WO2023097802A1 - 复波式活齿减速器内齿廓设计方法及两级减速器 - Google Patents

复波式活齿减速器内齿廓设计方法及两级减速器 Download PDF

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WO2023097802A1
WO2023097802A1 PCT/CN2021/139627 CN2021139627W WO2023097802A1 WO 2023097802 A1 WO2023097802 A1 WO 2023097802A1 CN 2021139627 W CN2021139627 W CN 2021139627W WO 2023097802 A1 WO2023097802 A1 WO 2023097802A1
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tooth
axis
coordinate system
movable
stage
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PCT/CN2021/139627
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French (fr)
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王家序
李俊阳
唐挺
曾星宇
韩彦峰
肖科
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重庆大学
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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
    • F16H57/00—General details of gearing
    • 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
    • F16H49/00—Other gearings
    • F16H49/001—Wave gearings, e.g. harmonic drive transmissions
    • 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]
    • G06F30/20—Design optimisation, verification or simulation
    • 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
    • F16H57/00—General details of gearing
    • F16H2057/0087—Computer aided design [CAD] specially adapted for gearing features; Analysis of gear systems
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/14—Force analysis or force optimisation, e.g. static or dynamic forces
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02T—CLIMATE CHANGE MITIGATION TECHNOLOGIES RELATED TO TRANSPORTATION
    • Y02T90/00—Enabling technologies or technologies with a potential or indirect contribution to GHG emissions mitigation

