Point contact surface worm gear transmission design method

Abstract

The invention provides a point contact surface worm gear transmission design method, which relates to the technical field of surface worm gear transmission, and is characterized in that a so-called mismatching mode is point contact surface worm gear transmission obtained by matching a conical surface enveloping cylindrical worm with an Archimedes surface worm gear transmission; according to the meshing principle of conical surface enveloping cylindrical worm-Archimedes point contact surface worm gear transmission, an instantaneous transmission ratio function is further deduced, the relation between the instantaneous transmission ratio function and grinding wheel generation surface parameters is determined, a point contact surface worm gear transmission tooth surface geometry-kinematics error coupling mechanism is revealed, and the grinding wheel generation surface geometry-kinematics error coupling mechanism is established. Therefore, a zero instantaneous transmission ratio function point contact surface worm gear transmission design method is provided, on the basis of the model, parameters such as instantaneous contact points and motion errors are calculated, and reasonable iteration initial values are screened out through the physical information neural network technology; finally, novel point contact surface worm gear transmission with zero transmission ratio error is provided.

Description

Technical Field

[0001] The present invention relates to the technical field of surface worm gear transmission, and in particular to a design method for a point contact surface worm gear transmission. Background Art

[0002] As production technology continues to evolve towards highly precise operations, high dynamic loads, and tolerance to extreme environments, the demand for high-precision, low-error-sensitivity precision worm gear transmissions continues to grow. Kinematic error is a key indicator of gear transmission operation. Therefore, improving error sensitivity is crucial to reducing gear transmission kinematic error. However, current research indicates that both trajectory-type and envelope-type point-contact worm gear transmissions face the same fundamental kinematic error. Therefore, analyzing the kinematic error characteristics of surface worm gear transmissions, establishing a kinematic error model, and ultimately developing surface worm gear transmissions with zero transmission ratio error are of great significance.

[0003] Existing surface worm gear transmissions all face the same problem: there is a fundamental motion error, and the motion error function is not sufficient to determine how to achieve a zero transmission ratio function. Therefore, it is necessary to further develop the surface worm gear transmission design method and propose a new type of surface worm gear transmission. Summary of the Invention

[0004] The present invention proposes a design method for a point contact surface worm gear transmission, aiming to solve the problem that the existing surface worm gear transmission has a fundamental motion error and the motion error function is not sufficient to determine how to achieve a zero transmission ratio function.

[0005] The present invention provides a method for designing a point contact worm gear transmission, comprising the following steps:

[0006] S1: Based on the Archimedes hob helicoid equation, determine the two basic quantities of the Archimedes hob helicoid, the helicoid normal vector, the mean curvature, the two orthogonal directions of the helicoid and the corresponding normal curvature and geodesic torsion;

[0007] S2: Establish the four coordinate systems of Archimedes hobbing surface worm gear hobbing process, which are and They respectively represent the initial position and current position of the Archimedes cylindrical hob and the initial position and current position of the Archimedes surface worm wheel, and their relative position relationship;

[0008] S3: Based on the four coordinate systems, the vector rotation method is used to determine the hob shape surface equation, shape surface normal vector, relative angular velocity vector, relative velocity vector, and hobbing meshing function when the Archimedes hob is machining the Archimedes surface worm gear;

[0009] S4: Through coordinate transformation σ o4 →σ2, combining the hob profile equation and the hobbing meshing function, the Archimedes surface worm gear tooth surface equation and its normal vector equation are obtained;

[0010] S5: Based on the gear hobbing meshing function, determine the gear hobbing meshing boundary function and the normal vector of the contact line between the Archimedes hob and the surface worm gear, the curvature interference boundary function, and the two directions along the worm gear tooth surface. Normal curvature and along the direction Geodesic torsion

[0011] S6: Based on the equation of the helical surface of the conical surface enveloping cylindrical worm and the tooth surface equation of the Archimedes surface worm gear, the underdetermined equations of the instantaneous contact point of the conical surface enveloping cylindrical worm-Archimedes surface worm gear transmission are established;

[0012] S7: Use elimination method to eliminate the underdetermined equations of the instantaneous contact point of the cone surface enveloping cylindrical worm-Archimedes surface worm gear transmission, and eliminate variables u3, and These five variables;

[0013] S8: along the worm gear tooth surface in two directions Normal curvature and along the direction Geodesic torsion Calculate the semi-major axis a of the contact ellipse e and semi-minor axis b e ;

[0014] S9: Based on the semi-major axis a of the contact ellipse e and semi-minor axis b e , determine the instantaneous contact ratio function, and use the vector rotation method to obtain the calculation equation of the instantaneous transmission ratio error function and the reference point of the mismatched surface worm gear transmission;

[0015] S10: Based on the instantaneous transmission ratio error function of the mismatched surface worm gear transmission, the physical information neural network technology is used to screen the iterative initial value to achieve a conical surface enveloping cylindrical worm-Archimedes surface worm gear transmission with zero transmission ratio error.

[0016] Furthermore, the specific method for determining the two basic quantities of the Archimedean hob helicoid, the helicoid normal vector, the mean curvature, the two orthogonal directions of the helicoid and the corresponding normal curvature and geodesic torsion based on the Archimedean hob helicoid equation in step S1 includes:

[0017] Establish a coordinate system fixed to the Archimedes hob for reflecting the current position of Archimedes hob; the origin O4 is located at the midpoint of Archimedes hob spiral length; the base vector coincides with the Archimedes hob axis; in the axial section of the hob angle is the Archimedes hob tooth angle; the straight line vector length is wherein the symbol represents the Archimedes hob spiral parameter, and θ4 represents the angle of rotation of the cutter around the base vector ;

[0018] According to the vector function and the vector sum relationship, the Archimedes hob spiral surface equation is:

[0019]

[0020] The first basic quantity of the Archimedes hob spiral surface is obtained as:

[0021] E4 ＝ 1,

[0022] The normal vector of the Archimedes hob spiral surface in the coordinate system σ4 is calculated as: :

[0023]

[0024] In the formula, the symbol and represent the circular vector function; the symbol represents the Archimedes hob spiral parameter; θ4 represents the angle of rotation of the cutter around the base vector ; u4 represents the distance from the coordinate origin O4 to a point on the Archimedes hob spiral surface; the superscript S = 1 or 2, when S = 1, the above formula represents the hob spiral surface meshing with the worm gear convex tooth surface; when S = 2, the above formula represents the hob spiral surface meshing with the worm gear concave tooth surface;

[0025] The second basic quantity of the Archimedes hob spiral surface is obtained as:

[0026] L4 = 0,

[0027] According to the calculation results, since the first basic quantity F4 ≠ 0 and the second basic quantity M4 ≠ 0, it is known that the parameter curves u4-line and θ4-line of the Archimedes hob spiral surface are not two principal directions; therefore, a right-hand unit orthogonal moving frame is established on the hob spiral surface After calculation, we get the two orthogonal directions of the hob helical surface and They are:

[0028]

[0029] Average curvature of the Archimedes hob helical surface It can be obtained from the first and second basic quantities:

[0030]

[0031] Get along Directional normal curvature He Yan Directional geodesic torsion for

[0032]

[0033] Furthermore, in step S3, based on the four coordinate systems, the specific method for determining the hob forming surface equation, the forming surface normal vector, the relative angular velocity vector, the relative velocity vector, and the hobbing meshing function when the Archimedes hob processes the Archimedes surface worm gear by the vector rotation method includes:

[0034] When machining a worm gear, when the Archimedes hob rotates around its axis, its helical surface can form a single-parameter surface family in the coordinate system; the equation of this single-parameter surface family can be obtained by coordinate transformation of the Archimedes hob helical surface as follows:

[0035]

[0036] Where,

[0037] Similarly, after the coordinate transformation σ4→σ o4 , normal vector of the hob production surface You can also use o4 is uniformly expressed as:

[0038]

[0039] Where,

[0040]

[0041] In the formula, the symbol and Represents a circular vector function; the symbol =Archimedes hob spiral parameter; θ4 represents the turning tool around the base vector The angle of rotation; u4 represents the distance from the coordinate origin O4 to a point on the Archimedes helical surface of the hob; superscript S = 1 or 2. When S = 1, the above formula represents the helical surface of the hob meshing with the convex tooth surface of the worm gear; when S = 2, the above formula represents the helical surface of the hob meshing with the concave tooth surface of the worm gear; Archimedes hob corner;

[0042] Based on this, the relative angular velocity vector of the worm gear on the hob machining surface is Can be in σ o4 In Chinese it is represented as:

[0043]

[0044] Furthermore, the relative velocity vector at the meshing point during gear hobbing is It can be expressed in σ o4 Zhongwei

[0045]

[0046] in,

[0047] Where z 42 Indicates the axial installation distance of the Archimedes hob, Σ 42 is the axis angle of the Archimedes hob, usually 90 degrees, i 42 is the transmission ratio of Archimedes hob when hobbing gears, L A is the Archimedes hob thread length, a 42 Indicates the process center distance of Archimedes hob installation;

[0048] Therefore, the meshing function Φ when the Archimedes hob is used to process the Archimedes surface worm gear is obtained. 42 for:

[0049]

[0050] Among them, the coefficient

[0051]

[0052] Where z 42 Indicates the axial installation distance of the Archimedes hob, Σ 42 is the axis angle of the Archimedes hob, usually 90 degrees, i 42 is the transmission ratio of Archimedes hob when hobbing gears, L AL is the thread length of Archimedes hob, a 42 is the process center distance of Archimedes hob installation. Symbol is the Archimedes hob spiral parameter; θ4 is the angle of rotation of the turning tool around the base vector; u4 is the distance from the coordinate origin O4 to a point on the Archimedes hob spiral surface; the superscript S = 1 or 2, when S = 1, the above formula represents the hob spiral surface meshing with the convex tooth surface of the face worm; when S = 2, the above formula represents the hob spiral surface meshing with the concave tooth surface of the worm; is the Archimedes hob rotation angle;

[0053] Further, the specific method for obtaining the Archimedes face worm tooth surface equation and its normal vector equation in step S4 by coordinate transformation σ o4 →σ2, in combination with the hob generative surface equation and the hobbing meshing function, includes:

[0054] After coordinate transformation σ o4 →σ2, the Archimedes face worm tooth surface equation and its normal vector equation are obtained;

[0055] The worm tooth surface equation is:

[0056]

[0057] wherein the coefficient

[0058]

[0059] In the formula, z 42 is the axial installation distance of Archimedes hob, Σ 42 is the axial intersection angle of Archimedes hob, which is usually 90 degrees, i 42 is the transmission ratio of Archimedes hob hobbing, L A L is the thread length of Archimedes hob, a 42 is the process center distance of Archimedes hob installation. Symbol is the Archimedes hob spiral parameter; θ4 is the angle of rotation of the turning tool around the base vector; u4 is the distance from the coordinate origin O4 to a point on the Archimedes hob spiral surface; the superscript S = 1 or 2, when S = 1, the above formula represents the hob spiral surface meshing with the convex tooth surface of the face worm; when S = 2, the above formula represents the hob spiral surface meshing with the concave tooth surface of the worm; is the Archimedes hob rotation angle;

[0060] ​​The normal vector component equation is:

[0061]

[0062] wherein z 42 represents the axial installation distance of the Archimedes hob, Σ 42 represents the axial intersection angle of the Archimedes hob, usually 90 degrees, i 42 represents the transmission ratio of the Archimedes hob during hobbing, L A represents the thread length of the Archimedes hob, a 42 represents the process center distance of the Archimedes hob installation. Symbol represents the Archimedes hob spiral parameter; θ4represents the angle of rotation of the hob around the base vector ; u4represents the distance from the coordinate origin O4to a point on the Archimedes hob spiral surface; the superscript S = 1 or 2, when S = 1, the above formula represents the hob spiral surface meshing with the convex tooth surface of the face worm; when S = 2, the above formula represents the hob spiral surface meshing with the concave tooth surface of the worm; Archimedes hob rotation angle.

[0063] Further, the specific method for determining the hobbing meshing boundary function, the normal vector of the contact line between the Archimedes hob and the face worm, the curvature interference boundary function, the normal curvature along two directions of the tooth surface of the worm and the geodesic torsion along the direction of the tooth surface of the worm based on the hobbing meshing function in step S5 comprises:

[0064] Taking the meshing function Φ 42 , the partial derivative with respect to the motion parameter , the hobbing meshing boundary function during machining of the face worm can be obtained as:

[0065]

[0066] wherein i 42 represents the transmission ratio of the Archimedes hob during hobbing, symbol represents the Archimedes hob spiral parameter; u4represents the distance from the coordinate origin O4to a point on the Archimedes hob spiral surface; Archimedes hob rotation angle; the superscript S = 1 or 2, when S = 1, the above formula represents the hob spiral surface meshing with the convex tooth surface of the face worm; when S = 2, the above formula represents the hob spiral surface meshing with the concave tooth surface of the worm;

[0067] ​When grinding the worm gear, an orthogonal movable frame can be established at any meshing point M4 on the hob tooth surface. Its o4 The two are located on the common tangent plane of the tooth surface and Internal unit basis vector and for

[0068]

[0069] Among them, after calculation, its component

[0070]

[0071] Thus, the normal vector of the contact line between the Archimy hob and the worm gear is obtained: Can be expressed in the active frame In Chinese:

[0072]

[0073] Among them, the coefficient

[0074]

[0075] Combined normal curvature geodesic torsion Relative angular velocity Relative velocity vector After calculation, the interference boundary function of the worm pair curvature is obtained as follows:

[0076]

[0077] When grinding the worm gear, an orthogonal movable frame can be established at any meshing point M4 on the hob tooth surface. Its o4 The two are located on the common tangent plane of the tooth surface and Internal unit basis vector and for

[0078]

[0079] Among them, after calculation, its component

[0080]

[0081] Finally, we can get the two directions along the worm gear tooth surface. Normal curvature and along the direction Geodesic torsion

[0082]

[0083] Where, is the normal curvature of the Archimedes hob helical surface; is the geodesic curvature of the Archimedes hob helicoid.

