A high-precision gear pair point contact elastohydrodynamic lubrication model solving method

Abstract

This invention discloses a method for solving a high-precision gear pair point contact elastohydrodynamic lubrication model. By constructing a high-precision gear pair tooth surface contact model, the oil film pressure and oil film thickness distribution of the gear pair during meshing are iteratively solved. Multigrids are used to realize information transfer and error correction between grids of different scales, which effectively improves the convergence speed and computational stability of the elastohydrodynamic lubrication numerical solution, while taking into account both computational accuracy and computational efficiency. This invention can characterize the lubrication characteristics and load-bearing capacity of gear pairs during meshing, providing technical support for the engineering application of high-precision gear pairs.

Description

Technical Field

[0001] This invention relates to the field of gear transmission technology, and in particular to a method for solving a high-precision point contact elastohydrodynamic lubrication model for gear pairs. Background Technology

[0002] High-precision gears, as crucial power transmission components, hold significant strategic importance in modern mechanical transmission systems. The transmission efficiency and service life of gear pairs are influenced by various factors, with lubrication performance being particularly critical. Good lubrication not only reduces friction and wear on the tooth surfaces but also effectively lowers the operating temperature of the gear pair, extending its service life and ensuring the stability and reliability of the transmission system.

[0003] To accurately extract the lubrication characteristics of gear pairs under use, this invention proposes a high-precision solution method for a point contact elastohydrodynamic lubrication model of gear pairs. By constructing a gear pair tooth surface contact model, the oil film pressure and oil film thickness distribution of the gear pair during meshing are iteratively solved. Multigrids are used to realize information transfer and error correction between grids of different scales, which effectively improves the convergence speed and computational stability of the elastohydrodynamic lubrication numerical solution, while taking into account both computational accuracy and computational efficiency. This provides theoretical support for the study of gear pair lubrication characteristics and promotes the development of related engineering technologies. Summary of the Invention

[0004] To overcome the shortcomings of existing technologies and fill related technical gaps, this invention provides a method for solving a high-precision gear pair point contact elastohydrodynamic lubrication model. The technical solution adopted by this invention to solve its technical problem is as follows:

[0005] A method for solving a high-precision gear pair point contact elastohydrodynamic lubrication model, characterized by the following steps:

[0006] Step (1): Input parameter settings. Set the basic input parameters required for gear pair and elastohydrodynamic lubrication calculation: geometric and meshing parameters of the gear pair, such as module m, pressure angle α, number of gear teeth N2, and number of gear shaper teeth N. s Etc; physical properties and operating parameters related to elastohydrodynamic lubrication calculations, specifically including parameters such as the dynamic viscosity η0, density ρ0, and viscosity-pressure coefficient of lubricating oil under standard atmospheric pressure;

[0007] Step (2): Preprocessing for elastohydrodynamic lubrication calculation. After setting the input parameters, perform preprocessing for elastohydrodynamic lubrication calculation on the gear pair; derive the gear pair tooth surface equation r based on the geometric parameters and meshing relationship of the gear pair. f and normal vector n f Based on the gear meshing principle, the gear pair contact equation is obtained, and the contact trace of the gear pair during the meshing process is extracted; the tooth surface overlap ε is calculated according to the meshing relationship of the gear pair, and the contact transition line is determined accordingly to realize the division of the tooth surface contact area;

[0008] Establish the elastic deformation equation δ of the tooth surface contact with respect to the tooth surface projection coordinates (x, y) and the time t experienced by the gear pair during the engagement to disengagement process. t The equations (x, y, t) and the pressure distribution equation p(x, y, t) can be expressed as follows:

[0009] ;

[0010] ;

[0011] Where A and B are the calculation parameters for the contact ellipse of the gear pair; E e K(e) is the equivalent elastic modulus of the gear pair. t ) and E(e t ) represents the first and second type elliptic integrals; a and b are half the lengths of the minor and major axes of the contact ellipse, respectively; e t Let be the eccentricity of the contact ellipse; a0 be the coordinate of the center of the contact ellipse in the x-direction; b0 be the coordinate of the center of the contact ellipse in the y-direction; p m (t) is the maximum Hertzian contact stress as a function of t; F m (t) is a function of the maximum meshing force on the tooth surface with respect to time t, and can be expressed as:

[0012] ;

[0013] Among them, c m (t) is the load distribution coefficient as a function of t; T is the meshing period of a single pair of gear teeth from engagement to disengagement; y c (t) and z c (t) represents the Y and Z axis coordinates of each contact point on the contact line of the cylindrical gear in the contact coordinate system;

[0014] The Reynolds equation describing a gear pair under isothermal contact elastohydrodynamic lubrication can be expressed as follows:

[0015] ;

[0016] Where ρ is the lubricant density; h is the lubricating oil film thickness; η is the lubricant viscosity; ∂(ρh) / ∂t is the oil film time constant; v e (t) and θ e (t) are functions of the entrainment velocity and the entrainment angle with respect to t, respectively;

[0017] The gear pair oil film thickness function h(x, y, t) with respect to the tooth surface position (x, y) and t can be expressed as:

[0018] ;

[0019] Where h0(t) is the thickness of the oil film center as a function of t, R ea (t) and R eb (t) is a function of t for the equivalent radii of the gear pair on the major and minor axes of the contact ellipse;

[0020] The effect of pressure on lubricant viscosity, described by the viscosity-pressure equation, can be viewed as a function η(x, y, t) of the tooth surface position (x, y) versus time t, and can be expressed as:

[0021] ;

[0022] ;

[0023] Where η0 is the viscosity of the lubricating oil under standard atmospheric pressure; α r c is the viscosity-pressure coefficient of the lubricating oil. p This is the lubricant pressure coefficient.

[0024] The effect of pressure on lubricant density can be viewed as a function ρ(x, y,t) of tooth surface position (x, y) and time t, and can be expressed as:

[0025] ;

[0026] Where ρ0 is the density of the lubricating oil under standard atmospheric pressure;

[0027] The pressure distribution equation p(x, y, t) within the solution domain Ω at any time t must satisfy the load balance equation, which can be expressed as:

[0028] ;

[0029] By combining the Reynolds equation, oil film thickness equation h(x, y, t), lubricant viscosity-pressure equation η(x, y, t), pressure-pressure equation ρ(x, y, t), and load balance equation of the gear pair, the elastohydrodynamic lubrication mathematical model of the gear pair is obtained.

[0030] Select dimensionless parameters: dimensionless coordinate X, dimensionless coordinate Y, dimensionless oil film thickness H, dimensionless oil film pressure P, and dimensionless lubricating oil viscosity. Dimensionless lubricating oil density Dimensionless time The dimensionless representation of the Reynolds equation can be expressed as:

[0031] ;

[0032] ;

[0033] Where, ε x and εy For equation operators;

[0034] The remaining elastohydrodynamic lubrication model governing equations, after being dimensionlessized, can be expressed as follows:

[0035] ;

[0036] ;

[0037] Where k is the contact ellipticity;

[0038] Step (3): Multigrid parameter setting and mesh initialization, constructing a multigrid calculation system for elastohydrodynamic lubrication numerical solution; setting the two-dimensional solution domain Ω for elastohydrodynamic lubrication calculation, which covers the instantaneous contact area of ​​the tooth surface and its necessary extended area; setting the number of mesh layers N required for multigrid, the number of nodes in the top layer mesh, and the mesh spacing ΔX and ΔY of each layer mesh in the dimensionless coordinate system; the node spacing of each upper-level network is twice that of the next network node; according to the mesh hierarchy, the mesh is generated layer by layer from coarse to fine, and the pressure field P(X, Y, ...) on each layer mesh is calculated. ) and oil film thickness field H(X, Y, Perform initialization settings;

[0039] Step (4): Solving the multigrid elastohydrodynamic lubrication problem. After the multigrid system is constructed, the elastohydrodynamic lubrication problem of the gear pair is solved. The initial dimensionless pressure P and the initial dimensionless oil film thickness H are given on the top grid, and the coefficients of each item in the discrete elastohydrodynamic lubrication equation are calculated accordingly.

