A gear design optimization method based on bayesian algorithm

Abstract

The application 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 the pure rolling relationship of a rack and a gear, a load stiffness coupling model is constructed by combining meshing partition and a tooth direction slicing method, contact and bending stress are obtained, and drive and non-drive side pressure angles, addendum height coefficients, dedendum fillet radius coefficients, radial variables and tooth widths are taken as design parameters. A multi-objective function of minimizing maximum dedendum bending stress, maximum contact stress and maximizing average overlapping degree is set. A Markov blanket structure is introduced to screen key variables, a Gaussian process is used to establish a proxy model, and an improved acquisition function based on an expected hypervolume is used for iterative optimization. The number of iterations, the improvement amplitude and the running time limit are taken as termination conditions. The optimal parameters are screened by combining a normalized comprehensive score. The gear design optimization method based on the Bayesian algorithm can improve the design accuracy of asymmetric high-overlapping gears and the efficiency of design parameter optimization.

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 the Bayesian algorithm has at least the following beneficial effects: a tooth direction slice modeling strategy is introduced on the basis of establishing a geometric mathematical model of the gear, a load distribution model reflecting the three-tooth and two-tooth meshing transition characteristics of the asymmetric high-coincidence gear is established, and the contact stress and the root bending stress of the gear are accurately calculated in cooperation.

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

[0008] According to some embodiments of the present application, the step S2 includes: a step S2.1 of discretizing the tooth into a plurality of independent slices in the tooth width direction, and dividing each slice into a three-tooth meshing area and a two-tooth meshing area in the meshing line direction;

[0009] A step S2.2 of establishing, in each slice, a tooth piece equivalent stiffness model including bending stiffness, shear stiffness and axial compression stiffness to form a tooth direction stiffness matrix;

[0010] A step S2.3 of introducing a coordinated deformation condition at the partition boundary of the three-tooth meshing area and the two-tooth meshing area, simultaneously solving the deformation equations of adjacent tooth pieces to obtain the coupling relationship between the meshing force and the deformation, and obtaining the normal stiffness and the load distribution of the tooth piece;

[0011] A step S2.4 of, at a position where there is tooth direction drum modification or tooth profile modification, superimposing a local contact offset caused by the modification as an additional deformation to the extrusion deformation to correct the local load distribution;

[0012] The step S3 includes: a step S3.1 of calculating the contact stress and the root bending stress of the tooth piece based on the Hertz contact theory and the dangerous cross-section bending theory, respectively, according to the load distribution and the stiffness matrix of the tooth piece, to form a stress database of the gear tooth surface.

[0013] According to some embodiments of the present application, in the step S4, the driving side pressure angle, the non-driving side pressure angle, the addendum coefficient, the dedendum radius coefficient, the radial variable and the tooth width are taken as design parameters, a multi-dimensional bounded design parameter space is constructed, multi-objective functions including minimization of the maximum root bending stress, minimization of the maximum contact stress and maximization of the average coincidence degree are set, and further, the average coincidence degree is taken as a negative value to participate in the minimization optimization.

[0014] According to some embodiments of the present application, in the step S4, a Markov blanket structure analysis mechanism is introduced to perform dependent screening on the design parameters of the gear, a Bayesian network structure is established for the target performance variable, a parameter subset having a direct or conditional dependence relationship with the target is identified, a redundant design parameter of the gear is removed, a Gaussian process model is used to perform proxy modeling to obtain a proxy model, in the Bayesian optimization process, a new sampling point is determined based on an expected hypervolume improved acquisition function, and a termination condition is set.

[0015] According to some embodiments of the present application, the sampling strategy of the proxy model adopts a multi-objective acquisition function based on expected hypervolume improvement, the multi-objective acquisition function calculates an expected improvement value according to the predicted mean and variance of the proxy model, a new sampling point is determined according to the maximum expected hypervolume after each iteration, the multi-objective acquisition function is actually evaluated and the proxy model is updated, and the Pareto front is continuously expanded under the condition of limited samples.

