CN114912216A - Method for generating variable involute-variable cycloid combined tooth profile - Google Patents
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Abstract
The invention discloses a method for generating a variable involute-variable cycloid combined tooth profile. The difficulty of designing the tooth profile of the existing gear for realizing the secondary unequal-amplitude transmission ratio is high. The invention divides the secondary unequal transmission ratio curve into four sections, wherein the two sections comprise two positions with the curvature radius smaller than the minimum curvature radius when the tooth profile is not undercut, thereby obtaining the whole secondary unequal transmission ratio curve expression by sectional fitting; then, determining curves of a driving wheel and a driven wheel, adopting variable involute tooth profiles at the tooth top part of the driving wheel and the tooth root part of the driven wheel, adopting variable cycloid tooth profiles at the tooth top part of the driving wheel and the tooth top part of the driven wheel, connecting the tooth top profile of the driving wheel with the tooth root profile, and connecting the tooth top profile of the driven wheel with the tooth root profile to form a variable involute and variable cycloid combined tooth profile. The invention reduces the pressure angle of the variable involute tooth profile at the pitch curve, and ensures that the pressure angle of the variable cycloid tooth profile and the variable involute tooth profile at the pitch curve is equal, so that the two tooth profiles are smoothly connected at the pitch curve.
Description
Technical Field
The invention belongs to the field of gear tooth profile design, and particularly relates to a method for generating a variable involute-variable cycloid combined tooth profile.
Background
In order to meet the requirement of secondary unequal-amplitude non-uniform transmission ratio of a plug seedling taking mechanism, a student provides a combined non-circular gear transmission mechanism which is formed by combining a driving wheel and a partial non-circular gear and a driven wheel, wherein the combined non-circular gear transmission mechanism is formed by combining a partial non-circular gear and an elliptic gear, the secondary unequal-amplitude transmission ratio is realized, the combined non-circular gear transmission mechanism is applied to the plug seedling taking mechanism, but a transmission ratio curve has a sharp point with too small curvature, a proper involute tooth profile cannot be formed at the sharp point, the involute tooth profile can generate a root cutting phenomenon when the curvature is too small, and the strength of the gear can be reduced or even the meshing is directly influenced.
In order to solve the problems in the transmission ratio curve, a new tooth profile is designed by combining an involute tooth profile with a cycloid tooth profile and replacing the unsmooth connecting section of the involute tooth profile and the cycloid tooth profile with a straight line, so that the transmission ratio generating sharp point can be continuous and the generated tooth profile is not undercut. However, the method has complex process algorithm for replacing unsmooth connecting sections of the involute tooth profile and the cycloid tooth profile by straight lines, increases the difficulty of tooth profile design, has poor meshing effect of the straight line sections, and reduces the transmission efficiency.
Disclosure of Invention
The invention aims to provide a method for generating a variable involute-variable cycloid combined tooth profile, aiming at the problem that the design difficulty of the tooth profile of the existing gear for realizing the secondary unequal-amplitude transmission ratio is high.
The technical scheme adopted by the invention is as follows:
the invention discloses a method for generating a variable involute-variable cycloid combined tooth profile, which comprises the following steps of:
the method comprises the following steps: according to the critical relation of undercut:
obtaining the minimum curvature radius rho of the transmission ratio curve of the non-circular gear when the tooth profile does not generate the undercut min (ii) a Wherein m is involute profile circular tooth for solving non-circular gear tooth profileThe modulus of the wheel; h is a * Solving the addendum coefficient of the involute tooth profile circular gear of the non-circular gear tooth profile; alpha is alpha 1 Solving the pressure angle of the involute tooth profile circular gear of the non-circular gear tooth profile; z is a radical of 0 Solving the tooth number of the involute tooth profile circular gear of the non-circular gear tooth profile; m, alpha 1 And h a * All take the standard value, z 0 Determining according to the transmission ratio of a certain position of the transmission ratio curve of the non-circular gear and the radial direction of the non-circular gear of the tooth profile to be solved at the position;
