Simulation method and device for strength of differential bevel gear

Abstract

The application relates to the field of automobile finite element simulation technology and discloses a simulation method and device for the strength of bevel gears of a differential mechanism, which comprises the following steps: dimension reduction and layered cutting are performed on a differential mechanism assembly geometric model; a finite element model distinguishing between a hardened layer and a non-hardened body is established and an elastic-plastic material attribute is distributed; a force transmission topology structure combining one-dimensional elastic units and multi-point constraint units is established to apply boundary conditions and loads; a calculation working condition matrix covering a whole working cycle is generated and solved in combination with extracted independent constraint points and discrete meshing positions; a sub-surface stress extrapolation method is used to remove contact artifacts in a stress field, and a real stress is used to calculate a static strength safety coefficient. The application truly restores the complex load state of bevel gears under surface hardening and dynamic meshing, effectively solves the problem of static safety coefficient distortion caused by local contact artifacts, improves the stress evaluation precision, and improves the simulation efficiency.

Description

Technical Field

[0001] This invention relates to the field of automotive finite element simulation technology, specifically to a simulation method and apparatus for the strength of differential bevel gears. Background Technology

[0002] The bevel gear in a differential is a crucial force transmission and differential component in an automobile. During operation, the differential housing supports the secondary driven gear, planetary gears, and axle gears, transmitting torque from the secondary driving gear to the axle gears to meet the vehicle's differential and torque transmission requirements. As a core load-bearing component of the differential assembly, efficient and accurate calculation of bevel gear strength during the product design phase can guide timely structural optimization, which is significant for saving development costs and shortening the R&D cycle.

[0003] Currently, obtaining the strength of differential bevel gears during product design mainly relies on two technical methods: experimental testing and simulation analysis. Among them, the method of conducting experiments using physical prototypes has the problems of long cycle and high manufacturing cost, and the testing stage often lags behind the early product development stage, which cannot fully meet the design requirements of rapid iteration.

[0004] To overcome the limitations of physical testing, the industry often uses finite element simulation technology to obtain strength data for bevel gears. This method involves building a finite element assembly model of the differential assembly and applying torque to calculate the stress distribution or safety factor of the structure, thereby determining whether the structure's load-bearing capacity meets the standards. However, the actual stress state of a differential is extremely complex, and existing simulation modeling methods have significant limitations in practical applications. To reduce modeling difficulty, existing simulation analyses often use simplified models, which frequently ignore the changes in the mechanical properties of bevel gears caused by surface hardening processes. Furthermore, existing models fail to fully consider the dynamic meshing position changes between different bevel gears, as well as the influence of the meshing position of the secondary driving gear and driven gear on the force transmission path of the internal bevel gears.

[0005] While some existing sophisticated simulation methods can obtain relatively accurate tooth root stress and guide tooth profile optimization, they also lack consideration for surface hardening characteristics and the comprehensive working conditions of multi-stage gears at different meshing positions. This deficiency caused by incomplete model definition makes it impossible for existing simulation methods to realistically reproduce the actual physical load state of the differential assembly, resulting in large fluctuations in the calculated differential bevel gear strength values ​​and poor overall calculation accuracy. Summary of the Invention

[0006] To address the shortcomings of existing technologies, this invention provides a simulation method and apparatus for the strength of differential bevel gears. This solves the problems in existing differential bevel gear simulation calculations, which suffer from low stress state restoration, poor stress calculation accuracy, and distorted static strength safety factor assessment due to the failure to comprehensively consider surface hardening characteristics, multi-gear dynamic meshing conditions, and the inability to eliminate contact artifacts.

[0007] To achieve the above objectives, the present invention provides the following technical solution: a simulation method for the strength of a differential bevel gear, characterized by comprising the following steps:

[0008] Geometric processing and meshing were performed on the geometric models of each component of the differential assembly to establish a finite element model of the differential assembly;

[0009] Define the material properties of each component in the finite element model and the assembly connection relationship. The material properties include linear elastic material properties and elastoplastic material properties.

[0010] Boundary conditions and force loads are applied to the assembled finite element model;

[0011] Based on the applied boundary conditions and loads, multiple calculation conditions covering the entire working cycle are defined and solved to obtain the structural stress analysis results of the differential assembly under each calculation condition.

[0012] Based on the structural stress analysis results, data processing is performed to remove contact artifacts, and the static strength safety factor of the differential assembly is calculated for strength evaluation.

[0013] Preferably, the steps of performing geometric processing and meshing on the geometric models of each component of the differential assembly to establish a finite element model of the differential assembly include:

[0014] Extract a 1 / 4 model of the differential housing and half a tooth model of the planetary gear and half-shaft gear to establish a dimensionality reduction calculation model;

[0015] Extract the edge line of the contact area at the meshing point of the planetary gear and the half-shaft gear;

[0016] A cutting line is established by setting an offset distance outward from the contact area in the normal direction of the edge of the contact area;

[0017] The geometric models of the planetary gear and the half-shaft gear are subjected to material layer cutting using the cutting lines to distinguish the outer hardened layer and the inner unhardened body, and then a mesh generation operation is performed.

[0018] Preferably, the step of defining the material properties and assembly connection relationships of each component in the finite element model includes:

[0019] The elastoplastic material properties are assigned to the outer hardened layer, the inner unhardened body, and the spool.

[0020] The nominal stress and nominal strain obtained from the tensile test are converted into real stress and real strain, and a plastic strain cutoff threshold is introduced in the strain conversion process.

[0021] When the calculated plastic strain is less than the plastic strain cutoff threshold, the plastic strain is forced to be zero.

[0022] Preferably, the step of applying boundary conditions and loads to the assembled finite element model includes:

[0023] By establishing an RBE3 unit to connect the outer ring surface and the bearing center, the front bearing and the rear bearing are fixed. The front bearing includes a front bearing inner ring, a front bearing rolling element and a front bearing outer ring, and the rear bearing includes a rear bearing inner ring, a rear bearing rolling element and a rear bearing outer ring.