[0016] According to some embodiments of the present application, the termination condition includes that the number of iterations reaches a preset upper limit, the target function improvement amplitude of continuous iterations for a plurality of rounds is lower than a minimum threshold, the expected hypervolume improvement function is less than a minimum threshold, or the optimization running time exceeds a predetermined upper limit, and the iteration is terminated and the candidate set of design parameters of the gear is output when any termination condition is met.

[0017] According to some embodiments of the present application, after the iteration is terminated, a comprehensive evaluation model is established for the candidate set of design parameters of the gear, the comprehensive evaluation model includes strength indicators, meshing performance indicators and structure manufacturability indicators, each indicator is linearly normalized and a weighted comprehensive score function is constructed, and each candidate design parameter of the gear is scored and sorted.

[0018] According to some embodiments of the present application, a constraint condition is set, the constraint condition includes material strength limit, machining tool interference, addendum height or corner radius overrun, the results not meeting the constraint condition are filtered based on the candidate design parameters of the gear, the remaining design parameters of the gear are sorted in descending order of score, and the combination of the design parameters of the gear is output.

[0019] Additional aspects and advantages of the present application will be given in part in the following description, will become apparent in part from the following description, or will be learned by practice of the present application. BRIEF DESCRIPTION OF DRAWINGS

[0020] The present application will be further described below in conjunction with the drawings and embodiments, in which:

[0021] Figure 1 It is a high coincidence gear meshing partition diagram in the embodiments of the present application;

[0022] Figure 2Figure 1 is a schematic diagram of an asymmetric gear slicing method in an embodiment of the present application;

[0023] Figure 3 Figure 2 is a schematic diagram of gear normal stiffness superposition in an embodiment of the present application;

[0024] Figure 4 Figure 3 is a schematic diagram of an asymmetric gear two-dimensional plate element model in an embodiment of the present application;

[0025] Figure 5 Figure 4 is a stress analysis diagram of an asymmetric gear bending stress in an embodiment of the present application;

[0026] Figure 6 Figure 5 is a schematic diagram of a Markov blanket structure of a target variable T in an embodiment of the present application. DETAILED DESCRIPTION

[0027] Embodiments of the present application are described in detail below with reference to the accompanying drawings, in which the same or similar elements or elements having the same or similar functions are denoted by the same or similar reference numerals throughout. The embodiments described below by reference to the accompanying drawings are exemplary and are intended only for the purpose of explaining the present application, and should not be understood as limiting the present application.

[0028] In the description of the present application, it should be understood that the orientation description, such as the orientation or position relationship indicated by up, down, etc., is based on the orientation or position relationship shown in the drawings, and is only for the purpose of facilitating the description of the present application and simplifying the description, and does not indicate or imply that the device or element referred to must have a particular orientation, be constructed and operated in a particular orientation, and therefore should not be understood as limiting the present application.

[0029] In the description of the present application, the plural refers to two or more. If there is a description of first, second, etc., it is only for the purpose of distinguishing technical features, and should not be understood as indicating or implying relative importance or implicitly indicating the number of technical features indicated or the sequence of technical features indicated.

[0030] In the description of the present application, unless otherwise explicitly limited, the words such as setting, mounting, connecting, etc. should be broadly understood, and the person skilled in the art can reasonably determine the specific meaning of the above words in the present application in combination with the specific content of the technical solution.

[0031] The gear design optimization method based on the Bayesian algorithm in an embodiment of the present application comprises:

[0032] Step S1, a geometric mathematical model of the rack cutter tooth surface is established based on the pure rolling relationship between the rack cutter and the gear division circle, the geometric mathematical model of the rack cutter tooth surface is mapped to the coordinate system of the gear, the motion envelope of the rack cutter tooth surface is solved, and a three-dimensional geometric mathematical model of the gear tooth surface is obtained;

[0033] Step S2, introduce the tooth direction slicing method to 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 piece including bending, shear 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, so as to obtain the normal stiffness and meshing force distribution of each tooth piece;

[0034] Step S3, according to the normal stiffness and meshing force distribution of each tooth piece, calculate the contact stress of each tooth piece based on the Hertz contact theory, combine the root geometry characteristics to calculate the root bending stress, and form the stress database of the gear tooth surface based on the obtained contact stress and root bending stress;

[0035] Step S4, according to the design parameters of the gear and the obtained stress of the gear tooth surface, establish a multi-objective Bayesian optimization model to cooperatively optimize the design parameters of the gear, and output the combination of the optimized design parameters of the gear. The output design parameter combination meets the balance of stress and coincidence degree.