calculating the curvature radius of each position of a secondary unequal-amplitude transmission ratio curve with two curve segments and the curvature radius smaller than the minimum curvature radius, and dividing the secondary unequal-amplitude transmission ratio curve into I according to the two positions with the curvature radius smaller than the minimum curvature radius 1 I 2 、I 2 I 3 、I 3 I 4 And I 4 I 5 Four sections of curve segments are provided, and the boundary points of the four sections of curve segments are respectively marked as I 1 、I 2 、I 3 、I 4 And I 5 The curve segment where the two positions with the curvature radius smaller than the minimum curvature radius are positioned is I 2 I 3 And I 4 I 5 ，I 2 I 3 The section of the driving wheel corresponding to the section has a rotation angle of [ theta ] 1 ,θ 2 -θ 1 ]，I 4 I 5 The section of the corresponding driving wheel has a rotation angle of [ theta ] 3 ,θ 4 -θ 3 ](ii) a Refitting the ratio curves I separately using Fourier functions 1 I 2 And I 3 I 4 Segments, respectively re-transitionally connected by polynomial curves I 2 I 3 Segment and I 4 I 5 Section, to ensure smooth connection of the whole transmission ratio curve, I must be satisfied 1 I 2 Section transmission ratio curve and I 2 I 3 Section ratio curve at intersection point I 2 Equal slope, I 2 I 3 Segment and I 3 I 4 Segment at intersection point I 3 The slope is equal, and the expression of the whole secondary unequal-amplitude transmission ratio curve is as follows:
in the formula (1), theta is the angular displacement of the driving wheel, a 11 、b 11 、c 11 、w 11 、a 22 、b 22 、c 22 And w 22 Is a coefficient of a Fourier function, I 1 I 2 Segment and I 3 I 4 After segment fitting, the known quantity is obtained; a is 1 、b 1 、c 1 、d 1 、a 2 、b 2 、c 2 And d 2 Is the coefficient of the polynomial curve, which is the quantity to be solved;
step two: i of the second unequal-amplitude transmission ratio curve 1 I 2 Segment is at 2 Point slope and I 3 I 4 Segment is at 3 The point slopes are respectively equal to I 2 I 3 Segment is at 2 Point slope and I 3 Point slope, I 1 I 2 Segment is at 2 The point slope is at I 2 Left derivative at Point i' - (θ 1 )，I 2 I 3 Segment is at 3 The point slope is at I 3 Left derivative at Point i' - (θ 2 )，I 2 I 3 Segment is at 2 The point slope is at I 2 Right derivative at Point i' + (θ 1 )，I 3 I 4 Segment is at 3 The point slope is at I 3 Right derivative at Point i' + (θ 2 ) To get i' + (θ 1 )＝i′ - (θ 1 ) And i' - (theta) 2 )＝i′ + (θ 2 ) And further obtaining:
and due to I 2 I 3 Transition curve through I 2 Points and I 3 Point, combined formula (2), solving to obtain I 2 I 3 Coefficient a in the equation of the section curve 1 、b 1 、c 1 And d 1 ；
Similarly, due to the second unequal-amplitude transmission ratio curve I 3 I 4 Segment is in I 4 Point slope and I 4 I 5 Segment is at 5 The point slopes are respectively equal to I 4 I 5 Segment is at 4 Point slope and I 1 I 2 Segment is in I 1 Point slope, again due to I 4 I 5 Transition curve through I 4 Points and I 5 Point, solve to obtain I 4 I 5 Coefficient a in the equation of the section curve 2 、b 2 、c 2 And d 2 ；
Then, determining pitch curves of the driving wheel and the driven wheel according to the secondary unequal transmission ratio curve expression and the given center distance, wherein the pitch curves are as follows:
by angular displacement of driven wheels
Obtaining:
for a given central distance a, according to a transmission ratio i, the angular displacement theta of a driving wheel and the angular displacement theta of a driven wheel c Calculating the pitch curve radial r of the driving wheel and the pitch curve radial r of the driven wheel c ：
Then, pitch curve coordinates (x, y) of the driving wheel with the origin of coordinates as the rotation center and pitch curve coordinates (x) of the driven wheel with the rotation center at (a,0) are obtained c ,y c )：
Step three: the tooth crest part outside the section curve of the driving wheel and the tooth root part inside the section curve of the driven wheel adopt variable involute tooth profiles, and the tooth root part inside the section curve of the driving wheel and the tooth crest part outside the section curve of the driven wheel adopt variable cycloid tooth profiles; then, connecting the tooth top profile of the driving wheel with the tooth root profile, and connecting the tooth top profile of the driven wheel with the tooth root profile to form the combined tooth profiles of the involute and the cycloid of the driving wheel and the driven wheel;
the analytic method for generating the variable involute tooth profile is as follows:
center of curvature coordinate (x) of driving wheel or driven wheel pitch curve q ,y q ) The calculation is as follows:
in the formula
(x j ,y j ) Is a driving wheel pitch curve coordinate (x, y) or a driven wheel pitch curve coordinate (x) c ,y c )；
Solving the pitch radius of the involute profile circular gear of the non-circular gear profile:
in the course of generating variable involute tooth profile, the pitch circle of involute tooth profile circular gear can be rolled around the driving wheel pitch curve or driven wheel pitch curve without sliding, the curvature centre of driving wheel pitch curve or driven wheel pitch curve and pitch circle meshing point M of driving wheel pitch curve or driven wheel pitch curve and involute tooth profile circular gear 2 The rotation center O of the involute tooth profile circular gear is collinear at three points, and the rotation center O coordinate (x) of the involute tooth profile circular gear when the driving wheel pitch curve or the driven wheel pitch curve protrudes outwards is obtained by utilizing the collinear property of the three points O ,y O )：
In the formula, rho is the curvature radius of a driving wheel pitch curve or a driven wheel pitch curve at a pitch circle meshing point of the involute tooth profile circular gear;
involute profile circular gear involute spread angle:
θ k ＝tan(α 1 )-α 1 (11)
the base radius of the involute tooth profile circular gear is as follows:
r b ＝r cos(α 1 ) (12)
the pitch circle of the involute tooth profile circular gear starts from the initial position and rolls along the curve of the driving wheel pitch or the curve of the driven wheel pitch along the arc length ds:
when the rolling arc length ds of the pitch circle of the involute tooth profile circular gear from the initial position is set, the tooth profile meshing point of the involute tooth profile circular gear and the driving wheel pitch curve or the driven wheel pitch curve is K, the intersection point of the generating line of the involute and the base circle of the involute tooth profile circular gear is N, and then the length KN of the generating line of the involute is set
The arc length between the initial position point M and the point N on the base circle of the involute tooth profile circular gear is represented as follows:
R kN ＝(θ k +α 2 +α 1 )r b (14)
the pressure angle α of the tooth profile meshing point is determined in Δ KON:
α＝tan -1 (θ k +α 2 +α 1 ) (15)
solving & lt KOM according to formula (16) 2 Comprises the following steps:
∠KOM 2 ＝α-α 1 (16)
in the delta KON, the distance R from the revolution center O of the involute tooth profile circular gear to the tooth profile meshing point is obtained K ：
Wherein f (mu) is a slip coefficient and is a given constant;
thus, the coordinates of the tooth profile meshing point K are determined:
wherein,
is a point M 2 The x-axis coordinate of (a) is,
is a point M 2 Y-axis coordinates of (a);
the analytical method for generating the variable cycloid tooth profile is as follows:
an elliptic gear rolls on a driving wheel pitch curve or a driven wheel pitch curve without sliding to form a variable cycloid tooth profile;
when the elliptic gear rolls on the curve of the driving wheel pitch or the curve of the driven wheel pitch without sliding, the rotation center O of the elliptic gear 2 The coordinates of (c) are calculated as follows:
wherein f is a concave-convex judging coefficient of a driving wheel pitch curve or a driven wheel pitch curve, and f is 1; x is the number of O2 For elliptic gearX-axis coordinate of the center of rotation, y O2 Is the y-axis coordinate of the revolution center of the elliptic gear; theta 0 The self-rotation angle displacement of the elliptic gear; theta m Setting theta for the initial included angle between the long axis of the elliptic gear and the x axis m 0; a is an elliptic pitch curve long semi-axis of the elliptic gear; b is an elliptic pitch curve short semi-axis of the elliptic gear; when (x) in the formula (19) j ,y j ) Theta is the coordinate (x, y) of the active wheel pitch curve 0 ＝θ c When (x) in the formula (19) j ,y j ) For the driven wheel pitch curve coordinate (x) c ,y c ) When theta is greater than theta 0 ＝θ；
The cycloid tooth profile coordinates of the driving wheel or the driven wheel are solved according to the formula (20):
in the formula, x b Is the x-axis coordinate, y, of the cycloid tooth profile of the driving wheel or the driven wheel b Is the y-axis coordinate of the cycloid tooth profile of the driving wheel or the driven wheel, and the spread angle theta 'of the elliptic gear' k ＝θ k 。
The invention has the following beneficial effects:
the invention aims at the problem of high design difficulty of the traditional gear tooth profile for realizing the secondary unequal-amplitude transmission ratio, and adopts a combined mode of variable involute and variable cycloid to generate the tooth profile, thereby avoiding the unsmooth phenomenon that the pressure angle of the involute tooth profile at a pitch curve is large, the pressure angle of the cycloid tooth profile at the pitch curve is small, and the involute tooth profile is directly connected with the cycloid tooth profile. Based on the design idea of the involute tooth profile, the invention adds the slip coefficient to form the variable involute tooth profile, so that the pressure angle of the variable involute tooth profile at the pitch curve is reduced, and based on the design idea of the cycloid tooth profile, an elliptic gear is used as a gear of a tool to roll out the variable cycloid tooth profile, so that the pressure angle of the variable cycloid tooth profile at the pitch curve is equal to the pressure angle of the variable involute tooth profile at the pitch curve, and the two tooth profiles can be directly and smoothly connected at the pitch curve.
