A method and system for designing the tooth root transition arc of a straight bevel gear.

Abstract

This invention relates to the field of gear transmission technology and discloses a design method for the root transition arc of a straight bevel gear. The method includes: using control points as input, calling a spherical linear interpolation function, and combining it with a Bézier curve algorithm to generate a smooth third-order Bézier transition curve located on a sphere; traversing different cone distances towards the small end of the gear to generate a series of spherical Bézier transition curves at different cone distance positions, discretizing them to form a three-dimensional point cloud of the root transition surface; interpolating or fitting this point cloud with the tooth surface point cloud at the same cone distance to form a smooth curve; and using a skinned surface modeling method in a CAD system based on these curves to generate continuous tooth surface sheets. The method designed in this invention restricts both the control points and the surface to a unit sphere, ensuring that the surface conforms to the spherical structure and does not deviate from the theoretical contact sphere; it also facilitates stable estimation of principal curvature and unit normal, making it suitable for high-precision LTCA or profile modeling.

Description

Technical Field

[0001] This invention relates to the field of gear transmission technology, and in particular to a design method and system for the tooth root transition arc of a straight bevel gear. Background Technology

[0002] For forged spherical involute bevel gears with a small number of teeth, the base cone angle is often larger than the root cone angle. Therefore, the transition surface between the base cone and the tooth root needs to be designed and optimized to improve the bending strength and load-bearing capacity of the tooth root. Although forged bevel gears are not constrained by the tool movement trajectory and have greater design freedom, this advantage is not utilized under the standard 0.3 times module transition fillet standard. Furthermore, the fixed fillet standard causes the starting position of the fillet to exceed the base cone angle of the bevel gear, resulting in meshing interference and affecting the contact transmission of the gear.

[0003] A spherical involute is a plane rolling around a cone, and the trajectory of any point on the line where the cone and the plane come into contact lies on a sphere with a corresponding cone distance.

[0004] Existing technologies, based on the standard spherical involute equation, cannot achieve coincidence of the modified tooth surface. Existing methods slice the tooth surface along the Z-axis, but the z-axis position of the tooth surface is too small for the spherical involute back cone. Therefore, it is necessary to increase the large end module and tooth width of the tooth surface to construct a complete transition surface sheet. Furthermore, the subsequent design involves cutting along the boundary of the spherical radius, which is a complex process. Summary of the Invention

[0005] This invention addresses the problem in existing technologies where increasing the large-end module and tooth width of the tooth surface is necessary to construct a complete transition surface sheet, and subsequent design involves cutting along the boundary of the spherical radius, resulting in a complex process. The invention provides a method for designing the transition arc at the tooth root of a straight bevel gear.

[0006] To solve the above-mentioned technical problems, the present invention provides the following technical solution:

[0007] A method for designing the root transition arc of a straight bevel gear, the method comprising:

[0008] Step 1: Determining the control points. The control points are determined by the spherical involute tooth surface of the straight bevel gear. The control points include the first control point P0, the second control point P1, the third control point P3, and the fourth control point P2.

[0009] Step 2: Using the four control points P0, P1, P2, and P3 as input, construct a smooth transition curve on the sphere based on the spherical parametric curve generation algorithm; and call the spherical linear interpolation function, combined with the Bézier curve algorithm, to generate a smooth third-order Bézier transition curve on the sphere.

[0010] Step 3: Traverse different cone distances to the small end of the straight bevel gear. Repeat steps 1 to 2 for each cone distance to generate a series of spherical transition curves at different cone distance positions. Discretize the spherical transition curves to form a three-dimensional point cloud of the tooth root transition surface.

[0011] Step 4: Interpolate or fit the 3D point cloud of the tooth root transition surface obtained in Step 3 with the tooth surface point cloud at the same cone distance to form a smooth curve. For the interpolated or fitted smooth curve, use the skin surface modeling method in the CAD system to generate a continuous tooth surface sheet.

[0012] Preferably, the spherical parametric curve generation algorithm includes the spherical Bézier curve algorithm or the spherical B-spline curve algorithm.

