A plastic worm gear metal worm static strength finite element analysis system and method

Abstract

This invention relates to a finite element analysis system and method for the static strength of plastic worm gears and metal worms, particularly for EPS plastics, and pertains to the fields of mechanical transmission design and computer-aided engineering technology. This invention systematically integrates six verification methods: bending strength, contact strength, limit (adhesive failure), plastic deformation exceeding limits, anisotropic failure, and thermal deformation center distance interference. This comprehensively covers the main static strength failure modes of EPS plastic worm gears and metal worms, filling a significant gap in existing tools that can only assess the bending and contact strength of metal materials. The limit verification is specifically designed for the high sliding speed characteristics of worm gear drives, and incorporates frictional heat analysis and steady-state / fully coupled thermo-mechanical analysis as mandatory analysis modules for EPS worm gear pairs. This automatically calculates the body temperature and flash temperature of the contact area and corrects material parameters in real time, overcoming the common problem of overestimation of the strength of plastic worm gears caused by neglecting thermal effects in existing metal worm gear analysis systems and general finite element software.

Description

Technical Field

[0001] This invention relates to the fields of mechanical transmission design and computer-aided engineering technology, specifically to a finite element analysis system and method for static strength analysis of plastic worm gears and metal worms. Background Technology

[0002] Worm gear drives, with their advantages of large transmission ratio, compact structure, smooth operation, and strong self-locking, are widely used in various drive mechanisms. In recent years, with the increasing demand for lightweight, low-cost, and maintenance-free systems, engineering plastics such as polyamide, polyoxymethylene (POM), and polyetheretherketone (PEEK) have been increasingly used in the manufacture of worm gears and even worms, forming plastic-plastic or steel-plastic worm gear pairings. Compared with metal worm gear drives, plastic worm gears exhibit significant differences in material mechanical behavior, frictional heat effects, and long-term performance degradation. These differences pose a serious challenge to existing analytical methods. First, the relative sliding speed of the tooth surfaces in worm gear drives is much higher than that of cylindrical gear drives of the same specifications. For plastic materials, this sliding speed, combined with the normal pressure on the tooth surfaces, easily results in a PV value (pressure-velocity product) that approaches or exceeds the material's allowable PV limit, causing a sharp rise in the contact area temperature (contact flash temperature effect), leading to localized softening or even melt adhesion failure of the plastic. Existing analytical formulas often neglect the flash temperature effect of plastic worm gears and cannot assess the risk of heat-induced strength degradation.

[0003] Secondly, the low elastic modulus of engineering plastics results in significantly greater contact deformation under the same load compared to metal pairs, leading to a fundamentally different contact stress distribution. Furthermore, the yield stress of plastics decreases significantly with increasing temperature, and moisture absorption further reduces the elastic modulus and strength of PA-type materials. These complex multi-field coupling effects introduce serious systematic errors into analytical formulas based on the assumption of a single linear elasticity.

[0004] Third, the worm gear tooth surface is a spatial curved surface generated by planar or helical envelope, and the worm wheel tooth profile is obtained by numerically solving the worm through envelope motion. Neither can be quickly generated using conventional parametric CAD commands. General-purpose finite element software lacks the ability to automatically model such complex surfaces. Manual modeling takes several to tens of hours, and the operation is extremely cumbersome in configuring nonlinear constitutive parameters for plastic materials, temperature-hygroscopic correction, and setting thermo-mechanical coupling boundary conditions, making it difficult for general engineering technicians to master.

[0005] Existing methods for analyzing the strength of plastic worm gears have the following fundamental shortcomings: Standard formulas such as GB / T10089 and ISO14521, as well as the VDI2736 plastic gear specification, do not establish a systematic analytical framework specifically for plastic worm gear-metal worm drives; directly applying analytical formulas for metal worm gears to plastic worm gears ignores key factors such as material nonlinearity, temperature softening, and the PV limit, leading to severe distortion in strength assessment; while general-purpose finite element software can theoretically address these issues, it has extremely high modeling barriers, long analysis cycles, and lacks an automatic evaluation mechanism for failure modes specific to plastic worm gears (flash adhesion failure, creep deformation exceeding limits, and moisture absorption dimensional changes).

[0006] In summary, there is currently no existing technology that can simultaneously consider the nonlinear characteristics of plastic materials, thermo-mechanical coupling effects, accurate modeling of the worm gear envelope tooth surface, analysis accuracy, ease of operation, and design optimization guidance capabilities for static strength finite element analysis of plastic worm gears. There is an urgent need to fill this technological gap. Summary of the Invention

[0007] The purpose of this invention is to provide a finite element analysis system for the static strength of plastic worm gears and metal worms. This system overcomes the core shortcomings of existing technologies, such as insufficient description of the nonlinear behavior of plastics and the thermal effects of high sliding speeds in worm gears by analytical formulas, high operational barriers of general finite element software and lack of ability to evaluate failure modes specific to plastic worms, and lack of closed-loop capability from analysis results to design optimization. This system enables automated and high-precision evaluation of the static strength of plastic worm gear and metal worm transmission pairs, and automatically provides comprehensive design suggestions including material optimization, tooth profile correction, and thermal management improvement when the strength does not meet the requirements.

[0008] To achieve the above objectives, the present invention adopts the following technical solution: On the one hand, the present invention provides a finite element analysis system for static strength of plastic material worm gear and metal worm, which includes seven functional modules connected in sequence: parametric modeling module, mesh generation module, constitutive configuration module, load and boundary condition application module, solution and post-processing module, verification module, and sensitivity analysis and optimization suggestion module.

