A method for obtaining the single tooth meshing stiffness of spiral bevel gears

Abstract

The present invention relates to a method for obtaining the single tooth meshing stiffness of a spiral bevel gear, and the method specifically comprises: obtaining the contact imprint range of the spiral bevel gear under the loaded contact state based on the Hertz contact model; discretizing the contact imprint range into a two-dimensional tooth profile boundary element model to form a plurality of pairs of large wheel tooth profile sheets and small wheel tooth profile sheets; obtaining the local contact force of each tooth profile sheet; combining the elastic boundary element method with the Hertz line contact theory to obtain the contact deformation and bending deformation of each pair of meshing large and small wheel tooth profile sheets, and then obtaining the meshing stiffness of the unit sheet; and superimposing all the meshing stiffnesses. Compared with the prior art, the present invention optimizes the calculation method and reduces the complexity of obtaining the single tooth meshing stiffness.

Description

Technical Field

[0001] The invention relates to the technical field of gears, and in particular to the technology of spiral bevel gears. Background Art

[0002] From the existing research methods, the analysis methods of the single tooth meshing stiffness of spiral bevel gears can be roughly divided into analytical method, semi-analytical method and full finite element method. Among them, the analytical method is generally based on the cantilever beam and Hertz contact model, the semi-analytical method is generally a hybrid finite element model and Hertz contact theory, and the full finite element analysis method is to complete the mechanical properties analysis by introducing contact units. Due to the complexity of the spiral bevel gear tooth surface itself, the complete analytical formula is generally difficult to reflect the complex microscopic tooth shape, and a certain safety factor needs to be added to ensure structural safety. The semi-analytical method and the full finite element method are generally based on a generalized finite element model. Due to the large amount of calculation of finite element analysis, although there are various improvements to the finite element model, it is still difficult to use it for coupling with complex systems and iterative optimization of arc bevel gears. Summary of the invention

[0003] The purpose of the present invention is to provide a method for obtaining the single tooth meshing stiffness of a spiral bevel gear, so as to solve the problem that the existing method is complex and difficult to be used for coupling with complex systems and iterative optimization of spiral bevel gears.

[0004] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0005] A method for obtaining the single tooth meshing stiffness of a spiral bevel gear, the method specifically comprising:

[0006] Based on the Hertz contact model, the contact imprint range of the spiral bevel gear under loaded contact state is obtained;

[0007] The contact mark range is discretized into a two-dimensional tooth profile boundary element model to form a number of pairs of large wheel tooth profile slices and small wheel tooth profile slices;

[0008] Obtain the local contact force of each tooth profile segment;

[0009] Combining the elastic boundary element method with the Hertz line contact, the contact deformation and bending deformation of each pair of meshing large and small gear tooth profiles are obtained, and then the meshing stiffness of the unit sheet is obtained;

[0010] Add up all meshing stiffnesses.

[0011] Furthermore, the method for obtaining the contact mark range of the spiral bevel gear under the loaded contact state is specifically: establishing a common tangent plane when the large wheel and the small wheel are meshing, projecting the major axis endpoints and the minor axis endpoints of the contact ellipse onto the common tangent plane, and then correcting the major axis vector and the minor axis vector projected onto the common tangent plane based on the Hertz point contact theory. The contact ellipse is formed by the deformation and extension of the two tooth surfaces on their common tangent plane under the action of torque when the large wheel and the small wheel are meshing.

[0012] Furthermore, the major axis direction vector of the contact ellipse on the common tangent plane can be expressed as:

[0013]

[0014] in: It is a point on the contact trace of the large wheel and the small wheel. The contact trace is the major axis of the contact ellipse. is the direction vector of the major axis of the contact ellipse on the common tangent plane; is the vector of the two end points of the major axis of the corrected ellipse on the common tangent plane; is the direction vector of the small wheel rotation axis; a is the length of the major axis of the contact ellipse.

[0015] Furthermore, the method of slicing the gear part in the contact deformation area at each meshing instant and discretizing it into a two-dimensional tooth profile boundary element model is as follows:

[0016] Acquire a plurality of two-dimensional tooth profile boundary element meshes of the large wheel and a corresponding two-dimensional tooth profile boundary element meshes of the small wheel along the major axis direction of the contact ellipse, wherein each slice corresponds one-to-one to the two-dimensional tooth profile boundary element mesh of the large wheel and the two-dimensional tooth profile boundary element mesh of the small wheel;

[0017] The intersection of the normal load direction and the neutral line of the boundary element is taken as the reference point for measuring the deformation of the tooth profile, and the reference point is deformed and corrected along the normal load direction to obtain the bending deformation of the tooth profile.

[0018] Further, the method for obtaining the local contact area S of the tooth profile sheet is:

[0019] A full ellipse is defined as a case where the contact ellipse does not exceed the tooth surface, and a half ellipse is defined as a case where the contact ellipse exceeds the tooth surface;

[0020] For the full ellipse case, if the intercept parameters a1 and a2 are both smaller than the major axis a of the contact ellipse, then

[0021] S = S1 + S2 + S3;

[0022]

[0023] in, are the intersection points of the left, right and middle sections of the sheet and the major axis of the contact ellipse, respectively. is the intersection of the required tooth profile section and the major axis of the contact ellipse, a is the length of the major axis of the contact ellipse, b is the length of the minor axis of the contact ellipse, P i(x) , P i(y) is the x value and y value of the i-th point;

[0024] If a1 or a2 is equal to the major axis a of the contact ellipse, then

[0025]

[0026] Where λ is the angle between the neutral plane of the tooth profile segment and the direction of the major axis vector of the contact ellipse.

[0027] Further, the method for obtaining the local contact area S of the tooth profile sheet is:

[0028] In the case of a semi-ellipse, the contact ellipse is trimmed;

[0029] Assume that the condition that the end point is within the range of the tooth top of the large gear tooth surface is C1; the condition that the end point is within the range of the tooth top of the small gear tooth surface is C2; the condition that the end point is within the range of the tooth length is C3; the distance from the end point to the tooth top on the large gear tooth surface is D1; ​​the distance from the end point to the tooth top on the small gear tooth surface is D2; the larger distance from the small end and the large end in the tooth length direction is D3; through θ tl As a variable, perform iterative calculation on the endpoints on one side of the ellipse, and let the conjugate contact line corresponding to the large wheel and the small wheel obtained by calculation be

[0030] Point P G , P P , the judgment condition can be obtained as follows:

[0031] Where: x pG is the distance in the x direction from any point P to the center of the tooth surface axis section, θ tl is the distance parameter from the contact point to the tool tip in the spiral bevel gear tooth surface equation, X o is the wheel crown, L is the distance from any point P on the tooth surface to the staggered point of the gear shaft, Γ a is the cone angle of the gear face, R or is the midpoint cone distance, F is the tooth surface width, and the subscripts G and P represent the large and small wheels.

