CN115186493A - Accurate modeling method for small-modulus involute cylindrical gear - Google Patents

Accurate modeling method for small-modulus involute cylindrical gear Download PDF

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CN115186493A
CN115186493A CN202210828692.3A CN202210828692A CN115186493A CN 115186493 A CN115186493 A CN 115186493A CN 202210828692 A CN202210828692 A CN 202210828692A CN 115186493 A CN115186493 A CN 115186493A
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周长江
夏宁伟
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Hunan University
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Abstract

The invention discloses a small-modulus involute cylindrical gear accurate modeling method, which comprises the following steps: the method comprises the steps of constructing a cutter-tooth profile mapping equation, solving the coordinates of involute tooth profile points, solving the coordinates of tooth root circular arc points, solving the coordinates of tooth addendum circular arc points, and performing programming calculation and software generation of a geometric model. The invention has the beneficial effects that: the precision of the geometric model of the small-modulus involute cylindrical gear is effectively improved, the three-dimensional coordinates of the target tooth surface control points are calculated and generated by programming software according to basic design parameters, the point cloud density is controllable, the method is suitable for various CAD and CAE software, the modeling efficiency can be further improved after the matched plug-in is written, and the industrial utility is directly exerted; the geometric model of the small module gear provided by the invention can be used for kinetic analysis, contact analysis, thermodynamic analysis and the like of the small module gear, can also be used for three-dimensional modeling and numerical simulation of a small module transmission system, effectively improves the related calculation precision, and promotes the technical development of the field of microminiature precision transmission.

Description

Accurate modeling method for small-modulus involute cylindrical gear
Technical Field
The invention relates to the technical field of gear modeling, in particular to a small-modulus involute cylindrical gear accurate modeling method.
Background
The small module (normal module is less than 1 mm) gear is used as a core element of a tiny precision transmission system and is widely applied in the fields of aerospace, instruments and meters, intelligent equipment and the like. Engineers often need to use three-dimensional models of mechanical parts to assist in further form design or in numerical simulations during product development. Therefore, an accurate small module gear three-dimensional model is established, and the key is the influence on the performance analysis and the optimization design of the small module transmission equipment.
At present, few three-dimensional modeling methods for small-modulus involute cylindrical gears exist in China, and most technicians directly substitute a modulus smaller than 1 millimeter into a normal-modulus gear program to generate a required gear model. In fact, the tooth profile of a normal-modulus gear follows the standard basic rack tooth profile of cylindrical gears for GB/T1356-2001 general machinery and heavy machinery, while the tooth profile of a small-modulus gear follows the basic tooth profile of GB/T2362-1990 small-modulus involute cylindrical gears, and the root fillet coefficient and the top clearance coefficient of the two basic tooth profiles are different, so that the existing modeling method ignores the principle difference of the small-modulus gear and the normal-modulus gear in the tooth profile design level.
In addition, the existing involute cylindrical gear modeling methods can be classified into a modeling method based on an ideal tooth profile and a modeling method based on a generating machining principle. The former modeling principle is simple and is mostly used for appearance representation in basic modeling software, and the latter model is more consistent with the actual tooth profile of most metal material gears and is mostly used for accurate slaving performance evaluation or finite element simulation analysis. In the three-dimensional modeling of the cylindrical helical gear, the mapping change from the normal tooth profile to the end tooth profile is considered in the two methods, but the situation that the circular arc of the tooth root is stretched into the elliptical arc in the mapping is ignored. In addition, in the hobbing of a small-module gear, a full-cutting type hob is used in some cases, in which the bottom edge of the hob is involved in cutting to form an addendum profile, and the dedendum fillet of the hob is also machined in accordance with the addendum fillet. In summary, the existing involute cylindrical gear modeling method cannot be completely applied to the small-modulus involute cylindrical gear, and the obtained result has necessary errors with the real tooth profile.
Disclosure of Invention
The invention discloses a small-modulus involute cylindrical gear accurate modeling method which is characterized in that a mapping relation between a hobbing cutter and a tooth profile equation is established based on a generating machining principle and a coordinate transformation method, the special characteristics of a top clearance, a tooth root and a tooth top fillet of a small-modulus gear and the arc flat rate change of an end face tooth root of a helical gear are fully considered, an accurate geometric model of the small-modulus involute cylindrical gear is established, and a good analysis basis is provided for the subsequent performance research and optimization design of the small-modulus gear and a microminiature transmission system, so that the technical problems related to the background technology can be effectively solved.
