CN110766801B

CN110766801B - Three-dimensional modeling method for involute line contact conical worm transmission pair

Info

Publication number: CN110766801B
Application number: CN201911010251.7A
Authority: CN
Inventors: 呼咏; 郭锦洲; 张学成; 于丽娟; 杜雨欣; 王宇; 祝智强
Original assignee: Jilin University
Current assignee: Jilin University
Priority date: 2019-10-23
Filing date: 2019-10-23
Publication date: 2023-04-07
Anticipated expiration: 2039-10-23
Also published as: CN110766801A

Abstract

The invention discloses a three-dimensional modeling method of an involute contact conical worm transmission pair, which solves the problems of larger error, complexity and easy tooth surface deformation when a model is introduced into other software in the prior art, and comprises the following steps: 1) Three-dimensional modeling of an involute line contact conical worm: (1) A part is newly built in CATIA software, the coordinate system of the part is a conical worm coordinate system, and the coordinate system is set as S ₁ I.e. O ₁ ‑X ₁ ,Y ₁ ,Z ₁ (ii) a (2) establishing an alpha surface; (3) establishing a beta surface; (4) creating a conical worm three-dimensional model; 2) Three-dimensional modeling of an involute line contact cone worm gear: (1) A part is newly built in CATIA software, the coordinate system of the part is a cone worm gear coordinate system, and the coordinate system is set as S ₂ I.e. O ₂ ‑X ₂ ,Y ₂ ,Z ₂ (ii) a (2) establishing an alpha' surface; (3) establishing a beta' surface; (4) establishing a three-dimensional model of the bevel worm gear; 3) The involute line contacts the bevel worm and worm gear assembly.

Description

Three-dimensional modeling method for involute line contact conical worm transmission pair

Technical Field

The invention relates to a method belonging to the technical field of mechanical transmission parts, in particular to a three-dimensional modeling method of an involute line contact conical worm transmission pair.

Background

The traditional bevel worm transmission has the problems that the tooth profile has theoretical error, the calculation is difficult and the tooth surface can not be ground. To address these problems, scientists have proposed involute spiroid drives, including involute point, line contact spiroid drives, and involute line contact spiroid drive pairs. The involute line contact conical worm transmission pair has higher strength, so that much attention is paid. The involute line contact conical worm transmission pair is a new type space staggered shaft transmission with large transmission ratio and multiple tooth line contact characteristics. Because the involute line contact conical worm transmission pair has a plurality of contact tooth numbers, compared with a common worm, the involute line contact conical worm transmission pair has the advantages of high transmission strength, stable transmission and good lubricating property, and is developed to a certain extent in China, but is at the laboratory level. The traditional conical worm transmission is already applied to agricultural machinery, hoisting machinery, machine tools, missiles and various military purposes, and the involute line contact conical worm transmission pair has the advantages of the traditional conical worm transmission, so that the application prospect of the involute line contact conical worm transmission is wide.

The numerical control machining technology makes a great breakthrough nowadays, and the importance of a three-dimensional model to machining is also highlighted. Because the existing five-axis linkage numerical control machine tool can process corresponding parts according to the three-dimensional model, the modeling precision has great influence on the quality of the parts. In the traditional modeling method of the worm, the tooth surface parameter equation is mostly utilized to calculate the coordinates of a large number of points and then the tooth surface of the worm and gear is formed; or the established worm entity is utilized to simulate hob processing, and the three-dimensional Boolean operation is utilized to remove the material of the worm wheel blank model to establish the worm wheel model. The models established by the methods have larger errors and are more complicated, and the problem of tooth surface deformation is easy to occur when the models are introduced into other software. A simple and accurate three-dimensional model greatly helps the study of the performance of the transmission pair, and the correct three-dimensional modeling process can provide reference for the machining method and improve the machining efficiency. Therefore, it is necessary to provide a simple and efficient modeling method for the involute contact conical worm gear pair.

Disclosure of Invention

The invention aims to solve the technical problems that the prior art has larger errors, is more complicated and complicated, and the tooth surface deformation is easy to occur when a model is introduced into other software, and provides a three-dimensional modeling method of an involute contact conical worm transmission pair.

In order to solve the technical problems, the invention adopts the following technical scheme: the three-dimensional modeling method of the involute line contact conical worm transmission pair comprises the following steps:

1) Three-dimensional modeling of an involute line contact conical worm:

(1) A part is newly built in CATIA software, the coordinate system of the part is a conical worm coordinate system, and the coordinate system is set as S ₁ I.e. O ₁ -X ₁ ,Y ₁ ,Z ₁ ；

(2) Establishing an alpha surface;

(3) Establishing a beta surface;

(4) Creating a conical worm three-dimensional model;

2) Three-dimensional modeling of an involute line contact cone worm gear:

(1) A part is newly built in CATIA software, the coordinate system of the part is a cone worm gear coordinate system, and the coordinate system is set as S ₂ I.e. O ₂ -X ₂ ,Y ₂ ,Z ₂ ；

(2) Establishing an alpha' surface;

(3) Establishing a beta' surface;

(4) Establishing a three-dimensional model of the bevel worm gear;

3) The involute line contacts the bevel worm and worm gear assembly.

The establishment of the alpha surface in the technical scheme is as follows:

(1) General formula (1)

Wherein i ₁ 、j ₁ 、k ₁ Are respectively coordinate axis X ₁ 、Y ₁ 、Z ₁ Direction vector of (u) _α Is a parameter of the alpha plane, theta _α Is the angle parameter of the alpha plane,

is the base radius of the tooth flank alpha of the spiroid worm>

Is the helix angle of the alpha base helix of the tooth surface of the conical worm _α Is the offset distance of the initial point of the alpha base circle spiral line of the tooth surface of the conical worm;

order to

Obtaining coordinate value of starting point>

0，-l _α Respectively correspond to X ₁ ,Y ₁ ,Z ₁ Coordinate values of the axes;

order to

Obtaining a linear parameter equation, wherein the slope of the straight line is the slope of the No. 1 straight line;

create a plane I, which is Y ₁ Z ₁ Offset plane of plane according to starting point X ₁ The coordinate values of the axes define the offset direction and the offset distance of the plane I, i.e. the offset direction is X ₁ In the positive direction, the offset distance is the base radius of the alpha surface of the worm

Drawing No. 1 straight line in the plane I, wherein the head end of the No. 1 straight line is Z ₁ Axial negative direction and in X ₁ Z ₁ On the plane with X ₁ Y ₁ A face distance of l _α -P _α ，P _α Is the base circle helix lead of the tooth flank alpha, according to

Determining the slope of line number 1, i.e. in Z ₁ The positive direction of the axis is from, the straight line is parallel to Z ₁ The clockwise included angle of the positive axial direction in the plane I is->

The length of line No. 1 is 2d _1ca ；

(2) Order to

The base circle helix equation can be obtained, and the rotation direction and the pitch P of the helix can be obtained according to the equation _α And coordinates of points on the spiral line, for drawing the spiral line;

creating No. 1 spiral line, the head end of No. 1 spiral line is coincident with the head end of No. 1 straight line, and the axis is Z ₁ Axis, pitch P _α Height of L +3 XP _α (ii) a According to

Determining the rotation direction of a No. 1 spiral line;

creating No. 2 spiral line, the tail end of No. 2 spiral line is coincided with the head end of No. 1 straight line, and the axis is Z ₁ Axis, pitch P _α Height of L +3 XP _α According to

Determining the rotation direction of No. 2 helix, wherein the extension direction of No. 2 helix is Z ₁ The axial negative direction;

(3) Drawing No. 2 straight line in the plane I, wherein the head end of the No. 2 straight line is Z ₁ Axial negative direction and in X ₁ Z ₁ On the plane, the distance from the head end of the No. 1 straight line is

Wherein->

Represents 0.5. Epsilon. Rounded up, where. Epsilon. Is the instantaneous contact tooth number; the head end of No. 2 straight line is far from X compared with the head end of No. 1 straight line ₁ Y ₁ Surface, slope of line No. 2 is the same as that of line No. 1, in Z ₁ The positive direction of the axis is the beginning, the No. 2 straight line and Z ₁ The clockwise included angle of the positive axial direction in the plane I is->

Length of 2d _1ca ，d _1ca The diameter of a tip cone at the small end of the conical worm is equal to the diameter of the tip cone, and the tail end of the No. 2 straight line is superposed with the No. 2 spiral line;

(4) The method comprises the steps of taking a No. 1 spiral line and a No. 2 spiral line as guide lines, taking a No. 1 straight line and a No. 2 straight line as outlines, creating a net-shaped curved surface I by utilizing a 'net-shaped curved surface' command in software, and obtaining an alpha surface by the cutting process of the step (3) in the step (4) of creating a conical worm three-dimensional model in 1) involute line contact conical worm three-dimensional modeling in the technical scheme.