[0032] When C1, C2, and C3 are all greater than 0, point P G , P P All are within the tooth surface range; when C1, C2, and C3 are not all greater than 0, it is necessary to determine the position relationship between the point and the tooth surface boundary, establish the gear axis section coordinate system, and obtain the distance between point P and the tooth.

[0033] The distance expression of the top is:

[0034] Where: Tooth width vector at pitch cone,

[0035] In the vector expression, γ is the pitch cone angle; is the axis section vector of any tooth surface point; F is the vector from the axis intersection point O to the wheel crown; a is the width of the tooth top line at the face cone position; γ a is the face cone angle;

[0036] The judgment conditions are arranged and combined as follows: When only C1 < 0, point P exceeds the tooth top line of the large wheel on the large wheel and exceeds the tooth root line of the small wheel on the small wheel. The distance between point P and the tooth top line of the large wheel is D1. By adjusting the θ in the tooth surface parameters tl Iterative calculation can find a point on the conjugate contact line that satisfies D1=0. This equation shows that the distance between a section on the conjugate contact line and the top line of the large gear tooth is 0, and it also shows that the point falls on the top of the tooth and is within the range of the tooth surface.

[0037] D2 and D3 are judged similarly. When D1 < 0 and D3 < 0, the endpoint is outside the top line of the large gear tooth and outside the tooth surface width range. D is taken as the smaller value between the two, and the tooth surface parameter θ is iterated. tl Until D=0, the point is within the tooth surface range, and the case of D2<0 and D3<0 is handled in the same way.

[0038] Finally, the calculation ends when the values ​​of both end points of the ellipse meet the tooth surface range conditions. The tooth surface range conditions are that C1, C2, and C3 are all greater than 0.

[0039] Furthermore, the method for obtaining the local contact force of each tooth profile sheet is:

[0040] Obtain the local contact force of the tooth profile sheet;

[0041]

[0042] Among them, P C1 , P C2 P is the two intersection points of the unit neutral layer line and the contact ellipse, and each pair of large wheel sheet and small wheel sheet is defined as a unit; C1(y) , P C2(y) are the y-axis coordinates of the upper and lower limits of the intersection, and the unit load f i is the local contact force of the tooth profile;

[0043] When the contact ellipse is a full ellipse, the above formula can be used for calculation. When the contact ellipse is a semi-ellipse,

[0044] And PC1 Located in P F On the left, the upper and lower limits of the integral are P C1(y) , P C2(y) , P C1 Located in P F On the right, the upper and lower limits of the integral are P F , P C2(y) ; Similarly, we can deduce situation; P F is a point on the tooth profile line, so that the center point P of the tooth profile fillet c To point P F The vector and point P F The tangent vectors on the tooth profile are orthogonal, and the center point P c To point P F The distance is the radius r of the tooth profile fillet, and the stress distribution on the neutral surface of the tooth profile sheet is P;

[0045] Furthermore, the method for obtaining the single tooth meshing stiffness is:

[0046] Based on the Hertz line contact theory, the Hertz deformation of each sheet is obtained, and the bending deformation of the large wheel sheet is obtained through the stress distribution P on the neutral surface. And the bending deformation of the small wheel Then, the single tooth meshing stiffness is obtained based on the following formula:

[0047]

[0048] Among them, K i is the meshing stiffness of each plate, and K is the meshing stiffness of a single tooth.

[0049] Beneficial effects of the present invention:

[0050] This chapter of the present invention calculates the contact mark and contact stress of the spiral bevel gear under the loaded contact state through the Hertz contact model. The gear teeth in the contact mark range are sliced ​​and discretized, and a two-dimensional tooth profile boundary element mesh model is constructed. The local contact force and the loaded stress area on the sheet are calculated according to the elliptical contact state. The meshing stiffness between the large and small wheel sheets is calculated by combining the elastic boundary element method and the Hertz line contact. Finally, the analytical calculation formula of the single tooth meshing stiffness is derived through the parallel model of multiple sheets. Compared with the prior art, the calculation method is optimized and the complexity of obtaining the single tooth meshing stiffness is reduced. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] Figure 1 It is a flow chart of the present invention;

[0052] Figure 2 is the positional relationship between the normal load direction and the tangential load direction;

[0053] Figure 3Schematic diagram of the contact area of ​​the discrete ellipse of the tooth profile method section;

[0054] Figure 4 It is a schematic diagram of the cross section of the gear meshing shaft;

[0055] Figure 5 This is the model diagram of the intersection of the normal section and the tooth profile;

[0056] Figure 6 Schematic diagram of tooth profile fillet radius error;

[0057] Figure 7 Approximate contact cylinder representing Hertzian line contact;

[0058] Figure 8 Represents the contact ellipse discrete element contact model;

[0059] Fig. 9 Indicates the contact ellipse shape of the tooth surface;

[0060] Fig.10 Indicates the relative position relationship between the sheet model and the contact ellipse;

[0061] Fig.11 Represents the intersection model of the sheet body and the elliptical contact area;

[0062] Fig.12 Represents the intersection model of the sheet body and the elliptical contact area;

[0063] Fig.13 Load handling flow chart showing full elliptical contact;

[0064] Fig.14 Represents the ellipse pruning flow chart;

[0065] Fig.15 It indicates that the conjugate contact point does not exceed the tooth top condition diagram;

[0066] Fig.16 Schematic diagram showing the coordinate position of the conjugate contact point on the tooth surface;

[0067] Fig.17 Represents the boundary conditions of the two-dimensional tooth profile boundary element mesh;

[0068] Fig.18 Indicates that the tooth profile is bent and deformed. DETAILED DESCRIPTION

[0069] The following will refer to the accompanying drawings and preferred embodiments to illustrate the implementation of the technical solution of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the contents disclosed in this specification. The present invention can also be implemented or applied through other different specific implementations, and the details in this specification can also be modified or changed in various ways based on different viewpoints and applications without departing from the spirit of the present invention. It should be understood that the preferred embodiments are only for illustrating the present invention, not for limiting the scope of protection of the present invention.

[0070] It should be noted that the illustrations provided in the following embodiments are only schematic illustrations of the basic concept of the present invention, and thus the drawings only show components related to the present invention rather than being drawn according to the number, shape and size of components in actual implementation. In actual implementation, the type, quantity and proportion of each component may be changed arbitrarily, and the component layout may also be more complicated.

