CN113761683B - Improved design method of heart-shaped non-circular gear pitch curve - Google Patents
Improved design method of heart-shaped non-circular gear pitch curve Download PDFInfo
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- 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
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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
- F16H55/08—Profiling
- F16H55/084—Non-circular rigid toothed member, e.g. elliptic gear
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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
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
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- G06F2111/00—Details relating to CAD techniques
- G06F2111/04—Constraint-based CAD
Abstract
The invention discloses an improved design method of a heart-shaped non-circular gear pitch curve, which provides an analysis type of the non-closed heart-shaped non-circular gear pitch curve according to the design principle of the non-circular gear pitch curve and the improvement of an original plane heart-shaped curve; further, by introducing the repaired pitch curve, a general expression of the closed heart-shaped non-circular gear pitch curve is obtained, and constraint conditions to be met by the repaired pitch curve are analyzed in order to ensure the smoothness of the repaired closed non-circular gear pitch curve; furthermore, the analytical formula of the repair pitch curve is given in the form of a polynomial through analysis of constraint conditions of the repair pitch curve, so that the analytical formula of the closed non-circular gear pitch curve can be obtained; finally, according to the meshing principle of the non-circular gear pair, a non-circular driven gear pitch curve conjugated with the heart-shaped non-circular driving gear pitch curve and constraint conditions thereof are provided. The improved heart-shaped non-circular gear pitch curve design method disclosed by the invention expands the design and application of the existing non-circular gear pitch curve, and has a certain guiding significance and practical value for designing a non-circular gear pair with any shape.
Description
Technical Field
The invention belongs to the field of non-circular gear design, and particularly relates to an improved design method of a heart-shaped non-circular gear pitch curve.
Background
With the development of non-circular gear products towards high speed, heavy load, light weight, high precision and automation, non-circular gears are one of important and commonly used transmission parts in mechanical products, and the design goal and the transmission performance of non-circular gears with arbitrary shapes are more and more complex and difficult to meet. The design target and the transmission performance of the non-circular gear are mainly determined by the design of the pitch curve, and early-stage design and research on the non-circular gear pitch curve mainly comprises the analysis of the geometric characteristics and the transmission performance of the eccentric circle, ellipse and other plane curves serving as the non-circular gear pitch curve due to the limitation of a computer technology and a processing machine tool.
At present, with the development planning of the German fourth industrial revolution of 'industrial 4.0', the 'two-way fusion' of automation and digitization and the breakthrough of new theories and new technologies such as modern computer technology, numerical control machine tools, intelligent manufacturing and the like, the design of non-circular gears with arbitrary shapes becomes possible.
Disclosure of Invention
The invention aims to expand the application of a non-circular gear in mechanical products, and discloses an improved design method of a heart-shaped non-circular gear pitch curve, which has a certain guiding significance and practical value for designing a non-circular gear design with any pitch curve shape.
In order to achieve the above purpose, the present invention provides the following technical solutions:
an improved design method of a heart-shaped non-circular gear pitch curve comprises the following steps:
step one, according to the design principle and constraint conditions of the non-circular gear pitch curve, an original plane heart-shaped curve is improved to obtain a non-closed heart-shaped non-circular gear pitch curve r u Polar equation of (θ), θ ε [0, θ t -θ s ]θ is the polar angle, θ s And theta t The start and stop angles are respectively the start and stop angles of the revolution of the heart-shaped non-circular gear pitch curve r (theta);
step two, obtaining a non-closed heart-shaped non-circular driving gear pitch curve r according to the meshing principle of the non-circular gear pair u (theta) conjugate non-circular driven gear pitch curve r m (θ m ) θ m Polar angle of non-closed non-circular driven gear pitch curve;
step three, by introducing a repair section curve r c (θ)，θ∈[θ t -θ s ,2π]In combination with non-closed heart-shaped non-circular gear pitch curve r u (θ) obtaining a general expression for a closed non-circular gear pitch curve R (θ)
Step four, according to the closed non-circular teethThe design principle of the wheel-section curve is that the repairing section curve R is obtained under the condition of ensuring the smoothness of the closed non-circular gear-section curve R (theta) c Constraint that (θ) should be satisfied
Step five, in order to obtain a uniquely determined repair pitch curve r c (θ) by repairing the section curve r c The constraint condition of (theta) is analyzed to obtain a third order polynomial with respect to the polar angle theta as a repair node curve r c General expression of (θ)
Wherein c 0 、c 1 、c 2 And c 3 Is a pending design parameter;
step six, combining non-closed heart-shaped non-circular gear pitch curve r u (θ) and repair section Curve r c The expression of (theta) and the constraint condition thereof are adopted to obtain a closed non-circular gear pitch curve R (theta) meeting the design principle of the closed non-circular gear pitch curve;
step seven, obtaining a non-circular driven gear pitch curve R conjugated with the closed non-circular gear pitch curve R (theta) according to the meshing principle of the closed non-circular gear c (θ c )，θ c To close the polar angle of the non-circular driven gear pitch curve.