[0009] The system's operation begins with the user inputting a complete set of design parameters for the plastic worm gear transmission pair through a parametric input interface, including: The set parameters of the worm gear include: normal module. Number of worm gear threads Lead angle Normal pressure angle axial pressure angle Pitch circle diameter Tooth tip circle diameter Root circle diameter Helix direction, worm type, displacement coefficient .

[0010] Worm gear geometry parameters include: number of teeth Tooth width displacement coefficient Hub Bore Spoke thickness Number of spokes .

[0011] Transmission parameters include: center distance Transmission ratio .

[0012] Material parameters include: worm gear material grade, worm wheel material grade, and moisture absorption status.

[0013] Thermal and humidity environmental parameters include: operating temperature Reference temperature relative humidity .

[0014] Load parameters include: transmitted power worm gear speed Usage coefficient Impact coefficient .

[0015] After the above parameters are entered, the system calls each functional module in the following order: parametric modeling → mesh generation → nonlinear constitutive configuration → multi-field coupled load application → nonlinear finite element solution → plastic worm gear and metal worm specific strength verification → sensitivity analysis.

[0016] Regarding the parametric modeling module, its core task is to automatically generate accurate three-dimensional geometric models of the worm gear helical tooth surface and the worm wheel envelope tooth surface based on the transmission parameters input by the user. First, let's discuss the parametric representation of the worm gear helical tooth surface. Taking the most commonly used Archimedes' worm as an example, its axial section tooth profile is a straight line, and the helical surface is formed by the helical motion of this straight line around the worm's axis. The spatial coordinates of any point on the right side of the worm's tooth surface... Described by the following parametric equation: Where r is the radial coordinate of each point on the tooth surface, and its value ranges from the root circle radius. to the tooth tip circle radius θ is the helical parameter, and the step size θ is usually taken as π / 180 (one section per degree); The worm lead parameter represents the axial distance the worm moves when it rotates a unit angle. The radius of the pitch circle This represents the axial pressure angle. For involute worms, the tooth surface equation is replaced with the involute helical surface equation; for ZK-type conical envelope worms, the corresponding conical envelope geometry is used. The system has built-in parametric equations for three types of worm tooth profiles, which automatically switch according to user selection.

[0017] Secondly, the numerical solution for the worm gear envelope tooth surface is required. The worm gear tooth surface is generated by the worm through envelope motion, and its shape depends on the shape of the worm's helical surface and the spatial relative motion relationship between the two. Let the worm rotation angle be... 1, corresponding to the worm gear rotation angle is Then the worm gear tooth surface is composed of a set of points that satisfy the following meshing equation: In the formula Let be the unit normal vector at the corresponding parameter point on the worm gear tooth surface. Let be the position vector of a point on the worm gear tooth surface in the worm wheel coordinate system. Describe the direction of the relative velocity of this point in the worm wheel coordinate system as the worm rotates. For given worm tooth surface parameters... The equation is about A univariate equation of 1 is solved using Newton's method of iteration: The iteration termination criterion is It typically converges within 5 iterations. (This involves iterating through the worm gear tooth surface parameters.) By solving the equation point by point for all discrete points, a complete discrete point cloud of the worm gear tooth surface is obtained. After B-spline surface fitting, an accurate three-dimensional surface of the worm gear tooth is generated.

[0018] The next step is to restore the root transition fillet of the plastic worm gear. Plastic materials lack the local stress redistribution ability of metals, and the stress concentration in the root transition fillet region has a more significant impact on the static strength results than in metal pairs. The basic radius of the transition fillet is determined by the following formula: For injection-molded plastic worm gears, the system supports two additional transition forms: optimized elliptical arc transition (which can reduce the stress concentration factor at the tooth root by approximately 15% to 25%) and user-defined transition curve (automatically fitted to a B-spline curve by importing discrete point clouds), to adapt to different mold design schemes. The coordinates of the tangent point between the transition arc and the tooth surface are automatically calculated by the system to ensure the geometric continuity of the tooth root transition region.

[0019] Finally, the plastic worm gear body structure is generated. Unlike solid metal worm gears, injection-molded plastic worm gears are typically designed as lightweight structures with spokes and webs to reduce warping caused by uneven cooling and to save material. The system generates the spokes based on the user-inputted thickness. Number of spokes Automatically generate the spoke web geometry and keyway structure at the hub. For large-tooth-width worm gears. The system automatically prompts whether to adopt a double-spoke structure to improve the uniformity of stiffness in the tooth width direction.

[0020] Regarding the mesh generation module, this module adopts a partitioned adaptive strategy, configuring the mesh density according to the differences in stress gradient in each region: The mesh size of the tooth contact area is determined based on the plastic modified Hertz contact half-width: In the formula and For temperature T and moisture absorption The corrected effective elastic modulus.

[0021] The contact area grid cell size should meet the following requirements: That is, at least 8 nodes should be arranged within the contact half-width range to ensure the accuracy of capturing the peak contact stress of plastic under large deformation.

[0022] The mesh size of the tooth root transition fillet area is controlled by the fillet radius: The contact area and tooth root area preferentially use hexahedral structured meshes, while complex geometric regions employ a hybrid mesh strategy with hexahedral meshes as the main type and tetrahedral meshes as a supplement. A prism transition layer is set between the structured and unstructured meshes. For worm gear materials reinforced with glass fiber (such as PA66-GF30), the system adaptively adjusts the mesh size according to the anisotropic stiffness ratio in the principal fiber direction and the perpendicular direction to ensure that the density of integration points in each direction meets the accuracy requirements.