[0036] In step S1, the tooth surface of the rack cutter includes a tooth crest straight line segment, a circular arc transition segment and a tooth root straight line segment. The geometric mathematical model of the tooth surface of the rack cutter is established by using the segmented modeling method, and the geometric continuity of adjacent segments is maintained when the equation of the gear tooth surface is constructed by pure rolling mapping, so that the conjugacy and machining feasibility of the gear tooth surface can be ensured.

[0037] In order to realize the fine modeling of the meshing load and the stiffness coupling calculation of the asymmetric high-coincidence gear, step S2 includes: step S2.1, discretize the tooth into several independent slices along the tooth width direction, and divide each slice into a three-tooth meshing area and a two-tooth meshing area in the meshing line direction; step S2.2, in each slice, establish a tooth piece equivalent stiffness model including bending stiffness, shear stiffness and axial compression stiffness to form a tooth stiffness matrix; step S2.3, at the partition boundary of the three-tooth meshing area and the two-tooth meshing area, introduce the coordination deformation condition to solve the coupling relationship between the meshing force and the deformation of the adjacent tooth pieces, and obtain the normal stiffness and load distribution of the tooth piece; step S2.4, at the position where there is tooth direction drum modification or profile modification, add the local contact offset caused by the modification as an additional deformation to the extrusion deformation to correct the local load distribution; step S3 includes: step S3.1, according to the obtained load distribution and stiffness matrix of the tooth piece, calculate the contact stress and root bending stress of the tooth piece based on the Hertz contact theory and the dangerous section bending theory respectively, and form the stress database of the gear tooth surface.

[0038] In order to realize the comprehensive balance of the strength and meshing performance of the gear, in step S4, the driving side pressure angle, the non-driving side pressure angle, the addendum coefficient, the root radius coefficient, the radial variable and the tooth width are taken as design parameters to construct a multi-dimensional bounded design parameter space, a multi-objective acquisition function including the minimization of the maximum root bending stress, the minimization of the maximum contact stress and the maximization of the average overlap is set, and the average overlap is taken as a negative value for minimization optimization.

[0039] In order 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 dependency of the design parameters of the gear, a Bayesian network structure is established for the target performance variable, a parameter subset having a direct or conditional dependency with the target is identified, the redundant design parameters of the gear are removed, a Gaussian process model is used for surrogate modeling to obtain a surrogate model, in the Bayesian optimization process, a new sampling point is determined based on the expected hypervolume improvement acquisition function, and a termination condition is set. The termination condition includes that the number of iterations reaches a preset upper limit, the target function improvement amplitude is lower than a minimum threshold for a continuous number of iterations, the expected hypervolume improvement function is less than a minimum threshold, and the optimization running time exceeds a predetermined upper limit. When any of the termination conditions is met, the iteration is terminated and the candidate set of design parameters of the gear is output. The multi-condition termination mechanism is used to prevent excessive calculation and ensure the convergence stability and controllability of the optimization process.

[0040] 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 according to the prediction mean and variance of the surrogate model. After each iteration, a new sampling point is determined according to the maximum expected hypervolume, the target function is actually evaluated, and the surrogate model is updated. Under the condition of limited samples, the Pareto front is continuously expanded, and the global convergence and search efficiency of the optimization result are improved.