[0071] This embodiment proposes a method for obtaining the single tooth meshing stiffness of a spiral bevel gear, specifically:

[0072] Based on the Hertz contact model, the contact mark range of the spiral bevel gear under loaded contact state is obtained. Specifically:

[0073] In order to calculate the contact state of the spiral bevel gear under load, it is necessary to know the local load contact model of the spiral bevel gear. Since the meshing of the spiral bevel gear belongs to the point contact condition, a local micro plane will be formed on the tooth surface where the large and small wheels contact due to elastic deformation. Due to the unequal curvature of the two tooth surfaces, an elliptical contact area will be formed on the tooth surface of the large and small wheels. In this embodiment, each moment of gear transmission contact is regarded as a quasi-static contact, and it is considered that the contact force of the large and small wheels at this moment is transmitted along the normal direction of the tooth surface of the conjugate contact point, and there is frictionless contact between the two teeth. Therefore, in this embodiment, the Hertz point contact theory model is used to solve the major and minor axes a, b of the contact ellipse and the contact stress P of the large and small wheels in the contact state.

[0074] Hertz's theory is based on the following assumptions:

[0075] (1) The surface is continuous and differentiable;

[0076] (2) The maximum deformation point is the initial conjugate contact point;

[0077] (3) Both contact bodies are elastic half spaces;

[0078] (4) The direction of the normal vector does not change during the deformation process;

[0079] In the gear model, when the large and small wheels are meshing, the two tooth surfaces will deform on their common tangent planes under the action of torque and extend into contact ellipses. The endpoints of the major and minor axes of the ellipse are projected onto the common tangent planes of the two tooth surfaces, and the contact state is corrected with the major and minor axes of the ellipse under loading. The size of the major and minor axes of the contact ellipse under load can be solved. Therefore, a local coordinate system ∑ H = {O H ; X, Y, Z}, the coordinates X, Y, and Z axes coincide with the major axis, minor axis, and normal vector of the instantaneous contact ellipse and the conjugate meshing point, respectively. Assume that the gear tooth elastic bodies S1 and S2 are under external load F N Contact occurs under the action of F N =0, S1, S2 in ∑ H Origin O H Contact occurs without deformation. H The relative curvature radii of the two elastic bodies near the point are γ S1 , γ S2 As the external load gradually increases, the elastic body deforms at the contact position, and the contact area gradually expands along the common tangent plane in its respective directions due to the difference in relative curvature to form a contact ellipse.

[0080] First, the tangent plane model is constructed, and the radial vector and normal vector of the tooth surface contact point are used to construct the tangent plane S V , project any point in space onto S V There is the following geometric relationship, formula (1):

[0081]

[0082] From formula (1), we can get the following relational expression (2):

[0083]

[0084] Set the ellipse endpoint vector Substitute into the above formula (2) to calculate the coordinates of the major axis of the ellipse in the Hertz contact model ∑ H The direction vector in the plane is rotated 90° around the plane normal vector to obtain the coordinates of the minor axis of the ellipse in the Hertz contact model ∑ H The direction vector in the image can be used to calculate the curvature K of the large and small wheels in the major and minor axis directions using the relationship between the basic types of surfaces in differential geometry. Gx , K Gy , K Px , K Py Then the relative curvature at the conjugate contact point is given by formula (3):

[0085]

[0086] The ratio of the major axis to the minor axis of the contact ellipse given by Hertz's point contact theory and relative curvature K x , K y The relationship between them is as follows:

[0087]

[0088] In the formula, is the eccentricity of the contact ellipse; is the complete elliptic integral of the first kind; is the complete elliptic integral of the second kind.

[0089] Since the Hertz point contact relationship is used in the range that the closer the ratio of the two relative curvatures is to 1, the more accurate the Hertz relationship is. When the research object is a spiral bevel gear, the curvature ratio of the two relative curvature directions of the gear tooth surface is much higher than 1. If the theoretical Hertz point contact model is used, the contact state calculation will have a large error. In order to avoid this result, the Hertz point contact supplementary formula is selected to correct the major and minor axis calculation error caused by the large relative curvature ratio in the object. The correction formula is shown in formula (5):

[0090]

[0091] In the formula, R e is the equivalent radius of curvature of the surface; F n is the normal load; E * is the equivalent elastic modulus of the large and small wheels; E and ν are the elastic modulus and Poisson's ratio; F1(e) is the Hertz contact correction coefficient.

[0092] Since the spiral bevel gear has a large helix angle in the tooth width direction, the normal force F has a large change in direction and magnitude at each moment. In order to use the Hertz contact correction model to more accurately calculate the contact ellipse of the spiral bevel gear under load, it is necessary to solve the magnitude and direction of the normal load at each meshing moment. Construct the spatial geometric relationship between the meshing point, the direction of the normal force, and the direction of the tangential force. The transmission of the spiral bevel gear mainly uses the small wheel as the driving wheel, and constructs the various vector relationships under the meshing state of the small wheel tooth surface, such as Figure 2 As shown:

[0093] Figure 2 In the figure, the orthogonal coordinate system is established at the vertex of the pitch cone of the small wheel, and the y-axis of the coordinate system is aligned with the rotation axis of the small wheel. coincide. The meshing point of the small wheel The direction vector of and The intersection of the two is the center of the circle, and the radius is from the instantaneous meshing point to the intersection of the two. The instantaneous meshing circle is constructed on the plane. The tangential direction of the instantaneous meshing circle at the meshing point is This is the direction vector of the tangential force. is the normal vector of the instantaneous meshing point, and is also the direction vector of the normal load at the meshing point θ is and The angular relationship in space, through which F n =F t / cosθ. Let the small wheel driving torque at this time be T p , then the tooth surface normal load F is solved by the following formula (6): n :

[0094]

[0095] F n Substituting into equation (6) and connecting with (6), a and b can be solved immediately. The maximum Hertzian contact stress P, the Hertzian contact deformation δ and the contact stress P at any point within the contact ellipse can be calculated by equation (7): (x,y) :

[0096]

[0097] The above formula is used to calculate the major and minor axes of the ellipse, stress distribution and the maximum stress value of the ellipse. The major axis parameter of the ellipse is also used to correct the major axis of the theoretical ellipse. The above formula (7) can obtain the projection point of the theoretical contact major axis on the common tangent plane: Then the major axis direction of the contact ellipse on the common tangent plane can be expressed by the following formula (8):

[0098]

[0099] In the formula, is a point on the contact trace; is the direction vector of the major axis of the ellipse on the common tangent plane; is the vector of the two end points of the major axis of the corrected ellipse on the common tangent plane. Finally, the vector of the two end points of the major axis of the loading contact ellipse is The normal vector at is projected onto the tooth surface to form a loaded contact ellipse range on the tooth surface. In this embodiment, the contact ellipse is the contact mark.