[0013] Preferably, the determination of the first control point P0, the second control point P1, the third control point P3, and the fourth control point P2 includes:

[0014] S1. Extract the large end base cone point of the spherical involute tooth surface of the straight bevel gear as the first control point P0. The coordinates of the first control point P0 are: ;

[0015] S2. Based on the coordinates of the first control point P0, calculate the tangent direction of the tooth surface at the first control point P0 along the sphere with the cone distance as the radius, and obtain the coordinates of the second control point P1 on the sphere by extending this tangent direction on the sphere. The coordinates of the second control point P1 are: ;

[0016] S3. Extract the point on the root cone of the large end of the straight bevel gear as the third control point P3. The coordinates of the third control point P3 are: ;

[0017] S4. Based on the coordinates of the third control point P3, calculate the relevant tangent direction at the third control point P3, and obtain the coordinates of the fourth control point P2 on the sphere by extending the spherical arc. The coordinates of the fourth control point P2 are: .

[0018] Preferably, S2, calculating the tangent direction of the tooth surface at the first control point P0 along the sphere based on the coordinates of the first control point P0, and obtaining the coordinates of the second control point P1 on the sphere by extending the tangent direction, includes:

[0019] Calculate the vector of the tooth surface along the tangent direction of the sphere at the first control point P0. The coordinates of the first control point P0 are: ,

[0020] ;

[0021] Where R is the cone distance, It is the base cone angle. It is the angle rotated by half the tooth thickness; determine the rotation axis used for spherical extension, and calculate the rotation axis vector u based on the position of point P0 and the tangent direction t3 obtained in the previous step; perform spherical rotation to obtain the coordinates of the second control point P1, and use the Rodrigues rotation formula to calculate the coordinates of the second control point P1. ;in, It is the angle of rotation around the u-axis.

[0022] As a preferred option, S3, extract the point on the root cone of the large end of the straight bevel gear as the third control point P3, and the coordinates of the third control point P3 are... It includes:

[0023] Based on the geometry of the cone and obtaining the coordinates of its intersection point with the sphere, construct a structure based on the cone angle. Define the conical surface and calculate the intersection line between the conical surface and the current cone distance sphere, then calculate the intersection point from the intersection line; the intersection point is the third control point P3, and the coordinates of the third control point P3 are... , Where R is the cone distance, , is the root cone angle, and γ is the pitch bevel thickness angle.

[0024] Preferably, in step S4, based on the coordinates of the third control point P3, the relevant tangent direction at the third control point P3 is calculated, and the coordinates of the fourth control point P2 on the sphere are obtained by extending the spherical arc: point P3 is rotated around the Z-axis by a predetermined angle φ to obtain the fourth control point P2, whose coordinates are... ;

[0025] Where R is the cone distance, γ is the root cone angle, and γ is the pitch bevel thickness angle. It is the distance rotated along the tangential direction.

[0026] Preferably, in step 2, four control points P0, P1, P2, and P3 are used as inputs. Based on the spherical parametric curve generation algorithm, a smooth transition curve located on the sphere is constructed. The spherical linear interpolation function is called, and combined with the Bézier curve algorithm, a smooth third-order Bézier transition curve located on the sphere is generated, which includes obtaining all interpolation points on the sphere through the SLERP function.

[0027] For the interpolation points obtained by the SLERP function, a series of discrete point sets are generated by recursively interpolating on the sphere using the Bézier algorithm. Let {B0(0), B0(0.01), ..., B0(1)} be the discrete points, and let B0(t) form a spherical Bézier curve.

[0028] As a preferred method, recursive interpolation calculations on the sphere using the Bézier algorithm include:

[0029] Three-layer spherical linear interpolation is recursively performed within the range of interpolation parameter t from 0 to 1;

[0030] The first layer of interpolation is performed between control points P0P1, P1P2, and P2P3 to obtain intermediate points Q0, Q1, and Q2.

[0031] The second-level interpolation is performed between intermediate points Q0Q1 and Q1Q2 to obtain secondary intermediate points A0 and A1.

[0032] The third-level interpolation is performed between the second-level intermediate points A0A1 to obtain the final curve point B0(t);

[0033] The set of all B0(t) points constitutes the spherical Bézier transition curve.

[0034] To address the aforementioned technical problems, the present invention also provides a straight bevel gear tooth root transition arc design system, which is used to implement the aforementioned straight bevel gear tooth root transition arc design method, comprising:

[0035] The control point determination module is used to determine four control points of the spherical Bézier curve. The four control points include the first control point P0, the second control point P1, the third control point P3, and the fourth control point P2.