In the first step, the first step is to perform,
1) Measuring original planar cardioid curve r in counter-clockwise direction along x-axis o Polar angle θ of (θ), θ ε [0,2π]And obtain corresponding original plane heart curve r o Polar equation for (θ):
r o (θ)＝a(1-cosθ),θ∈[0,2π]wherein a is the design parameter of the original plane heart curve, and a is equal to 0;
2) Obtaining an original plane heart curve r o Constraint of (θ):
wherein θ is s And theta t The start and stop angles are respectively the start and stop angles of the revolution of the heart-shaped non-circular gear pitch curve r (theta);
3) In order to obtain a non-circular gear with a turning motion starting from a polar angle of 0, a non-closed heart-shaped non-circular gear pitch curve r is obtained in combination with steps 1) and 2) u Polar equation for (θ):
in the second step, according to the meshing principle of the non-circular gear, when the non-circular gear pitch curve r is satisfied u (θ) as a non-circular drive gear pitch curve in a pair of conjugate non-circular gear pairs, a non-circular driven gear pitch curve r conjugated thereto m (θ m ) The method comprises the following steps:
wherein A is the center distance of a conjugated non-circular gear pair, and the sign is thatThe upper middle minus sign "-" is applicable to the external-engagement non-circular gear pair and the lower plus sign "+" is applicable to the internal-engagement non-circular gear pair,
and obtaining the transmission relation of non-closed heart-shaped non-circular gear pair
In the fourth step, the condition of smoothness of the closed non-circular gear segment curve R (θ) is that no non-minutiae exist on the closed non-circular gear segment curve R (θ).
In step six, the determined non-closed heart-shaped non-circular gear pitch curve r u (θ) is a known condition; second, through the repair node curve r in the fourth step c The constraint that (θ) should be satisfied and the third order polynomial in step five regarding the polar angle θ are used as the repair pitch curve r c General expression of (θ), solving for the repair section curve r c Pending parameter c of (θ) 0 、c 1 、c 2 And c 3 And expressions thereof; finally, the solved repair section curve r c And substituting the (theta) into the closed non-circular gear pitch curve R (theta) to obtain the repaired closed non-circular gear pitch curve R (theta).
In step seven, according to the closed condition of non-circular gear meshing, the external meshing and internal meshing non-circular gear pitch curves R (theta) conjugated with the closed non-circular gear pitch curves R (theta) c (θ c )：
Wherein A is c Is the conjugate closed non-circular gear pair center distance throughObtained N c Is the number of leaves of the conjugate closed non-circular driven gear pitch curve, and N c Is an integer, and is preset by a designer according to the application condition of the closed non-circular gear pair.
The invention has the beneficial effects that: in order to further expand the application of the non-circular gear in mechanical products, the invention discloses an improved heart-shaped non-circular gear pitch curve design method, which starts from an original plane heart-shaped curve and provides a non-closed heart-shaped non-circular gear pitch curve design method and a closed heart-shaped non-circular gear pitch curve design method according to the design characteristics of the non-circular gear pitch curve and the meshing principle of a non-circular gear pair.