Drawings
FIG. 1 is a schematic diagram of a second unequal-amplitude transmission ratio curve according to an embodiment of the invention.
Fig. 2 is a schematic diagram of the curvilinear engagement of the driving wheel and the driven wheel segments in the embodiment of the invention.
FIG. 3 is a schematic diagram of generating a variable involute profile in an embodiment of the present invention.
Fig. 4 is a schematic diagram of the generation of a variable cycloid tooth profile in an embodiment of the present invention.
Fig. 5 is a schematic view of an overall tooth profile in an embodiment of the present invention.
Detailed Description
The embodiments of the present invention will be further described with reference to the accompanying drawings, which are intended to be illustrative of the present invention and not to be construed as limiting the invention, and any modifications, equivalent substitutions or improvements made within the spirit and principle of the present invention shall be included in the scope of the appended claims, and all technical solutions which are not described in detail in the present invention shall be known in the art.
A method for generating a variable involute-variable cycloid combined tooth profile comprises the following steps:
the method comprises the following steps: according to the critical relation of undercut:
obtaining the minimum curvature radius rho of the transmission ratio curve of the non-circular gear when the tooth profile does not generate the undercut min (ii) a Wherein m is the modulus of an involute profile circular gear used for solving a non-circular gear profile; h is a * Solving the addendum coefficient of the involute tooth profile circular gear of the non-circular gear tooth profile; alpha is alpha 1 Solving the pressure angle of the involute tooth profile circular gear of the non-circular gear tooth profile; z is a radical of 0 Solving the tooth number of the involute tooth profile circular gear of the non-circular gear tooth profile; m, alpha 1 And h a * All take the standard value, z 0 The gear ratio of a position of the transmission ratio curve of the non-circular gear is determined according to the radial direction of the non-circular gear of the tooth profile to be solved at the position.
Calculating the second unequal amplitude transmission of two curve segments with the curvature radius smaller than the minimum curvature radiusDividing the curve of the second unequal transmission ratio into I according to the curvature radius of each position of the curve of the dynamic ratio and the two positions of which the curvature radius is smaller than the minimum curvature radius 1 I 2 、I 2 I 3 、I 3 I 4 And I 4 I 5 Four sections of curve segments are provided, and the boundary points of the four sections of curve segments are respectively marked as I 1 、I 2 、I 3 、I 4 And I 5 The curve segment where the two positions with the curvature radius smaller than the minimum curvature radius are positioned is I 2 I 3 And I 4 I 5 ，I 2 I 3 The section of the corresponding driving wheel has a rotation angle of [ theta ] 1 ,θ 2 -θ 1 ]，I 4 I 5 The section of the corresponding driving wheel has a rotation angle of [ theta ] 3 ,θ 4 -θ 3 ]As shown in fig. 1; refitting the ratio curves I separately using Fourier functions 1 I 2 And I 3 I 4 Segment, re-transition-connecting I respectively by polynomial curves 2 I 3 Segment and I 4 I 5 Section, to ensure smooth connection of the whole transmission ratio curve, I must be satisfied 1 I 2 Section transmission ratio curve and I 2 I 3 Section ratio curve at intersection point I 2 Equal slope, I 2 I 3 Segment and I 3 I 4 Segment at intersection point I 3 The slope is equal, and the expression of the whole secondary unequal-amplitude transmission ratio curve is as follows:
in the formula (1), theta is the angular displacement of the driving wheel, a 11 、b 11 、c 11 、w 11 、a 22 、b 22 、c 22 And w 22 Is a coefficient of a Fourier function, I 1 I 2 Segment and I 3 I 4 After segment fitting, the known quantity is obtained; a is 1 、b 1 、c 1 、d 1 、a 2 、b 2 、c 2 And d 2 Is the coefficient of the polynomial curve and is the quantity to be solved.
