CN114800032B - Non-circular gear design method - Google Patents

Non-circular gear design method Download PDF

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CN114800032B
CN114800032B CN202210446684.2A CN202210446684A CN114800032B CN 114800032 B CN114800032 B CN 114800032B CN 202210446684 A CN202210446684 A CN 202210446684A CN 114800032 B CN114800032 B CN 114800032B
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circular
circular gear
tooth profile
coordinate system
cutter
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CN114800032A (en
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李大伟
刘永平
李泽宇
龚俊
魏永峭
任忠涛
董长斌
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Lanzhou University of Technology
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B23—MACHINE TOOLS; METAL-WORKING NOT OTHERWISE PROVIDED FOR
    • B23Q—DETAILS, COMPONENTS, OR ACCESSORIES FOR MACHINE TOOLS, e.g. ARRANGEMENTS FOR COPYING OR CONTROLLING; MACHINE TOOLS IN GENERAL CHARACTERISED BY THE CONSTRUCTION OF PARTICULAR DETAILS OR COMPONENTS; COMBINATIONS OR ASSOCIATIONS OF METAL-WORKING MACHINES, NOT DIRECTED TO A PARTICULAR RESULT
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    • G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
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    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/17—Mechanical parametric or variational design
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    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
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    • G06F30/20—Design optimisation, verification or simulation

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Abstract

The embodiment of the application discloses a non-circular gear design method, and relates to the field of gear design. The method comprises the following steps: establishing a two-dimensional motion relation between the non-circular gear and the circular cutter; obtaining a tooth profile envelope equation of the non-circular gear according to the two-dimensional motion relation; establishing a tooth profile curve equation of the circular cutter, and obtaining a tooth profile meshing equation of the non-circular teeth and the circular cutter; and obtaining a tooth profile curve equation of the non-circular gear according to the tooth profile envelope equation, the tooth profile curve equation of the circular cutter and the tooth profile meshing equation. By the method, the accuracy of obtaining the tooth profile curve of the non-circular gear can be improved, and the obtaining process is simpler.

Description

Non-circular gear design method
Technical Field
The application relates to the field of gear design, in particular to a non-circular gear design method.
Background
The gear transmission is one of the most important transmission forms in mechanical transmission, and a common circular gear can only transmit with a fixed transmission ratio, and a non-circular gear can transmit with a variable transmission ratio. The actual model of the non-circular gear is obtained by machining, and the most effective method is to obtain the tooth profile curve of the non-circular gear according to the machining principle to establish an accurate non-circular gear model.
In the prior art, the method for obtaining the tooth profile curve of the non-circular gear has the problems of low precision, complex obtaining process and the like. For example, in patent CN106151452a, a geometric model of a non-circular gear tooth profile is obtained by establishing a normal equidistant curve family of a non-circular gear pitch curve, extracting tooth socket boundary points according to polar angle positions and distances of points on the equidistant curve family, and connecting the extracted points. The technology can only obtain the geometric model of the non-circular gear tooth profile, and cannot obtain the mathematical model of the non-circular gear tooth profile, so that the precision of the non-circular gear tooth profile obtained by the technology is low. For another example, in patent CN108916318B, the tool is used to envelop a non-circular gear to form an envelope curve group of the tooth profile of the non-circular gear, and then a numerical algorithm is used to screen the boundary points of the envelope curve group to obtain the tooth profile of the non-circular gear. The process of obtaining the non-circular gear tooth profile by the patent technology is very complex, the calculation amount is large, the design period is long, although the obtained tooth profile is more accurate than that in the patent CN106151452A, the mathematical model of the non-circular gear tooth profile cannot be obtained, and the accuracy is still lower. As in patent CN105889456B, the envelope equation and the meshing equation are obtained based on the tooth profile envelope method, and the non-circular gear tooth profile with high accuracy can be obtained theoretically, but in this method, the envelope equation is very complex and it is difficult to obtain the specific form of the meshing equation, and thus it is difficult to obtain the specific form of the non-circular gear tooth profile equation.
Therefore, the traditional method for obtaining the tooth profile curve of the non-circular gear has the problems of low precision, complex obtaining process and the like.
Disclosure of Invention
The application provides a non-circular gear design method, a circular cutter rotates around a fixed rotation center, the non-circular gear performs rotation and translation motion, a node of the circular cutter and the non-circular gear is fixed, a two-dimensional motion relation between the non-circular gear and the circular cutter is established, a tooth profile envelope equation with a simpler form is obtained, a tooth profile curve equation of the circular cutter is established, a specific form of a tooth profile meshing equation of the non-circular gear and the circular cutter is obtained, and finally a mathematical model of a non-circular gear tooth profile with higher precision is obtained. Therefore, the accuracy of obtaining the tooth profile curve of the non-circular gear can be improved, and the obtaining process is simpler.