[0040] Discretized at the grid nodes, the Reynolds equation can be expressed as:

[0041] ;

[0042] Where L is the coefficient matrix; P is the dimensionless pressure P at the multigrid node (i, j). i, j The column vector is composed of the Reynolds equations; f is the discrete vector composed of the right side of the Reynolds equations; the superscript h indicates the grid level (h=1, 2, 3, …, N).

[0043] The dimensionless film thickness discretization form H at the multi-grid node (i, j) i, j It can be represented as,

[0044] ;

[0045] Among them, X i and Y j P represents the dimensionless coordinates at node (i, j); m, n D represents the dimensionless pressure at node (m, n);m, n i, j For P m, n The deformation influence coefficient at node (i, j);

[0046] The load balance equation at the multigrid node (i, j) can be discretized as follows:

[0047] ;

[0048] On a fine mesh h, the initial solution P of the Reynolds equations h Perform v1 iterations to reduce high-frequency errors; calculate the residual r based on the current solution. h Using the restricted operator I h H The residual is transferred from the fine grid h to the coarse grid H, and the pressure distribution on the top grid is updated row by row. The viscosity, density, and oil film thickness distribution of the lubricating oil are simultaneously corrected based on the updated pressure field. A correction equation L is constructed on the coarse grid H. H e H = r H , here e H For the correction term of the solution, if the bottom layer of the mesh has been reached, the solution is directly obtained; the correction term of the coarse mesh H is passed through operator I. h H Interpolate back to the finer mesh h, update the solution on the finer mesh, and perform v2 iterations on the finer mesh to further reduce residual error;

[0049] During the iteration process, the convergence of the calculation results is judged by setting the pressure residual convergence criterion ΔP and the load residual convergence criterion ΔF. The pressure residual convergence value Δp and the load residual convergence value ΔF used to determine whether the target network has converged can be expressed as follows:

[0050] ;

[0051] ;

[0052] Among them, P i, j n and P i, j n 'W represents the dimensionless pressure at node (i, j) before and after iteration on the Nth layer mesh. p δ is the relaxation factor. i, j For the calculation parameters of the discrete Reynolds equations, Δp and ΔF can be adjusted according to the requirements of calculation accuracy and efficiency.

[0053] When the convergence conditions of both pressure residual and load residual are met simultaneously, the current loop calculation is considered complete; if the convergence conditions are not met, the next loop iteration continues until the calculation results converge.

[0054] Step (5): Post-processing of elastohydrodynamic lubrication calculation. After completing the multigrid elastohydrodynamic lubrication solution, post-processing analysis is performed on the calculation results. The dimensionless oil film thickness H and pressure P distribution are restored to actual values ​​with physical dimensions. A distribution relationship diagram between dimensionless coordinates and oil film thickness h and contact pressure p is plotted within the solution domain Ω to characterize the elastohydrodynamic lubrication characteristics of the gear pair during meshing. Physical quantity data, such as minimum oil film thickness, maximum contact pressure, and load-bearing capacity, are extracted from the calculation results. Attached Figure Description

[0055] Figure 1 This is a flowchart illustrating the implementation of a high-precision gear pair point contact elastohydrodynamic lubrication model solution method proposed in this invention.

[0056] Figure 2 This is a schematic diagram of the contact between the rough surfaces of the gear pair of the present invention;

[0057] Figure 3 This is a schematic diagram of the contact trace of the gear tooth surface in this invention;

[0058] Figure 4 This is a schematic diagram of the oil film pressure under elastohydrodynamic lubrication of the gear pair according to the present invention;

[0059] Figure 5 This is a schematic diagram of the oil film thickness under elastohydrodynamic lubrication of the gear pair according to the present invention. Detailed Implementation

[0060] Embodiments of the present invention will be described with reference to the accompanying drawings, which will be further described below. Figure 1 — Figure 5 The specific embodiments of the present invention will be described in detail below.