[0023] After the mesh is generated, the system automatically performs a quality check on each element: aspect ratio. Jacobi ratio Quadrilateral warp When the proportion of units with any quality indicator exceeding the standard exceeds 5% of the total number of units, the system automatically triggers a local re-partitioning.

[0024] Regarding the constitutive configuration module, this module is a distinguishing feature of this system compared to metal worm gear analysis systems and general finite element software. It is specifically designed to handle the complex mechanical behavior of plastic materials under the influence of temperature, humidity, and strain rate. First, the changes in the elastic modulus and yield stress of plastics with temperature are respectively corrected using the following functions: in and Reference temperature (Usually taken as the elastic modulus and yield stress at 23℃) (unit 1 / ℃) and (Unit: 1 / ℃) represents the material-dependent temperature coefficient, read from the built-in material library. When the operating temperature exceeds the glass transition temperature... When the material reaches 85%, the system automatically issues a warning, indicating that the material may have entered the softening zone.

[0025] Contact flash temperature generated by high sliding speed in worm gear pair The impact on localized temperature on the tooth surface requires special consideration. The system uses the Blok flash temperature formula to calculate the peak temperature rise in the contact area: In the formula This is the Blok correction factor (taken as 1.11). The value is Hertzian contact line load (unit: N / mm). The relative sliding speed of the tooth surface (unit: m / s). and These are combined parameters of density, specific heat capacity, and thermal conductivity of the worm gear and worm wheel materials (automatically retrieved from the material library). , This refers to the tangential velocity at each contact point on the tooth surfaces of the worm and worm wheel. If the contact flash temperature... If the temperature exceeds the heat distortion temperature (HDT) of the worm gear material (read from the material library), the system will automatically issue an adhesion failure risk warning and trigger a special check for heat-induced failure in the strength verification module.

[0026] For polyamide worm gear materials such as PA6, PA66, and PA12, the moisture absorption... (Mass fraction %) affects the elastic modulus and yield stress through the following correction factors: moisture absorption Determined by the relative humidity (RH) input by the user and the material equilibrium moisture absorption curve. The effective elastic modulus after superimposing temperature and moisture absorption corrections is: Moisture absorption also causes the PA type worm gear to expand in size (hygroscopic expansion coefficient). Approximately 0.15%~0.30% / 1% The system automatically calculates the impact of moisture absorption and expansion on the pitch circle diameter and tooth thickness of the worm gear, and combines this with thermal expansion to assess the actual center distance change.

[0027] For short glass fiber reinforced worm gear materials such as PA66-GF30, the fibers preferentially orient along the flow direction during injection molding, resulting in orthotropic anisotropy. The system employs an orthotropic elastic constitutive model with nine independent elastic constants. The fiber orientation distribution is estimated using the Halpin-Tsai micromechanical model, either from fiber volume fraction and fiber orientation distribution, or by direct input of measured values ​​by the user. Fiber orientation information can be exported from mold flow analysis software (Moldflow / Moldex3D) and imported into this system to achieve accurate anisotropic field distribution, which is crucial for understanding the influence of fiber orientation at the critical section of the worm gear tooth root on bending strength.

[0028] Regarding the load and boundary condition application module, this module is responsible for converting engineering load parameters into finite element boundary conditions and handling the high sliding speed frictional heat generation and thermo-mechanical coupling effects unique to plastic worm gear pairs: First, the three-dimensional meshing force of the worm gear pair is calculated. Based on the input transmitted power P (unit: kW) and worm speed n1 (unit: r / min), the system first calculates the working torque of the worm wheel shaft: In the formula, η is the efficiency of the worm gear drive, which is determined by the lead angle. With equivalent friction angle It is determined that the initial estimate can be taken as η≈0.7~0.9 (plastic pairs have a higher coefficient of friction, and their efficiency is usually lower than that of metal pairs). In the three-dimensional meshing mechanics analysis of worm gear drives, taking the worm wheel as the reference body, the axial force, tangential force, and radial force borne by the worm wheel tooth surface are as follows: In the formula The temperature-dependent coefficient of friction is given by... Calculations are performed by reading from the built-in material pairing library. and Steel-PA66 matching Approximately 0.10~0.15, PA66-PA66 pairing. (Approximately 0.20~0.35). The three components are combined to form the total normal load on the tooth surface: Plastic worm gear and metal worm drive ( For speeds ≤3 m / s, the value is taken as 1.0~1.2 (considering the large error in the plastic tooth profile), and the tooth load distribution coefficient is... Take a value of 1.0 to 1.2. Maximum equivalent load in static strength analysis. .

[0029] Secondly, there is the thermo-mechanical coupling load; the system provides two thermo-mechanical coupling analysis modes: a. Steady-state thermo-mechanical decoupling analysis: First, a steady-state thermal analysis is performed to calculate the temperature field distribution (body temperature) of the gear under stable operating conditions. Then, the temperature field is used as the initial field for static analysis, and the material parameters of each element are corrected in real time using equations 8 to 13. The heat flux density of the worm gear meshing contact area is calculated using the Blok flash temperature model: in This represents the Hertzian contact pressure distribution. The local relative sliding velocity at each point within the contact ellipse (varying along the contact line direction). The boundary condition coefficient for convective heat transfer on the gear's outer surface. Select from the built-in recommended value library based on the lubrication method; for unlubricated natural convection, select [the appropriate value]. ; Selection of grease lubrication .

[0030] b. Fully Coupled Thermo-Mechanical Analysis: The heat conduction equation and the elastoplastic mechanics equation are solved simultaneously within the same finite element model. These two equations are bidirectionally coupled through temperature-dependent material parameters and frictional heat source terms, employing a Staggered iterative strategy. This method is suitable for applications with significant thermal effects. If the flash temperature prediction exceeds 90% of the HDT, or if the operating condition requires assessment of thermal fatigue.