The establishment of the beta surface in the technical scheme is as follows:

(1) From formula (2)

Wherein u is _β Is a parameter of the beta plane, θ _β Is the angular parameter of the beta-plane,

is the base radius of the tooth flank beta of the spiroid worm>

Is the helix angle of the base helix of the tooth surface beta of the conical worm _β Is the offset distance of the starting point of the tooth surface beta base circle helix;

order to

Can obtain the coordinate value of the starting point->

0，-l _β Respectively correspond to X ₁ ,Y ₁ ,Z ₁ Coordinate values of the axes; order to

A linear parameter equation can be obtained, and the slope of the straight line is the slope of the No. 3 straight line;

create plane II, which is Y ₁ Z ₁ Offset plane of plane according to starting point X ₁ The coordinate values of the axes define the offset direction and the offset distance of the plane II, i.e. the offset direction is X ₁ In the positive direction, the offset distance is the base radius of a beta surface on the conical worm

Then drawing No. 3 straight line in the plane II, wherein the head end of the No. 3 straight line is Z ₁ In the negative axial direction, with X ₁ Y ₁ A face distance of l _β -P _β ，P _β Is the base circle helix lead of the tooth flank beta, based on the determination of the slope of the line number 3, i.e. in Z ₁ The positive direction of the axis is the beginning, the No. 3 straight line and Z ₁ The clockwise included angle of the positive axial direction in the plane II is->

The length of No. 3 straight line is 2d _1ca ，d _1ca The diameter of the tip cone at the small end of the conical worm;

(2) Order to

The base circle helix equation can be obtained, and the rotation direction and the pitch P of the helix can be known according to the equation _β And the coordinates of points on the spiral line can be used for drawing the spiral line;

creating No. 3 spiral line, the head end of No. 3 spiral line is coincident with the head end of No. 3 straight line, and the axis is Z ₁ Axis, pitch P _β Height of L +3 XP _α According to

Determining the rotation direction of a No. 3 spiral line; the extension direction of No. 3 helical line is Z ₁ The negative axis direction;

creating No. 4 spiral line, the head end of No. 4 spiral line is coincided with the tail end of No. 3 straight line, and the axis is Z ₁ Axis, pitch P _β Height of L +3 XP _α ，P _α Is the base circle helix lead of the tooth flank alpha, according to

The rotating direction of No. 4 spiral line is determined, and the extending direction of No. 4 spiral line is Z ₁ The negative axis direction;

(3) Creating a No. 4 straight line which is a tangent of the No. 3 spiral line, wherein the head end of the No. 4 straight line is superposed with the tail end of the No. 3 spiral line, and the length of the No. 4 straight line is 2d _1ca The tail end of the No. 4 straight line is superposed with the tail end of the No. 4 spiral line;

(4) And (3) taking the No. 3 spiral line and the No. 4 spiral line as guide lines, taking the No. 3 straight line and the No. 4 straight line as outlines, creating a reticular curved surface II by utilizing a command of 'a reticular curved surface' in software, and obtaining a beta surface by the cutting process of the step (3) in the step (4) of creating the conical worm three-dimensional model in 1) involute line contact conical worm three-dimensional modeling in the technical scheme.

The technical scheme is that the establishment of the conical worm three-dimensional model refers to the following steps:

(1) Creating a conic surface of a spiroid

The axis of the conical surface of the top of the conical worm is Z ₁ Shaft, half cone angle delta ₁ At Z ₁ The diameter of the conical surface of the conical worm at position = -E is d _1ca E is an offset distance and integrally penetrates through the net-shaped curved surface I and the net-shaped curved surface II;

(2) Creating a root cone of a spiroid worm

The axis of the conical surface of the top of the conical worm is Z ₁ Axial, half-cone angle delta ₁ At Z ₁ The diameter of the conical surface section of the worm root at the position of = -E is d _1cf The whole body passes through the net-shaped curved surface I and the net-shaped curved surface II;

(3) Cutting curved surface

Cutting the net-shaped curved surface I and the net-shaped curved surface II to enable the head end and the tail end of the two curved surfaces to be positioned in the plane I, and filling the plane by utilizing the boundary line of the head end and the tail end of the two curved surfaces to form a filling plane I and a filling plane II; keeping the parts of the two curved surfaces between the conical surface of the top of the conical worm and the conical surface of the root of the conical worm, wherein the curved surfaces cut by the reticular curved surface I and the reticular curved surface II are the alpha surface and the beta surface;

(4) Generating a spiroid entity

Cutting the conical worm top conical surface and conical worm root conical surface, reserving the parts of alpha surface, beta surface, filling plane I and filling plane II, namely conical worm top conical cutting surface and conical worm root conical cutting surface, using 'joining' command in software to join alpha surface, beta surface, filling plane I, filling plane II, conical worm top conical cutting surface and conical worm root conical cutting surface, creating joining curved surface I, using 'closed curved surface' command in software to fill solid in joining curved surface I to form conical worm gear tooth solid I, using conical worm gear tooth solid I as object, using Z as object ₁ In a circular array with axis as center, the number of arrays being z ₁ ，z ₁ The number of the heads of the conical worm is obtained to obtain a gear tooth entity II of the conical worm; establishing a root cone model of the conical worm, wherein the half cone angle of the root cone is delta ₁ At Z ₁ Diameter of cross section at = -E is d _1cf The integral length is greater than the gear tooth entity I and the gear tooth entity II of the awl worm; finally cut off Z ₁ > -E and Z ₁ And the entity at < -E-L is completed by modeling the conical worm.

The establishment of the alpha' plane in the technical scheme is as follows:

(1) From formula (3)

Wherein i ₂ 、j ₂ 、k ₂ Are respectively coordinate axis X ₂ 、Y ₂ 、Z ₂ Direction vector of (u) _α' Is a parameter of the alpha' plane, theta _α' Is the angle parameter of the a' plane,

is the base radius of the bevel gear flank alpha>

Is the helix angle of the alpha' base helix of the tooth surface of the worm gear _α' Is the offset distance of the initial point of the spiral line of the basic circle of the tooth surface alpha' of the bevel gear;

order to

Obtaining coordinate value of starting point>

0，l _α' Respectively correspond to X ₂ ,Y ₂ ,Z ₂ Coordinate values of the axes; order to

A linear parameter equation can be obtained, and the slope of the straight line is the slope of the No. 5 straight line;

establishing a plane III, plane III being Y ₂ Z ₂ Offset from the plane by X from the starting point ₂ The values of the axes determine the direction and distance of the offset of the plane III, i.e. the direction of the offset is X ₂ Negative direction, offset distance of

Drawing No. 5 straight line in the plane III, wherein the head end of the No. 5 straight line is Z ₂ In the positive direction of the axis, and at X ₂ Z ₂ On the plane, the head end of No. 5 straight line and X ₂ Y ₂ Distance of face is l _α' +P _α' ，P _α' Is a bevel gear tooth surface alpha' base circle helical lead;

by

Determining the slope of line No. 5, in Z ₂ The positive direction of the axis is the beginning, the No. 5 straight line and Z ₂ The clockwise included angle of the positive axial direction on the plane III is->

The helix being an alpha-base helix of the tooth surface of the spiroidLift angle, length of line # 5->

P _α Is the tooth surface of the conic worm, alpha base circle helix lead, epsilon is the instantaneous contact tooth number, z ₁ Number of heads of conical worm, /) _α Is the offset distance of the initial point of the alpha base circle spiral line of the tooth surface of the conical worm;

(2) Order to

Obtaining a base circle spiral line equation, and obtaining the rotation direction and the pitch P of the spiral line according to the equation _α' And coordinates of points on the spiral line, for drawing the spiral line;

creating No. 5 spiral line, the head end of No. 5 spiral line is coincided with the head end of No. 5 straight line, and the axis is Z ₂ Axis, direction of extension Z ₂ Axial negative direction, pitch P _α' Height of 2P _α' According to

Determining the rotation direction of No. 5 spiral line;

creating No. 6 helix, the head end of No. 6 helix is coincident with the tail end of No. 5 helix, and the axis is Z ₂ Axis, direction of extension Z ₂ Axial negative direction, pitch P _α' Height of 2P _α' According to

Determining the rotation direction of No. 6 spiral line;

(3) Creating No. 6 straight line, the straight line is tangent to No. 5 spiral line, the head end of No. 6 straight line is coincided with the tail end of No. 5 spiral line, and the length is

The tail end of the No. 6 straight line is positioned on the No. 6 spiral line;

(4) And (3) taking the No. 5 spiral line and the No. 6 spiral line as guide lines and the No. 5 straight line and the No. 6 straight line as outlines, creating a net-shaped curved surface III by utilizing a ' net-shaped curved surface ' command in software, and obtaining an alpha ' surface by the cutting process of the step (4) in the step (4) of creating the three-dimensional model of the bevel worm gear in 2) involute line contact three-dimensional modeling of the bevel worm gear in the technical scheme.

The establishment of the beta' plane in the technical scheme is as follows:

(1) General formula (4)

Wherein u is _β' Is a parameter of the beta' plane, theta _β' Is the angular parameter of the beta' plane,

is the base radius of the tooth surface beta' of the bevel gear,

is the helix angle of the base helix of the bevel gear tooth surface beta _β' Is the offset distance of the initial point of the base circle spiral line of the tooth surface beta' of the bevel gear;

order to

Get the coordinate value of the starting point->

0，-l _β' Respectively correspond to X ₂ ,Y ₂ ,Z ₂ Coordinate values of the axes; order to

A linear parameter equation can be obtained, and the slope of the straight line is the slope of the No. 7 straight line;

create plane IV, which is Y ₂ Z ₂ Using the starting point X ₂ Coordinate values determine the offset direction and offset distance of plane IV, and the offset direction is X ₂ In the negative direction by an offset distance of

Drawing No. 7 straight line in plane IV, no. 7Head end of straight line is at Z ₂ Axial negative semi-axis, and X ₂ Y ₂ Distance of plane l _β' +P _β' ，P _β' Is the lead of the base circle helix of the bevel gear tooth surface beta', according to

The slope of line 7 can be determined, i.e. in Z ₂ The positive direction of the axis is from, the straight line is parallel to Z ₂ The clockwise included angle of the positive axial direction on the plane IV is->

Is the helix angle of a beta base circle helix of the tooth surface of the conical worm, and the length of the No. 7 straight line is->

