Contact-bending performance collaborative optimization design method of gear pair under consideration of spatial constraint

Abstract

The invention discloses a contact-bending performance collaborative optimization design method of a face gear pair under consideration of spatial constraint, which comprises the following steps: step 1, defining a design variable feasible region: determining a design variable optimization space of the face gear pair based on spatial constraint conditions, design variables including continuous optimization variables and discrete optimization variables; 2, a multi-objective optimization objective function is constructed, wherein the minimum tooth surface maximum contact stress and the minimum tooth root maximum bending stress serve as optimization objectives; step 3, configuring a multi-objective optimization algorithm, realizing global performance optimization of the face gear pair through collaborative optimization of spatial constraint and bearing performance, and solving to obtain optimal design parameters; and 4, designing the face gear pair based on the optimal design parameters. According to the contact-bending performance collaborative optimization design method of the face gear pair under the consideration of the spatial constraint, the global performance optimization of the face gear pair design is realized by establishing a collaborative optimization mechanism of the spatial geometric constraint and the multi-target bearing performance.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of face gear design, and particularly relates to a contact-bending performance collaborative optimization design method for a face gear pair under space constraints. BACKGROUND

[0002] Compared with bevel gear transmission, face gear transmission has a larger power-weight ratio, and the pinion gear engaged with the face gear has no axial load, so the supporting structure is simple, and meanwhile, the transmission performance of the face gear is not sensitive to the axial installation error of the pinion gear, and the face gear can achieve more stable transmission effect. These characteristics make the face gear have great potential in severe space environment and high load demand applications. The design parameters of the face gear are key factors affecting the transmission stability, load capacity and service life of the face gear, and therefore, reasonable selection of the design parameters is particularly important.

[0003] In actual engineering, the structural space constraints and the inner and outer diameter limits of the face gear bring great challenges to the determination of the design parameters. At present, there are few design methods for the face gear considering the space constraints and the contact-bending comprehensive performance, and the existing design methods for the face gear cannot take into account the space constraints and the load capacity requirements of the face gear, so that the load capacity of the face gear cannot be fully optimized under the given space boundary conditions. SUMMARY

[0004] Therefore, the purpose of the present application is to provide a contact-bending performance collaborative optimization design method for a face gear pair under space constraints, which establishes a collaborative optimization mechanism for multi-objective load capacity under space geometric constraints, and realizes the global performance optimization of the face gear pair design.

[0005] To achieve the above purpose, the present application provides the following technical scheme: A contact-bending performance collaborative optimization design method for a face gear pair under space constraints, comprising the following steps: Step 1: defining the feasible region of design variables: determining the design variable optimization space of the face gear pair based on the space constraint conditions, wherein the design variables include continuous optimization variables and discrete optimization variables; Step 2: constructing a multi-objective optimization objective function: the multi-objective optimization objective function takes the minimization of the maximum contact stress of the tooth surface and the maximum bending stress of the tooth root as the optimization objective; Step 3: configuring a multi-objective optimization algorithm, and through collaborative optimization of the contact-bending performance of the face gear pair, the global performance optimization of the face gear pair is realized, and the optimal design parameters are obtained; Step 4: designing the face gear pair based on the optimal design parameters.

[0006] Further, the continuous optimization variables include the modulus of the face gear , the inner diameter of the face gear , and the outer diameter of the face gear and the displacement coefficient ; the discrete optimization variables include pinion teeth number , face gear teeth number , and pressure angle .

[0007] Further, the spatial constraints include self-geometry constraints, actual spatial constraints, and contact position constraints; The self-geometry constraints include: , ; The actual spatial constraints include: , ; The contact position constraints include: , ; Wherein: is the minimum inner radius of the face gear to prevent undercut; is the maximum outer radius to prevent addendum undercut; is the minimum inner radius of the face gear allowed by the actual assembly space; is the maximum outer radius of the face gear allowed by the actual assembly space; is the average position of each point on the contact path on the tooth width; is the tooth width of the face gear.

