CN108006193B - Ideal gear surface model modeling method based on hobbing simulation - Google Patents

Ideal gear surface model modeling method based on hobbing simulation Download PDF

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CN108006193B
CN108006193B CN201711253601.3A CN201711253601A CN108006193B CN 108006193 B CN108006193 B CN 108006193B CN 201711253601 A CN201711253601 A CN 201711253601A CN 108006193 B CN108006193 B CN 108006193B
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gear
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tooth surface
coordinate system
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CN108006193A (en
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陈洪芳
陈猛
石照耀
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Beijing University of Technology
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    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16H—GEARING
    • F16H55/00—Elements with teeth or friction surfaces for conveying motion; Worms, pulleys or sheaves for gearing mechanisms
    • F16H55/02—Toothed members; Worms
    • F16H55/17—Toothed wheels
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16H—GEARING
    • F16H55/00—Elements with teeth or friction surfaces for conveying motion; Worms, pulleys or sheaves for gearing mechanisms
    • F16H55/02—Toothed members; Worms
    • F16H55/08—Profiling
    • F16H55/0806—Involute profile
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16H—GEARING
    • F16H55/00—Elements with teeth or friction surfaces for conveying motion; Worms, pulleys or sheaves for gearing mechanisms
    • F16H55/02—Toothed members; Worms
    • F16H55/22—Toothed members; Worms for transmissions with crossing shafts, especially worms, worm-gears
    • G—PHYSICS
    • 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
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/30—Circuit design
    • G06F30/36—Circuit design at the analogue level
    • G06F30/367—Design verification, e.g. using simulation, simulation program with integrated circuit emphasis [SPICE], direct methods or relaxation methods

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Abstract

The invention discloses an ideal gear surface model modeling method based on hobbing simulation. Based on the gear surface model modeling method, a gear tooth surface simulation model is firstly established, then a hob model is established, a gear coordinate system is transformed into the same coordinate system with the hob, a hob and gear meshing model is simulated, a point on a gear tooth surface, which is closest to the hob along the normal direction of the gear tooth surface, is calculated to calculate the hobbing cutting thickness, and finally, a gear tooth surface model can be obtained. The method can be used for quantitatively evaluating the influence of the shafting errors of the hob and the gear on the surface quality of the processed gear, provides a theoretical basis for tracing the gear processing errors and provides theoretical experimental data and guidance for improving the gear processing precision.

Description

Ideal gear surface model modeling method based on hobbing simulation
Technical Field
The invention relates to a gear surface model modeling method, in particular to an ideal gear surface model modeling method based on gear hobbing simulation. Belongs to the technical field of gear machining surface model modeling.
Background
With respect to the modeling problem of the hobbing simulation, the parameterized design of the hob and the gear is necessary, and the precise modeling of the hob and the gear is realized by adjusting the parameters of the hob and the gear to control the geometric shape of the built model. At present, most of researches adopt a method of obtaining a gear tooth profile curve by an envelope curve, and finally, the establishment of a gear three-dimensional model is realized. The problems with these studies are: the gear tooth profile is obtained by adopting an envelope curve method instead of directly aiming at the gear hobbing process. The hob shape and the machining movement mechanism in gear hobbing are much more complex.
Therefore, it is necessary to invent an ideal gear surface model modeling method based on gear hobbing simulation, which is based on space meshing and contact transition simulation of a hob and a spur gear, calculates a point on a gear tooth surface closest to the hob along a normal direction of the gear tooth surface to calculate a gear hobbing cutting thickness, and finally obtains a gear tooth surface model.
Disclosure of Invention
The invention provides an ideal gear surface model modeling method based on gear hobbing simulation, which can be used for quantitatively evaluating the influence of shafting errors of a hob and a gear on the surface quality of a machined gear, providing a theoretical basis for tracing the gear machining errors and providing theoretical experimental data and guidance for improving the gear machining precision.