[0024] An RBE3 element is established to connect the inner surface to the geometric center of the half-shaft gear, which is used to constrain the half-shaft gear.

[0025] Extract the nodes of the RBE3 element, and establish the load application path by combining the meshing nodes, RBE3 element, SPRING element, and SPRING element nodes.

[0026] Preferably, the step of applying boundary conditions and loads to the assembled finite element model further includes:

[0027] The direction of force is determined using a local coordinate system, with the meshing node as the origin and the radial, tangential, and axial directions of the gear as the coordinate axes.

[0028] Based on the input power parameters of the differential system, the circumferential force, radial force, and axial force are derived and mapped to the load application path along the determined force direction.

[0029] The torque of the half-shaft gear is applied to the corresponding node, and the preload of the bolt is applied to the preload section.

[0030] Preferably, the steps for defining multiple computational conditions covering a complete work cycle include:

[0031] Extract the complete meshing cycle of a single tooth profile, divide the meshing range of a tooth into multiple discrete meshing positions along the axis of the local coordinate system, and perform spatial geometric assembly of the finite element model according to the meshing positions to generate multiple independent initial assembly models.

[0032] Multiple equally divided meshing nodes are evenly distributed and extracted on the circumference of the secondary driven gear as independent constraint force points;

[0033] The selected independent constraint stress points are combined with the initial assembly model in a full permutation and cross-combination to generate a discretized calculation working condition matrix for the entire working cycle.

[0034] Preferably, the step of removing contact artifacts includes:

[0035] Extract the maximum Mises equivalent stress value of all solid mesh nodes in the structural stress analysis results under all calculation conditions, and construct a full-cycle stress envelope field;

[0036] Extract the spatial stress gradient of the grid nodes. When the stress gradient between adjacent grid nodes exceeds the set stress gradient threshold, the corresponding local area is determined to be a contact artifact area.

[0037] Preferably, after determining that the corresponding local area is a contact artifact region, the step of calculating the corrected true surface stress of the contact artifact region using the subsurface stress extrapolation method includes:

[0038] Along the normal direction of the component surface, extract the Mises equivalent stress values ​​of adjacent mesh nodes into the material interior;

[0039] A polynomial fitting curve was established between the extracted subsurface nodal stress values ​​and the corresponding normal depth coordinates using the least squares method.

[0040] The surface intercept value is calculated by setting the normal depth of the polynomial fitting curve to zero, and the constant term is used as the corrected true surface contact stress to replace the original non-physical stress peak value of the contact artifact area.

[0041] Preferably, the step of calculating the static strength safety factor of the differential assembly for strength evaluation includes:

[0042] The maximum structural stress of the component is obtained based on the extreme value data in the corrected full-cycle stress envelope field.

[0043] For components with the linear elastic material properties, the yield strength of the corresponding material is taken as the ultimate tensile strength; for components with the elastoplastic material properties, the tensile strength of the corresponding material is taken as the ultimate tensile strength.

[0044] The static strength safety factor is obtained by calculating the ratio of the ultimate strength of the material to the maximum stress of the structure.

[0045] Preferably, a simulation device for the strength of a differential bevel gear is characterized by comprising:

[0046] The model processing module is used to perform geometric processing on the geometric models of various components of the differential assembly and establish finite element models. It distinguishes the outer hardened layer and the inner unhardened body of the components and performs mesh generation operations independently.

[0047] The attribute and assembly module is used to define the material properties and assembly connection relationships of the finite element model output by the model processing module, distinguish between linear elastic materials and elastoplastic materials, and read contact surface information to complete system assembly.

[0048] The boundary and load application module is used to establish constraints and stress environment for the assembled finite element model, and to establish load application paths to apply torque to the half-shaft gear and preload to the bolts.

[0049] The working condition and solution module is used to generate multiple calculation working conditions covering the complete working cycle and perform solution calculations, and output the structural stress distribution data of the finite element model under the calculation working conditions;

[0050] The strength evaluation module is used to perform evaluation operations on the structural stress distribution data output by the working condition and the solution module, remove contact artifacts, extract the maximum stress of the structure to calculate the static strength safety factor, and output the strength evaluation results.

[0051] The above solution achieves the following beneficial technical effects:

[0052] This application simulates the contact state of gears at different meshing positions by geometrically layering the half-shaft gear and planetary gear and assigning them corresponding material properties. It combines the full-cycle discrete calculation working condition matrix with the application of force load in the local coordinate system. The physical characteristics of bevel gear surface hardening and the direction of reaction force at the meshing point are incorporated into the model, which realistically reproduces the stress state of the differential assembly during actual operation. This minimizes the calculation deviation between the simulation and the real physical entity and improves the calculation accuracy of structural stress.

[0053] This application effectively obtains the true surface contact stress by extracting the full-cycle stress envelope field and using the subsurface stress extrapolation method to correct the set contact artifact area. This avoids the distortion of the static strength safety factor caused by excessive local compressive stress extrema in conventional contact nonlinear calculations. It also solves the problem of blindly increasing the strength reserve coefficient in traditional design to prevent pseudo-stress. Under the premise of ensuring that the components pass the test on the first try, it effectively guides the lightweight design of the differential and reduces the manufacturing cost.

[0054] This application employs a symmetric dimensionality reduction model for basic mesh generation and utilizes the topological structure of one-dimensional Spring elements to replace the complex three-dimensional solid contact relationships between gears, thereby reducing the computational scale of the finite element model while ensuring the accuracy of mechanical transmission. This standardized modeling process not only reduces human preprocessing time and subjective judgment bias, but also allows the constructed finite element model to be directly reused for stiffness and durability calculations of components such as differential housings and planetary gears, avoiding redundant modeling, saving computational cycles, and accelerating product design iteration. Attached Figure Description

[0055] Figure 1 This is a schematic diagram of the differential assembly structure;

[0056] Figure 2 This is a schematic diagram of the geometric cutting of a half-shaft gear;

[0057] Figure 3 This is a schematic diagram of the geometric cutting of the meshing contact area of ​​the half-shaft gear;

[0058] Figure 4 This is a schematic diagram of a 1 / 4 scale model of the differential housing;

[0059] Figure 5 This is a schematic diagram of half a tooth of a half-shaft gear.