Drawings
Fig. 1 shows an original planar cardioid curve r with design parameter a=1 in an embodiment of the present invention o (θ)。
FIG. 2 is a flow chart of a non-closed heart-shaped non-circular gear pitch curve design in an embodiment of the invention.
FIG. 3a shows an embodiment of the present invention with r u (0)＝r u (θ t -θ s ) Characteristic non-closed heart-shaped non-circular gear pitch curve r u (θ) design example.
FIG. 3b shows an embodiment of the present invention with r u (0)≠r u (θ t -θ s ) Characteristic non-closed heart-shaped non-circular gear pitch curve r u (θ) design example.
FIG. 4a is a graph of a pitch curve r of a non-closed heart-shaped non-circular drive gear according to an embodiment of the present invention and as illustrated in FIG. 3a u (θ)(r u (0)＝r u (θ t -θ s ) Conjugate external engagement non-circular driven gear pitch curve r m (θ m )。
FIG. 4b shows a non-closed heart-shaped non-circular driving gear pitch curve r according to an embodiment of the present invention and as shown in FIG. 3a u (θ)(r u (0)＝r u (θ t -θ s ) Conjugated internal engaged non-circular driven gear pitch curve r m (θ m )。
FIG. 5 shows the driving relationship i corresponding to the conjugate non-circular gear pitch curve shown in FIG. 4 in an embodiment of the present invention m (θ)。
FIG. 6 is a flow chart of a closed non-circular gear pitch curve design after repair in an embodiment of the present invention.
FIG. 7a is a graph r of a non-closed heart-shaped non-circular gear pitch curve according to an embodiment of the present invention and FIG. 3a u (θ) (r u (0)＝r u (θ t -θ s ) A corresponding closed non-circular gear pitch curve R (θ).
FIG. 7b is a graph r of a non-closed heart-shaped non-circular gear pitch curve according to an embodiment of the present invention and FIG. 3b u (θ) (r u (0)≠r u (θ t -θ s ) A corresponding closed non-circular gear pitch curve R (θ).
FIG. 8a is a schematic diagram of an external mesh non-circular driven gear pitch curve R conjugated with the closed non-circular driving gear pitch curve R (θ) shown in FIG. 7b in accordance with an embodiment of the present invention c (θ c )。
FIG. 8b is a section R of an internally engaged non-circular driven gear conjugated to the section R (θ) of the closed non-circular drive gear described in FIG. 7b in accordance with an embodiment of the present invention c (θ c )。
Detailed Description
The following description of the embodiments of the present invention will be made with reference to the accompanying drawings, in which it is evident that the embodiments described are only some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by one of ordinary skill in the art based on the embodiments of the invention without undue burden, are within the scope of the invention.
It should be noted that all directional indicators (such as up, down, left, right, front, and rear … …) in the embodiments of the present invention are merely used to explain the relative positional relationship, movement, etc. between the components in a particular posture (as shown in the drawings), and if the particular posture is changed, the directional indicator is changed accordingly.
Fig. 1 is an original planar cardioid curve r with design parameter a=1 o (θ). In the figure, Γ (o-xy) is the plane rectangular coordinate system, o is the origin of coordinates, assuming the original plane heart curve r o Polar angle θ (θ ε [0,2π)]) Measured in a counter-clockwise direction along the x-axis, a planar cardioid curve r o The polar equation for (θ) is:
r o (θ)＝a(1-cosθ),θ∈[0,2π] (1)
where a (a. Noteq.0) is a planar cardioid design parameter.
For convenience of solving and calculating, the value of the design parameter a in the present invention is 1 unless otherwise specified, and the design parameter a=1 in this embodiment should not be considered as limiting the present invention, but any modification based on the spirit of the present invention (any value of the design parameter a other than zero) should be within the scope of the present invention.