The embodiment of the application provides a non-circular gear design method, which comprises the following steps: establishing a two-dimensional motion relation between the non-circular gear and the circular cutter; obtaining a tooth profile envelope equation of the non-circular gear according to the two-dimensional motion relation; establishing a tooth profile curve equation of the circular cutter, and obtaining a tooth profile meshing equation of the non-circular teeth and the circular cutter; and obtaining a tooth profile curve equation of the non-circular gear according to the tooth profile envelope equation, the tooth profile curve equation of the circular cutter and the tooth profile meshing equation.
In summary, the present application has at least the following technical effects:
1. this application carries out the rotation through making circular cutter around fixed centre of revolution, makes non-circular gear carry out rotation and translation motion to make circular cutter and non-circular gear's node fixed, establish the motion relation of non-circular gear and circular cutter, obtain the profile of tooth envelope equation that the form is comparatively simple, and then make the process of obtaining the profile of tooth curve equation of non-circular gear simpler.
2. According to the method, the specific form of the tooth profile meshing equation of the non-circular gear and the circular cutter is obtained, and the subsequent steps are carried out by combining the tooth profile envelope equation, so that a large number of processes for solving the boundary of the envelope curve family are avoided, and the process for obtaining the tooth profile curve equation of the non-circular gear is simpler.
3. According to the method, the specific mathematical model of the tooth profile of the non-circular gear is obtained through the tooth profile envelope equation, the tooth profile curve equation of the circular cutter and the tooth profile meshing equation, and the accuracy of the obtained tooth profile curve of the non-circular gear is further improved.
4. This application is through utilizing circular cutter to process the non-circular gear, can design the non-circular gear of pitch curve indent, can also design the non-circular gear of inner gearing, has higher adaptability and flexibility in the non-circular gear design of multiple shape.
5. This application is fixed through the centre of a circle that makes node and circular cutter, can make circular cutter follow same direction and move back the sword to avoid moving back the sword and interfering, it is more convenient when being applied to gear machining operation.
Therefore, the scheme provided by the application can relieve the problems of low precision, complex acquisition process and the like of the method for acquiring the tooth profile curve of the non-circular gear.
Drawings
In order to more clearly illustrate the technical solutions in the embodiments of the present application, the drawings needed to be used in the description of the embodiments are briefly introduced below, and it is obvious that the drawings in the following description are only some embodiments of the present application, and it is obvious for those skilled in the art to obtain other drawings based on these drawings without creative efforts.
FIG. 1 is a schematic flow chart illustrating a non-circular gear design method provided in embodiment 1 of the present application;
FIG. 2 is a schematic view showing the machining of a non-circular gear provided in embodiment 1 of the present application;
FIG. 3 is a schematic pitch curve diagram of a non-circular gear and a circular cutter provided in embodiment 1 of the present application at an initial time;
FIG. 4 is a schematic view showing a pitch curve of a non-circular gear and a circular cutter provided in embodiment 1 of the present application at a certain moment during the movement;
FIG. 5 shows a schematic view of a tooth profile of a circular cutter provided in example 1 of the present application;
FIG. 6 shows a schematic view of the tooth profile engagement provided in example 1 of the present application;
fig. 7 shows a schematic tooth profile of a non-circular gear provided in embodiment 1 of the present application;
fig. 8 shows a schematic view of an internally meshing non-circular gear provided in embodiment 1 of the present application.
Detailed Description
In order to make the technical solutions of the present application better understood, the technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the drawings in the embodiments of the present application, and it is obvious that the described embodiments are only a part of the embodiments of the present application, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present application.
At present, in the method for obtaining the tooth profile curve of the non-circular gear in the prior art, only a geometric model of the tooth profile of the non-circular gear can be obtained generally, and a mathematical model of the tooth profile of the non-circular gear cannot be obtained, so that the problem of low precision exists. In the method for obtaining the curve equation of the tooth profile of the non-circular gear through the envelope equation and the mesh equation, the envelope equation is complex, and the specific form of the mesh equation is difficult to obtain, so that the problem that the process of obtaining the tooth profile of the non-circular gear is complex exists.
Therefore, in order to solve the above-mentioned drawbacks, an embodiment of the present application provides a method for designing a non-circular gear, the method including: establishing a two-dimensional motion relation between the non-circular gear and the circular cutter, obtaining a tooth profile envelope equation of the tooth profile of the non-circular gear according to the two-dimensional motion relation, establishing a tooth profile curve equation of the circular cutter, obtaining a tooth profile meshing equation of the non-circular gear and the circular cutter, and obtaining the tooth profile curve equation of the non-circular gear according to the tooth profile envelope equation, the tooth profile curve equation of the circular cutter and the tooth profile meshing equation.
The circular cutter is enabled to rotate, the rotation center is fixed, and the non-circular gear is enabled to rotate and move in a translation mode, so that the nodes of the circular cutter and the non-circular gear are fixed, on the basis, the two-dimensional motion relation of the non-circular gear and the circular cutter is established, the tooth profile envelope equation with a simpler form is obtained, the tooth profile curve equation of the circular cutter is established, the specific form of the tooth profile meshing equation of the non-circular gear and the circular cutter is obtained, the mathematical model of the tooth profile of the non-circular gear with higher precision is obtained, the precision of obtaining the tooth profile curve of the non-circular gear is improved, and the obtaining process is simpler. The following describes a method of designing a non-circular gear to which the present application relates.