The invention is realized by adopting the following technical scheme:
an ideal gear surface model modeling method based on hobbing simulation comprises the following steps:
the method comprises the following steps: and establishing a gear tooth surface simulation model. Gear coordinate system denoted Sg(Og:Xg,Yg,Zg) The coordinate system takes the center of the involute straight toothed spur gear as the origin of coordinates OgWith gear axis as YgAxis, taking as Z the ray from origin through the theoretical nodal position on the tooth surfacegThe axis is positive, and finally X is determined according to a right-hand rulegA shaft. Gear coordinate system Sg(Og:Xg,Yg,Zg) Wherein y is 0, the gear end section, N is the tangent point of the normal line of the tooth profile passing through the node C on the base circle, and the end face pressure angle at the node C is α0The pressure angle of the end face at any point on the tooth profile is α, and the point E is the tangent point of the normal line of the tooth profile passing through the point F on the base circle.
Any point (Y, α) on the tooth surface of the gear, where Y denotes that point is at YgCoordinate of direction, α represents the face pressure angle at that pointgThe shaft rotates by an angle
Figure BDA0001492242910000011
And (4) showing. The worm and the gear are in point contact. In a gear coordinate system SgIn which the angle of the gear about its axis is specified when the theoretical point of engagement coincides with a pitch point on the tooth surface
Figure BDA0001492242910000012
A point on the tooth flank of the gear is defined by a given set of parameters
Figure BDA0001492242910000013
Unique determination, y ∈ [ -b/2, b/2]，α∈(0,αta)，
Figure BDA0001492242910000014
b is the tooth width of the gear, αtaThe face tooth top pressure angle.
Let l0Represents the spread arc length at node C, then
Figure BDA0001492242910000021
Wherein r isbIs the base radius of the gear.
Let lαRepresenting the developed arc length at the tooth profile point with the pressure angle of α
lα＝rbtan(α) (2)
Theta is the normal to the F point of the tooth surface and XgAngle between positive axes if at any point F on the tooth face, the face pressure angle is α, YgThe direction coordinate is y, and the tooth surface where F is located is rotated from the initial position
Figure BDA0001492242910000022
Angle, the clockwise rotation takes a negative value, then angle θ is:
Figure BDA0001492242910000023
wherein inv (α)0) Tan (α) - α, inv () is an involute function, point E (x)E,yE,zE) And point F (x)F,yF,zF) Is expressed as:
Figure BDA0001492242910000024
Figure BDA0001492242910000025
from this, point F (x)F,yF,zF) With arbitrariness, equation (5) is the parametric equation for the tooth flanks of the gear.
Step two: and establishing a model of the hob. Establishing a worm coordinate system Sw(Ow:Xw,Yw,Zw). In the gear hobbing process, a hob and a gear are transmitted by a pair of worm gears, the hob is equivalent to a worm in meshing, and a plurality of grooves are formed in worm teeth to form cutting edges, so that a worm model is built to replace the hob. Taking worm axis as ZwThe position of the starting point A of the base circle involute is selected as XwIn the forward direction, the cross section of the end where A is located is ZwThe axis intersection point is the origin O of the worm coordinate systemw,YwThe axial direction is determined by the right-handed cartesian coordinate system rule.
λb1Is the base circle helix angle of the worm, rb1The base radius of the worm. When lambda isb1、rb1And the coordinates of the point A are determined, the space position of the whole involute helicoid tooth surface is determined.
Involute helicoid of point A and XwOwYwThe intersection line of the planes is AB, and AB is an involute of one end face of the worm. Let M be any point on the tooth surface of the worm, and F be the point M on XwOwYwUpper edge Z of planewLet the polar coordinates OF point F be (ρ, θ), where the radius ρ is the length OF the segment OF, and the polar angle θ be ∠ XwOwF. A tangent plane BCD passing through the M point and serving as a base cylinder, wherein the tangent line between the tangent plane BCD and the base cylinder is DC, and XwOwYwThe intersection of the planes is BC, then the line segment DC is parallel to ZwA shaft. Making line segment ME parallel to plane XwOwYw。
Coordinate (x) of any point M of worm tooth surfacew,yw,zw) Is uniquely determined by the polar coordinates (ρ, θ) of point F, so point M is denoted as M (ρ, θ). ∠ AOC (ρ, θ) as:
∠AOC(ρ,θ)＝θ+cos(rb1/ρ) (6)
Figure BDA0001492242910000031
expressed as:
Figure BDA0001492242910000032
m (ρ, θ) is:
Figure BDA0001492242910000033
formula (8) requires correction, andwthe origin of coordinates of the axis is set at the node P, so Z at the node P in equation (8) is first calculatedwThe axis coordinates are:
zP＝zM(rw1,0) (9)
wherein r isw1The radius of the working pitch circle of the worm.
