Improved method, medium and electronic device for load-bearing contact analysis of spiral bevel gears

Abstract

This invention belongs to the field of mechanical dynamics technology, and discloses an improved method, medium, and electronic device for load-bearing contact analysis of spiral bevel gears. Based on the tooth contact analysis method, the geometric parameters of the tooth surfaces of the large and small gears are obtained, a load-bearing contact analysis model of the spiral bevel gear pair is constructed, and the overall compliance matrix, overall contact compliance matrix, and local contact compliance matrix of the tooth surface are calculated. A deformation compatibility iterative equation is established, its ill-conditioned nature is analyzed, and a preprocessing matrix is ​​established to improve the deformation compatibility iterative equation. After improvement, the time-varying meshing stiffness and tooth surface load are solved. Based on the tooth surface load, the principal curvature of the tooth surface of the large and small gears is calculated using the Euler analytical method and the principal direction is determined. The contact stress is then solved using Hertz contact theory. This method improves upon the problems of traditional tooth load-bearing contact analysis methods, which suffer from low accuracy and efficiency due to the ill-conditioned nature of the contact compliance matrix under low load conditions and the reliance on numerical differences for curvature calculation.

Description

Technical Field

[0001] This invention relates to the field of mechanical dynamics, specifically to an improved method, medium, and electronic device for analyzing the load-bearing contact of spiral bevel gears. Background Technology

[0002] Gear tooth bearing contact analysis is a common method for calculating time-varying meshing stiffness. It is based on iteratively solving the normal load and compliance matrix of the deformation compatibility equation to obtain the meshing stiffness. Hertz contact theory is a classic theory for solving contact stress. Its core is to simplify the contact problem into an elliptical contact area and establish analytical relationships between contact size, maximum pressure and load, material and relative curvature.

[0003] However, compared to parallel shaft gears, existing methods for analyzing the contact under load in spiral bevel gears are less efficient and accurate. They do not consider the ill-conditioned nature of the contact compliance matrix under low load conditions, nor do they address the accuracy loss caused by numerical difference calculations for curvature. Currently, there is a lack of a method that can balance computational efficiency and accuracy and is suitable for calculating the time-varying meshing stiffness and contact stress of spiral bevel gear pairs. Summary of the Invention

[0004] In view of this, the purpose of this invention is to propose an improved method, medium, and electronic device for analyzing the bearing contact of spiral bevel gears, in order to solve the technical problems of ill-conditioned contact compliance matrix and accuracy loss caused by numerical difference curvature calculation under low load conditions.

[0005] The technical solution adopted in this invention is as follows: an improved method for analyzing the bearing contact of spiral bevel gears, comprising the following steps:

[0006] S1. Obtain the geometric parameters of the tooth surfaces of the large and small gears based on the tooth contact analysis method;

[0007] S2. Based on the geometric parameters of the tooth surfaces of the large and small gears, construct a bearing contact analysis model for the spiral bevel gear pair, and then calculate the overall compliance matrix, overall contact compliance matrix, and local contact compliance matrix of the tooth surface.

[0008] S3. Establish deformation coordination iterative equations, analyze the ill-conditioned nature of deformation coordination iterative equations, and establish a preprocessing matrix to improve deformation coordination iterative equations; after improvement, solve for time-varying meshing stiffness and tooth surface load.

[0009] S4. Based on the tooth surface load, the principal curvature of the gear tooth surface is calculated and the principal direction is determined using the Euler analytical method, and the contact stress is solved by combining the Hertz contact theory.

[0010] S1 specifically includes the following steps:

[0011] S11. Based on the transformation of the machining coordinate system and the principle of tool-tooth surface conjugation, the gear tooth surface parameter equation is derived.

[0012] For spiral bevel gear pairs, the parameter equations of the large gear tooth surface are derived based on the transformation of the machining coordinate system and the principle of tool-tooth surface conjugation. The meshing equations of the large gear tooth surface and the tool and the gear rotation projection surface equations are combined to obtain the node coordinates of the large gear tooth surface.

[0013] Large gear tooth surface S g Envelope surface function r 2 is represented as:

[0014]

[0015] In the formula, M 2g This represents the coordinate transformation matrix from the tool coordinate system to the blank coordinate system. s g , i g and P 2 represents the straight cutting edge length, tool tip rotation angle, and large gear rotation angle of the machining tool. r g For the tooth surface of the large gear S g The tooth surface vector function;

[0016] Then, the meshing equation between the large gear tooth surface and the cutting tool is obtained. :

[0017]

[0018] In the formula, Represents the differential form;

[0019] The equation for the rotational projection plane of the gear is expressed as:

[0020]

[0021] In the formula, r 2( x ), r 2( y ), r 2( z These are the coordinates of the gear tooth surface points in the blank coordinate system. x , y , z about s g , i g The function; X M for M Point on the rotational projection plane of the gear X Axis coordinates; YM for M Point on the rotational projection plane of the gear Y Axis coordinates; M The point is any node on the tooth surface of the large gear;

[0022] S12. Solve for the coordinates of the nodes on the tooth surface of the small gear using the method for solving the coordinates of the nodes on the tooth surface of the large gear.

[0023] S13. Establish and assemble the geometric model of the tooth surface of the large and small gears;

[0024] Based on the coordinates of the nodes on the tooth surface of the large gear and the small gear, a geometric model of the tooth surfaces of both gears is established. According to the assembly conditions of the spiral bevel gear pair, the tooth surfaces of the large gear and the small gear must have a common contact point. The large gear and the small gear are rotated around their respective axes by a certain angle until their tooth surfaces contact each other. In the assembly coordinate system, the tooth surface function and normal vector of the large gear and the tooth surface function and normal vector of the small gear are expressed as follows:

[0025]

[0026] In the formula, f p The pinion rotation angle at different meshing moments; f g To correspond to the rotation angle of the large gear at the moment of meshing, M z1 , M z2 These represent the transformation matrices from the blank coordinate system to the assembly coordinate system for the pinion and from the blank coordinate system to the assembly coordinate system for the gear, respectively. Both are 4-row, 4-column matrices. L z1 , L z2 They are respectively M z1 The matrix consists of 3 rows and 3 columns. M z2 A matrix consisting of 3 rows and 3 columns; s p , i p and P 1 represents the straight cutting edge length of the pinion machining tool, the tool tip rotation angle, and the pinion rotation angle, respectively. r z1 , r z2 These are the tooth surface functions of the pinion tooth surface and the gear tooth surface in the assembly coordinate system, respectively. , These are the unit normal vectors of the envelope surface function of the pinion gear and the envelope surface function of the gear, respectively. , These are the unit normal vectors of the pinion tooth surface function in the assembly coordinate system and the unit normal vectors of the gear tooth surface function in the assembly coordinate system, respectively.