P _β Is the tooth surface beta base circle helix lead of the conical worm, epsilon is the instantaneous contact tooth number, z ₁ Number of heads of conical worm, /) _β Is the offset distance of the initial point of the beta base circle spiral line of the tooth surface of the conical worm, d _1ca The diameter of the tip cone at the small end of the conical worm is measured by taking the No. 7 straight line as Z ₂ Rotating angle based on the axis>

To obtain No. 7 rotation line, z ₂ The number of teeth of the bevel gear is Z ₂ The positive direction of the axis looks at anticlockwise rotation;

(2) Order to

The base circle helix equation can be obtained, and the rotation direction and the pitch P of the helix can be known according to the equation _β And coordinates of points on the spiral line, for drawing the spiral line;

creating No. 7 spiral line, the head end of No. 7 spiral line is superposed with the head end of No. 7 rotating straight line, and the axis is Z ₂ Axis, direction of extension Z ₂ Positive axial direction, pitch P _β' Height of 2P _β' According to

Determining the rotation direction of a No. 7 spiral line;

creating No. 8 spiral line, the head end of No. 8 spiral line is coincided with the tail end of No. 7 rotating straight line, and the axis is Z ₂ Axis, direction of extension Z ₂ Positive axial direction, pitch P _β' Height of 2P _β' According to

Determining the rotation direction of the No. 8 spiral line;

(3) Creating No. 8 straight line, wherein the No. 8 straight line is a tangent line of the No. 7 spiral line, the head end of the No. 8 straight line is coincided with the tail end of the No. 7 spiral line, and the length of the No. 8 straight line is

The tail end of the No. 8 straight line is superposed with the No. 8 spiral line;

(4) And (3) taking the No. 7 spiral line and the No. 8 spiral line as guide lines, taking the No. 7 rotating straight line and the No. 8 straight line as outlines, creating a net-shaped curved surface IV by utilizing a ' net-shaped curved surface ' command in software, and obtaining a beta ' surface by the cutting process of the step (4) in the step (4) of creating the three-dimensional model of the bevel worm gear in 2) involute line contact three-dimensional modeling in the technical scheme.

The technical scheme is that the establishment of the three-dimensional model of the bevel worm gear comprises the following steps:

(1) Creating a tapered worm gear tip cone

The axis of the top cone surface of the cone worm wheel is Z ₂ Shaft, half cone angle delta ₂ With vertex at Z ₂ Axial negative semi-axis and X ₂ Y ₂ Plane distance of | Δ l _2ca The radius of the big end of the top conical surface of the conical worm wheel is R _α The opening direction is Z ₂ The shaft is in the positive direction;

(2) Creating a root cone of a bevel worm gear

The axis of the root and the cone of the bevel gear is Z ₂ Shaft, half cone angle delta ₂ (ii) a The normal distance between the root conical surface of the bevel worm wheel and the top conical surface of the bevel worm wheel is h, and the offset direction is Z ₂ In the positive axial direction, the radius of the big end of the conical surface of the root of the conical worm wheel is R _α Direction of openingIs Z ₂ The shaft is in the positive direction;

(3) Creating a curved surface of an inner ring of a bevel worm wheel and a curved surface of an outer ring of the bevel worm wheel

Respectively create a radius of R _i And a radius R _a The cylindrical curved surfaces are the inner ring curved surface of the bevel worm wheel and the outer ring curved surface of the bevel worm wheel, and the axes are Z ₂ The shaft and the two cylindrical curved surfaces need to pass through the reticular curved surface III and the reticular curved surface IV, and the radiuses are R respectively _α And R _i ；

(4) Cutting curved surface

Cutting the net-shaped curved surface III and the net-shaped curved surface IV, and reserving the parts of the two curved surfaces in the middle of the conical surface at the top of the bevel worm wheel, the conical surface at the root of the bevel worm wheel, the curved surface at the inner ring of the bevel worm wheel and the curved surface at the outer ring of the bevel worm wheel, namely an alpha 'surface and a beta' surface;

cutting the conical surface of the top of the bevel worm wheel and the conical surface of the root of the bevel worm wheel, and reserving the parts of the conical surface of the top of the bevel worm wheel and the conical surface of the root of the bevel worm wheel between an alpha 'surface and a beta' surface, namely the conical cutting surface of the top of the bevel worm wheel and the conical cutting surface of the root of the bevel worm wheel; cutting the curved surface of the inner ring of the bevel worm wheel and the curved surface of the outer ring of the bevel worm wheel, and reserving parts among a cutting surface of a tip cone of the bevel worm wheel, a cutting surface of a root cone of the bevel worm wheel, an alpha 'surface and a beta' surface, namely the cutting curved surface of the inner ring of the bevel worm wheel and the cutting curved surface of the outer ring of the bevel worm wheel;

jointing an alpha 'surface, a beta' surface, a bevel gear top cone cutting surface, a bevel gear root cone cutting surface, a bevel gear inner ring cutting curved surface and a bevel gear outer ring cutting curved surface by using a jointing command in software to create a jointing curved surface II;

(5) Generating bevel worm gear tooth surface entity

Filling the joining curved surface II with a solid by means of a software 'closed curved surface' command to complete a bevel worm gear tooth, which is taken as an object, Z ₂ Circular array z with axis as axis of rotation ₂ The entity establishes a three-dimensional model of the root cone of the bevel worm gear by utilizing a rotator command in software, wherein the axis of the root cone model of the bevel worm gear is Z ₂ Shaft, half cone angle delta ₂ (ii) a The normal distance between the root conical surface of the bevel worm wheel and the top conical surface of the bevel worm wheel is h, and the offset direction is Z ₂ In the positive axial direction, the radius of the large end is R _α The opening direction is Z ₂ The shaft is positive, and finally, the entity in the curved surface of the inner ring of the bevel worm wheel, namely the bevel worm wheel is removedThe modeling is completed by this point.

The assembly of the involute line contact bevel worm and worm gear in the technical scheme is as follows:

(1) Newly creating an assembly file, and combining a part coordinate system S displayed in software ₁ And S ₂ Assembling, inserting the built three-dimensional models of the worm wheel and the worm, giving fixed constraint to the worm wheel, and giving Y to the worm ₁ Z ₁ Plane and conic worm wheel Y ₂ Z ₂ Plane offset restriction, wherein the offset is A, and A is the center distance between the bevel worm wheel and the bevel worm;

(2) Given conical worm X ₁ Y ₁ Flat and tapered worm gear Z ₂ X ₂ Plane coincidence constraint, giving to conic worm Z ₁ X ₁ Flat and tapered worm gear X ₂ Y ₂ Plane coincidence constraint;

(3) Satisfy the conical worm Y ₁ Z ₁ Planar on-cone worm gear X ₂ Negative axle shaft, conic worm Y ₁ Positive axial direction and bevel worm gear Z ₂ The positive direction of the axis is opposite, the conical worm Z ₁ Positive axial direction and conic worm wheel Y ₂ The positive directions of the axes are the same, and the conical worm X ₁ Shaft and bevel worm gear X ₂ The positive direction of the axis is the same.

Compared with the prior art, the invention has the beneficial effects that:

1. the three-dimensional modeling method of the involute line contact conical worm transmission pair can avoid parametric modeling and improve modeling efficiency and modeling precision;

2. because the tooth surfaces of the worm and the worm wheel in the three-dimensional modeling method of the involute line contact conical worm transmission pair are involute helicoids, the modeling methods of the worm and the worm wheel are similar, the penetration can be realized only by understanding the modeling method of one part, and the technical requirement of modeling is reduced;

3. according to the three-dimensional modeling method of the involute contact conical worm transmission pair, the assembly of the involute contact worm gear and the involute contact worm is completed by using the coordinate plane and the coordinate system in software, manual subjective adjustment is not needed, and the method is beneficial to subsequent programming.

Drawings

The invention is further described with reference to the accompanying drawings in which:

FIG. 1 is a block diagram of a modeling flow of a three-dimensional modeling method of an involute contact conical worm transmission pair according to the invention;

FIG. 2 is a block diagram of a three-dimensional modeling process of a conical worm in the three-dimensional modeling method of the involute contact conical worm transmission pair;

FIG. 3 is a graph of a curve position relationship required for modeling an alpha surface of a conical worm in the three-dimensional modeling method of the involute line contact conical worm transmission pair of the invention;

FIG. 4 is a graph of a curve position relationship required for modeling a beta surface of a conical worm in the three-dimensional modeling method of the involute contact conical worm transmission pair of the invention;

FIG. 5 is an axonometric view of the conical worm structure in the three-dimensional modeling method of the involute contact conical worm transmission pair according to the invention;

FIG. 6 is a graph of the relationship between the positions of curves required for modeling the alpha' surface of the bevel worm gear in the three-dimensional modeling method of the involute contact bevel worm gear pair;

FIG. 7-1 is a graph of the relationship between the positions of curves required for modeling the beta' plane of the bevel worm gear in the three-dimensional modeling method of the involute contact bevel worm gear pair according to the present invention;

FIG. 7-2 is an enlarged partial view taken at A in FIG. 7-1;

FIG. 8 is an axonometric view of the teeth structure of the worm wheel of the bevel gear in the three-dimensional modeling method of the involute contacting bevel worm gear pair of the present invention;

FIG. 9 is an axonometric view of the conical worm gear structure in the three-dimensional modeling method of the involute contact conical worm gear transmission pair according to the invention;

FIG. 10 is a perspective view of the bevel worm gear pair in the three-dimensional modeling method of the involute line contact bevel worm gear pair of the present invention;