[0008] Further, the multi-objective optimization objective function is represented as: Wherein: is the multi-objective optimization objective function; is the maximum contact stress function of the tooth surface; is the maximum bending stress function of the tooth root; The maximum contact stress function of the tooth surface is defined as: Wherein: represents the maximum contact stress of the tooth surface of the face gear; wherein: represents the design parameter module ; represents the inner diameter of the face gear ; represents the outer diameter of the face gear ; represents the teeth number of the face gear ; represents the pinion radius ; represents the pressure angle ; represents the displacement coefficient ; The maximum bending stress function at the tooth root is defined as: in: This represents the maximum root bending stress on the tooth surface of a face gear.

[0009] Furthermore, the equation for the contact trace on the face gear is: in: The base circle radius of the pinion; For the face gear rotation angle; The meshing angle between the gear cutter and the pinion; The angle of rotation of the small wheel; The initial angle from the center of the pinion tooth groove to the starting point of the involute; ; Number of teeth of pinion Number of teeth on the kneading gear The ratio; This is the distance between the axis of the gear shaping tool and the axis of the cylindrical gear; Let be the base circle radius of the gear shaping cutter.

[0010] Furthermore, the maximum contact stress on the tooth surface of the face gear. for: in: For normal loads; To contact the major semi-axis of the ellipse; To contact the minor semi-axis of the ellipse.

[0011] Furthermore, the bending stress at the tooth root is: in: It is the tooth profile coefficient; This is the stress correction factor; For tooth width; Modulus; This is the normal load.

[0012] Furthermore, the multi-objective optimization algorithm employs the NSGA-II algorithm, which uses an adaptive mutation operator to apply a differentiated mutation mechanism to continuous and discrete variables, including: A Gaussian perturbation is applied to a continuous variable, and the perturbation amplitude is dynamically adjusted by a scaling factor; A uniform sampling random integer and index value jumping strategy is adopted for discrete variables.

[0013] Furthermore, a scaled Gaussian perturbation is applied to the continuous variables. , represented as: in: Represents the parent generation; Represents a standard normal distribution; This is a scaling factor to ensure that the disturbance magnitude matches the dimensions of the design variables; It is a linear decay factor, and: in: Indicates the current algebra; Represents the maximum algebra; Number of teeth on the opposite gear and pinion teeth By uniformly sampling random integers within a defined interval, it can be expressed as: in: and These represent the number of teeth on the pinion. Number of teeth on the kneading gear Random integers are sampled uniformly within a defined interval; and These represent the number of teeth on the pinion. The minimum and maximum values ​​within the specified interval; and These represent the number of teeth on the face gear. The minimum and maximum values ​​within the specified interval; This indicates that integers are randomly selected with equal probability within the interval; Pressure angle The index value jumps randomly in {1,2,3}, corresponding to the standard angles {20°,22.5°,25°}.

[0014] Furthermore, a dynamic normalization method is employed for the multi-objective optimization objective function to ensure that each objective has a balanced contribution weight during the evolutionary selection process. Based on the distribution of objective function values ​​in each generation of the population, the dynamic normalization method calculates the extreme values ​​of each objective and maps the objective values ​​to a unified interval through a linear transformation, expressed as: in: Indicates the first The normalized value of each objective; Indicates the first The original function values ​​of each objective; and They represent the first and second digits in the current population, respectively. The minimum and maximum values ​​of each target.

[0015] The beneficial effects of this invention are as follows: This invention presents a design method for the coordinated optimization of the contact-bending performance of gear pairs under spatial constraints, which has the following technical advantages: (1) Overcoming the contradiction between spatial constraints and load-bearing performance: Traditional face gear design cannot take into account both geometric boundaries and load-bearing performance in a narrow installation space, which leads to an increased risk of tooth root fracture or tooth surface pitting; This invention establishes a parameter feasible domain driven by spatial constraints and constructs a contact trace position control mechanism to force the contact trace to be stable in the middle region of the tooth surface, thereby avoiding local stress concentration under a given spatial boundary. (2) Achieve global collaborative optimization of contact-bending performance: A single optimization objective (such as minimizing only the contact stress) will lead to the deterioration of the tooth root bending stress, and vice versa; This invention can simultaneously optimize the maximum contact stress on the tooth surface and the maximum bending stress at the tooth root by constructing a multi-objective optimization objective function. (3) Solving the problem of mixed variable optimization: The parameters of the face gear contain continuous and discrete variables, and traditional algorithms are prone to getting trapped in local optima. This invention designs an adaptive mutation operator to apply Gaussian perturbation to the continuous variables and integer sampling / index jump to the discrete variables. This can coordinate the optimization of spatial constraints and load-bearing performance, achieve the global performance optimization of the face gear pair, and solve for the optimal design parameters. Attached Figure Description