[0027] S2 specifically includes the following steps:

[0028] S21. Establish a bearing contact analysis model for spiral bevel gear pairs and obtain the overall compliance matrix of the tooth surface;

[0029] Based on the tooth surface function and normal vector of the large gear and the small gear, a bearing contact analysis model of the spiral bevel gear pair is constructed using the tooth bearing contact analysis method. Fixed constraints are applied to the inner hole nodes of the spiral bevel gear pair, and unit forces are sequentially applied to each point on the tooth surface. The displacement of the tooth surface points after each loading is recorded, and the overall tooth surface compliance matrix corresponding to each meshing moment is constructed. , represented as:

[0030]

[0031] In the formula, superscript K Indicates the moment of engagement; subscript j Indicates the point of application of the unit force; subscript i Indicates the displacement extraction point; n This represents the total number of nodes on all gear contact surfaces;

[0032] S22. Solve for the overall contact compliance matrix and the local contact compliance matrix;

[0033] The displacement of the inner layer nodes on the tooth surface characterizes the overall displacement of the spiral bevel gear pair; the deformation generated on the gear tooth surface includes the overall contact compliance matrix of the gear. Local contact compliance matrix of tooth surface ;

[0034] Based on the overall compliance matrix of the large and small gears The nodes on the tooth surfaces of the large and small gears are densified using cubic spline interpolation. Then, the distances between potential contact points and all nodes on the tooth surfaces of the large and small gears are calculated. The interpolation point closest to the potential contact point, or the compliance of the large or small gear tooth surface node, is used as the compliance of the potential contact point, thus obtaining the compliance matrix of the potential contact point of the large gear. l G Potential contact point compliance matrix of pinion l P The final potential contact point overall contact compliance matrix Represented as:

[0035]

[0036] In the formula, the superscripts P and G represent the pinion and gear, respectively; This indicates the potential contact point compliance of the large gear at the moment of meshing. This indicates the potential contact point compliance of the pinion at the moment of meshing; k This indicates the potential contact point for extracting displacement. l Indicates the potential point of contact where force is applied; m This represents the number of potential contact points discrete along the long axis of the contact at the moment of meshing.

[0037] Local contact deformation of the tooth surface is caused by the contact between the tooth surfaces due to elastic deformation, and the corresponding local contact compliance matrix is... , represented as:

[0038]

[0039] In the formula, E It is the elastic modulus; L The distance between potential contact points; F k It is the first k Normal meshing force at each potential contact point; l ck Indicates the first k The contact compliance value at the potential contact point for extracting displacement.

[0040] S3 specifically includes the following steps:

[0041] S31. Establishment of deformation coordination iterative equations and solution of meshing stiffness and tooth surface load;

[0042] The deformation compatibility iterative equation is expressed as:

[0043]

[0044] In the formula, and These represent the radial row distance matrix from the initial contact point of the pinion to the axis of rotation and the column distance matrix from the initial contact point of the pinion to the axis of rotation, respectively. This represents the force distribution matrix in the normal direction of the potential contact point; The gap matrix represents the potential contact points; i f This indicates the angle of rotation of the pinion under load after it receives torque; T This indicates the input torque of the pinion;

[0045] After solving the deformation compatibility iterative equations, the normal load vector corresponding to each meshing position is obtained. F b and the normal compliance matrix at the actual contact point l bThe normal deformation of a spiral bevel gear pair at any contact point is expressed as:

[0046]

[0047] The tooth surface load corresponding to the discrete potential contact points along the long axis at the moment of meshing is expressed as:

[0048]

[0049] In the formula, d 1 represents the cone angle of the pinion; α n Indicates the normal pressure angle; β Indicates the average helix angle. R Indicates the cone distance; F i It is the first i Normal meshing force at each potential contact point;

[0050] The meshing stiffness of a spiral bevel gear pair at any contact position is expressed as:

[0051]

[0052] In the formula, u bi For the first i Normal deformation at each potential contact point;

[0053] S32. Establishment and application of preprocessing matrix;

[0054] Input torque of the pinion T Under the condition of ≤400 N·m, the original coefficient matrix is ​​defined. A for:

[0055]

[0056] Construct a preprocessing matrix P , represented as:

[0057]

[0058] In the formula, I It is the identity matrix;

[0059] The improved deformation compatibility iterative equation is expressed as:

[0060]

[0061] After obtaining and solving the improved deformation coordination iterative equation, the meshing stiffness of the spiral bevel gear pair at any contact position is... k mesh The expression remains unchanged;

[0062] Input torque of the pinion T Under working conditions >400 N·m, the deformation coordination iterative equation of the spiral bevel gear pair remains unchanged, and the meshing stiffness of the spiral bevel gear pair at any contact position is... k mesh The expression remains unchanged.

[0063] S4 specifically includes the following steps:

[0064] S41, Euler's analytical method: Foundations of differential geometry;

[0065] At the contact point M m Construct a common tangent plane at point A that is tangent to both the tooth surfaces of the large gear and the small gear. S pg The common tangent plane along any tangent vector t normal curvature The degree of curvature of the gear tooth surfaces along this direction is represented as:

[0066]

[0067] In the formula, and These represent the large gear along the tangent vector. t Normal curvature, pinion along tangent vector t The normal curvature;

[0068] Using the tooth bearing contact analysis method, three adjacent tooth surface nodes along the minor axis of the contact ellipse are obtained, denoted as follows: P 1( x 1, y 1, z 1) P 2( x 2, y 2, z 2) and P 3( x 3, y 3, z 3);

[0069] The tooth surfaces of large and small gears are P 1. P 2 and P The first-order partial derivative expression at point 3 is considered equivalent, only indicating the tooth surface at... P The first-order partial derivative at point 1 is:

[0070]

[0071] In the formula, r u They represent the tooth surfaces of the large and small gears respectively.u Tangent vector whose coordinate direction changes; r v Indicates the tooth surface of the large and small gears along v Tangent vector whose coordinate direction changes;

[0072] The first and second basic form coefficients are expressed as follows:

[0073]

[0074] In the formula, s Let be the unit normal vector of the two gear tooth surfaces; O , P , G The coefficients are the first fundamental form coefficients; Q , M , N The coefficients are the second fundamental form.