[0031] Following this is the real-time calculation and early warning of PV values. After the load calculation is completed, the system automatically calculates the local PV values ​​at each contact point of the worm gear pair and compares them with the temperature-related PV limit of the worm wheel material: In the formula The PV limit value of the worm gear material at the local contact temperature is retrieved from a material library (which stores PV limit curves for common plastics such as PA66 and POM at different temperatures). Exceed If the load exceeds 80%, the system will issue a yellow warning during the load application phase; if it exceeds... If the error occurs, a red warning will be issued and an adhesion failure check will be triggered in the strength check module.

[0032] Finally, constraints are applied, restricting all linear displacement degrees of freedom and releasing only the rotational degree of freedom about the worm axis to simulate the transmission input. For plastic worm gear shaft holes, the system provides two constraint forms: keyway connection (applying fixed constraints on the keyway contact surface and radial displacement-only constraints on the remaining inner wall surfaces) and interference fit connection (applying a radial preload field to the inner wall of the shaft hole, the preload value being automatically calculated from the interference and the material's elastic modulus), to accommodate different engineering schemes for plastic worm gear shaft connections. Frictional contact pairs are automatically established between the contact surfaces, and tangential friction follows a temperature-dependent Coulomb friction model.

[0033] Regarding the solution and post-processing modules, the first aspect is the nonlinear solution strategy. The Newton-Raphson iterative method is used to solve the composite equations involving material nonlinearity, contact nonlinearity, and thermo-mechanical coupling. The tangent stiffness matrix consists of the material tangent stiffness matrix and the contact stiffness matrix. For bilinear elastoplastic constitutive models, a radial return mapping algorithm is used to process the plastic constitutive integral at each integration point. The normal contact constraint employs a penalty function method, with the penalty stiffness automatically determined based on the equivalent stiffness of the contact body. In the formula The equivalent elastic modulus after temperature and humidity correction is derived from... Sure, This is the penalty stiffness factor (default value is 10). The iterative convergence criterion is the relative tolerance of the residuals. When the iteration fails to converge, the load splitting strategy is automatically activated.

[0034] Secondly, the post-processing extraction indexes are used. Based on the failure characteristics of plastic worm gear pairs, the system extracts the following six evaluation indexes: maximum principal stress. von Mises equivalent stress Maximum contact pressure Equivalent plastic strain Tsai-Hill damage index, maximum nodal thermal displacement For fiber-reinforced plastic worm gears, the system also extracts the stress tensor in the local coordinate system of each element and compares it with the fiber direction strength and the strength perpendicular to the fiber direction, respectively, and applies the Tsai-Hill failure criterion: In the formula, F1 and F2 are the allowable stresses in the fiber direction and perpendicular direction, respectively, and F 12 The allowable shear stress within the layer is read from the material library.

[0035] Regarding the verification module, in addition to the two basic verifications of bending strength and contact strength, this module adds three special verifications specific to plastic worm gear pairs, forming a complete verification system covering the main failure modes of plastic worm gear drives.

[0036] Table 1: Verification System Table

[0037] First, there are the allowable values ​​for bending and contact strength, and the allowable static bending stress. Allowable contact static stress All require temperature and moisture correction: The value is typically taken into account that plastic materials have a higher dispersibility than metals, and is usually between 1.5 and 2.5. Typically, it is taken as 1.2 to 1.5. Material strength limit. and The data is retrieved from the built-in material library, which contains the ultimate tensile strength data of PA66 (dry / wet), PA66-GF30 (dry / wet), POM, PBT, PEEK and their modified varieties at different temperatures. The data sources cover technical manuals of mainstream material suppliers and publicly available academic literature.

[0038] followed by Limit check, Limit checks are the most unique aspect of the strength system for plastic worm gear pairs. While typically negligible in the analysis of ordinary plastic cylindrical gears, this check becomes a critical control factor in worm gear drives due to the high sliding speeds. The system considers local variations within the entire contact ellipse. The value distribution is evaluated by integration, excluding the peak value. In addition, the average contact area is calculated. value: Next is the check of the heat distortion center distance. The coefficient of linear thermal expansion (CTE) of plastics is usually 3 to 6 times that of metals (PA66 is about 80 × 10⁻⁶). -6 / ℃, steel approximately 12×10 -6 / ℃), the effect of thermal deformation on the center distance cannot be ignored. A comprehensive system calculation of the effects of thermal expansion and hygroscopic expansion on the center distance is as follows: In the formula , Here is the coefficient of linear thermal expansion of the worm and worm wheel materials. , For their respective warming, , The coefficient of hygroscopic expansion is 1 / 3. , Each represents its moisture absorption capacity. When Exceeding the nominal tooth flank clearance When the temperature reaches 50%, the system issues an interference warning and automatically calculates the required margin for the center distance of thermal-humidity compensation.

[0039] Regarding the sensitivity analysis and optimization suggestion module, this module is automatically activated when any strength check conclusion fails or the safety factor is lower than 1.2. It then uses systematic sensitivity calculations to identify stress or... The parameters with the most significant impact are identified, and design improvement suggestions with clear priorities are generated in conjunction with engineering constraints.

