameters are screened in combination with normalized comprehensive scores. According to the gear design optimization method based on the Bayesian algorithm, the design precision of the asymmetric high-overlap-ratio gear and the design parameter optimization efficiency can be improved.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of gear design, in particular to a gear design optimization method based on a Bayesian algorithm. BACKGROUND

[0002] In high-performance mechanical transmission systems, asymmetric high-contact ratio cylindrical gears are gradually applied in aerospace, high-end equipment and new energy fields due to their higher load-carrying capacity and better load distribution performance. However, the complex structure design and variable meshing state of such gears bring significant challenges to strength analysis and parameter optimization.

[0003] The traditional gear design method mainly has two technical bottlenecks when facing asymmetric tooth shape and high-contact ratio characteristics. The first aspect is the insufficient modeling accuracy of meshing load, which is not coupled with stiffness variation. The second aspect is the low calculation efficiency of the optimization method, which is difficult to adapt to complex design space. SUMMARY

[0004] The present application aims to at least solve one of the technical problems existing in the prior art. To this end, the present application proposes a gear design optimization method based on a Bayesian algorithm, which can improve the design accuracy of asymmetric high-contact ratio gears and improve the efficiency of design parameter optimization.

[0005] The gear design optimization method based on the Bayesian algorithm according to the embodiment of the present application comprises: step S1, establishing a geometric mathematical model of the rack cutter tooth surface based on the pure rolling relationship between the rack cutter and the gear division circle, mapping the geometric mathematical model of the rack cutter tooth surface to the coordinate system of the gear, solving the motion envelope of the rack cutter tooth surface, and obtaining a three-dimensional geometric mathematical model of the gear tooth surface; step S2, introducing a tooth direction slicing method to the three-dimensional geometric mathematical model of the gear tooth surface, discretizing the tooth into a plurality of independent slices along the tooth width direction, establishing an equivalent stiffness model of the bending, shear and axial deformation of the tooth piece, and then combining the meshing partition theory to construct the load distribution model of the three-tooth meshing area and the double-tooth meshing area, to obtain the normal stiffness and meshing force distribution of each tooth piece; step S3, according to the normal stiffness and meshing force distribution of each tooth piece, calculating the contact stress of each tooth piece based on the Hertz contact theory, and combining the tooth root geometric characteristics to calculate the tooth root bending stress, and forming a stress database of the gear tooth surface based on the obtained contact stress and tooth root bending stress; step S4, according to the design parameters of the gear and the obtained stress of the gear tooth surface, establishing a multi-objective Bayesian optimization model, and cooperatively optimizing the design parameters of the gear, and outputting the combination of the optimized design parameters of the gear.

[0006] The gear design optimization method based on Bayesian algorithm according to embodiments of the present invention has at least the following beneficial effects: Based on the established geometric mathematical model of the gear, a tooth-axis slicing modeling strategy is introduced to establish a load distribution model that reflects the transition characteristics of three-tooth and two-tooth meshing in asymmetric gears with high overlap ratios, achieving coordinated and accurate calculation of gear contact stress and tooth root bending stress. Furthermore, the introduction of the Bayesian optimization algorithm significantly reduces the design space dimension, improves the prediction accuracy of the surrogate model, and achieves global equilibrium optimization of gear bending strength, contact strength, and overlap ratio performance.

[0007] According to some embodiments of the present invention, in step S1, the tooth surface of the rack cutter includes a straight line segment at the tooth tip, a circular arc transition segment, and a straight line segment at the tooth root. A geometric mathematical model of the rack cutter tooth surface is established by segmented modeling, and the geometric continuity of adjacent segments is maintained when constructing the equation of the gear tooth surface through pure rolling mapping.

[0008] According to some embodiments of the present invention, step S2 includes: step S2.1, discretizing the tooth into several independent slices along the tooth width direction, each slice being divided into a three-tooth meshing area and a two-tooth meshing area in the meshing line direction; Step S2.2: In each slice, establish an equivalent stiffness model of the tooth slice that includes bending stiffness, shear stiffness and axial compressive stiffness, and form a stiffness matrix in the tooth direction. Step S2.3: At the boundary between the three-tooth meshing zone and the two-tooth meshing zone, introduce the coordinated deformation conditions to solve the deformation equations of adjacent tooth plates, solve the coupling relationship between meshing force and deformation, and obtain the normal stiffness and load distribution of the tooth plates. Step S2.4: In the area where there is tooth-direction drum-shaped modification or tooth profile modification, the local contact offset caused by the modification is added as an additional deformation amount to the extrusion deformation to correct the local load distribution. Step S3 includes: Step S3.1, based on the load distribution and stiffness matrix of the gear tooth, calculating the contact stress and root bending stress of the gear tooth according to Hertzian contact theory and critical section bending theory respectively, forming a database of stress on the gear tooth surface.