Step two: i of the second unequal-amplitude transmission ratio curve 1 I 2 Segment is at 2 Point slope and I 3 I 4 Segment is in I 3 The point slopes are respectively equal to I 2 I 3 Segment is at 2 Point slope and I 3 Point slope, I 1 I 2 Segment is in I 2 The point slope is at I 2 Left derivative i' - (theta) at a point 1 )，I 2 I 3 Segment is at 3 The point slope is at I 3 Left derivative at point i' - (θ) 2 )，I 2 I 3 Segment is at 2 The point slope is at I 2 Right derivative at Point i' + (θ 1 )，I 3 I 4 Segment is at 3 The point slope is at I 3 Right derivative at point i' + (θ 2 ) To get i' + (θ 1 )＝i′ _ (θ 1 ) And i' _ (θ 2 )＝i′ + (θ 2 ) And further obtaining:
and due to I 2 I 3 Transition curve through I 2 Points and I 3 Point, combined formula (2), solving to obtain I 2 I 3 Coefficient a in the equation of the section curve 1 、b 1 、c 1 And d 1 。
Similarly, due to the second unequal-amplitude transmission ratio curve I 3 I 4 Segment is at 4 Point slope and I 4 I 5 Segment is at 5 The point slopes are respectively equal to I 4 I 5 Segment is at 4 Point slope and I 1 I 2 Segment is at 1 Point slope, again due to I 4 I 5 Transition curve through I 4 Points and I 5 Point, solve to obtain I 4 I 5 Coefficient a in the equation of the section curve 2 、b 2 、c 2 And d 2 。
Then, determining pitch curves of the driving wheel and the driven wheel according to the secondary unequal transmission ratio curve expression and the given center distance, wherein the pitch curves are as follows:
by angular displacement of driven wheels
Obtaining:
for a given central distance a, according to a transmission ratio i, the angular displacement theta of a driving wheel and the angular displacement theta of a driven wheel c Calculating the pitch curve radius r of the driving wheel and the pitch curve radius r of the driven wheel c ：
Then, pitch curve coordinates (x, y) of the driving wheel with the origin of coordinates as the rotation center and pitch curve coordinates (x) of the driven wheel with the rotation center at (a,0) are obtained c ,y c )：
The driving and driven wheel segments are in curvilinear engagement as shown in figure 2.
Step three: the tooth top part outside the section curve of the driving wheel and the tooth root part inside the section curve of the driven wheel adopt variable involute tooth profiles, and the tooth root part inside the section curve of the driving wheel and the tooth top part outside the section curve of the driven wheel adopt variable cycloid tooth profiles; then, the tooth crest outline of the driving wheel is connected with the tooth root outline, the tooth crest outline of the driven wheel is connected with the tooth root outline, and the combined tooth profiles of the involute and the cycloid of the driving wheel and the driven wheel are formed, as shown in an ABCDEF curve section in fig. 5, a part I in fig. 5 shows that the tooth profile connecting section EGD has unsmooth curve when the involute and cycloid combined tooth profiles are directly adopted in a partially enlarged manner, and the connecting section EHD of the combined tooth profiles of the involute and the cycloid is smooth.
The analytic method for generating the variable involute tooth profile (according to the tooth profile meshing principle, solving the variable involute tooth profile by solving the meshing point) is as follows:
center of curvature coordinate (x) of driving wheel or driven wheel pitch curve q ,y q ) The calculation is as follows:
in the formula
(x j ,y j ) Is a driving wheel pitch curve coordinate (x, y) or a driven wheel pitch curve coordinate (x) c ,y c )。
Solving the pitch radius of the involute profile circular gear of the non-circular gear profile:
in the generating process of the variable involute tooth profile, the pitch circle 2 of the involute tooth profile circular gear rolls around the driving wheel pitch curve 1 or the driven wheel pitch curve without sliding, and the rotary center of the involute tooth profile circular gear generates a deviation R relative to the driving wheel pitch curve or the driven wheel pitch curve K A curved path of (2), wherein R K The distance from the revolution center O of the involute tooth profile circular gear to the tooth profile meshing point is a variable; the curvature center of the driving wheel pitch curve or the driven wheel pitch curve, the pitch circle meshing point M of the driving wheel pitch curve or the driven wheel pitch curve and the involute tooth profile circular gear 2 The rotation center O of the involute tooth profile circular gear is collinear at three points, and the rotation center O coordinate (x) of the involute tooth profile circular gear when the driving wheel pitch curve or the driven wheel pitch curve protrudes outwards is obtained by utilizing the collinear property of the three points O ,y O )：
In the formula, rho is the curvature radius of a driving wheel pitch curve or a driven wheel pitch curve at a pitch circle meshing point of the involute tooth profile circular gear;
involute profile circular gear involute spread angle:
θ k ＝tan(α 1 )-α 1 (11)
the base radius of the involute tooth profile circular gear is as follows:
r b ＝r cos(α 1 ) (12)
the pitch circle of the involute tooth profile circular gear starts from the initial position and rolls along the curve of the driving wheel pitch or the curve of the driven wheel pitch along the arc length ds:
as shown in fig. 3, when the rolling arc length ds starts from the initial position of the pitch circle of the involute tooth profile circular gear, the tooth profile meshing point of the involute tooth profile circular gear and the driving wheel pitch curve or the driven wheel pitch curve is K, the intersection point of the generating line of the involute and the base circle of the involute tooth profile circular gear is N, and the length KN of the generating line of the involute is N
The arc length between the initial position point M and the point N on the base circle of the involute tooth profile circular gear is represented as follows:
R kN ＝(θ k +α 2 +α 1 )r b (14)
the pressure angle α of the tooth profile meshing point is determined in Δ KON:
α＝tan -1 (θ k +α 2 +α 1 ) (15)
solving & lt KOM according to formula (16) 2 Comprises the following steps:
∠KOM 2 ＝α-α 1 (16)
in the delta KON, the distance R from the rotation center O of the involute tooth profile circular gear to the tooth profile meshing point is obtained K ：
Wherein f (mu) is a slip coefficient and is a given constant;
thus, the coordinates of the tooth profile meshing point K are determined:
wherein,
is a point M 2 The x-axis coordinate of (a) is,
is a point M 2 Y-axis coordinate of (a).