Example 1
Referring to fig. 1, fig. 1 is a schematic flow chart illustrating a method for designing a non-circular gear according to embodiment 1 of the present application. In this embodiment, the method for designing a non-circular gear may include the steps of:
step S110: and establishing a two-dimensional motion relation between the non-circular gear and the circular cutter.
Wherein, the non-circular gear is also called as a special-shaped gear, and the pitch curve of the non-circular gear is non-circular. After the non-circular gear and another gear form a gear pair, the instantaneous angular speed ratio is changed according to a certain established motion law in the meshing process. In the embodiment of the application, the non-circular gear can be a gear with an elliptic pitch curve, can also be a gear with a concave pitch curve, and can also be a gear with gear teeth engaged inside and a pitch curve of any shape.
At present, when a non-circular gear is designed, the non-circular gear can be machined mainly through cutters such as a hobbing cutter, a gear shaping cutter and a gear milling cutter. In the embodiment of the application, the circular cutter can be a gear shaping cutter which is simple and convenient to operate.
In an exemplary embodiment, the two-dimensional motion relationship may include: the non-circular gear is wound around a rotation center O on the end surface thereof g The circular cutter rotates around the center O of a circle on the end surface of the non-circular gear and simultaneously translates c And the end face of the non-circular gear and the end face of the circular cutter are positioned on the same plane in a self-rotation motion mode.
Wherein, as shown in fig. 2, the non-circular gear and the circular cutter are in the process of moving, but the center O of the circle on the end surface of the circular cutter c Fixed, the circular tool only surrounds the centre of a circle O c The rotation movement is not carried out, the translation movement is not carried out, and the revolution movement is not carried out around the non-circular gear. The non-circular gear can make translational motion on a plane parallel to the end face thereof, and the rotation center O on the end face of the non-circular gear g Also make translational motion on the plane, while the non-circular gear is wound around the rotation center O g Doing self-rotation movement.
In an exemplary embodiment, step S110 may further include sub-step S111 to sub-step S114.
Substep S111: at the initial moment, the node P of the pitch curve of the non-circular gear and the pitch curve of the circular cutter and the rotation center O of the non-circular gear g And the center O of the circular cutter c On the same straight line with the center O of the circular cutter c Is an origin O 0 To do so by
Figure BDA0003617196280000051
In the direction of x 0 Positive axial direction with said x 0 The positive direction of the shaft rotates clockwise by 90 degrees in the direction of y 0 In the positive axial direction, a fixed coordinate system S is established on the end face of the circular cutter 0 (O 0 -x 0 -y 0 ) And said node P is in the coordinate system S 0 Coordinates of (5)Is fixed and not changed.
As shown in fig. 3, fig. 3 is a pitch curve diagram of the non-circular gear and the circular cutter at the initial time. Specifically, the node P of the circular cutter and the non-circular gear is fixed on the coordinate system S 0 Coordinate of (5) (-r) g 0) at r g Equal to the pitch radius of a circular cutter.
Sub-step S112: at the initial moment, the center O of the circular cutter is used c Is an origin O 1 To in order to
Figure BDA0003617196280000052
In the direction of x 1 Positive axial direction, with said x 1 The positive direction of the shaft rotates clockwise by 90 degrees in the direction of y 1 In the positive axial direction, a moving coordinate system S is established on the end surface of the circular cutter 1 (O 1 -x 1 -y 1 ) Said coordinate system S 1 Origin O of 1 Fixed, x 1 Axes and y 1 The shaft makes autorotation motion along with the circular cutter.
In the embodiment of the present application, as shown in fig. 3, the center O is located at the initial time and during the movement c And origin O 0 Origin O 1 Overlapping and fixed in this plane.
At an initial time, x 0 Axis and x 1 The axes coincide, y 0 Axis and y 1 The axes coincide.
During the movement, x 0 Axis and y 0 Fixing a shaft; x is the number of 1 Axes and y 1 The shaft follows the circular cutter around the center O c Do autorotation motion and the rotation angle of the round cutter is equal to x 1 Axis and y 1 The angle of rotation of the shaft.
Substep S113: at the initial time, the rotation center O of the non-circular gear is set g Is an origin O p To in order to
Figure BDA0003617196280000061
In the direction of x p Positive axial direction, with said x p The positive direction of the shaft rotates anticlockwise by 90 degrees in the direction of y p In the positive axial direction, a moving coordinate system S is established on the end face of the non-circular gear p (O p -x p -y p ) Said coordinate system S p Origin O of p 、x p Axis and y p The shaft makes translational motion along with the non-circular gear.
Substep S114: at the initial time, the rotation center O of the non-circular gear is set g Is an origin O 2 To in order to
Figure BDA0003617196280000062
In the direction of x 2 Positive axial direction with said x 2 The direction of the positive direction anticlockwise rotating by 90 degrees is the positive direction of the y axis, and a moving coordinate system S is established on the end face of the non-circular gear 2 (O 2 -x 2 -y 2 ) Said coordinate system S 2 Origin O of 2 The non-circular gear does translational motion along with the non-circular gear, and the coordinate system S 2 X of 2 Axis and y 2 The shaft makes self-rotation movement and translation movement along with the non-circular gear.