In order to achieve the purpose, the technical scheme of the invention is as follows:
a small module involute cylindrical gear accurate modeling method comprises the following steps:
analyzing the relative position and motion rule of a cutter and a cut workpiece in the hobbing process, and constructing a mapping equation of a cutter surface point to a tooth profile surface point;
solving a mapping equation of the side edge of the hob to the involute tooth profile, and calculating coordinates of involute tooth profile points;
solving a mapping equation of the top edge of the hob to the arc of the tooth root, and calculating the coordinates of the arc point of the tooth root;
solving a mapping equation of the hob bottom edge to the addendum arc, and calculating addendum arc point coordinates;
establishing a complete end face tooth profile curve equation including an involute tooth profile, a tooth root circular arc and a tooth top circular arc based on MATLAB (matrix laboratory) mathematical software, and sweeping an end face tooth profile tooth line along a spiral angle to obtain a three-dimensional surface equation of the small-modulus involute cylindrical gear;
and sixthly, inputting gear design parameters based on the obtained three-dimensional surface equation of the small-modulus involute cylindrical gear, calculating to obtain three-dimensional coordinates of gear surface points, importing the three-dimensional coordinates into three-dimensional software CATIA (computer-graphics aided three-dimensional Interactive application) to generate a tooth surface point cloud, and performing spline curve sweeping and sewing operations to obtain a three-dimensional geometric model of the small-modulus involute cylindrical gear.
As a preferable improvement of the present invention, the step one specifically comprises the following steps:
according to the generating machining principle, in an end face view of a workpiece to be cut, a hobbing cutter and the workpiece to be cut do rack-gear forced meshing motion, namely, a cutter pitch line does pure rolling relative to a tooth blank reference circle, and a workpiece coordinate system XOY and a cutter coordinate system X are constructed 1 PY 1 Let the longitudinal axes OY and PY 1 The initial position is when parallel;
as the tool rotates around the workpiece to be cut
Figure BDA0003747298840000021
At radian, the tool pitch line PY 1 Contact point N with reference circle of gear blank in tool coordinate system X 1 PY 1 From the origin P (0, 0) to a point
Figure BDA0003747298840000022
The engagement point of the tool and the target tooth profile is denoted as M (X, y) in the workpiece coordinate system XOY and is denoted as X (X, y) in the tool coordinate system X 1 PY 1 Denoted as M' (x) 1 ,y 1 ) Then point M (M') is in the tool coordinate system X 1 PY 1 And the object coordinate system XOY:
Figure BDA0003747298840000023
wherein r is a reference circle radius, X is the position of the engagement point of the tool and the target tooth profile on the X axis in the workpiece coordinate system XOY, Y is the position of the engagement point of the tool and the target tooth profile on the Y axis in the workpiece coordinate system XOY, and X is the radius of the reference circle 1 For the engagement point of the tool with the target tooth profile in the tool coordinate system X 1 PY 1 In X 1 Position on axis, y 1 For the tool-to-target tooth-profile engagement point in the tool coordinate system X 1 PY 1 Middle Y 1 The position on the shaft is such that,
Figure BDA0003747298840000031
the arc through which the tool is rotated around the workpiece being cut.
As a preferable improvement of the present invention, the second step specifically comprises:
analyzing the forming effect of a single-side cutting edge of a hob tooth on a single-side tooth profile of a gear, and dividing the cutting edge into a top edge, a side edge and a bottom edge, wherein the top edge generates a tooth root circular arc in a cutting mode, the side edge generates an involute tooth profile in a cutting mode, and the bottom edge generates an tooth top circular arc in a cutting mode;
shaft PY is used to produce involute tooth profile when hob side edge is cutting 1 The coordinate of the intersection point of the side edge and the pitch line is (0 0 ) Mesh point M' (x) 1 ,y 1 ) In the tool coordinate system X 1 PY 1 The position of (1) with the angle of rotation
Figure BDA0003747298840000035
The coordinate function of the changes is:
Figure BDA0003747298840000032
wherein α is the pressure angle, y 0 Is the intersection point of the side edge and the pitch line in the tool coordinate system X 1 PY 1 Middle Y 1 A position on the shaft;
coordinate parameter x of mesh point M 1 、y 1 Substituting in formula (1) to obtain the coordinate of involute tooth profile point M in static coordinate system XOY.