[0040] First, a parameter sensitivity analysis is performed. The optimal parameter vector for the plastic worm gear drive is defined as follows: There are a total of 8 parameter dimensions. For the i-th parameter p... i With ±10% perturbation applied respectively, the sensitivity coefficient is defined as follows: In the formula, For the first Initial design values ​​of the parameters, and Apply respectively and The parameter values ​​after the disturbance; The reference stress response values ​​are obtained from the finite element method under the initial parameters. and The corresponding parameters are set as follows: and The stress response value is obtained by resolving it. A negative value indicates that increasing this parameter is beneficial to improving strength; The larger the value, the more significant the adjustment effect. For Limit check, the response quantity of sensitivity analysis is replaced with Identify the parameter that has the greatest impact on frictional heat (usually the lead angle). relative sliding speed with tooth surface (Related parameters).

[0041] Due to each parameter perturbation (such as For any changes in the stiffness matrix, the complete nonlinear finite element equations need to be resolved. To balance computational accuracy and efficiency, this system employs a fast reanalysis algorithm based on the combinatorial approximation (CA) using Neumann series expansion. This algorithm does not directly iterate the high-dimensional nonlinear equations globally, but instead iterates the perturbed unknown displacement field... It can be approximated as a linear combination of a series of basis vectors: In the formula, The number of basis vectors selected; It is a basis vector matrix; Let be the vector of reduced combination coefficients to be determined. The basis vector matrix is... Based on the recursive generation of Neumann series expansion terms: the first basis vector is the displacement solution under the initial equilibrium state. Subsequent basis vectors are calculated using a recursive relation: ,(in Let be the initial tangent stiffness matrix. The stiffness matrix variable is caused by parameter perturbation. ).

[0042] Subsequently, the high-dimensional equation is projected onto the equation derived from... In Zhang Cheng's low-dimensional subspace, the coefficients are obtained by solving the reduction equation. Then, the displacement field after perturbation can be obtained. Engineering practice shows that when the number of basis vectors... When the values ​​are 5 to 8, the approximation error is usually less than 1%, while the amount of calculation per operation is only 10% to 20% of that of a complete nonlinear reanalysis. This allows the entire set of sensitivity calculations, which involves two perturbations for each of the eight design parameters (a total of 16 reanalysis operations), to be completed efficiently within minutes.

[0043] Secondly, multi-dimensional optimization suggestions are proposed. After the sensitivity coefficient calculation is completed, the system will proceed according to... Based on the size sorting and combined with engineering constraints to filter infeasible adjustment directions, priority ranking optimization suggestions are generated in the following four dimensions: a. The minimum parameter adjustment required to achieve the target safety factor (default S≥1.25) based on linear extrapolation estimation. It provides the adjustment direction and expected adjustment amount for parameters such as module, lead angle, and tooth root fillet, and automatically verifies whether the adjusted parameters meet the preset geometric constraints such as fixed center distance and upper limit of tooth width.

[0044] b. Retrieve candidate materials that meet the strength requirements from the built-in plastic material library and rank them according to their overall cost-effectiveness (strength improvement, material cost increase percentage, and increase in injection molding process difficulty). The typical material upgrade path for plastic worm gears is: unreinforced PA66 → PA66-GF15 → PA66-GF30 → POM → PEEK. At the same time, evaluate whether the PV limit of each candidate material meets the requirements under the current working conditions.

[0045] c. When PV exceeds limits or thermal deformation issues are prominent, the system provides engineering recommendations to reduce the contact area temperature: improve the lubrication method (evaluate the impact of switching from no lubrication to grease lubrication or oil lubrication). Limit increase and Increased composite effect), increased gearbox heat dissipation area (quantitative assessment of the effect of increased heat dissipation fins on steady-state body temperature) The reduction amount) and the reduction in speed or load (calculated reduction) or make Value dropped to Minimum adjustment required below 80% of the limit.

[0046] d. To address the issue of thermal deformation, it is recommended to increase the number of spokes or adjust the spoke arrangement to improve heat dissipation uniformity; to address the issue of injection molding warpage, it is recommended to adjust the spoke thickness ratio to reduce wall thickness difference; to address the risk of anisotropic failure of fiber-reinforced materials, it is recommended to adjust the gate position (with the support of mold flow analysis tools) to optimize the orientation of fibers at the critical section of the tooth root.

[0047] This invention also provides a computational device for implementing the aforementioned finite element analysis system for static strength of worm gears. The computational device includes: at least one processor, a memory, an input / output interface, and a communication bus. The communication bus is used to enable connection and communication between these components. The memory, as a non-volatile computer-readable storage medium, is used to store non-volatile software programs, executable modules, and data structures. In this embodiment, the memory specifically stores: a built-in material and environment database, including but not limited to nonlinear constitutive parameters, heat distortion temperature (HDT), temperature-dependent PV limit curves, and elastic constants of plastic materials such as PA66, POM, PEEK, and their glass fiber reinforced modified varieties under different temperatures and humidity conditions. System program modules: namely, the computer program instructions for the aforementioned parametric modeling module, mesh generation module, constitutive configuration module, load and boundary condition application module, solution and post-processing module, verification module, and sensitivity analysis and optimization suggestion module. The processor executes the various complex calculations and control logics of this analysis system by running the non-volatile software programs and instructions stored in the memory. Specifically, the processor is configured to: execute the Newton-Raphson iteration method to numerically solve the envelope tooth surface of the worm gear; invoke a nonlinear solver (such as one based on the Newton-Raphson iteration method) to process the complex equations involving material nonlinearity, contact nonlinearity, and thermo-mechanical bidirectional coupling; and run a combined approximation (CA) algorithm based on Neumann series expansion to perform multi-dimensional parameter sensitivity reanalysis with minute-level computational efficiency. The processor can be a general-purpose central processing unit, graphics processing unit, microprocessor, or application-specific integrated circuit. The input interface receives various parameters entered by engineers through a parametric interactive interface, including the complete set of geometric parameters of the worm gear, transmission parameters, material grade, thermal and humidity environmental parameters (such as operating temperature and relative humidity), and external load parameters. The output interface is typically connected to a display terminal for visualizing the precise 3D geometric model generated by the processor, mesh generation results, local PV value distribution cloud maps, contact flash warning information, and the final multi-dimensional priority optimization suggestions including material upgrades, tooth profile corrections, and thermal management improvements.

