Optimization method of mesh stiffness and vibration characteristics of spiral bevel gears and verification method thereof

Abstract

This invention discloses a method for optimizing the meshing stiffness and vibration characteristics of spiral bevel gears and its verification method, comprising: 1. determining the operating parameters of the spiral bevel gear; 2. establishing a geometric model of the spiral bevel gear and calculating its geometric parameters; 3. applying multiphysics analysis to simulate the thermo-mechanical coupling effect of the spiral bevel gear; 4. calculating the maximum stress and time-varying meshing stiffness of the spiral bevel gear; 5. analyzing and identifying the key design parameters affecting the performance of the spiral bevel gear based on the vibration analysis results; 6. performing sensitivity analysis on each parameter and optimizing the spiral bevel gear design using a multi-objective optimization algorithm; 7. obtaining the optimized vibration analysis results through harmonic response analysis based on the maximum stress and time-varying meshing stiffness of the optimized spiral bevel gear geometric model. This invention has the advantages of improving meshing stiffness, extending service life, and meeting the high-performance requirements of automotive transmission systems and aerospace fields.

Description

Technical Field

[0001] This invention belongs to the field of spiral bevel gear technology, specifically a method for optimizing the meshing stiffness and vibration characteristics of spiral bevel gears and its verification method. Background Technology

[0002] With the continuous advancement of automotive transmission technology and the aerospace industry, the performance requirements for automotive transmission systems and aircraft are becoming increasingly stringent, especially in terms of improving meshing stiffness and reducing system vibration. As a key component in automotive transmission systems and aircraft, the design and manufacturing quality of spiral bevel gears directly affects the efficiency and reliability of the entire transmission system. However, existing technologies have a series of shortcomings in the design and optimization of spiral bevel gears, particularly in accurately simulating stress distribution under actual working conditions, optimizing meshing stiffness, and controlling vibration.

[0003] In existing technologies, the design of spiral bevel gears often relies on traditional empirical formulas and simplified models. These methods cannot fully consider the complex load conditions and dynamic changes encountered by gears in actual operation, resulting in significant deficiencies in meshing stiffness and vibration control. Specifically, existing technologies have significant shortcomings in the following aspects:

[0004] 1. Inaccurate prediction of stress distribution and meshing stiffness: Due to the lack of effective stress analysis techniques and calculation tools, existing methods are unable to accurately predict the stress distribution and meshing stiffness of spiral bevel gears under actual working conditions, which in turn affects the transmission efficiency and system stability of the gears.

[0005] 2. Limited system vibration control capability: Existing technologies mainly rely on passive vibration reduction measures, such as adding damping materials or changing gear geometry parameters. These measures often cannot fundamentally solve the vibration problem and may increase the system weight and complexity.

[0006] 3. Lack of consideration for special working conditions: Ensuring the performance of automotive transmission systems and aircraft under special working conditions such as extreme temperatures, high-speed rotation and varying loads is a key aspect of design. Existing technologies often neglect these special requirements, resulting in gear systems failing to meet the high standards of this field in terms of reliability and durability.

[0007] Therefore, how to provide a method for optimizing the meshing stiffness and vibration characteristics of spiral bevel gears through comprehensive stress distribution technology is a problem that urgently needs to be solved by those skilled in the art. Summary of the Invention

[0008] This invention addresses the shortcomings of existing technologies by proposing a method for optimizing the meshing stiffness and vibration characteristics of spiral bevel gears, along with its verification method. The aim is to improve the meshing accuracy and operational stability of spiral bevel gears, reduce fatigue damage caused by stress concentration, thereby extending their service life and enhancing the reliability of automotive transmission systems and aircraft performance.

[0009] To achieve the above-mentioned objectives, the present invention adopts the following technical solution:

[0010] The present invention provides a method for optimizing the meshing stiffness and vibration characteristics of spiral bevel gears, characterized by the following steps:

[0011] S1. Determine the operating parameters of the spiral bevel gear, and collect the operating parameters of the spiral bevel gear under actual working conditions by real-time monitoring of the spiral bevel gear's operation. Then, calculate the average stress σ based on the operating parameters and the operating parameters. avg and maximum stress σ max ;

[0012] S2. Based on the aforementioned operating parameters, establish a geometric model of the spiral bevel gear and calculate its geometric parameters; use finite element software to mesh the spiral bevel gear geometric model, and determine the mesh's shape factor Q and density D. mesh Evaluate the quality of the mesh to obtain the stress distribution in the key stress areas of the spiral bevel gear;

[0013] S3. Perform multiphysics analysis on the geometric model of the spiral bevel gear and simulate the thermo-mechanical coupling effect of the spiral bevel gear in the actual working process. Then, use equations (10) and (11) to calculate the temperature distribution T(x,y,z) and thermal stress distribution σ of the spiral bevel gear at different positions (x,y,z). thermal (x,y,z):

[0014]

[0015] σ thermal (x,y,z)=E·α T ·(T(x,y,z)-T ref (11)

[0016] In equations (10)-(11), T0 represents the initial temperature of the spiral bevel gear during actual operation, j represents the total number of heat sources, and q i Let f(x,y,z,i) represent the heat power density of the i-th heat source, f(x,y,z,i) represent the influence function of the i-th heat source at position (x,y,z), E represent the Young's modulus of the gear material, and α represent the heat power density of the i-th heat source. T T represents the coefficient of thermal expansion of the gear material. refThis indicates the reference temperature of the spiral bevel gear during actual operation.