[0036] The spherical Bézier curve generation module takes four control points P0, P1, P2, and P3 as input, constructs a smooth transition curve on the sphere based on the spherical parametric curve generation algorithm, and calls the spherical linear interpolation function in combination with the Bézier curve algorithm to generate a smooth third-order Bézier transition curve on the sphere.

[0037] The 3D point cloud generation module for the tooth root transition surface traverses different cone distances to the small end of the straight bevel gear. For each cone distance, the module determines the control point and generates a series of spherical transition curves at different cone distance positions through the control point determination module and the spherical Bézier curve generation module. The spherical transition curves are then discretized, and the discretized curves form the 3D point cloud of the tooth root transition surface.

[0038] The continuous tooth surface sheet generation module interpolates or fits the 3D point cloud of the tooth root transition surface obtained by the tooth root transition surface generation module with the tooth surface point cloud at the same cone distance into a smooth curve. The interpolated or fitted smooth curve is then used in the CAD system to generate a continuous tooth surface sheet using the skin surface modeling method.

[0039] Preferably, the first control point P0 is the large end base cone point of the spherical involute tooth surface of the straight bevel gear; the second control point P1 is obtained by extending the tangent direction of the tooth surface at the first control point P0 along the spherical surface with the cone distance as the radius.

[0040] The third control point P3 is a point on the root cone of the large end of the straight bevel gear; the fourth control point P2 is obtained based on the relevant tangent direction at the third control point P3 and by extending it through a spherical arc.

[0041] This invention, by adopting the above technical solutions, has significant technical effects:

[0042] The method designed in this invention restricts the control points and curved surfaces to a unit spherical surface, ensuring that the curved surface conforms to the spherical structure and does not deviate from the theoretical contact spherical surface.

[0043] The method designed in this invention is beneficial for the stable estimation of principal curvature and unit normal, and is suitable for high-precision LTCA or shape-modification modeling.

[0044] This invention constructs a Bézier curve / surface on a sphere for geometric transition design of the gear meshing region; it is natively nested in a spherical manifold; and it solves the problems of tangential jump and normal deviation that exist when the bevel gear tooth surface is modified and transitioned.

[0045] This invention proposes a method for constructing a third-order spherical Bézier curve. Control points include two endpoints located on the spherical surface; two intermediate control points are constructed along the tangential direction of the spherical surface; all control points satisfy spherical constraints; and construction conditions ensuring G2 continuity are provided. The fitted spherical Bézier curve or surface is used to connect involute curves and modified surfaces, maintaining normal / curvature continuity. Combined with existing forging processes, this improves surface smoothness and reduces stress concentration. Attached Figure Description

[0046] Figure 1 This is a flowchart of Embodiment 1 of the present invention.

[0047] Figure 2 This is a schematic diagram of the Bezier curve generation method of the present invention.

[0048] Figure 3 This is a schematic diagram of the control point coordinates of the present invention.

[0049] Figure 4This is a schematic diagram of the spherical Bezier curve of the present invention.

[0050] Figure 5 This is a schematic diagram of surface point cloud generation according to the present invention.

[0051] Figure 6 This is a schematic diagram of the complete transition arc generation of the present invention.

[0052] Among them, CAD (Computer-Aided Design); LTCA (Load Tooth Contact Analysis); Bézier (Bézier curve); and SLERP (Spherical Linear Interpolation) are used for spherical linear interpolation. Detailed Implementation

[0053] The present invention will now be described in further detail with reference to the accompanying drawings and embodiments.

[0054] Example 1

[0055] A method for designing the root transition arc of a straight bevel gear. Figure 1 The methods include:

[0056] Step 1: Determining the control points. The control points are determined by the spherical involute tooth surface of the straight bevel gear. The control points include the first control point P0, the second control point P1, the third control point P3, and the fourth control point P2.

[0057] Step 2: Using the four control points P0, P1, P2, and P3 as input, construct a smooth transition curve on the sphere based on the spherical parametric curve generation algorithm; and call the spherical linear interpolation function, combined with the Bézier curve algorithm, to generate a smooth third-order Bézier transition curve on the sphere.

[0058] Step 3: Traverse different cone distances to the small end of the straight bevel gear. Repeat steps 1 to 2 for each cone distance to generate a series of spherical transition curves at different cone distance positions. Discretize the spherical transition curves to form a three-dimensional point cloud of the tooth root transition surface.