The analytical method for generating the variable cycloid tooth profile is as follows:
the elliptic gear is adopted to roll on the curve of the driving wheel section or the curve of the driven wheel section without sliding to form a variable cycloid tooth profile, as shown in figure 4.
When the elliptic gear rolls on the curve of the driving wheel pitch or the curve of the driven wheel pitch without sliding, the rotation center O of the elliptic gear 2 The coordinates of (c) are calculated as follows:
wherein f is a driving wheel pitch curve or a driven wheel pitchA curve concave-convex judging coefficient, wherein f is 1 (when the pitch curve is non-circular, f is 1 when the pitch curve is convex, and f is-1 when the pitch curve is concave); x is the number of O2 X-axis coordinate, y, of the center of revolution of an elliptical gear O2 Is the y-axis coordinate of the revolution center of the elliptic gear; theta 0 The self-rotation angle displacement of the elliptic gear; theta m Setting theta for the initial included angle between the long axis of the elliptic gear and the x axis m 0; a is an elliptic pitch curve long semi-axis of the elliptic gear; and b is an elliptic pitch curve short semi-axis of the elliptic gear.
When the elliptic gear rolls on the curve of the driving wheel pitch or the curve of the driven wheel pitch without sliding, the self-rotation angle displacement theta is generated along with the rotation of the elliptic gear 0 The calculation formula is as follows:
when (x) in the formula (19) j ,y j ) Theta is the coordinate (x, y) of the active wheel pitch curve 0 ＝θ c When (x) in the formula (19) j ,y j ) For the driven wheel pitch curve coordinate (x) c ,y c ) When theta is greater than theta 0 ＝θ；
The cycloid tooth profile coordinates of the driving wheel or the driven wheel are solved according to the formula (20):
in the formula, x b Is the x-axis coordinate, y, of the cycloid tooth profile of the driving wheel or the driven wheel b Is the y-axis coordinate of the cycloid tooth profile of the driving wheel or the driven wheel, and the spread angle theta 'of the elliptic gear' k ＝θ k 。
The invention aims at the problems that the involute tooth profile has large pressure angle at the pitch curve and the cycloid tooth profile has small pressure angle at the pitch curve, and the unsmooth phenomenon exists when the involute tooth profile is directly connected with the cycloid tooth profile, and adds a slip coefficient to form a variable involute tooth profile (in figure 3, the positions of a meshing point K' when the tooth profile I of an involute tooth profile circular gear and a driving wheel or a driven wheel are involute tooth profiles II and the tooth profile I of the involute tooth profile circular gear are compared with the meshing point K when the driving wheel or the driven wheel is the variable involute tooth profile III) are shown, so that the pressure angle of the variable involute tooth profile at the pitch curve is reduced, based on the design idea of cycloid tooth profile, the elliptic gear is used as a cutter gear to roll out a variable cycloid tooth profile, the pressure angle of the variable cycloid tooth profile at the pitch curve is equal to that of the variable involute tooth profile at the pitch curve, so that the two tooth profiles can be directly and smoothly connected at the pitch curve.