In the embodiment of the present application, as shown in fig. 3, the center of rotation O is formed at the initial time and during the movement g And origin O p Origin O 2 And the translation tracks of the non-circular gears are the same as the translation tracks of the non-circular gears.
At an initial time, x 0 Axis and x 1 Axis, x p Axis, x 2 Axis coincidence, y 0 Axis and y 1 Axis coincidence, y p Axis and y 2 The axes coincide.
During the movement, x p Axis and y p The shaft only does translational motion along with the non-circular gear, and the translational track of the shaft is the same as that of the non-circular gear; x is a radical of a fluorine atom 2 Axes and y 2 The shaft not only does translational motion along with the non-circular gear, but also winds around a rotation center O along with the non-circular gear g Make autorotation movement, and the rotation angle of the non-circular gear is equal to x 2 Axes and y 2 The angle of rotation of the shaft.
Step S120: and obtaining a tooth profile envelope equation of the non-circular gear according to the two-dimensional motion relation.
In the embodiment of the application, the tooth profile envelope equation is an envelope equation formed by the tooth profile of the non-circular gear and the tooth profile of the circular cutter, and the tooth profile envelope equation is used for calculating the tooth profile curve equation of the non-circular gear.
In an exemplary embodiment, step S120 may further include sub-steps S121 through S126.
Substep S121: establishing a pitch curve equation of the non-circular gear
Figure BDA0003617196280000071
Wherein +>
Figure BDA0003617196280000072
For the gear ratio of the non-circular gear>
Figure BDA0003617196280000073
For said node P in said coordinate system S 2 Is radially based on>
Figure BDA0003617196280000074
And said x 2 And the included angle of the shafts, a, is the center distance of the non-circular gear pair.
In the embodiment of the present application, the non-circular gear and the circular cutter are in the process of moving, as shown in fig. 4, fig. 4 is a schematic pitch curve diagram of the non-circular gear and the circular cutter at a certain time in the process of moving.
Substep S122: establishing a coordinate system S 1 Into a coordinate system S 0 Transformation relation M of 01 Wherein S is 0 ＝M 01 S 1 ，
Figure BDA0003617196280000075
φ 1 Is a corner of the round knife and>
Figure BDA0003617196280000076
wherein it is present>
Figure BDA0003617196280000077
Is equal to the initial moment>
Figure BDA0003617196280000078
Value r g Is the pitch radius of the circular cutter.
As shown in FIG. 4, φ 1 Being the corner of said circular tool, i.e. y 0 Axis and y 1 The angle of the axes.
Figure BDA0003617196280000079
Is radially based on>
Figure BDA00036171962800000710
And x 2 Included angle of the shaft>
Figure BDA00036171962800000711
Equal to initial instant>
Figure BDA00036171962800000712
Value, optionally, in the present embodiment>
Figure BDA00036171962800000713
Equal to 0 deg..
In the embodiment of the application, the pitch curve of the non-circular gear and the pitch curve of the circular cutter are in a pure rolling relationship, namely the rolling arc length of the pitch curve of the non-circular gear is equal to the rolling arc length of the pitch curve of the circular cutter in unit time length. The rolling arc length of the non-circular gear is
Figure BDA00036171962800000714
The rolling arc length of the circular cutter is phi 1 r g Thus there are
Figure BDA00036171962800000715
Substep S123: establishing a coordinate system S 0 Conversion to coordinate system S p Transformation relation M of p0 Wherein S is p ＝M p0 S 0 ，
Figure BDA0003617196280000081
Mu is the radial direction of the tangent line at the node P and the non-circular gear>
Figure BDA0003617196280000082
And is/are included angle of>
Figure BDA0003617196280000083
As shown in fig. 4, t is a tangent line of the pitch curve of the non-circular gear and the pitch curve of the circular cutter at a node P, and μ is a tangent line t and a radial direction
Figure BDA0003617196280000084
The included angle of (a).
In the embodiment of the application, the moving track of the non-circular gear in the plane is in the fixed coordinate system S 0 The expression in (3) can be derived directly from the geometric relationship shown in fig. 4:
Figure BDA0003617196280000085
the coordinate system S can thus be obtained 0 Conversion to coordinate system S p Transformation relation of (2)
Figure BDA0003617196280000086
Substep S124: establishing a coordinate system S p Into a coordinate system S 2 Transformation relation M of 2p Wherein S is 2 ＝M 2p S p ，
Figure BDA0003617196280000087
φ 2 Is the angle of rotation of the non-circular gear, and
Figure BDA0003617196280000088
wherein, mu 0 Equal to the value of μ at the initial instant.