[0084] Furthermore, after the coordinate transformation σ2→σ o2 , the tooth surface equations and normal vectors of the worm gear in S4 are respectively o2 Expressed as

[0085]

[0086] in,

[0087]

[0088] The underdetermined equations for the instantaneous contact point of the cone-surface-enveloped cylindrical worm-Archimedes surface worm gear transmission in step S6 are:

[0089]

[0090] Where θ4 represents the turning tool around the base vector The angle of rotation; u4 represents the distance from the coordinate origin O4 to a point on the spiral surface of the Archimedes hob; Hob corner for Archimedes; It is the rotation angle of the cylindrical worm when grinding with a conical grinding wheel; The angle of rotation of the cylindrical worm around its axis is enveloping the conical surface; The worm wheel rotation angle is Archimedes plane; Φ 42 is the meshing function when Archimedes hob is used to machine Archimedes surface worm gear; Φ 31 is the meshing function of the conical grinding wheel enveloping the cylindrical worm; and The cone surface envelops the cylindrical worm in the coordinate system σ o2 Tooth surface equation and its normal vector in; and The Archimedes surface worm gear in the coordinate system σ o2 Tooth surface equation and its normal vector in; is the vector between the coordinate system origin O2 and O1

[0091] Furthermore, in step S7, the variables u3, and These five variables then become functions of the three variables θ3, u4, and θ4, and the equations are:

[0092]

[0093] Furthermore, in step S8, the worm gear tooth surface is moved in two directions along the worm gear tooth surface. Normal curvature and along the direction Geodesic torsion Calculate the semi-major axis a of the contact ellipse e and semi-minor axis b e Specific methods include:

[0094] According to the meshing theory, the point conjugate tooth surface couple Along direction and The relative curvature parameters can be calculated along the worm gear tooth surface in two directions. and Normal curvature and And the direction geodesic torsion get:

[0095]

[0096] Among them, the symbol Refers to the worm helical surface At the contact point P i Along the direction and Curvature parameter; combined with the generalized Euler and Betrand formulas, along the direction and Normal curvature Along direction geodesic torsion Can be obtained as:

[0097]

[0098] Where, and are the two normal curvatures of the worm helical surface along the main directions; is the geodesic torsion along the main direction of the worm helical surface; β is the direction of the worm helical surface Direction of worm gear tooth surface Angle;

[0099] Then according to the generalized Euler formula, the point conjugate tooth surface couple can be obtained At the contact point P i The two relative principal curvatures are:

[0100]

[0101] Where, and The point conjugate tooth surface couple composed of the worm helical surface and the worm wheel tooth surface Along direction and The relative curvature parameter, is the relative geodesic torsion; v t is the vector measured along the common normal To two tooth surface pairs Relative main direction and Angle between

[0102] Using the relative principal curvature, the semi-major axis a of the contact ellipse can be easily obtained e and semi-minor axis b e for:

[0103]

[0104] in, is a constant; for this worm pair, its value is This is based on the coating particle size obtained in the rolling test; after determining a sufficient contact ellipse, a complete mismatched surface worm gear pair contact pattern can be formed.

[0105] Furthermore, in step S9, based on the semi-major axis a of the contact ellipse e and semi-minor axis b e , determine the instantaneous contact ratio function, use the vector rotation method to obtain the calculation equation of the instantaneous transmission ratio error function and the reference point of the mismatched surface worm gear transmission. The specific method includes:

[0106] Based on the general expression of transmission ratio error and the tooth surface equation of Archimedes surface worm gear, the instantaneous transmission ratio of mixed mismatch Archimedes surface worm gear pair can be obtained: for:

[0107]

[0108] In order to achieve the instantaneous transmission ratio equal to the process transmission ratio, the grinding wheel normal vector guarantees the following relationship:

[0109]

[0110] According to the mismatch profile, the instantaneous transmission ratio error of the worm gear transmission Formula, so that the instantaneous transmission ratio error is 0, that is:

[0111]

[0112] Where, u4 represents the distance from the coordinate origin O4 to a point on the Archimedes hob spiral surface; p is the spiral parameter of the worm, n 3z (θ3) is the component of the worm normal vector, Archimedes hob tooth profile inclination, is the Archimedes hob spiral parameter;

[0113] Finally, a new type of worm gear transmission with zero transmission ratio error is realized.

[0114] Beneficial effects of the present invention:

[0115] This method improves the transmission ratio error and error sensitivity existing in the design of traditional surface worm gear transmission, and can design a new surface worm gear transmission with zero transmission ratio error in principle; by comparing the tooth surface contact analysis results with those of the classic surface worm gear transmission, the new surface worm gear transmission has zero transmission ratio error, lower error sensitivity, and better contact performance.

[0116] Based on the implementation methods provided in the above aspects, this application can also be further combined to provide more implementation methods. BRIEF DESCRIPTION OF THE DRAWINGS

[0117] The above and other objects, features and advantages of the exemplary embodiments of the present invention will become readily understood by reading the detailed description below with reference to the accompanying drawings. In the accompanying drawings, several embodiments of the present invention are shown in an exemplary and non-limiting manner, and the same or corresponding reference numerals represent the same or corresponding parts, wherein:

[0118] Figure 1 Setting up a graph for the vector function of the present invention;

[0119] Figure 2 This is a diagram showing the coordinate system setup when the hybrid mismatched Archimedes surface worm gear transmission of the present invention is engaged;

[0120] Figure 3 This is a comparison chart of the kinematic error values ​​of the conical surface enveloping cylindrical worm-Archimedes surface worm gear transmission of the present invention and the classic surface worm gear transmission. DETAILED DESCRIPTION

[0121] The exemplary embodiments disclosed in the present application will be described in more detail below with reference to the accompanying drawings. Although exemplary embodiments of the present application are shown in the accompanying drawings, it should be understood that the present application can be implemented in various forms and should not be limited by the embodiments described herein. On the contrary, these embodiments are provided to enable a more thorough understanding of the present application and to fully convey the scope of the present application to those skilled in the art. Unless otherwise specified, the technical means used in the examples are conventional means well known to those skilled in the art.