[0017] S4. Calculate the maximum stress σ of the spiral bevel gear using equations (12) and (13) respectively. max And time-varying meshing stiffness K:

[0018] σ max =max (x,y,z)∈V σ(x,y,z) (12)

[0019]

[0020] In equations (12)-(13), V represents the total volume of the geometric model of the spiral bevel gear, σ(x,y,z) is the stress at point (x,y,z) within volume V, and F applied It is the external force applied to the spiral bevel gear, and δ is the amount of deformation caused by the meshing of the spiral bevel gear;

[0021] S5. Based on the stress distribution and time-varying meshing stiffness K, analyze the influence of each geometric parameter in S2 on the stress distribution and time-varying meshing stiffness K in order to evaluate the fatigue life of the spiral bevel gear and thus identify the key geometric parameters that affect the performance of the spiral bevel gear.

[0022] S6. Perform sensitivity analysis on each key geometric parameter, and optimize the key geometric parameters of the spiral bevel gear using a multi-objective optimization algorithm based on the optimization objectives shown in equations (14) and (15). The optimized key geometric parameters are obtained using equations (16)-(20), including the optimized module m. opt Optimized number of teeth z opt Optimized helix angle β opt Optimized tooth tip height h a,opt Optimized tooth width b opt And the optimized time-varying meshing stiffness K opt and the optimized maximum stress σ max,opt :

[0023]

[0024] minσ max,opt =min{σ opt (x,y,z)},(x,y,z)∈V opt (15)

[0025] m opt =m inital +Δm·λ m (16)

[0026] z opt =z inital +Δz·λz (17)

[0027] β opt =β inital +Δβ·λ β (18)

[0028]

[0029] b opt =b inital +Δb·λ b (20)

[0030] In equations (14)-(20), F applied,opt It is the external force applied to the optimized spiral bevel gear, δ opt It is the deformation amount generated by the optimized spiral bevel gear meshing, σ opt (x, y, z) represents the volume V of the optimized spiral bevel gear. opt The stress at the point at the intrinsic location (x,y,z), m inital z inital β inital h a,inital b inital The initial module, initial number of teeth, initial helix angle, initial addendum, and initial tooth width of the spiral bevel gear are represented by Δm, Δz, Δβ, and Δh, respectively. a Δb represents the changes in the module, number of teeth, helix angle, addendum, and width of the spiral bevel gear, respectively. λ m , λ z , λ β , λ ha , λ b These represent the adjustment coefficients for the module, number of teeth, helix angle, addendum, and width of the spiral bevel gear, respectively.

[0031] S7. The maximum stress σ based on the optimized spiral bevel gear geometric model. max,opt and time-varying meshing stiffness K opt The vibration characteristics of the optimized spiral bevel gear system were analyzed using harmonic response analysis.

[0032] The method for optimizing the meshing stiffness and vibration characteristics of spiral bevel gears based on comprehensive stress distribution, as described in this invention, is also characterized by the following working parameters in S1: working speed v, average load F. avg and operating temperature range T range The average contact area A of the spiral bevel gear tooth surface contact The minimum contact area A of the spiral bevel gear tooth surface min,contact ;

[0033] The operating parameters include: the rotational speed n of the spiral bevel gear under different operating conditions, and the maximum load F. max and minimum load F min and the frequency of load changes;

[0034] The mean stress σ is calculated using equations (1) and (2) respectively. avg and maximum stress σ max ;

[0035]

[0036] S2 includes:

[0037] S2.1 The geometric parameters of the spiral bevel gear include: the tip circle diameter D a Root circle diameter D f Tooth tip height h a Tooth root height h f , helix angle β, tooth width b, module m; where D is calculated using equations (3)-(7). a D f h a h f β:

[0038] D a =m·(Z+2) (3)

[0039] D f =D a -2·m·cosa (4)

[0040] h a =1.25×m (5)

[0041] h f =m+b (6)

[0042]

[0043] In equations (3)-(7), D m Z is the mean diameter of the spiral bevel gear, L is the number of teeth, b is the axial length of the tooth surface, and b is the distance between the tooth tips.

[0044] S2.2 Based on geometric parameters, the geometric model of the spiral bevel gear is meshed using finite element software to obtain the mesh.