Then, equation (8) is modified to:
Figure BDA0001492242910000034
equation (10) is the parameter equation of one side of the involute worm gear, wherein the parameter rho ∈ [ r ]f1,ra1]，θ∈[-π,π]，rf1Radius of root of worm gear ra1The radius of the worm tooth top circle.
Step three: and (5) coordinate transformation. In order to simulate the meshing and contacting process of the worm tooth surface and the gear tooth surface, the gear tooth surface and the worm tooth surface must be placed in the same coordinate system, wherein the gear coordinate system is converted into the worm coordinate system, so that a coordinate conversion matrix needs to be obtained. Wherein the worm axis and the gear axis are in a non-coplanar linear relationship. During the machining process, the axis ZwAnd axis YgThe intersection angle of the working axes is recorded as ∑wExpressed as:
∑w＝β1±β2+π/2+eΣ(11)
β therein1Pitch angle of reference circle for worm β2For the pitch angle of the gear, eΣIs the working axis intersection angle ∑wMounting error of (2).
The selection of the positive sign and the negative sign in the formula (11) is related to the spiral directions of the worm and the gear. When the spiral directions of the worm and the gear are opposite, a minus sign is taken, and when the spiral directions of the worm and the gear are the same, a plus sign is taken.
Axis ZwAnd axis YgThe working center distance between them is marked as awExpressed as:
aw＝rw1+rw2＝r1+r2+ea(12)
wherein r isw2Is the working pitch radius of the gear, r1Radius of reference circle, r, of worm2For the gear pitch radius, eaIs a center distance awMounting error of (2).
From the gear coordinate system Sg(Og:Xg,Yg,Zg) To the worm coordinate system Sw(Ow:Xw,Yw,Zw) Has a position coordinate transformation matrix of Mwg，MwgIs recorded as:
Figure BDA0001492242910000041
the gear tooth surface is expressed in the worm coordinate system as:
Fw(xFw,yFw,zAw)＝Mwg·F(xF,yF,zF) (14)
worm coordinate system Sw(Ow:Xw,Yw,Zw) By Fw(xFw,yFw,zAw) And (4) showing.
Step four: and obtaining an ideal gear tooth surface model.
The tooth profile normal equation at the gear F point is as follows:
Figure BDA0001492242910000042
gear coordinate system Sg(Og:Xg,Yg,Zg) Point E (x) in (1)E,yE,zE) And point F (x)F,yF,zF) To the worm coordinate system Sw(Ow:Xw,Yw,Zw) In (1).
Figure BDA0001492242910000043
Figure BDA0001492242910000044
The gear tooth surface normal equation in the worm coordinate system is expressed as:
Figure BDA0001492242910000045
simultaneous equations (10) and (18) can be solved by (ρ, θ) as an unknown, i.e., a set of coordinates of the intersection of the gear tooth surface normal and the worm, represented by the parameters (ρ, θ). Defining the intersection point in a worm coordinate system Sw(Ow:Xw,Yw,Zw) Has the coordinates of
Figure BDA0001492242910000046
The worm tooth surface is limited within a certain range, theta ∈ [ -pi, pi [ -phi [ ]]，ρ∈[rf1,ra1]The intersection coordinates (ρ, θ) are the unique solution.
At the initial worm and gear rotation angle position, the distance between a point on the gear tooth surface and the intersection point of the worm tooth surface and the selected gear tooth surface along the normal direction is represented as D1. At the corner position, the distances between other points on the tooth surface of the gear and the intersection point of the worm tooth surface along the normal direction of the tooth surface and the point are respectively represented as D2,D3,D4,······,DnAnd n represents the number of points on the tooth surface of the gear and is a positive integer. Finding the shortest distance Dmin1Expressed as:
Dmin1＝Min[D2,D3,D4,······,Dn](19)
the shortest distance D obtainedmin1The coordinate of the intersection point of (a), i.e. the position of the gear and the worm at presentThe actual meshing point of the worm and the gear.