[0009] According to some embodiments of the present invention, in step S4, a multidimensional bounded design parameter space is constructed using the driving side pressure angle, non-driving side pressure angle, tooth tip height coefficient, tooth root fillet radius coefficient, radial variable, and tooth width as design parameters. A multi-objective function is set, including minimizing the maximum tooth root bending stress, minimizing the maximum contact stress, and maximizing the average overlap. Furthermore, the average overlap is taken as a negative value to participate in the minimization optimization.

[0010] According to some embodiments of the present invention, in step S4, a Markov blanket structure analysis mechanism is introduced to screen the gear design parameters for dependency, a Bayesian network structure is established for the target performance variables, a subset of parameters that have direct or conditional dependencies with the target are identified, redundant gear design parameters are removed, and a Gaussian process model is used for surrogate modeling to obtain a surrogate model. In the Bayesian optimization process, new sampling points are determined based on the expected hypervolume improvement acquisition function, and termination conditions are set.

[0011] According to some embodiments of the present invention, the sampling strategy of the surrogate model adopts a multi-objective acquisition function based on the expected hypervolume improvement. The multi-objective acquisition function calculates the expected improvement value based on the predicted mean and variance of the surrogate model. After each iteration, a new sampling point is determined based on the maximum value of the expected hypervolume. The multi-objective acquisition function is realistically evaluated and the surrogate model is updated, and the Pareto front is continuously expanded under the condition of limited samples.

[0012] According to some embodiments of the present invention, the termination conditions include the number of iterations reaching a preset upper limit, the improvement of the objective function in several consecutive iterations being lower than a minimum threshold, the expected supervolume improvement function being less than a minimum threshold, or the optimization running time exceeding a predetermined upper limit. When any of the termination conditions is met, the iteration is terminated and a set of design parameters for the candidate gears is output.

[0013] According to some embodiments of the present invention, after the iteration is terminated, a comprehensive evaluation model is established for the set of design parameters of the candidate gears. The comprehensive evaluation model includes strength index, meshing performance index and structural manufacturability index. Each index is linearly normalized and a weighted comprehensive scoring function is constructed to score and rank the design parameters of each candidate gear.

[0014] According to some embodiments of the present invention, constraints are set, including material strength limit, machining tool interference, tooth tip height or fillet radius exceeding the limit. Based on the design parameters of the candidate gears, the results that do not meet the constraints are filtered out, and the remaining gear design parameters are sorted from high to low according to the score, and the combination of gear design parameters is output.

[0015] Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0016] The present invention will be further described below with reference to the accompanying drawings and embodiments, wherein: Figure 1 This is a schematic diagram of the high overlap ratio gear meshing zone in an embodiment of the present invention; Figure 2 This is a schematic diagram of the asymmetric gear slicing method in an embodiment of the present invention; Figure 3 This is a schematic diagram of the superposition of gear normal stiffness in an embodiment of the present invention; Figure 4 This is a schematic diagram of an asymmetric gear two-dimensional plate unit model in an embodiment of the present invention; Figure 5 This is a stress analysis diagram of the bending stress of the asymmetric gear in an embodiment of the present invention; Figure 6 This is a schematic diagram of the Markov blanket structure of the target variable T in an embodiment of the present invention. Detailed Implementation

[0017] Embodiments of the present invention are described in detail below. Examples of these embodiments are shown in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present invention, and should not be construed as limiting the present invention.

[0018] In the description of this invention, it should be understood that the orientation descriptions, such as up, down, etc., are based on the orientation or positional relationship shown in the drawings and are only for the convenience of describing this invention and simplifying the description, and are not intended to indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation of this invention.

[0019] In the description of this invention, "multiple" refers to two or more. The use of "first" and "second" is for distinguishing technical features only and should not be construed as indicating or implying relative importance, or implicitly indicating the number of indicated technical features or their sequential relationship.

[0020] In the description of this invention, unless otherwise explicitly defined, terms such as "set up," "install," and "connect" should be interpreted broadly, and those skilled in the art can reasonably determine the specific meaning of the above terms in this invention in conjunction with the specific content of the technical solution.