[0122] The present invention proposes a new type of surface worm gear transmission, the purpose of which is to propose a design method that can eliminate the instantaneous transmission ratio error of the surface worm gear transmission, achieve low error sensitivity, and reduce the production cost. By establishing the meshing theory of the conical surface enveloping cylindrical worm-Archimedes surface worm gear transmission, and further revealing the tooth surface geometry-kinematic error coupling mechanism of the point contact surface worm gear transmission, a simpler instantaneous transmission error expression is obtained, overcoming the inherent characteristic of the traditional point contact surface worm gear transmission that there is an instantaneous transmission ratio error, thereby further improving the motion accuracy of the point contact transmission.

[0123] Combine Figures 1 to 3 As shown, an optional embodiment of the present invention provides a method for designing a point contact worm gear transmission, the specific steps of the method include:

[0124] S1: Based on the Archimedes hob helicoid equation, determine the two basic quantities of the Archimedes hob helicoid, the helicoid normal vector, the mean curvature, the two orthogonal directions of the helicoid and the corresponding normal curvature and geodesic torsion;

[0125] S2: If Figure 2 As shown in the figure, ① is the center distance between the Archimedes hob and the surface worm gear when cutting teeth; ② is the axial installation distance of the Archimedes hob; ③ is the axial angle between the axis of the Archimedes hob and the axis of the surface worm gear; ④ is a schematic diagram of the Archimedes hob; ⑤ is a schematic diagram of the surface worm gear;

[0126] The four coordinate systems for the Archimedes hobbing process of worm gear hobbing are established. and They respectively represent the initial position and current position of the Archimedes cylindrical hob and the initial position and current position of the Archimedes surface worm wheel, and their relative position relationship;

[0127] S3: Based on the four coordinate systems, the vector rotation method is used to determine the hob shape surface equation, shape surface normal vector, relative angular velocity vector, relative velocity vector, and hobbing meshing function when the Archimedes hob is machining the Archimedes surface worm gear;

[0128] S4: Through coordinate transformation σ o4 →σ2, combining the hob profile equation and the hobbing meshing function, the Archimedes surface worm gear tooth surface equation and its normal vector equation are obtained;

[0129] S5: Based on the gear hobbing meshing function, determine the gear hobbing meshing boundary function and the normal vector of the contact line between the Archimedes hob and the surface worm gear, the curvature interference boundary function, and the two directions along the worm gear tooth surface. Normal curvature and along the direction Geodesic torsion

[0130] S6: Based on the equation of the helical surface of the conical surface enveloping cylindrical worm and the tooth surface equation of the Archimedes surface worm gear, the underdetermined equations of the instantaneous contact point of the conical surface enveloping cylindrical worm-Archimedes surface worm gear transmission are established;

[0131] S7: Use elimination method to eliminate the underdetermined equations of the instantaneous contact point of the cone surface enveloping cylindrical worm-Archimedes surface worm gear transmission, and eliminate variables u3, and These five variables;

[0132] S8: along the worm gear tooth surface in two directions Normal curvature and along the direction Geodesic torsion Calculate the semi-major axis a of the contact ellipse e and semi-minor axis b e ;

[0133] S9: Based on the semi-major axis a of the contact ellipse e and semi-minor axis b e , determine the instantaneous contact ratio function, and use the vector rotation method to obtain the calculation equation of the instantaneous transmission ratio error function and the reference point of the mismatched surface worm gear transmission;

[0134] S10: Based on the instantaneous transmission ratio error function of the mismatched surface worm gear transmission, the relationship between the normal vectors of the conical grinding wheel generating surface used for grinding the worm and the hob is determined, and the grinding wheel generating surface is designed to achieve a conical surface enveloping cylindrical worm-Archimedes surface worm gear transmission with zero transmission ratio error; the problem is solved by using the physical information neural network technology, and the iterative initial value and design parameters are used as part of the physical information neural network loss function, thereby solving the iterative initial value problem of the high-dimensional nonlinear equation group encountered, and solving the inverse problem of multiple mismatch parameters based on meshing performance.

[0135] Optionally, the specific method of determining two basic quantities of the Archimedean hob helicoid, the helicoid normal vector, the mean curvature, the two orthogonal directions of the helicoid and the corresponding normal curvature and geodesic torsion based on the Archimedean hob helicoid equation in step S1 includes:

[0136] Establish a coordinate system fixed to the Archimedes hob Used to reflect the current position of the Archimedes hob; the origin O4 is located at the midpoint of the Archimedes hob spiral length; the base vector Coincident with the Archimedes hob axis; within the axial section of the hob angle is the Archimedes hob tooth profile inclination; Length is The symbols represents the Archimedes hob spiral parameter, θ4 represents the turning tool around the base vector Angle of rotation;

[0137] According to the vector function Figure 1 As shown in the figure, ① is the length of the Archimedes hob's spiral surface; ② is the initial position of the Archimedes hob; ③ is the Archimedes hob tooth profile inclination; ④ is the displacement of a point on the hob's spiral surface after the Archimedes hob rotates by an angle of θ4, and its displacement is equal to The symbols

[0138] represents the Archimedes hob spiral parameter, θ4 represents the turning tool around the base vector The angle of rotation; ⑤ is the current position of the Archimedes hob after rotation; ⑥ is the distance from the origin of the coordinate system to a point on the Archimedes hob helical surface; and according to the vector sum relationship, the Archimedes hob helical surface equation is:

[0139]

[0140] The first basic quantity of the Archimedes cutter helicoid is:

[0141] E4=1,

[0142] Calculate the Archimedes hob helicoid normal vector in coordinate system σ4 for:

[0143]

[0144] In the formula, the symbol and Represents a circular vector function; the symbol =Archimedes hob spiral parameter; θ4 represents the turning tool around the base vector The angle of rotation; u4 represents the distance from the coordinate origin O4 to a point on the Archimedes helical surface of the hob; superscript S = 1 or 2. When S = 1, the above formula represents the helical surface of the hob meshing with the convex tooth surface of the worm gear; when S = 2, the above formula represents the helical surface of the hob meshing with the concave tooth surface of the worm gear;

[0145] The second basic quantity of the Archimedes hob helicoid is obtained as:

[0146] L4=0,

[0147] According to the calculation results, since the first basic quantity F4≠0 and the second basic quantity M4≠0, it can be seen that the parameter curves u4-line and θ4-line directions of the Archimedes hob helical surface are not the two main directions; therefore, a right-handed unit orthogonal moving frame is established on the hob helical surface. After calculation, we get the two orthogonal directions of the hob helical surface and They are:

[0148]

[0149] Average curvature of the Archimedes hob helical surface It can be obtained from the first and second basic quantities:

[0150]

[0151] Get Along Directional normal curvature He Yan Directional geodesic torsion for

[0152]

[0153] Optionally, in step S3, based on the four coordinate systems, the specific method of determining the hob forming surface equation, the forming surface normal vector, the relative angular velocity vector, the relative velocity vector, and the hobbing meshing function when the Archimedes hob processes the Archimedes surface worm gear by the vector rotation method includes:

[0154] When machining a worm gear, when the Archimedes hob rotates around its axis, its helical surface can form a single-parameter surface family in the coordinate system; the equation of this single-parameter surface family can be obtained by coordinate transformation of the Archimedes hob helical surface as follows:

[0155]

[0156] Where,

[0157] Similarly, after the coordinate transformation σ4→σ o4 , normal vector of the hob production surface You can also use o4 is uniformly expressed as:

[0158]

[0159] Where,

[0160]

[0161] In the formula, the symbol and Represents a circular vector function; the symbol =Archimedes hob spiral parameter; θ4 represents the turning tool around the base vector The angle of rotation; u4 represents the distance from the coordinate origin O4 to a point on the Archimedes helical surface of the hob; superscript S = 1 or 2. When S = 1, the above formula represents the helical surface of the hob meshing with the convex tooth surface of the worm gear; when S = 2, the above formula represents the helical surface of the hob meshing with the concave tooth surface of the worm gear; Archimedes hob corner;

[0162] Based on this, the relative angular velocity vector of the worm gear on the hob machining surface is Can be in σ o4 In Chinese it is represented as:

[0163]

[0164] Furthermore, the relative velocity vector at any meshing point during gear hobbing is It can be expressed in σ o4 Zhongwei

[0165]

[0166] in,

[0167] Where z 42 Indicates the axial installation distance of the Archimedes hob, Σ 42 is the axis angle of the Archimedes hob, usually 90 degrees, i 42 is the transmission ratio of Archimedes hob when hobbing gears, L A is the Archimedes hob thread length, a42 Indicates the process center distance of Archimedes hob installation;

[0168] Therefore, the meshing function of Archimedes hob machining Archimedes surface worm gear can be obtained as follows:

[0169]

[0170] Among them, the coefficient

[0171] Where z 42 Indicates the axial installation distance of the Archimedes hob, Σ 42 is the axis angle of the Archimedes hob, usually 90 degrees, i 42 is the transmission ratio of Archimedes hob when hobbing gears, L A is the Archimedes hob thread length, a 42 Indicates the process center distance of Archimedes hob installation. Symbol =Archimedes hob spiral parameter; θ4 represents the turning tool around the base vector The angle of rotation; u4 represents the distance from the coordinate origin O4 to a point on the Archimedes helical surface of the hob; superscript S = 1 or 2. When S = 1, the above formula represents the helical surface of the hob meshing with the convex tooth surface of the worm gear; when S = 2, the above formula represents the helical surface of the hob meshing with the concave tooth surface of the worm gear; Archimedes hob corner.

[0172] Optionally, the coordinate transformation σ in step S4 is performed o4 →σ2, combining the hob profile equation and the hobbing meshing function, the specific method to obtain the Archimedes surface worm gear tooth surface equation and its normal vector equation includes:

[0173] After coordinate transformation σ o4 →σ2, we can get the Archimedes surface worm gear tooth surface equation and its normal vector equation;

[0174] The worm gear tooth surface equation is:

[0175]

[0176] Among them, the coefficient

[0177]

[0178] Where z 42 Indicates the axial installation distance of the Archimedes hob, Σ42 is the axis angle of the Archimedes hob, usually 90 degrees, i 42 is the transmission ratio of Archimedes hob when hobbing gears, L A is the Archimedes hob thread length, a 42 Indicates the process center distance of Archimedes hob installation. Symbol =Archimedes hob spiral parameter; θ4 represents the turning tool around the base vector The angle of rotation; u4 represents the distance from the coordinate origin O4 to a point on the Archimedes helical surface of the hob; superscript S = 1 or 2. When S = 1, the above formula represents the helical surface of the hob meshing with the convex tooth surface of the worm gear; when S = 2, the above formula represents the helical surface of the hob meshing with the concave tooth surface of the worm gear; Archimedes hob corner;

[0179] The normal vector component equation is:

[0180]

[0181] Where z 42 Indicates the axial installation distance of the Archimedes hob, Σ 42 is the axis angle of the Archimedes hob, usually 90 degrees, i 42 is the transmission ratio of Archimedes hob when hobbing gears, L A is the Archimedes hob thread length, a 42 Indicates the process center distance of Archimedes hob installation. Symbol =Archimedes hob spiral parameter; θ4 represents the turning tool around the base vector The angle of rotation; u4 represents the distance from the coordinate origin O4 to a point on the spiral surface of the Archimedes hob; is the Archimedes cutter angle; superscript S = 1 or 2. When S = 1, the above formula represents the helical surface of the cutter meshing with the convex tooth surface of the worm gear; when S = 2, the above formula represents the helical surface of the cutter meshing with the concave tooth surface of the worm gear.

[0182] Optionally, in step S5, based on the gear hobbing meshing function, the gear hobbing meshing boundary function and the normal vector of the contact line between the Archimedes hob and the worm gear, the curvature interference boundary function, and the two directions along the worm gear tooth surface are determined. Normal curvature and along the direction Geodesic torsion Specific methods include:

[0183] Take the meshing function Φ 42 About Motion Parameters Partial derivatives can be used to obtain the tooth meshing boundary function when machining the worm gear for:

[0184]

[0185] Where i 42 is the transmission ratio of Archimedes hobbing, symbol Indicates the Archimedes hob spiral parameter; u4 indicates the distance from the coordinate origin O4 to a point on the Archimedes hob spiral surface; Archimedes hob angle; superscript S = 1 or 2, when S = 1, the above formula represents the hob helical surface meshing with the convex tooth surface of the worm gear; when S = 2, the above formula represents the hob helical surface meshing with the concave tooth surface of the worm gear;

[0186] When grinding the worm gear, an orthogonal movable frame can be established at any meshing point M4 on the hob tooth surface. Its o4 The two are located on the common tangent plane of the tooth surface and Internal unit basis vector and for

[0187]

[0188] Among them, after calculation, its component

[0189]

[0190] Thus, the normal vector of the contact line between the Archimy hob and the worm gear is obtained: Can be expressed in the active frame In Chinese:

[0191]

[0192] Among them, the coefficient

[0193]

[0194] Combined normal curvature geodesic torsion Relative angular velocity Relative velocity vector After calculation, the interference boundary function of the worm pair curvature is obtained as follows:

[0195]

[0196] Where, is the contact line normal vector, is the relative velocity vector at any meshing point during gear hobbing, is the tooth engagement boundary function. and For two o4 Common tangent plane of the center hob and the surface worm gear and Internal unit basis vectors;

[0197] Finally, we can get the two directions along the worm gear tooth surface. Normal curvature and along the direction Geodesic torsion

[0198]

[0199] Where, is the normal curvature of the Archimedes hob helical surface; is the geodesic curvature of the Archimedes hob helicoid.