[0016] To make the objectives, technical solutions, and beneficial effects of this invention clearer, the following figures are provided for illustration: Figure 1 A flowchart illustrating the collaborative optimization design method for the contact-bending performance of gear pairs under spatial constraints according to the present invention; Figure 2 This is a diagram of the involute tooth profile of a gear shaping cutter; Figure 3 This is a schematic diagram showing the meshing relationship between the face gear, the gear shaper, and the pinion. Figure 4 The contact trace of the face gear; Figure 5 Contact trace diagrams at different engagement angles; Figure 6 This is a graph showing the change in the principal curvature of the tooth surface. Figure 7 A schematic diagram of the contact ellipse on the tooth surface of a face gear; Figure 8 This is a schematic diagram of the critical section at the root of a face gear. Figure 9 The flowchart for optimizing the algorithm. Detailed Implementation

[0017] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, so that those skilled in the art can better understand and implement the present invention. However, the embodiments described are not intended to limit the present invention.

[0018] like Figure 1 As shown, this embodiment considers a collaborative optimization design method for the contact-bending performance of a gear pair under spatial constraints. First, the optimization space of the design variables is determined, classifying continuous and discrete optimization variables. Second, an objective function model for meshing performance optimization is established, and the constraints are analyzed. Then, the optimization algorithm is configured to address the coexistence of continuous and discrete variables. Finally, the optimal solution for the gear pair parameters is obtained.

[0019] Specifically, such as Figure 1 As shown in the figure, this embodiment considers the contact-bending performance co-optimization design method of gear pairs under spatial constraints, including the following steps.

[0020] Step 1: Define the feasible region of design variables: Based on spatial constraints, determine the optimization space of the design variables for the face gear pair. The design variables include continuous optimization variables and discrete optimization variables. Specifically, the continuous optimization variables include the module of the face gear. inner diameter of face gear outer diameter of face gear and displacement coefficient The discrete optimization variables include the number of teeth on the pinion. Number of teeth on a face gear and pressure angle .

[0021] Within a confined engineering assembly space, the inner and outer diameters of the face gear are strictly limited by adjacent components. This spatial constraint directly determines the geometric boundary of the face gear and indirectly induces risks such as structural interference. In view of this, it is necessary to establish a spatial constraint system for the face gear: (1) Based on the inherent characteristics of the face gear, the inner diameter of the face gear is designed... It should satisfy a value greater than the minimum inner radius required to prevent root cutting. , outer diameter It should be less than the limiting outer radius to prevent the tooth tip from becoming pointed. (2) Based on the spatial constraints in practical applications, the boundary conditions of the gear dimensions are derived, and then a feasible parameter design space for engineering is constructed; (3) Based on the contact characteristics analysis, when the contact trace is close to the inner or outer diameter end, it will cause edge contact and asymmetric load distribution, resulting in local stress concentration in the tooth surface area. Therefore, the position of the contact trace should be as close as possible to the middle position of the tooth surface. That is, the spatial constraints include the geometric constraints of the device itself, the actual spatial constraints and the contact position constraints; The geometric constraints themselves include: , ; Actual spatial constraints include: , ; Contact position constraints include: , ; in: The minimum inner radius to prevent undercutting of the face gear; The maximum outer radius to prevent the tooth tip from becoming pointed; The minimum inner radius of the face gear allowed by the actual assembly space; This is the maximum outer radius of the face gear allowed by the actual assembly space. This represents the average position of each point on the contact trace along the tooth width. The tooth width of the face gear.

[0022] The aforementioned theoretical constraints are quantified into specific feasible domains of design variables, which are reflected in setting numerical ranges for each key parameter based on actual installation space limitations. The following parameter space serves as an example.