[0021] An embodiment of the present invention provides a gear design optimization method based on a Bayesian algorithm, comprising: Step S1: Based on the pure rolling relationship between the rack cutter and the gear pitch circle, establish the geometric mathematical model of the rack cutter tooth surface, map the geometric mathematical model of the rack cutter tooth surface to the coordinate system of the gear, solve the motion envelope of the rack cutter tooth surface, and obtain the three-dimensional geometric mathematical model of the gear tooth surface. Step S2: Introduce the tooth slicing method into the three-dimensional geometric mathematical model of the gear tooth surface. Discretize the tooth into several independent slices along the tooth width direction, establish the equivalent stiffness model of the tooth slice for bending, shearing and axial deformation, and then combine the meshing partition theory to construct the load distribution model of the three-tooth meshing area and the two-tooth meshing area to obtain the normal stiffness and meshing force distribution of each tooth slice. Step S3: Based on the normal stiffness and meshing force distribution of each gear, calculate the contact stress of each gear based on Hertz contact theory, and calculate the tooth root bending stress in combination with the tooth root geometric characteristics. Based on the obtained contact stress and tooth root bending stress, form a database of stress on the gear tooth surface. Step S4: Based on the gear design parameters and the obtained gear tooth surface stress, establish a multi-objective Bayesian optimization model, perform collaborative optimization on the gear design parameters, and output the optimized combination of gear design parameters. The output combination of design parameters satisfies the balance between stress and overlap.

[0022] In step S1, the tooth surface of the rack cutter includes a straight line segment at the tooth tip, a circular arc transition segment, and a straight line segment at the tooth root. A segmented modeling method is used to establish the geometric mathematical model of the rack cutter tooth surface. When constructing the equation of the gear tooth surface through pure rolling mapping, the geometric continuity of adjacent segments is maintained, thereby ensuring the conjugacy of the gear tooth surface and the feasibility of machining.

[0023] To achieve precise modeling and stiffness coupling calculation of the meshing load of asymmetric high overlap gears, step S2 includes: Step S2.1, discretizing the tooth into several independent slices along the tooth width direction, with each slice divided into a three-tooth meshing region and a two-tooth meshing region along the meshing line direction; Step S2.2, establishing an equivalent stiffness model of the tooth slice in each slice, including bending stiffness, shear stiffness, and axial compressive stiffness, forming a tooth-direction stiffness matrix; Step S2.3, introducing coordinated deformation conditions at the boundary between the three-tooth meshing region and the two-tooth meshing region to connect adjacent teeth. The deformation equation of the gear tooth is obtained by solving the coupling relationship between the meshing force and the deformation, and the normal stiffness and load distribution of the gear tooth are obtained. Step S2.4: In the part where there is tooth bulging or tooth profile modification, the local contact offset caused by the modification is superimposed as an additional deformation amount into the extrusion deformation to correct the local load distribution. Step S3 includes: Step S3.1: Based on the obtained load distribution and stiffness matrix of the gear tooth, the contact stress and tooth root bending stress of the gear tooth are calculated based on Hertz contact theory and critical section bending theory, respectively, to form a database of stress on the gear tooth surface.

[0024] To achieve a comprehensive balance between gear strength and meshing performance, in step S4, a multidimensional bounded design parameter space is constructed using the driving side pressure angle, non-driving side pressure angle, tooth tip height coefficient, tooth root fillet radius coefficient, radial variable, and tooth width as design parameters. A multi-objective acquisition function is set, including minimizing the maximum tooth root bending stress, minimizing the maximum contact stress, and maximizing the average overlap. Furthermore, the average overlap is taken as a negative value to participate in the minimization optimization.

[0025] To further improve the modeling efficiency and prediction accuracy of the surrogate model, in step S4, a Markov blanket structure analysis mechanism is introduced to screen the gear design parameters for dependency. A Bayesian network structure is established for the target performance variables to identify a subset of parameters that have a direct or conditional dependency relationship with the target. After eliminating redundant gear design parameters, a Gaussian process model is used for surrogate modeling to obtain the surrogate model. During the Bayesian optimization process, new sampling points are determined based on the expected hypervolume improvement sampling function, and termination conditions are set. Termination conditions include the number of iterations reaching a preset upper limit, the improvement of the objective function in several consecutive iterations falling below a minimum threshold, the expected hypervolume improvement function being less than a minimum threshold, and the optimization running time exceeding a predetermined upper limit. When any of the termination conditions is met, the iteration terminates and a set of candidate gear design parameters is output. The multi-condition termination mechanism is used to prevent overcomputation and ensure the convergence stability and computational controllability of the optimization process.

[0026] The sampling strategy of the surrogate model adopts a multi-objective acquisition function based on the expected hypervolume improvement. The multi-objective acquisition function calculates the expected improvement value based on the predicted mean and variance of the surrogate model. After each iteration, a new sampling point is determined based on the maximum value of the expected hypervolume. The objective function is evaluated in a realistic way and the surrogate model is updated. Under the condition of limited samples, the Pareto front is continuously expanded to improve the global convergence and search efficiency of the optimization results.