[0060] Figure 6 This is a schematic diagram of the mesh division of the transition area of ​​the half-shaft gear;

[0061] Figure 7 This is a schematic diagram of different meshing assembly positions of the half-shaft gear;

[0062] Figure 8 This is a schematic diagram of a bearing;

[0063] Figure 9 This is a schematic diagram of the RBE3 unit built on the outer ring of the rear bearing;

[0064] Figure 10 This is a schematic diagram of an RBE3 unit built on the inner surface of the half-shaft gear;

[0065] Figure 11 This is a schematic diagram of establishing SPRING and RBE3 units on the secondary reduction driven gear;

[0066] Figure 12 This is a schematic diagram showing the distribution of the gear meshing force at 36 loading points;

[0067] Figure 13 This is a flowchart illustrating a simulation method for the strength of a differential bevel gear.

[0068] The components are as follows: 1. Differential housing; 2. Secondary driven gear; 3. Planetary gear; 4. Half-shaft gear; 5. Slotted shaft; 6. Front bearing; 7. Rear bearing; 8. Bolt; 9. Local coordinate system; 11. Outer hardened layer; 12. Inner unhardened body; 13. Contact area edge line; 14. Cutting line; 15. 1 / 4 model; 16. Half-tooth model; 21. Meshing position; 31. Front bearing inner ring; 32. Front bearing rolling element; 33. Front bearing outer ring; 34. Rear bearing inner ring; 35. Rear bearing rolling element; 36. Rear bearing outer ring; 41. First RBE3 element; 42. Outer ring surface; 43. Bearing center; 44. Second RBE3 element; 45. Inner surface; 46. Geometric center of half-shaft gear; 47. RBE3 element node; 51. Meshing node; 52. Third RBE3 element; 53. SPRING element; 54. SPRING element node. Detailed Implementation

[0069] The technical solution of the present invention will now be clearly and completely described with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0070] See attached document Figure 1 To be continued Figure 12 This invention provides a simulation device for the strength of differential bevel gears, applicable to applications such as... Figure 1 The differential assembly shown is an example. The simulation device includes a model processing module, a property and assembly module, a boundary and load application module, a working condition and solution module, and a strength evaluation module.

[0071] The model processing module is used to perform geometric processing on the geometric models of each component of the differential assembly and establish finite element models. This module performs structural cutting operations on planetary gear 3 and half-shaft gear 4, distinguishing the outer hardened layer 11 and the inner non-hardened body 12, and establishing cutting lines 14 based on the contact area edge lines 13; further, this module extracts the 1 / 4 model 15 and half-tooth model 16 for dimensionality reduction calculations, and independently performs mesh generation operations on the differential housing 1, the second-stage driven gear 2, planetary gear 3, half-shaft gear 4, slotted shaft 5, front bearing 6 and its included front bearing inner ring 31, front bearing rolling elements 32 and front bearing outer ring 33, rear bearing 7 and its included rear bearing inner ring 34, rear bearing rolling elements 35 and rear bearing outer ring 36, and bolt 8.

[0072] The Properties and Assembly module defines the material properties and assembly connections of the finite element model output by the Model Processing module. This module distinguishes between linear elastic materials and elastoplastic materials, assigns elastic modulus, Poisson's ratio, and actual stress-strain data to the outer hardened layer 11, the inner unhardened body 12, and the mesh models of each component, and reads contact surface information to define the contact relationships and common node connections between components, thus completing the system assembly of the finite element model.

[0073] The boundary and load application module is used to establish constraints and stress environment for the assembled finite element model. This module establishes the first RBE3 element 41 to connect the outer ring surface 42 and the bearing center 43 to fix the front bearing 6 and the rear bearing 7; establishes the second RBE3 element 44 to connect the inner surface 45 and the geometric center 46 of the half-shaft gear to constrain the half-shaft gear 4; extracts the RBE3 element node 47, uses the local coordinate system 9 to determine the force direction, and establishes the load application path by combining the meshing node 51, the third RBE3 element 52, the SPRING element 53 and the SPRING element node 54, and applies the torque of the half-shaft gear 4 and the preload of the bolt 8 to the corresponding nodes.

[0074] The working condition and solution module is used to generate simulation matrices and perform solution calculations. This module generates independent calculation working conditions based on different meshing positions 21 of the secondary driven gear 2, and outputs stress distribution data of the finite element model under the calculation working conditions.

[0075] The strength evaluation module is used to perform evaluation operations on the stress distribution data output by the working condition and solution modules. This module extracts the maximum structural stress at a specified node, calculates the static strength safety factor, and outputs the strength evaluation results.

[0076] Corresponding to the above simulation device, refer to the appendix. Figure 13 This invention also provides a simulation method for the strength of a differential bevel gear. This simulation method is executed by the aforementioned simulation device and includes the following steps:

[0077] S1. Perform geometric processing and mesh generation on the geometric models of each component of the differential assembly to establish a finite element model of the differential assembly. Perform structural cutting on the planetary gear 3 and half-shaft gear 4 in the differential assembly, and perform mesh generation on the differential housing 1, secondary driven gear 2, planetary gear 3, half-shaft gear 4, slotted shaft 5, front bearing 6, rear bearing 7 and bolt 8.

[0078] S2 defines the material properties and assembly connection relationships of each component in the established finite element model. It assigns elastic modulus, Poisson's ratio, and actual stress-strain data to each component's finite element model, and defines the contact relationships and common node connections between contacting parts, thus completing the assembly of the finite element model.