[0061] Reference Figure 1 A method for solving a high-precision gear pair point contact elastohydrodynamic lubrication model includes the following steps:

[0062] Step (1): Input parameter settings. Set the basic input parameters required for gear pair and elastohydrodynamic lubrication calculations. These input parameters include, but are not limited to: the geometric and meshing parameters of the gear pair, such as module m, pressure angle α, number of gear teeth N2, and number of gear shaper teeth N. s Etc; physical properties and operating parameters related to elastohydrodynamic lubrication calculations, specifically including parameters such as the dynamic viscosity η0, density ρ0, and viscosity-pressure coefficient of the lubricating oil under standard atmospheric pressure.

[0063] Step (2): Pre-processing for elastohydrodynamic lubrication calculation. After setting the input parameters, pre-processing for elastohydrodynamic lubrication calculation is performed on the gear pair. Based on the geometric parameters and meshing relationship of the gear pair, the gear tooth surface equation r is derived. f (φ f ,, θ g, u g ) and normal vector n f (φ f , θ g ), can be represented as:

[0064] (1);

[0065] Where, r g (θ g , u g ) represents the tooth surface equation of the gear producing gear; n g (θ g ) is the normal vector of the gear-generating gear; φ f θ is the angle through which the gear rotates about its own axis. g For the tooth profile parameters of the gear; u g M represents the tooth profile parameter of the gear. fg (φ f L is the coordinate transformation matrix from the gear coordinate system to the gear coordinate system; fg (φ f ) is M fg (φ f Remove the homogeneous coordinates from the 3×3 matrix.

[0066] like Figure 2 As shown, solve the tooth surface equation r of the gear pair. f This can generate discrete point cloud data of the gear tooth surface. Based on the gear meshing principle, the gear pair contact equation is obtained, which can be expressed as:

[0067] (2);

[0068] Where, r f (a) (φ2, φ g , θ g ) is in the contact coordinate system S a The gear tooth surface equation in n; f (a) (φ g , θ g ) is in the contact coordinate system S a The gear normal vector in the image; r c (a) (φ1, θ c , u c ) is in the contact coordinate system S a The equation for the tooth surface of a cylindrical gear in n; c (a) (θ c ) is in the contact coordinate system S aThe normal vector in the coordinate system; φ1 is the angle through which the cylindrical gear rotates about its own axis in the contact coordinate system; φ2 is the angle through which the cylindrical gear rotates about its own axis in the contact coordinate system; θ c For the tooth profile parameters of a cylindrical gear; u c For the tooth direction parameters of the cylindrical gear.

[0069] Solving the contact equations of the gear pair allows us to extract the contact traces of the gear pair during meshing, such as... Figure 3 As shown. The tooth surface overlap ratio ε is calculated based on the meshing relationship of the gear pair, and the contact transition line is determined accordingly, thereby realizing the division of the tooth surface contact area.

[0070] Under the assumption of no lubrication, we establish the elastic deformation equation for the tooth surface contact δ, which relates to the tooth surface projection coordinates (x, y) and the time t experienced by the gear pair during engagement to disengagement. t The equations (x, y, t) and the pressure distribution equation p(x, y, t) can be expressed as follows:

[0071] (3);

[0072] (4);

[0073] Where A and B are the calculation parameters for the contact ellipse of the gear pair; E e K(e) is the equivalent elastic modulus of the gear pair. t ) and E(e t ) represents the first and second type elliptic integrals; a and b are half the lengths of the minor and major axes of the contact ellipse, respectively; e t Let be the eccentricity of the contact ellipse; a0 be the coordinate of the center of the contact ellipse in the x-direction; b0 be the coordinate of the center of the contact ellipse in the y-direction; p m (t) is the maximum Hertzian contact stress as a function of t; F m (t) is a function of the maximum meshing force on the tooth surface with respect to time t, and can be expressed as:

[0074] (5);

[0075] Among them, c m (t) is the load distribution coefficient as a function of t; T is the meshing period of a single pair of gear teeth from engagement to disengagement; y c (t) and z c (t) represents the Y and Z axis coordinates of each contact point on the contact line of the cylindrical gear in the contact coordinate system.

[0076] The Reynolds equation describing a gear pair under isothermal contact elastohydrodynamic lubrication can be expressed as follows:

[0077] (6);

[0078] Where ρ is the lubricant density; h is the lubricating oil film thickness; η is the lubricant viscosity; ∂(ρh) / ∂t is the oil film time constant; v e (t) and θ e (t) are functions of the entrainment velocity and entrainment angle with respect to t, respectively.