[0027] To ensure the strength and reliability of gears while also considering meshing performance and manufacturing feasibility, a comprehensive evaluation model is established for the set of design parameters of candidate gears after the iteration is terminated. The comprehensive evaluation model includes strength indicators, meshing performance indicators, and structural manufacturability indicators. Each indicator is linearly normalized and a weighted comprehensive scoring function is constructed to score and rank the design parameters of each candidate gear.

[0028] Constraints are set, including material strength limits, tool interference, and exceeding limits for addendum or fillet radius. Based on the design parameters of candidate gears, results that do not meet the constraints are filtered out. The remaining candidate gear design parameters are then sorted from highest to lowest score, and the combination of gear design parameters is output. Furthermore, a three-dimensional geometric model of the gear and strength prediction results can be generated based on the combination of gear design parameters, realizing rapid parameter optimization and strength co-design of asymmetric high overlap gears.

[0029] Using the gear's module, number of teeth, driving-side pressure angle, non-driving-side pressure angle, addendum coefficient, and root fillet radius as basic design parameters, a geometric parameter system for the gear pair is constructed to ensure high overlap ratio and transmission continuity during transmission. When establishing a geometric mathematical model for the rack cutter tooth surface, a segmented modeling approach can be adopted, specifically including the addendum straight segment, transition arc segment, and root straight segment. This accurately expresses the asymmetric tooth profile characteristics and meets machining requirements, providing the basic geometric conditions for gear tooth surface generation. Based on the pure rolling relationship between the rack cutter and the gear pitch circle, a tooth surface mapping equation is established to realize the transfer of the cutter tooth profile in the gear coordinate system, ensuring the conjugacy and continuity of the gear tooth surface. By solving the motion envelope of the cutter tooth surface, the actual tooth profile of the gear is obtained. The cutter addendum, transition segment, and root curve sequentially form different regions of the corresponding gear tooth profile, thus obtaining a complete three-dimensional geometric model. Through the aforementioned steps, the constructed gear tooth surface can not only accurately reflect the asymmetric tooth profile and high overlap characteristics, but also serve as input for meshing partitioning, slice stiffness modeling, and strength co-optimization.

[0030] In gear transmissions, to increase gear contact performance and load-bearing capacity, the pressure angle on the primary load-bearing side of the gear is typically increased during gear design to effectively thicken the tooth root region. This structural feature results in different base circle radii for the left and right tooth profiles, leading to an asymmetrical distribution of the contact ratio during meshing. The contact ratio is defined as the ratio of the distance the contact point on the tooth surface moves along the line of action to the base pitch length when the driving gear rotates through a tooth pitch angle. High contact ratio gears refer to gears with a high contact ratio... This means that at any given time, two or three pairs of teeth are engaged simultaneously, thereby significantly improving the continuity of tooth surface contact and the uniformity of load transfer.

[0031] During meshing of asymmetric high-overlap gears, if the overlap ratio on the meshing side meets the following conditions... , combined Figure 1 As shown, along the meshing line, from the meshing start point F to the tooth apex A, the gear pair successively experiences the three-tooth meshing region and the two-tooth meshing region. The three-tooth meshing region is segment FE, containing tooth pairs DC and BA; the two-tooth meshing region is segment ED, containing tooth pair CB. Here, A represents the position of the tooth apex on the meshing side. The lengths of the three-tooth and two-tooth meshing regions are expressed as functions... and The function expression is based on the base circle tooth pitch. ,get: (1).

[0032] This allows us to further determine the range of pressure angle variation within different meshing zones, laying the foundation for subsequent deformation analysis and stiffness modeling.

[0033] For gear structures with tooth profile or tooth contour modifications, traditional concentrated load or uniformly distributed load models are insufficient to accurately represent the local load concentration effect. Therefore, it is preferable to introduce the coupling effect of the modification geometry into the slicing method modeling process. That is, on each slice, the local meshing clearance and contact displacement are calculated by combining the modification parameters corresponding to that location, such as the amount of tooth bulging. The contact offset caused by the modification is then considered. Add to total extrusion deformation In the middle, the corrected extrusion amount is By adding a local meshing error term caused by the shape modification parameter to the stiffness calculation and deformation compatibility equation, a consistent mapping between the shape modification geometry and the local load response is achieved.

[0034] To accurately analyze the gear tooth deformation behavior under multi-tooth meshing, a modeling strategy combining the slicing method and the potential energy method is further adopted. First, the gear teeth are divided into N independent slices of equal width along the tooth direction, such as... Figure 2 As shown, ignoring the coupling effect between individual slices, only the local contact on the theoretical tooth surface is considered on each slice, such as... Figure 3 As shown, the stiffness components, including bending, shear, and axial stiffness, are calculated based on the potential energy method. The stiffness expression for each slice (i.e., the toothed slice) is obtained: (2), in, Let n be the total stiffness of the nth toothed plate; , , These represent the bending, shearing, and axial stiffness of the gear, respectively, with p and g referring to the driving and driven gears, respectively.