[0079] S3. Apply boundary conditions and loads to the assembled finite element model. Fix the front bearing 6, rear bearing 7, half-shaft gear 4, and secondary driven gear 2 in the finite element model, and apply the torque of half-shaft gear 4 and the preload of bolt 8 to the corresponding nodes.

[0080] S4 defines the calculation conditions and performs the calculation based on the applied boundary conditions and loads. Multiple calculation conditions are defined according to different constraint points of the second-stage driven gear 2, and the structural stress analysis results of the differential assembly under each calculation condition are calculated and obtained.

[0081] S5. Based on the structural stress analysis results, calculate and evaluate the static strength safety factor of the differential assembly. Extract the maximum structural stress at a specified node of the differential assembly, calculate the static strength safety factor of the differential bevel gear, and evaluate the strength requirements of planetary gear 3 and half-shaft gear 4 based on this static strength safety factor.

[0082] See attached document Figure 1 Appendix Figure 2 and attached Figure 3 The geometric model of each component of the differential assembly is geometrically processed and meshed to establish a finite element model of the differential assembly, which includes the following steps.

[0083] Specifically, the geometric models of planetary gear 3 and half-shaft gear 4 undergo material layering and cutting. Differential bevel gears typically employ surface hardening processes in actual manufacturing, resulting in differences in the mechanical properties of the gear surface and internal materials. To reflect this physical characteristic in the finite element model, this embodiment, based on preset gear surface hardening layer depth parameters, cuts the solid geometric models of planetary gear 3 and half-shaft gear 4 into an outer hardened layer 11 and an inner unhardened body 12. Through geometric cutting operations, a single solid is split into two independent geometric domains, which are then used to subsequently assign corresponding depths of elastoplastic material properties.

[0084] When performing material layer cutting, a cutting line 14 with an offset distance is established based on the contact area edge 13. During the establishment of the cutting line 14, the contact area edge 13 at the meshing point of the planetary gear 3 and the half-shaft gear 4 is extracted. Since the contact area involves nonlinear contact calculations in subsequent solutions, if the cutting line 14 directly coincides with the contact area edge 13, node adhesion or mesh distortion may easily occur during mesh node processing. Therefore, an offset distance is set on the outer extension surface outside the contact area in the normal direction of the contact area edge 13 to establish the offset cutting line 14, which serves as the interface between the outer hardened layer 11 and the inner unhardened body 12. Setting the offset distance eliminates the spatial overlap between the material interface and the physical contact surface. In a specific embodiment, the offset distance is set to 0.2 mm. The principle behind determining this offset distance value is that the contact stress concentration area under load is concentrated in a very small local area, and an offset distance of 0.2 mm can move the abrupt boundary of material properties away from the contact area where the stress gradient changes most drastically. The offset distance is usually taken as one-fifth to one-half of the basic size of the contact surface mesh to ensure numerical stability during finite element iterative calculations.

[0085] Furthermore, a reduced-dimensional geometric model is extracted based on the structural symmetry of the differential assembly components. The differential housing 1 exhibits a uniformly symmetrical structure in the circumferential direction, and a quarter model 15, representing one-quarter of the overall geometric model, is extracted as the basic computational unit. The teeth of the planetary gear 3 and the half-shaft gear 4 are periodically symmetrically distributed in the circumferential direction, and a half-tooth model 16, representing one side of the symmetry plane of a single tooth, is extracted as the basic computational model. By extracting the quarter model 15 and the half-tooth model 16, periodically symmetrical boundary conditions are used to replace the overall solid model, reducing the node size of the finite element model while preserving the structural geometric features.

[0086] See attached document Figure 5 and attached Figure 6The geometric models of each component after geometric processing are meshed. In this embodiment, the meshing objects include the dimensionally reduced differential housing 1, the secondary driven gear 2, the planetary gear 3, the half-shaft gear 4, as well as the slotted shaft 5, the front bearing 6, the rear bearing 7, and the bolts 8. Hexahedral mesh elements are used for discretization of the outer hardened layer 11, the inner unhardened body 12, and the stress concentration area at the tooth root. Second-order tetrahedral mesh elements are used for discretization of irregular geometric areas such as the differential housing 1. A transition envelope mesh is established at the boundary between the hexahedral mesh and the second-order tetrahedral mesh. By setting common node conditions or establishing binding constraints at the boundary of the envelope mesh, the transfer of physical quantities between mesh nodes of different shapes is realized. Furthermore, the front bearing 6 consists of the inner ring 31, the rolling elements 32, and the outer ring 33, and the rear bearing 7 consists of the inner ring 34, the rolling elements 35, and the outer ring 36. Solid meshing is performed independently on each internal component of the front bearing 6 and the rear bearing 7. After the mesh generation operation is completed, the finite element model of the differential assembly is output. The established finite element model is used as the basis model for subsequent attribute definition and assembly.

[0087] See attached document Figure 7 After establishing the finite element model, the material properties and assembly connection relationships of each component are defined, which includes the following steps.

[0088] Specifically, linear elastic and elastoplastic material properties are distinguished and assigned to the finite element model. Planetary gear 3 and half-shaft gear 4 experience extremely high contact stress during operation, and may locally enter a plastic yielding state. Therefore, elastoplastic material properties are assigned to the outer hardened layer 11, the inner unhardened body 12, and the slotted shaft 5 of planetary gear 3 and half-shaft gear 4. To accurately reflect the physical characteristics at different depths, material parameters with different yield strengths are input to the outer hardened layer 11 and the inner unhardened body 12 in this embodiment. Furthermore, the differential housing 1, the secondary driven gear 2, the front bearing 6, the rear bearing 7, and the bolts 8 exhibit minimal deformation during the operating cycle; therefore, linear elastic material properties are assigned to these components, defining only the elastic modulus and Poisson's ratio parameters.