[0079] The gear pair oil film thickness function h(x, y, t) with respect to the tooth surface position (x, y) and t can be expressed as:

[0080] (7);

[0081] Where h0(t) is the thickness of the oil film center as a function of t, R ea (t) and R eb (t) is the equivalent radius of the gear pair on the major and minor axes of the contact ellipse as a function of t.

[0082] The effect of pressure on lubricant viscosity, described by the viscosity-pressure equation, can be viewed as a function η(x, y, t) of the tooth surface position (x, y) versus time t, and can be expressed as:

[0083] (8);

[0084] (9);

[0085] Where η0 is the viscosity of the lubricating oil under standard atmospheric pressure; α r c is the viscosity-pressure coefficient of the lubricating oil. p This is the lubricant pressure coefficient.

[0086] The effect of pressure on lubricant density can be viewed as a function ρ(x, y,t) of tooth surface position (x, y) and time t, and can be expressed as:

[0087] (10);

[0088] Where ρ0 is the density of the lubricating oil under standard atmospheric pressure.

[0089] The pressure distribution equation p(x, y, t) within the solution domain Ω at any time t must satisfy the load balance equation, which can be expressed as:

[0090] (11);

[0091] By combining the Reynolds equation, oil film thickness equation h(x, y, t), lubricant viscosity-pressure equation η(x, y, t), viscosity-pressure equation ρ(x, y, t), and load balance equation of the gear pair, the elastohydrodynamic lubrication mathematical model of the gear pair can be obtained.

[0092] Select dimensionless parameters: dimensionless coordinate X, dimensionless coordinate Y, dimensionless oil film thickness H, dimensionless oil film pressure P, and dimensionless lubricating oil viscosity. Dimensionless lubricating oil density Dimensionless time The dimensionless representation of the Reynolds equation can be expressed as:

[0093] (12);

[0094] (13);

[0095] Where, ε x and ε y For equation operators.

[0096] The remaining elastohydrodynamic lubrication model governing equations, after being dimensionlessized, can be expressed as follows:

[0097] (14);

[0098] (15);

[0099] Where k is the contact ellipticity.

[0100] Step (3): Multigrid parameter setting and mesh initialization. After completing the preprocessing for elastohydrodynamic lubrication, a multigrid calculation system for numerical solution of elastohydrodynamic lubrication is constructed. The two-dimensional solution domain Ω for elastohydrodynamic lubrication calculation is set, covering the instantaneous contact region of the tooth surface and its necessary extended region. The number of mesh layers N required for the multigrid method, the number of nodes in the top layer mesh, and the mesh spacing ΔX and ΔY of each layer mesh in the dimensionless coordinate system are set. The number of nodes in the top layer mesh is (2... N+5 +1)×(2 N+5 +1), the next level (Level N-1) is (2 N+4 +1)×(2 N+4 +1), and so on. The spacing between nodes in each higher-level network is twice that of the nodes in the next network. According to the hierarchical relationship of the multigrid method, the grid is generated layer by layer from coarse to fine, and the pressure field P(X, Y, ...) on each layer of the grid is... ) and oil film thickness field H(X, Y, Perform initialization settings.

[0101] Step (4): Solving the multigrid elastohydrodynamic lubrication problem. After the multigrid system is constructed, the multigrid method is used to solve the elastohydrodynamic lubrication problem of the gear pair. The initial dimensionless pressure P and the initial dimensionless oil film thickness H are given on the top-level grid, and the coefficients of each term in the discrete elastohydrodynamic lubrication equation are calculated accordingly.

[0102] Discretized at the grid nodes, the Reynolds equation can be expressed as:

[0103] (16);

[0104] Where L is the coefficient matrix; P is the dimensionless pressure P at the multigrid node (i, j). i, j The column vector is composed of the Reynolds equation; f is the discrete vector composed of the right side of the Reynolds equation; the superscript h indicates the grid level (h=1, 2, 3, …, N).