[0035] The reciprocal expression for axial stiffness: (3); The reciprocal expression for bending stiffness: (4); in, , .

[0036] The reciprocal expression for shear stiffness: (5), In the above expression, It is the elastic modulus; , This represents the effective shear cross-sectional area of ​​the toothed plates on both sides of the contact line; This is the shear correction factor, typically taken as 0.83; , These are the angles between the tooth profiles of the driving and driven gears and the base circle when they begin to mesh; , These are the starting angles at which the load begins to be carried on the tooth profiles of the driving and driven gears, respectively. , The rate of change of tooth profile along the meshing line is used for deformation integral; , The moment of inertia of the tooth cross section relative to the direction of meshing; This represents the polar angle position of the tooth surface meshing point; It is the lateral coordinate projection of the point of application of the contact force on the normal line of action.

[0037] The stiffness analysis of the active gear tooth slice can be uniformly expressed as: (6); The corresponding total meshing force is: (7), in, , , , They represent the first i The stiffness, meshing deformation, initial error, and meshing force of the gear teeth.

[0038] Considering the deformation coordination relationship of the driving wheel, Hertzian contact, and driven wheel, we get: (8); The total transformation is the sum of three parts: (9); Substituting the deformation of each part into a unified force-stiffness expression, we get: (10) in, , The driving wheel and the driven wheel are respectively the first j Local contact stiffness of the gear teeth; Let J be the overall deformation mass of the j-th tooth; , These represent the local contact deformations of the j-th tooth of the driving and driven gears, respectively. , These represent the contact stiffness and contact deformation of the bearing system under the load of the j-th gear, respectively. is the meshing force acting on the j-th tooth.

[0039] When performing finite element modeling and compliance extraction, planar four-node plate elements are used for finite element modeling. The gear body is modeled as an elastic body, while the gear teeth are considered as rigid bodies. Fixed boundaries are applied to the inner hole nodes of the gear to simulate support conditions. A unit force is applied at different meshing positions, and the displacements of the meshing nodes are extracted to construct the compliance matrix.Figure 4 As shown.

[0040] A unit load is sequentially applied to each meshing node of the gear plate element model. At each node r, the applied load component is set to... , Its composition is the total meshing force at the nodes. The deflection vector of node r can be represented as the two-dimensional displacement response of that point under load. No load is applied to any of the nodes except node r.

[0041] Along the meshing line, the deformation response of the node can be expressed as: (11), in, Let r be the compliance column vector. , These are respectively represented as node r at... shaft and The load component of the shaft.

[0042] Within a complete meshing cycle, a unit load is applied sequentially along the contact line from the start to the end of the meshing, and the deflection at the corresponding nodes is extracted to construct the compliance matrix C of the entire gear body: (12).

[0043] Based on the aforementioned results of solving for slice stiffness and load distribution, the load force vector on each toothed plate can be obtained. Contact stress at each tooth plate Based on Hertzian contact theory, we can calculate the following: (13) in, Let J be the load borne by the j-th tooth. The width of the j-th tooth; , The radius of curvature at the contact point between the driving wheel and the driven wheel; and The Poisson ratios of the driving and driven wheels are respectively. and These are the elastic moduli of the driving wheel and the driven wheel, respectively.

[0044] To calculate the bending stress at the gear tooth root, the influence of friction is ignored, and the tooth surface load is shifted along the meshing line to the gear's axis of symmetry, then decomposed into tangential and radial components. It's important to understand that the tangential component primarily induces bending and shear stresses, while the radial component primarily induces compressive stresses. Since gear fatigue failure typically begins on the bending-tensile side, the tooth root bending strength is assessed based on the maximum tensile stress at the critical section. For the bending stress analysis of asymmetric gears, please refer to [reference needed].Figure 5 As shown.

[0045] Bending stress at the tooth root of asymmetric high overlap gears The calculation formula is as follows: (14) In the formula, Let be the load angle at the upper boundary of the single-tooth meshing zone, B be the tooth width, m be the module, and F be the load force on the meshing side. and For the tooth profile and stress correction factors of asymmetric gears: (15); (16) In the formula, The correction factor is a value between 0.7 and 0.8. The chordal tooth thickness is given by any plane section at the tooth root, in mm. The chordal tooth thickness is given by any plane section on the meshing side, in mm. The chordal tooth thickness is given by any plane section on the non-meshing side, in mm. The distance from any plane section at the root of the gear tooth to the intersection point of the load centerline is given in mm; it can be known that... , For the driving side pressure angle, The radius of curvature of the intersection point of any plane section at the root of the gear tooth and the transition curve on the meshing side is expressed in mm.