[0200] Furthermore, after the coordinate transformation σ2→σ o2 , the tooth surface equation and normal vector of the worm gear in step S4 are respectively o2 Expressed as

[0201]

[0202] in,

[0203]

[0204] The underdetermined equations for the instantaneous contact point of the cone-surface-enveloped cylindrical worm-Archimedes surface worm gear transmission in step S6 are:

[0205]

[0206] After coordinate transformation σ2→σ o2 , the tooth surface equation and normal vector of the worm gear in step S4 are respectively o2 Expressed as

[0207]

[0208] in,

[0209]

[0210] Optionally, the variables u3, u4, and u6 are eliminated in step S7. and These five variables then become functions of the three variables θ3, u4, and θ4, and the equations are:

[0211]

[0212] Specifically, the solution to a system of nonlinear equations usually relies on iterative methods, such as Newton's method; however, if the elimination method is not used: 1. The use of direct iterative methods is sensitive to initial values. If the initial values ​​are not selected appropriately, it may result in failure to converge or convergence to an incorrect solution; 2. For high-dimensional nonlinear equations, the direct solution method is computationally intensive, such as direct solution (Gaussian elimination, matrix decomposition, etc.), which requires more time and computing resources; 3. The direct solution of high-dimensional nonlinear equations may encounter numerical stability problems, especially when dealing with floating-point calculations.

[0213] To this end, by using elimination to gradually eliminate unknowns, a complex multivariate system of equations is transformed into a lower-dimensional system. This allows subsequent iterations to be performed on a smaller system of equations, thereby improving solution efficiency. Furthermore, the elimination of nonlinear equations makes it easier to determine whether the system has a solution, determine the number of solutions, and select initial values ​​for iteration. Furthermore, elimination can reduce the complexity of the problem, helping to improve the stability and accuracy of the numerical solution.

[0214] Optionally, in step S8, the worm gear tooth surface is moved in two directions along the worm gear tooth surface. Normal curvature and along the direction Geodesic torsion Calculate the semi-major axis a of the contact ellipse e and semi-minor axis b e Specific methods include:

[0215] According to the meshing theory, the point conjugate tooth surface couple Along direction and The relative curvature parameters can be calculated along the worm gear tooth surface in two directions. and Normal curvature and And the direction geodesic torsion get:

[0216]

[0217] Among them, the symbol Refers to the worm helical surface At the contact point P i Along the direction and Curvature parameter; combined with the generalized Euler and Betrand formulas, along the direction and Normal curvature Along direction geodesic torsion Can be obtained as:

[0218]

[0219] Where, and are the two normal curvatures of the worm helical surface along the main directions; is the geodesic torsion along the main direction of the worm helical surface; β is the direction of the worm helical surface Direction of worm gear tooth surface Angle;

[0220] Then according to the generalized Euler formula, the point conjugate tooth surface couple can be obtained At the contact point P i The two relative principal curvatures are:

[0221]

[0222] Where, and The point conjugate tooth surface couple composed of the worm helical surface and the worm wheel tooth surface Along direction and The relative curvature parameter, is the relative geodesic torsion; v t is the vector measured along the common normal To two tooth surface pairs Relative main direction and Angle between

[0223] Using the relative principal curvature, the semi-major axis a of the contact ellipse can be easily obtained e and semi-minor axis b e for:

[0224]

[0225] in, is a constant; for this worm pair, its value is This is based on the coating particle size obtained in the rolling test; after determining a sufficient contact ellipse, a complete mismatched surface worm gear pair contact pattern can be formed.

[0226] Optionally, in step S9, based on the semi-major axis a of the contact ellipse e and semi-minor axis be The specific method for determining the instantaneous contact ratio function, obtaining the instantaneous transmission ratio error function of the mismatched profile worm transmission and the calculation equation of the reference point by using the vector rotation method comprises:

[0227] The instantaneous transmission ratio of the mixed mismatched Archimedes face worm pair is obtained from the general expression of the transmission ratio error and the tooth surface equation of the Archimedes face worm is:

[0228]

[0229] To realize that the instantaneous transmission ratio is equal to the process transmission ratio, the following relationship is ensured by the vector of the grinding wheel method:

[0230]

[0231] According to the instantaneous transmission ratio error of the mismatched profile worm transmission Formula, the instantaneous transmission ratio error is 0, that is:

[0232]

[0233] In the formula, u4 represents the distance from the coordinate origin O4 to a point on the Archimedes hob spiral surface; p is the spiral parameter of the worm, n 3z (θ3) is the component of the worm method vector, the tooth profile inclination angle of the Archimedes hob, is the Archimedes hob spiral parameter;

[0234] Finally, a new type of face worm transmission with zero transmission ratio error is realized.

[0235] Verification example

[0236] As Figure 3 shown, Figure 3 in FIG. a, the left side is the instantaneous transmission ratio error of the classic face worm transmission, and the right side is the instantaneous transmission ratio error of the new type of face worm transmission proposed by the application; Figure 3 in FIG. b, the left side is the instantaneous transmission ratio error of the classic face worm transmission, and the right side is the instantaneous transmission ratio error of the new type of face worm transmission proposed by the application; Figure 3The left side of the middle figure c is the rotation angle error of the classic face worm gear transmission, and the right side is the rotation angle error of the novel face worm gear transmission; the novel point contact face worm gear pair instantaneous transmission ratio error and angular velocity error of the application are all 0 according to the above design, which shows that the design method proposed in the application can effectively improve the meshing performance of the traditional point contact face worm gear transmission; and the rotation angle error curve is parabolic, and its movement error curve is intersected on the adjacent movement period, which can reduce the impact and noise of the point contact face worm gear transmission; it shows that the meshing performance of the novel point contact face worm gear pair is good.

Claims

1. A method for designing a point contact worm gear transmission, characterized in that: The steps include: S1: Based on the Archimedes hob helicoid equation, determine the two basic quantities of the Archimedes hob helicoid, the helicoid normal vector, the mean curvature, the two orthogonal directions of the helicoid and the corresponding normal curvature and geodesic torsion; S2: Establish the four coordinate systems of Archimedes hobbing surface worm gear hobbing process, which are and They respectively represent the initial position and current position of the Archimedes cylindrical hob and the initial position and current position of the Archimedes surface worm wheel, and their relative position relationship; S3: Based on the four coordinate systems, the vector rotation method is used to determine the hob shape surface equation, shape surface normal vector, relative angular velocity vector, relative velocity vector, and hobbing meshing function when the Archimedes hob is machining the Archimedes surface worm gear; S4: Through coordinate transformation σ o4 →σ2, combining the hob profile equation and the hobbing meshing function, the Archimedes surface worm gear tooth surface equation and its normal vector equation are obtained; S5: Based on the gear hobbing meshing function, determine the gear hobbing meshing boundary function and the normal vector of the contact line between the Archimedes hob and the surface worm gear, the curvature interference boundary function, and the two directions along the worm gear tooth surface. Normal curvature and along the direction Geodesic torsion S6: Based on the equation of the helical surface of the conical surface enveloping cylindrical worm and the tooth surface equation of the Archimedes surface worm gear, the underdetermined equations of the instantaneous contact point of the conical surface enveloping cylindrical worm-Archimedes surface worm gear transmission are established; S7: Use elimination method to eliminate the underdetermined equations of the instantaneous contact point of the cone surface enveloping cylindrical worm-Archimedes surface worm gear transmission, and eliminate variables u3, and These five variables; S8: along the worm gear tooth surface in two directions Normal curvature and along the direction Geodesic torsion Calculate the semi-major axis a of the contact ellipse e and semi-minor axis b e ; S9: Based on the semi-major axis a of the contact ellipse e and semi-minor axis b e , determine the instantaneous contact ratio function, and use the vector rotation method to obtain the calculation equation of the instantaneous transmission ratio error function and the reference point of the mismatched surface worm gear transmission; S10: Based on the instantaneous transmission ratio error function of the mismatched surface worm gear transmission, the physical information neural network technology is used to screen the iterative initial value to achieve a conical surface enveloping cylindrical worm-Archimedes surface worm gear transmission with zero transmission ratio error.