[0105] The dimensionless film thickness discretization form H at the multi-grid node (i, j) i, j It can be represented as,

[0106] (17);

[0107] Among them, X i and Y j P represents the dimensionless coordinates at node (i, j); m, n D represents the dimensionless pressure at node (m, n); m, n i, j For P m, n The deformation influence coefficient at node (i, j).

[0108] The load balance equation at the multigrid node (i, j) can be discretized as follows:

[0109] (18);

[0110] On a fine mesh h, the initial solution P of the Reynolds equations h Perform v1 iterations to reduce high-frequency errors; calculate the residual r based on the current solution. h Using the restricted operator I h H The residual is transferred from the fine grid h to the coarse grid H, and the pressure distribution on the top grid is updated row by row iteratively. Based on the updated pressure field, the viscosity, density, and oil film thickness distribution of the lubricating oil are simultaneously corrected. A correction equation L is constructed on the coarse grid H. H e H = r H , here e H For the correction term of the solution, if the bottom layer of the mesh has been reached, the solution is directly obtained; the correction term of the coarse mesh H is passed through operator I.h H Interpolate back to the finer mesh h, update the solution on the finer mesh, and perform v2 iterations on the finer mesh to further reduce residual error.

[0111] During the iteration process, the convergence of the calculation results is judged by setting the pressure residual convergence criterion ΔP and the load residual convergence criterion ΔF. The pressure residual convergence value Δp and the load residual convergence value ΔF used to determine whether the target network has converged can be expressed as follows:

[0112] (19);

[0113] (20);

[0114] Among them, P i, j n and P i, j n 'W represents the dimensionless pressure at node (i, j) before and after iteration on the Nth layer mesh. p δ is the relaxation factor. i, j These are the calculation parameters for the discrete Reynolds equations. In this embodiment, ΔP is set to 0.00001 and ΔF is set to 0.0001, which can be adjusted according to the requirements of calculation accuracy and efficiency.

[0115] When the convergence conditions of both pressure residual and load residual are met simultaneously, the current loop calculation is considered complete; if the convergence conditions are not met, the next loop iteration continues until the calculation results converge.

[0116] Step (5): Post-processing of elastohydrodynamic lubrication calculation. After completing the multigrid elastohydrodynamic lubrication solution, post-processing analysis is performed on the calculation results; the dimensionless oil film thickness H and pressure P distribution are restored to actual values ​​with physical dimensions; such as... Figure 4 and Figure 5 As shown, a distribution diagram of dimensionless coordinates and oil film thickness h and contact pressure p is plotted within the solution domain Ω to characterize the elastohydrodynamic lubrication characteristics of the gear pair during meshing. Key physical quantities, such as minimum oil film thickness, maximum contact pressure, and load-bearing capacity, are extracted from the calculation results to provide a reliable calculation basis for the lubrication condition assessment, structural design optimization, and engineering applications of the gear pair.

[0117] The above description is merely a preferred embodiment of the invention and does not constitute any limitation on the invention. Any modifications, alterations, or equivalent changes made to the above embodiments based on the essence of the invention shall still fall within the protection scope of the invention.