[0041] In order to ensure the reliability of the gear strength while considering the meshing performance and manufacturing feasibility, after the iteration is terminated, a comprehensive evaluation model is established for the candidate set of design parameters of the gear. The comprehensive evaluation model includes strength indicators, meshing performance indicators and structure manufacturability indicators. Each indicator is linearly normalized and a weighted comprehensive score function is constructed to score and sort each candidate design parameter of the gear.

[0042] Constraint conditions are set, including material strength limit, machining tool interference, addendum or corner radius overrun. Based on the candidate design parameters of the gear, the results that do not meet the constraint conditions are filtered, and the remaining candidate design parameters of the gear are sorted in descending order of score, and the combination of the design parameters of the gear is output. Further, a three-dimensional geometric model of the gear and a strength prediction result can be generated based on the combination of the design parameters of the gear, realizing the rapid parameter optimization and strength collaborative design of the asymmetric high-overlap gear.

[0043] With the modulus, the number of teeth, the driving side pressure angle, the non-driving side pressure angle, the addendum coefficient and the dedendum radius as the basic design parameters, the geometric parameter system of the gear pair is constructed, so as to ensure that the gear has high coincidence degree and transmission continuity in the transmission process. When the tooth profile of the rack cutter is established, the modeling method of segmentation can be adopted, including the addendum straight line segment, the transition arc segment and the dedendum straight line segment, which can accurately express the asymmetric tooth profile characteristics and meet the processing requirements, and provides the basic geometric conditions for the generation of the gear tooth surface. Based on the pure rolling relationship between the rack cutter and the gear division circle, the tooth surface mapping equation is established, the transmission of the cutter tooth profile in the gear coordinate system is realized, and the conjugacy and continuity of the gear tooth surface are ensured. By solving the motion envelope of the cutter tooth surface, the actual tooth profile of the gear is obtained, wherein the cutter addendum, the transition segment and the root curve form different regions of the corresponding gear tooth profile in turn, so that the complete three-dimensional geometric model is obtained. Through the foregoing steps, the gear tooth surface constructed can not only accurately reflect the asymmetric tooth profile and high coincidence degree, but also can be used as the input of meshing partition division, slice stiffness modeling and strength collaborative optimization.

[0044] In the gear transmission, in order to increase the contact performance and carrying capacity of the gear, the pressure angle of the main load side of the gear is usually increased during the design of the gear, so as to effectively thicken the dedendum area. This structural feature makes the left and right tooth profile base circle radii of the gear different, and further makes the coincidence degree of the gear in the meshing process asymmetrically distributed. The meshing coincidence degree is defined as the ratio of the distance of the tooth surface meshing point along the meshing line to the base pitch length when the driving gear rotates through a pitch angle. The high coincidence degree gear refers to the gear whose meshing coincidence degree is greater than 1.5, that is, at any moment, two or three teeth are involved in the meshing, so as to significantly improve the continuity of the tooth surface contact and the uniformity of the load transmission.

[0045] In the meshing process of the asymmetric high coincidence degree gear, if the meshing side coincidence degree satisfies , and is shown in Figure 1 , then along the meshing line direction, from the meshing starting point F to the addendum point A, the gear pair successively experiences the three-tooth meshing area and the double-tooth meshing area. The three-tooth meshing area, that is, the segment FE, contains the tooth pairs DC, BA; the double-tooth meshing area, that is, the segment ED, contains the tooth pair CB. Among them, A represents the meshing side addendum position. The lengths of the three-tooth meshing area and the double-tooth meshing area are respectively represented as functions and , and the function expressions are based on the base circle pitch , and the following is obtained:

[0046] (1).

[0047] ​The pressure angle variation range in different meshing zones can be further determined, which lays a foundation for subsequent deformation analysis and stiffness modeling.

[0048] For gear structures with helical tooth profile modification or profile modification features, traditional concentrated load or uniformly distributed load models are difficult to accurately represent the local load concentration effect. Therefore, it is preferred to introduce the coupling effect of modification geometry in the modeling process of the slice method, that is, on each slice, the local meshing gap and contact displacement are calculated in combination with the modification parameters corresponding to the position, such as the amount of drum shape. The contact displacement amount caused by modification is added to the total extrusion deformation amount. Add the total extrusion deformation amount The corrected extrusion amount is In the stiffness calculation and deformation coordination equation, the local meshing error term caused by the modification parameter is added to realize the consistent mapping of the modification geometry and the local load response.