[0059] Step 4: Interpolate or fit the 3D point cloud of the tooth root transition surface obtained in Step 3 with the tooth surface point cloud at the same cone distance to form a smooth curve. For the interpolated or fitted smooth curve, use the skin surface modeling method in the CAD system to generate a continuous tooth surface sheet.

[0060] In this embodiment, the algorithm for generating the spherical parametric curve is the spherical Bézier curve algorithm. The Bezier curve is a vector curve proposed by Rierre Bezier. Taking the third-order Bezier curve as an example (see...),... Figure 2 Points A, B, and C are the base points. Find point D on line segment AB and point E on BC such that AD / AB = BE / BC. Then find point F on DE such that DF / FE = AD / AB = BE / BC. There are infinitely many points F on the plane. The curve formed by connecting all points F is the Bezier curve. It can be seen that the starting point and ending point of the Bezier curve are tangent to line segments AB and BC respectively, that is, they transition smoothly with points A and B.

[0061] The determination of the first control point P0, the second control point P1, the third control point P3, and the fourth control point P2 includes: S1, extracting the large end base cone point of the spherical involute tooth surface of the straight bevel gear as the first control point P0, and the coordinates of the first control point P0 are: ;

[0062] S2. Based on the coordinates of the first control point P0, calculate the tangent direction of the tooth surface at the first control point P0 along the sphere with the cone distance as the radius, and obtain the coordinates of the second control point P1 on the sphere by extending this tangent direction on the sphere. The coordinates of the second control point P1 are: ;

[0063] S3. Extract the point on the root cone of the large end of the straight bevel gear as the third control point P3. The coordinates of the third control point P3 are: ;

[0064] S4. Based on the coordinates of the third control point P3, calculate the relevant tangent direction at the third control point P3, and obtain the coordinates of the fourth control point P2 on the sphere by extending the spherical arc. The coordinates of the fourth control point P2 are: .

[0065] S2. Based on the coordinates of the first control point P0, the implementation of calculating the tangent direction of the tooth surface at the first control point P0 along the sphere, and obtaining the coordinates of the second control point P1 on the sphere by extending this tangent direction includes:

[0066] Calculate the vector of the tooth surface along the tangent direction of the sphere at the first control point P0. The coordinates of the first control point P0 are: ,

[0067] ;

[0068] Where R is the cone distance, It is the base cone angle. It is the angle rotated by half the tooth thickness; determine the rotation axis used for spherical extension, and calculate the rotation axis vector u based on the position of point P0 and the tangent direction t3 obtained in the previous step; perform spherical rotation to obtain the coordinates of the second control point P1, and use the Rodrigues rotation formula to calculate the coordinates of the second control point P1. ;in, It is the angle of rotation around the u-axis.

[0069] S3. Extract the point on the root cone of the large end of the straight bevel gear as the third control point P3. The coordinates of the third control point P3 are: It includes:

[0070] Based on the geometry of the cone and obtaining the coordinates of its intersection point with the sphere, construct a structure based on the cone angle. Define the conical surface and calculate the intersection line between the conical surface and the current cone distance sphere, then calculate the intersection point from the intersection line; the intersection point is the third control point P3, and the coordinates of the third control point P3 are... , Where R is the cone distance, , is the root cone angle, and γ is the pitch bevel thickness angle.

[0071] S4. Based on the coordinates of the third control point P3, calculate the relevant tangent direction at the third control point P3, and obtain the coordinates of the fourth control point P2 on the sphere by extending the spherical arc: Rotate point P3 around the Z-axis by a predetermined angle φ to obtain the fourth control point P2. The coordinates of the fourth control point are... ;

[0072] ;

[0073] Where R is the cone distance, γ is the root cone angle, and γ is the pitch bevel thickness angle. It is the distance rotated along the tangential direction.

[0074] In step 2, four control points P0, P1, P2, and P3 are used as inputs. Based on the spherical parametric curve generation algorithm, a smooth transition curve located on the sphere is constructed. The spherical linear interpolation function is called, and combined with the Bézier curve algorithm, a smooth third-order Bézier transition curve located on the sphere is generated, which includes obtaining all interpolation points on the sphere through the SLERP function.