[0023] Given that the installation space for face gears in practical engineering strictly limits their inner and outer radius dimensions, this embodiment optimizes the design by treating the inner and outer radii as key constraints. Therefore, the outer radius is the design variable. The value range is [90, 110] to satisfy the external installation boundary and the inner radius. The value range is [80, 100] to avoid structural interference and dynamic balance deterioration caused by excessive tooth width; module The value range is selected as [2.5, 4.5]; number of teeth on the face gear. The range is [20, 170], and the number of pinion teeth meshing with it is... The value range is selected as [17, 40]; pressure angle The range is the common 20 degrees, 22.5 degrees, and 25 degrees; displacement coefficient The value range is [-0.8, 0.8]. Table 1 shows the upper and lower limits of the optimization parameters.

[0024] Step 2: Construct a multi-objective optimization objective function: The multi-objective optimization objective function takes minimizing the maximum contact stress on the tooth surface and the maximum bending stress at the tooth root as the optimization objectives.

[0025] By quantifying performance parameters, a multi-objective optimization objective function oriented towards practical engineering applications is established. The specific method is as follows: First, considering the structural load-bearing requirements, evaluation indicators are established with the extreme values ​​of tooth surface contact stress and tooth root bending stress as the core, providing a quantitative benchmark for subsequent optimization. Based on spatial meshing differential geometry theory, a contact trace equation for the face gear is established, and the position of the face gear contact trace line is analyzed. Using Hertzian contact theory combined with differential geometry theory, the principal curvatures of the face gear and pinion tooth surfaces are calculated, thereby deriving an analytical calculation model for the face gear contact stress. Finally, the 30° tangent method is used to find the critical section at the tooth root, constructing an analytical calculation model for the face gear tooth root bending stress. This objective function analysis framework lays the theoretical foundation for subsequent parameter optimization.

[0026] (1) Multi-objective optimization objective function Variations in parameters such as the pressure angle and module significantly affect the distribution of the contact trace on the tooth surface, potentially causing it to shift towards the inner or outer diameter end. This shift leads to asymmetrical load distribution, resulting in abnormally high local contact stress and tooth root bending stress. By systematically optimizing these parameters, the contact trace can be effectively controlled to stably distribute in the central region of the tooth surface, thereby eliminating the risk of stress concentration and achieving a comprehensive optimal solution for both contact stress and tooth root bending stress within a given parameter space.

[0027] In the design of face gear parameters, the selection of the objective function must satisfy the constraint of ensuring structural strength. Research has found that maximum contact stress dominates pitting failure on the tooth surface, while maximum tooth root bending stress is the core cause of tooth breakage. These two factors together constitute key influencing factors on transmission system performance. Therefore, minimizing maximum contact stress and maximum tooth root bending stress is used as the optimization objective for the selection and design of face gear pair parameters.

[0028] In this embodiment, the maximum contact stress function of the tooth surface is defined as: in: This represents the maximum contact stress on the tooth surface of a face gear; where: Represents the design parameter module ; Indicates the inner diameter of the face gear ; Indicates the outer diameter of the face gear ; Indicates the number of teeth on a face gear ; Represents the radius of the small wheel ; Indicates pressure angle ; Displacement coefficient .

[0029] The maximum bending stress function at the tooth root is defined as: in: This represents the maximum root bending stress on the tooth surface of a face gear.

[0030] This embodiment proposes a multi-objective optimization design method that uses the face gear parameters as design variables and the contact-bending performance of the face gear pair as the optimization objective. The multi-objective optimization objective function is expressed as follows: in: Optimize the objective function for multiple objectives; This is the function of the maximum contact stress on the tooth surface; This is the function for the maximum bending stress at the tooth root.

[0031] (2) Location of contact trace Let the tooth surface of the gear shaping tool be... The tooth surface of the pinion is The tooth surface of the face gear is When machining face gears with a gear shaper, the tooth profile of the shaper and the tooth profile of the face gear are conjugate. The tool processes the gear blank along a continuous cutting line, thereby obtaining the ideal face gear tooth profile. When the face gear meshes with the pinion, because the pinion has 1 to 3 fewer teeth than the shaper, the two gears can only contact each other in a very small area or theoretically at a single point at any given moment. This point contact moves continuously during transmission, enabling smooth and accurate power transmission.