[0075] in, r uu for r u The second-order partial derivative; r uv for r u and r v The second-order partial derivative; r vv for r v The second-order partial derivatives are expressed as:

[0076]

[0077] S42. Calculate the principal curvature and determine the principal direction;

[0078] Principal curvature k 1 and k 2. Solve using the following formula;

[0079]

[0080] In the formula, the eigenvector V Corresponding to the main direction, k 1≥ k 2; k This indicates the solution for curvature;

[0081] Any principal curvature k f Substituting into the equation, we can solve for the corresponding tangent vector. t = ξr u + ther v The coefficient ratio in x : or ;

[0082]

[0083] In the formula, f =1, 2 represent the principal curvatures, respectively. k 1 and principal curvature k 2;

[0084] Tangent vector t 1 and t 2. The process is determined as follows: First, on the common tangent plane... S pg The above indicates the directions of the major and minor axes of the contact ellipse, followed by the common tangent plane. S pg The main direction is identified to determine the major axis direction of the contact ellipse of the large gear. The minor axis direction of the ellipse in contact with the large gear The pinion contacts the major axis of the ellipse. The minor axis direction of the ellipse in contact with the pinion ;

[0085] male tangent S pg The direction on represents the direction when any tangent vector is used. t The extreme curvature values ​​that occur when the direction changes, the principal curvature k 1 and k Both are used with arbitrary tangent vectors along a given surface. t Normal curvature representation:

[0086]

[0087]

[0088] In the formula, f yes and The angle between them l 1. l 2 are respectively and t Angle between 1, and t The angles between 2 and 3, both of which are measured in the positive direction in the counterclockwise direction. l Then it is and any vector t The angle between them; , , , The pinion tooth surface edge t The principal curvature in direction 1, along the pinion tooth surface tPrincipal curvature in 2 directions, along the tooth surface of the large gear t The principal curvature in direction 1, along the tooth surface of the large gear t Principal curvature in two directions;

[0089] Along vector t The relative curvature is expressed as:

[0090]

[0091] To determine the representation of the common tangent S pg upper contact point M m The principal direction at the contact line is obtained through calculation. f and l The relationship between them:

[0092]

[0093] S43, Solving for contact stress;

[0094] Sure and Along vector t After determining the relative curvature, the contact stress at the potential contact point is determined by Hertz contact theory. s H Represented as:

[0095]

[0096] In the formula, m 2 and m 1 represents the Poisson's ratio of the large gear and the Poisson's ratio of the small gear, respectively; E 2 and E 1 represents the elastic modulus of the tooth surface material of the large gear and the elastic modulus of the tooth surface material of the small gear, respectively.

[0097] A storage medium includes a stored program, wherein, when the program is executed, an improved method for analyzing the bearing contact of spiral bevel gears is performed.

[0098] An electronic device includes a memory, a processor, and a computer program stored in the memory and running on the processor, wherein the processor executes an improved spiral bevel gear load contact analysis method via the computer program.

[0099] Compared with the prior art, the present invention has the following beneficial effects:

[0100] This invention addresses the convergence problem caused by ill-conditioned compliance matrices under low-load conditions by introducing a preprocessing matrix to improve the iterative convergence of the equation system and enhance the stability and speed of the solution.

[0101] This invention abandons the traditional numerical difference approach and directly derives the analytical expression of the principal curvature of the tooth surface and determines the principal direction based on the Euler equation in differential geometry, thereby fundamentally improving the calculation accuracy of curvature parameters.

[0102] This invention improves the solution speed of the spiral bevel gear tooth bearing contact analysis method by comparing the results with those of the finite element method and the traditional gear tooth bearing contact analysis method (hereinafter referred to as the traditional method), while ensuring the reliability of the time-varying meshing stiffness and contact stress analysis results. This achieves a synergistic improvement in computational efficiency and accuracy. Attached Figure Description

[0103] Figure 1 This is a flowchart of the improved spiral bevel gear bearing contact analysis method of the present invention;

[0104] Figure 2 A schematic diagram of the coordinate system for machining spiral bevel gears;

[0105] Figure 3 A schematic diagram of the rotating projection surface of the large gear;

[0106] Figure 4 For illustration purposes, (a) is a schematic diagram of an ideal element in the natural coordinate system, (b) is a schematic diagram of an actual element in the absolute coordinate system, and (c) is a finite element model diagram.

[0107] Figure 5 This is a diagram showing the contact trajectory and potential contact points under the rotating projection plane.

[0108] Figure 6 Here is a flowchart of the preprocessing method;

[0109] Figure 7 (a) is a schematic diagram of the meshing plane and the transmission error under load; (b) is a schematic diagram of the transmission error under load.

[0110] Figure 8 Torque Comparison diagram of time-varying meshing stiffness of the lower gear pair;

[0111] Figure 9 Torque Comparison diagram of time-varying meshing stiffness of the lower gear pair;

[0112] Figure 10 A local coordinate system diagram of the contact instantaneous surface;

[0113] Figure 11 (a) is a schematic diagram of contact stress; (b) is the torque of the traditional method. (a) Contact stress diagram of the lower arc bevel gear pair; (b) Torque of this method Contact stress diagram of lower arc bevel gear pair;

[0114] Figure 12 This is a comparison diagram of the contact stress areas of spiral bevel gear pairs using the traditional method and the method described in this paper. Detailed Implementation

[0115] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0116] It should be noted that the terms "first," "second," etc., in the specification, claims, and accompanying drawings of this invention are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate so that the embodiments of the invention described herein can be implemented in orders other than those illustrated or described herein. Furthermore, the terms "comprising" and "having," and any variations thereof, are intended to cover a non-exclusive inclusion; for example, a process, method, system, product, or apparatus that comprises a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.