According to the design principle of the non-circular gear pitch curve, in order to design a heart-shaped non-circular gear pitch curve, a plane heart-shaped curve r o (θ) the following constraints should be satisfied:
in θ s And theta t The start and stop angles of the revolution of the heart-shaped non-circular gear pitch curve r (theta) are respectively shown.
Further, in order to make the non-circular gear start to rotate from the polar angle of 0, the non-closed heart-shaped non-circular gear pitch curve r can be obtained by combining the formula (1) and the formula (2) u Polar equation for (θ):
furthermore, according to the meshing principle of the non-circular gear, when the non-circular gear pitch curve r of the formula (3) is satisfied u (θ) as a non-circular drive gear pitch curve in a pair of conjugate non-circular gear pairs, a non-circular driven gear pitch curve r conjugated thereto m (θ m ) The method comprises the following steps:
wherein A is the center distance of a conjugated non-circular gear pair, and the symbolThe upper middle minus sign "-" applies to the external-gear non-circular gear bless and the lower plus sign "+" applies to the internal-gear non-circular gear pair.
FIG. 2 is a flow chart of non-closed heart-shaped non-circular gear pitch curve design. Non-closed heart-shaped non-circular gear pitch curve r depicted in the figure u Start-stop rotation angle θ of (θ) s 、θ t And the center distance A of the conjugate non-circular gear pair is a known condition; will start and stop the rotation angle theta s 、θ t Substituting into (3) to obtain non-closed heart-shaped non-circular gear pitch curve r u (θ); finally, the center distance A of the non-circular gear pair is solved to obtain a node curve r u And (θ) is substituted into the formula (4) to obtain a pair of conjugate non-circular gear Fu node curves.
FIGS. 3a and 3b are two exemplaryNon-closed heart-shaped non-circular gear pitch curve r u (θ) design example. Non-closed heart-shaped non-circular gear pitch curve r depicted in the figure u The design parameters for (θ) are set forth in Table 1, where all parameters are in International units. In particular, the units of all parameters according to the invention are of the international system of units.
Table 1 two typical non-closed heart-shaped non-circular gear pitch curves r u Design parameters of (θ)
FIGS. 4a and 4b are graphs of non-closed heart-shaped non-circular drive gear pitch curves r, respectively, as described in FIG. 3a u (θ)(r u (0)＝r u (θ t -θ s ) Conjugated non-circular driven gear pitch curve r m (θ m ). Non-circular driven gear pitch curve r depicted in the figure m (θ m ) The polar equations of (c) and their associated design parameters are listed in table 2.
Table 2 and non-closed heart-shaped non-circular drive gear pitch curve r depicted in fig. 3a u (θ)(r u (0)＝r u (θ t -θ s ) Non-circular driven gear pitch curve r) of the conjugate m (θ m ) Equation and design parameters thereof
FIG. 5 is a transmission relationship i corresponding to the conjugated non-circular gear secondary pitch curve depicted in FIG. 4 m (θ). The transmission relation i depicted in the figure m (θ) can be determined by the formula (5).
Further, on the basis of equation (3), the repair section curve r can be introduced c (θ) (θ∈[θ t -θ s ,2π]) Obtaining a closed non-circular gear pitch curve R (θ):
according to the design principle of the closed non-circular gear pitch curve, in order to ensure the smoothness of the closed non-circular gear pitch curve R (theta) (i.e. ensure that no non-micropoints exist on the closed non-circular gear pitch curve R (theta)), the pitch curve R is repaired c (θ)(θ∈[θ t -θ s ,2π]) The following constraints should be satisfied:
further, by analyzing the constraint condition in the formula (7), it is known that the heart-shaped non-circular gear pitch curve r u Start-stop rotation angle θ of (θ) s 、θ t Given, repair section curve r c The expression of (θ) should contain at least four undetermined design parameters for the unique determination of the repair curve r c (θ), the embodiment of the invention takes a cubic polynomial as shown in the formula (8) about the polar angle θ as a repair node curve r c And (θ) completing the design of the closed non-circular gear pitch curve R (θ).