[0032] similar Figure 2 The process of establishing the equation for the tooth surface of the gear shaper is carried out in the fixed coordinate system of the pinion. Below, the equation of the pinion tooth surface and the unit normal vector can be established: in: The base circle radius of the pinion; The initial angle from the center of the pinion tooth groove to the starting point of the involute; The angle of development corresponding to a point on the involute line; Let be the axial parameter of the pinion, where "+" and "-" represent the left and right tooth surfaces of the pinion, respectively.

[0033] The meshing relationship between the face gear, the gear shaper, and the pinion is as follows: Figure 3 As shown, during the process of shaping a gear, the tooth surface of the gear shaper cutter... Tooth surface of face gear Maintain line contact at all times. Similarly, during the simulated meshing process between the gear shaper and the cylindrical gear, its tooth surface... With the tooth surface of the cylindrical gear They are always in line contact. However, these two contact lines are not parallel and both must pass through node P.

[0034] like Figure 3 As shown, the expression for the distance B between the axis of the gear shaper and the axis of the cylindrical gear is: in: Indicates the pitch circle radius of the gear shaping cutter; This indicates the pitch circle radius of the smaller gear.

[0035] From the equation of the pinion tooth surface and the unit normal vector of the tooth surface, we can derive the coordinate system... The equation for the contact trajectory on the tooth surface of a medium cylindrical gear is: in: The meshing angle between the gear cutter and the pinion; The angle of rotation of the small wheel; Number of teeth of pinion Number of teeth on the kneading gear The ratio; This indicates the base circle radius of the gear shaping cutter.

[0036] Similar to the coordinate transformation from a gear shaper to a face gear, the contact trajectory is transformed from the pinion's motion coordinate system to the face gear's motion coordinate system. Therefore, the contact trace on the face gear can be represented as: in: The base circle radius of the pinion; The rotation angle of the face gear; The meshing angle between the gear cutter and the pinion; The angle of rotation of the small wheel; The initial angle from the center of the pinion tooth groove to the starting point of the involute; Number of teeth of pinion Number of teeth on the kneading gear The ratio; This is the distance between the axis of the gear shaping tool and the axis of the cylindrical gear.

[0037] Using the parameters shown in Table 2 as an example, a face gear tooth surface model is created using programming software, and the contact traces on it are drawn. Figure 4 As shown, the contact trace is located on the working tooth surface and is a line... The spatial curve is a parameter. Its position on the tooth surface and the meshing angle are related. It is very relevant, such as Figure 5As shown, the size of the meshing angle affects the overall position of the contact trace in the tooth width direction.

[0038] (3) Contact stress on the tooth surface of the face gear Assume the input power of the face gear pair is The angular velocity of the driving wheel is Then the input torque It can be represented as: Normal load The following is derived from the gear geometry parameters and input torque: in: Modulus; This refers to the number of teeth on the pinion. This is the pressure angle.

[0039] Inter-tooth load distribution factor This is used to correct the uneven load distribution problem in multi-tooth meshing. The load on a single tooth pair is then expressed as: The elastic moduli of the pinion and the face gear are respectively and Poisson's ratios are respectively and Its comprehensive elastic modulus It can be represented as: According to differential geometry theory, the principal curvatures of a surface can be calculated using the first and second fundamental forms. Given a parametric surface... Its principal curvature , Satisfies the quadratic equation: in: For the mean curvature, It represents Gaussian curvature.

[0040] The principal curvature solution is: The formula for calculating principal curvature can be expanded as follows: in: , , It is the first fundamental quantity of a surface. , , It is the second fundamental quantity of the surface.

[0041] From this, the principal curvatures of the pinion tooth surface and the face gear tooth surface can be obtained. , and , Taking the parameters in Table 2 as an example, the curves showing the variation of the principal curvature of the pinion and face gear with the pinion's aspect ratio θ1 are plotted, as follows. Figure 6 As shown.

[0042] Assume the surface parametric equation is The first fundamental quantity of a surface can be expressed as: The second fundamental quantity of a surface can be expressed as: in: Representing the parametric equations of the surface right The partial derivative; Representing the parametric equations of the surface right The partial derivative; Representing the parametric equations of the surface right The second partial derivative; Representing the parametric equations of the surface right and The second partial derivative; Representing the parametric equations of the surface right The second partial derivative; The normal vector is represented as: The above calculation , , , , , Substituting these values ​​into the curvature formula yields the principal curvature of the pinion tooth surface. and Similarly, the principal curvature of the tooth surface of a face gear can also be derived. and .