[0117] like Figure 1 As shown, this invention provides an improved method for load-bearing contact analysis of spiral bevel gears, specifically including the following steps:

[0118] S1. Obtain the geometric parameters of the tooth surfaces of the large and small gears based on the tooth contact analysis method;

[0119] S11. Derive the gear tooth surface parameter equations based on machining coordinate system transformation and the tool-tooth surface conjugate principle; the specific process is as follows:

[0120] Figure 2 This is a schematic diagram of the coordinate system for machining spiral bevel gears; the diagram shows... S g ( x g , y g , z g ( ) is the coordinate system of the large gear cutting tool. O g is the origin of the coordinate system for the large gear cutting tool; S c ( x c , yc , z c ( ) represents the local coordinate system of the rocking table. O c Let be the origin of the local coordinate system of the rocking table; S m ( x m , y m , z m ( ) is the machine tool coordinate system. O m This is the origin of the machine tool coordinate system; S d ( x d , y d , z d ), S e ( x e , y e , z e ( ) is an auxiliary coordinate system. O d , O e All of these are the origins of the auxiliary coordinate system; S 2( x 2, y 2, z 2) is the coordinate system of the large gear blank. O 2 is the origin of the coordinate system of the large gear blank.

[0121] The machining of spiral bevel gears is essentially the process of machining the tooth surface of spiral bevel gears, which is the transformation of the tool from the tool coordinate system to the blank coordinate system.

[0122] Modeling the tooth surface of spiral bevel gears is the process of establishing the mapping relationship between machining parameters and the tooth surface. For Gleason spiral bevel gears, based on machining coordinate system transformation and the tool-tooth surface conjugation principle, the parameter equations of the large gear tooth surface are derived, and the meshing equations of the large gear tooth surface and the tool, along with the gear rotation projection plane equation, are combined to obtain the nodal coordinates of the large gear tooth surface.

[0123] Large gear tooth surface S g Envelope surface function r 2 can be represented as:

[0124]

[0125] In the formula, M2g This represents the coordinate transformation matrix from the tool coordinate system to the blank coordinate system. s g , i g and P 2 represents the straight cutting edge length, the blade tip rotation angle, and the large gear rotation angle, respectively.

[0126] Furthermore, the meshing equation between the large gear tooth surface and the cutting tool can be expressed as:

[0127]

[0128] The equation of the rotational projection plane of the gear can be expressed as:

[0129]

[0130] In the formula, r 2( x ), r 2( y ), r 2( z These are the coordinates of the tooth surface points of the spiral bevel gear in the blank coordinate system. x , y , z about s g , i g The function; X M for M Point on the rotational projection plane of the gear X Axis coordinates; Y M for M Point on the rotational projection plane of the gear Y Axis coordinates; M Point A represents any node on the tooth surface of the large gear; a schematic diagram of the rotating projection surface of the large gear is shown below. Figure 3 As shown;

[0131] S12. Solve for the coordinates of the nodes on the tooth surface of the pinion;

[0132] The machining process of pinions is similar to that of gears. The solution process for the coordinates of the nodes on the tooth surface of pinions can be derived from the solution method for the coordinates of the nodes on the tooth surface of gears.

[0133] S13. Establish and assemble the geometric model of the tooth surface of the large and small gears;

[0134] Based on the nodal coordinates of the large gear tooth surface and the small gear tooth surface, a geometric model of the large and small gear tooth surfaces is established. According to the gear assembly conditions, the tooth surfaces of the two gears must have a common contact point. Therefore, the large and small gears need to be rotated around their respective axes by a certain angle until the tooth surfaces contact each other. In the assembly coordinate system, the tooth surface functions and normal vectors of the large and small gears can be expressed as:

[0135]

[0136] In the formula, f p The pinion rotation angle is given at different meshing times. f g This is the rotation angle of the large gear at the moment of meshing. M z1 , M z2 These represent the transformation matrices of the pinion from the blank coordinate system to the assembly coordinate system and the transformation matrix of the gear from the blank coordinate system to the assembly coordinate system, respectively. L z1 , L z2 They are respectively M z1 , M z2 A matrix consisting of 3 rows and 3 columns;

[0137] S2. Based on the geometric parameters of the tooth surfaces of the large and small gears, construct a bearing contact analysis model for the spiral bevel gear pair, and then calculate the overall compliance matrix, overall contact compliance matrix, and local contact compliance matrix of the tooth surface.

[0138] S21. Establish a bearing contact analysis model for spiral bevel gear pairs and obtain the overall compliance matrix of the tooth surface;

[0139] A finite element model is established using eight-node hexahedral isoparametric elements, such as... Figure 4 (a) and Figure 4 As shown in (b), in the figure A - H The model consists of 8 nodes of isoparametric elements. Based on the geometric parameters of the gear tooth surfaces, a bearing contact analysis model for the spiral bevel gear pair is constructed using the gear tooth bearing contact analysis method. The bearing contact analysis model for the spiral bevel gear pair is as follows: Figure 4 As shown in (c);

[0140] Subsequently, fixed constraints were applied to the inner hole nodes of the spiral bevel gear pair, and unit forces were applied sequentially to each point on the tooth surface. The displacement of the tooth surface points after each loading was recorded, thereby constructing the overall tooth surface compliance matrix corresponding to each meshing moment. , can be represented as:

[0141]

[0142] In the formula, superscript K Indicates the moment of engagement; subscript j Indicates the point of application of the unit force; subscript i Indicates the displacement extraction point; n This represents the total number of nodes on all gear contact surfaces;

[0143] S22. Solve for the overall contact compliance matrix and the local contact compliance matrix;

[0144] The displacement of the inner layer nodes on the tooth surface characterizes the overall displacement of the spiral bevel gear pair; the deformation generated on the gear tooth surface includes the overall contact compliance matrix of the gear. Local contact compliance matrix of tooth surface ;

[0145] Based on the overall compliance matrix of the large and small gears The nodes on the tooth surfaces of the large and small gears are densified using cubic spline interpolation. Then, the distances between potential contact points and all nodes on the tooth surfaces of the large and small gears are calculated. The interpolation point closest to the potential contact point, or the compliance of the large or small gear tooth surface node, is used as the compliance of the potential contact point, thus obtaining the compliance matrix of the potential contact point of the large gear. l G Potential contact point compliance matrix of pinion l P The final potential contact point overall contact compliance matrix Represented as:

[0146]

[0147] In the formula, the superscripts P and G represent the pinion and gear, respectively; and These represent the potential contact point compliance at the meshing moment of the large gear and the potential contact point compliance at the meshing moment of the small gear, respectively. k This indicates the potential contact point for extracting displacement. l Indicates the potential point of contact where force is applied; m This represents the number of potential contact points discrete along the long axis of the contact at the moment of meshing.