[0045] S2.3 Calculate the density D of the divided grid using equation (8). mesh :

[0046]

[0047] In equation (8), N is the total number of elements in the mesh, and V is the total volume of the helical bevel gear geometric model;

[0048] The shape factor Q of the divided mesh is calculated using equation (9):

[0049]

[0050] In equation (9), h max h is the longest side length in the grid. min It is the minimum side length in the grid;

[0051] S2.4, If the shape factor Q and density D mesh If the mesh is within the set threshold range, it means that the mesh meets the quality requirements; otherwise, return to S2.2 to re-mesh.

[0052] S2.5 Use post-processing tools to extract stress data of key stress areas from the meshed spiral bevel gear geometric model in order to evaluate the stress distribution of key stress areas of the spiral bevel gear under different working conditions.

[0053] S7 includes:

[0054] S7.1 Using the harmonic response analysis method shown in equations (21)-(22), calculate the natural frequency ω of the optimized spiral bevel gear system. opt The damping ratio ζ of the optimized spiral bevel gear system opt :

[0055]

[0056] In equations (21)-(22), K opt It is the time-varying meshing stiffness of the optimized spiral bevel gear system, g opt It is the mass of the optimized spiral bevel gear system, c opt It is the damping coefficient of the optimized spiral bevel gear system;

[0057] S7.2, according to ω opt and ζ opt Under external excitation, the response amplitude and phase of the spiral bevel gear system, as well as the response characteristics of the spiral bevel gear system at various frequencies, are calculated using harmonic response analysis and frequency domain analysis. These results are then used as the analysis results of the vibration characteristics of the optimized spiral bevel gear system.

[0058] The verification method of this invention is characterized by verifying the analysis results of the vibration characteristics obtained by the optimization method for the meshing stiffness and vibration characteristics of the spiral bevel gear according to the following steps:

[0059] S8. Using equation (23), construct the loss function L to minimize the vibration response of the system containing the spiral bevel gear. opt After solving, the loss function L of the vibration response is obtained. opt The weighting coefficients w1, w2, and w3 of the vibration response of the spiral bevel gear system are minimized to meet the vibration characteristic requirements of the spiral bevel gear.

[0060] L opt =w1·(σ max,opt -σ target ) 2 +w2·(K opt -K target ) 2 +w3·(ζ opt -ζ target ) 2 (twenty three)

[0061] In equation (23), σ target It is the target maximum stress, K target It is the target time-varying meshing stiffness, ζ target It is the damping ratio of the target system;

[0062] S9. Taking into account the interaction of multiple physical fields, and performing multi-physics coupling analysis on the system containing the spiral bevel gear, the total force F of the system is calculated using equations (24)-(25). system Then, a system-level optimization objective function L is constructed. system :

[0063] F system =F mechanical +F thermal +F dynamic (twenty four)

[0064] L system =w1·(σ system -σ target ) 2 +w2·(K system -K target ) 2 +w3·(ζ system -ζ target ) 2 (25)

[0065] In equation (24), F mechanical F thermal and F dynamic These represent mechanical force, thermal force, and dynamic response force, respectively.

[0066] In equation (25), σ systemRepresents the system-level stress distribution, ζ system K represents the system-level damping ratio. system This represents the time-varying meshing stiffness at the system level.

[0067] S10. Perform parameter sensitivity analysis on system-level parameters and analyze the optimized system-level stress distribution σ. system Dynamic response force F dynamic And system-level damping ratio ζ system Perform calculations and verification:

[0068] S10.1, Using equation (26) to set the sensitivity function S(θ) j ):

[0069]

[0070] In equation (26), θ j This represents the j-th adjustment parameter at the system level;

[0071] S10.2, Construct the optimized system-level objective function value L using equation (27) system,opt Used to adjust parameters and observe the effect on the system-level objective function L system The impact;

[0072] L system,opt =min{L system (θ1,θ2,...,θ j ,...,θ p )}(27)

[0073] In equation (27), p represents the total number of adjustment parameters in the system;

[0074] S10.3 Calculate the system-level stress distribution σ using equations (28)-(30). system Dynamic response force F dynamic And system-level damping ratio ζ system :

[0075] σ system =max{σ(x′,y′,z′)},(x′,y′,z′)∈V system (28)

[0076] F dynamic =g system ·a system +c system ·v system +K system ·x system (29)

[0077]

[0078] In equations (28)-(30), V system Let V be the volume of the system containing the spiral bevel gear, and σ(x′,y′,z′) represent the volume V. system The stress at the point (x′, y′, z′) within the interior, g system c system K system These are the system-level mass, damping coefficient, and stiffness, a system v system x system These are system-level acceleration, velocity, and displacement;

[0079] S10.4, σ system The optimized design was compared with the material's yield strength to verify whether it met the strength requirements; F dynamic The system response under dynamic loads is compared with that under actual operating conditions to evaluate whether it is within an acceptable range. Harmonic response analysis or time-domain analysis is performed on the optimized spiral bevel gear system to verify ζ. system Effectiveness in actual operation.