[0049] In order to accurately analyze the gear tooth deformation behavior under multi-tooth meshing, a modeling strategy combining the slice method and the potential energy method is further adopted. First, the gear tooth is divided into N independent slices along the helical direction, as shown in Figure 2 Ignoring the coupling effect between the slices, only the local contact on the theoretical tooth surface is considered on each slice, as shown in Figure 3 Based on the potential energy method, the stiffness components are calculated, including bending, shear and axial stiffness. The stiffness expression of each slice (i.e. tooth slice) is obtained:

[0050] (2),

[0051] wherein, is the total stiffness of the nth tooth slice; , , are the bending, shear and axial stiffness of the tooth slice, respectively, and p and g denote the driving wheel and the driven wheel, respectively.

[0052] The inverse expression of the axial stiffness is:

[0053] (3);

[0054] The inverse expression of the bending stiffness is:

[0055] (4);

[0056] wherein, , .

[0057] The inverse expression of the shear stiffness is:

[0058] (5),

[0059] In the above expressions, is the elastic modulus; , is the shear effective cross-sectional area of the tooth flanks on both sides of the contact line; is the shear correction factor, usually taken as 0.83; , are the angles between the tooth profile line and the base circle when the driving wheel and the driven wheel start to mesh, respectively; , are the start angles of the driving wheel and the driven wheel tooth profile under load, respectively; , is the tooth profile variation rate in the direction of the meshing line, used for deformation integration; , is the moment of inertia of the tooth cross section relative to the direction of the meshing normal line; is the polar angular position of the tooth surface meshing point; is the horizontal coordinate projection of the contact force point on the normal action line.

[0060] The driving wheel tooth slice stiffness analysis can be uniformly represented as:

[0061] (6);

[0062] The corresponding total meshing force is:

[0063] (7),

[0064] wherein, , , , respectively represent the stiffness, meshing deformation, initial error and meshing force of the first i tooth slice.

[0065] Considering the deformation coordination relationship of the driving wheel, the Hertz contact and the driven wheel, we obtain:

[0066] (8);

[0067] The total deformation is the sum of the three parts:

[0068] (9);

[0069] Substitute the deformation of each part with the unified force-stiffness expression to obtain:

[0070] (10),

[0071] wherein, , are the local contact stiffnesses of the first j tooth slices of the driving wheel and the driven wheel, respectively. is the overall deformation quality of the jth tooth slice; , is the local contact deformation of the jth tooth slice respectively for driving wheel and driven wheel; , is the contact stiffness and contact deformation of the bearing system under the load of the jth tooth slice respectively, is the meshing force acting on the jth tooth slice.

[0072] When performing finite element modeling and flexibility extraction, a plane four-node plate element is used for finite element modeling, the wheel body part of the gear is modeled as an elastic body, and the tooth part is regarded as a rigid body. The inner hole nodes of the gear are fixedly bounded to simulate the support condition, and a unit force is loaded at different meshing positions to extract the meshing node displacement and construct the flexibility matrix. As shown in Figure 4 .

[0073] 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 as , and the sum is the total meshing force of the node . The deflection vector of node r can be expressed as the two-dimensional displacement response of the point under the action of the load. No load is applied to the remaining nodes except node r.

[0074] In the direction of the meshing line, the deformation response of the node can be expressed as:

[0075] (11),

[0076] wherein, is the flexibility column vector at node r, , respectively represent the load components at node r in the axis and axis.

[0077] In a complete meshing cycle, a unit load is sequentially applied along the contact line from the meshing start point to the end point, and the deflection of the corresponding node is extracted to construct the flexibility matrix C of the entire gear wheel body:

[0078] (12).