[0148] Local contact deformation of the tooth surface is caused by the contact between the tooth surfaces due to elastic deformation, and the corresponding contact compliance matrix is... , can be represented as:

[0149]

[0150] In the formula, E It is the elastic modulus; L It is the distance between potential points of contact;F k It is the first k Normal meshing force at each potential contact point;

[0151] S3. Establish deformation coordination iterative equations, analyze the ill-conditioned nature of deformation coordination iterative equations, and establish a preprocessing matrix to improve deformation coordination iterative equations; after improvement, solve for time-varying meshing stiffness and tooth surface load; after improvement, solve for time-varying meshing stiffness and perform finite element verification, and the verification results show that the method is effective.

[0152] S31. Establishment of deformation coordination iterative equations and solution of meshing stiffness and tooth surface load;

[0153] The deformation compatibility iterative equation is expressed as:

[0154]

[0155] In the formula, and These represent the radial row distance matrix from the initial contact point of the pinion to the axis of rotation and the column distance matrix from the initial contact point of the pinion to the axis of rotation, respectively. This represents the force distribution matrix in the normal direction of the potential contact point; The gap matrix represents the potential contact points; i f This indicates the angle of rotation of the pinion under load after it receives torque; T This indicates the input torque of the pinion;

[0156] After solving the deformation compatibility iterative equations, the normal load vector corresponding to each meshing position is obtained. F b and the normal compliance matrix at the actual contact point l b The normal deformation of a spiral bevel gear pair at any contact point is expressed as:

[0157]

[0158] The tooth surface load corresponding to the discrete potential contact points along the long axis at the moment of meshing is expressed as:

[0159]

[0160] In the formula, d 1 represents the cone angle of the pinion; α n Indicates the normal pressure angle; β Indicates the average helix angle. R Indicates the cone distance; F i It is the first i Normal meshing force at each potential contact point;

[0161] The meshing stiffness of a spiral bevel gear pair at any contact position can be expressed as:

[0162]

[0163] In the formula, u bi For the first i Normal deformation of each potential contact point.

[0164] S32. Establishment and application of preprocessing matrix;

[0165] Input torque of the pinion T Under the condition of ≤400 N·m, the original coefficient matrix is ​​defined. A for:

[0166]

[0167] Construct a preprocessing matrix P , represented as:

[0168]

[0169] In the formula, I It is the identity matrix;

[0170] The improved deformation compatibility iterative equation is expressed as:

[0171]

[0172] After obtaining and solving the improved deformation coordination iterative equation, the meshing stiffness of the spiral bevel gear pair at any contact position is... k mesh The expression remains unchanged;

[0173] Input torque of the pinion Under the operating conditions, the deformation coordination iterative equation of the spiral bevel gear pair remains unchanged, and the meshing stiffness of the spiral bevel gear pair at any contact position is... k mesh The expression remains unchanged.

[0174] Under low load conditions, the actual load-bearing contact area decreases, resulting in extremely uneven distribution of contact stress and causing the compliance matrix to exhibit an ill-conditioned state.

[0175] Specifically, this means: under low load conditions, the local contact compliance matrix... The local flexibility values ​​at various points vary greatly in magnitude: the flexibility at some actual contact points approaches zero, while the flexibility at non-contact points theoretically approaches infinity. This leads to... It approaches a singularity. When the matrix is... Superposition constitutes the system equationsA 11 =-( + )hour, The singularity of the coefficient matrix dominates the system characteristics, causing the eigenvalues ​​of the coefficient matrix to be extremely dispersed, which seriously undermines the numerical stability of the iterative solution and leads to convergence oscillations or even divergence.

[0176] The flowchart of the pretreatment method is as follows: Figure 6 As shown, the preprocessing method essentially involves constructing a preprocessing matrix. P Ill-conditioned nature of the control coefficient matrix.

[0177] S33. Establish a finite element model of the spiral bevel gear pair and calculate the time-varying meshing stiffness for verification;

[0178] The basic geometric parameters of the spiral bevel gear pair are shown in Table 1:

[0179] Table 1 Basic geometric parameters of spiral bevel gear pairs

[0180]

[0181] Meshing stiffness is expressed as:

[0182]

[0183] In the formula, i pa This indicates the rotation angle of the pinion under load at different meshing positions; d 1 indicates the cone angle of the pinion; α n Indicates the normal pressure angle; β Indicates the average helix angle. R Indicates the cone distance; Figure 7 (a) is a schematic diagram of the meshing plane. Figure 7 (b) is the diagram of the transmission error under load. LTE This refers to the error transmitted under load.

[0184] The time-varying meshing stiffness error is expressed as:

[0185]

[0186] In the formula, k mesh Let be the meshing stiffness of a bevel gear pair at any contact position. k 0 represents the time-varying meshing stiffness value in the ANSYS finite element model;

[0187] The model of this invention in torque and The time-varying meshing stiffness comparison diagram of the lower gear pair is shown below. Figure 8 and Figure 9As shown, the comparison results are presented in Tables 2 and 3:

[0188] Table 2 Comparison of Time-Variation Meshing Stiffness Errors Using Different Calculation Methods

[0189]

[0190] Table 3 Comparison of calculation time for time-varying meshing stiffness using the method in this paper and the traditional method.

[0191]

[0192] When the input torque is and Under different operating conditions, the time-varying meshing stiffness of the model of this invention was calculated, and the results are as follows: Figure 8 and Figure 9 As shown, the time-varying meshing stiffness curves calculated by the three methods (finite element method, traditional method, and the method in this paper) exhibit highly consistent trends.

[0193] As shown in Table 2, the relative error between this method and the finite element method results is relatively small. Under operating conditions, its maximum relative error is only 1.99%, lower than the 3.61% of the traditional method; Under operating conditions, the error further decreased to 1.83%, demonstrating the effectiveness of this method in improving the accuracy of stiffness calculation. Especially in The accuracy improvement is more significant under low torque conditions, thanks to the fact that the preprocessing matrix effectively suppresses the numerical ill-conditioning of the system equations.