[0046] Under multi-objective, strongly nonlinear coupling, and multi-constraint conditions, a Bayesian optimization algorithm is employed to optimize the geometric parameters of asymmetric gears with high contact ratios, aiming to simultaneously minimize bending stress and contact stress while improving key performance indicators such as contact ratio. This method boasts advantages such as high convergence efficiency and high sample utilization, making it suitable for design space searches in complex gear systems.

[0047] Based on the structural characteristics of asymmetric high overlap gears, the following design parameters are selected to form an optimization parameter space, including the drive-side pressure angle. Non-driving side pressure angle Tooth tip height coefficient Tooth root fillet radius coefficient radial variable Tooth width .

[0048] Each design parameter has a set range of values ​​based on past engineering experience, forming a multi-dimensional bounded design space: (17).

[0049] In the formula, Design space for multidimensional boundedness; , This represents the value space for the nth design parameter; , These are the lower and upper limits of the first design parameter.

[0050] The objective function takes into account the gear bending strength, contact strength, and meshing performance, and can be set as a multi-objective form as follows.

[0051] Minimize maximum tooth root bending stress: (18); Minimize maximum contact stress: (19); Maximize the average overlap, then minimize it after taking the negative value: (20) in, Indicates the combination of design parameters. This indicates the meshing phase. The stress values ​​mentioned above can be calculated using the slice method, which is characterized by nonlinearity and high evaluation cost.

[0052] Bayesian optimization is used to globally optimize the objective function. First, a Gaussian Process (GP) model is used to surrogate the objective function, allowing for nonparametric modeling and uncertainty quantification of the relationship between design parameters and performance response, given a limited sample size. To improve the modeling efficiency and prediction accuracy of the surrogate model, a Markov blanket structure analysis mechanism is introduced into the design parameter space to analyze the target performance variables. (e.g., tooth root bending stress or contact stress) Construct the corresponding Bayesian network structure and identify Markov blanket sets. This allows us to obtain a subset of design parameters that have a direct or conditional dependency on the target variable. By using this structural identification result, redundant variables with weak or conditionally independent influence on the target variable can be effectively eliminated, thereby improving the computational efficiency and predictive stability of the model.

[0053] Markov blanket Represented as: (twenty one), In the formula, Let T be the parent node. A child node of T. Let T be the spouse node of T, i.e., the other parent nodes of the child nodes. The Markov blanket structure of the target variable T is illustrated as follows: Figure 6 As shown.

[0054] After determining the subset of key variables, a Gaussian process is used to model this subset of variables to obtain the predicted mean of the objective function at each design point. with standard deviation This forms the posterior probability distribution: (twenty two), in, Input the known sample points. For the corresponding objective function value, Let the covariance vector be the new point and the training samples. The covariance matrix between training samples, To observe the noise variance, It is an identity matrix.

[0055] Next, the Expected Improvement (EI) function or Upper Confidence Bound (UCB) is used as a sampling strategy to guide the selection of new sample points, balancing exploration and utilization between the mean and uncertainty.

[0056] (twenty three), In the formula, The optimal target value among the currently known samples, such as minimum stress; For new observation points The objective function value, The expected value is the average of the predicted distribution of GP.

[0057] In Bayesian optimization, to ensure the generalization and global search capabilities of the initial model, Latin Hypercube Sampling (LHS) can be used to generate an initial sample set covering the entire design parameter space. LHS ensures uniform distribution across each variable dimension, avoiding bias in the initial model.

[0058] Specifically, let the initial sample points be: (twenty four), in, This represents the true value of the objective function calculated using the slice method and evaluated by the stress analysis model. This dataset was used to train the initial Gaussian process model. .

[0059] In the proxy model Lower evaluation acquisition function , such as the expected improvement function The optimal sampling point is obtained: (25); New point A realistic target assessment yielded the following results: (26); Update dataset and model: (27).

[0060] In the design of asymmetric high overlap gears, multiple performance indicators (such as bending stress, contact stress, and overlap) often need to be considered simultaneously. The Expected Hypervolume Improvement (EHVI) multi-objective Bayesian optimization strategy is used to address these issues.

[0061] EHVI is a classic acquisition function in multi-objective Bayesian optimization, aiming to maximize hypervolume improvement based on the current Pareto front. (28) in, Indicates at the sampling point The Pareto solution set after adding the predicted values ​​relative to the reference point Supervolume increment: (29) Where P is the current Pareto set, and H is the hypervolume function. As a reference point, the direction of the inferior solution is usually chosen for all objective function values.

[0062] By maximizing EHVI, the Pareto boundary can be continuously expanded with limited sample points to obtain a better combination of gear parameters, achieving a multi-objective balance between bending strength, contact strength, and meshing performance.