If the worm flanks are considered as ideal involute helicoids. The effect of the worm rotation and the axial translation of the worm on the meshing process is the same, so the translation of the worm flanks along their axes replaces the rotation of the worm. In the case of a single-headed worm, the gear rotates by one tooth for each revolution of the worm, and the worm correspondingly translates along the axial direction thereof by one pitch pwThe distance of (c). Divide the angle of a tooth into m equal parts, the corresponding worm translates by a tooth pitch pwIs also divided by m. At each angular position of worm and gear, D is obtainedmin1,Dmin2,······,DminmThe coordinates of the corresponding intersection points of the shortest distances are respectively
Figure BDA0001492242910000051
The coordinates of the intersection points obtained can be fitted to the gear tooth surface obtained by hobbing under the condition of no error.
And in the fourth step, obtaining an ideal gear tooth surface model simulated by gear hobbing processing.
Compared with the prior art, the invention has the following beneficial technical effects:
the novel beneficial effects of the invention are as follows: by providing the gear surface model modeling method based on the gear hobbing simulation, a worm and gear meshing model can be conveniently and quickly established, a theoretical basis is provided for tracing the gear machining error, theoretical experimental data and guidance are provided for improving the gear machining precision, and the method has practical significance for researching the influence rule of the shafting errors of the hob and the gear on the gear surface precision of the machined gear.
Drawings
FIG. 1 is a coordinate system establishing a gear tooth surface and a normal to the gear tooth surface.
Fig. 2 is a diagram for establishing a worm coordinate system.
Fig. 3 is a simulated gear tooth surface.
Fig. 4 is a simulated worm tooth flank.
Fig. 5 shows a simulated engagement of the worm and the gear.
FIG. 6 is a flow chart of the method.
Detailed Description
The present invention is further described in detail below with reference to the attached drawings so that those skilled in the art can implement the invention by referring to the description text.
An ideal gear surface model modeling method based on hobbing simulation is characterized by comprising the following steps:
the method comprises the following steps: and establishing a gear tooth surface simulation model. See fig. 1, the gear coordinate system is denoted Sg(Og:Xg,Yg,Zg) The coordinate system takes the center of the involute straight toothed spur gear as the origin of coordinates OgWith gear axis as YgAxis, taking as Z the ray from origin through the theoretical nodal position on the tooth surfacegThe axis is positive, and finally X is determined according to a right-hand rulegA shaft. Gear coordinate system Sg(Og:Xg,Yg,Zg) Wherein y is 0, the gear end section, N is the tangent point of the normal line of the tooth profile passing through the node C on the base circle, and the end face pressure angle at the node C is α0The pressure angle of the end face at any point on the tooth profile is α, and the point E is the tangent point of the normal line of the tooth profile passing through the point F on the base circle.
Any point (Y, α) on the tooth surface of the gear, where Y denotes that point is at YgCoordinate of direction, α represents the face pressure angle at that pointgThe shaft rotates by an angle
Figure BDA0001492242910000061
And (4) showing. The worm and the gear are in point contact. In a gear coordinate system SgIn which the angle of the gear about its axis is specified when the theoretical point of engagement coincides with a pitch point on the tooth surface
Figure BDA0001492242910000062
The simulations are now performed in conjunction with the values of the parameters of the specific worm and gear, which are shown in table 1.
TABLE 1 Worm and Gear parameters
Figure BDA0001492242910000063
Initial position of gear tooth surface
Figure BDA0001492242910000064
One point on the tooth surface of the gear is defined by a given set of parameters
Figure BDA0001492242910000065
Referring to the parameters in table 1, the developed arc length at the tooth profile point at which the pressure angle on the tooth surface of the gear is α is obtained from equation (2):
lα＝37.2545×tan(α) (20)
theta is the normal to the F point of the tooth surface and XgAngle between positive axes if at any point F on the tooth face, the face pressure angle is α, YgThe direction coordinate is y, and the tooth surface where F is located is rotated from the initial position
Figure BDA0001492242910000066
Angle, clockwise rotation takes a negative value. From the formula (3), the normal line and X of the F point of the tooth surface are obtainedgAngle θ between shaft positive directions:
Figure BDA0001492242910000067
obtaining a parameter equation of the gear according to the formula (5):
Figure BDA0001492242910000071
wherein, y ∈ [ -b/2, b/2]，α∈(0,αta)，αtaα pressure angle for face teeth of gearta＝arccos(rb/ra) Arccos (37.2545/45) ═ 34.1188 degrees. One gear tooth flank was drawn in Mathematica software: parametricplot3D [ { x [ ]F,yF,zF},{y,-b/2,b/2},{α,0,αta}]The resulting entire gear tooth surface is plotted in fig. 3.