As a preferable improvement of the present invention, the third step specifically includes:
when the hob top edge cuts to generate a tooth root circular arc and the spiral angle of the target gear is not 0, the projection of the hob teeth on the end face of the tooth blank is elongated along a pitch line, the projection of the tool nose fillet is an elliptical arc, and therefore the projection is expressed by an elliptical parameter equation:
Figure BDA0003747298840000033
in the formula, x top 、y top Respectively representing the tool coordinate system X 1 PY 1 The transverse and longitudinal distances of the lower point M' from the center C of the ellipse, a,b is the length of the transverse half shaft and the longitudinal half shaft of the ellipse respectively, gamma is the parameter of the ellipse angle, and the radius of the tool nose fillet is rho 0 With a helix angle β, a = ρ 0 ，b＝ρ 0 /cosβ；
Setting the center C of the ellipse in the tool coordinate system X 1 PY 1 Has the coordinate of (x) c ,y c ) Then mesh point M' (x) 1 ,y 1 ) In the tool coordinate system X 1 PY 1 The coordinates in (1) are:
Figure BDA0003747298840000034
the tooth crest height, the top clearance and the displacement are considered to obtain the ellipse center C in the cutter coordinate system X 1 PY 1 Coordinates (c) of (a):
Figure BDA0003747298840000041
in the formula, h a * Is the coefficient of crest height, c * Is the coefficient of the head clearance, x is the coefficient of deflection, m is the modulus of the end face, rho 0 * Root fillet coefficient, m n Is the normal modulus, α t Is an end face pressure angle;
the over-engagement point M' is normal to the arc of the tooth root, and the axis PY 1 Meet at a point
Figure BDA0003747298840000042
Thereby the coordinates and the rotation angle of the meshing point M' are obtained
Figure BDA0003747298840000043
Is associated with x 1 、y 1 And (3) substituting the formula (1) to obtain the coordinate of the tooth root circular arc point M in the workpiece coordinate system XOY.
As a preferred improvement of the invention, let c be as defined in GB/T2362-1990 * ＝0.35，ρ 0 * =0.2, the remaining coefficient values being the same as the normal-modulus gear standard.
As a preferable improvement of the present invention, the step four specifically includes:
when the bottom edge of the hob is used for cutting to generate an addendum circular arc and the helix angle of the target gear is not 0, the projection of the hob teeth on the end surface of the gear blank is elongated along a pitch line, and the projection of the root fillet of the hob teeth is an elliptic arc, so that the projection is also expressed by an elliptic parameter equation:
Figure BDA0003747298840000044
wherein x is root 、y root Respectively representing the tool coordinate system X 1 PY 1 Lower mesh point M' spaced from ellipse center C 1 The distance a and the distance b are respectively the length of a transverse half shaft and a longitudinal half shaft of the ellipse, gamma is an ellipse angle parameter, and the radius of a cutter root fillet is also rho 0 Then a = ρ 0 ，b＝ρ 0 /cosβ；
Setting the center C of an ellipse 1 In the tool coordinate system X 1 PY 1 Has the coordinate of (x) c1 ,y c1 ) Then mesh point M' (x) 1 ,y 1 ) In the tool coordinate system X 1 PY 1 The coordinates in (1) are:
Figure BDA0003747298840000045
the center C of the ellipse is obtained by considering the tooth crest height and the displacement 1 In the tool coordinate system X 1 PY 1 Coordinates (c) of (a):
Figure BDA0003747298840000046
the over-meshing point M' is taken as the normal of the tooth crest circular arc and the shaft PY 1 Cross over at the point
Figure BDA0003747298840000047
Thereby the coordinates and the rotation angle of the meshing point M' are obtained
Figure BDA0003747298840000048
Is associated with x 1 、y 1 Substituting the formula (1) to obtain the coordinate of the tooth top circular arc point M in the workpiece coordinate system XOY.
As a preferable improvement of the present invention, in step six, the gear design parameters include the number of teeth, the module, the pressure angle, the tooth width, the helix angle, the profile modification coefficient, the root fillet coefficient, the crest height coefficient, and the tip clearance coefficient.
The invention has the following beneficial effects:
1. the precision of the geometric model of the small-modulus involute cylindrical gear is effectively improved, the three-dimensional coordinates of the target tooth surface control points are calculated and generated by programming software according to basic design parameters, the point cloud density is controllable, the method is suitable for various CAD and CAE software, the modeling efficiency can be further improved after the matched plug-in is written, and the industrial utility is directly exerted;
2. the geometric model of the small module gear provided by the invention can be used for dynamics analysis, contact analysis, thermodynamic analysis and the like of the small module gear, and also can be used for three-dimensional modeling and numerical simulation of a small module transmission system, the related calculation precision is effectively improved, and the technical development in the field of microminiature precision transmission is promoted.