2. The point contact worm gear transmission design method according to claim 1, characterized in that: The specific method for determining two basic quantities of the Archimedean hob helicoid, the helicoid normal vector, the mean curvature, the two orthogonal directions of the helicoid and the corresponding normal curvature and geodesic torsion based on the Archimedean hob helicoid equation in step S1 includes: Establish a coordinate system fixed to the Archimedes hob Used to reflect the current position of the Archimedes hob; the origin O4 is located at the midpoint of the Archimedes hob spiral length; the base vector Coincident with the Archimedes hob axis; within the axial section of the hob angle is the Archimedes hob tooth profile inclination; Length is The symbols represents the Archimedes hob spiral parameter, θ4 represents the turning tool around the base vector Angle of rotation; According to the vector function and the vector sum relationship, the Archimedes hob helicoid equation is: The first basic quantity of the Archimedes cutter helicoid is: E4＝1， Calculate the Archimedes hob helicoid normal vector in coordinate system σ4 for: In the formula, the symbol and represents a circular vector function; the symbol =Archimedes hob spiral parameter; θ4 represents the turning tool around the base vector The angle of rotation; u4 represents the distance from the coordinate origin O4 to a point on the Archimedes helical surface of the hob; superscript S = 1 or 2. When S = 1, the above formula represents the helical surface of the hob meshing with the convex tooth surface of the worm gear; when S = 2, the above formula represents the helical surface of the hob meshing with the concave tooth surface of the worm gear; The second basic quantity of the Archimedes hob helicoid is obtained as: L4＝0， According to the calculation results, since the first basic quantity F4≠0 and the second basic quantity M4≠0, it can be seen that the parameter curves u4-line and θ4-line directions of the Archimedes hob helical surface are not the two main directions; therefore, a right-handed unit orthogonal moving frame is established on the hob helical surface. After calculation, we get the two orthogonal directions of the hob helical surface and They are: Average curvature of the Archimedes hob helical surface From the first and second basic quantities we can get: Get along Directional normal curvature He Yan Directional geodesic torsion for 3. The point contact worm gear transmission design method according to claim 1, characterized in that: In step S3, based on the four coordinate systems, the specific method for determining the hob production surface equation, the production surface normal vector, the relative angular velocity vector, the relative velocity vector, and the hobbing meshing function when the Archimedes hob processes the Archimedes surface worm gear by the vector rotation method includes: When machining a worm gear, when the Archimedes hob rotates around its axis, its helical surface can form a single-parameter surface family in the coordinate system; the equation of this single-parameter surface family can be obtained by coordinate transformation of the Archimedes hob helical surface as follows: Where, Similarly, after the coordinate transformation σ4→σ o4 , normal vector of the hob production surface In σ o4 is uniformly expressed as: Where, In the formula, the symbol and represents a circular vector function; the symbol =Archimedes hob spiral parameter; θ4 represents the turning tool around the base vector The angle of rotation; u4 represents the distance from the coordinate origin O4 to a point on the Archimedes helical surface of the hob; superscript S = 1 or 2. When S = 1, the above formula represents the helical surface of the hob meshing with the convex tooth surface of the worm gear; when S = 2, the above formula represents the helical surface of the hob meshing with the concave tooth surface of the worm gear; Archimedes hob corner; Based on this, the relative angular velocity vector of the worm gear on the hob machining surface is In σ o4 In Chinese it is represented as: Furthermore, the relative velocity vector at any meshing point during gear hobbing is Indicates that in σ o4 Zhongwei in, Where z 42 Indicates the axial installation distance of the Archimedes hob, Σ 42 is the axis angle of the Archimedes hob, usually 90 degrees, i 42 is the transmission ratio of Archimedes hob when hobbing gears, L A is the Archimedes hob thread length, a 42 Indicates the process center distance of Archimedes hob installation; Therefore, the meshing function Φ when the Archimedes hob is used to process the Archimedes surface worm gear is obtained. 42 for: Among them, the coefficient Where z 42 Indicates the axial installation distance of the Archimedes hob, Σ 42 is the axis angle of the Archimedes hob, usually 90 degrees, i 42 is the transmission ratio of Archimedes hob when hobbing gears, L A is the Archimedes hob thread length, a 42 Indicates the process center distance of Archimedes hob installation; symbol =Archimedes hob spiral parameter; θ4 represents the turning tool around the base vector The angle of rotation; u4 represents the distance from the coordinate origin O4 to a point on the Archimedes helical surface of the hob; superscript S = 1 or 2. When S = 1, the above formula represents the helical surface of the hob meshing with the convex tooth surface of the worm gear; when S = 2, the above formula represents the helical surface of the hob meshing with the concave tooth surface of the worm gear; Archimedes hob corner.

4. The point contact worm gear transmission design method according to claim 1, characterized in that: The coordinate transformation σ in step S4 is o4 →σ2, combining the hob profile equation and the hobbing meshing function, the specific method to obtain the Archimedes surface worm gear tooth surface equation and its normal vector equation includes: After coordinate transformation σ o4 →σ2, we get the Archimedes surface worm gear tooth surface equation and its normal vector equation; The worm gear tooth surface equation is: Among them, the coefficient Where z 42 Indicates the axial installation distance of the Archimedes hob, Σ 42 is the axis angle of the Archimedes hob, usually 90 degrees, i 42 is the transmission ratio of Archimedes hob when hobbing gears, L A is the Archimedes hob thread length, a 42 Indicates the process center distance of Archimedes hob installation; symbol =Archimedes hob spiral parameter; θ4 represents the turning tool around the base vector The angle of rotation; u4 represents the distance from the coordinate origin O4 to a point on the Archimedes helical surface of the hob; superscript S = 1 or 2. When S = 1, the above formula represents the helical surface of the hob meshing with the convex tooth surface of the worm gear; when S = 2, the above formula represents the helical surface of the hob meshing with the concave tooth surface of the worm gear; Archimedes hob corner; The normal vector component equation is: Where z 42 Indicates the axial installation distance of the Archimedes hob, Σ 42 is the axis angle of the Archimedes hob, usually 90 degrees, i 42 is the transmission ratio of Archimedes hob when hobbing gears, L A is the Archimedes hob thread length, a 42 Indicates the process center distance of Archimedes hob installation; symbol =Archimedes hob spiral parameter; θ4 represents the turning tool around the base vector The angle of rotation; u4 represents the distance from the coordinate origin O4 to a point on the Archimedes hob spiral surface; is the Archimedes cutter angle; superscript S = 1 or 2. When S = 1, the above formula represents the helical surface of the cutter meshing with the convex tooth surface of the worm gear; when S = 2, the above formula represents the helical surface of the cutter meshing with the concave tooth surface of the worm gear.