[0194] As shown in Table 3, this method exhibits a significant advantage in computational efficiency. Compared to the finite element method, its computational efficiency is improved by nearly two orders of magnitude; the computation time is reduced by approximately 50% compared to this method, mainly due to the preprocessing method effectively optimizing the iterative convergence process. The computer configuration used was: Core i7-14700 3.32GHz 20-core 28-thread CPU, 32GB RAM.

[0195] S4. Based on the tooth surface load, the principal curvature of the gear tooth surface is calculated and the principal direction is determined using the Euler analytical method. The contact stress is then solved using Hertz contact theory. The calculation results are verified, demonstrating the effectiveness of this method.

[0196] S41, Euler's analytical method: Foundations of differential geometry;

[0197] The local coordinate system diagram of the contact instantaneous surface is as follows: Figure 10 As shown, to determine the direction of the principal axis of the loaded contact ellipse, it is necessary to [do something] at the contact point. M m Construct a common tangent plane at point A that is tangent to both the tooth surfaces of the large gear and the small gear. S pg Along any tangent vector on the common tangent plane t normal curvature The degree of curvature of the gear tooth surfaces along this direction, characterizing the size of the gear, can be expressed as:

[0198]

[0199] In the formula, and This indicates that the large gear is along the tangent vector. t Normal curvature, pinion along tangent vector t The normal curvature;

[0200] Using the tooth bearing contact analysis method, three adjacent tooth surface nodes along the minor axis of the contact ellipse are obtained, denoted as follows: P 1( x 1, y 1, z 1) P 2( x 2, y 2, z 2) and P 3( x 3, y 3, z 3).

[0201] curved surfaces P 1. P 2 and P The expression for the first-order partial derivative (tangent vector) at point 3 is similar, therefore it only represents the surface at... P The first-order partial derivative at point 1 is:

[0202]

[0203] The first and second basic form coefficients are expressed as follows:

[0204]

[0205] In the formula, s Let be the unit normal vector of the two gear tooth surfaces; O , P , G The coefficients are the first fundamental form coefficients; Q , M , N These are the coefficients of the second fundamental form; where, r uu for r u The second-order partial derivative; r uv for r uand r v The second-order partial derivative; r vv for r v The second-order partial derivatives are expressed as:

[0206]

[0207] S42. Calculate the principal curvature and determine the principal direction;

[0208] Principal curvature k 1 and k 2 can be solved using the following formula;

[0209]

[0210] In the formula, the eigenvector V Corresponding to the main direction, k 1≥ k 2; k This indicates the solution for curvature.

[0211] Any principal curvature k f Substituting into the equation, the corresponding principal direction vector can be obtained. t = ξr u + ther v The coefficient ratio in x : or .

[0212]

[0213] In the formula, f =1, 2 represent the principal curvatures, respectively. k 1 and principal curvature k 2;

[0214] Tangent vector t 1 and t 2. The process is determined as follows: First, on the common tangent plane... S pg The above indicates the directions of the major and minor axes of the contact ellipse, followed by the common tangent plane. S pg The main direction is identified to determine the major axis direction of the contact ellipse of the large gear. The minor axis direction of the ellipse in contact with the large gear The pinion contacts the major axis of the ellipse. The minor axis direction of the ellipse in contact with the pinion ;

[0215] male tangent S pgThe direction above indicates that when any vector t The extreme curvature values ​​that occur when the direction changes, the principal curvature k 1 and k Both can be used on a given surface along any vector. t Normal curvature representation:

[0216]

[0217]

[0218] In the formula, f yes and The angle between them l 1. l 2 are respectively and t Angle between 1, and t The angles between 2 and 3, both of which are measured in the positive direction in the counterclockwise direction. l Then it is and any vector t The angle between them; , , , The pinion tooth surface edge t The principal curvature in direction 1, along the pinion tooth surface t Principal curvature in 2 directions, along the tooth surface of the large gear t The principal curvature in direction 1, along the tooth surface of the large gear t Principal curvature in two directions;

[0219] Along vector t The relative curvature is expressed as:

[0220]

[0221] To determine the representation of the common tangent S pg upper contact point M m The principal direction of the contact line can be obtained through calculation. f and l The relationship between them:

[0222]

[0223] S43, Solving for contact stress;

[0224] Determine the large gear and small gear along the vector t normal curvature and After considering relative curvature, the contact stress at potential contact points is determined using Hertz contact theory. s H It can be represented as:

[0225]

[0226] In the formula, m 2 and m 1 represents the Poisson's ratio of the large gear and the Poisson's ratio of the small gear, respectively; E 2 and E 1 represents the elastic modulus of the tooth surface material of the large gear and the elastic modulus of the tooth surface material of the small gear, respectively.

[0227] S44, Contact stress verification;

[0228] Traditional gear tooth contact analysis methods combined with Hertz contact theory are used to calculate the contact stress at potential contact points. s H1 for:

[0229]

[0230] In the formula, , denoted as the surface normal curvature of the potential contact point of the large and small gears, respectively.

[0231] Figure 11 (a) Torque in the traditional method Contact stress diagram of lower arc bevel gear pair; Figure 11 (b) is the torque of this method. Contact stress diagram of lower arc bevel gear pair; Figure 12 This is a comparison diagram of the contact stress areas of spiral bevel gear pairs using the traditional method and the method described in this paper;

[0232] The results of the maximum contact stress obtained by this method and the traditional method are shown in Table 4:

[0233] Table 4. Maximum contact stress results of this method and the traditional method.

[0234]

[0235] Depend on Figure 11 (a) Figure 11 (b) Figure 12 As shown in Table 4, in and Under torque conditions, the contact elliptical region shape obtained by the model of this invention is basically consistent, but the stress distribution obtained by this method is smoother. Especially under low torque conditions, the maximum contact stress value calculated by this method is lower, which is consistent with the actual situation that elastic deformation makes the load distribution more uniform. This avoids the stress overestimation caused by the curvature approximation error of the traditional method. It is worth noting that although the traditional method has a certain degree of stress overestimation, the maximum stress calculated by it is still within the allowable limit range.

[0236] Contact stress values ​​obtained using the Euler analytical method Furthermore, the more uniform distribution of contact stress within the contact area proves the effectiveness of the Euler analytical method. Traditional methods can lead to stress overestimation due to curvature approximation errors, while the Euler analytical method, based on rigorous differential geometry derivation, theoretically avoids such errors.