[0080] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0081] 1. This invention, through the comprehensive application of stress analysis and multiphysics coupling analysis, can more accurately simulate and analyze the stress distribution of spiral bevel gears under actual working conditions, thereby optimizing the design to achieve higher time-varying meshing stiffness and better vibration characteristics. Compared with traditional design methods, this method effectively improves the transmission efficiency and dynamic stability of the spiral bevel gear system, while reducing noise and improving the overall performance and reliability of automotive transmission systems and aircraft.

[0082] 2. This invention employs a system-level optimization method, comprehensively considering the interactions of all components within the spiral bevel gear system. Through multiphysics coupling analysis, it achieves active control of the vibration of the spiral bevel gear system. This not only effectively reduces the vibration of the spiral bevel gear system during the design phase but also avoids the increased weight and design complexity that may result from traditional passive vibration reduction measures, thus meeting the stringent requirements of this field for high efficiency and lightweight design.

[0083] 3. This invention specifically considers special environments and requirements, such as extreme temperatures, high-speed rotation, and varying loads. Through targeted design and manufacturing measures, it ensures the high reliability and durability of the spiral bevel gear system. This method guarantees that the spiral bevel gear system maintains excellent performance during long-term operation and harsh working environments in automotive transmission systems or aircraft, significantly improving its safety and reliability.

[0084] 4. This invention ensures the practical applicability and effectiveness of the optimized design scheme through system-level verification and experimental testing. This method not only provides a means to comprehensively evaluate and optimize the performance of the system containing the spiral bevel gear, but also provides accurate technical guidance for subsequent manufacturing and experimental verification, greatly shortening the product development cycle and reducing R&D costs. Attached Figure Description

[0085] Figure 1 This is a flowchart of the method of the present invention;

[0086] Figure 2 The maximum stress distribution σ before and after optimization by the method of this invention. system Comparison chart;

[0087] Figure 3 The dynamic response force F before and after optimization of the method of the present invention dynamic Comparison chart;

[0088] Figure 4 The damping ratio ζ of the system before and after optimization by the method of this invention. system Comparison chart. Detailed Implementation

[0089] In this embodiment, as Figure 1 As shown: A method for optimizing the meshing stiffness and vibration characteristics of spiral bevel gears based on comprehensive stress distribution, comprising the following steps:

[0090] S1. Determine the operating parameters of the spiral bevel gear, including: operating speed v, average load F. avg and operating temperature range T range By monitoring the operation of the spiral bevel gear in real time, the operating parameters of the spiral bevel gear under actual working conditions are collected, including: the rotational speed n and the maximum load F of the spiral bevel gear under different working conditions. max and minimum load F min The load variation frequency; and thus the average stress σ is calculated using equations (1) and (2) respectively. avg and maximum stress σ max ;

[0091]

[0092] In equations (1) and (2), A contact A represents the average contact area of ​​the tooth surface of a spiral bevel gear. min,contact This represents the minimum contact area of ​​the spiral bevel gear tooth surface.

[0093] Mean stress σ avg and maximum stress σ max This represents the stress distribution of the spiral bevel gear under different loads.

[0094] At the same time, analyze the operating temperature T range The influence of different operating temperatures on the material properties of spiral bevel gears is that they can potentially affect the operating efficiency and lifespan of spiral bevel gears.

[0095] S2. Based on the operating parameters, establish a geometric model of the spiral bevel gear, and use equations (3)-(7) to calculate and measure the geometric parameters of the spiral bevel gear, including: the tip circle diameter D. a Root circle diameter D f Tooth tip height h a Tooth root height h f Helix angle β, tooth width b, module m;

[0096] D a =m·(z+2) (3)

[0097] D f =D a -2·m·cosa (4)

[0098] h a =1.25m (5)

[0099] h f =m+d (6)

[0100]

[0101] In equations (3)-(7), D m is the mean diameter of the spiral bevel gear, z is the number of teeth, L is the axial length of the tooth surface, and d is the distance between the tooth tips.

[0102] The geometric model of the spiral bevel gear was meshed using finite element software, and the mesh density D was calculated using equation (8). mesh Mesh density directly affects the accuracy of the analysis and the computation time. High-density meshes are suitable for the critical stress areas of gears. The shape factor Q of the mesh is calculated using equation (9) to evaluate and adjust the mesh density and quality, ensuring that the mesh shape is suitable for accurate analysis, so as to obtain the stress distribution in the critical stress areas of the spiral bevel gear.

[0103]

[0104] In equations (8)-(9), N is the total number of elements in the mesh, V is the total volume of the helical bevel gear geometric model, and h max h is the longest side length in the grid. min It is the minimum side length in the grid.

[0105] If the shape factor Q and density D meshIf the mesh is within the set threshold range, it means that the mesh meets the quality requirements; otherwise, the mesh needs to be adjusted by re-meshing, adjusting the mesh generation parameters, or changing the element type.