Step two: and establishing a model of the hob. See fig. 2, establishing a worm coordinate system Sw(Ow:Xw,Yw,Zw). In the gear hobbing process, a hob and a gear are transmitted by a pair of worm gears, the hob is equivalent to a worm in meshing, and a plurality of grooves are formed in worm teeth to form cutting edges, so that a worm model is built to replace the hob. Taking worm axis as ZwThe position of the starting point A of the base circle involute is selected as XwIn the forward direction, the cross section of the end where A is located is ZwThe axis intersection point is the origin O of the worm coordinate systemw,YwThe axial direction is determined by the right-handed cartesian coordinate system rule. In FIG. 2, λb1Is the base circle helix angle of the worm, rb1The base radius of the worm. When lambda isb1、rb1And the coordinates of the point A are determined, the space position of the whole involute helicoid tooth surface is determined.
Involute helicoid of point A and XwOwYwThe intersection line of the planes is AB, and AB is an involute of one end face of the worm. Let M be any point on the tooth surface of the worm, and F be the point M on XwOwYwUpper edge Z of planewLet the polar coordinates OF point F be (ρ, θ), where the radius ρ is the length OF the segment OF, and the polar angle θ be ∠ XwOwF. A tangent plane BCD passing through the M point and serving as a base cylinder, wherein the tangent line between the tangent plane BCD and the base cylinder is DC, and XwOwYwThe intersection of the planes is BC, then the line segment DC is parallel to ZwA shaft. Making line segment ME parallel to plane XwOwYw。
Coordinate (x) of any point M of worm tooth surfacew,yw,zw) Since it is uniquely determined by the polar coordinates (ρ, θ) of the F point, the M point is referred to as M (ρ, θ). According to the parameters of the worm in Table 1, the base circle helix angle λ of the wormb129.2468 degrees, the base radius r of the wormb1＝8.1698mm。
∠ AOC (ρ, θ) is calculated as:
∠AOC(ρ,θ)＝θ+cos(8.1698/ρ) (23)
Figure BDA0001492242910000072
and (4) calculating to obtain:
Figure BDA0001492242910000081
from equation (10), the parametric equation for one flank of the worm is:
Figure BDA0001492242910000082
taking parameter rho ∈ [ r ]f1,ra1]，θ∈[-3π,3π]. One-sided worm tooth flanks were drawn in Mathematica software: parametricplot3D [ { x [ ]M,yM,zM},{ρ,rf1,ra1},{θ,-3π,3π}]The resulting one-sided worm flank is plotted in fig. 4.
Step three: and (5) coordinate transformation. In order to simulate the meshing and contacting process of the worm tooth surface and the gear tooth surface, the gear tooth surface and the worm tooth surface must be placed in the same coordinate system, wherein the gear coordinate system is converted into the worm coordinate system, so that a coordinate conversion matrix needs to be obtained. Wherein the worm axis and the gear axis are in a non-coplanar linear relationship. During the machining process, the axis ZwAnd axis YgThe intersection angle of the working axes is recorded as ∑wExpressed as:
∑w＝β1±β2+π/2+eΣ(26)
β therein1Pitch angle of reference circle for worm β2For the pitch angle of the gear, eΣIs the working axis intersection angle ∑wMounting error of (2).
The selection of the positive sign and the negative sign in the formula (11) is related to the spiral directions of the worm and the gear. When the spiral directions of the worm and the gear are opposite, a minus sign is taken, and when the spiral directions of the worm and the gear are opposite, a plus sign is taken.
Irrespective of working-axis intersection angle ∑wMounting error e ofΣAnd center distance awMounting error e ofaFrom the formula (11), the intersection angle of the working axes is ∑w169.63 degrees. From the formula (12), the center is obtainedDistance: a isw67 mm. From the gear coordinate system S, by the formula (13)g(Og:Xg,Yg,Zg) To the worm coordinate system Sw(Ow:Xw,Yw,Zw) The position coordinate transformation matrix of (a) is:
Figure BDA0001492242910000083
from equation (15), the worm coordinate system Sw(Ow:Xw,Yw,Zw) The arbitrary point of the tooth surface of the gear is Fw(xFw,yFw,zAw). After the gear tooth surface is subjected to coordinate transformation, a three-dimensional simulation of meshing of the worm and the gear is generated and is shown in figure 5.