FIG. 11-1 is a schematic diagram of the meshing between the alpha surface and the alpha' surface of the bevel worm transmission pair in the three-dimensional modeling method of the involute contacting bevel worm transmission pair of the present invention;

FIG. 11-2 is a schematic diagram of the engagement between the beta surface and the beta' surface of the bevel worm transmission pair in the three-dimensional modeling method of the involute contact bevel worm transmission pair according to the present invention;

in the figure: 1. a conic worm, no. 101.1 straight line, no. 10101.1 straight line head end, no. 10102.1 straight line tail end, no. 102.1 spiral line, no. 10201.1 spiral line head end, no. 103.2 spiral line, no. 10301.2 spiral line head end, no. 104.2 straight line, no. 10401.2 straight line head end, no. 10402.2 straight line tail end, no. 105.3 straight line, no. 10501.3 straight line head end, no. 10502.3 straight line tail end, no. 106.3 spiral line, no. 10601.3 spiral line head end, no. 10602.3 spiral line tail end, no. 107.4 spiral line, no. 10701.4 spiral line head end, no. 10702.4 spiral line tail end, no. 108.4 straight line, no. 10801.4 straight line head end, no. 10802.4 straight line tail end, no. 109. Alpha face, no. 110. Beta face, no. 2 conic worm wheel, no. 201.5 straight line, no. 20101.5 straight line head end, 20102.5 number straight line terminal, 202.5 number spiral line, 20201.5 number spiral line head end, 20202.5 number spiral line terminal, 203.6 number spiral line, 20301.6 number spiral line head end, 204.6 number straight line, 20401.6 number straight line head end, 20402.6 number straight line terminal, 205.7 number straight line, 20501.7 number straight line head end, 206.7 number rotary straight line, 20601.7 number rotary straight line head end, 20602.7 number rotary straight line terminal end, 207.7 number spiral line, 20701.7 number spiral line head end, 20702.7 number spiral line terminal end, 208.8 number spiral line, 20801.8 number spiral line head end, 209.8 number straight line, 20901.8 number straight line head end, 20902.8 number straight line terminal end, 210. Bevel worm gear teeth, 211. Alpha 'face, 212. Beta' face.

Detailed Description

The invention is described in detail below with reference to the attached drawing figures:

for purposes of making the objects, aspects and advantages of the present invention more apparent, the following detailed modeling and assembling process is set forth in conjunction with the drawings and one embodiment, it being understood that the specific embodiments described herein are for purposes of illustration only and are not intended to be limiting. The specific parameters of the involute contact conical worm gear pair of the embodiment are shown in the following table

TABLE 1 three-dimensional modeling parameters of example involute line contact conical worm gear pair

The left tooth surface and the right tooth surface of the involute line contact conical worm are respectively an alpha surface and a beta surface, wherein the vector expressions of any point on the alpha surface and the beta surface are respectively as follows:

in the formula: i.e. i ₁ 、j ₁ And k ₁ Respectively is the coordinate axis X of the coordinate system where the involute conical worm is positioned ₁ ,Y ₁ ,Z ₁ Direction vector of (u) _α And u _β Is a parameter of alpha surface and beta surface and is greater than or equal to 0, the geometric meaning is the length of a straight generatrix of a spiral surface, theta _α And theta _β Is the angle parameter of alpha surface and beta surface, and the geometric meaning is the tangent point of the straight generatrix of the spiral surface and the spiral line of the base circle and the origin O of the coordinate system ₁ Connecting line with X ₁ Angle in positive direction, positive value being defined as from Z ₁ Observing in the positive direction, wherein the anticlockwise direction is positive; the tooth surfaces of the involute line contact bevel worm gear matched with the involute bevel worm are respectively an alpha 'surface and a beta' surface, wherein the vectors of any point on the alpha 'surface and the beta' surface are respectively expressed as

In the formula: i.e. i ₂ 、j ₂ And k ₂ Respectively is the coordinate axis X of the coordinate system where the involute cone worm gear is positioned ₂ ,Y ₂ ,Z ₂ Direction vector of (u) _α' And u _β' Is a parameter of an alpha 'surface and a beta' surface, the geometric meaning is the length of a straight generatrix of a spiral surface, theta _α' And theta _β' Is the angle parameter of the alpha 'surface and the beta' surface, and has the geometric meaning of the tangent point of the straight generatrix of the spiral surface and the spiral line of the base circle and the origin O of the coordinate system ₂ Connecting line with X ₂ The angle in the positive direction, positive being defined as from Z ₂ Viewed in the positive direction, the counterclockwise direction is positive.

The three-dimensional modeling example steps of the involute line contact conical worm transmission pair are carried out under CATIA V5R 21 software, and the environment language of the software is simplified Chinese.

Referring to fig. 1, the three-dimensional modeling method of the involute contact conical worm transmission pair includes: modeling of an involute contact spiroid, modeling of an involute contact spiroid gear, and assembly of the involute contact spiroid gear.

1. Involute line contact conical worm three-dimensional modeling

Referring to fig. 2, the three-dimensional modeling of the involute contact conical worm comprises the following steps:

1) A part is newly built in CATIA software, a part coordinate system in the software is a conical worm coordinate system, and the coordinate system is set as S ₁ I.e. O ₁ -X ₁ ,Y ₁ ,Z ₁ ；

2) Establishing alpha plane 109

Referring to fig. 2 and 3, the step of establishing the α -plane 109 is as follows:

(1) General formula (1)

Order to

Obtaining coordinate value of starting point>

0，-l _α Respectively correspond to X ₁ ,Y ₁ ,Z ₁ Coordinate values of the axes; order to

A linear parameter equation can be obtained, and the slope of the straight line is the slope of the No. 1 straight line 101;

create a plane I, which is Y ₁ Z ₁ Offset plane of plane according to starting point X ₁ The coordinate values of the axes define the offset direction and the offset distance of the plane I, i.e. the offset direction is X ₁ Positive direction, offset distance is the base radius of alpha surface of worm

Examples of the embodiments

Drawing a No. 1 straight line 101 in a plane I, and drawing a No. 1 straight line head end 10101 in a Z direction ₁ Axial negative direction and in X ₁ Z ₁ On the plane, with X ₁ Y ₁ A face distance of l _α -P _α Examples of the examples _α -P _α =18.8864mm, according to

Determining the slope of line 101 No. 1, i.e. in Z ₁ The positive direction of the axis is from, the straight line is parallel to Z ₁ The clockwise included angle in the plane I in the positive direction of the axis is->

In embodiment->

The length of the No. 1 straight line 101 is 2d _1ca Example 2d _1ca ＝83.5180mm；

(2) Order to

The base circle helical equation can be obtainedThe rotation direction and the pitch P of the spiral line can be known according to the equation _α And coordinates of points on the spiral line can be used for drawing the spiral line, in the embodiment, the rotation direction of the base circle spiral line is right-handed, and the thread pitch is P _α ＝26.7171mm；

Creating a No. 1 helix 102, the No. 1 helix head end 10201 coinciding with the No. 1 straight line head end 10101, the axis being Z ₁ Axis, pitch P _α Examples P _α =26.7171mm and has a height of L +3 XP _α Examples L +3 XP _α =173.6614mm; according to

Determining the rotation direction of a No. 1 spiral line, wherein the rotation direction is right rotation in the embodiment; the extending direction of No. 1 spiral line 102 is Z ₁ The negative axis direction;

creating No. 2 spiral line 103, leading end 10301 of No. 2 spiral line and tail end 10102 of No. 1 straight line, and the axis is Z ₁ Axis, pitch P _α Examples P _α =26.7171mm and has a height of L +3 XP _α Examples L +3 XP _α =173.6614mm, according to

Determining the rotation direction of a No. 2 spiral line 103, wherein the rotation direction is right rotation; no. 2 helix 103 extends in the direction Z ₁ The negative axis direction;

(3) Drawing No. 2 straight line 104 in the plane I, and No. 2 straight line head 10401 in the plane Z ₁ Axial negative direction and in X ₁ Z ₁ On the plane, the distance from the head end 10101 of No. 1 straight line is

Wherein->

Represents a rounding of 0.5 epsilon up, in the examples

No. 2 straight line head end 10401 is far away from X compared with No. 1 straight line head end 10101 ₁ Y ₁ Surface, slope of line No. 2 104 and line No. 1Lines 101 are identical, with Z ₁ The positive direction of the axis is the start, the No. 2 straight line and Z ₁ The clockwise included angle of the positive axial direction in the plane I is->

Length of 2d _1ca In the embodiment->

Length of 2d _1ca =83.5180mm, no. 2 straight line end 10402 coincides with No. 2 spiral line 103;

(4) With the No. 1 spiral line 102 and the No. 2 spiral line 103 as guide lines and the No. 1 straight line 101 and the No. 2 straight line 104 as outlines, a reticular curved surface I is created by using a command of a 'reticular curved surface' in software, and the alpha surface 109 is obtained by the cutting process of the step (3) in the step (4) of creating the three-dimensional model of the conical worm in the 1) involute line contact conical worm three-dimensional modeling in the technical scheme;

3) Build up of beta surface 110

Referring to fig. 2 and 4, the steps for establishing the beta plane 110 are as follows:

(1) General formula (2)

Order to

Obtaining coordinate value of starting point>

A linear parameter equation can be obtained, and the slope of the straight line is the slope of the No. 3 straight line 105;

create plane II, which is Y ₁ Z ₁ Offset plane of plane according to starting point X ₁ The coordinate values of the axes define the offset direction and the offset distance of the plane II, i.e. the offset direction is X ₁ In the positive direction, the offset distance is the base radius of the beta surface on the conical worm 1