Alternatively, as shown in FIG. 3, in the embodiment of the present application, μ 0 Equal to 90 deg.. From the geometrical relationships in FIG. 4, one can obtain
Figure BDA0003617196280000089
Substep S125: according to the pitch curve equation of the non-circular gear
Figure BDA00036171962800000810
Transformation relation M 01 The transformation relation M p0 And the transformation relation M 2p Determining a coordinate system S of the circular cutter tooth profile 1 Coordinate system S converted into the non-circular gear tooth profile 2 Transformation relation M of 21 Wherein S is 2 ＝M 21 S 1 ，M 21 ＝M 2p M p0 M 01 。
Substep S126: according to the transformation relation M 21 Obtaining a tooth profile envelope equation of the non-circular gear:
Figure BDA0003617196280000091
wherein (x) 1 ，y 1 ) For the circular tool tooth profile in the coordinate system S 1 The coordinates of (a) are (b), (x) 2 ，y 2 ) For the non-circular gear tooth profile in the coordinate system S 2 Of (2) is calculated.
According to the non-circular gear design method, the circular cutter rotates around the fixed circle center, the non-circular gear performs rotation and translation motion, and the node of the circular cutter and the non-circular gear is fixed on the S 0 In a coordinate system, a two-dimensional motion relation between the non-circular gear and the circular cutter is established based on the method, a tooth profile envelope equation with a simpler form is obtained, and the process of obtaining the tooth profile curve of the non-circular gear is simpler.
Step S130: and establishing a tooth profile curve equation of the circular cutter, and obtaining a tooth profile meshing equation of the non-circular gear and the circular cutter.
In the embodiment of the application, the tooth profile meshing equation is the meshing equation of the tooth profile of the non-circular gear and the tooth profile of the circular cutter, and the tooth profile meshing equation is used for calculating the tooth profile curve equation of the non-circular gear.
In an exemplary embodiment, step S130 may further include sub-step S131 and sub-step S132.
Substep S131: establishing a tooth profile curve equation of the circular cutter:
Figure BDA0003617196280000092
wherein r is b Is the base radius of the circular tool u s Is the involute variable parameter delta of the circular cutter 0 And the involute initial point angle of the circular cutter is obtained.
As an alternative embodiment, x is shown in FIG. 5 1 The first complete tooth on the upper side of the axle is a first tooth, the second complete tooth is a second tooth, and the Nth complete tooth is an Nth tooth. Point E is the starting point of the involute of the first gear tooth, delta 01 For the involute start point angle of a first tooth, the involute start point angle of each tooth constitutes the variable delta 0 。
Point F is a point on the first gear tooth, line PQ is a normal to point F, and line PQ is tangent to base circle at point G, line O 1 G and line O 1 E included angle is variable u s 。
Substep S132: when the coordinate of the meshing point of the circular cutter tooth profile and the non-circular gear tooth profile in the coordinate system S1 is (x) 1 ，y 1 ) And then, obtaining a tooth profile meshing equation of the non-circular gear and the circular cutter:
Figure BDA0003617196280000101
where Ψ is equal to the pressure angle of the pitch circle of the circular cutter, γ is the x and the tangent at the point of engagement of the circular cutter tooth profile and the non-circular gear tooth profile 1 Angle of axis and->
Figure BDA0003617196280000102
As shown in fig. 6, a solid line aa indicates the tooth profile of the circular cutter at the initial time, a solid line bb indicates the tooth profile of the non-circular gear at the initial time, and M point is the tooth profile at the initial timePoint of engagement, m 1 The point is a point on the solid line aa. The dotted line aa is the tooth profile of the round tool at the next moment, the dotted line bb is the tooth profile of the non-round gear at the next moment, M 1 The point is the mesh point at the next time instant. The coordinate axes shown in FIG. 6 are x of the initial time 1 Axis and y 1 Axis, x at the next moment not shown in FIG. 6 1 Axis and y 1 A shaft.
When the circular cutter rotates phi 1 When the solid line aa moves to the dotted line aa, the solid line bb moves to the dotted line bb; m of initial time 1 M of point movement to next time 1 A point becomes a meshing point of the next moment; p of the initial moment 1 P of point movement to next moment 2 And become nodes.
t m1 Is m 1 Tangent of point circular cutter tooth profile, passing through origin O 1 Making a tangent t m1 Parallel line O of 1 N, and parallel line O 1 N and x at the initial time 1 The included angle of the axes is psi, and optionally psi is constant in the embodiments of the present application, and may be equal to the pressure angle of the pitch circle of the circular tool, e.g., may be 20 °.
According to the non-circular gear design method, the specific form of the tooth profile meshing equation is obtained by establishing the tooth profile meshing equation of the non-circular gear and the circular cutter and fixing the node P of the non-circular gear and the circular cutter, and the subsequent steps are carried out by combining the tooth profile enveloping equation, so that a large number of envelope curve family boundary solving processes are avoided, and the process of obtaining the tooth profile curve equation of the non-circular gear is simpler.
Step S140: and obtaining a tooth profile curve equation of the non-circular gear according to the tooth profile envelope equation, the tooth profile curve equation of the circular cutter and the tooth profile meshing equation.