[0048] In another aspect, the present invention provides a computer-readable storage medium storing a computer program thereon, which, when executed by a processor, implements all the method steps described above for the automated, high-precision evaluation of the static strength of plastic worm gears and metal worms, and for providing design optimization suggestions. Those skilled in the art will understand that all or part of the processes implementing the system functions of the above embodiments can be accomplished by a computer program instructing related hardware (such as servers, workstations, or local high-configuration computing nodes).

[0049] In summary, compared with the prior art, the present invention has the following beneficial effects: 1. This invention integrates bending strength, contact strength, The system integrates six verification methods: limit (adhesion failure), excessive plastic deformation, anisotropic failure, and interference of center distance during hot deformation. This comprehensive approach covers the main static strength failure modes of plastic worm gears and metal worms, filling a significant gap in existing tools that can only assess bending and contact strength. The limit check is specifically designed for the high sliding speed characteristics of worm gear drives.

[0050] 2. This invention incorporates frictional heat analysis and steady-state / fully coupled thermo-mechanical analysis as mandatory analysis modules for plastic worm gear metal worm pairs. It automatically calculates the body temperature and flash temperature of the contact area and corrects material parameters in real time, overcoming the common problem of overestimation of the strength of plastic worm gears caused by neglecting thermal effects in existing metal worm gear analysis systems and general finite element software.

[0051] 3. This invention combines the parametric equations of the worm gear helical surface, the numerical solution of the worm wheel envelope meshing, and the interface for generating the spoke structure of the plastic worm wheel injection molding, pre-compensation for warpage deformation, and importing mold flow analysis data to form a dedicated modeling process for plastic molding.

[0052] 4. The sensitivity analysis and optimization suggestion module of this invention covers four dimensions: geometric parameter adjustment, material upgrade, thermal management improvement, and wheel structure optimization. It upgrades static strength analysis from a simple verification tool to a design aid tool, fundamentally changing the previous iterative design mode of plastic worm gears that relied on repeated trial and error based on personal experience, and effectively shortening the product design cycle. Detailed Implementation

[0053] To enable those skilled in the art to better understand the present application, the technical solutions in specific embodiments of the present application will be clearly and completely described below. Unless otherwise defined, the technical or scientific terms used in this invention should have the ordinary meaning understood by those skilled in the art.

[0054] Example 1 This embodiment uses a steel-worm gear-plastic worm wheel transmission pair applied in an EPS (Electric Power Steering) system as an example to demonstrate the complete analysis process of the system of the present invention. The worm material is 42CrMo steel (quenched and tempered), and the worm wheel material is short glass fiber reinforced polyamide PA66-GF30 (30% fiber mass fraction, injection molded). This transmission pair operates under long-term conditions of in-vehicle temperature (up to 70℃ in summer) and certain humidity, and has clear requirements for static strength, thermal deformation, and PV limit. The analysis objective is to verify that all static strength indicators of the worm wheel meet the requirements under rated operating conditions, and to output optimization suggestions when approaching the limit.

[0055] Users input the following complete set of design parameters through the parametric input interface: worm geometry: ; ; ; ; Worm gear geometry: ; ; ; The worm gear is made of 42CrMo steel; the worm wheel is made of PA66-GF30. ; Relative humidity Transmission efficiency = 0.85, worm speed Use coefficient .

[0056] The second step involves the system automatically generating an accurate three-dimensional geometric model based on the above parameters, following these steps: Generating a linear tooth profile based on the Archimedean worm's axial section, with a step size of π / 180, resulting in a 360×30 discrete point matrix; solving the meshing equation point-by-point using the Newton-Raphson iteration method, with iterative convergence criteria. The average convergence time is 4; the basic fillet radius is... Optimized elliptical arc transition is used, based on user input. , The system automatically generates spokes and keyways at the hub. During the injection molding process of the spoke web, the system detects the tooth width. ,satisfy The single-sided spoke criterion is used to generate a complete three-dimensional geometric model of the worm gear with keyways by adopting a single-sided six-spoke plate design.

[0057] The third step involves calculating the corrected Hertzian contact half-width of the plastic based on the elastic modulus corrected for temperature and moisture absorption, and then determining the mesh density for each region accordingly. Corrected Hertzian contact half-width calculation: The mesh size configuration for each region is obtained using the size control formula, including the tooth surface contact area. Actual size 0.10mm, approximately 12400 units; tooth root transition fillet area The actual size is 0.09mm, with approximately 8600 units; the actual size of the worm gear body area is 0.5~1.5mm, with approximately 35000 units; the actual size of the worm tooth body area is 0.3~1.0mm, with approximately 18000 units.

[0058] The fourth step involves the system retrieving the nonlinear constitutive parameters of PA66-GF30 from the built-in material library and performing temperature-hygroscopic superposition correction: temperature-corrected elastic modulus. Temperature-corrected yield stress When RH=50%, the equilibrium moisture absorption of PA66 is approximately 1.5% (read from the equilibrium curve in the material library). ; No risk of adhesion; The fifth step involves calculating the load on the worm gear using the following steps: Worm gear shaft working torque Efficiency η is taken as 0.85); worm gear tangential force Considering the temperature-dependent friction coefficient μ(70℃) = 0.12 × [1 + 0.003 × (70 - 23)] ≈ 0.137, calculate the axial force. Radial force Total normal load Maximum equivalent load .