Claims (1)
1. A method for generating a variable involute-variable cycloid combined tooth profile is characterized in that: the method comprises the following steps:
the method comprises the following steps: according to the critical relation of undercut:
obtaining the minimum curvature radius rho of the transmission ratio curve of the non-circular gear when the tooth profile does not generate the undercut min (ii) a Wherein m is the modulus of the involute profile circular gear used for solving the non-circular gear profile; h is a * Solving the addendum coefficient of the involute tooth profile circular gear of the non-circular gear tooth profile; alpha is alpha 1 Solving the pressure angle of the involute tooth profile circular gear of the non-circular gear tooth profile; z is a radical of 0 Solving the tooth number of the involute tooth profile circular gear of the non-circular gear tooth profile; m, alpha 1 And h a * All take the standard value, z 0 Determining the transmission ratio of a position of the transmission ratio curve of the non-circular gear and the radial direction of the non-circular gear of the tooth profile to be solved at the position;
calculating the curvature radius of each position of a secondary unequal-amplitude transmission ratio curve with two curve segments and the curvature radius smaller than the minimum curvature radius, and dividing the secondary unequal-amplitude transmission ratio curve into I according to the two positions with the curvature radius smaller than the minimum curvature radius 1 I 2 、I 2 I 3 、I 3 I 4 And I 4 I 5 Four sections of curve segments are provided, and the boundary points of the four sections of curve segments are respectively marked as I 1 、I 2 、I 3 、I 4 And I 5 The curve section where the two positions with the curvature radius smaller than the minimum curvature radius are positioned is I 2 I 3 And I 4 I 5 ，I 2 I 3 The section of the corresponding driving wheel has a rotation angle of [ theta ] 1 ，θ 2 -θ 1 ]，I 4 I 5 The section of the corresponding driving wheel has a rotation angle of [ theta ] 3 ，θ 4 -θ 3 ](ii) a Refitting the ratio curves I separately using Fourier functions 1 I 2 And I 3 I 4 Segment, re-transition-connecting I respectively by polynomial curves 2 I 3 Segment and I 4 I 5 Section, to ensure smooth connection of the whole transmission ratio curve, I must be satisfied 1 I 2 Section transmission ratio curve and I 2 I 3 Section ratio curve at intersection point I 2 Equal in slope, I 2 I 3 Segment and I 3 I 4 Segment at intersection point I 3 The slope is equal, and the expression of the whole secondary unequal-amplitude transmission ratio curve is as follows:
in the formula (1), theta is the angular displacement of the driving wheel, a 11 、b 11 、c 11 、w 11 、a 22 、b 22 、c 22 And w 22 Is a coefficient of a Fourier function, I 1 I 2 Segment and I 3 I 4 After segment fitting, the known quantity is obtained; a is 1 、b 1 、c 1 、d 1 、a 2 、b 2 、c 2 And d 2 Is the coefficient of the polynomial curve, which is the quantity to be solved;
step two: i of the second unequal-amplitude transmission ratio curve 1 I 2 Segment is at 2 Slope of point and I 3 I 4 Segment is at 3 The point slopes are respectively equal to I 2 I 3 Segment is at 2 Point slope and I 3 Point slope, I 1 I 2 Segment is at 2 The point slope is at I 2 Left derivative at Point i' - (θ 1 )，I 2 I 3 Segment is at 3 The point slope is at I 3 Left derivative at Point i' - (θ 2 )，I 2 I 3 Segment is at 2 The point slope is at I 2 Right derivative at Point i' + (θ 1 )，I 3 I 4 Segment is in I 3 The point slope is at I 3 Right derivative at Point i' + (θ 2 ) To get i' + (θ 1 )＝i′ - (θ 1 ) And i' - (θ 2 )＝i′ + (θ 2 ) Further, obtaining:
and due to I 2 I 3 Transition curve through I 2 Points and I 3 Point, combined formula (2), solving to obtain I 2 I 3 Coefficient a in the equation of the section curve 1 、b 1 、c 1 And d 1 ；
Similarly, due to the second unequal-amplitude transmission ratio curve I 3 I 4 Segment is at 4 Point slope and I 4 I 5 Segment is at 5 The point slopes are respectively equal to I 4 I 5 Segment is at 4 Point slope and I 1 I 2 Segment is at 1 Point slope, again due to I 4 I 5 Transition curve through I 4 Points and I 5 Point, solve to obtain I 4 I 5 Coefficient a in the equation of the section curve 2 、b 2 、c 2 And d 2 ；
Then, determining pitch curves of the driving wheel and the driven wheel according to the secondary unequal transmission ratio curve expression and the given center distance, wherein the pitch curves are as follows:
by angular displacement of driven wheels
Obtaining:
for a given center distancea, according to the transmission ratio i, the angular displacement theta of the driving wheel and the angular displacement theta of the driven wheel c Calculating the pitch curve radius r of the driving wheel and the pitch curve radius r of the driven wheel c ：
Then, pitch curve coordinates (x, y) of the driving wheel with the origin of coordinates as the rotation center and pitch curve coordinates (x) of the driven wheel with the rotation center at (a,0) are obtained c ，y c )：