[0075] For the interpolation points obtained by the SLERP function, a series of discrete point sets are generated by recursively interpolating on the sphere using the Bézier algorithm. Let {B0(0), B0(0.01), ..., B0(1)} be the discrete points, and let B0(t) form a spherical Bézier curve.

[0076] In three-dimensional space, let V1 and V2 be any two non-collinear unit direction vectors on a sphere. Then any point on the sphere can be linearly represented by these two vectors: V = aV1 + bV2; where a and b are coefficients, and V is an arbitrary unit interpolation point. Since arc length equals angle multiplied by radius, arc length is directly proportional to angle. By the principles of calculus, the sine represents the projection component of the angle onto the circle, which is proportional to the arc length distribution. and This can be used to represent the proportional distribution of arc length at both ends; therefore, the coefficients a and b can be expressed as... , .

[0077] All interpolation points on the sphere can be obtained through the SLERP function. Then, following the same principle as the Bezier curve, Q0, Q1, and Q2 are the proportional points of arcs P0P1, P1P2, and P2P3, respectively; A0 and A1 are the proportional points of arcs Q0Q1 and Q1Q2, respectively; and B0 is the proportional point of A0 and A1.

[0078] ;

[0079] ;

[0080] ;

[0081] ;

[0082] ;

[0083] ;

[0084] For the interpolation points obtained by the SLERP function, the Bézier algorithm is used to perform recursive interpolation calculations on the sphere to generate a series of discrete point sets B0(t), where B0(t) is {B0(0), B0(0.01), ..., B0(1)}. The set of discrete points B0(t) constitutes the spherical Bézier curve.

[0085] Recursive interpolation calculations are performed on the sphere using the Bézier algorithm, including:

[0086] Three-layer spherical linear interpolation is recursively performed within the range of interpolation parameter t from 0 to 1;

[0087] The first layer of interpolation is performed between control points P0P1, P1P2, and P2P3 to obtain intermediate points Q0, Q1, and Q2.

[0088] The second-level interpolation is performed between intermediate points Q0Q1 and Q1Q2 to obtain secondary intermediate points A0 and A1.

[0089] The third-level interpolation is performed between the second-level intermediate points A0A1 to obtain the final curve point B0(t);

[0090] The set of all B0(t) points constitutes the spherical Bézier transition curve.

[0091] All points B0 form the transition curve on the sphere. Figure 4 The red line represents a transition curve that can be obtained for each R. By iterating over R, the surface point cloud can be obtained. Figure 5 As shown. This yields the point cloud of the transition surface, which can then be used with a skinning method to obtain the complete surface, as shown. Figure 6 A complete bevel gear can be obtained through mirroring and arraying operations using CAD software. The bending stress can then be optimized using finite element analysis on the bevel gear model.

[0092] Example 2

[0093] The difference from Example 1 is in step 1: the determination of control points. The control points are determined by the spherical involute tooth surface of the straight bevel gear. The control points include the first control point P0, the second control point P1, the third control point P3 and the fourth control point P2.

[0094] Step 2: Using the four control points P0, P1, P2, and P3 as input, construct a smooth transition curve on the sphere based on the spherical parametric curve generation algorithm, which is a spherical B-spline curve algorithm; and call the spherical linear interpolation function, combined with the Bézier curve algorithm, to generate a smooth third-order Bézier transition curve on the sphere.

[0095] Step 3: Traverse different cone distances to the small end of the straight bevel gear. Repeat steps 1 to 2 for each cone distance to generate a series of spherical transition curves at different cone distance positions. Discretize the spherical transition curves to form a three-dimensional point cloud of the tooth root transition surface.

[0096] Step 4: Interpolate or fit the 3D point cloud of the tooth root transition surface obtained in Step 3 with the tooth surface point cloud at the same cone distance to form a smooth curve. For the interpolated or fitted smooth curve, use the skin surface modeling method in the CAD system to generate a continuous tooth surface sheet.

[0097] Example 3

[0098] Based on the above embodiments, this embodiment first constructs a planar Bézier curve, and then generates a spherical curve by projecting it onto the gear sphere (with the center of the sphere being O), thereby obtaining the modified transition segment.