[0063] To ensure that the Bayesian optimization process achieves convergence while avoiding overcomputation, multiple termination conditions can be set to determine when to complete the optimization iteration. Through a multi-condition joint termination mechanism, the number of optimization rounds and computational resource consumption are effectively limited while ensuring convergence stability, thus improving the algorithm's engineering controllability and deployment efficiency.

[0064] The first method is to set a maximum number of iteration rounds. When the number of sample iterations When the time is right, the optimization process is terminated to prevent oversampling or getting caught in local oscillations in the complex high-dimensional design parameter space.

[0065] The second method is to set a minimum boost threshold. When the improvement in the objective function value of several consecutive sampling points is less than a threshold If the condition is met, the optimization is considered to have approximately converged, and the iteration is stopped early. This condition applies to single-objective scenarios or scenarios where convergence is prioritized for a specific performance metric.

[0066] The third method is to set a minimum threshold for the hypervolume increment. When the EHVI value is less than the threshold in a certain round, it is determined that the Pareto front has stabilized and optimization stops.

[0067] The fourth method is to set an upper limit on the iteration calculation time by setting a maximum calculation time. This mechanism controls overall computing resources, terminating the optimization process if the actual computation time exceeds a certain value. It is suitable for optimization strategy control in actual engineering deployments or resource-constrained environments.

[0068] Furthermore, based on the set of candidate gear design parameters obtained from multi-objective optimization, a comprehensive evaluation model is constructed to quantitatively score and rank the structural strength and performance indicators of each design scheme, and select the optimal gear design combination that is structurally reasonable, has balanced performance, and is engineering feasible.

[0069] First, an evaluation index system is constructed, including strength indices, meshing performance indices, and structural manufacturability indices. Strength indices include the maximum bending stress at the tooth root. Maximum contact stress on tooth surface Meshing performance indicators include average axial overlap. Structural manufacturability indicators include pressure angle difference, addendum coefficient, and root fillet limit parameters. Controlling the pressure angle difference... To avoid excessive manufacturing difficulty, the tooth tip height coefficient is limited to avoid being too large, and the tooth root fillet is limited to avoid being too small.

[0070] To facilitate unified evaluation of indicators across different dimensions, a linear normalization method is used to map each indicator to... For the interval, inverse normalization is used for indices where "smaller is better" (such as bending stress): (30); For indicators that are considered "the larger the better" (such as overlap), positive normalization is used: (31); Construct a comprehensive scoring function after normalization: (32), in, This is the normalized result of the i-th "smaller is better" category indicator; This is the normalized result of the j-th "larger is better" category indicator; , The maximum values ​​of the i-th and j-th performance indicators; , Let be the minimum value of the i-th and j-th performance indicators; , These are the original values ​​of the i-th and j-th performance indicators; For comprehensive evaluation functions; The weight of the k-th indicator is... .

[0071] Furthermore, an engineering feasibility filtering rule is proposed to screen out unreasonable design solutions. Unreasonable design solutions include those where any stress index exceeds the material limit, such as... An excessively small or large pressure angle can cause interference with the cutting tool. After processing according to the above filtering rules, the remaining solutions are evaluated using a scoring function. Sort the solutions from highest to lowest quality and output the optimal solution or several alternative solutions. To enhance the interpretability of the optimization results in engineering practice, the following should also be output: the parameter configuration for each candidate solution; the corresponding bending / contact stress values ​​and overlap data.

[0072] The most suitable combination of gear parameters can be selected based on the scoring ranking and actual needs to achieve the optimal design that combines strength performance, meshing stability and manufacturing feasibility.

[0073] The embodiments of the present invention have been described in detail above with reference to the accompanying drawings. However, the present invention is not limited to the above embodiments. Within the scope of knowledge possessed by those skilled in the art, various changes can be made without departing from the spirit of the present invention.

Claims

1. A gear design optimization method based on Bayesian algorithm, used for the design optimization of asymmetric gears with high overlap ratio, characterized in that, include: Step S1: Based on the pure rolling relationship between the rack cutter and the gear pitch circle, establish a geometric mathematical model of the rack cutter tooth surface, map the geometric mathematical model of the rack cutter tooth surface to the coordinate system of the gear, solve the motion envelope of the rack cutter tooth surface, and obtain a three-dimensional geometric mathematical model of the gear tooth surface. Step S2: Introduce the tooth slicing method into the three-dimensional geometric mathematical model of the gear tooth surface, discretize the tooth into several independent slices along the tooth width direction, establish the equivalent stiffness model of the tooth slice for bending, shearing and axial deformation, and then combine the meshing partition theory to construct the load distribution model of the three-tooth meshing area and the two-tooth meshing area to obtain the normal stiffness and meshing force distribution of each tooth slice. Step S3: Based on the normal stiffness and meshing force distribution of each gear, calculate the contact stress of each gear based on Hertz contact theory, and calculate the tooth root bending stress in combination with the tooth root geometric characteristics. Based on the obtained contact stress and tooth root bending stress, form a database of stress on the gear tooth surface. Step S4: Based on the gear design parameters and the obtained database of stress on the gear tooth surface, establish a multi-objective Bayesian optimization model to collaboratively optimize the gear design parameters and output the optimized combination of gear design parameters.