[0079] According to the foregoing slice stiffness and load distribution solving results, the load force vector on each tooth slice can be obtained, and the contact stress at each tooth slice can be calculated according to the Hertz contact theory, and is obtained:

[0080] (13),

[0081] wherein, Load on the jth tooth Width of the jth tooth , Curvature radius at the contact point of the driving wheel and the driven wheel and Poisson's ratio of the driving wheel and the driven wheel, respectively and Elastic modulus of the driving wheel and the driven wheel, respectively

[0082] To calculate the bending stress of the gear tooth, the effect of friction is ignored, and the tooth surface load force is moved along the meshing line to the symmetry axis of the gear, and then decomposed into tangential and radial components. It should be understood that the tangential component is mainly to cause bending stress and shear stress; the radial component is mainly to cause tooth pressure stress. Since tooth fatigue failure usually starts from the bending tensile side, the bending strength of the tooth root is evaluated according to the maximum tensile stress on the dangerous cross section. The stress analysis of the bending stress of the asymmetric gear can refer to the formula shown in Figure 5 .

[0083] Bending stress of the tooth root of the asymmetric high-coincidence gear The calculation formula is as follows:

[0084] (14),

[0085] In the formula, is the load angle of the upper boundary point of the single tooth meshing area, B is the tooth width, m is the modulus, F is the load force received on the meshing side, and are the tooth profile and stress correction coefficients of the asymmetric gear:

[0086] (15);

[0087] (16),

[0088] In the formula, is the correction coefficient, which takes a value between 0.7 and 0.8; is the chordal thickness of an arbitrary cross section of the tooth root, in mm; is the chordal thickness of an arbitrary cross section of the meshing side, in mm; is the chordal thickness of an arbitrary cross section of the non-meshing side, in mm, is the distance from the load center line intersection point of an arbitrary cross section of the tooth root, in mm; it can be known , is the pressure angle on the driving side, is the curvature radius of the intersection point of an arbitrary cross section of the tooth root and the meshing side transition curve, in mm.

[0089] Under the conditions of multi-objective, strong nonlinear coupling and multi-constraint, the geometric parameters of asymmetric high-contact-ratio gears are optimized by using the Bayesian optimization algorithm to realize the synchronous minimization of bending stress and contact stress, and to improve the key performance indicators such as contact ratio. This method has the advantages of high convergence efficiency and strong sample utilization, and is suitable for design space search of complex gear systems.

[0090] Combined with the structural characteristics of asymmetric high-contact-ratio gears, the following design parameters are selected to form the optimization parameter space, including the drive side pressure angle , the non-drive side pressure angle , the addendum coefficient , the root fillet radius coefficient , the radial variable , and the tooth width .

[0091] Each design parameter is set according to past engineering experience to form a multi-dimensional bounded design space:

[0092] (17).

[0093] wherein, is a multi-dimensional bounded design space; , is the value space of the nth design parameter; , is the lower limit and upper limit of the first design parameter.

[0094] The objective function considers the bending strength, contact strength and meshing performance of the gear, and can be set in the following multi-objective form.

[0095] Minimization of maximum root bending stress:

[0096] (18);

[0097] Minimization of maximum contact stress:

[0098] (19);

[0099] Maximization of average contact ratio, after taking the negative value, minimization:

[0100] (20),

[0101] wherein, represents the design parameter combination, represents the meshing phase. The stress values mentioned above can be calculated by the slice method, which has the characteristics of nonlinearity and high evaluation cost.

[0102] The above objective function is globally optimized using Bayesian optimization. A Gaussian Process (GP) model is used to proxy model the objective function to non-parametrically model the relationship between design parameters and performance responses and quantify uncertainty under limited sample size. To improve the modeling efficiency and prediction accuracy of the proxy model, a Markov blanket structure analysis mechanism is first introduced in the design parameter space to construct the corresponding Bayesian network structure for the target performance variable (such as the root bending stress or contact stress) and identify the Markov blanket set , thereby obtaining a subset of design parameters that have direct or conditional dependencies on the target variable. Through the structure identification result, redundant variables that have weak or conditional independent effects on the target variable can be effectively eliminated, thereby improving the computational efficiency and prediction stability of the model.