[0099] Example 4

[0100] Based on the above embodiments, this embodiment is a design system for the tooth root transition arc of a straight bevel gear, which includes:

[0101] The control point determination module is used to determine four control points of the spherical Bézier curve. The four control points include the first control point P0, the second control point P1, the third control point P3, and the fourth control point P2.

[0102] The spherical Bézier curve generation module takes four control points P0, P1, P2, and P3 as input, constructs a smooth transition curve on the sphere based on the spherical parametric curve generation algorithm, and calls the spherical linear interpolation function in combination with the Bézier curve algorithm to generate a smooth third-order Bézier transition curve on the sphere.

[0103] The 3D point cloud generation module for the tooth root transition surface traverses different cone distances to the small end of the straight bevel gear. For each cone distance, the module determines the control point and generates a series of spherical transition curves at different cone distance positions through the control point determination module and the spherical Bézier curve generation module. The spherical transition curves are then discretized, and the discretized curves form the 3D point cloud of the tooth root transition surface.

[0104] The continuous tooth surface sheet generation module interpolates or fits the 3D point cloud of the tooth root transition surface obtained by the tooth root transition surface generation module with the tooth surface point cloud at the same cone distance into a smooth curve. The interpolated or fitted smooth curve is then used in the CAD system to generate a continuous tooth surface sheet using the skin surface modeling method.

[0105] The first control point P0 is the large end base cone point of the spherical involute tooth surface of the straight bevel gear; the second control point P1 is obtained by extending the tangent direction of the tooth surface at the first control point P0 along the spherical surface with the cone distance as the radius.

[0106] The third control point P3 is a point on the root cone of the large end of the straight bevel gear; the fourth control point P2 is obtained based on the relevant tangent direction at the third control point P3 and by extending it through a spherical arc.

Claims

1. A design method of a straight bevel gear tooth root transition arc, characterized by, The method comprises: Step 1: determination of control points, determination of control points is carried out through a spherical involute tooth surface of a straight bevel gear, and the control points comprise a first control point P0, a second control point P1, a third control point P3 and a fourth control point P2; The determination of the first control point P0, the second control point P1, the third control point P3 and the fourth control point P2 comprises: S1, extract the big end base cone point of the spherical involute tooth surface of the straight bevel gear as the first control point P0, the coordinates of the first control point P0 are ; S2, based on the coordinates of the first control point P0, calculate the tangent direction of the tooth surface along the spherical surface with the cone distance as the radius at the first control point P0, and obtain the coordinates of the second control point P1 on the spherical surface by extending the tangent direction on the spherical surface, the coordinates at the second control point P1 are ; S3, extract the point on the big end root cone of the straight bevel gear as the third control point P3, the coordinates of the third control point P3 are ; S4, based on the coordinates of the third control point P3, the relevant tangent direction at the third control point P3 is calculated, and the coordinates of the fourth control point P2 on the spherical surface are obtained by extending the spherical arc, the coordinates of the fourth control point P2 being ; Step 2: taking the four control points P0, P1, P2 and P3 as inputs, generating a smooth transition curve on a spherical surface based on a spherical parameter curve generation algorithm, and calling a spherical linear interpolation function to generate a smooth third-order Bézier transition curve on the spherical surface in combination with a Bézier curve algorithm; Step 3: traversing different cone distances to the small end of the straight bevel gear, repeatedly executing steps 1 to 2 for each cone distance, generating a series of spherical transition curves at different cone distance positions, and discretizing the spherical transition curves to form a three-dimensional point cloud of the root transition surface after discretization; Step 4: interpolating or fitting the three-dimensional point cloud of the root transition surface obtained in step 3 with the tooth surface point cloud at the same cone distance into a smooth curve, and generating a continuous tooth surface sheet body in a CAD system by using a skin surface modeling method for the smooth curve after interpolation or fitting.

2. The design method of the fillet of straight bevel gear according to claim 1, characterized in that, The spherical parameter curve generation algorithm comprises a spherical Bézier curve algorithm or a spherical B-spline curve algorithm.