Drawings
In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the drawings needed to be used in the description of the embodiments are briefly introduced below, it is obvious that the drawings in the following description are only some embodiments of the present invention, and other drawings can be obtained by those skilled in the art without inventive efforts, wherein:
FIG. 1 is an initial position view of a tool relative to a workpiece to be cut during machining according to the present invention;
FIG. 2 is a view showing the position of the tool after rotating with respect to the workpiece to be cut by a radian;
FIG. 3 is a sectional view of the hob teeth of the present invention;
FIG. 4 is a graph of the side edge points of the hob of the present invention;
FIG. 5 is a graph of the coordinates of the point of the top edge of the hob of the present invention;
FIG. 6 is a graph of the coordinates of the bottom edge points of the hob in accordance with the present invention;
FIG. 7 is a single side two dimensional end profile view of the present invention;
FIG. 8 is a diagram of the envelope of the hob of the present invention;
FIG. 9 is a comparison of the modeling method provided by the present invention with the profile modeling of an existing method;
FIG. 10 is a graph of tooth profiles for the present invention with a respective index of-0.5;
FIG. 11 is a graph of tooth profiles for the present invention with respective index values of 0;
FIG. 12 is a graph of tooth profiles for the profile modification coefficients of 0.5, respectively, according to the present invention;
FIG. 13 is a three-dimensional tooth surface map in the mathematical software MATLAB of the present invention;
FIG. 14 is a cloud point diagram of the three-dimensional CATIA software of the present invention;
FIG. 15 is a model diagram of a small module involute cylindrical gear in the three-dimensional software CATIA of the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be obtained by a person skilled in the art without making any creative effort based on the embodiments in the present invention, belong to the protection scope of the present invention.
It should be noted that all directional indicators (such as up, down, left, right, front, back \8230;) in the embodiments of the present invention are only used to explain the relative positional relationship between the components, the motion situation, etc. in a specific posture (as shown in the attached drawings), and if the specific posture is changed, the directional indicator is changed accordingly.
In addition, the descriptions related to "first", "second", etc. in the present invention are only for descriptive purposes and are not to be construed as indicating or implying relative importance or implicitly indicating the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one such feature. In the description of the present invention, "a plurality" means at least two, e.g., two, three, etc., unless specifically limited otherwise.
In the present invention, unless otherwise expressly stated or limited, the terms "connected," "secured," and the like are to be construed broadly, and for example, "secured" may be a fixed connection, a removable connection, or an integral part; can be mechanically or electrically connected; they may be directly connected or indirectly connected through intervening media, or they may be interconnected within two elements or in a relationship where two elements interact with each other unless otherwise specifically limited. The specific meanings of the above terms in the present invention can be understood by those skilled in the art according to specific situations.
In addition, the technical solutions in the embodiments of the present invention may be combined with each other, but it must be based on the realization of the technical solutions by those skilled in the art, and when the technical solutions are contradictory to each other or cannot be realized, such a combination of the technical solutions should not be considered to exist, and is not within the protection scope of the present invention.
The invention provides a small-modulus involute cylindrical gear accurate modeling method, which comprises the following steps:
analyzing the relative position and motion rule of a cutter and a cut workpiece in the hobbing process, and constructing a mapping equation of the surface point of the cutter to the surface point of the tooth profile, wherein the mapping equation specifically comprises the following steps:
according to the generating machining principle, in an end face view of a workpiece to be cut, a hobbing cutter and the workpiece to be cut do rack-gear forced meshing motion, namely, a cutter pitch line does pure rolling relative to a tooth blank reference circle, and a workpiece coordinate system XOY and a cutter coordinate system X are constructed 1 PY 1 Let the longitudinal axes OY and PY 1 The parallel is the initial position, as shown in fig. 1, where z is the number of teeth, m is the module of the end face, and r is the radius of the reference circle.
As the tool rotates around the workpiece to be cut
Figure BDA0003747298840000061
At the time of radian, the cutter sectionLine PY 1 Contact point N with reference circle of gear blank in tool coordinate system X 1 PY 1 From the origin P (0, 0) to a point
Figure BDA0003747298840000062
As shown in particular in fig. 2. The engagement point of the tool and the target tooth profile is denoted as M (X, y) in the workpiece coordinate system XOY and is denoted as X (X, y) in the tool coordinate system X 1 PY 1 Denoted as M' (x) 1 ,y 1 ) Then point M (M') is in the tool coordinate system X 1 PY 1 And the object coordinate system XOY:
Figure BDA0003747298840000071
wherein r is a reference circle radius, X is the position of the engagement point of the tool and the target tooth profile on the X axis in the workpiece coordinate system XOY, Y is the position of the engagement point of the tool and the target tooth profile on the Y axis in the workpiece coordinate system XOY, and X is the radius of the reference circle 1 For the engagement point of the tool with the target tooth profile in the tool coordinate system X 1 PY 1 In X 1 Position on the axis, y 1 For the tool-to-target tooth-profile engagement point in the tool coordinate system X 1 PY 1 Middle Y 1 The position on the shaft is such that,
Figure BDA0003747298840000072
the arc through which the tool rotates about the workpiece being cut.