[0103] Markov blanket is represented as:

[0104] (21),

[0105] wherein, is the parent node of T, is the child node of T, is the spouse node of T, i.e., the other parent node of the child node. The Markov blanket structure of the target variable T is shown in Figure 6 .

[0106] After determining the key variable subset, a Gaussian process is used to model the variable subset to obtain the predicted mean and standard deviation of the objective function at each design point, forming the posterior probability distribution:

[0107] (22),

[0108] wherein, is the known sample point input, is the corresponding objective function value, is the covariance vector of the new point and the training samples, is the covariance matrix between the training samples, is the observation noise variance, is the identity matrix.

[0109] Next, the Expected Improvement (EI) or Upper Confidence Bound (UCB) is used as the sampling strategy to guide the selection of new sample points, balancing exploration and exploitation between the mean and uncertainty.

[0110] ​ (23),

[0111] where, is the optimal objective value in the current known samples, e.g. the minimum stress; is the objective function value of the observed new point, is the expectation (average of the predictive distribution of the GP).

[0112] In the Bayesian optimization process, to ensure the generalization ability and global search ability of the initial model, Latin Hypercube Sampling (LHS) can be used to generate an initial sample set covering the entire design parameter space. LHS can ensure uniform distribution of each variable dimension and avoid initial model bias.

[0113] Specifically, let the initial sample points be:

[0114] (24),

[0115] where, represents the true value of the objective function evaluated by the slice method load calculation and stress analysis model. This dataset is used to train the initial Gaussian process model .

[0116] Under the proxy model , the acquisition function such as the expected improvement function is evaluated to obtain the optimal sampling point:

[0117] (25);

[0118] The real target of the new point is evaluated to obtain:

[0119] (26);

[0120] Update the dataset and model:

[0121] (27).

[0122] In the design of asymmetric high contact ratio gears, multiple performance indicators (such as bending stress, contact stress, and contact ratio) often need to be considered simultaneously. The Expected Hypervolume Improvement (EHVI) multi-objective Bayesian optimization strategy is used for processing.

[0123] EHVI is a classic acquisition function in multi-objective Bayesian optimization, which aims to maximize the hypervolume improvement based on the current Pareto front:​

[0124] (28),

[0125] where, represents the hyper-volume increment of the Pareto solution set after adding the prediction value at the sampling point to the reference point .

[0126] (29),

[0127] where, P is the current Pareto set, H is the hyper-volume function, is the reference point, usually selected in the direction of inferior solution of all objective function values.

[0128] By maximizing EHVI, the Pareto boundary can be continuously expanded with limited sample points, obtaining better gear parameter combinations and achieving multi-objective balance between bending strength, contact strength and meshing performance.

[0129] To ensure the performance convergence of the Bayesian optimization process while avoiding excessive computation, multiple termination conditions can be set to determine the completion time of optimization iteration. Through the joint termination mechanism of multiple conditions, the optimization round number and the consumption of computing resources are effectively limited while ensuring the stability of convergence, improving the controllability and deployment efficiency of the algorithm.

[0130] The first is to set the maximum iteration round When the number of sample iterations , terminate the optimization process to prevent excessive sampling or falling into local oscillation in complex high-dimensional design parameter space.

[0131] The second is to set the minimum improvement amplitude threshold When the improvement amplitude of the sampling point on the objective function value is less than the threshold for several rounds in a row, it is considered that the optimization has approximately converged, and the iteration is stopped in advance. This condition is suitable for single-objective or priority convergence of a certain performance indicator.

[0132] The third is to set the minimum threshold of hyper-volume increment When the EHVI value of a certain round is less than the threshold, it is determined that the Pareto frontier has stabilized, and the optimization is stopped.