Claims

1. A method for solving a high-precision gear pair point contact elastohydrodynamic lubrication model, characterized in that: Step (1): Input parameter settings, set the basic input parameters required for gear pair and elastohydrodynamic lubrication calculation: the geometric and meshing parameters of the gear pair, the physical property parameters and operating condition parameters related to elastohydrodynamic lubrication calculation; Step (2): Preprocessing for elastohydrodynamic lubrication calculation. After setting the input parameters, perform preprocessing for elastohydrodynamic lubrication calculation on the gear pair; derive the gear pair tooth surface equation r. f and normal vector n f The contact equation of the gear pair is obtained, and the contact trace of the gear pair during the meshing process is extracted; Establish the elastic deformation equation δ of the tooth surface contact with respect to the tooth surface projection coordinates (x, y) and the time t experienced by the gear pair during the engagement to disengagement process. t (x, y, t) and the pressure distribution equation p(x, y, t); The Reynolds equation describing a gear pair under isothermal contact elastohydrodynamic lubrication can be expressed as follows: ； Where ρ is the lubricant density, h is the lubricating oil film thickness, η is the lubricant viscosity, ∂(ρh) / ∂t is the oil film time constant, and v e (t) and θ e (t) are functions of the entrainment velocity and the entrainment angle with respect to t, respectively; By combining the Reynolds equation, oil film thickness equation h(x, y, t), lubricant viscosity-pressure equation η(x, y, t), pressure-pressure equation ρ(x, y, t), and load balance equation of the gear pair, the elastohydrodynamic lubrication mathematical model of the gear pair is obtained. Select dimensionless parameters: dimensionless coordinate X, dimensionless coordinate Y, dimensionless oil film thickness H, dimensionless oil film pressure P, and dimensionless lubricating oil viscosity. Dimensionless lubricating oil density Dimensionless time The control equations of the elastohydrodynamic lubrication model are dimensionless. Step (3): Multigrid parameter setting and mesh initialization, constructing a multigrid calculation system for elastohydrodynamic lubrication numerical solution; setting the two-dimensional solution domain Ω for elastohydrodynamic lubrication calculation, which covers the instantaneous contact area of ​​the tooth surface and its necessary extended area; setting the number of mesh layers N required for multigrid, the number of nodes in the top layer mesh, and the mesh spacing ΔX and ΔY of each layer mesh in the dimensionless coordinate system; the node spacing of each upper-level network is twice that of the next network node; according to the mesh hierarchy, the mesh is generated layer by layer from coarse to fine, and the pressure field P(X, Y, ...) on each layer mesh is calculated. ) and oil film thickness field H(X,Y, Perform initialization settings; Step (4): Solving the multigrid elastohydrodynamic lubrication problem. After the multigrid system is constructed, the elastohydrodynamic lubrication problem of the gear pair is solved. The initial dimensionless pressure P and the initial dimensionless oil film thickness H are given on the top grid, and the coefficients of each item in the discrete elastohydrodynamic lubrication equation are calculated accordingly. Discretized at the grid nodes, the Reynolds equation can be expressed as: ； Where L is the coefficient matrix; P is the dimensionless pressure P at the multigrid node (i, j). i, j The column vector is composed of the Reynolds equations; f is the discrete vector composed of the right side of the Reynolds equations; the superscript h indicates the grid level (h=1, 2, 3, …, N). On a fine mesh h, the initial solution P of the Reynolds equations h Perform v1 iterations to reduce high-frequency errors; calculate the residual r based on the current solution. h Using the restricted operator I h H The residual is transferred from the fine grid h to the coarse grid H, and the pressure distribution on the top grid is updated row by row. The viscosity, density, and oil film thickness distribution of the lubricating oil are simultaneously corrected based on the updated pressure field. A correction equation L is constructed on the coarse grid H. H e H = r H , here e H For the correction term of the solution, if the bottom layer of the mesh has been reached, the solution is directly obtained; the correction term of the coarse mesh H is passed through operator I. h H Interpolate back to the finer mesh h, update the solution on the finer mesh, and perform v2 iterations on the finer mesh to further reduce residual error; During the iteration process, the convergence of the calculation results is judged by setting the pressure residual convergence criterion ΔP and the load residual convergence criterion ΔF. The pressure residual convergence value Δp and the load residual convergence value ΔF used to determine whether the target network has converged can be expressed as follows: ； ； Among them, P i, j n and P i, j n 'W represents the dimensionless pressure at node (i, j) before and after iteration on the Nth layer mesh. p δ is the relaxation factor. i, j For the calculation parameters of the discrete Reynolds equations, Δp and ΔF can be adjusted according to the requirements of calculation accuracy and efficiency. When the convergence conditions of both pressure residual and load residual are met simultaneously, the current loop calculation is considered complete; if the convergence conditions are not met, the next loop iteration continues until the calculation results converge. Step (5): Post-processing of elastohydrodynamic lubrication calculation. After completing the multigrid elastohydrodynamic lubrication solution, post-processing analysis is performed on the calculation results. The distribution relationship between dimensionless coordinates and oil film thickness h and contact pressure p is plotted in the solution domain Ω.

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