[0100] The contact mark range is discretized into a two-dimensional tooth profile boundary element model to form several pairs of large wheel tooth profile slices and small wheel tooth profile slices. Specifically:

[0101] In view of the technical problem, the force process analysis of the gear teeth shows that local contact deformation, gear tooth bending deformation from the root to the top of the tooth, and torsional deformation on the spokes are generated at the moment of meshing of the large and small wheels. For bending stiffness and torsional stiffness, the two-dimensional boundary element method is introduced to solve the problem of efficient calculation of the linear part of the meshing stiffness of two-dimensional asymmetric arbitrary tooth profiles. The boundary conditions are similar to those of finite elements, and only the displacement constraints and load constraints of the nodes need to be added.

[0102] When using the boundary element method to calculate bending deformation, it is first necessary to determine the unit type that characterizes the research domain. Different unit types will have differences in quantity and characterization accuracy when characterizing the tooth profile shape. After determining the unit type, the calculation target domain is meshed according to the unit type. In this study, the contact area of ​​the spiral bevel gear is discretized, and the bending deformation of the discrete sheet model can be equivalently processed through the center plane of the model. The contour of the neutral surface of the sheet can be represented by a boundary element mesh, and the mesh is pre-processed with boundary conditions through force analysis when the gear teeth are meshed. Finally, the boundary element model is calculated and the bending deformation is obtained through data post-processing. The boundary element method has unique advantages over the cantilever beam model and the finite element method.

[0103] The boundary element mesh unit type selected in this embodiment constructs a boundary element model of the two-dimensional tooth profile of the spiral bevel gear in the form of boundary element mesh model construction. In order to make the calculation object more consistent with the actual model, it is necessary to fill in the rest of the parts except the mesh points of the single tooth profile of the gear. During the meshing process of the gear teeth, the gear teeth on the left and right of the meshing teeth may have a certain influence on the gear teeth currently under stress. The introduction of the left and right half tooth profiles makes it closer to the overall situation of the real single tooth meshing. In order to include the torsional deformation of the gear tooth matrix during calculation, the gear tooth matrix of the target tooth profile needs to be supplemented. Therefore, the complete boundary element gear model is defined as three main parts, 1) is the main loaded tooth profile part. This part can be discretized by constructing a normal section along the normal direction of the tooth surface in the elliptical area calculated by the single tooth loading contact analysis. 2) is the fillet position. On the one hand, the fillet has a great influence on the stress concentration of the tooth root. Like the finite element, the fillet in the boundary element can also better avoid the stress concentration phenomenon, making the calculation more accurate; on the other hand, adding fillets in the boundary element also conforms to the morphological characteristics of the real gear tooth root position. 3) is the tooth profile boundary. The tooth profile boundary is added to include the torsional deformation of the gear tooth matrix in the calculation and increase the calculation accuracy. The tooth profile part and the fillet part are both free boundary conditions, while the tooth profile boundary is a fixed boundary condition.

[0104] Through the analysis of the meshing force at the moment of contact under the action of gear tooth load, the direction of the meshing force is along the normal direction of the tooth profile on the tooth surface, and the forces of the large and small wheels are just in opposite directions. When the tooth surface is meshing, there is only one instantaneous contact ellipse at each moment. The size of the ellipse changes with the load size, and the size of the ellipse usually does not extend to the entire tooth surface. The Saint-Venant principle points out that if the resultant force of the load is correct, the stress effect in the area far away from the load is almost zero. Therefore, the current discretization of the gear tooth segment only needs to consider the stress distribution range under the load, that is, the slicing of a single gear tooth only needs to be within the range of its instantaneous contact ellipse domain. Therefore, the position of the instantaneous contact ellipse on the tooth surface can be used to construct several tooth profile normal sections in the direction of the major axis, such as Figure 3 As shown in the figure, the intersection line between the solution section and the two sides of the tooth profile is obtained and the intersection line is discretized into several nodes, which finally form the discrete points of the tooth profile.

[0105] The meshing state of the shaft sections of the large and small wheels is as follows Figure 4 As shown in the figure, the contact ellipse range on the tooth surface is discretized along the vertical direction of the root cone. The normal section of the meshing point is established here, and the concave and convex tooth surfaces S1 and S2 of the gear tooth are constructed using the fitting surface. Figure 5 The normal vector of the conjugate contact point P0 on S1 is Calculate the tangent vector orthogonal to the normal vector at the root cone position of tooth surface S1 The cut vector It is also the normal vector of the normal section through P0. According to the vector Perpendicular to Also perpendicular to And two points P0 and P v0 They are all located at the tooth roots of the two tooth surfaces. In the dual parameters of the fitting surface, the parameters v in the tooth height direction are all 0. The above relationship can be used to obtain formula (9):

[0106]

[0107] Where: A, B, C are vectors Components in the x, y, and z directions; is the vector of any point on the normal section plane. The two tooth surface points P are iterated through equation (9): v0 , P v1 The v parameters of surfaces S1 and S2 are discretized, and the u parameters are iterated in formula (9) to finally obtain all the discrete points where the normal section intersects with the surfaces S1 and S2.

[0108] The two-dimensional grid of the boundary element is defined in the xy plane of the coordinate system. In order to facilitate point numbering, the origin of the xoy plane is defined here as the first point of the concave tooth root position. Since the normal section is intercepted in the gear coordinate system, the tooth profile points obtained are located in the same plane, but are still in the space. The two-dimensional boundary element calculation cannot be performed directly, and the intercepted tooth profile points need to be transformed into the defined plane xoy. In order to more simply process the tooth profile points, choose to write the normal section changes into the xy plane, then all tooth profile points can be written in the same way. Let the normal vector of the xoy plane be Then the normal vector of the normal section Need to transfer to the normal vector According to the geometric relationship of vector space, the normal vector of formula (10) and the transformation relationship of each coordinate point can be obtained.

[0109]

[0110] Will Then you can get the law vector Rotary axis at this time and the normal vector of the xoy plane Angles available Indicates. Around Rotation The two can be overlapped, and the tooth profile point set is located in a certain plane in the airspace with the x-axis as the normal, and the x-axis coordinates of the tooth profile point set are all equal. Therefore, rotating the current tooth profile point 90° around the y-axis can obtain the position with the same z-coordinate on the tooth profile point, and marking the z-coordinates of all tooth profile points as 0, so that all tooth profile points can be written on the plane xoy to obtain the tooth profile vector set. The root of the tooth profile point is located on the y-axis, and the concave tooth root of the tooth surface is used as the origin of the boundary element grid. All tooth profile vectors are subtracted from the current tooth profile vector set. The vector set of tooth profile points located at the origin can be obtained. Finally, the two tooth root positions of the two tooth profile points are calculated. The angle σ with the x-axis is obtained by rotating the tooth profile vector set around the z-axis by σ, and finally obtaining the tooth profile section points on the plane xoy that conform to the boundary element mesh rules.