In embodiment->

Then drawing No. 3 straight line 105 in plane II, no. 3 straight line head 10501 in Z ₁ In the negative axial direction, with X ₁ Y ₁ A face distance of l _β -P _β Example l _β -P _β =9.5851mm, according to the determination of the slope of line No. 3 105, i.e. in Z ₁ Line 105 and Z in No. 3 with the positive direction of the axis as the start ₁ The clockwise included angle in the plane II in the positive direction of the axis is->

In embodiment->

The length of line 105 No. 3 is 2d _1ca Example 2d _1ca ＝83.5180mm；

(2) Order to

The base circle helix equation can be obtained, and the rotation direction and the pitch P of the helix can be known according to the equation _β And coordinates of points on the spiral line can be used for drawing the spiral line, in the embodiment, the rotation direction of the base circle spiral line is right-handed, and the thread pitch is P _β ＝23.1472mm；

Creating No. 3 helix 106, no. 3 helix head 10601 coinciding with No. 3 straight head 10501, with axis Z ₁ Axis, pitch P _β Examples P _β =23.1472mm and a height of L +3 × P _α Examples of L +3 XP _α =173.6614mm, according to

Determining number 3 spiral 106The rotation direction is right-handed in the embodiment; the extending direction of the No. 3 spiral line 106 is Z ₁ The axial negative direction;

creating No. 4 helix 107, no. 4 helix head 10701 coinciding with No. 3 straight line end 10502, axis Z ₁ Axis, pitch P _β Examples P _β =23.1472mm and a height of L +3 × P _α Examples of L +3 XP _α =173.6614mm, according to

Determining the rotation direction of a No. 4 spiral line 107, wherein the rotation direction is right rotation; the extending direction of No. 4 spiral line 107 is Z ₁ The axial negative direction;

(3) Creating a No. 4 straight line 108 tangent to the No. 3 helix 106 with a head 10801 of the No. 4 straight line coincident with the end 10602 of the No. 3 helix and a length of 2d _1ca Example 2d _1ca =83.5180mm, no. 4 straight line end 10802 coincides with No. 4 spiral line end 10702;

(4) Taking a No. 3 spiral line 106 and a No. 4 spiral line 107 as guide lines, taking a No. 3 straight line 105 and a No. 4 straight line 108 as outlines, creating a net-shaped curved surface II by utilizing a command of 'a net-shaped curved surface' in software, and obtaining a beta surface 110 by the cutting process of the step (3) in the step (4) of creating a three-dimensional model of the conical worm in 1) involute line contact conical worm three-dimensional modeling in the technical scheme;

4) Creating a three-dimensional model of a spiroid

Referring to fig. 5, the steps for creating the conical worm three-dimensional model are as follows:

(1) Creating a conic surface of a spiroid

The axis of the conical surface of the top of the conical worm is Z ₁ Axial, half-cone angle delta ₁ Example δ ₁ =7.2477 °; at Z ₁ The diameter of the conical surface at the tip of the conical worm at the position of = -E is d _1ca Examples Z ₁ ＝-E＝-38.8567，d _1ca =41.7590mm, and the whole passes through the reticular curved surface I and the reticular curved surface II;

(2) Creating a root cone of a spiroid worm

The axis of the conical surface of the top of the conical worm is Z ₁ Axial, half-cone angle delta ₁ Examples of the embodimentsδ ₁ =7.2477 °; at Z ₁ The diameter of the conical surface section of the worm root at the position of = -E is d _1cf Examples Z ₁ ＝-E＝-38.8567，d _1cf =23.8509mm, and integrally penetrates through the reticular curved surface I and the reticular curved surface II;

(3) Cutting curved surface

Cutting the mesh curved surface I and the mesh curved surface II to enable the head end and the tail end of the two curved surfaces to be positioned in the plane I, and filling the plane by utilizing the boundary line of the head end and the tail end of the two curved surfaces to form a filling plane I and a filling plane II; keeping the parts of the two curved surfaces between the top conical surface of the worm and the conical surface of the worm root, wherein the curved surfaces cut by the reticular curved surface I and the reticular curved surface II are the alpha surface 109 and the beta surface 110;

(4) Generating conic worm entity

Cutting the conical worm top conical surface and conical worm root conical surface, reserving the parts of alpha surface 109, beta surface 110, filling plane I and filling plane II, namely conical worm top conical cutting surface and conical worm root conical cutting surface, using 'joining' command in software to join alpha surface 109, beta surface 110, filling plane I, filling plane II, conical worm top conical cutting surface and conical worm root conical cutting surface, creating joining curved surface I, using 'closed curved surface' command in software to fill solid in the joining curved surface I to form conical worm gear tooth solid I, using the conical worm gear tooth solid I as object, using Z as object ₁ In a circular array with axis as center, the number of arrays being z ₁ In the example z ₁ =2 to obtain a cone worm gear tooth entity II; establishing a root cone model of the conical worm, wherein the half cone angle of the root cone is delta ₁ At Z ₁ The diameter of the cross section at = -E = -38.8567mm is d _1cf The overall length is greater than the gear tooth entity I and the gear tooth entity II of the conical worm; finally cut off Z ₁ > -E and Z ₁ And the entity at < -E-L is completed by modeling the conical worm.

2. Involute line contact cone worm gear three-dimensional modeling

Referring to fig. 2, since the bevel worm wheel 2 and the bevel worm 1 are both involute helicoids, the main three-dimensional modeling steps of the bevel worm wheel and the bevel worm are similar, and the involute line contact bevel worm wheel three-dimensional modeling steps are as follows:

1) A part is newly built in CATIA softwareThe coordinate system is a bevel worm gear coordinate system, and the coordinate system is set as S ₂ I.e. O ₂ -X ₂ ,Y ₂ ,Z ₂ ；

2) Establishing an alpha' face 211

Referring to fig. 6, the steps for establishing the α' plane are as follows:

(1) General formula (3)

Order to

Can obtain the coordinate value of the starting point->

A linear parameter equation can be obtained, and the slope of the straight line is the slope of the No. 5 straight line 201;

Examples of the embodiments

Drawing No. 5 straight line 201 in a plane III, and drawing the head end 20101 of the No. 5 straight line in Z ₂ Positive axial direction, and in X ₂ Z ₂ On the plane, the head end of No. 5 straight line 20101 and X ₂ Y ₂ Distance of face is l _α' +P _α' Examples of the examples _α' +P _α' =1077.6594mm in diameter, by>

Determine the slope of line 201 # 5 in Z ₂ Line 201 and Z in No. 5 with the positive axial direction as the start ₂ The clockwise included angle of the positive axial direction on the plane II I is->

In embodiment->

Line 5 has a length ^ 5>

In an embodiment>

(2) Order to

The base circle helix equation can be obtained, and the rotation direction and the pitch P of the helix can be known according to the equation _α' And coordinates of points on the spiral line can be used for drawing the spiral line, in the embodiment, the spiral direction of the base circle is left-handed, and the thread pitch is P _α' ＝990.2444mm；

Creating No. 5 helix 202, with No. 5 helix head 20201 coinciding with No. 5 helix head 20101, with axis Z ₂ Axis, direction of extension Z ₂ Axial negative direction, pitch P _α' Examples P _α' =990.2444mm and 2P height _α' Example 2P _α' =1980.4888mm, according to

Determining the rotation direction of the No. 5 spiral line 202, wherein the rotation direction is left rotation in the embodiment;

creating No. 6 helix 203, the head end 20301 of No. 6 helix coinciding with the end 20102 of No. 5 helix, the axis being Z ₂ Axis, direction of extension Z ₂ Axial negative direction, pitch P _α' Examples P _α' =990.2444mm and 2P height _α' "ShiExample 2P _α' =1980.4888mm, according to

Determining the rotation direction of a No. 6 spiral line 203, wherein the rotation direction is left rotation;

(3) A No. 6 straight line 204 is created, the straight line is a tangent of the No. 5 spiral line 202, the head end 20401 of the No. 6 straight line is superposed with the tail end 20202 of the No. 5 spiral line, and the length of the head end is equal to that of the tail end 20202 of the No. 5 spiral line

Examples of the embodiments

Number 6 straight end 20402 should be located on number 6 spiral 203.

(4) Taking the No. 5 spiral line 202 and the No. 6 spiral line 203 as guide lines, taking the No. 5 straight line 201 and the No. 6 straight line 204 as outlines, creating a net-shaped curved surface III by utilizing a ' net-shaped curved surface ' command in software, and obtaining an alpha ' surface 211 by the cutting process of the step (4) in the step (4) of creating a three-dimensional model of the bevel worm gear in 2) involute line contact three-dimensional modeling in the technical scheme;

3) Build up of beta' face 212

Referring to fig. 7-1 and 7-2, the steps for establishing the β' plane are as follows:

(1) General formula (4)

Order to

Get the coordinate value of the starting point->

Can be prepared byObtaining a linear parameter equation, wherein the slope of the linear is the slope of the No. 7 straight line 205;

create plane IV, which is Y ₂ Z ₂ Using the starting point X ₂ Coordinate values determine the offset direction and offset distance of plane IV, and the offset direction is X ₂ Negative direction, offset distance of

Examples of the embodiments

Drawing No. 7 straight line 205 in plane IV, no. 7 straight line head end 20501 in Z ₂ Axial negative semi-axis, and X ₂ Y ₂ Distance of plane l _β' +P _β' Examples of the examples _β' +P _β' =1308.5095mm, according to

The slope of line 205 # 7 can be determined, i.e., in Z ₂ The positive direction of the axis is from, the straight line is parallel to Z ₂ The clockwise included angle of the positive axial direction on the plane IV is->