[0106] Post-processing tools were used to extract stress data from the key stress areas of the meshed spiral bevel gear geometry model in order to evaluate the stress distribution of the key stress areas of the spiral bevel gear under different working conditions.

[0107] S3. Perform multiphysics analysis on the geometric model of the spiral bevel gear, import the set boundary conditions and material properties, apply radial and axial loads, as well as torque, and simulate the thermo-mechanical coupling effect of the spiral bevel gear in the actual working process. Then, use equations (10) and (11) to calculate the temperature distribution T(x,y,z) and thermal stress distribution σ of the spiral bevel gear at different positions (x,y,z). thermal (x,y,z):

[0108]

[0109] σ thermal (x,y,z)=E·α T ·(T(x,y,z)-T ref (11)

[0110] In equations (10)-(11), T0 represents the initial temperature of the spiral bevel gear during actual operation, j represents the total number of heat sources, and q i Let f(x,y,z,i) represent the heat power density of the i-th heat source, f(x,y,z,i) represent the influence function of the i-th heat source at position (x,y,z), E represent the Young's modulus of the gear material, and α represent the heat power density of the i-th heat source. T T represents the coefficient of thermal expansion of the gear material. ref This indicates the reference temperature of the spiral bevel gear during actual operation.

[0111] S4. Calculate the maximum stress σ of the spiral bevel gear using equations (12) and (13) respectively. max And time-varying meshing stiffness K:

[0112] σ max =max (x,y,z)∈V σ(x,y,z) (12)

[0113]

[0114] In equations (12)-(13), V represents the volume of the spiral bevel gear, σ(x,y,z) is the stress at point (x,y,z) within volume V, and F applied It is the external force applied to the spiral bevel gear, and δ is the amount of deformation caused by the meshing of the spiral bevel gear;

[0115] Based on the stress analysis results, the maximum stress σ of the spiral bevel gear max To evaluate the fatigue life of the gear and compare it with the fatigue limit of the material: if the maximum stress σ of the spiral bevel gear max If the stress is less than the material's fatigue limit, it means that the spiral bevel gear can withstand the maximum cyclic stress during operation without fatigue failure.

[0116] S5. Based on the stress distribution and time-varying meshing stiffness K, analyze the influence of each geometric parameter in S2 on the stress distribution and time-varying meshing stiffness K to evaluate the fatigue life of the spiral bevel gear, thereby identifying the key geometric parameters affecting the performance of the spiral bevel gear, including: module m, number of teeth z, helix angle β, and addendum h. a Tooth width b.

[0117] S6. Perform sensitivity analysis on each key geometric parameter, and optimize the key geometric parameters of the spiral bevel gear using a multi-objective optimization algorithm based on the optimization objectives shown in equations (14) and (15). The optimized key geometric parameters are obtained using equations (16)-(20), including the optimized module m. opt Optimized number of teeth z opt Optimized helix angle β opt Optimized tooth tip height h a,opt Optimized tooth width b opt And the optimized time-varying meshing stiffness K opt and the optimized maximum stress σ max,opt :

[0118]

[0119] minσ max,opt =min{σ opt (x,y,z)},(x,y,z)∈V opt (15)

[0120] m opt =m inital +Δm·λ m (16)

[0121] z opt =z inital +Δz·λ z (17)

[0122] β opt =β inital +Δβ·λ β (18)

[0123]

[0124] b opt =b inital +Δb·λ b (20)

[0125] In equations (14)-(20), F applied,opt It is the external force applied to the optimized spiral bevel gear, δ opt It is the deformation amount generated by the optimized spiral bevel gear meshing, σ opt (x, y, z) represents the volume V of the optimized spiral bevel gear. opt The stress m at the point at the intrinsic location (x,y,z) inital z inital β inital h a,inital b inital The initial module, initial number of teeth, initial helix angle, initial addendum, and initial tooth width of the spiral bevel gear are represented by Δm, Δz, Δβ, and Δh, respectively. a Δb represents the changes in the module, number of teeth, helix angle, addendum, and width of the spiral bevel gear, respectively. λ m , λ z , λ β , λ ha , λ b These represent the adjustment coefficients for the module, number of teeth, helix angle, addendum, and width of the spiral bevel gear, respectively. The adjustment coefficients are used to determine the direction and magnitude of adjustment for each parameter.

[0126] S7. Based on the optimized key geometric parameters, stress distribution and time-varying meshing stiffness analysis are performed on the optimized spiral bevel gear geometric model to obtain vibration analysis results, which are used to verify the optimized stress distribution σ. opt and the optimized time-varying meshing stiffness K opt The effectiveness.

[0127] Based on the optimized time-varying meshing stiffness K opt Using the harmonic response analysis method shown in equations (21)-(22), the natural frequency ω of the optimized spiral bevel gear system is calculated. opt The damping ratio ζ of the optimized spiral bevel gear system opt :

[0128]

[0129] In equations (21)-(22), K opt It is the meshing stiffness of the optimized spiral bevel gear system, g opt It is the mass of the optimized spiral bevel gear system, c optIt is the damping coefficient of the optimized spiral bevel gear system.