[0133] The fourth is to set the upper limit of the iteration calculation time, control the overall computing resources by setting the maximum calculation time , once the actual calculation time exceeds this value, the optimization process is terminated. This mechanism is suitable for optimization strategy control in actual engineering deployment or resource-limited environment.

[0134] Further, based on the candidate gear design parameter set obtained by multi-objective optimization, a comprehensive evaluation model is constructed to quantitatively score and sort the structural strength and performance indicators of each design scheme, and the optimal gear design combination with reasonable structure, balanced performance and engineering feasibility is selected from the candidate gear design parameter set.

[0135] Firstly, an evaluation index system is constructed, including strength indicators, meshing performance indicators and structure manufacturability indicators. The strength indicators include the maximum bending stress of the tooth root , the maximum contact stress of the tooth surface . The meshing performance indicators include the average axial coincidence degree . The structure manufacturability indicators include the pressure angle difference value, the addendum coefficient and the tooth root fillet limit parameter. The pressure angle difference value is controlled to avoid excessive manufacturing difficulty, the addendum coefficient is limited to avoid being too large, and the tooth root fillet is limited to avoid being too small.

[0136] In order to facilitate unified evaluation of indicators of different dimensions, linear normalization method is adopted to map each indicator to the interval, and reverse normalization is adopted for the "the smaller the better" type indicators (such as bending stress):

[0137] (30);

[0138] Normalization is adopted for the "the larger the better" type indicators (such as coincidence degree):

[0139] (31);

[0140] After normalization, the comprehensive score function is constructed:

[0141] (32),

[0142] wherein, is the normalized result of the i-th "the smaller the better" type indicator; is the normalized result of the j-th "the larger the better" type indicator; , is the maximum value of the i-th and j-th performance indicators; , is the minimum value of the i-th and j-th performance indicators; , is the original value of the i-th and j-th performance indicators; is the comprehensive evaluation function; is the weight of the k-th indicator, .

[0143] Further, engineering feasibility filtering rules are proposed to filter unreasonable design solutions, including any stress indicator exceeding the material limit, such as ; pressure angle too small or too large causes interference of cutting tool. After the above filtering rule processing, the remaining solutions are sorted from high to low according to the scoring function , and the optimal solution or several alternative solutions are output. To enhance the explainability of the optimization results in engineering practice, the parameter configuration of each candidate solution, the corresponding bending / contact stress value and the coincidence degree data should be output simultaneously.

[0144] The most suitable gear parameter combination can be selected according to the scoring and ranking combined with actual requirements, so as to realize the comprehensive optimal design of strength performance, meshing stability and manufacturing feasibility.

[0145] The embodiments of the application are described in detail above in combination with the drawings, but the application is not limited to the above embodiments, and various changes can be made within the knowledge range possessed by those skilled in the art without departing from the purpose of the application.

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 S2.1: The tooth slicing method is introduced into the three-dimensional geometric mathematical model of the gear tooth surface. The tooth is discretized 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: Combine the meshing partition theory to construct the load distribution model of the three-tooth meshing zone and the two-tooth meshing zone. That is, at the partition boundary of the three-tooth meshing zone and the two-tooth meshing zone, introduce the coordinated deformation condition to solve the deformation equation of the adjacent tooth plates, and then solve the coupling relationship between meshing force and deformation to 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: Based on the load distribution and stiffness matrix of the gear tooth, calculate the contact stress and tooth root bending stress of the gear tooth according to Hertzian contact theory and critical section bending theory respectively, and 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, a multi-objective Bayesian optimization model is established. Specifically, 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 for surrogate modeling to obtain a surrogate model. During the Bayesian optimization process, new sampling points are determined based on the expected hypervolume improvement acquisition function, and termination conditions are set 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, 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.

4. The gear design optimization method based on Bayesian algorithm according to claim 1, 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.

5. The gear design optimization method based on Bayesian algorithm according to claim 1, 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.

6. The gear design optimization method based on Bayesian algorithm according to claim 1, 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.

7. The gear design optimization method based on Bayesian algorithm according to claim 6, 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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