Step two, solving a mapping equation of the side edge of the hob to the involute tooth profile, and calculating coordinates of involute tooth profile points, wherein the method specifically comprises the following steps:
analyzing the forming effect of the one-side cutting edge of the hob tooth on the one-side tooth profile of the gear, and dividing the cutting edge into a top edge, a side edge and a bottom edge as shown in fig. 3, wherein the top edge generates a tooth root circular arc in cutting, the side edge generates an involute tooth profile in cutting, and the bottom edge generates an tooth top circular arc in cutting. Where the involute profile is the main portion of the overall tooth profile, calculation is preferred.
When the side edge of the hob is cut to generate an involute tooth profile, the meshing point M' (x) 1 ,y 1 ) In-motion coordinate systemX 1 PY 1 Is shown in FIG. 4, the axis PY 1 The coordinate of the intersection point of the side edge and the pitch line is (0 0 ) Mesh point M' (x) 1 ,y 1 ) In the tool coordinate system X 1 PY 1 The position of (1) with the angle of rotation
Figure BDA0003747298840000073
The coordinate function that varies is:
Figure BDA0003747298840000074
wherein α is the pressure angle, y 0 The intersection point of the side edge and the pitch line is in the tool coordinate system X 1 PY 1 Middle Y 1 A position on the shaft;
coordinate parameter x of mesh point M 1 、y 1 And (3) substituting the involute profile point M in the formula (1) to obtain the coordinate of the involute profile point M in a static coordinate system XOY.
Step three, solving a mapping equation of the top edge of the hob to the arc of the tooth root, and calculating the coordinates of the arc point of the tooth root, wherein the method specifically comprises the following steps:
mesh point M' (x) when the hob top edge is cutting to create a root arc 1 ,y 1 ) In a moving coordinate system X 1 PY 1 When the helix angle of the target gear is not 0, the projection of the hob teeth on the end face of the gear blank is elongated along the pitch line, the projection of the tip fillet is an elliptical arc, and therefore, the position is expressed by an elliptical parameter equation:
Figure BDA0003747298840000075
in the formula, x top 、y top Respectively representing the tool coordinate system X 1 PY 1 The transverse and longitudinal distances from the lower point M' to the center C of the ellipse, a and b are the lengths of the transverse and longitudinal semiaxes of the ellipse respectively, gamma is the parameter of the ellipse angle, and the radius of the tool nose fillet is rho 0 With a helix angle β, a = ρ 0 ，b＝ρ 0 /cosβ；
Setting the center C of the ellipse in the tool coordinate system X 1 PY 1 Has the coordinate of (x) c ,y c ) Then mesh point M' (x) 1 ,y 1 ) In the tool coordinate system X 1 PY 1 The coordinates in (1) are:
Figure BDA0003747298840000081
the tooth crest height, the top clearance and the displacement are considered to obtain the ellipse center C in the cutter coordinate system X 1 PY 1 Coordinates (c) of (a):
Figure BDA0003747298840000082
in the formula, h a * Is the crest coefficient of tooth, c * Is the coefficient of the head clearance, x is the coefficient of deflection, m is the modulus of the end face, rho 0 * Root fillet coefficient, m n Is the normal modulus, α t Is the end face pressure angle. According to the provisions of GB/T2362-1990, let c * ＝0.35，ρ 0 * =0.2, the remaining coefficient values being the same as the normal-modulus gear standard.
The over-engagement point M' being normal to the arc of the tooth root, with the axis PY 1 Cross over at the point
Figure BDA0003747298840000083
Thereby the coordinates and the rotation angle of the meshing point M' are obtained
Figure BDA0003747298840000084
Is associated with x 1 、y 1 Substituting the formula (1) to obtain the coordinate of the tooth root circular arc point M in the workpiece coordinate system XOY.