[0089] Furthermore, for components endowed with elastoplastic material properties, a data conversion from nominal engineering stress and strain to actual physical stress and strain is performed. The basic data obtained from tensile testing are nominal stress and nominal strain, calculated based on the initial cross-sectional area of ​​the specimen. Under stress conditions where the structure undergoes significant plastic deformation, the cross-sectional area of ​​the material shrinks, and the nominal stress cannot reflect the true stress level within the material. To meet the constitutive input requirements of the Mises yield criterion in the finite element method software, it is necessary to convert the data to actual stress and actual strain. In this embodiment, the data conversion is performed using the following formula:

[0090] ;

[0091] ;

[0092] ;

[0093] ;

[0094] In the formula, For actual stress, To respond realistically, For nominal stress, In order to respond nominally, For plastic strain, For elastic strain, It is the elastic modulus.

[0095] A plastic strain cutoff threshold is introduced during the strain transformation process to ensure the convergence of nonlinear calculations. The plastic strain data calculated by the formula often exhibits extremely small numerical fluctuations near the yield point, causing singularities in the finite element solver when calculating the tangent stiffness matrix. In this embodiment, the plastic strain cutoff threshold is set to 1×10⁻⁶. -5 When the calculated plastic strain Less than 1×10 -5 At that time, a mandatory order The corresponding stress stage is treated as a purely elastic stage. Typically, considering the truncation error of the finite element solver's double-precision floating-point calculation, the truncation threshold is set within a range of 10. -6 Up to 10 -4 By filtering out minute plastic strain fluctuations through a truncation threshold, the solution efficiency of multi-contact nonlinear models is improved without affecting the overall strength calculation accuracy.

[0096] After defining the material properties, the contact relationships and common node connections between the various components of the differential assembly are set. In the finite element model, the tooth surface contact pairs between planetary gear 3 and half-shaft gear 4, the cylindrical surface contact pairs between the inner hole of planetary gear 3 and the surface of the slotted shaft 5, and the contact pairs between the two ends of the slotted shaft 5 and the mounting holes of the differential housing 1 are established. The contact behavior adopts a face-to-face penalty function contact algorithm, and the static friction coefficient and dynamic friction coefficient in the range of 0.05 to 0.15 are input according to the physical state of the steel components under actual lubrication conditions. For fixed connection parts such as bolt 8 and differential housing 1, binding constraints or common node connections are established.

[0097] A multi-state static assembly model covering the dynamic meshing cycle is constructed. During operation, the differential bevel gear exhibits a continuous dynamic meshing state, with tooth surface contact stiffness and load distribution changing in real time with the rotation angle. To utilize static implicit calculations to cover the dynamic process, this embodiment extracts the complete meshing cycle of a single tooth profile, dividing the meshing range of a tooth into six discrete meshing positions 21 along the 9Z axis of the local coordinate system. The principle of dividing a single tooth pitch into six equal parts is that the end face overlap of common differential bevel gears is typically between 1.0 and 2.0. Six discrete positions can cover key stress characteristic positions such as the highest point of single-tooth meshing and the alternating load distribution points in the double-tooth meshing area with sufficient spatial resolution. The differential assembly finite element model is spatially geometrically assembled according to the six discrete meshing positions 21, generating six independent initial assembly models. Each assembly model corresponds to a static physical position of the gear from engagement, alternating single and double tooth meshing to disengagement, enabling subsequent finite element analysis to capture the extreme stress conditions throughout the entire working cycle.

[0098] See attached document Figure 8 Appendix Figure 9 and attached Figure 10 To establish constraints and stress environment for the finite element model and achieve accurate application of loads and boundary conditions, the specific steps include:

[0099] Specifically, boundary constraints for the front bearing 6, rear bearing 7, and half-shaft gear 4 are established using multi-point constraint units. In the displacement constraint definitions for the front bearing 6 and rear bearing 7, the outer ring surface 42 is interpolated and connected to the corresponding bearing center 43 by establishing a first RBE3 unit 41. A full-degree-of-freedom fixed constraint is applied to the bearing center 43 to simulate the supporting effect of the bearing housing on the entire differential system. In this embodiment, the constraint scheme can evenly distribute the rigid support effect of the bearing housing to the outer ring surface, avoiding non-physical stress artifacts caused by single-node constraints. For the half-shaft gear 4, the inner surface 45 is connected to the geometric center 46 of the half-shaft gear by establishing a second RBE3 unit 44. The remaining four degrees of freedom, excluding the rotational degree of freedom around the axis and the axial displacement degree of freedom, are constrained on the geometric center 46 of the half-shaft gear, ensuring that the half-shaft gear 4 can truly transmit torque and generate axial displacement tendency, thereby completely restoring the axial thrust characteristics of the half-shaft gear under load due to the bevel tooth effect.

[0100] Furthermore, an elastic force transmission component is established to construct the load transmission path between gears. A spatial topology network is established between the geometric center and contact position of the planetary gear 3 and the half-shaft gear 4. In this embodiment, based on the previously established first RBE3 unit 41 and second RBE3 unit 44, the corresponding RBE3 unit node 47 is extracted, and a one-dimensional SPRING unit 53 is established between the selected meshing node 51 on the gear tooth surface and the corresponding third RBE3 unit 52, combined with the SPRING unit node 54. Through the stiffness matrix and spatial geometric configuration of the one-dimensional SPRING unit 53, discrete nodes are physically associated to form a topology structure that can adaptively adjust deformation transmission, thereby avoiding non-physical stress concentration caused by concentrated forces being directly applied to local nodes. To ensure that the load does not undergo non-physical attenuation during transmission and to avoid singularities in the finite element stiffness matrix, the stiffness of the one-dimensional SPRING unit 53 is set between 1×10^5 N / mm and 1×10^7 N / mm in this embodiment. The principle behind the range setting is that high stiffness can ensure the rigidity of force transmission, while upper limit control can effectively prevent non-convergence during iterative calculations by the solver.