In an exemplary embodiment, the engagement point of the circular cutter tooth profile and the non-circular gear tooth profile is in the coordinate system S 1 The track equation in (1) is the tooth profile curve equation of the circular cutter, and the meshing point of the tooth profile of the circular cutter and the tooth profile of the non-circular gear is in the coordinate system S 2 The trajectory equation in (1) is a tooth profile curve equation of the non-circular gear.
In an exemplary embodiment, step S140 may further include sub-step S141.
Substep S141: obtaining a tooth profile curve equation of the non-circular gear according to the tooth profile envelope equation, the tooth profile curve equation of the circular cutter and the tooth profile meshing equation:
Figure BDA0003617196280000111
wherein, by
Figure BDA0003617196280000112
Therefore, the following steps are carried out: γ may be represented by u s And delta 0 Is obtained by the expression of 1 = π - γ + Ψ: phi is a 1 Can be derived from the expression of γ, so it can be seen that: phi is a 1 Can be composed of s And delta 0 The expression of (c) is obtained.
By
Figure BDA0003617196280000113
Therefore, the following steps are carried out: />
Figure BDA0003617196280000114
Can be made of 1 Is obtained, thus->
Figure BDA0003617196280000115
Can be composed of s And delta 0 The expression of (c) is obtained.
By
Figure BDA0003617196280000116
Therefore, the following steps are carried out: mu can be determined by>
Figure BDA0003617196280000117
Is obtained by>
Figure BDA0003617196280000118
Figure BDA0003617196280000119
Therefore, the following steps are carried out: phi is a 2 Can be selected from>
Figure BDA00036171962800001110
Is given by the expression of (a) thus phi 2 Can be composed of s And delta 0 The expression of (c) is obtained.
Thus, non-circular gear tooth profile coordinates (x) 2 ，y 2 ) Can be represented by the curve equation of s And delta 0 The expression of (c) is obtained.
As shown in fig. 7, each tooth of the non-circular gear corresponds to a left-side tooth profile and a right-side tooth profile, and the left-side tooth profile and the right-side tooth profile of each tooth form a complete involute of one tooth. Optionally, in fig. 7, the tooth profile located to the right of the dotted line L1 and to the left of the dotted line L2 is the left tooth profile of the gear tooth, and the tooth profile located to the right of the dotted line L2 and to the left of the dotted line L3 is the right tooth profile of the gear tooth.
In the tooth profile curve equation, when
Figure BDA00036171962800001111
And the obtained tooth profile curve equation is the left tooth profile of each tooth of the non-circular gear.
In the tooth profile curve equation, when
Figure BDA00036171962800001112
And the obtained tooth profile curve equation is the right tooth profile of each tooth of the non-circular gear.
The application provides a method for designing a non-circular gear, which takes a non-circular gear with a known transmission ratio as an example, and designs the tooth profile of the non-circular gear according to the method. The design parameters of the non-circular gear are shown in table 1.
TABLE 1 non-circular Gear parameters
Figure BDA0003617196280000121
According to the specific steps, the equation of the pitch curve of the non-circular gear is as follows:
Figure BDA0003617196280000122
pitch radius r of circular cutting tool g =17, base radius r b =15.97, using the above data in combination with the profile curve equation for circular tools, with the parameter u of the involute of the circular tool s And the involute start angle delta 0 As independent variables, the tooth profile curve equation of the non-circular gear can be obtained, and specifically, the simulation diagram is shown in fig. 7.
According to the non-circular gear design method, the specific mathematical model of the non-circular gear tooth profile is obtained through the tooth profile envelope equation, the tooth profile curve equation of the circular cutter and the tooth profile meshing equation, and the accuracy of the obtained non-circular gear tooth profile is further improved.
According to the non-circular gear design method, the non-circular gear is machined through the circular cutter, the non-circular gear with the concave pitch curve can be designed, the non-circular gear with the gear teeth meshed with the gear teeth can be designed, and the non-circular gear design method is shown in figure 8.
The non-circular gear design method provided by the application also fixes the node and the circle center of the circular cutter, so that the circular cutter can retreat from the same direction, the cutter retreating interference is avoided, and the operation is more convenient and faster when the non-circular gear design method is applied to gear machining.
Example 2
The embodiment 2 of the present application also provides a non-circular gear design method, and different from embodiment 1, in this embodiment, a method for establishing a two-dimensional motion relationship between a non-circular gear and a circular cutter may be:
fixed coordinate system S 0 (O 0 -x 0 -y 0 ) In, y 0 The positive direction of the axis is opposite to that in example 1, i.e. by said x 0 Positive direction of axis is anticlockwiseDirection of rotation by 90 deg. is y 0 The positive direction of the axis.
Moving coordinate system S 1 (O 1 -x 1 -y 1 ) In, y 1 The positive direction of the axis is opposite to that in example 1, i.e. by said x 1 The positive direction of the axis rotates counterclockwise by 90 degrees in the direction y 1 The positive direction of the axis.
Moving coordinate system S p (O p -x p -y p ) In, y p The positive axial direction is opposite to that in example 1, namely, by the x p The positive direction of the shaft rotates clockwise by 90 degrees in the direction y p The positive direction of the axis.