[0059] Given that the operating temperature of 70℃ is higher than the reference temperature of 47℃ (ΔT>40℃), the system automatically recommends using the steady-state thermo-mechanical decoupling analysis mode. The heat flux density in the contact area is calculated using the Blok flash temperature model, and the convective heat transfer coefficient of the outer surface is taken as assuming grease lubrication conditions. PV real-time calculation results (< ) The sixth step employs the Newton-Raphson iterative method to solve the complex equation system containing material nonlinearity, contact nonlinearity, and thermo-mechanical coupling. The plastic constitutive integrals at each integration point are processed using the radial back-mapping algorithm. The solution process involves three load steps, with an average of seven iterations per step to achieve convergence (residual tolerance). Extract the following indicators: maximum principal stress (At the tooth root transition fillet) 61.3; von Mises equivalent stress (Critical section at tooth root) 58.7; Maximum contact pressure (Contact ellipse center) 89.4; Equivalent plastic strain 0.24%; Tsai-Hill damage index 0.61; maximum nodal thermal displacement thermal0.038 The seventh step involves the system verifying each of the six specific intensity indicators, and the results are summarized below: Table 2: Verification Results of Example 1

[0060] Verification conclusion: The static strength safety factor for tooth root bending is SF=1.04, which is lower than the target safety factor of 1.25; the change in the center distance of thermal deformation is close to the limit (reaching 89.6%). Based on this, the system automatically activates the sensitivity analysis and optimization suggestion module.

[0061] The system optimizes the parameter vector. Each perturbation was applied by ±10%, and a fast reanalysis algorithm based on Neumann series expansion (CA) was used (basis vector number s=6, approximation error <0.8%). All 16 reanalysis runs took approximately 3.2 minutes. The sensitivity coefficients are as follows: Table 3: Sensitivity coefficient table.

[0062] Step 8 provides optimization suggestions: First, the geometric parameter adjustment recommendations are as follows: with the bending strength safety factor reaching the target value SF≥1.25, the minimum adjustment amount is calculated by linear extrapolation. Increasing the normal module from m=2.0mm to m=2.25mm (+12.5%) is expected to increase SF to approximately 1.31; increasing the root transition fillet radius from ρf=0.612mm to ρf=0.76mm, combined with optimized elliptical arc transition, is expected to increase SF to approximately 1.26, without affecting the center distance, making it the preferred low-cost improvement solution.

[0063] Secondly, material upgrade suggestions: search the built-in plastic material library for candidate materials that meet the current static strength and PV requirements. For example, switching to PA66-GF40 would increase σFlim by about 18% (about 213MPa). Increased to 5.2 MPa·m / s; cost increased by approximately 15%, while the manufacturing difficulty remained essentially unchanged; alternatively, POM could be used, with similar flexural modulus. It is superior to PA66, with minimal moisture absorption and expansion (which can significantly improve the problem of center distance in heat deformation); the strength improvement is limited, and the cost is similar.

[0064] Following this are suggestions for improving thermal management, including the current change in the center distance of thermal deformation. The value has reached 89.6% of the limit. The following thermal management measures are recommended: switch from grease lubrication to circulating oil lubrication. It is expected that... It can be increased to 150~200W / (m 2 ·K), steady-state body temperature This can reduce the temperature by approximately 8-12℃, corresponding to a reduction of about 15% in thermal deformation due to center distance; adding aluminum alloy heat dissipation fins to the outer wall of the gearbox can quantitatively reduce the steady-state temperature by approximately 6℃, improving the center distance variation by approximately 11%; reducing n1 from 1500r / min to 1350r / min Reduce by 10%, The contact flash temperature dropped to 2.88 MPa·m / s. The temperature dropped to about 24°C. When the temperature is reduced to about 94°C, the thermal deformation at the center distance can be reduced by about 6%.

[0065] Finally, regarding wheel structure optimization suggestions, increasing the number of spokes from 6 to 8 can improve heat dissipation uniformity, reduce circumferential temperature non-uniformity by approximately 22%, and reduce center distance deviation caused by thermal warpage; adjusting the ratio of spoke thickness to rim thickness to between 0.55 and 0.65 can reduce warpage deformation caused by uneven cooling by approximately 15% to 20%; and adjusting the gate position with the support of mold flow analysis tools to control the fiber orientation angle at the critical section of the tooth root to ≤20° (along the direction of bending stress at the tooth root) can further improve the bending strength in this area by approximately 15% to 25%.

[0066] Finally, it should be noted that the described embodiments are merely some, not all, of the embodiments of the present invention. Those skilled in the art will understand that various changes, modifications, substitutions, and variations can be made to these embodiments without departing from the principles and spirit of the present invention. The scope of the present invention is defined by the claims and their equivalents; that is, all other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.

Claims

1. A finite element analysis system for the static strength of a plastic worm gear and a metal worm, characterized in that, The system comprises the following modules connected in sequence: The parametric modeling module generates three-dimensional geometric models of the worm helical tooth surface and the worm wheel envelope tooth surface based on the parameters of the worm gear transmission pair. The mesh generation module divides the model by configuring the mesh density according to the stress gradient differences in each region of the three-dimensional geometric model. The constitutive configuration module handles the mechanical behavior of plastic materials under the influence of temperature, humidity and strain rate, and corrects the elastic modulus and yield stress of the material in real time. The load and boundary condition application module converts engineering load parameters into finite element boundary conditions and handles the frictional heat generation and thermo-mechanical coupling effect at high sliding speeds in plastic worm gear pairs. The solution and post-processing module uses an iterative method to solve the composite finite element equation containing material nonlinearity, contact nonlinearity and thermo-mechanical coupling, and extracts multiple indices for static strength evaluation. The verification module performs intensity verification based on the aforementioned indicators; The sensitivity analysis and optimization suggestion module is activated when any strength verification conclusion is unqualified or the safety factor is lower than the target value. It identifies significant parameters through sensitivity calculation and generates design improvement suggestions.