Wherein, c 0 、c 1 、c 2 And c 3 Is a pending design parameter.
FIG. 6 is a flow chart of the closed non-circular gear pitch curve design after repair. In the figure, a certain non-closed heart-shaped non-circular gear pitch curve r is used u (θ) is a known condition; next, a repair curve r is solved by the formulas (7) and (8) c Pending parameter c of (θ) 0 、c 1 、c 2 And c 3 And expressions thereof; finally, the solved repair section curve r c Substitution of (θ)And (6) obtaining the repaired closed non-circular gear pitch curve R (theta).
FIGS. 7a and 7b are graphs of non-closed heart-shaped non-circular gear pitch curves r corresponding to FIGS. 3a and 3b, respectively u (θ) a corresponding closed non-circular gear pitch curve R (θ). The design parameters for the closed non-circular gear pitch curve R (θ) depicted in the figures are listed in Table 3.
Table 3 non-closed heart shaped non-circular gear pitch curve r in fig. 3 u Closed non-circular gear pitch curve R (theta) equation corresponding to (theta) and design parameters thereof
Further, according to the closed condition of the non-circular gear meshing, the external meshing and internal meshing non-circular gear pitch curves R (theta) conjugated with the closed non-circular gear pitch curves R (theta) c (θ c ) Can be expressed as:
wherein A is c The center distance of the closed non-circular gear pair which is conjugate can be calculated by the formula (10).
Wherein N is c Is the number of leaves of the conjugate closed non-circular driven gear pitch curve, and N c Is an integer, and is preset by a designer according to the application condition of the closed non-circular gear pair.
FIGS. 8a and 8b are non-circular driven gear pitch curves R conjugated with the closed non-circular drive gear pitch curve R (θ) described in FIG. 7b, respectively c (θ c ). Non-circular driven gear pitch curve R as depicted c (θ c ) The polar equations and their associated design parameters are listed in table 4.
Table 4 and closed non-circular main body as described in fig. 7bNon-circular driven gear pitch curve R with conjugated driven gear pitch curve R (θ) c (θ c ) Equation and design parameters thereof
The examples should not be construed as limiting the invention, but any modifications based on the spirit of the invention should be within the scope of the invention.
Claims (6)
1. An improved design method of a heart-shaped non-circular gear pitch curve is characterized by comprising the following steps of: the method comprises the following steps:
step one, according to the design principle and constraint conditions of the non-circular gear pitch curve, an original plane heart-shaped curve is improved to obtain a non-closed heart-shaped non-circular gear pitch curver u (θ) Is used as a reference to the polar equation of (a),，θin the form of a polar angle, the angle of the polar,θ s andθ t respectively heart-shaped non-circular gear pitch curvesr(θ) A start-stop angle of the rotation;
step two, obtaining a non-closed heart-shaped non-circular driving gear pitch curve according to the meshing principle of the non-circular gear pairr u (θ) Conjugate non-circular driven gear pitch curver m (θ m )
，θ m Is the polar angle of the non-closed non-circular driven gear pitch curve,Ais the center distance of the conjugated non-circular gear pair;
step three, by introducing a repair section curver c (θ)，In combination with non-closed heart-shaped non-circular gear pitch curvesr u (θ) Obtaining a closed non-circular gear pitch curveR(θ) General expression->；
Step four, guaranteeing the closed non-circular gear pitch curve according to the design principle of the closed non-circular gear pitch curveR(θ) Under the condition of smoothness of (2) obtaining a repair joint curver c (θ) Constraint to be satisfied
；
Step five, in order to obtain a uniquely determined repair pitch curver c (θ) By repairing the pitch curver c (θ) The constraint condition is analyzed to obtain a value related to the polar angleθThird degree polynomial as repair node curver c (θ) General expression of (2)
Wherein, the method comprises the steps of, wherein,c 0 、c 1 、c 2 andc 3 is a pending design parameter;
step six, combining non-closed heart-shaped non-circular gear pitch curvesr u (θ) And repairing the section curver c (θ) The expression and the constraint condition of the (a) to obtain the closed non-circular gear pitch curve meeting the design principle of the closed non-circular gear pitch curveR(θ)；
Step seven, obtaining a pitch curve of the closed non-circular gear according to the meshing principle of the closed non-circular gearR(θ) Conjugated non-circular driven gear pitch curveR c (θ c )，θ c To close the polar angle of the non-circular driven gear pitch curve.