5. The point contact worm gear transmission design method according to claim 1, characterized in that: In step S5, based on the gear hobbing meshing function, the gear hobbing meshing boundary function and the normal vector of the contact line between the Archimedes hob and the worm gear, the curvature interference boundary function, and the two directions along the worm gear tooth surface are determined. Normal curvature and along the direction Geodesic torsion The specific methods include: Take the meshing function Φ 42 About Motion Parameters Partial derivatives can be used to obtain the tooth meshing boundary function when machining the worm gear for: Where i 42 is the transmission ratio of Archimedes hobbing, symbol Indicates the Archimedes hob spiral parameter; u4 indicates the distance from the coordinate origin O4 to a point on the Archimedes hob spiral surface; Archimedes hob angle; superscript S = 1 or 2, when S = 1, the above formula represents the hob helical surface meshing with the convex tooth surface of the worm gear; when S = 2, the above formula represents the hob helical surface meshing with the concave tooth surface of the worm gear; When grinding the worm gear, an orthogonal movable frame can be established at any meshing point M4 on the hob tooth surface. Its o4 The two are located on the common tangent plane of the tooth surface and Internal unit basis vector and for Among them, after calculation, its component Thus, the normal vector of the contact line between the Archimy hob and the worm gear is obtained: Indicates that the active frame In Chinese: Among them, the coefficient Combined normal curvature geodesic torsion Relative angular velocity Relative velocity vector After calculation, the worm pair curvature interference boundary function Ψ is obtained 42 for: Where, is the contact line normal vector, is the relative velocity vector at any meshing point during gear hobbing, is the cutting tooth engagement boundary function; and For two o4 Common tangent plane of the center hob and the surface worm gear and Internal unit basis vectors; Finally, we can get the two directions along the worm gear tooth surface. and Normal curvature and and along the direction Geodesic torsion Where, is the normal curvature of the Archimedes hob helical surface; is the geodesic curvature of the Archimedes hob helicoid.

6. The method for designing a point contact worm gear transmission according to claim 1, characterized in that: After coordinate transformation σ2→σ o2 , the tooth surface equation and normal vector of the worm gear in step S4 are respectively o2 Expressed as in, The underdetermined equations for the instantaneous contact point of the cone-surface-enveloped cylindrical worm-Archimedes surface worm gear transmission in step S6 are: Where θ4 represents the turning tool around the base vector The angle of rotation; u4 represents the distance from the coordinate origin O4 to a point on the Archimedes hob spiral surface; Hob corner for Archimedes; It is the rotation angle of the cylindrical worm when grinding with a conical grinding wheel; The angle of rotation of the cylindrical worm around its axis is enveloping the conical surface; The worm wheel rotation angle is Archimedes plane; Φ 42 is the meshing function when Archimedes hob is used to machine Archimedes surface worm gear; Φ 31 is the meshing function of the conical grinding wheel enveloping the cylindrical worm; and The cone surface envelops the cylindrical worm in the coordinate system σ o2 Tooth surface equation and its normal vector in; and The Archimedes surface worm gear in the coordinate system σ o2 Tooth surface equation and its normal vector in; is the vector between the coordinate system origin O2 and O1.

7. The method for designing a point contact worm gear transmission according to claim 1, wherein: The variables u3, u4, and u6 are eliminated in step S7. and These five variables then become functions of the three variables θ3, u4, and θ4, and the equations are:

8. The method for designing a point contact worm gear transmission according to claim 1, wherein: In step S8, the worm gear tooth surface is moved in two directions along the worm gear tooth surface. Normal curvature and along the direction Geodesic torsion Calculate the semi-major axis a of the contact ellipse e and semi-minor axis b e The specific methods include: According to the meshing theory, the point conjugate tooth surface couple Along direction and The relative curvature parameters can be obtained from the two directions along the worm gear tooth surface and Normal curvature and and along the direction geodesic torsion get: Among them, the symbol Refers to the worm helical surface At the contact point P i Along the direction and Curvature parameter; combined with the generalized Euler-Bertrand formula, along the direction and Normal curvature Along direction geodesic torsion The result is: Where, and are the two normal curvatures of the worm helical surface along the main directions; is the geodesic torsion along the main direction of the worm helical surface; β is the direction of the worm helical surface Direction of worm gear tooth surface Angle; Then according to the generalized Euler formula, the point conjugate tooth surface couple can be obtained At the contact point P i The two relative principal curvatures are: Where, and The point conjugate tooth surface couple composed of the worm helical surface and the worm wheel tooth surface Along direction and The relative curvature parameter, is the relative geodesic torsion; v t is the vector measured along the common normal To two tooth surface pairs Relative main direction and Angle between Using the relative principal curvature, it is easy to get the semi-major axis a of the contact ellipse e and semi-minor axis b e for: in, is a constant; for this worm pair, its value is This is based on the coating particle size obtained in the rolling test; after determining a sufficient contact ellipse, a complete mismatched surface worm gear pair contact pattern is formed.

9. The method for designing a point contact worm gear transmission according to claim 1, wherein: In step S9, the semi-major axis a of the contact ellipse is e and semi-minor axis b e , determine the instantaneous contact ratio function, use the vector rotation method to obtain the calculation equation of the instantaneous transmission ratio error function and the reference point of the mismatched surface worm gear transmission. The specific method includes: Based on the universal expression of transmission ratio error and the tooth surface equation of Archimedes surface worm gear, the instantaneous transmission ratio of mixed mismatch Archimedes surface worm gear pair is obtained. for: In order to achieve the instantaneous transmission ratio equal to the process transmission ratio, the grinding wheel normal vector guarantees the following relationship: According to the mismatch profile, the instantaneous transmission ratio error of the worm gear transmission Formula, so that the instantaneous transmission ratio error is 0, that is: Where, u4 represents the distance from the coordinate origin O4 to a point on the Archimedes hob spiral surface; p is the spiral parameter of the worm, n 3z (θ3) is the component of the worm normal vector, Archimedes hob tooth profile inclination, is the Archimedes hob spiral parameter; Finally, a new type of worm gear transmission with zero transmission ratio error is realized.

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