Step three: the tooth crest part outside the section curve of the driving wheel and the tooth root part inside the section curve of the driven wheel adopt variable involute tooth profiles, and the tooth root part inside the section curve of the driving wheel and the tooth crest part outside the section curve of the driven wheel adopt variable cycloid tooth profiles; then, connecting the tooth top profile of the driving wheel with the tooth root profile, and connecting the tooth top profile of the driven wheel with the tooth root profile to form the combined tooth profiles of the involute and the cycloid of the driving wheel and the driven wheel;
the analytic method for generating the variable involute tooth profile is as follows:
center of curvature coordinate (x) of driving wheel or driven wheel pitch curve q ，y q ) The calculation is as follows:
in the formula
(x j ，y j ) Is a driving wheel pitch curve coordinate (x, y) or a driven wheel pitch curve coordinate (x) c ，y c )；
Solving the pitch radius of the involute profile circular gear of the non-circular gear profile:
in the course of generating variable involute tooth profile, the pitch circle of involute tooth profile circular gear can be rolled around the driving wheel pitch curve or driven wheel pitch curve without sliding, the curvature centre of driving wheel pitch curve or driven wheel pitch curve and pitch circle meshing point M of driving wheel pitch curve or driven wheel pitch curve and involute tooth profile circular gear 2 The rotation center O of the involute tooth profile circular gear is collinear at three points, and the rotation center O coordinate (x) of the involute tooth profile circular gear when the driving wheel pitch curve or the driven wheel pitch curve protrudes outwards is obtained by utilizing the collinear property of the three points O ，y O )：
In the formula, rho is the curvature radius of a driving wheel pitch curve or a driven wheel pitch curve at a pitch circle meshing point of the circular gear with the involute tooth profile;
involute profile circular gear involute spread angle:
θ k ＝tan(α 1 )-α 1 (11)
the base radius of the involute tooth profile circular gear is as follows:
r b ＝r cos(α 1 ) (12)
the pitch circle of the involute tooth profile circular gear starts from the initial position and rolls along the curve of the driving wheel pitch or the curve of the driven wheel pitch along the arc length ds:
when the rolling arc length ds of the pitch circle of the involute tooth profile circular gear from the initial position is set, the tooth profile meshing point of the involute tooth profile circular gear and the driving wheel pitch curve or the driven wheel pitch curve is K, the intersection point of the generating line of the involute and the base circle of the involute tooth profile circular gear is N, and then the length KN of the generating line of the involute is set
The arc length between the initial position point M and the point N on the base circle of the involute tooth profile circular gear is represented as follows:
R kN ＝(θ k +α 2 +α 1 )r b (14)
the pressure angle α of the tooth profile meshing point is determined in Δ KON:
α＝tan -1 (θ k +α 2 +α 1 ) (15)
solving & lt KOM according to formula (16) 2 Comprises the following steps:
∠KOM 2 ＝α-α 1 (16)
in the delta KON, the distance R from the revolution center O of the involute tooth profile circular gear to the tooth profile meshing point is obtained K ：
Wherein f (mu) is a slip coefficient and is a given constant;
thus, the coordinates of the tooth profile meshing point K are determined:
wherein,
is a point M 2 The x-axis coordinate of (a) is,
is a point M 2 Y-axis coordinates of (a);
the analytical method for generating the variable cycloid tooth profile is as follows:
an elliptic gear rolls on a driving wheel pitch curve or a driven wheel pitch curve without sliding to form a variable cycloid tooth profile;
when the elliptic gear rolls on the curve of the driving wheel pitch or the curve of the driven wheel pitch without sliding, the rotation center O of the elliptic gear 2 The coordinates of (c) are calculated as follows:
wherein f is a concave-convex judging coefficient of a driving wheel pitch curve or a driven wheel pitch curve, and f is 1; x is the number of O2 X-axis coordinate, y, of the center of revolution of an elliptical gear O2 Is the y-axis coordinate of the revolution center of the elliptic gear; theta 0 The self-rotation angle displacement of the elliptic gear; theta m Setting theta as the initial included angle between the long axis of the elliptic gear and the x axis m 0; a is an elliptic pitch curve long semi-axis of the elliptic gear; b is an elliptic pitch curve short semi-axis of the elliptic gear; when (x) in the formula (19) j ，y j ) Theta is the coordinate (x, y) of the active wheel pitch curve 0 ＝θ c When (x) in the formula (19) j ，y j ) For the driven wheel pitch curve coordinate (x) c ，y c ) When theta is greater than theta 0 ＝θ；
The cycloid tooth profile coordinates of the driving wheel or the driven wheel are solved according to the formula (20):
in the formula, x b Is the x-axis coordinate, y, of the cycloid tooth profile of the driving wheel or the driven wheel b Is the y-axis coordinate of the cycloid tooth profile of the driving wheel or the driven wheel, and the spread angle theta 'of the elliptic gear' k ＝θ k 。
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