[0043] The principal direction vector is obtained by solving the characteristic equation of the curvature matrix: in: If the tangent direction of a curve on a surface always remains consistent with the principal direction, then this curve is called a curvature line. When selecting a curvature line as a parametric curve, the parameterization condition must satisfy... At this point, the main direction can be directly from and express.

[0044] Substituting the parameter equations of the pinion tooth surface and the face gear tooth surface into the above formula yields the principal direction vector of the pinion tooth surface. , Principal direction vector of the gear tooth surface , .

[0045] Direct the main direction of the small wheel By transforming to the face gear coordinate system, the angle between the principal curvature directions of the pinion and the face gear can be calculated using the vector dot product: Solving for the contact ellipse parameters using elliptic integrals: Introducing the eccentricity of the ellipse Solving through numerical optimization: In the formula, and These are the first and second type of complete elliptic integrals, respectively. and The calculation formula is as follows: Finally contacting the major semi-axis of the ellipse With short half shaft Determined by the following formula: in: Taking the parameters in Table 2 as an example, based on the calculated semi-major and semi-minor axis values ​​of the contact ellipse, a schematic diagram of the contact ellipse on the tooth surface of the face gear is drawn, as shown below. Figure 7 As shown.

[0046] According to Hertzian contact theory, the maximum contact stress on the tooth surface can be derived. for: in: For normal loads; To contact the major semi-axis of the ellipse; To contact the minor semi-axis of the ellipse.

[0047] (4) Bending stress at the root of the face gear As a non-standard gear, face gears currently lack a standard method for accurately calculating tooth root bending stress. This paper references the calculation method for cylindrical gear tooth root bending stress and constructs an analytical calculation model for face gear tooth root bending stress: in: It is the tooth profile coefficient; This is the stress correction factor; For tooth width; Modulus; This is the normal load.

[0048] Calculate the normal vector of the transition surface of the gear based on the principles of differential geometry. And establish the contact point search equation using the 30° tangent direction vector. For example... Figure 8 As shown, numerical optimization is used to solve for the coordinates of the tangent point. Determine the tooth thickness S at the critical section. f for: Define the lever arm for: in, y is the axial coordinate of the load application point.

[0049] Calculate the radius of curvature of the transition curve using symbolic differentiation. : in: The equation of the transition surface of the gear for the gear shaper angle The first derivative, The equation of the transition surface of the gear for the gear shaper angle The second derivative of .

[0050] The tooth form factor can be calculated based on the geometric parameters of the critical section. : in, For the force arm, The angle of application of the load. The tooth thickness is for the critical section.

[0051] Introducing load distribution factor Curvature Influence Factor The nonlinear stress correction coefficient model is constructed as follows: in: Finally, based on the parameters obtained above and the formula for maximum tooth root bending stress, the maximum bending stress at the tooth root of the face gear can be calculated.

[0052] Step 3: Configure a multi-objective optimization algorithm to achieve the global optimal performance of the face gear pair by co-optimizing spatial constraints and load-bearing performance, and solve for the optimal design parameters.

[0053] In this embodiment, the NSGA-II algorithm is selected for optimizing the meshing performance of the face gear pair. Figure 9 To establish the optimization algorithm flowchart, the NSGA-II algorithm parameters are set as follows: initial population size of 150, number of iterations of 200, crossover probability of 0.9, and tolerance of 1e-6. The algorithm employs an adaptive mutation operator for mixed-variable optimization, using a differentiated mutation mechanism to address the coexistence of continuous and discrete variables in the gear multi-objective optimization problem. By dynamically adjusting the Gaussian perturbation intensity of the continuous variables, combined with a random jump strategy for the discrete variables, the algorithm balances its global exploration and local exploitation capabilities. A linear decay factor, scale, is constructed to reduce the mutation intensity as the number of iterations increases. In this embodiment, the multi-objective optimization algorithm uses the NSGA-II algorithm. The NSGA-II algorithm employs a differentiated mutation mechanism for continuous and discrete variables through an adaptive mutation operator, including: applying Gaussian perturbation to the continuous variables, with the perturbation amplitude dynamically adjusted by a scaling factor; and using a uniform sampling random integer and index value jump strategy for the discrete variables.