[0237] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. An improved method for analyzing the bearing contact of spiral bevel gears, characterized in that, Includes the following steps: S1. Obtain the geometric parameters of the tooth surfaces of the large and small gears based on the tooth contact analysis method; S2. Construct a bearing contact analysis model for the spiral bevel gear pair based on the geometric parameters of the tooth surfaces of the large and small gears, and then calculate the overall compliance matrix of the tooth surfaces. Overall contact compliance matrix and local contact compliance matrix ; S3. Establish deformation coordination iterative equations, analyze the ill-conditioned nature of deformation coordination iterative equations, and establish a preprocessing matrix to improve deformation coordination iterative equations; after improvement, solve for time-varying meshing stiffness and tooth surface load. S3 specifically includes the following steps: S31. Establishment of deformation coordination iterative equations and solution of meshing stiffness and tooth surface load; The deformation compatibility iterative equation is expressed as: In the formula, and These represent the radial row distance matrix from the initial contact point of the pinion to the axis of rotation and the column distance matrix from the initial contact point of the pinion to the axis of rotation, respectively. This represents the force distribution matrix in the normal direction of the potential contact point; The gap matrix represents the potential contact points; θ f This indicates the angle of rotation of the pinion under load after it receives torque; T This indicates the input torque of the pinion; After solving the deformation compatibility iterative equations, the normal load vector corresponding to each meshing position is obtained. F b and the normal compliance matrix at the actual contact point λ b The normal deformation of a spiral bevel gear pair at any contact point is expressed as: The tooth surface load corresponding to the discrete potential contact points along the long axis at the moment of meshing is expressed as: In the formula, δ 1 represents the cone angle of the pinion; α n Indicates the normal pressure angle; β Indicates the average helix angle. R Indicates the cone distance; F i It is the first i Normal meshing force at each potential contact point; The meshing stiffness of a spiral bevel gear pair at any contact position is expressed as: In the formula, u bi For the first i Normal deformation at each potential contact point; S32. Establishment and application of preprocessing matrix; Input torque of the pinion T Under the condition of ≤400 N·m, the original coefficient matrix is ​​defined. A for: Construct a preprocessing matrix P , represented as: In the formula, I It is the identity matrix; The improved deformation compatibility iterative equation is expressed as: After obtaining and solving the improved deformation coordination iterative equation, the meshing stiffness of the spiral bevel gear pair at any contact position is... k mesh The expression remains unchanged; Input torque of the pinion T Under working conditions >400 N·m, the deformation coordination iterative equation of the spiral bevel gear pair remains unchanged, and the meshing stiffness of the spiral bevel gear pair at any contact position is... k mesh The expression remains unchanged; S4. Based on the tooth surface load, the principal curvature of the gear tooth surface is calculated and the principal direction is determined using the Euler analytical method, and the contact stress is solved by combining the Hertz contact theory.

2. The improved spiral bevel gear bearing contact analysis method according to claim 1, characterized in that, S1 specifically includes the following steps: S11. Based on the transformation of the machining coordinate system and the principle of tool-tooth surface conjugation, the gear tooth surface parameter equation is derived. For spiral bevel gear pairs, the parameter equations of the large gear tooth surface are derived based on the transformation of the machining coordinate system and the principle of tool-tooth surface conjugation. The meshing equations of the large gear tooth surface and the tool and the gear rotation projection surface equations are combined to obtain the node coordinates of the large gear tooth surface. Large gear tooth surface Σ g Envelope surface function r 2 is represented as: In the formula, M 2g This represents the coordinate transformation matrix from the tool coordinate system to the blank coordinate system. s g , θ g and Ψ 2 represents the straight cutting edge length, tool tip rotation angle, and large gear rotation angle of the machining tool. r g For the tooth surface of the large gear Σ g The tooth surface vector function; Then, the meshing equation between the large gear tooth surface and the cutting tool is obtained. : In the formula, Represents the differential form; The equation for the rotational projection plane of the gear is expressed as: In the formula, r 2( x ), r 2( y ), r 2( z These are the coordinates of the gear tooth surface points in the blank coordinate system. x , y , z about s g , θ g The function; X M for M Point on the rotational projection plane of the gear X Axis coordinates; Y M for M Point on the rotational projection plane of the gear Y Axis coordinates; M The point is any node on the tooth surface of the large gear; S12. Solve for the coordinates of the nodes on the tooth surface of the small gear using the method for solving the coordinates of the nodes on the tooth surface of the large gear. S13. Establish and assemble the geometric model of the tooth surface of the large and small gears; Based on the coordinates of the nodes on the tooth surface of the large gear and the small gear, a geometric model of the tooth surfaces of both gears is established. According to the assembly conditions of the spiral bevel gear pair, the tooth surfaces of the large gear and the small gear must have a common contact point. The large gear and the small gear are rotated around their respective axes by a certain angle until their tooth surfaces contact each other. In the assembly coordinate system, the tooth surface function and normal vector of the large gear and the tooth surface function and normal vector of the small gear are expressed as follows: In the formula, φ p The pinion rotation angle at different meshing moments; φ g To correspond to the rotation angle of the large gear at the moment of meshing, M z1 , M z2 These represent the transformation matrices from the blank coordinate system to the assembly coordinate system for the pinion and from the blank coordinate system to the assembly coordinate system for the gear, respectively. Both are 4-row, 4-column matrices. L z1 , L z2 They are respectively M z1 The matrix consists of 3 rows and 3 columns. M z2 A matrix consisting of 3 rows and 3 columns; s p , θ p and Ψ 1 represents the straight cutting edge length of the pinion machining tool, the tool tip rotation angle, and the pinion rotation angle, respectively. r z1 , r z2 These are the tooth surface functions of the pinion tooth surface and the gear tooth surface in the assembly coordinate system, respectively. , These are the unit normal vectors of the envelope surface function of the pinion gear and the envelope surface function of the gear, respectively. , These are the unit normal vectors of the pinion tooth surface function in the assembly coordinate system and the unit normal vectors of the gear tooth surface function in the assembly coordinate system, respectively.