Solving a mapping equation of the hob bottom edge to the addendum arc, and calculating the addendum arc point coordinate, which specifically comprises the following steps:
mesh point M' (x) when the hob bottom edge is cutting to create a top radius 1 ,y 1 ) In a moving coordinate system X 1 PY 1 In the position shown in fig. 6, when the target gear is in a spiralWhen the angle is not 0, the projection of the hob cutter tooth on the end surface of the gear blank is elongated along a pitch line, the projection of the cutter root fillet is an elliptical arc, and therefore, the projection is also expressed by an elliptical parameter equation:
Figure BDA0003747298840000085
wherein x is root 、y root Respectively representing the tool coordinate system X 1 PY 1 Lower mesh point M' spaced from ellipse center C 1 The distance a and the distance b are respectively the length of a transverse half shaft and a longitudinal half shaft of the ellipse, gamma is an ellipse angle parameter, and the radius of a cutter root fillet is also rho 0 Then a = ρ 0 ，b＝ρ 0 /cosβ；
Setting the center C of an ellipse 1 In the tool coordinate system X 1 PY 1 Has the coordinate of (x) c1 ,y c1 ) Then mesh point M' (x) 1 ,y 1 ) In the tool coordinate system X 1 PY 1 The coordinates in (1) are:
Figure BDA0003747298840000086
the center C of the ellipse is obtained by considering the tooth crest height and the displacement 1 In the tool coordinate system X 1 PY 1 Coordinates (c) of (a):
Figure BDA0003747298840000091
the over-meshing point M' is taken as the normal of the tooth crest arc and the shaft PY 1 Meet at a point
Figure BDA0003747298840000094
Thereby the coordinates and the rotation angles of the meshing point M' are measured
Figure BDA0003747298840000092
Is associated with x 1 、y 1 Substituting the formula (1) to obtain the coordinate of the tooth top circular arc point M in the workpiece coordinate system XOY.
Establishing a complete end face tooth profile curve equation including an involute tooth profile, a tooth root circular arc and a tooth top circular arc based on mathematical software MATLAB, and sweeping an end face tooth profile tooth line along a spiral angle to obtain a small-modulus involute cylindrical gear three-dimensional surface equation;
and sixthly, inputting gear design parameters based on the obtained three-dimensional surface equation of the small-modulus involute cylindrical gear, calculating to obtain three-dimensional coordinates of gear surface points, importing the three-dimensional coordinates into three-dimensional software CATIA (computer-graphics aided three-dimensional Interactive application) to generate a tooth surface point cloud, and performing spline curve sweeping and sewing operations to obtain a three-dimensional geometric model of the small-modulus involute cylindrical gear.
Specifically, the gear design parameters include the number of teeth, the module, the pressure angle, the tooth width, the helix angle, the profile factor, the root fillet factor, the tip height factor, and the tip clearance factor.
The precise modeling method of the small-modulus involute cylindrical gear provided by the invention is verified through the embodiment 1.
Example 1
According to the design parameters in the table 1, the method provided by the invention is used for establishing a geometric model of the small-modulus involute cylindrical gear.
TABLE 1 design parameters for small modulus involute cylindrical gears
Figure BDA0003747298840000093
The one-sided two-dimensional face profile calculated and generated in MATLAB is shown in fig. 7.
The hob paths were calculated and generated, enveloped and verified for correctness of the resulting tooth profile as shown in fig. 8.
The tooth profile obtained by the conventional constant modulus gear modeling method is obviously different from that obtained by the conventional constant modulus gear modeling method at the root and the tip of a tooth as shown in fig. 9.
The tooth profiles when the displacement coefficients of the driving wheel are-0.5, 0 and 0.5 are respectively shown in figures 10-12, and it can be seen that when the displacement is large enough, the bottom edge of the hob does not participate in cutting, and the circular arc of the tooth top disappears.
Further generation of three-dimensional tooth surfaces in MATLAB is shown in fig. 13.
The coordinates of the tooth surface points are imported into the CATIA to generate point clouds as shown in FIG. 14, and finally, a three-dimensional model is obtained as shown in FIG. 15.
The invention has the following beneficial effects: further generation of three-dimensional tooth surfaces in MATLAB is shown in fig. 10.
The tooth surface point coordinates are led into the CATIA generated point cloud as shown in figure 11, and finally a three-dimensional model is obtained as shown in figure 12, so that the accurate modeling method of the small-modulus involute cylindrical gear provided by the invention is adopted, the establishment of the accurate geometric model of the small-modulus involute cylindrical gear is realized, and a good analysis basis is provided for the subsequent performance research and optimization design of the small-modulus gear and the microminiature transmission system.
1. The precision of the geometric model of the small-modulus involute cylindrical gear is effectively improved, the three-dimensional coordinates of the target tooth surface control points are calculated and generated by programming software according to basic design parameters, the point cloud density is controllable, the method is suitable for various CAD and CAE software, the modeling efficiency can be further improved after the method is written into a matched plug-in, and the industrial utility can be directly exerted;
2. the geometric model of the small module gear provided by the invention can be used for kinetic analysis, contact analysis, thermodynamic analysis and the like of the small module gear, can also be used for three-dimensional modeling and numerical simulation of a small module transmission system, effectively improves the related calculation precision, and promotes the technical development of the field of microminiature precision transmission.
While embodiments of the invention have been described above, it is not intended to be limited to the details shown herein, and to the particular embodiments shown, but it is to be understood that all changes and modifications that come within the spirit and scope of the invention are desired to be protected by the teachings herein.