[0111] After completing the tooth profile processing, the remaining positions of the tooth profile mesh need to be supplemented. The left and right tooth profiles are shifted to the right and left by one pitch to fill the mesh of the remaining two half teeth. The thickness of the gear tooth matrix is ​​1.5 times the tooth height, and the matrix mesh is constructed through the tooth profile points of the two half teeth. The mesh of the tooth profile fillet position must first determine the fillet radius. According to the national standard GB / T1356-2001

[91] , the fillet radius of the tooth root is selected as 0.38 times the module.

[0112] When constructing the tooth profile fillet, due to the particularity of the spiral bevel gear tooth profile, the pressure angles of the left and right tooth profiles are different, resulting in unequal fillet radii within the same tooth profile range. Figure 6 .

[0113] Figure 6 It can be seen that since the pressure angle on the left side of the tooth profile is greater than that on the right side, the root fillet of the same radius at the left root point cannot be located on the same root line as the root fillet on the right side. In the figure, δ is the distance from the bottom of the fillet to the root line. At this time, the left tooth profile needs to be extended, and the tooth profile points of the left tooth profile are fitted with a polynomial fitting equation to obtain the following expression (11):

[0114] Ax 3 +Bx 2 +Cx+Dy=0 Formula (11)

[0115] In the formula, A, B, C, and D are the fitting parameters of the polynomial. Figure 6 The geometric relationship between the fillet, tooth profile, and tooth root line can be established. Due to the condition that the tooth profile line is tangent to the fillet, it can be obtained that there is a point P on the tooth profile line. F The center of the fillet is P c To point P F The vector and point P F The tangent vectors on the tooth profile are orthogonal, and the center point P c To point P F The distance is the fillet radius r, and P can be calculated F The tangent vector angle is obtained by rotating it 90° to obtain P F The angle of the normal vector at is

[0116] Combine this with the radius and P F The vector at Perform vector calculation to get the center vector of the tooth root fillet The specific relationship is shown in formula 12:

[0117]

[0118] The following conditions must also be met:

[0119]

[0120] In the formula, is the tooth root line vector, (y) represents the y-direction value in the vector.

[0121] According to the above deduction, a rounded corner with smooth transition can be constructed.

[0122] After completing the boundary meshes of the large and small wheels, the two-dimensional tooth profile boundary element meshes of the large and small wheels can be obtained according to the conditional output.

[0123] Then, the 2D tooth profile boundary element mesh of the large wheel and the corresponding 2D tooth profile boundary element mesh of the small wheel are modified by the deformation of the tooth profile. Specifically:

[0124] The two-dimensional tooth profile boundary element mesh model of the tooth profile sheet unit obtained by the above method now needs to set boundary conditions for the mesh of the boundary element for the tooth profile bending deformation. The tooth edge contour point and the fillet point are both free boundary ends, while the gear tooth base position is a fixed boundary end. The load needs to be applied in the normal direction of the force point at the tooth profile point. The specific boundary conditions are as follows: Fig.17 shown.

[0125] The direction of the normal load can be calculated based on the normal line of the unit surface, but the bending deformation of the tooth profile section cannot be directly measured by the curved tooth profile shape. Similar to the deflection treatment of a cantilever beam, a single tooth profile is regarded as a cantilever beam, the midpoint between the two points on the tooth profile vertex and the root point is taken as the tooth profile neutral line, and the deflection deformation of the tooth profile neutral line is taken as the tooth profile bending deformation, see Fig.18 .

[0126] Fig.18 Where F is the magnitude of the normal force acting on the tooth profile; α is the angle at which the normal force F acts on the tooth profile; the dotted line I is defined as the line connecting the midpoints of the left and right points of the tooth top and the left and right points of the tooth root. This line is the neutral line of the gear tooth profile, and the intersection point P of the normal force F and the neutral line I is the reference point for measuring the bending deformation of the tooth profile. When this point deforms as the tooth profile is loaded, the deformation direction of this point is not necessarily consistent with the direction of the normal load. According to the calculation principle of stiffness, the force divided by the deformation of the object in the direction of the force is defined as the stiffness of the object to resist deformation. Therefore, it is necessary to correct the deformation of the reference point P in the direction of the normal force F, and then according to the Somigliana identity, the displacement of any point in the domain can be obtained by all the known surface forces and displacements on the boundary, and then the original deformation P of the reference point P can be obtained. x , P y The total deformation of point P after correction is solved according to the following formula: xy :

[0127]

[0128] Get the local contact force of each tooth profile. Specifically:

[0129] As mentioned above, when calculating the bending deformation of a single tooth, the gear tooth is discretized into several slices along the major axis of the contact ellipse, and the neutral plane of the slice is equivalent to the bending deformation of the entire slice. In order to solve the deformation of the slice, it is first necessary to solve the magnitude of the contact force on a single slice. For example, the contact stress in the entire meshing ellipse area satisfies the point contact Hertz distribution model, but considering that the thickness of a single slice model is small, the stress change in the thickness direction is almost negligible, while the stress gradient change in the height direction is still large. Therefore, in this case, the two slices of the large and small wheels at the corresponding meshing positions can be regarded as two contacting cylinders, and the acting force is the distribution of the entire elliptical contact stress on the action area of ​​the slice.

[0130] Through the above analysis and the discretization of the sheet, each sheet on each instantaneous contact ellipse area of ​​the large and small wheel teeth is now assumed to be a pair of meshing cylindrical gears. When the hypothetical gears are in meshing state, not only will the tooth profile bend and deform, but also a local contact deformation will occur at the tooth profile contact position. For each sheet gear, it is assumed that it is in contact along the major axis of the contact ellipse. The nonlinear contact deformation δ caused by the normal load F applied to the sheet cross section is H It can be described by the Hertzian line contact formula derived by Weber and Banaschek. See formula (14):

[0131]

[0132] Where: H is the Hertz deformation, F is the normal load acting on the sheet segment, b is the thickness of the section, ν is the Poisson's ratio, E is the elastic modulus, and h c is the distance from the contact point to the neutral layer of the tooth profile along the load direction, a represents the minor axis length of the contact, R * is the equivalent radius of curvature, R is the radius of curvature, K n is the normal curvature of the contact point. The subscripts 1 and 2 in the symbol represent the large wheel and the small wheel respectively.