3. The design method of the fillet of straight bevel gear according to claim 1, characterized in that, S2, based on the coordinates of the first control point P0, the tangent direction of the tooth surface at the first control point P0 along the spherical surface is calculated, and the coordinates of the second control point P1 on the spherical surface are obtained by extending the tangent direction, and the implementation comprises: calculating the tooth surface at the first control point P0 in the tangential direction vector of the spherical surface ; for the coordinates at the first control point P0 , ； ; where R is the cone distance, is the base cone angle, is the angle of half tooth thickness turned; determine the rotation axis for the spherical extension, calculate the rotation axis vector u according to the position of the P0 point and the tangent direction t3 obtained in the last step; perform spherical rotation to obtain the coordinates of the second control point P1, and calculate the coordinates of the second control point P1 using the Rodrigues rotation formula, ; where, is the angle of rotation around the u axis.

4. The design method of the fillet of straight bevel gear according to claim 1, characterized in that, S3, extract the point on the big end root cone of the straight bevel gear as the third control point P3, the coordinates of the third control point P3 are It comprises: The intersection point coordinates of the root cone and the spherical surface are obtained by the root cone geometry The intersection line of the conical surface and the spherical surface is calculated, and the intersection point is calculated by the intersection line; the intersection point is the third control point P3, and the coordinates of the third control point P3 are , ; wherein R is the cone distance, , is the root cone angle, and γ is the addendum angle.

5. The design method of the fillet of straight bevel gear according to claim 1, characterized in that, S4, based on the coordinates of the third control point P3, the relevant tangent direction at the third control point P3 is calculated, and the coordinates of the fourth control point P2 on the spherical surface are obtained by extending the spherical arc: rotating the point P3 around the Z axis by a predetermined angle φ to obtain the fourth control point P2, and the coordinates of the fourth control point are ; ; where R is the pitch cone, is the root cone angle and γ is the angle of the addendum cone. is the distance along the tangential rotation.

6. The design method of the fillet of straight bevel gear according to claim 1, characterized in that, In step 2, taking the four control points P0, P1, P2 and P3 as inputs, generating a smooth transition curve on a spherical surface based on a spherical parameter curve generation algorithm, and calling a spherical linear interpolation function to generate a smooth third-order Bézier transition curve on the spherical surface in combination with a Bézier curve algorithm, which comprises: obtaining all interpolation points on the spherical surface through a SLERP function; For the interpolation point obtained by the SLERP function, a series of discrete point sets are generated by recursive interpolation calculation on the spherical surface through the Bézier algorithm {B0(0), B0(0.01),..., B0(1)}, the discrete point set B0(t) constitutes a spherical Bézier curve.

7. The design method of a straight bevel gear tooth root transition circular arc according to claim 6, characterized in that, Recursive interpolation calculation on the spherical surface through the Bézier algorithm comprises: Recursive three-layer spherical linear interpolation is carried out in the range of the interpolation parameter t from 0 to 1; The first layer interpolation is carried out between the control points P0P1, P1P2 and P2P3 to obtain intermediate points Q0, Q1 and Q2; The second layer interpolation is carried out between the intermediate points Q0Q1 and Q1Q2 to obtain secondary intermediate points A0 and A1; The third layer interpolation is carried out between the secondary intermediate points A0A1 to obtain the final curve point B0(t); The set of all B0(t) points constitutes a spherical Bézier transition curve.

8. A design system for the root transition arc of a straight bevel gear, characterized in that, The straight bevel gear root transition arc design method according to any one of claims 1-7 comprises A control point determination module is configured to determine four control points of a spherical Bézier curve, and the four control points comprise a first control point P0, a second control point P1, a third control point P3 and a fourth control point P2; The spherical Bézier curve generation module takes four control points P0, P1, P2, P3 as input, constructs a smooth transition curve on the spherical surface based on the spherical parameter curve generation algorithm, and calls the spherical linear interpolation function to generate a smooth third-order Bézier transition curve on the spherical surface in combination with the Bézier curve algorithm; The three-dimensional point cloud generation module of the dedendum transition surface traverses different cone distances to the small end of the straight bevel gear, generates a series of spherical transition curves at different cone distance positions for each cone distance through the control point determination module and the spherical Bézier curve generation module, and discretizes the spherical transition curves to form a three-dimensional point cloud of the dedendum transition surface; The continuous tooth surface sheet generation module interpolates or fits the three-dimensional point cloud of the dedendum transition surface obtained by the three-dimensional point cloud generation module of the dedendum transition surface and the tooth surface point cloud at the same cone distance into a smooth curve, and generates a continuous tooth surface sheet in the CAD system by using the skin surface modeling method for the smooth curve.

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