Step four: and obtaining an ideal gear tooth surface model. Transforming the coordinates into a matrix MwgSubstituting the formula (17) and the formula (18) to obtain the middle point of the worm coordinate system
Figure BDA0001492242910000084
And point
Figure BDA0001492242910000085
The coordinate of the intersection point of the normal line of the tooth surface of the gear and the worm represented by the parameter [ rho ], theta ] is obtained by combining the equation (25) and the equation (19). Defining the intersection point in a worm coordinate system Sw(Ow:Xw,Yw,Zw) Has the coordinates of
Figure BDA0001492242910000086
The worm tooth surface is limited within a certain range, theta ∈ [ -pi, pi [ -phi [ ]]，ρ∈[rf1,ra1]The intersection coordinates (ρ, θ) are the unique solution.
At the initial worm and gear rotation angle position, the distance between a point on the gear tooth surface and the intersection point of the worm tooth surface and the selected gear tooth surface along the normal direction is represented as D1. At the corner position, the distances between other points on the tooth surface of the gear and the intersection point of the worm tooth surface along the normal direction of the tooth surface and the point are respectively represented as D2,D3,D4,······,DnAnd n represents the number of points on the tooth surface of the gear and is a positive integer. Finding the shortest distance Dmin1Expressed as:
Dmin1＝Min[D2,D3,D4,······,Dn](28)
the shortest distance D obtainedmin1The coordinate of the intersection point of the worm and the gear is the actual meshing point of the worm and the gear at the current position of the gear and the worm.
If the worm flanks are considered as ideal involute helicoids. The effect of the worm rotation and the axial translation of the worm on the meshing process is the same, so the translation of the worm flanks along their axes replaces the rotation of the worm. In the case of a single-headed worm, the gear rotates by one tooth for each revolution of the worm, and the worm correspondingly translates along the axial direction thereof by one pitch pwThe distance of (c). Divide the angle of a tooth into m equal parts, the corresponding worm translates by a tooth pitch pwIs also divided by m. At each angular position of worm and gear, D is obtainedmin1,Dmin2,······,DminmThe coordinates of the corresponding intersection points of the shortest distances are respectively
Figure BDA0001492242910000091
The coordinates of the intersection points obtained can be fitted to the gear tooth surface obtained by hobbing under the condition of no error.
The resulting gear tooth surfaces can be used for the following purposes:
the method has practical significance for researching the influence rule of the shafting errors of the hob and the gear on the tooth surface roughness and the cross section appearance of the processed gear, provides a theoretical basis for tracing the gear processing errors and provides theoretical experimental data and guidance for improving the gear processing precision.

Claims (2)

1. An ideal gear surface model modeling method based on hobbing simulation is characterized in that: the method comprises the following steps:
the method comprises the following steps: establishing a gear tooth surface simulation model; gear coordinate system denoted Sg(Og:Xg,Yg,Zg) The coordinate system takes the center of the involute straight toothed spur gear as the origin of coordinates OgWith gear axis as YgAxis, taking as Z the ray from origin through the theoretical nodal position on the tooth surfacegThe axis is positive, and finally X is determined according to a right-hand rulegA shaft; gear coordinate system Sg(Og:Xg,Yg,Zg) Wherein y is 0, the section of the gear end, N is the tangent point of the normal line of the tooth profile passing through the node C on the base circle, and the pressure angle of the end face at the node C is α0The pressure angle of the end face at any point on the tooth profile is α, and the point E is the tangent point of the normal line of the tooth profile passing through the point F on the base circle;
any point (Y, α) on the tooth surface of the gear, where Y denotes that point is at YgDirectional coordinate α represents the end face pressure angle of the point, gear tooth flank around YgThe shaft rotates by an angle
Figure FDA0002411825620000011
Represents; the worm and the gear are in point contact; in a gear coordinate system SgIn which the angle of the gear about its axis is specified when the theoretical point of engagement coincides with a pitch point on the tooth surface