Claims (7)

1. A small-modulus involute cylindrical gear accurate modeling method is characterized by comprising the following steps:
analyzing the relative position and motion rule of a cutter and a cut workpiece in the hobbing process, and constructing a mapping equation of a cutter surface point to a tooth profile surface point;
solving a mapping equation of the side edge of the hob to the involute tooth profile, and calculating coordinates of involute tooth profile points;
solving a mapping equation of the top edge of the hob to the arc of the tooth root, and calculating the coordinates of the arc point of the tooth root;
solving a mapping equation of the hob bottom edge to the addendum arc, and calculating addendum arc point coordinates;
establishing a complete end face tooth profile curve equation including an involute tooth profile, a tooth root circular arc and a tooth top circular arc based on MATLAB (matrix laboratory) mathematical software, and sweeping an end face tooth profile tooth line along a spiral angle to obtain a three-dimensional surface equation of the small-modulus involute cylindrical gear;
and sixthly, inputting gear design parameters based on the obtained three-dimensional surface equation of the small-modulus involute cylindrical gear, calculating to obtain three-dimensional coordinates of gear surface points, importing the three-dimensional coordinates into three-dimensional software CATIA (computer-graphics aided three-dimensional Interactive application) to generate a tooth surface point cloud, and performing spline curve sweeping and sewing operations to obtain a three-dimensional geometric model of the small-modulus involute cylindrical gear.
2. The accurate modeling method for the small-modulus involute cylindrical gear of claim 1, characterized in that: the first step specifically comprises the following steps:
according to the generating machining principle, in an end face view of a workpiece to be cut, a hobbing cutter and the workpiece to be cut do rack-gear forced meshing motion, namely, a cutter pitch line does pure rolling relative to a tooth blank reference circle, and a workpiece coordinate system XOY and a cutter coordinate system X are constructed 1 PY 1 Let the longitudinal axes OY and PY 1 The initial position is when parallel;
when the tool is rotated around the workpiece to be cut
Figure FDA0003747298830000012
At radian, the tool pitch line PY 1 Contact point N with reference circle of gear blank in tool coordinate system X 1 PY 1 From the origin P (0, 0) to a point
Figure FDA0003747298830000013
The engagement point of the tool and the target tooth profile is denoted as M (X, y) in the workpiece coordinate system XOY and is denoted as X (X, y) in the tool coordinate system X 1 PY 1 In the notation of M' (x) 1 ,y 1 ) Then point M (M') is in the tool coordinate system X 1 PY 1 And the object coordinate system XOY:
Figure FDA0003747298830000011
wherein r is a reference circle radius, X is the position of the engagement point of the tool and the target tooth profile on the X axis in the workpiece coordinate system XOY, Y is the position of the engagement point of the tool and the target tooth profile on the Y axis in the workpiece coordinate system XOY, and X is the radius of the reference circle 1 For the engagement point of the tool with the target tooth profile in the tool coordinate system X 1 PY 1 In (C) X 1 Position on axis, y 1 For the tool-to-target tooth-profile engagement point in the tool coordinate system X 1 PY 1 Middle Y 1 The position on the shaft is such that,
Figure FDA0003747298830000014
the arc through which the tool rotates about the workpiece being cut.
3. The accurate modeling method for the small-modulus involute cylindrical gear of claim 2, characterized by comprising the following steps: the second step specifically comprises:
analyzing the forming effect of a single-side cutting edge of a hob tooth on a single-side tooth profile of a gear, and dividing the cutting edge into a top edge, a side edge and a bottom edge, wherein the top edge generates a tooth root circular arc in a cutting mode, the side edge generates an involute tooth profile in a cutting mode, and the bottom edge generates an tooth top circular arc in a cutting mode;
shaft PY is used to produce involute tooth profile when hob side edge is cutting 1 The coordinate of the intersection point of the side edge and the pitch line is (0 0 ) Mesh point M' (x) 1 ,y 1 ) In the tool coordinate system X 1 PY 1 The position of (1) with the angle of rotation
Figure FDA0003747298830000024
The coordinate function that varies is:
Figure FDA0003747298830000021
wherein α is the pressure angle, y 0 Is the intersection point of the side edge and the pitch line in the tool coordinate system X 1 PY 1 Middle Y 1 A position on the shaft;
coordinate parameter x of mesh point M 1 、y 1 Substituting in formula (1) to obtain the coordinate of involute tooth profile point M in static coordinate system XOY.