Moving coordinate system S 2 (O 2 -x 2 -y 2 ) In, y 2 The positive axial direction is opposite to that in example 1, namely, by the x 2 The positive direction of the shaft rotates clockwise by 90 degrees in the direction y 2 The positive direction of the axis.
Those skilled in the art can know that the specific method and principle for establishing the two-dimensional motion relationship between the non-circular gear and the circular cutter by using the method and subsequently obtaining the tooth profile curve equation of the non-circular gear are the same as those in embodiment 1, and are not described herein again.

Claims (3)

1. A method of designing a non-circular gear, the method comprising:
s110, establishing a two-dimensional motion relation between a non-circular gear and a circular cutter;
s120, obtaining a tooth profile envelope equation of the non-circular gear according to the two-dimensional motion relation;
s130, establishing a tooth profile curve equation of the circular cutter, and obtaining a tooth profile meshing equation of the non-circular gear and the circular cutter;
s140, obtaining a tooth profile curve equation of the non-circular gear according to the tooth profile envelope equation, the tooth profile curve equation of the circular cutter and the tooth profile meshing equation;
the step S110 includes:
at the initial moment, the node P of the pitch curve of the non-circular gear and the pitch curve of the circular cutter, and the rotation center O of the non-circular gear g And the center O of the circular cutter c On the same straight line with the center O of the circular cutter c Is an origin O 0 To do so by
Figure FDA0004055181960000011
In the direction of x 0 Positive axial direction with said x 0 The positive direction of the shaft rotates clockwise by 90 degrees in the direction of y 0 In the positive axial direction, a fixed coordinate system S is established on the end face of the circular cutter 0 (O 0 -x 0 -y 0 ) And said node P is in the coordinate system S 0 The coordinates in (1) are fixed and unchanged;
at the initial moment, the center O of the circular cutter is used c Is an origin O 1 To do so by
Figure FDA0004055181960000012
In the direction of x 1 Positive axial direction with said x 1 The positive direction of the shaft rotates clockwise by 90 degrees in the direction of y 1 In the positive axial direction, a moving coordinate system S is established on the end surface of the circular cutter 1 (O 1 -x 1 -y 1 ) Said coordinate system S 1 Origin O of 1 Fixed, x 1 Axis and y 1 The shaft rotates along with the circular cutter;
at the initial moment, the rotation center O of the non-circular gear is used g Is an origin O p To do so by
Figure FDA0004055181960000013
In the direction of x p Positive axial direction with said x p The positive direction of the shaft rotates anticlockwise by 90 degrees in the direction of y p In the positive axial direction, a moving coordinate system S is established on the end face of the non-circular gear p (O p -x p -y p ) Said coordinate system S p Origin O of p 、x p Axis and y p The shaft makes translational motion along with the non-circular gear;
at the initial moment, the rotation center O of the non-circular gear is used g Is an origin O 2 To do so by
Figure FDA0004055181960000014
In the direction of x 2 Positive axial direction with said x 2 The positive direction of the shaft rotates anticlockwise by 90 degrees in the direction of y 2 In the positive axial direction, a moving coordinate system S is established on the end face of the non-circular gear 2 (O 2 -x 2 -y 2 ) Said coordinate system S 2 Origin O of 2 The coordinate system S moves in translation along with the non-circular gear 2 X of 2 Axis and y 2 The shaft does self-rotation movement and translation movement along with the non-circular gear;
the step S120 includes:
establishing a pitch curve equation of the non-circular gear
Figure FDA0004055181960000021
Wherein it is present>
Figure FDA0004055181960000022
Is the transmission ratio of the non-circular gear>
Figure FDA0004055181960000023
For node P in said coordinate system S 2 Is radially based on>
Figure FDA0004055181960000024
And said x 2 The included angle of the shaft, a, is the center distance of the non-circular gear pair;
establishing a coordinate system S 1 Conversion to coordinate system S 0 Transformation relation M of 01 Wherein S is 0 ＝M 01 S 1 ，
Figure FDA0004055181960000025
φ 1 Is a corner of the round knife and>
Figure FDA0004055181960000026
Figure FDA0004055181960000027
wherein it is present>
Figure FDA0004055181960000028
Is equal to the initial moment>
Figure FDA0004055181960000029
Value r g The pitch radius of the circular cutter;
establishing a coordinate system S 0 Conversion to coordinate system S p Transformation relation M of p0 Wherein S is p ＝M p0 S 0 ，
Figure FDA00040551819600000210
Mu is a tangent line of the pitch curve of the non-circular gear and the pitch curve of the circular cutter at a node P and the radial direction of the non-circular gear>
Figure FDA00040551819600000211
And is/are included angle of>
Figure FDA00040551819600000212
Establishing a coordinate system S p Conversion to coordinate system S 2 Transformation relation M of 2p Wherein S is 2 ＝M 2p S p ，
Figure FDA00040551819600000213
φ 2 Is the angle of rotation of the non-circular gear and->
Figure FDA00040551819600000214
Figure FDA00040551819600000215
Wherein, mu 0 Is equal toμ value at the initial time;