In embodiment->

Line # 7 has a length of 205 @>

Examples of the embodiments

Line No. 7 205 is marked with Z ₂ Rotation angle of shaft as axis

In an embodiment>

Obtain No. 7 rotation line 206, the rotation direction is along Z ₂ The positive direction of the axis looks at anticlockwise rotation; />

(2) Order to

The base circle helix equation can be obtained, and the rotation direction and the pitch P of the helix can be known according to the equation _β And coordinates of points on the spiral line can be used for drawing the spiral line, in the embodiment, the rotation direction of the base circle spiral line is right-handed, and the thread pitch is P _β' ＝1240.9817mm；

Creating No. 7 helix 207, no. 7 helix head end 20701 coinciding with No. 7 rotation straight line head end 20601, the axis being Z ₂ Axis, direction of extension Z ₂ Positive axial direction, pitch P _β' Examples P _β' =1240.9817mm, height 2P _β' Example 2P _β' =2481.9634mm, according to

Determining the rotation direction of a No. 7 spiral line, wherein the rotation direction is right rotation;

creating No. 8 helix 208, the head end 20801 of No. 8 helix is coincident with the tail end 20602 of No. 7 rotating straight line, and the axis is Z ₂ Axis, direction of extension Z ₂ Positive axial direction, pitch P _β' Examples P _β' =1240.9817mm and a height of 2P _β' Example 2P _β' =2481.9634mm, according to

Determining the rotation direction of a No. 8 spiral line, wherein the rotation direction is right rotation;

(3) Creating line No. 8 209, line No. 8 209 being a tangent to helix No. 7 207, the head end of line No. 8 20901 coinciding with the end 20702 of the helix No. 7 and having a length of

Examples of the embodiments

Number 8 straight end 20902 coincides with number 8 helix 208;

(4) Taking the No. 7 spiral line 207 and the No. 8 spiral line 208 as guide lines, taking the No. 7 rotating straight line 206 and the No. 8 straight line 209 as outlines, creating a net-shaped curved surface IV by utilizing a ' net-shaped curved surface ' command in software, and obtaining a beta ' surface 212 by the cutting process of the step (4) in the step (4) of creating a three-dimensional model of the bevel worm gear in 2) involute line contact three-dimensional modeling in the technical scheme;

4) Creating a three-dimensional model of a pyramidal worm gear

The steps for creating the three-dimensional model of the worm gear cone are as follows:

(1) Creating a tapered worm gear tip cone

The axis of the top conical surface of the conical worm wheel is Z ₂ Shaft, half cone angle delta ₂ With vertex at Z ₂ Axial negative semi-axis and X ₂ Y ₂ Plane distance of | Δ l _2ca |, example | Δ l _2ca The diameter | =4.2998mm, and the radius of the large end of the top conical surface of the cone worm wheel is R _α Examples of R _α =169.7689mm, opening direction is Z ₂ The shaft is in the positive direction;

(2) Creating a root cone of a bevel worm gear

The axis of the conical surface of the conical worm gear root is Z ₂ Shaft, half cone angle delta ₂ Example δ ₂ =79.9132 °; the normal distance between the root conical surface and the top conical surface of the bevel worm wheel is h, h =8.8825mm in the embodiment, and the offset direction is Z ₂ In the positive axial direction, the radius of the big end of the conical surface of the root of the conical worm wheel is R _α Examples of R _α =169.7689mm, opening direction is Z ₂ The shaft is in the positive direction;

Respectively creating a radius of R _i And a radius R _a The cylindrical curved surfaces are the inner ring curved surface of the bevel worm wheel and the outer ring curved surface of the bevel worm wheel, and the axes are Z ₂ The shaft and the two cylindrical curved surfaces need to pass through the reticular curved surface III and the reticular curved surface IV, and the radiuses are R respectively _α And R _i In the examples, R is created separately _i =106.3041mm and R _α =169.7689mm cylinderA curved surface;

(4) Cutting curved surface

Cutting the net-shaped curved surface III and the net-shaped curved surface IV, and keeping the parts of the two curved surfaces in the middle of the conical surface at the top of the bevel worm wheel, the conical surface at the root of the bevel worm wheel, the curved surface at the inner ring of the bevel worm wheel and the curved surface at the outer ring of the bevel worm wheel, namely an alpha 'surface 211 and a beta' surface 212; cutting the conical surface at the top of the bevel worm wheel and the conical surface at the root of the bevel worm wheel, and reserving the parts of the conical surface at the top of the bevel worm wheel and the conical surface at the position between an alpha 'surface 211 and a beta' surface 212, namely the cutting surface of the conical top cone of the bevel worm wheel and the cutting surface of the conical root cone of the bevel worm wheel; cutting the bevel worm wheel inner ring curved surface and the bevel worm wheel outer ring curved surface, and reserving parts among a bevel worm wheel top bevel cutting surface, a bevel worm wheel root bevel cutting surface, an alpha 'surface 211 and a beta' surface 212, namely the bevel worm wheel inner ring cutting curved surface and the bevel worm wheel outer ring cutting curved surface; jointing an alpha 'surface 211, a beta' surface 212, a bevel gear top cone cutting surface, a bevel gear root cone cutting surface, a bevel gear inner ring cutting curved surface and a bevel gear outer ring cutting curved surface by using a 'jointing' command in software to create a jointing curved surface II;

(5) Tooth surface entity of generating bevel worm wheel

Referring to FIGS. 8 and 9, the filling of the mating curved surface II with entities using the software "closed surface" command completes a worm gear tooth 210, which is the target of Z ₂ Circular array z with axis as axis of rotation ₂ Individual entities, examples z ₂ =50, all gear teeth of the bevel gear are obtained, then a three-dimensional model of the bevel gear root cone is established by utilizing a 'rotator' command in software, and the axis of the bevel gear root cone model is Z ₂ Axial, half-cone angle delta ₂ Example δ ₂ =79.9132 °; the normal distance between the root conical surface of the bevel worm wheel and the top conical surface of the worm wheel is h, h =8.8825mm in the embodiment, and the offset direction is Z ₂ In the positive axial direction, the radius of the large end is R _α Examples of R _α =169.7689mm, opening direction is Z ₂ The shaft is forward, finally, the entity in the curved surface of the inner ring of the bevel worm gear is removed, and the modeling of the bevel worm gear is completed;

3. involute line contact taper worm and cone worm gear assembly

Referring to fig. 10, the assembling steps of the involute contact conical worm and worm gear are as follows:

1) New construction assemblyBody file, part coordinate system S displayed in combination with software ₁ And S ₂ Assembling, inserting the built three-dimensional model of the conical worm wheel, and using 'fixed constraint' in software to give fixed constraint to the conical worm wheel, limit the whole freedom of the conical worm wheel and give Y-shaped conical worm for convenient assembly ₁ Z ₁ Plane and cone worm wheel Y ₂ Z ₂ Plane offset constraint, the offset is A, in the embodiment, A = -100mm;

2) Given conical worm X ₁ Y ₁ Plane and conic worm gear Z ₂ X ₂ Plane coincidence constraint, giving to conic worm Z ₁ X ₁ Flat and tapered worm gear X ₂ Y ₂ Plane coincidence constraint;

3) At the same time, it should also be satisfied that the conical worm Y ₁ Z ₁ Planar on-cone worm gear X ₂ Negative axle shaft, conic worm Y ₁ Positive axial direction and bevel worm gear Z ₂ The positive direction of the axis is opposite, the conical worm Z ₁ Positive axial direction and conic worm wheel Y ₂ The positive directions of the axes are the same, and the conical worms X ₁ Shaft and bevel worm gear X ₂ The positive direction of the axis is the same.

And (3) verifying modeling accuracy:

referring to fig. 11-1 and 11-2, by using a "cutting" command in software, a plane I and a plane II are respectively used as geometric targets to cut the assembly body, and if an α plane and an α 'plane, and a β plane and a β' plane are respectively engaged with each other by using a plurality of straight lines as instantaneous contact lines at the same time, it is proved that modeling is correct; so far, the assembly of the involute line contact bevel worm gear is completed.

According to the method, the property that the tooth surface of the bevel worm gear of the bevel worm is in contact with the involute line is a spiral involute surface is utilized, a tooth surface bus is used for sweeping around a base circle spiral line to generate the tooth surface based on CATIA software, and the assembly is completed by utilizing a CATIA coordinate system. The modeling principles of the worm gear and the worm cone are the same, the modeling method is similar, the modeling method does not depend on parametric modeling, the property of the tooth surface is tightly combined according to a curved surface equation, the modeling and assembling method is simple, efficient and systematic, the auxiliary processing and manufacturing can be realized, and a good foundation can be laid for the subsequent research on the aspects of mechanical property, efficiency and the like.

Claims

1. A three-dimensional modeling method of an involute contact conical worm transmission pair is characterized by comprising the following steps of:

1) Three-dimensional modeling of an involute line contact conical worm:

(2) Establishing an alpha plane (109);

(3) Establishing a beta plane (110);

(4) Creating a conical worm three-dimensional model;

2) Three-dimensional modeling of an involute line contact cone worm gear:

(2) Establishing an alpha' plane (211);

(3) Establishing a beta' plane (212);

(4) Establishing a three-dimensional model of the bevel worm gear;

3) The involute line contacts the bevel worm and worm gear assembly.