[0130] The vibration characteristics of the optimized spiral bevel gear system were evaluated to obtain vibration analysis results:

[0131] The natural frequency ω of the optimized spiral bevel gear system, calculated according to equations (21) and (22). opt And damping ratio ζ opt Under external excitation, the response amplitude and phase of the system are calculated through harmonic response analysis and frequency domain analysis to examine the response characteristics of the system at various frequencies.

[0132] Based on the vibration analysis results, the effectiveness was verified:

[0133] The optimized stress distribution is compared with the unoptimized stress distribution to evaluate the changes in stress peak value and stress concentration area, verifying whether the expected stress reduction target has been achieved. Simultaneously, the time-varying meshing stiffness K of the spiral bevel gear system before and after optimization is compared. opt and natural frequency ω opt Check whether the design requirements are met;

[0134] Based on the above analysis results, a comprehensive evaluation is conducted on the optimized spiral bevel gear system to confirm its improvement in stress distribution, time-varying stiffness, and vibration characteristics. If the improvement meets the design requirements, it indicates the effectiveness of the optimization scheme; if the improvement does not meet expectations, further iterative optimization of the design parameters is needed.

[0135] In this embodiment, the vibration characteristic analysis results obtained are verified based on the optimization method of helical bevel gear meshing stiffness and vibration characteristics. The verification steps are as follows:

[0136] S8. Using equation (23), construct the loss function L to minimize the vibration response of the system containing the spiral bevel gear. opt After solving, the loss function L of the vibration response is obtained. opt The weighting coefficients w1, w2, and w3 of the vibration response of the spiral bevel gear system are minimized to meet the vibration characteristic requirements of the spiral bevel gear.

[0137] L opt =w1·(σ max,opt -σ target ) 2 +w2·(K opt -K target ) 2 +w3·(ζ opt -ζ target ) 2 (twenty three)

[0138] In equation (23), σ target It is the target maximum stress, K target It is the target time-varying meshing stiffness, ζ target It is the damping ratio of the target system;

[0139] S9. Taking into account the interaction of multiple physical fields, and performing multi-physics coupling analysis on the system containing the spiral bevel gear, the total force F of the system is calculated using equations (24)-(25). system Then, a system-level optimization objective function L is constructed. system :

[0140] F system =F mechanical +F thermal +F dynamic (twenty four)

[0141] L system =w1·(σ system -σ target ) 2 +w2·(K system -K target ) 2 +w3·(ζ system -ζ target ) 2 (25)

[0142] In equation (24), F mechanical F thermal and F dynamic These represent mechanical force, thermal force, and dynamic response force, respectively.

[0143] In equation (25), σ system Represents the system-level stress distribution, ζ system K represents the system-level damping ratio. system This represents the time-varying meshing stiffness at the system level.

[0144] S10. Perform parameter sensitivity analysis on system-level parameters and analyze the optimized system-level stress distribution σ. system Dynamic response force F dynamic And system-level damping ratio ζ system Perform calculations and verification:

[0145] The sensitivity function S(θ) is set using equation (26). j ):

[0146]

[0147] In equation (26), θ j This represents the j-th adjustment parameter at the system level;

[0148] The optimized system-level objective function value L is constructed using equation (27). system,opt Used to adjust parameters and observe the effect on the system-level objective function L system The impact;

[0149] L system,opt =min{L system (θ1,θ2,...,θ j ,...,θ p )} (27)

[0150] In equation (27), p represents the total number of adjustment parameters in the system;

[0151] The stress distribution σ at the system level is calculated using equations (28)-(30). system Dynamic response force F dynamic And system-level damping ratio ζ system :

[0152] σ system =max{σ(x′,y′,z′)},(x′,y′,z′)∈V system (28)

[0153] F dynamic =g system ·a system +c system ·v system +K system ·x system (29)

[0154]

[0155] In equations (28)-(30), V system Let V be the volume of the system containing the spiral bevel gear, and σ(x′,y′,z′) represent the volume V. system The stress at the point (x′, y′, z′) within the interior, g system c system K system These are the system-level mass, damping coefficient, and stiffness, a system v system x system These are system-level acceleration, velocity, and displacement.

[0156] σ system The optimized design is compared with the material's yield strength to verify whether it meets the strength requirements. If the maximum stress value is lower than the material's yield strength, the design is considered safe. dynamicThe system response under dynamic loads is compared with that under actual operating conditions to evaluate whether it is within an acceptable range. Harmonic response analysis or time-domain analysis is performed on the optimized spiral bevel gear system to verify ζ. system Its effectiveness in practical operation ensures that the system will not experience excessive vibration due to insufficient damping, nor will it reduce response efficiency due to excessive damping.