[0101] Based on this, the theoretical forces on the gear are decomposed in three dimensions using local coordinate system 9 and mapped to the elastic force transmission component. When establishing local coordinate system 9, the gear meshing point is taken as the origin, and the radial, tangential, and axial directions of the gear are used as the coordinate axes. Based on the input power parameters of the differential system, the circumferential force, radial force, and axial force are derived. In this embodiment, the force decomposition calculation is performed using the following formula:

[0102] ;

[0103] ;

[0104] ;

[0105] In the formula, , , These are the circumferential force, radial force, and axial force of the gear, respectively. The torque transmitted by the gears. The pitch circle diameter of the gear. The normal pressure angle of the gear. The helix angle at the gear pitch circle. The physical essence of circumferential force, radial force, and axial force lies in transforming the macroscopic torque load into concentrated force components in three orthogonal directions based on the spatial geometric contact relationship of the gear. Spatial geometric characteristic parameters such as diameter, pressure angle, and helix angle are directly and quantitatively input based on the design drawing parameters of the bevel gear in the differential assembly. The calculated triaxial forces are further mapped to the corresponding force direction and magnitude of the SPRING element 53 in the local coordinate system 9, thereby transforming the macroscopic dynamic load into the adaptive input boundary of the microscopic finite element nodes.

[0106] In addition, the torque of the half-shaft gear 4 and the bolt preload are quantitatively applied based on the mechanical characteristics of the external power source. In this embodiment, the loads in the finite element model include two types: the torque of the half-shaft gear 4 and the bolt preload. For the torque of the half-shaft gear 4, a torque in the z-axis direction of the local coordinate system 9 is applied at node 47 of the RBE3 element. The magnitude of the torque is half of the differential output torque to simulate the reaction of the half-shaft on the half-shaft gear 4. For the simulation of the stress environment of the bolt 8, in order to restore the preload assembly state between the differential housing 1 and the secondary driven gear 2, a preload section is established by cutting the middle section of the bolt 8 screw. The bolt preload is calculated using the following formula, with the direction of action along the axial direction of the bolt 8:

[0107] ;

[0108] In the formula, For bolt preload, The bolt tightening torque, This is the bolt tightening torque coefficient. The bolt diameter is [value]. In this embodiment, the bolt tightening torque coefficient is [value]. The value is set to 0.2. The introduction of the calculation formula can accurately reproduce the local clamping stress generated by the bolts due to pre-tightening assembly on the differential housing 1, avoiding deviations in the overall strength assessment caused by ignoring assembly stress. The calculated bolt pre-tightening force is applied axially to the pre-tightening section to complete the configuration of the complete load and boundary conditions of the differential assembly finite element model.

[0109] See attached document Figure 11 and attached Figure 12 According to the different definitions of the constraint points of the secondary driven gear 2, multiple calculation conditions are defined, and the structural stress analysis results of the differential assembly under each calculation condition are calculated and obtained. Specifically, the following steps are included.

[0110] Specifically, 36 equally spaced meshing nodes are evenly distributed and extracted on the circumference of the secondary driven gear 2 as independent constraint force points. During actual vehicle operation, the differential housing 1 rotates around the half-shaft axis under the drive of the secondary driven gear 2, and the relative spatial position of the power input position of the external main reduction gear with respect to the planetary gear 3 and the slotted shaft 5 inside the differential constantly changes. In this embodiment, a node is extracted every 10° along the pitch circle circumference of the secondary driven gear 2, for a total of 36 constraint points. The basis for setting 36 constraint points is that the 10° rotation step size can provide sufficient phase angle resolution, avoid missing the local stress mutation points generated when the gear meshing position and the cross spatial structure of the slotted shaft 5 alternate, and at the same time, it can control the overall finite element calculation scale within a reasonable engineering acceptable range. In the finite element model, each calculation applies full degree of freedom fixation to only a single constraint point to simulate the support position of the power input at the current rotation angle, covering various extreme force conditions during gear operation.

[0111] Furthermore, the selected 36 circumferential constraint points are combined and intersected with the 6 independently constructed assembly models to generate a discretized calculation condition matrix for the entire working cycle. The microscopic alternating single and double tooth meshing of the bevel gears inside the differential and the macroscopic rotation of the differential assembly together determine the actual force state of the gears. In this embodiment, the 36 macroscopic rotational constraint positions and 6 microscopic dynamic meshing positions 21 are fully permuted and combined to construct a complete calculation matrix. The total number of calculation conditions is determined by the following formula:

[0112] ;

[0113] In the formula, The total number of calculation conditions. The number of equally divided constraint points on the circumference of the second-stage driven gear 2 is 36; The number of discrete assembly positions for a single gear tooth meshing cycle is set to 6. The 216 independent calculation conditions determined by the above formula constitute a condition matrix covering the complete working cycle of the differential, realizing the coverage of the position state of the dynamic operation process by static calculation.

[0114] After setting the working condition matrix, the defined boundary conditions, material nonlinear parameters, and load matrix are input into the solver to perform nonlinear iterative calculations. Due to the introduction of nonlinear contact states between tooth surfaces and material nonlinearities in the high-stress region as the material enters the plastic yielding stage in the finite element model, conventional linear solving algorithms cannot obtain the true mechanical distribution. In this embodiment, the Newton-Raphson iterative algorithm is used for implicit nonlinear calculations. During the calculation process, the solver performs analysis step by step according to the set working condition matrix, continuously updating the tangent stiffness matrix in each increment step until the contact state is stable and the residual force is less than the set convergence tolerance. To ensure the accuracy of the nonlinear contact iteration and the numerical stability of the solution process, in this embodiment, the force convergence tolerance is set to 0.5% of the load increment, and the displacement convergence tolerance is set to 0.1% of the displacement increment. After convergence, the structural stress analysis results of the differential assembly under each calculated working condition are obtained and extracted, mainly including the stress tensor and strain tensor of each solid mesh node.