2. The static strength finite element analysis system according to claim 1, characterized in that, When the parametric modeling module generates the worm gear envelope tooth surface through numerical solution, it sets the worm rotation angle as follows: The worm gear tooth surface is composed of a set of points satisfying the following meshing equation. In the formula This is the unit normal vector at the corresponding parameter point on the worm gear tooth surface, transformed to the worm wheel coordinate system. Let be the position vector of the point on the worm gear tooth surface in the worm wheel coordinate system. The system describes the direction of the relative velocity of a point in the worm gear coordinate system during worm gear rotation. Newton's iteration method is used to solve for the discrete point cloud of the worm gear tooth surface point by point. The system also automatically reconstructs the transition fillet at the root of the plastic worm gear tooth. The basic radius formula for the transition fillet is... .

3. The static strength finite element analysis system according to claim 2, characterized in that, The mesh generation module uses the following criteria when configuring mesh density: Calculate the corrected Hertzian contact half-width for plastic after temperature and moisture absorption correction. ; Controlling the mesh cell size of the tooth surface contact area To ensure that at least 8 nodes are arranged within the contact half-width range; Control the mesh size of the tooth root transition fillet area .

4. The static strength finite element analysis system according to claim 1, characterized in that, In the constitutive configuration module, the temperature T and moisture absorption are considered. The expression for the effective elastic modulus after superposition correction is: In addition, the module uses the Blok flash temperature formula to calculate the peak temperature rise in the contact area. If the sum of the contact flash temperature and the working body temperature exceeds the heat distortion temperature of the worm gear material, the system will automatically issue a warning of adhesion failure risk.

5. The static strength finite element analysis system according to claim 1, characterized in that, The load and boundary condition application module is configured with the following two thermo-mechanical coupling analysis modes for use: Steady-state thermo-mechanical decoupling analysis: First, steady-state thermal analysis is performed to calculate the stable operating temperature of the gear, and then the temperature field is imported into static analysis to correct the material parameters in real time; Fully Coupled Thermo-Mechanical Analysis: Solving the heat conduction equation and the elastoplastic mechanics equation simultaneously in the same model, achieving bidirectional coupling through temperature-dependent material parameters and frictional heat source terms; After the load calculation is completed, the system automatically calculates the local load. value If the largest Materials with values ​​exceeding the current contact temperature If the limit is reached, a special check for adhesion failure will be triggered.

6. The static strength finite element analysis system according to claim 1, characterized in that, The solution and post-processing module uses the Newton-Raphson iterative method to solve the composite equations and extracts the following six key evaluation indicators for EPS worm gear pairs: Maximum principal stress, von Mises equivalent stress, maximum contact pressure, equivalent plastic strain, Tsai-Hill failure index, and maximum nodal thermal displacement.

7. The static strength finite element analysis system according to claim 1 or 6, characterized in that, The verification module performs the following six specific judgments: The verification includes: root bending static strength, tooth surface contact static strength, PV limit, plastic deformation exceeding limit, anisotropic failure, and hot deformation center distance. Among them, the thermal deformation center distance verification comprehensively calculates the influence of thermal expansion and hygroscopic expansion on the center distance. An interference warning is issued when the clearance exceeds 50% of the nominal tooth flank clearance.

8. The static strength finite element analysis system according to claim 1, characterized in that, The sensitivity analysis and optimization suggestion module will apply a ±10% perturbation to the optimization parameter dimension and use a combined approximate fast reanalysis algorithm based on Neumann series expansion to calculate the sensitivity coefficient. The multi-dimensional design improvement suggestions include the following four dimensions: estimated minimum geometric parameter adjustment amount, candidate plastic material upgrade path with comprehensive cost performance, thermal management improvement suggestions to reduce contact area temperature, and hub / spoke structure optimization suggestions for thermal deformation and warping.

9. A finite element method for static strength analysis of plastic worm gear and metal worm, characterized in that, The method includes the following steps: Based on the complete set of design parameters of the worm gear transmission pair entered by the user, the meshing equation is solved using the Newton-Raphson iteration method to generate an accurate three-dimensional geometric model and automatically generate an adaptive mesh. Configure a nonlinear constitutive model of plastic with temperature and moisture absorption superposition correction, and apply multi-field load boundary conditions with thermo-mechanical coupling effect; The Newton-Raphson iterative method was used to solve the composite nonlinear finite element equations and extract six evaluation indicators. The system sequentially performs six specific strength checks: root bending static strength, tooth surface contact static strength, PV limit, plastic deformation exceeding limit, anisotropic failure, and thermal deformation center distance, and outputs the check conclusions.

10. The method according to claim 9, characterized in that, When any strength verification result is unqualified or the safety factor is lower than the target value, the following steps are also included: apply ±10% perturbation to each of the optimization parameters such as module, tooth root fillet, and lead angle; use a combined approximation algorithm based on Neumann series expansion to quickly calculate the sensitivity coefficient; and generate geometric parameter adjustment amount, candidate material upgrade path, thermal management improvement measures to reduce contact area temperature, and wheel spoke structure optimization suggestions in order of sensitivity coefficient sorting to form a multi-dimensional priority ranking design improvement scheme.

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