[0157] Example:

[0158] 1. Taking a pair of spiral bevel gears as an example, the larger gear has 10 teeth and the smaller gear has 41 teeth. The basic geometric parameters are shown in Table 1:

[0159] Table 1 Geometric parameters of spiral bevel gear pair

[0160]

[0161] Figure 2 The maximum stress distribution σ before and after optimization by the method of this invention. system Comparison chart, by Figure 2 We can obtain: the maximum stress distribution σ system The stress was 100 MPa before optimization and 70 MPa after optimization, with the maximum stress value reduced by 30%. This indicates that the maximum stress borne by the spiral bevel gear during meshing has been effectively reduced, which can extend the life of the spiral bevel gear, improve its transmission efficiency, reduce vibration and noise, and improve its load-bearing capacity.

[0162] Figure 3 The dynamic response force F before and after optimization of the method of the present invention dynamic Comparison chart, by Figure 3 We can obtain: Dynamic response force F dynamic The original value was 500N, which was reduced to 300N after optimization. This indicates that the system containing the spiral bevel gear responds more smoothly and the vibration amplitude is reduced when facing external dynamic loads, thus improving the reliability of the system.

[0163] Figure 4 The damping ratio ζ of the system before and after optimization by the method of this invention. system Comparison chart, by Figure 4 We can obtain: the system damping ratio ζ system The increase from 0.05 before optimization to 0.1 after optimization means that the system can absorb and dissipate vibration energy more quickly, thereby improving the stability of the system.

[0164] In summary, this method optimizes the maximum stress distribution σ of the spiral bevel gear system. system Dynamic response force F dynamic The damping ratio ζ of the system systemThe effects are quite significant. This is of great importance for improving the fatigue life and reliability of spiral bevel gears, enhancing the durability and stability of the system, and reducing maintenance costs.

Claims

1. A method of optimizing the meshing stiffness and vibration characteristics of a spiral bevel gear, characterized by, Includes the following steps: S1. Determine the operating parameters of the spiral bevel gear, and collect the operating parameters of the spiral bevel gear under actual working conditions by real-time monitoring of the spiral bevel gear's operation. Then, calculate the average stress based on the operating parameters and the operating parameters. and maximum stress ; The operating parameters in S1 include: operating speed v, average load. and operating temperature range Average contact area of ​​spiral bevel gear tooth surface Minimum contact area of ​​spiral bevel gear tooth surface ; The operating parameters include: the rotation speed n of the spiral bevel gear under different working conditions, the maximum load and the minimum load , and the load change frequency; The average stress and the maximum stress are calculated using formula (1) and formula (2), respectively and formula (2) ; (1) (2) S2. Based on the aforementioned operating parameters, establish a geometric model of the spiral bevel gear and calculate its geometric parameters; use finite element software to mesh the spiral bevel gear geometric model, and determine the mesh's shape factor Q and density... Evaluate the quality of the mesh to obtain the stress distribution in the key stress areas of the spiral bevel gear; S2 includes: S2.1 The geometric parameters of the spiral bevel gear include: the tip circle diameter. Root circle diameter Tooth tip height , tooth root height helix angle Tooth width b and module m; where, the tooth width b and module m are calculated using equations (3)-(7). , , , , : (3) (4) (5) (6) (7) in the formulae (3) to (7), is the pitch diameter of the spiral bevel gear, Z is the number of teeth, L is the axial length of the tooth surface, and b is the addendum spacing; S2.2 Based on geometric parameters, the geometric model of the spiral bevel gear is meshed using finite element software to obtain the mesh. S2.3, calculating the density of the divided grid using formula (8) : (8) In equation (8), N is the total number of elements in the mesh, and V is the total volume of the helical bevel gear geometric model; The shape factor Q of the divided mesh is calculated using equation (9): (9) In formula (9), is the maximum side length in the grid, is the minimum side length in the grid; S2.4, If the shape factor Q and density If the mesh is within the set threshold range, it means that the mesh meets the quality requirements; otherwise, return to S2.2 to re-mesh. S2.5 Use post-processing tools to extract stress data of key stress areas from the meshed spiral bevel gear geometric model in order to evaluate the stress distribution of key stress areas of the spiral bevel gear under different working conditions. S3. Perform multiphysics analysis on the geometric model of the spiral bevel gear and simulate the thermo-mechanical coupling effect of the spiral bevel gear in the actual working process. Then, use equations (10) and (11) to calculate the spiral bevel gear at different positions. Temperature distribution below and thermal stress distribution : (10) (11) In equations (10)-(11), This represents the initial temperature of the spiral bevel gear during actual operation, and j represents the total number of heat sources. This represents the heat power density of the i-th heat source. Indicates the location of the i-th heat source. The influence function at the location, where E represents the Young's modulus of the gear material. This represents the coefficient of thermal expansion of the gear material. This indicates the reference temperature of the spiral bevel gear during actual operation. S4. Calculate the maximum stress of the spiral bevel gear using equations (12) and (13) respectively. And time-varying meshing stiffness K: (12) (13) In equations (12)-(13), V represents the total volume of the geometric model of the spiral bevel gear. Position within volume V Stress at point It is the external force applied to the spiral bevel gear. It is the amount of deformation caused by the meshing of spiral bevel gears; S5. Based on the stress distribution and time-varying meshing stiffness K, analyze the influence of each geometric parameter in S2 on the stress distribution and time-varying meshing stiffness K in order to evaluate the fatigue life of the spiral bevel gear and thus identify the key geometric parameters that affect the performance of the spiral bevel gear. S6. Perform sensitivity analysis on each key geometric parameter, and optimize the key geometric parameters of the spiral bevel gear using a multi-objective optimization algorithm based on the optimization objectives shown in equations (14) and (15). The optimized key geometric parameters are obtained using equations (16)-(20), including: the optimized module. Optimized number of teeth Optimized helix angle Optimized tooth tip height Optimized tooth width and optimized time-varying meshing stiffness and the optimized maximum stress : (14) (15) (16) (17) (18) (19) (20) In equations (14)-(20), It is the external force applied to the optimized spiral bevel gear. It refers to the deformation generated by the optimized meshing of the spiral bevel gears. The volume of the optimized spiral bevel gear Inner position Stress at the point, , , , , These represent the initial module, initial number of teeth, initial helix angle, initial addendum, and initial tooth width of the spiral bevel gear, respectively. , , , , These represent the changes in the module, number of teeth, helix angle, addendum, and width of the spiral bevel gear, respectively. , , , , These represent the adjustment coefficients for the module, number of teeth, helix angle, addendum, and width of the spiral bevel gear, respectively. S7、the maximum stress of the optimized spiral bevel gear geometry model and time-varying mesh stiffness an analysis result of vibration characteristics of a system in which the optimized spiral bevel gear is located is obtained by a harmonic response analysis method. S7.1, calculating the natural frequency of the system in which the optimized spiral bevel gear is located by using the harmonic response analysis method shown in formula (21) - formula (22) and the damping ratio of the system in which the optimized spiral bevel gear is located : (21) (22) In equations (21)-(22), It is the time-varying meshing stiffness of the optimized spiral bevel gear system. It refers to the mass of the optimized spiral bevel gear system. It is the damping coefficient of the optimized spiral bevel gear system; S7.2, according to and Under external excitation, the response amplitude and phase of the spiral bevel gear system, as well as the response characteristics of the spiral bevel gear system at various frequencies, are calculated using harmonic response analysis and frequency domain analysis. These results are then used as the analysis results of the vibration characteristics of the optimized spiral bevel gear system.