[0133] Since the curvature of the spiral bevel gear tooth surface is not constant along the contact line direction, there are infinitely many different curvature radii at each instantaneous position. However, since the contact area is formed by the meshing of two tooth surfaces, the long axis of the contact area under load can be regarded as the theoretical contact line. Therefore, when solving the curvature of each cross-section segment on the contact line, in order to simplify the processing, only the curvature perpendicular to the contact line direction is solved. To better illustrate this point, as shown in Figure 7As shown, it is assumed here that the sheet segment model of a single gear tooth is divided, and a pseudo-contact cylinder is constructed inside the tooth profile of each sheet segment. The axis of the cylinder must be in the same direction as the contact line on the sheet segment. R1 and R2 in formula 14 are the radii of the meshing cylinder on the cross-sectional segment. By correcting the contact area ellipse, the direction vector of the major axis of the ellipse on the surface is obtained. The direction vector of the minor axis direction is obtained by rotating the vector 90° around the meshing point normal vector. According to the "Principle of Gear Meshing", the angle η between the two directions can be calculated. In differential geometry, the parameters R1 and R2 can be obtained through Euler's formula 15.

[0134]

[0135] Where: K I , K II are the two principal curvatures, K n is the normal curvature in the direction with an angle difference of η from the main curvature direction.

[0136] The quasi-contact cylinder model of the sheet contact model is established above. When the model is divided, the stress distribution area is also divided. Therefore, after dividing the contact area, the following can be obtained: Figure 8 The contact model of the discrete unit shown in the figure. The local coordinate system is established with the conjugate contact point as the origin, the x direction is consistent with the major axis direction of the contact ellipse, and the y direction is consistent with the minor axis direction of the contact ellipse. Figure 8 Where Δl is the width of the discrete unit, and also the thickness of the discrete sheet; f i is the normal load on each discrete element within the contact range. The normal loads of all elements should satisfy Where n is the number of cross-sectional segments.

[0137] The above model is applied to the sheet body in the single tooth contact area to calculate the force area on each sheet body micro segment, and then the load size on each sheet body micro segment can be calculated through the stress distribution model in the Hertz point contact model. First, the unit area of ​​the contact ellipse is divided.

[0138] During the meshing process, the major axis of the ellipse forms a certain angle with the tooth height. When the meshing force increases, the contact ellipse becomes larger. When the load reaches a certain level, the ellipse may exceed the tooth surface. Fig. 9 shown. Fig. 9The diagram is a schematic diagram of the three states of the contact ellipse on the theoretical tooth surface of the large wheel. The dotted line in the figure represents the projection of the tooth top line of the small wheel meshing with the large wheel on the large wheel. The two ellipse endpoints of the actually calculated conjugate contact area should not exceed the edge of the actual working tooth surface. Taking the large wheel as an example, the tooth top direction should not exceed the tooth top line of the large wheel; the tooth root direction should not exceed the theoretical projection line position of the tooth top line of the small wheel in the meshing state with the large wheel; and the tooth length direction should not exceed the boundary of the large end or the small end. The elliptical contact state that exceeds this constraint is collectively referred to as a "semi-ellipse" in this article, and the complete contact ellipse state is collectively referred to as a "full ellipse".

[0139] The complete elliptical state of the contact area fully satisfies the Hertz contact model, and the stress distribution also conforms to the elliptical distribution form of Hertz stress. Geometrically, there is no need to consider the influence of the tooth top line and the tooth root line on the ellipse. Compared with the trimmed ellipse, the geometric state is more regular.

[0140] This embodiment first analyzes and calculates the local contact area of ​​the full ellipse position, then establishes a semi-ellipse trimming model for the semi-ellipse situation, solves the major axis endpoint of the trimmed ellipse within the tooth surface range, and then calculates the local contact area. In the case of semi-ellipse contact, a load correction model is proposed to calculate the local contact force under semi-ellipse contact, and in the case of full ellipse contact, the local contact force is solved by integrating the stress distribution within the contact area.

[0141] Solution of local contact area of ​​full elliptical contact

[0142] The processing flow of the local load area at the full ellipse is as follows Fig.13 As shown in the figure, when processing the complete ellipse, first divide the complete ellipse into two upper and lower regions along the minor axis from the conjugate contact point position, as shown above Fig. 9 , the left and right regions of the ellipse are processed in the same way. Let the radial vector of the major axis endpoint of the contact area be The conjugate contact point radius vector is The law vector is The number of segments on the left and right regions is n respectively. along Projecting onto the common tangent plane gives the vector The step length between the sheets on the contact ellipse is Direction Vector The intersection of the two end faces of the sheet body, the neutral plane and the major axis of the ellipse is calculated by the following formula (16). The relationship between the parameters of the contact ellipse and the sheet body is Fig.10 :

[0143]

[0144] Where: They are the intersection points of the left, right and middle sections of the sheet and the major axis of the contact ellipse, is the intersection point between the required sheet cross section and the major axis of the contact ellipse, and i represents the current sheet segment number. Fig.10 Where λ is the angle between the neutral surface of the sheet segment and the direction of the major axis vector of the contact ellipse, and the neutral surface is the boundary element mesh constructed above. Project the two end points of the boundary element of the neutral surface on the tooth profile onto the common tangent plane, and the line connecting the two points can be regarded as the projection of the normal section on the plane. Let the normal section projection line vector be but Intercept parameter

[0145] Secondly, judging the size of the intercept parameters a1, a2 and the contact ellipse major axis parameter a, it is easy to get a1.OR.a2≤a. This is divided into two cases for discussion.

[0146] 1) When the two intercept parameters a1 and a2 are both smaller than the major axis a of the contact ellipse, the micro-segment of the sheet and the contact ellipse are Fig.11 Position state, at this time, there are 4 intersection points between the two end lines of the sheet and the ellipse, and the force area of ​​the sheet segment is the area range of the intersection of the sheet segment and the contact ellipse. The parameter λ is used to write the straight line equation of the cross-section end line, which is convenient for finding the intersection point with the ellipse. This part of the area can be divided into Fig.11 The three parts in are solved, where S1 and S3 are the parts containing the elliptical arc segments, and S2 is the quadrilateral part composed of 4 intersection points. The total area is S = S1 + S2 + S3. The partial area is solved according to formula (17):

[0147]

[0148] Where P i(x) , P i(y) The x and y values ​​of the i-th point.