[0054] Apply scaled Gaussian perturbation to continuous variables , represented as: in: Represents the parent generation; Represents a standard normal distribution; This is a scaling factor to ensure that the disturbance magnitude matches the dimensions of the design variables; It is a linear decay factor, and: in: Indicates the current algebra; This represents the maximum algebra.

[0055] Number of teeth on the opposite gear and pinion teeth By uniformly sampling random integers within a defined interval, it can be expressed as: in: and These represent the number of teeth on the pinion. Number of teeth on the kneading gear Random integers are sampled uniformly within a defined interval; and These represent the number of teeth on the pinion. The minimum and maximum values ​​within the specified interval; and These represent the number of teeth on the face gear. The minimum and maximum values ​​within the specified interval; This indicates that integers are randomly selected with equal probability within the interval.

[0056] Pressure angle The index value jumps randomly in {1,2,3}, corresponding to the standard angles {20°,22.5°,25°}.

[0057] In this algorithm, to address the search bias problem caused by differences in the dimensions or numerical ranges of the objective functions in multi-objective optimization, this embodiment employs dynamic normalization of the objective functions. By adjusting the scale of the objective functions in real time, heterogeneity among objective functions is eliminated, ensuring that each objective has a balanced contribution weight in the evolutionary selection process, thereby improving the uniformity of the Pareto front distribution and the convergence accuracy. Dynamic normalization calculates the extreme values ​​of each objective based on the distribution of objective function values ​​in each generation of the population (e.g., ...). and ), and through linear transformation, the target value is mapped to a unified interval, which is mathematically expressed as: in: Indicates the first The normalized value of each objective; Indicates the first The original function values ​​of each objective; and They represent the first and second digits in the current population, respectively. The minimum and maximum values ​​of each target.

[0058] Dynamic normalization can reduce the algorithm's dependence on the target scale sensitivity in multi-objective optimization problems, make non-dominated sorting and diversity preservation mechanisms more accurately reflect the true distribution characteristics of the solution set, alleviate the local frontier stagnation caused by the nonlinear coupling of the objective function, and enhance the algorithm's global exploration ability in complex problems.

[0059] Step 4: Design the face gear pair based on the optimal design parameters.

[0060] After multi-objective optimization and corresponding decision-making processes, the optimal parameter combination is finally obtained as shown in Table 3.

[0061] The above-described embodiments are merely preferred embodiments provided to fully illustrate the present invention, and the scope of protection of the present invention is not limited thereto. Equivalent substitutions or modifications made by those skilled in the art based on the present invention are all within the scope of protection of the present invention. The scope of protection of the present invention is defined by the claims.

Claims

1. A method for co-optimizing the contact-bending performance of gear pairs under spatial constraints, characterized in that: Includes the following steps: Step 1: Define the feasible region of design variables: Determine the optimization space of design variables for the face gear pair based on spatial constraints. The design variables include continuous optimization variables and discrete optimization variables. Step 2: Construct a multi-objective optimization objective function: The multi-objective optimization objective function takes minimizing the maximum contact stress on the tooth surface and the maximum bending stress at the tooth root as the optimization objectives; Step 3: Configure a multi-objective optimization algorithm to achieve the global optimal performance of the face gear pair by co-optimizing the contact-bending performance of the face gear pair, and solve for the optimal design parameters; Step 4: Design the face gear pair based on the optimal design parameters.

2. The method for co-optimization design of contact-bending performance of gear pairs considering spatial constraints according to claim 1, characterized in that: The continuous optimization variables include the module of the face gear. inner diameter of face gear outer diameter of face gear and displacement coefficient The discrete optimization variables include the number of teeth on the pinion. Number of teeth on a face gear and pressure angle .

3. The method for co-optimization design of contact-bending performance of gear pairs considering spatial constraints according to claim 2, characterized in that: The spatial constraints include their own geometric constraints, actual spatial constraints, and contact position constraints. The geometric constraints themselves include: , ; Actual spatial constraints include: , ; Contact position constraints include: , ; in: The minimum inner radius to prevent undercutting of the face gear; The maximum outer radius to prevent the tooth tip from becoming pointed; The minimum inner radius of the face gear allowed by the actual assembly space; This is the maximum outer radius of the face gear allowed by the actual assembly space. This represents the average position of each point on the contact trace along the tooth width. The tooth width of the face gear.