2. A method for three-dimensional modelling of an involute contact spiroid transmission pair according to claim 1, wherein said establishing an α -plane (109) is:

(1) General formula (1)

is the base radius of the tooth flank alpha of the bevel worm>

order to

Obtaining coordinate value of starting point>

order to

Obtaining a linear parameter equation, wherein the slope of the straight line is the slope of the No. 1 straight line (101);

Drawing a No. 1 straight line (101) in a plane I, wherein the head end (10101) of the No. 1 straight line is arranged in Z ₁ Axial negative direction and in X ₁ Z ₁ On the plane with X ₁ Y ₁ A face distance of l _α -P _α ，P _α Is the base circle helix lead of the tooth flank alpha, according to

Determining the slope of line No. 1 (101), i.e. in Z ₁ The positive direction of the axis is from, the straight line is parallel to Z ₁ The clockwise included angle of the positive direction of the axis in the plane I is/>

The length of the No. 1 straight line (101) is 2d _1ca ；

(2) Order to

creating a No. 1 spiral line (102), wherein the head end (10201) of the No. 1 spiral line is superposed with the head end (10101) of the No. 1 straight line, and the axis is Z ₁ Axis, pitch P _α Height of L +3 XP _α (ii) a According to

Determining the rotation direction of a No. 1 spiral line;

creating No. 2 spiral line (103), wherein the head end (10301) of the No. 2 spiral line is superposed with the tail end (10102) of the No. 1 straight line, and the axis is Z ₁ Axis, pitch P _α Height of L +3 XP _α According to

The rotation direction of No. 2 spiral line (103) is determined, and the extension direction of No. 2 spiral line (103) is Z ₁ The axial negative direction;

(3) Drawing a No. 2 straight line (104) in a plane I, wherein the head end (10401) of the No. 2 straight line is arranged in a Z direction ₁ Axial negative direction and in X ₁ Z ₁ On the plane, the distance from the head end (10101) of No. 1 straight line is (1)

Wherein->

Represents 0.5. Epsilon. Rounded up, ε being the instantaneous contact tooth number; the head end of No. 2 straight line (10401) is far from X than the head end of No. 1 straight line (10101) ₁ Y ₁ Surface, slope of line No. 2 (104) and line No. 1 (101)) Same as in Z ₁ The positive direction of the axis is the beginning, the No. 2 straight line and Z ₁ The clockwise included angle of the positive direction of the axis in the plane I is

Length of 2d _1ca ，d _1ca The diameter of a tip cone at the small end of the conical worm is equal to the diameter of the tip cone, and the tail end (10402) of the No. 2 straight line is superposed with the No. 2 spiral line (103);

(4) And (2) taking the No. 1 spiral line (102) and the No. 2 spiral line (103) as guide lines, taking the No. 1 straight line (101) and the No. 2 straight line (104) as outlines, creating a net-shaped curved surface I by utilizing a command of 'net-shaped curved surface' in software, and obtaining an alpha surface (109) by the cutting process of the step (3) in the step (4) of creating the conical worm three-dimensional model in the 1) involute line contact conical worm three-dimensional modeling.

3. The method for three-dimensional modeling of an involute contact spiroid drive pair of claim 1, wherein said establishing a β plane (110) comprises:

(1) From formula (2)

is the base radius of the flank beta of the spiroid worm>

Is the helix angle of the base helix of the tooth surface beta of the conical worm _β Is the offset distance of the starting point of the tooth surface beta base circle spiral line;

order to

Can obtain the coordinate value of the starting point->

A linear parameter equation can be obtained, and the slope of the linear is the slope of the No. 3 linear (105);

create plane II, which is Y ₁ Z ₁ Offset plane of plane according to starting point X ₁ The coordinate values of the axes define the offset direction and the offset distance of the plane II, i.e. the offset direction is X ₁ In the positive direction, the offset distance is the base radius of a beta surface on the conical worm (1)

Then drawing No. 3 straight line (105) in the plane II, wherein the head end (10501) of the No. 3 straight line is arranged in Z ₁ In the negative axial direction, with X ₁ Y ₁ A face distance of l _β -P _β ，P _β Is the base circle helix lead of the tooth surface beta, according to the determined slope of the No. 3 straight line (105), namely in Z ₁ Line No. 3 (105) and Z in the positive direction of the axis ₁ The clockwise included angle of the positive axial direction in the plane II is->

The length of line 105 No. 3 is 2d _1ca ，d _1ca The diameter of the tip cone at the small end of the conical worm;

(2) Order to

creating a No. 3 spiral line (106), wherein the head end (10601) of the No. 3 spiral line is superposed with the head end (10501) of the No. 3 straight line, and the axis is Z ₁ Axis, pitch P _β Height of L +3 XP _α ，P _α Is the base circle helix lead of the tooth flank alpha, according to

Determining the rotation direction of a No. 3 spiral line (106); the extension direction of the No. 3 spiral line (106) is Z ₁ The negative axis direction;

creating a No. 4 spiral line (107), wherein the head end (10701) of the No. 4 spiral line is superposed with the tail end (10502) of the No. 3 straight line, and the axis is Z ₁ Axis, pitch P _β Height of L +3 XP _α According to

The rotating direction of the No. 4 spiral line (107) is determined, and the extending direction of the No. 4 spiral line (107) is Z ₁ The axial negative direction;

(3) Creating a No. 4 straight line (108) which is a tangent of the No. 3 helix (106), the head end (10801) of the No. 4 straight line being coincident with the tail end (10602) of the No. 3 helix, and the length of the head line being 2d _1ca The No. 4 straight line end (10802) is superposed with the No. 4 spiral line end (10702);

(4) And (3) taking a No. 3 spiral line (106) and a No. 4 spiral line (107) as guide lines, taking a No. 3 straight line (105) and a No. 4 straight line (108) as outlines, creating a net-shaped curved surface II by utilizing a command of 'net-shaped curved surface' in software, and obtaining a beta surface (110) by the cutting process of the step (3) in the step (4) of creating the conical worm three-dimensional model in the 1) involute line contact conical worm three-dimensional modeling.

4. The method for three-dimensional modeling of an involute contact spiroid transmission pair according to claim 1, wherein the creating of the spiroid three-dimensional model is:

(1) Creating a conic surface of a spiroid

(2) Creating a root cone of a spiroid worm

The axis of the conical surface of the top of the conical worm is Z ₁ Shaft, half cone angle delta ₁ At Z ₁ The diameter of the conical surface section of the worm root at the position of = -E is d _1cf The whole body passes through the net-shaped curved surface I and the net-shaped curved surface II;

(3) Cutting curved surface

Cutting the net-shaped curved surface I and the net-shaped curved surface II to enable the head end and the tail end of the two curved surfaces to be positioned in the plane I, and filling the plane by utilizing the boundary line of the head end and the tail end of the two curved surfaces to form a filling plane I and a filling plane II; the parts of the two curved surfaces between the conical surface of the top of the conical worm and the conical surface of the root of the conical worm are reserved, and the curved surfaces cut by the reticular curved surface I and the reticular curved surface II are an alpha surface (109) and a beta surface (110);

(4) Generating conic worm entity

Cutting the conical worm top conical surface and the conical worm root conical surface, reserving parts of an alpha surface (109), a beta surface (110), a filling plane I and a filling plane II, namely the conical worm top conical cutting surface and the conical worm root conical cutting surface, joining the alpha surface (109), the beta surface (110), the filling plane I, the filling plane II, the conical worm top conical cutting surface and the conical worm root conical cutting surface by using a 'joining' command in software, creating a joining curved surface I, filling a solid in the joining curved surface I to form a conical worm gear tooth solid I by using a 'closed curved surface' command in the software, taking the conical worm gear tooth solid I as an object, and taking Z as an object ₁ In a circular array with axis as center, the number of arrays being z ₁ ，z ₁ The number of the heads of the conical worm is obtained to obtain a gear tooth entity II of the conical worm; establishing a root cone model of the conical worm, wherein the half cone angle of the root cone is delta ₁ At Z ₁ Diameter of cross section at = -E is d _1cf The overall length is greater than the gear tooth entity I and the gear tooth entity II of the conical worm; finally cut off Z ₁ > -E and Z ₁ And the entity at < -E-L is completed by modeling the conical worm.

5. The method for three-dimensional modeling of an involute contact spiroid drive pair according to claim 1, wherein said establishing an α' plane (211) is:

(1) General formula (3)

Wherein i ₂ 、j ₂ 、k ₂ Are respectively coordinate axis X ₂ 、Y ₂ 、Z ₂ Direction vector of (u) _α' Is a parameter of the alpha' plane, theta _α' Is the angular parameter of the alpha' face,

is the base radius of the tooth flank α' of the bevel gear in combination with a gear in the cylinder>

Is the helix angle of the alpha' base helix of the tooth surface of the worm gear _α' Is the offset distance of the initial point of the spiral line of the alpha' base circle of the tooth surface of the bevel worm wheel;

order to

Obtaining coordinate value of starting point>

0，l _α' Respectively correspond to X ₂ ,Y ₂ ,Z ₂ Coordinate values of the axes; make/combine>

A linear parameter equation can be obtained, and the slope of the linear is the slope of the No. 5 linear (201);

establishing a plane III, plane III being Y ₂ Z ₂ Offset from the plane by X from the starting point ₂ The values of the axes coordinate the direction and distance of the offset of the plane III, i.e. the direction of the offset is X ₂ Negative direction, offset distance of

Drawing a No. 5 straight line (201) in a plane III, wherein the head end (20101) of the No. 5 straight line is arranged in Z ₂ In the positive direction of the axis, and at X ₂ Z ₂ On the plane, the head end of No. 5 straight line (20101) and X ₂ Y ₂ Distance of face is l _α' +P _α' ，P _α' Is a bevel worm gearA tooth surface alpha' base circle helix lead;

by

Determining the slope of line No. 5 (201) in Z ₂ The positive direction of the axis is the beginning, the No. 5 straight line (201) and the Z ₂ The clockwise included angle of the positive axial direction on the plane III is->