[0149] 2) When the two intercept parameters a1 or a2 are equal to the major axis a of the contact ellipse, it means that one end face of the sheet segment is located at the end point of the major axis of the ellipse, while the other end face is still within the range of the ellipse. Fig.12 State, at this time, the two end lines of the sheet body and the ellipse have only two intersection points. At this time, the loaded part is the area enclosed by one end of the ellipse and the end line of the cross section within the ellipse. The area S is solved according to formula (18):

[0150]

[0151] Solution of Local Contact Area of ​​Semi-Elliptical Contact

[0152] like Fig.14As shown in the figure, when discussing the load-bearing area of ​​the semi-ellipse segment, we must first trim its edges according to the tooth profile boundary characteristics. Based on the above description of the semi-ellipse, we can conclude that the conditions for the ellipse endpoint to fall within the tooth surface range are: 1) The endpoint P does not exceed the top edge of the large wheel tooth; 2) The endpoint P does not exceed the top edge of the small wheel tooth; 3) The endpoint P must be within the tooth width in the direction of the tooth length. Based on the above conditions, we can determine whether any point is within the tooth surface range. Fig.15 The relationship diagram of any point P in the tooth surface range, in which L and R are the projection parameters of any point on the gear shaft section; X o The intersection point O between the gear crown and the shaft r The distance R or is the outer radius of the gear, P e is the intersection of point P and the cone line along the axial direction, and the distance from point O r The distance is X or Combining conditions 1 and 2 with their geometric relationship, the calculation condition that point P does not exceed the edge of the tooth top can be expressed as formula (19):

[0153]

[0154] After rearranging the above formula (19), we can get formula (20):

[0155]

[0156] X o , R or According to the calculation, the wheel crown distance of the large and small wheels changes accordingly; Γ a is the face cone angle.

[0157] According to condition 3) Fig.15 and Fig.16 The position relationship is established, and the gear shaft section tooth frame is established, and the root cone generatrix is ​​taken as the horizontal coordinate x, and the straight line passing through the midpoint of the tooth surface and perpendicular to the root cone line is taken as the vertical coordinate y. Fig.15 It can be seen that, measured along the root cone generatrix direction, from the staggered point O on the gear tooth axis r The distance R to the y axis or It can be calculated by the following formula (21):

[0158] R go =R mr sinΓ r +L mr cosΓ r Formula (21)

[0159] Depend on Fig.15 The coordinates x of any point P in the coordinate system xoy can also be derived p ,y p The calculation formula is (22):

[0160]

[0161] In the formula, Z R is the distance from the root cone vertex to the axis intersection point; Γ r is the root cone angle.

[0162] At this time, point P needs to be within the tooth surface range in the tooth length direction, and only needs to satisfy 0.5F-|x p |>0. F is the tooth surface width. When the tooth surface point satisfies both equation (22) and equation 0.5F-|x p |>0, the point is within the tooth surface.

[0163] Based on the principle of solving the contact ellipse endpoints using conjugate difference surfaces, the calculation process for determining whether the endpoint falls within the tooth surface range and the intersection with the tooth surface boundary when it falls outside the tooth surface is as follows.

[0164] First, let the condition that the end point is within the range of the tooth top of the large gear tooth surface be C1; the condition that the end point is within the range of the tooth top of the small gear tooth surface be C2; ​​the condition that the end point is within the range of the tooth length be C3; the distance from the end point to the tooth top on the large gear tooth surface is D1; ​​the distance from the end point to the tooth top on the small gear tooth surface is D2; the larger distance from the small end and the large end in the tooth length direction is D3. tl As a variable, the endpoints of one side of the ellipse are iteratively calculated to obtain several points on the same conjugate contact line. Let the point on the conjugate contact line corresponding to the large and small wheels be P G , P P , we can get conditional judgment (23) as:

[0165]

[0166] Where: x pG When C1, C2, and C3 are all greater than 0, point P G , P P All are within the tooth surface range; when C1, C2, and C3 are not all greater than 0, it is necessary to determine the position relationship between the point and the tooth surface boundary, establish the gear axis section coordinate system, and obtain the distance expression (24) between point P and the tooth top:

[0167]

[0168] D1 and D2 are solved by equation (24), and the expression of D3 is equation (25):

[0169]

[0170] Where: Tooth width vector at the pitch cone, where γ is the pitch cone angle; is the axis section vector of any tooth surface point; F is the vector from the axis intersection point O to the wheel crown; a is the width of the tooth top line at the face cone position; γ a is the cone angle. 1(2) Convert to solve and The angle between them, D3 is converted into the solution vector Cone vector in section The projection length D in the direction 31 With D 32 The larger value in .

[0171] Then, due to the different points where the endpoints fall on the tooth surface, the judgment conditions can be arranged and combined into the following 5 types of judgments. When only C1 < 0, the point exceeds the tooth top line of the large wheel on the large wheel and exceeds the tooth root line of the small wheel on the small wheel. The distance from the tooth top line of the large wheel is D1. tl Iterative calculation can find a point on the conjugate contact line that satisfies D1=0. This equation shows that the distance from the large wheel tooth top line on the conjugate contact line is 0, and it also shows that the point falls on the tooth top and is within the tooth surface range. Similarly, D2 and D3 can be judged similarly. When D1<0 and D3<0 exist, the endpoint is outside the large wheel tooth top line and outside the tooth surface width range. Take D as the smaller value between the two, and iterate the tooth surface parameter θ tl Until D = 0, the point is within the tooth surface. The same is true for the case where D2 < 0 and D3 < 0.

[0172] Solution of local contact loads

[0173] After the unit area is divided, the normal load force borne by each sheet under the current load distribution needs to be calculated. When the number of units is large enough, it can be considered that the load distribution of the unit in the width direction remains unchanged, and the load distribution in the length direction obeys the Hertz distribution theory or the dimensionality reduction load distribution method. The load on the unit area can be averaged according to the Hertz distribution, and the neutral layer line of the unit section is taken as the average reference. The load distribution on the unit is consistent with the load distribution on the neutral layer line. Let the two intersection points of the neutral layer line and the contact ellipse be P C1 , P C2 ; The maximum stress value of the contact ellipse is P0; The angle between the cross section and the major axis of the contact ellipse is α; The intercept is t; The area of ​​the cross section is S i According to the Hertz stress distribution model, the unit stress distribution on the neutral layer line and the normal load f on the contact unit surface can be obtained. i As shown in formula (25):

[0174]

[0175] Where: P C1(y) , P C2(y)are the y-axis coordinates of the upper and lower limits of the intersection respectively. Considering the positional relationship between the neutral plane projection line and the tooth top line, it is necessary to C1 , P C2 The intersection point P of the neutral layer line and the tooth top line F When the contact ellipse is complete, the calculation can be performed directly according to formula (25). When the contact ellipse is a semi-ellipse, And P C1 Located in P F On the left, the upper and lower limits of the integral are P C1(y) , P C2(y) , P C1 Located in P F On the right, the upper and lower limits of the integral are P F , P C2(y) . Similarly, we can deduce If the load distribution is subject to dimensionality reduction, the unit force can be obtained by integrating equation (25) within the load range of the cross-section segment. The calculation of the area and the intersection point is the same as above.