2. The gear design optimization method based on Bayesian algorithm according to claim 1, characterized in that, In step S1, the rack cutter tooth surface includes a straight line segment at the tooth tip, a circular arc transition segment, and a straight line segment at the tooth root. The geometric mathematical model of the rack cutter tooth surface is established by segmented modeling, and the geometric continuity of adjacent segments is maintained when constructing the equation of the gear tooth surface through pure rolling mapping.

3. The gear design optimization method based on Bayesian algorithm according to claim 1, characterized in that, Step S2 includes: Step S2.1: Discretize the tooth into several independent slices along the tooth width direction. Each slice is divided into a three-tooth meshing area and a two-tooth meshing area in the meshing line direction. Step S2.2: In each slice, establish an equivalent stiffness model of the tooth slice including bending stiffness, shear stiffness and axial compressive stiffness, and form a stiffness matrix in the tooth direction. Step S2.3: At the boundary between the three-tooth meshing area and the two-tooth meshing area, introduce the deformation equations of adjacent tooth plates by coordinating deformation conditions, solve the coupling relationship between meshing force and deformation, and obtain the normal stiffness and load distribution of the tooth plates. Step S2.4: In the area where there is tooth-direction drum-shaped modification or tooth profile modification, the local contact offset caused by the modification is added as an additional deformation amount to the extrusion deformation to correct the local load distribution. Step S3 includes: Step S3.1: Based on the load distribution and stiffness matrix of the gear teeth, calculate the contact stress and root bending stress of the gear teeth according to Hertzian contact theory and critical section bending theory respectively, and form a database of stress on the gear tooth surface.

4. The gear design optimization method based on Bayesian algorithm according to claim 1, characterized in that, In step S4, the driving side pressure angle, non-driving side pressure angle, tooth tip height coefficient, tooth root fillet radius coefficient, radial variable, and tooth width are used as the design parameters of the gear to construct a space of multidimensional bounded design parameters, and set a multi-objective function including minimizing the maximum tooth root bending stress, minimizing the maximum contact stress, and maximizing the average overlap.

5. The gear design optimization method based on Bayesian algorithm according to claim 1, characterized in that, In step S4, a Markov blanket structure analysis mechanism is introduced to screen the gear design parameters for dependency. A Bayesian network structure is established for the target performance variables to identify a subset of parameters that have a direct or conditional dependency relationship with the target. After removing redundant gear design parameters, a Gaussian process model is used to perform surrogate modeling to obtain a surrogate model. During the Bayesian optimization process, a new sampling point is determined based on the expected hypervolume improvement acquisition function, and a termination condition is set.

6. The gear design optimization method based on Bayesian algorithm according to claim 5, characterized in that, The sampling strategy of the surrogate model adopts a multi-objective acquisition function based on the expected hypervolume improvement. The multi-objective acquisition function calculates the expected improvement value based on the predicted mean and variance of the surrogate model. After each iteration, a new sampling point is determined based on the maximum expected hypervolume. The multi-objective acquisition function is evaluated in a realistic way and the surrogate model is updated. Under the condition of limited samples, the Pareto front is continuously expanded.

7. The gear design optimization method based on Bayesian algorithm according to claim 5, characterized in that, The termination conditions include the number of iterations reaching a preset upper limit, the improvement of the objective function being lower than a minimum threshold for several consecutive iterations, the expected supervolume improvement function being less than a minimum threshold, or the optimization running time exceeding a predetermined upper limit. When any of the termination conditions is met, the iteration is terminated and a set of candidate design parameters for the gear is output.

8. The gear design optimization method based on Bayesian algorithm according to claim 5, characterized in that, After the iteration is terminated, a comprehensive evaluation model is established for the set of design parameters of the candidate gears. The comprehensive evaluation model includes strength index, meshing performance index and structural manufacturability index. Each index is linearly normalized and a weighted comprehensive scoring function is constructed to score and rank the design parameters of each candidate gear.

9. The gear design optimization method based on Bayesian algorithm according to claim 8, characterized in that, Set constraints, including material strength limit, machining tool interference, and exceeding limits for tooth tip height or fillet radius. Based on the candidate gear design parameters, filter out results that do not meet the constraints, and then sort the remaining gear design parameters from high to low scores, and output the combination of gear design parameters.

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