Is the lead angle of an alpha base spiral line of the tooth surface of the awl worm, and the length of a No. 5 straight line (201) is->

(2) Order to

creating a No. 5 spiral line (202), wherein the head end (20201) of the No. 5 spiral line is superposed with the head end (20101) of the No. 5 straight line, and the axis is Z ₂ Axis, direction of extension Z ₂ Axial negative direction, pitch P _α' Height of 2P _α' According to

Determining the rotation direction of a No. 5 spiral line (202);

creating a No. 6 spiral line (203), wherein the head end (20301) of the No. 6 spiral line is superposed with the tail end (20102) of the No. 5 straight line, and the axis is Z ₂ Axis, direction of extension Z ₂ Axial negative direction, pitch P _α' Is high and highDegree of 2P _α' According to

Determining the rotating direction of a No. 6 spiral line (203);

(3) Creating a No. 6 straight line (204) which is a tangent of the No. 5 spiral line (202), wherein the head end (20401) of the No. 6 straight line is superposed with the tail end (20202) of the No. 5 spiral line, and the length of the straight line is equal to that of the No. 5 spiral line

The No. 6 straight line end (20402) is positioned on the No. 6 spiral line (203);

(4) And (2) taking the No. 5 spiral line (202) and the No. 6 spiral line (203) as guide lines, taking the No. 5 straight line (201) and the No. 6 straight line (204) as outlines, creating a net-shaped curved surface III by utilizing a ' net-shaped curved surface ' command in software, and obtaining an alpha ' surface (211) through the cutting process of the step (4) in the step (4) of creating the three-dimensional tapered worm gear model in the 2) involute line contact three-dimensional tapered worm gear modeling.

6. The method for three-dimensional modeling of an involute contact spiroid drive pair of claim 1, wherein said establishing a β' plane (212) comprises:

(1) General formula (4)

is the base radius of the bevel gear flank beta>

Is the helix angle of the base helix of the bevel gear tooth surface beta'. The _β' Is the offset distance of the initial point of the base circle spiral line of the tooth surface beta' of the bevel gear; />

Order to

Get the coordinate value of the starting point->

A linear parameter equation can be obtained, and the slope of the straight line is the slope of the No. 7 straight line (205);

Drawing No. 7 straight line (205) in plane IV, and the head end (20501) of No. 7 straight line in Z ₂ Axial negative semi-axis, and X ₂ Y ₂ Distance of plane l _β' +P _β' ，P _β' Is the lead of the base circle helix of the bevel gear tooth surface beta', according to

The slope of line # 7 (205) can be determined, i.e., in Z ₂ The positive direction of the axis is from, the straight line is parallel to Z ₂ The clockwise included angle of the positive axial direction on the plane IV is->

Is the helix angle of a beta base circle helix of a bevel worm tooth surface, and the length of a No. 7 straight line (205) is

P _β Is the tooth surface beta base circle spiral lead of the conical worm, epsilon is the instantaneous contact tooth number, z ₁ Number of heads of conical worm, /) _β Is the offset of the initial point of the beta base circle spiral line of the tooth surface of the conical worm, d _1ca The diameter of the tip cone at the small end of the conical worm is measured by taking the No. 7 straight line (205) as Z ₂ Rotating angle based on the axis>

Obtaining No. 7 rotating straight line (206), z ₂ The number of teeth of the bevel gear is Z ₂ The positive direction of the axis looks at anticlockwise rotation;

(2) Order to

creating No. 7 spiral line (207), wherein the head end (20701) of the No. 7 spiral line is superposed with the head end (20601) of the No. 7 rotating straight line, and the axis is Z ₂ Axis, direction of extension Z ₂ Positive axial direction, pitch P _β ', height of 2P _β ', based on

Determining the rotation direction of a No. 7 spiral line;

creating a No. 8 spiral line (208), wherein the head end (20801) of the No. 8 spiral line is superposed with the tail end (20602) of the No. 7 rotating straight line, and the axis is Z ₂ Axis, direction of extension Z ₂ Positive axial direction, pitch P _β' Height of 2P _β' According to

Determining the rotation direction of the No. 8 spiral (208);

(3) Creating a No. 8 straight line (209), wherein the No. 8 straight line (209) is a tangent line of the No. 7 spiral line (207), the head end (20901) of the No. 8 straight line is superposed with the tail end (20702) of the No. 7 spiral line, and the length of the head end and the tail end is equal to

The 8 # straight line end (20902) is superposed with the 8 # spiral line (208);

(4) And (2) taking the No. 7 spiral line (207) and the No. 8 spiral line (208) as guide lines, taking the No. 7 rotating straight line (206) and the No. 8 straight line (209) as outlines, creating a net-shaped curved surface IV by utilizing a ' net-shaped curved surface ' command in software, and obtaining a beta ' surface (212) through the cutting process of the step (4) in the step (4) of creating the three-dimensional tapered worm gear model in the 2) involute line contact tapered worm gear three-dimensional modeling.

7. The method for three-dimensional modeling of an involute contact spiroid transmission pair according to claim 1, wherein the creating of the spiroid wheel three-dimensional model is:

(1) Creating a tapered worm gear tip cone

The axis of the top conical surface of the conical worm wheel is Z ₂ Shaft, half cone angle delta ₂ With vertex at Z ₂ Axial negative semi-axis and X ₂ Y ₂ Plane distance of | Δ l _2ca The radius of the big end of the top conical surface of the conical worm wheel is R _α The opening direction is Z ₂ The shaft is in the positive direction;

(2) Creating a root cone of a spiroid gear

The axis of the conical surface of the conical worm gear root is Z ₂ Shaft, half cone angle delta ₂ (ii) a The normal distance between the root conical surface of the bevel worm wheel and the top conical surface of the bevel worm wheel is h, and the offset direction is Z ₂ In the positive axial direction, the radius of the big end of the conical surface of the root of the conical worm wheel is R _α The opening direction is Z ₂ The shaft is in the positive direction;

(4) Cutting curved surface

Cutting the net-shaped curved surface III and the net-shaped curved surface IV, and reserving the parts of the two curved surfaces in the middle of the conical surface at the top of the bevel worm wheel, the conical surface at the root of the bevel worm wheel, the curved surface at the inner ring of the bevel worm wheel and the curved surface at the outer ring of the bevel worm wheel, namely an alpha 'surface (211) and a beta' surface (212);

cutting the conical surface at the top of the conical worm wheel and the conical surface at the root of the conical worm wheel, and reserving the parts of the conical surface at the top of the conical worm wheel and the conical surface at the root of the conical worm wheel between an alpha 'surface (211) and a beta' surface (212), namely the conical worm wheel top conical cutting surface and the conical worm wheel root conical cutting surface; cutting the curved surface of the inner ring of the bevel worm wheel and the curved surface of the outer ring of the bevel worm wheel, and reserving parts among a cutting surface of a tip cone of the bevel worm wheel, a cutting surface of a root cone of the bevel worm wheel, an alpha 'surface (211) and a beta' surface (212), namely the cutting curved surface of the inner ring of the bevel worm wheel and the cutting curved surface of the outer ring of the bevel worm wheel;

jointing an alpha 'surface (211), a beta' surface (212), a bevel worm gear top cone cutting surface, a bevel worm gear root cone cutting surface, a bevel worm gear inner ring cutting curved surface and a bevel worm gear outer ring cutting curved surface by using a 'jointing' command in software to create a jointing curved surface II;

(5) Generating bevel worm gear tooth surface entity

By means of a software "closed curve" command, the joining curve II is filled with a solid body to complete a bevel worm gear tooth (210), subject to this tooth, Z ₂ Circular array z with axis as axis of rotation ₂ The entity establishes a three-dimensional model of the root cone of the bevel worm gear by utilizing a 'rotator' command in software, and the axis of the root cone model of the bevel worm gear is Z ₂ Shaft, half cone angle delta ₂ (ii) a The normal distance between the root conical surface of the bevel worm wheel and the top conical surface of the bevel worm wheel is h, and the offset direction is Z ₂ The positive direction of the axis, the radius of the big end is R _α The opening direction is Z ₂ And (5) the shaft is forward, and finally, the entity in the curved surface of the inner ring of the bevel worm gear is removed, and the modeling of the bevel worm gear is completed.

8. The three-dimensional modeling method of the involute contact spiroid transmission pair of claim 1, wherein the involute contact spiroid and worm gear assembly means:

(1) Newly creating an assembly file, and combining a part coordinate system S displayed in software ₁ And S ₂ Assembling, inserting the built three-dimensional models of the worm wheel and the worm, giving fixed constraint to the worm wheel, and giving Y to the worm ₁ Z ₁ Flat and conical wormWheel Y ₂ Z ₂ Plane offset constraint, wherein the offset is A, and A is the center distance between the bevel worm wheel and the bevel worm;

(2) Given conical worm X ₁ Y ₁ Plane and conic worm gear Z ₂ X ₂ Plane coincidence constraint, giving to conic worm Z ₁ X ₁ Flat and tapered worm gear X ₂ Y ₂ Plane coincidence constraint;

(3) Satisfy the conical worm Y ₁ Z ₁ Planar on-cone worm gear X ₂ Negative axis semi-axis, conic worm Y ₁ Positive axial direction and bevel worm gear Z ₂ The positive direction of the axis is opposite, the conical worm Z ₁ Positive axial direction and conic worm wheel Y ₂ The positive directions of the axes are the same, and the conical worms X ₁ Shaft and bevel worm gear X ₂ The positive direction of the axes is the same.