4. The method for co-optimization design of contact-bending performance of gear pairs considering spatial constraints according to claim 1, characterized in that: The multi-objective optimization objective function is expressed as follows: in: Optimize the objective function for multiple objectives; This is the function of the maximum contact stress on the tooth surface; This is the function for the maximum bending stress at the tooth root; The maximum contact stress function of the tooth surface is defined as: in: This represents the maximum contact stress on the tooth surface of a face gear; where: Indicates the design parameter modulus ; Indicates the inner diameter of the face gear ; Indicates the outer diameter of the face gear ; Indicates the number of teeth on a face gear ; Represents the radius of the small wheel ; Indicates pressure angle ; Displacement coefficient ; The maximum bending stress function at the tooth root is defined as: in: This represents the maximum root bending stress on the tooth surface of a face gear.

5. The method for co-optimization design of contact-bending performance of gear pairs considering spatial constraints according to claim 4, characterized in that: The equation of the contact trace on the face gear is: in: The base circle radius of the pinion; For the face gear rotation angle; The meshing angle between the gear cutter and the pinion; The angle of rotation of the small wheel; The initial angle from the center of the pinion tooth groove to the starting point of the involute; ; Number of teeth of pinion Number of teeth on the kneading gear The ratio; This is the distance between the axis of the gear shaping tool and the axis of the cylindrical gear; Let be the base circle radius of the gear shaping cutter.

6. The method for co-optimization design of contact-bending performance of gear pairs considering spatial constraints according to claim 4, characterized in that: Maximum contact stress on the tooth surface of a face gear for: in: For normal loads; To contact the major semi-axis of the ellipse; To contact the minor semi-axis of the ellipse.

7. The method for co-optimization design of contact-bending performance of gear pairs considering spatial constraints according to claim 4, characterized in that: The bending stress at the tooth root is: in: It is the tooth profile coefficient; This is the stress correction factor; For tooth width; Modulus; This is the normal load.

8. The method for co-optimization design of contact-bending performance of gear pairs considering spatial constraints according to claim 1, characterized in that: The multi-objective optimization algorithm employs the NSGA-II algorithm, which uses an adaptive mutation operator to apply a differentiated mutation mechanism to continuous and discrete variables, including: A Gaussian perturbation is applied to a continuous variable, and the perturbation amplitude is dynamically adjusted by a scaling factor; A uniform sampling random integer and index value jumping strategy is adopted for discrete variables.

9. The method for co-optimization design of contact-bending performance of gear pairs considering spatial constraints according to claim 8, characterized in that: Apply scaled Gaussian perturbation to continuous variables , represented as: in: Represents the parent generation; Represents a standard normal distribution; This is a scaling factor to ensure that the disturbance magnitude matches the dimensions of the design variables; It is a linear decay factor, and: in: Indicates the current algebra; Represents the maximum algebra; Number of teeth on the opposite gear and pinion teeth By uniformly sampling random integers within a defined interval, it can be expressed as: in: and These represent the number of teeth on the pinion. Number of teeth on the kneading gear Random integers are sampled uniformly within a defined interval; and These represent the number of teeth on the pinion. The minimum and maximum values ​​within the specified interval; and These represent the number of teeth on the face gear. The minimum and maximum values ​​within the specified interval; This indicates that integers are randomly selected with equal probability within the interval; Pressure angle The index value jumps randomly in {1,2,3}, corresponding to the standard angles {20°,22.5°,25°}.

10. The method for co-optimization design of contact-bending performance of gear pairs considering spatial constraints according to claim 1, characterized in that: For multi-objective optimization, a dynamic normalization method is used to ensure that each objective has a balanced contribution weight in the evolutionary selection process. The dynamic normalization method calculates the extreme values ​​of each objective based on the distribution of objective function values ​​in each generation of the population, and maps the objective values ​​to a unified interval through a linear transformation, expressed as: in: Indicates the first The normalized value of each objective; Indicates the first The original function values ​​of each objective; and They represent the first and second digits in the current population, respectively. The minimum and maximum values ​​of each target.

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