[0115] The obtained structural stress analysis results are processed to remove contact artifacts and evaluate the static strength safety factor of the differential assembly. The specific steps include the following steps.

[0116] Stress data from mesh nodes are extracted from the computational load case matrix to construct a full-cycle stress envelope field. The differential's stress state changes in real time throughout its entire operating cycle. In this embodiment, all stress analysis results for 216 independent computational load cases are traversed, and the maximum Mises equivalent stress value under all load cases is extracted for each solid mesh node in the finite element model. The maximum Mises equivalent stress values ​​of each node are spatially combined to form a full-cycle stress envelope field covering the entire geometric domain. By constructing this stress envelope field, discrete dynamic transient results are transformed into a single static extreme value distribution, providing a unified data foundation for subsequent static strength evaluation.

[0117] This paper identifies and addresses non-physical stress peaks in contact artifact regions within the stress envelope field. When finite element analysis (FEM) deals with nonlinear contacts on complex curved surfaces such as gears, the high local stress concentrations at contact boundaries and small geometric chamfers, influenced by mesh discretization accuracy and penalty function contact stiffness, often result in contact artifacts. To avoid interference from artifacts in static strength evaluation, this embodiment extracts the spatial stress gradient of mesh nodes and sets a stress gradient threshold of 500 MPa / mm. When the stress gradient between adjacent mesh nodes exceeds 500 MPa / mm, the corresponding local area is identified as a contact artifact region. The principle behind setting the gradient threshold is that the yield strength of conventional gear steel is typically in the range of hundreds of megapascals to gigapascals. In the real continuous medium mechanics transmission law, a real stress step exceeding 500 MPa is unlikely to occur within a 1 mm physical distance. Extremely high gradients exceeding the conventional numerical range are usually non-physical phenomena caused by numerical calculation truncation errors or contact stiffness penetration.

[0118] The subsurface stress extrapolation method is used to calculate the true surface stress after correction for contact artifact areas. For the identified contact artifact areas, Mises equivalent stress values ​​of 3 to 5 adjacent mesh nodes are extracted into the material along the normal direction of the part surface. The reason for limiting the extraction to 3 to 5 mesh layers is that too few extraction layers will still be affected by the noise in the surface contact penalty function calculation, while too many extraction layers will penetrate into the inner unhardened body 12, deviating from the true physical stress gradient distribution law of the surface. The least squares method is used to establish a polynomial fitting curve between the extracted subsurface node stress values ​​and the corresponding normal depth coordinates. In this embodiment, the stress extrapolation calculation is performed using the following formula:

[0119] ;

[0120] In the formula, Normal depth Fitting stress at the point, The distance from the grid node to the normal depth of the surface. , , These are the polynomial fitting coefficients obtained by calculating using the least squares method. The physical principle behind using quadratic polynomials for fitting is that, according to Hertzian contact mechanics theory, when a metal surface is subjected to pressure, the equivalent stress in the subsurface layer exhibits a nonlinear parabolic attenuation characteristic as it propagates inward. Quadratic polynomials can best reproduce this continuous medium mechanics transmission law. Let... The surface intercept value was calculated. , the constant term The corrected real surface contact stress replaces the original non-physical stress peak in the contact artifact region. The extrapolation method utilizes the physical characteristic that the subsurface mesh is less affected by the contact stiffness algorithm, thus restoring the real surface stress state that conforms to the laws of continuous medium mechanics.

[0121] The static strength safety factor of each component of the differential assembly is calculated based on the corrected stress envelope field. The corresponding material failure criteria are selected according to the linear elastic and elastoplastic material properties of different components. In this embodiment, the static strength safety factor of the differential bevel gear is calculated based on the strength analysis results of the finite element model of the differential assembly, and the strength of the differential bevel gear is evaluated. The static strength safety factor is calculated using the following formula:

[0122] ;

[0123] In the formula, For the static strength safety factor of the component, The ultimate strength of the material This represents the maximum stress of the structure. For the material's ultimate strength... For the differential housing 1 and components such as the front bearing 6 and rear bearing 7, which are made of linearly elastic materials, the yield strength of the corresponding material is taken as the [value / value]. The input value; for planetary gear 3, half-shaft gear 4, and slotted shaft 5, which are assigned elastic-plastic material properties, slight plastic deformation is allowed, and the tensile strength of the corresponding material is taken as the input value. The input values. The above basic data on yield strength and tensile strength were obtained through standard material tensile tests. Maximum stress of the structure. The value is directly derived from the extreme value data in the stress envelope field after contact artifact correction. Based on the calculated static strength safety factor of the components, it is determined whether the differential assembly meets the design safety requirements under extreme torque input.

[0124] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A simulation method for the strength of a differential bevel gear, characterized in that, Includes the following steps: Geometric processing and meshing were performed on the geometric models of each component of the differential assembly to establish a finite element model of the differential assembly; Define the material properties of each component in the finite element model and the assembly connection relationship. The material properties include linear elastic material properties and elastoplastic material properties. Boundary conditions and force loads are applied to the assembled finite element model; Based on the applied boundary conditions and loads, multiple calculation conditions covering the entire working cycle are defined and solved to obtain the structural stress analysis results of the differential assembly under each calculation condition. Based on the structural stress analysis results, data processing is performed to remove contact artifacts, and the static strength safety factor of the differential assembly is calculated for strength evaluation.