4. The accurate modeling method for the small-modulus involute cylindrical gear of claim 3, characterized by comprising the following steps: the third step specifically comprises:
when the top edge of the hob is cut to generate a tooth root circular arc and the spiral angle of a target gear is not 0, the projection of the hob teeth on the end surface of the gear blank is elongated along a pitch line, the projection of a tool nose fillet is an elliptical arc, and the elliptical parameter equation is used for representing:
Figure FDA0003747298830000022
in the formula, x top 、y top Respectively representing the tool coordinate system X 1 PY 1 The transverse and longitudinal distances from the lower point M' to the center C of the ellipse, a and b are the lengths of the transverse and longitudinal semiaxes of the ellipse respectively, gamma is the parameter of the ellipse angle, and the radius of the rounded corner of the tool nose is rho 0 With a helix angle β, a = ρ 0 ，b＝ρ 0 /cosβ；
Setting the ellipse center C in the tool coordinate system X 1 PY 1 Has the coordinate of (x) c ,y c ) Then mesh point M' (x) 1 ,y 1 ) In the tool coordinate system X 1 PY 1 The coordinates in (1) are:
Figure FDA0003747298830000023
the tooth crest height, the top clearance and the displacement are considered to obtain the ellipse center C in the cutter coordinate system X 1 PY 1 Coordinates (c) of (a):
Figure FDA0003747298830000031
in the formula, h a * Is the crest coefficient of tooth, c * Is the coefficient of the tip clearance, x is the coefficient of deflection, m is the modulus of the end face, rho 0 * Root fillet coefficient, m n Is the normal modulus, α t Is an end face pressure angle;
the over-engagement point M' is normal to the arc of the tooth root, and the axis PY 1 Cross over at the point
Figure FDA0003747298830000035
Thereby the coordinates and the rotation angle of the meshing point M' are obtained
Figure FDA0003747298830000036
Is associated with x 1 、y 1 And (3) substituting the formula (1) to obtain the coordinate of the tooth root circular arc point M in the workpiece coordinate system XOY.
5. The accurate modeling method for the small-modulus involute cylindrical gear of claim 4, characterized by comprising the following steps: according to the provisions of GB/T2362-1990, let c * ＝0.35，ρ 0 * =0.2, and the remaining coefficient values are the same as the normal-modulus gear standard.
6. The small-modulus involute cylindrical gear accurate modeling method according to claim 4 is characterized in that: the fourth step specifically comprises:
when the bottom edge of the hob is used for cutting to generate an addendum circular arc and the helix angle of the target gear is not 0, the projection of the hob teeth on the end surface of the gear blank is elongated along a pitch line, and the projection of the root fillet of the hob teeth is an elliptic arc, so that the projection is also expressed by an elliptic parameter equation:
Figure FDA0003747298830000032
wherein x is root 、y root Respectively representing the tool coordinate system X 1 PY 1 Lower mesh point M' from ellipse center C 1 The distance a and the distance b are respectively the length of a transverse half shaft and a longitudinal half shaft of the ellipse, gamma is an ellipse angle parameter, and the radius of a cutter root fillet is also rho 0 Then a = ρ 0 ，b＝ρ 0 /cosβ；
Setting the center C of an ellipse 1 In the tool coordinate system X 1 PY 1 Has a coordinate of (x) c1 ,y c1 ) Then mesh point M' (x) 1 ,y 1 ) In the tool coordinate system X 1 PY 1 The coordinates in (1) are:
Figure FDA0003747298830000033
the center C of the ellipse is obtained by considering the tooth crest height and the displacement 1 In the tool coordinate system X 1 PY 1 Coordinates (c) of (a):
Figure FDA0003747298830000034
the over-meshing point M' is taken as the normal of the tooth crest arc and the shaft PY 1 Meet at a point
Figure FDA0003747298830000037
Thereby the coordinates and the rotation angle of the meshing point M' are obtained
Figure FDA0003747298830000038
Is associated with x 1 、y 1 And (3) substituting the formula (1) to obtain the coordinate of the tooth top circular arc point M in the workpiece coordinate system XOY.
7. The accurate modeling method for the small-modulus involute cylindrical gear of claim 1, characterized in that: in step six, the gear design parameters comprise tooth number, modulus, pressure angle, tooth width, helix angle, profile index, root fillet index, crest height index and tip clearance index.
CN202210828692.3A 2022-07-15 2022-07-15 Accurate modeling method for small-modulus involute cylindrical gear Pending CN115186493A (en)

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Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115630454A (en) * 2022-10-21 2023-01-20 南京工业大学 Precise modeling method for Grarison spiral bevel gear

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115630454A (en) * 2022-10-21 2023-01-20 南京工业大学 Precise modeling method for Grarison spiral bevel gear
CN115630454B (en) * 2022-10-21 2023-10-20 南京工业大学 Method for accurately modeling Glison spiral bevel gear

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