according to the pitch curve equation of the non-circular gear
Figure FDA00040551819600000216
Transformation relation M 01 The transformation relation M p0 And the transformation relation M 2p Determining the coordinate system S of the circular cutter tooth profile 1 Coordinate system S converted into the non-circular gear tooth profile 2 Transformation relation M of 21 Wherein S is 2 ＝M 21 S 1 ，M 21 ＝M 2p M p0 M 01 ；
According to the transformation relation M 21 Obtaining a tooth profile envelope equation of the non-circular gear:
Figure FDA0004055181960000031
wherein (x) 1 ，y 1 ) For the circular tool tooth profile in the coordinate system S 1 The coordinates of (a) are (b), (x) 2 ，y 2 ) For the non-circular gear tooth profile in the coordinate system S 2 Coordinates of (5);
the step S130 includes:
establishing a tooth profile curve equation of the circular cutter:
Figure FDA0004055181960000032
wherein r is b Is the base radius of the circular tool u s Is the involute variable parameter delta of the circular cutter 0 The involute initial point angle of the circular cutter is obtained;
the step S130 further includes:
when the coordinate of the meshing point of the circular cutter tooth profile and the non-circular gear tooth profile in the coordinate system S1 is (x) 1 ，y 1 ) And then, obtaining a tooth profile meshing equation of the non-circular gear and the circular cutter:
Figure FDA0004055181960000033
where Ψ is equal to the pressure angle of the circular-cutter pitch circle, γ is the tangent at the point of engagement of the circular-cutter tooth profile and the non-circular-gear tooth profile and x 1 Angle of axis and->
Figure FDA0004055181960000034
The meshing point of the circular cutter tooth profile and the non-circular gear tooth profile is in the coordinate system S 1 The track equation in (2) is a tooth profile curve equation of the circular cutter, and the meshing point of the tooth profile of the circular cutter and the tooth profile of the non-circular gear is in the coordinate system S 2 The trajectory equation in (1) is a tooth profile curve equation of the non-circular gear;
the step S140 includes:
obtaining a tooth profile curve equation of the non-circular gear according to the tooth profile envelope equation, the tooth profile curve equation of the circular cutter and the tooth profile meshing equation:
Figure FDA0004055181960000035
2. the non-circular gear design method of claim 1, wherein the two-dimensional motion relationship comprises: the non-circular gear is wound around a rotation center O on the end surface thereof g The circular cutter rotates around the center O of a circle on the end surface of the non-circular gear and simultaneously translates c And the end face of the non-circular gear and the end face of the circular cutter are positioned on the same plane in a self-rotation motion mode.
3. The non-circular gear design method according to claim 1, wherein the step S110 further comprises:
at the initial moment, the node P of the pitch curve of the non-circular gear and the pitch curve of the circular cutter and the revolution center O of the non-circular gear g And the center O of the circular cutter c On the same straight line with the center O of the circular cutter c Is an origin O 0 To do so by
Figure FDA0004055181960000041
In the direction of x 0 Positive axial direction with said x 0 The positive direction of the shaft rotates anticlockwise by 90 degrees in the direction of y 0 In the positive axial direction, a fixed coordinate system S is established on the end face of the circular cutter 0 (O 0 -x 0 -y 0 ) And said node P is in the coordinate system S 0 The coordinates in (1) are fixed and unchanged;
at the initial moment, the center O of the circular cutter is used c Is an origin O 1 To do so by
Figure FDA0004055181960000042
In the direction of x 1 Positive axial direction with said x 1 The positive direction of the shaft rotates anticlockwise by 90 degrees in the direction of y 1 In the positive axial direction, a moving coordinate system S is established on the end surface of the circular cutter 1 (O 1 -x 1 -y 1 ) Said coordinate system S 1 Origin O of 1 Fixed, x 1 Axis and y 1 The shaft rotates along with the circular cutter;
at the initial moment, the rotation center O of the non-circular gear is used g Is an origin O p To do so by
Figure FDA0004055181960000043
In the direction of x p Positive axial direction with said x p The positive direction of the shaft rotates clockwise by 90 degrees in the direction of y p In the positive axial direction, a moving coordinate system S is established on the end face of the non-circular gear p (O p -x p -y p ) Said coordinate system S p Origin O of p 、x p Axis and y p The shaft makes translational motion along with the non-circular gear;
at the initial moment, the rotation center O of the non-circular gear is used g Is an origin O 2 To do so by
Figure FDA0004055181960000044
In the direction of x 2 Positive axial direction with said x 2 The positive direction of the shaft rotates clockwise by 90 degrees in the direction of y 2 In the positive axial direction, a moving coordinate system S is established on the end face of the non-circular gear 2 (O 2 -x 2 -y 2 ) Said coordinate system S 2 Origin O of 2 The coordinate system S moves in translation along with the non-circular gear 2 X of 2 Axis and y 2 The shaft makes self-rotation movement and translation movement along with the non-circular gear. />
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