2. The simulation method for the strength of a differential bevel gear according to claim 1, characterized in that, The steps for geometric processing and meshing of the geometric models of each component of the differential assembly to establish a finite element model of the differential assembly include: Extract a 1 / 4 model (15) of the differential housing (1) and a half-tooth model (16) of the planetary gear (3) and the half-shaft gear (4) to establish a dimension reduction calculation model; Extract the contact area edge line (13) at the meshing point of the planetary gear (3) and the half-shaft gear (4); In the normal direction of the edge line (13) of the contact area, a cutting line (14) is established by setting an offset distance outward from the contact area. The geometric models of the planetary gear (3) and the half-shaft gear (4) are processed by material layer cutting using the cutting line (14) to distinguish the outer hardened layer (11) and the inner unhardened body (12), and then a mesh generation operation is performed.

3. The simulation method for the strength of a differential bevel gear according to claim 2, characterized in that, The steps for defining the material properties of each component and the assembly connection relationships in the finite element model include: The elastoplastic material properties are distributed to the outer hardened layer (11), the inner unhardened body (12), and the spool (5); The nominal stress and nominal strain obtained from the tensile test are converted into real stress and real strain, and a plastic strain cutoff threshold is introduced in the strain conversion process. When the calculated plastic strain is less than the plastic strain cutoff threshold, the plastic strain is forced to be zero.

4. The simulation method for the strength of a differential bevel gear according to claim 1, characterized in that, The steps of applying boundary conditions and loads to the assembled finite element model include: By establishing the first RBE3 unit (41) to connect the outer ring surface (42) and the bearing center (43), the front bearing (6) and the rear bearing (7) are fixed. The front bearing (6) includes the front bearing inner ring (31), the front bearing rolling element (32) and the front bearing outer ring (33), and the rear bearing (7) includes the rear bearing inner ring (34), the rear bearing rolling element (35) and the rear bearing outer ring (36). The inner surface (45) and the geometric center (46) of the half-shaft gear are connected by establishing a second RBE3 unit (44) to constrain the half-shaft gear (4). Extract the RBE3 element node (47), and combine the meshing node (51), the third RBE3 element (52), the SPRING element (53), and the SPRING element node (54) to establish the load application path.

5. The simulation method for the strength of a differential bevel gear according to claim 4, characterized in that, The steps of applying boundary conditions and loads to the assembled finite element model further include: The direction of force is determined using a local coordinate system (9), with the meshing node (51) as the origin and the radial, tangential and axial directions of the gear as the coordinate axes. Based on the input power parameters of the differential system, the circumferential force, radial force, and axial force are derived and mapped to the load application path along the determined force direction. The torque of the half-shaft gear (4) is applied to the corresponding node, and the preload of the bolt (8) is applied to the preload section.

6. The simulation method for the strength of a differential bevel gear according to claim 2, characterized in that, Define the steps for multiple computational conditions covering the complete work cycle, including: Extract the complete meshing cycle of a single tooth profile, divide the meshing range of a tooth into multiple discrete meshing positions (21) along the axis of the local coordinate system (9), and perform spatial geometric assembly of the finite element model according to the meshing positions (21) to generate multiple independent initial assembly models. Multiple equally divided meshing nodes are evenly distributed and extracted on the circumference of the secondary driven gear (2) as independent constraint force points; The selected independent constraint stress points are combined with the initial assembly model in a full permutation and cross-combination to generate a discretized calculation working condition matrix for the entire working cycle.

7. The simulation method for the strength of a differential bevel gear according to claim 1, characterized in that, The step of removing contact artifacts includes: Extract the maximum Mises equivalent stress value of all solid mesh nodes in the structural stress analysis results under all calculation conditions, and construct a full-cycle stress envelope field; Extract the spatial stress gradient of the grid nodes. When the stress gradient between adjacent grid nodes exceeds the set stress gradient threshold, the corresponding local area is determined to be a contact artifact area.

8. The simulation method for the strength of a differential bevel gear according to claim 7, characterized in that, After determining that the corresponding local area is a contact artifact region, the steps for calculating the corrected true surface stress of the contact artifact region using the subsurface stress extrapolation method include: Along the normal direction of the component surface, extract the Mises equivalent stress values ​​of adjacent mesh nodes into the material interior; A polynomial fitting curve was established between the extracted subsurface nodal stress values ​​and the corresponding normal depth coordinates using the least squares method. The surface intercept value is calculated by setting the normal depth of the polynomial fitting curve to zero, and the constant term is used as the corrected true surface contact stress to replace the original non-physical stress peak value of the contact artifact area.

9. The simulation method for the strength of a differential bevel gear according to claim 8, characterized in that, The steps for calculating the static strength safety factor of the differential assembly and conducting a strength evaluation include: The maximum structural stress of the component is obtained based on the extreme value data in the corrected full-cycle stress envelope field. For components with the linear elastic material properties, the yield strength of the corresponding material is taken as the ultimate tensile strength; for components with the elastoplastic material properties, the tensile strength of the corresponding material is taken as the ultimate tensile strength. The static strength safety factor is obtained by calculating the ratio of the ultimate strength of the material to the maximum stress of the structure.

10. A simulation device for the strength of a differential bevel gear, used to implement a simulation method for the strength of a differential bevel gear according to any one of claims 1 to 9, characterized in that, include: The model processing module is used to perform geometric processing on the geometric models of various components of the differential assembly and establish finite element models. It distinguishes the outer hardened layer and the inner unhardened body of the components and performs mesh generation operations independently. The attribute and assembly module is used to define the material properties and assembly connection relationships of the finite element model output by the model processing module, distinguish between linear elastic materials and elastoplastic materials, and read contact surface information to complete system assembly. The boundary and load application module is used to establish constraints and stress environment for the assembled finite element model, and to establish load application paths to apply torque to the half-shaft gear and preload to the bolts. The working condition and solution module is used to generate multiple calculation working conditions covering the complete working cycle and perform solution calculations, and output the structural stress distribution data of the finite element model under the calculation working conditions; The strength evaluation module is used to perform evaluation operations on the structural stress distribution data output by the working condition and the solution module, remove contact artifacts, extract the maximum stress of the structure to calculate the static strength safety factor, and output the strength evaluation results.

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