2. A method of verification, characterized by, The vibration characteristic analysis results obtained by the optimization method for the meshing stiffness and vibration characteristics of the spiral bevel gear described in claim 1 are verified by following these steps: S8. Using equation (23), construct a loss function to minimize the vibration response of the system containing the spiral bevel gear. After solving, the loss function of the vibration response is obtained. Weighting coefficients of the stress distribution corresponding to minimization Weighting coefficient for time-varying meshing stiffness The weighting coefficient of the vibration response of the system containing the spiral bevel gear This satisfies the vibration characteristic requirements of spiral bevel gears; (23) In formula (23), is the target maximum stress, is the target time-varying mesh stiffness, is the target system damping ratio; S9, considering the interaction of multiple physical fields, and performing multi-physical field coupling analysis on the system where the spiral bevel gear is located, so as to calculate the total force of the system by using formula (24)-formula (25) , and further constructing a system-level optimization objective function : (24) (25) In formula (24), , and respectively represent mechanical force, thermal force and dynamic response force; In equation (25), This represents the stress distribution at the system level. This represents the system-level damping ratio. This represents the time-varying meshing stiffness at the system level; S10. Perform parameter sensitivity analysis on system-level parameters and analyze the stress distribution of the optimized system-level parameters. Dynamic response Damping ratio at the system level Perform calculations and verification: S10.1, Set the sensitivity function with formula (26) : (26) In formula (26), represents the jth adjustment parameter at the system level; S10.2 Construct the optimized system-level objective function value using equation (27) Used to adjust parameters and observe the effect on the system-level objective function The impact; (27) In equation (27), p represents the total number of adjustment parameters in the system; S10.3, Calculate stress distribution at system level using equations (28) - (30) , Dynamic response forces and damping ratios at system level : (28) (29) (30) In equations (28)-(30), Let V be the volume of the system containing the spiral bevel gear. Representing volume Inner position Stress at point , , These are the system-level mass, damping coefficient, and stiffness. , , These are system-level acceleration, velocity, and displacement; S10.4, will The optimized design was compared with the material's yield strength to verify whether it met the strength requirements; The system's response under dynamic loads is compared with that under actual operating conditions to evaluate whether it is within an acceptable range. Harmonic response analysis or time-domain analysis is performed on the optimized spiral bevel gear system to verify its performance. Effectiveness in actual operation.

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