2. The method for designing an improved heart-shaped non-circular gear pitch curve according to claim 1, wherein: in the first step, the first step is to perform,
1) Edge of the framexMeasuring original planar cardioid curve in anticlockwise direction in axial directionr o (θ) Polar angle of (2)θ，θ∈[0，2π]And obtain corresponding original plane heart curver o (θ) Polar equation of (2):
whereinaParameters are designed for the original planar cardioid curve,a ≠ 0；
2) Obtaining an original plane heart-shaped curver o (θ) Is a constraint on (c):
whereinθ s Andθ t respectively heart-shaped non-circular gear pitch curvesr(θ) A start-stop angle of the rotation;
3) In order to obtain a non-circular gear with a turning motion starting from a polar angle of 0, a non-closed heart-shaped non-circular gear pitch curve is obtained in combination with steps 1) and 2)r u (θ) Polar equation of (2):
。
3. the method for designing an improved heart-shaped non-circular gear pitch curve according to claim 2, wherein: in the second step, according to the meshing principle of the non-circular gear, when the non-circular gear pitch curve is satisfiedr u (θ) As the non-circular driving gear pitch curve in a pair of conjugated non-circular gear pairs, the non-circular driven gear pitch curve conjugated therewithr m (θ m ) The method comprises the following steps:
，
wherein the symbol is'"upper middle minus">The plus sign "+" below is applicable to an external-engagement non-circular gear pair,
and obtaining the transmission relation of non-closed heart-shaped non-circular gear pair
。
4. The method for designing an improved heart-shaped non-circular gear pitch curve according to claim 1, wherein: in step four, the closed non-circular gear pitch curveR(θ) Provided that the smoothness of (2) is a closed non-circular gear pitch curveR(θ) No non-minutiae are present thereon.
5. The method for designing an improved heart-shaped non-circular gear pitch curve according to claim 1, wherein: in the sixth step, the determined non-closed heart-shaped non-circular gear pitch curver u (θ) Is a known condition; second, through the repair section curve in the fourth stepr c (θ) Constraints to be met and in step five regarding polar angleθThird degree polynomial as repair node curver c (θ) Solving for the repair section line by the general expression of (2)r c (θ) Is to be determined by the parameters ofc 0 、c 1 、c 2 Andc 3 and expressions thereof; finally, the solved repair node curver c (θ) Substituting closed non-circular gear pitch curveR(θ) The repaired closed non-circular gear pitch curve can be obtainedR(θ)。
6. The method for designing an improved heart-shaped non-circular gear pitch curve according to claim 1, wherein: in the seventh step, according to the closed condition of the non-circular gear meshing, the non-circular gear pitch curve is closedR(θ) Conjugate external and internal engagement non-circular gear pitch curvesR c (θ c )：
Wherein the method comprises the steps ofA c Is the conjugate closed non-circular gear pair center distance throughThe process is carried out in a manner that,N c is the number of the leaves of the conjugate closed non-circular driven gear pitch curve, andN c is an integer, and is preset by a designer according to the application condition of the closed non-circular gear pair.
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| CN110425377A (en) * | 2019-08-16 | 2019-11-08 | 燕山大学 | A kind of inertia pipe robot of bidirectional-movement |
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| US6644947B2 (en) * | 2002-03-14 | 2003-11-11 | Tuthill Corporation | Wave tooth gears using identical non-circular conjugating pitch curves |
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