[0176] Combining the elastic boundary element method with the Hertz line contact, the meshing stiffness of each large wheel tooth profile sheet and small wheel tooth profile sheet is obtained, and all meshing stiffnesses are superimposed. Specifically:

[0177] The unit load f i Substituting into equation (14), the stress distribution P on the neutral surface can be used in the boundary element preprocessing to calculate the Hertz contact deformation δ hi And the bending deformation of the gear teeth The instantaneous meshing stiffness at the current rotation angle can be solved by equation (26), and the single tooth meshing stiffness can be obtained by calculating the entire single tooth meshing cycle by equation (1).

[0178]

[0179] The above embodiments are only preferred embodiments for fully illustrating the present invention, and the protection scope of the present invention is not limited thereto. Any equivalent substitution or change made by a person skilled in the art based on the present invention is within the protection scope of the present invention.

Claims

1. A method for obtaining the single tooth meshing stiffness of a spiral bevel gear, characterized in that: The method is specifically as follows: Based on the Hertz contact model, the contact imprint range of the spiral bevel gear under loaded contact state is obtained; The contact mark range is discretized into a two-dimensional tooth profile boundary element model to form a number of pairs of large wheel tooth profile slices and small wheel tooth profile slices; Obtain the local contact force of each tooth profile segment; Combining the elastic boundary element method with the Hertz line contact, the contact deformation and bending deformation of each pair of meshing large and small gear tooth profiles are obtained, and then the meshing stiffness of the unit sheet is obtained; Add up all meshing stiffnesses; Based on the Hertz line contact theory, the Hertz deformation of each sheet is obtained by hi , through the stress distribution on the neutral surface P Get the bending deformation of the large wheel sheet And the bending deformation of the small wheel sheet Then, the single tooth meshing stiffness is obtained based on the following formula: Among them, K i is the meshing stiffness of each sheet, K is the meshing stiffness of a single tooth; unit load f i is the local contact force of the tooth profile sheet; is the bending deformation of the sheet of the large wheel; is the bending deformation of the small wheel sheet.

2. The method for obtaining the single tooth meshing stiffness of a spiral bevel gear according to claim 1, characterized in that: The method for obtaining the contact mark range of the spiral bevel gear under the loaded contact state is specifically: establishing a common tangent plane when the large wheel and the small wheel are meshing, projecting the major axis endpoint and the minor axis endpoint of the contact ellipse onto the common tangent plane, and then correcting the major axis vector and the minor axis vector projected onto the common tangent plane based on the Hertz point contact theory. The contact ellipse is formed by the deformation and extension of the two tooth surfaces on their common tangent plane under the action of torque when the large wheel and the small wheel are meshing.

3. The method for obtaining the single tooth meshing stiffness of a spiral bevel gear according to claim 2, characterized in that: The major axis direction vector of the contact ellipse on the common tangent plane can be expressed as: in: It is a point on the contact trace of the large wheel and the small wheel. The contact trace is the major axis of the contact ellipse. is the direction vector of the major axis of the contact ellipse on the common tangent plane; is the vector of the two end points of the major axis of the corrected ellipse on the common tangent plane; is the direction vector of the small wheel rotation axis; a is the length of the major axis of the contact ellipse.

4. The method for obtaining the single tooth meshing stiffness of a spiral bevel gear according to claim 3, characterized in that: The method of slicing the gear part in the contact deformation area at each meshing moment and discretizing it into a two-dimensional tooth profile boundary element model is as follows: Acquire a plurality of two-dimensional tooth profile boundary element meshes of the large wheel and a corresponding two-dimensional tooth profile boundary element meshes of the small wheel along the major axis direction of the contact ellipse, wherein each slice corresponds one-to-one to the two-dimensional tooth profile boundary element mesh of the large wheel and the two-dimensional tooth profile boundary element mesh of the small wheel; The intersection of the normal load direction and the neutral line of the boundary element is taken as the reference point for measuring the deformation of the tooth profile, and the reference point is deformed and corrected along the normal load direction to obtain the bending deformation of the tooth profile.

5. The method for obtaining the single tooth meshing stiffness of a spiral bevel gear according to claim 4, characterized in that: The method for obtaining the local contact area S of the tooth profile sheet is: Wherein, the full ellipse is defined as the case where the contact ellipse does not exceed the tooth surface, and the half ellipse is defined as the case where the contact ellipse exceeds the tooth surface; For the full ellipse case, if the intercept parameters a1 and a2 are both smaller than the major axis a of the contact ellipse, then S = S1 + S2 + S3; in, are the intersection points of the left, right and middle sections of the sheet and the major axis of the contact ellipse, respectively. is the intersection of the required tooth profile section and the major axis of the contact ellipse, a is the length of the major axis of the contact ellipse, b is the length of the minor axis of the contact ellipse, P i(x) , P i(y) is the x value and y value of the i-th point; If a1 or a2 is equal to the major axis a of the contact ellipse, then Where λ is the angle between the neutral plane of the tooth profile segment and the direction of the major axis vector of the contact ellipse.

6. The method for obtaining the single tooth meshing stiffness of a spiral bevel gear according to claim 4, characterized in that: The method for obtaining the local contact force of each tooth profile segment is: Obtain the local contact force of the tooth profile sheet; Among them, P C1 , P C2 P is the two intersection points of the unit neutral layer line and the contact ellipse, and each pair of large wheel sheet and small wheel sheet is defined as a unit; C1(y) , P C2(y) are the y-axis coordinates of the upper and lower limits of the intersection, and the unit load f i is the local contact force of the tooth profile sheet; When the contact ellipse is a full ellipse, the above formula can be used for calculation. When the contact ellipse is a semi-ellipse, And P C1 Located in P F On the left, the upper and lower limits of the integral are P C1(y) , P C2(y) , P C1 Located in P F On the right, the upper and lower limits of the integral are P F , P C2(y) ; Similarly, we can deduce situation; P F is a point on the tooth profile line, so that the center point P of the tooth profile fillet c To point P F The vector and point P F The tangent vectors on the tooth profile are orthogonal, and the center point P c To point P F The distance is the radius r of the tooth profile fillet, and P is the stress distribution on the neutral surface of the tooth profile sheet.

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