Figure FDA0002411825620000012
A point on the tooth surface of the gear is defined by a given set of parameters (y, α,
Figure FDA0002411825620000013
) Unique determination, y ∈ [ -b/2, b/2]，α∈(0,αta)，
Figure FDA0002411825620000014
b is the tooth width of the gear, αtaIs the end face tooth top pressure angle;
let l0Represents the spread arc length at node C, then
Figure FDA0002411825620000015
Wherein r isbBeing toothed gearsThe radius of the base circle;
let lαRepresenting the developed arc length at the tooth profile point with the pressure angle of α
lα＝rbtan(α) (2)
Theta is the normal to the F point of the tooth surface and XgThe included angle between the positive directions of the axes, if at any point F on the tooth surface, the pressure angle of the end face is α, YgThe direction coordinate is y, and the tooth surface where F is located is rotated from the initial position
Figure FDA0002411825620000016
Angle, the clockwise rotation takes a negative value, then angle θ is:
Figure FDA0002411825620000017
wherein inv (α)0) Tan (α) - α, inv () is an involute function, point E (x)E,yE,zE) And point F (x)F,yF,zF) Is expressed as:
Figure FDA0002411825620000018
Figure FDA0002411825620000019
from this, point F (x)F,yF,zF) The formula (5) is a parameter equation of the gear tooth surface;
step two: establishing a model of the hob; establishing a worm coordinate system Sw(Ow:Xw,Yw,Zw) (ii) a In the gear hobbing process, a hob and a gear are transmitted by a pair of worm gears, the hob is equivalent to a worm in meshing, and a plurality of grooves are formed in worm teeth to form cutting edges, so that a worm model is established to replace the hob; taking worm axis as ZwThe position of the starting point A of the base circle involute is selected as XwIn the forward direction, the cross section of the end where A is located isZwThe axis intersection point is the origin O of the worm coordinate systemw,YwThe axial direction is determined by the right-handed cartesian coordinate system rule;
λb1is the base circle helix angle of the worm, rb1Is the base radius of the worm; when lambda isb1、rb1After the coordinates of the point A and the point A are determined, the space position of the whole involute helicoid tooth surface is also determined;
involute helicoid of point A and XwOwYwThe intersection line of the planes is AB, and AB is an involute of one end face of the worm; let M be any point on the tooth surface of the worm, and F be the point M on XwOwYwUpper edge Z of planewLet the polar coordinates OF point F be (rho, theta), where the polar diameter rho is the length OF line segment OF and the polar angle theta is ∠ XwOwF; a tangent plane BCD passing through the M point and serving as a base cylinder, wherein the tangent line between the tangent plane BCD and the base cylinder is DC, and XwOwYwThe intersection of the planes is BC, then the line segment DC is parallel to ZwA shaft; making line segment ME parallel to plane XwOwYw；
Coordinate (x) of any point M of worm tooth surfacew,yw,zw) Is uniquely determined by the polar coordinates (rho, theta) of point F, so point M is denoted as M (rho, theta); ∠ AOC (rho, theta) is expressed as:
∠AOC(ρ,θ)＝θ+cos(rb1/ρ) (6)
Figure FDA0002411825620000021
expressed as:
Figure FDA0002411825620000022
m (ρ, θ) is:
Figure FDA0002411825620000031
formula (8) requires correction, andwthe origin of coordinates of the axes being set at the nodeP, so Z at node P in equation (8) is first calculatedwThe axis coordinates are:
zP＝zM(rw1,0) (9)
wherein r isw1The radius of a working pitch circle of the worm;
then, equation (8) is modified to:
Figure FDA0002411825620000032
equation (10) is a parameter equation of one side tooth surface of the involute worm, wherein the parameter rho ∈ [ r ]f1,ra1]，θ∈[-π,π]，rf1Radius of root of worm gear ra1The radius of the worm gear top circle;
step three: transforming coordinates; in order to realize the meshing and contact process simulation of the worm tooth surface and the gear tooth surface, the gear tooth surface and the worm tooth surface are required to be placed in the same coordinate system, and the gear coordinate system is converted into a worm coordinate system, so that a coordinate conversion matrix is required to be solved; wherein the axes of the worm and the gear form a non-coplanar linear relationship; during the machining process, the axis ZwAnd axis YgThe intersection angle of the working axes is recorded as ∑wExpressed as:
∑w＝β1±β2+π/2+eΣ(11)
β therein1Pitch angle of reference circle for worm β2For the pitch angle of the gear, eΣIs the working axis intersection angle ∑wMounting error of (2);
the selection of the positive sign and the negative sign in the formula (11) is related to the spiral directions of the worm and the gear; when the spiral directions of the worm and the gear are opposite, a minus sign is taken, and when the spiral directions of the worm and the gear are the same, a plus sign is taken;
axis ZwAnd axis YgThe working center distance between them is marked as awExpressed as:
aw＝rw1+rw2＝r1+r2+ea(12)
wherein r isw2Is the working pitch radius of the gear, r1Radius of reference circle, r, of worm2For the gear pitch radius, eaIs a center distance awMounting error of (2);
from the gear coordinate system Sg(Og:Xg,Yg,Zg) To the worm coordinate system Sw(Ow:Xw,Yw,Zw) Has a position coordinate transformation matrix of Mwg，MwgIs recorded as:
Figure FDA0002411825620000041
the gear tooth surface is expressed in the worm coordinate system as:
Fw(xFw,yFw,zAw)＝Mwg·F(xF,yF,zF) (14)
worm coordinate system Sw(Ow:Xw,Yw,Zw) By Fw(xFw,yFw,zAw) Represents;
step four: obtaining an ideal gear tooth surface model;
the tooth profile normal equation at the gear F point is as follows:
Figure FDA0002411825620000042
gear coordinate system Sg(Og:Xg,Yg,Zg) Point E (x) in (1)E,yE,zE) And point F (x)F,yF,zF) To the worm coordinate system Sw(Ow:Xw,Yw,Zw) Performing the following steps;
Ew(xEw,yEw,zEw)＝Mwg·E(xE,yE,zE) (16)
Fw(xFw,yFw,zFw)＝Mwg·E(xF,yF,zF) (17)
the gear tooth surface normal equation in the worm coordinate system is expressed as:
Figure FDA0002411825620000043
simultaneous equations (10) and (18) can be solved by using (rho, theta) as an unknown number, namely a set of coordinates of the intersection point of the normal line of the gear tooth surface and the worm, which are expressed by parameters (rho, theta); defining the intersection point in a worm coordinate system Sw(Ow:Xw,Yw,Zw) Has the coordinates of
Figure FDA0002411825620000044
The worm tooth surface is limited within a certain range, theta ∈ [ -pi, pi [ -phi [ ]]，ρ∈[rf1,ra1]The intersection point coordinates (rho, theta) are unique solutions;
at the initial worm and gear rotation angle position, the distance between a point on the gear tooth surface and the intersection point of the worm tooth surface and the selected gear tooth surface along the normal direction is represented as D1(ii) a At the corner position, the distances between other points on the tooth surface of the gear and the intersection point of the worm tooth surface along the normal direction of the tooth surface and the point are respectively represented as D2,D3,D4,……,DnN represents the number of points on the tooth surface of the gear, and is a positive integer; finding the shortest distance Dmin1Expressed as:
Dmin1＝Min[D2,D3,D4,……,Dn](19)
the shortest distance D obtainedmin1The intersection point coordinate of the gear is the actual meshing point of the worm and the gear at the current position of the gear and the worm;
if the worm tooth surface is regarded as an ideal involute helicoid; the effect of the influence of the rotation of the worm and the axial translation of the worm on the meshing process is the same, so the translation of the tooth surface of the worm along the axial line of the tooth surface of the worm replaces the rotation of the worm; when the worm is a single-head worm, the gear rotates by one tooth every time the worm rotates by one circle, and the worm correspondingly translates along the axial direction of the worm by one tooth pitchpwThe distance of (d); divide the angle of a tooth into m equal parts, the corresponding worm translates by a tooth pitch pwIs also m equal parts; at each angular position of worm and gear, D is obtainedmin1,Dmin2,……,DminmThe coordinates of the corresponding intersection points of the shortest distances are respectively
Figure FDA0002411825620000051
Figure FDA0002411825620000052
The coordinates of the intersection points obtained can be fitted to the gear tooth surface obtained by hobbing under the condition of no error.
2. The ideal gear surface model modeling method based on hobbing process simulation according to claim 1, characterized in that: and obtaining an ideal gear tooth surface model obtained in the fourth step by the gear hobbing process simulation.
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