3. The improved spiral bevel gear bearing contact analysis method according to claim 1, characterized in that, S2 specifically includes the following steps: S21. Establish a bearing contact analysis model for spiral bevel gear pairs and obtain the overall compliance matrix of the tooth surface; Based on the tooth surface function and normal vector of the large gear and the small gear, a bearing contact analysis model of the spiral bevel gear pair is constructed using the tooth bearing contact analysis method. Fixed constraints are applied to the inner hole nodes of the spiral bevel gear pair, and unit forces are sequentially applied to each point on the tooth surface. The displacement of the tooth surface points after each loading is recorded, and the overall tooth surface compliance matrix corresponding to each meshing moment is constructed. , represented as: In the formula, superscript K Indicates the moment of engagement; subscript j Indicates the point of application of the unit force; subscript i Indicates the displacement extraction point; n This represents the total number of nodes on all gear contact surfaces; S22. Solve for the overall contact compliance matrix and the local contact compliance matrix; The displacement of the inner layer nodes on the tooth surface characterizes the overall displacement of the spiral bevel gear pair; the deformation generated on the gear tooth surface includes the overall contact compliance matrix of the gear. Local contact compliance matrix of tooth surface ; Based on the overall compliance matrix of the large and small gears The nodes on the tooth surfaces of the large and small gears are densified using cubic spline interpolation. Then, the distances between potential contact points and all nodes on the tooth surfaces of the large and small gears are calculated. The interpolation point closest to the potential contact point, or the compliance of the large or small gear tooth surface node, is used as the compliance of the potential contact point, thus obtaining the compliance matrix of the potential contact point of the large gear. λ G Potential contact point compliance matrix of pinion λ P The final potential contact point overall contact compliance matrix Represented as: In the formula, the superscripts P and G represent the pinion and gear, respectively; This indicates the potential contact point compliance of the large gear at the moment of meshing. This indicates the potential contact point compliance of the pinion at the moment of meshing; k This indicates the potential contact point for extracting displacement. l Indicates the potential point of contact where force is applied; m This represents the number of potential contact points discrete along the long axis of the contact at the moment of meshing. Local contact deformation of the tooth surface is caused by the contact between the tooth surfaces due to elastic deformation, and the corresponding local contact compliance matrix is... , represented as: In the formula, E It is the elastic modulus; L The distance between potential contact points; F k It is the first k Normal meshing force at each potential contact point; λ ck Indicates the first k The contact compliance value at the potential contact point for extracting displacement.

4. The improved spiral bevel gear bearing contact analysis method according to claim 1, characterized in that, S4 specifically includes the following steps: S41, Euler's analytical method: Foundations of differential geometry; At the contact point M m Construct a common tangent plane at point A that is tangent to both the tooth surfaces of the large gear and the small gear. Σ pg The common tangent plane along any tangent vector t normal curvature The degree of curvature of the gear tooth surfaces along this direction is represented as: In the formula, and These represent the large gear along the tangent vector. t Normal curvature, pinion along tangent vector t The normal curvature; Using the tooth bearing contact analysis method, three adjacent tooth surface nodes along the minor axis of the contact ellipse are obtained, denoted as follows: P 1( x 1, y 1, z 1) P 2( x 2, y 2, z 2) and P 3( x 3, y 3, z 3); The tooth surfaces of large and small gears are P 1. P 2 and P The first-order partial derivative expression at point 3 is considered equivalent, only indicating the tooth surface at... P The first-order partial derivative at point 1 is: In the formula, r u They represent the tooth surfaces of the large and small gears respectively. u Tangent vector whose coordinate direction changes; r v Indicates the tooth surface of the large and small gears along v Tangent vector whose coordinate direction changes; The first and second basic form coefficients are expressed as follows: In the formula, s Let be the unit normal vector of the two gear tooth surfaces; O , P , G The coefficients are the first fundamental form coefficients; Q , M , N The coefficients are the second fundamental form. in, r uu for r u The second-order partial derivative; r uv for r u and r v The second-order partial derivative; r vv for r v The second-order partial derivatives are expressed as: S42. Calculate the principal curvature and determine the principal direction; Principal curvature κ 1 and κ 2. Solve using the following formula; In the formula, the eigenvector V Corresponding to the main direction, κ 1≥ κ 2; κ This indicates the solution for curvature; Any principal curvature κ f Substituting into the equation, we can solve for the corresponding tangent vector. t = ξr u + ηr v The coefficient ratio in ξ : η ; In the formula, f =1, 2 represent the principal curvatures, respectively. κ 1 and principal curvature κ 2; Tangent vector t 1 and t 2. The process is determined as follows: First, on the common tangent plane... Σ pg The above indicates the directions of the major and minor axes of the contact ellipse, followed by the common tangent plane. Σ pg The main direction is identified to determine the major axis direction of the contact ellipse of the large gear. The minor axis direction of the ellipse in contact with the large gear The pinion contacts the major axis of the ellipse. The minor axis direction of the ellipse in contact with the pinion ; male tangent Σ pg The direction on represents the direction when any tangent vector is used. t The extreme curvature values ​​that occur when the direction changes, the principal curvature κ 1 and κ Both are used with arbitrary tangent vectors along a given surface. t Normal curvature representation: In the formula, φ yes and The angle between them λ 1. λ 2 are respectively and t Angle between 1, and t The angles between 2 and 3, both of which are measured in the positive direction in the counterclockwise direction. λ Then it is and any vector t The angle between them; , , , The pinion tooth surface edge t The principal curvature in direction 1, along the pinion tooth surface t Principal curvature in 2 directions, along the tooth surface of the large gear t The principal curvature in direction 1, along the tooth surface of the large gear t Principal curvature in two directions; Along vector t The relative curvature is expressed as: To determine the representation of the common tangent Σ pg upper contact point M m The principal direction at the contact line is obtained through calculation. φ and λ The relationship between them: S43, Solving for contact stress; Sure and Along vector t After determining the relative curvature, the contact stress at the potential contact point is determined by Hertz contact theory. σ H Represented as: In the formula, μ 2 and μ 1 represents the Poisson's ratio of the large gear and the Poisson's ratio of the small gear, respectively; E 2 and E 1 represents the elastic modulus of the tooth surface material of the large gear and the elastic modulus of the tooth surface material of the small gear, respectively.

5. A storage medium, characterized in that, The storage medium includes a stored program, wherein, when the program is executed, the improved spiral bevel gear bearing contact analysis method according to any one of claims 1-4 is performed.

6. An electronic device, characterized in that, It includes a memory, a processor, and a computer program stored in the memory and running on the processor, wherein the processor executes the improved spiral bevel gear load contact analysis method according to any one of claims 1-4 through the computer program.

Images