Definitions

  • the invention relates to the field of transmissions, in particular to a method for designing an internal tooth profile of a compound wave movable tooth reducer and a two-stage reducer.
  • Harmonic and RV reducers achieve high precision through the error averaging effect of multi-tooth meshing, and increase the number of meshing teeth to improve carrying capacity, and are widely used in industrial robot joints.
  • the harmonic reducer transforms the gear index circle into an ellipse through the deformation of the thin-walled flexible wheel to achieve double-wave multi-tooth meshing.
  • the rigidity is poor, and the precise design and modification of the time-variable pitch curve meshing pair are difficult, the tooth surface is prone to wear, and the accuracy retention is poor.
  • the processing technology of the flexible spline is poor, and it is prone to fatigue fracture under the condition of periodic alternating stress.
  • the RV reducer achieves multi-tooth meshing through contact elastic deformation, so it requires extremely high machining accuracy.
  • Micron-level errors will turn multi-tooth meshing into multi-tooth interference, resulting in reliability problems such as wear, vibration, and noise.
  • the structure is very complicated, and the difficulty of ensuring the machining accuracy is further improved after the assembly dimension chain is lengthened.
  • the movable tooth transmission realizes the multi-tooth meshing of the time-varying curve by releasing the constraints of the gear and the ring gear, and avoids the limitation of the multi-tooth meshing between the harmonic and the RV reducer, so the structure is simple and the manufacturability is good.
  • the design of the tooth profile of the internal ring gear requires complex calculations, and the accuracy is not ideal, which makes the subsequent processing relatively complicated and increases the processing cost.
  • the present invention provides a method for designing the internal tooth profile of a compound-wave movable tooth reducer and a two-stage reducer, which can obtain the internal tooth profile of the internal ring gear through simple calculations compared with the prior art, reducing the subsequent Processing complexity, thereby reducing processing costs.
  • the present invention provides a method for designing the internal tooth profile of a compound wave movable tooth reducer.
  • the movable tooth reducer includes an inner ring gear, a movable tooth assembly and an elliptical cam.
  • the movable tooth assembly includes a movable tooth and a movable tooth cage.
  • the movable tooth Under the action of the elliptical cam, the teeth mesh and roll or disengage with the ring gear; wherein the design method of the internal tooth profile of the ring gear includes the following steps:
  • M 21 is the coordinate transformation matrix from the active tooth coordinate system S 1 to the internal tooth coordinate system S 2
  • W 21 is the base vector transformation matrix from the active tooth coordinate system S 1 to the internal tooth coordinate system S 2
  • r 2 is The radial vector of the tooth profile coordinates of the active tooth in the internal tooth coordinate system S 2
  • n 2 is the normal vector in the internal tooth coordinate system S 2 of the tooth profile coordinates of the active tooth
  • v 2 (12) is the relative velocity of the movable tooth surface and the internal tooth surface at the contact point O f of the two conjugate surfaces;
  • step d Turn the conjugate angle in step d Substitute into formula (1) to obtain the internal tooth profile.
  • step c
  • ⁇ is the angle between the normal line of the contact point O f and the Y axis; ⁇ is the radial deformation of the elliptical cam; is the rotation angle of the movable tooth relative to the major axis of the cam; the above parameters with superscript "'" are derivatives.
  • step b
  • s is the arc length parameter of the movable tooth
  • n x is the component of the normal vector n 1 on the X axis
  • n y is the component of the normal vector n 1 on the Y axis
  • ⁇ is the angle between the contact point O f vector and the Y axis
  • the invention also discloses a double-stage compound wave movable tooth reducer, which includes a first-stage deceleration assembly and a second-stage deceleration assembly that is in transmission cooperation with the first-stage deceleration assembly, and the first-stage deceleration assembly and the second-stage deceleration assembly
  • the deceleration components are compound wave movable tooth reducers, and the internal tooth profile of the internal ring gear is obtained by the above-mentioned design method.
  • the elliptical cam of the first-stage deceleration assembly is used to input power
  • the ring gear of the first-stage deceleration assembly is fixed
  • the movable tooth cage of the first-stage deceleration assembly is rigid to the movable tooth cage of the second-stage deceleration assembly.
  • transmission, and the rigid transmission between the elliptical cam of the first-stage reduction assembly and the elliptical cam of the second-stage reduction assembly, and the ring gear of the second-stage reduction assembly is used to output power.
  • the design method of the internal tooth profile of the compound wave movable tooth reducer and the two-stage reducer of the present invention aiming at the meshing form of the time-varying pitch curve of the elliptical cam profile, the B matrix kinematics method is used to solve the two-stage transmission respectively Since the meshing matrix has uniqueness that does not change with the change of the form of the conjugate surface, the calculation of the B matrix has nothing to do with the geometric parameters of the conjugate surface, and the same expression can be used regardless of the expression of the conjugate surface.
  • a B matrix is used for calculation and analysis, which simplifies the calculation process, improves design efficiency and design accuracy, reduces the complexity of subsequent processing, and reduces processing costs compared with the traditional design method of movable tooth profile based on envelope theory.
  • Fig. 1 is a schematic diagram of the movement principle of the compound wave movable tooth transmission
  • Figure 2 is a schematic diagram of the mechanism of the two-stage compound wave movable tooth reducer
  • Figure 3 is a schematic cross-sectional view of the first-stage reduction assembly (the structure of the second-stage reduction assembly is the same, but the parameters may be different).
  • Movable teeth are generally cylindrical or bead-shaped, and the center refers to the center of the cross-section.
  • the center of the elliptical cam also refers to the center of the ellipse of the cross-section, which will not be repeated here;
  • the cam ellipse is a standard ellipse, According to the parameter equation of the standard elliptic cam, the elliptic cam curve
  • the vector radius ⁇ is
  • r b is the radius of the base circle of the cam
  • r is the radius of the cylindrical movable tooth
  • i is the transmission ratio of the single-stage movable tooth
  • the method for designing the internal tooth profile of the compound wave movable tooth reducer in this embodiment includes the following steps:
  • the determination of the number of movable teeth and internal teeth (fixed internal teeth or output internal teeth) of the two stages needs to be determined according to the principle of the NN transmission mechanism; for example, a two-stage reducer, as shown in Figure 2 Shown, the movable tooth 1 and internal gear 2 of the first-stage reduction assembly are fixed gears; the movable teeth 2 and internal gear 4 of the second-stage reduction assembly are output internal teeth.
  • the transmission ratio i H4 of the mechanism can be obtained as
  • n H and n 4 are the rotational speeds of the elliptical cam and the output internal teeth respectively
  • Z 1 and Z 3 are the number of movable teeth of the first-stage reduction assembly and the second-stage reduction assembly respectively
  • Z 2 and Z 4 are the number of active teeth of the first-stage reduction assembly and the second-stage reduction assembly Fixed number of internal teeth and output internal teeth for primary and secondary reduction assemblies.
  • the new type of movable tooth transmission is a symmetrical multi-teeth meshing
  • the number of movable teeth and internal teeth and the difference between the meshing teeth of the two stages Z 2 -Z 1 , Z 4 -Z 3 should be even numbers to avoid the occurrence of unbalanced force when the cam is running at high speed.
  • the number of teeth Z 1 and Z 3 of the two-stage movable teeth can be obtained by rounding off.
  • a. Establish a coordinate system including:
  • the elliptical cam is The input device, the internal teeth are fixed, and the cage is output; the cam rotates clockwise, and the movable tooth rotates counterclockwise under the action of the cam. During the rotation process, it is always tangent to the cam, and the fixed internal tooth is always in contact with the movable tooth. Based on Relative kinematic relationship, using the following B-matrix kinematics method to solve the internal tooth profile;
  • M 21 is the coordinate transformation matrix from the active tooth coordinate system S 1 to the internal tooth coordinate system S 2
  • W 21 is the base vector transformation matrix from the active tooth coordinate system S 1 to the internal tooth coordinate system S 2
  • r 2 is The radial vector of the tooth profile coordinates of the active tooth in the internal tooth coordinate system S 2
  • n 2 is the normal vector in the internal tooth coordinate system S 2 of the tooth profile coordinates of the active tooth
  • the radial vector r 1 and the normal vector n 1 of the tooth profile coordinates of the active tooth are respectively
  • s is the arc length parameter of the movable tooth
  • n x is the component of the normal vector n 1 on the X axis
  • n y is the component of the normal vector n 1 on the Y axis
  • ⁇ is the angle between the normal line of the contact point O f and the Y axis; ⁇ is the radial deformation of the elliptical cam; is the rotation angle of the movable tooth relative to the major axis of the cam; the above parameters with superscript "'" are derivatives.
  • v 2 (12) is the relative velocity of the movable tooth surface and the internal tooth surface at the contact point O f of the two conjugate surfaces;
  • step d Substitute into formula (1), and combine formulas (4) and (5) to obtain the profile of the internal tooth profile;
  • the meshing matrix has uniqueness that does not change with the change of the form of the conjugate surface
  • the calculation of the B matrix has nothing to do with the geometric parameters of the conjugate surface, and the same B matrix can be used for calculation and analysis regardless of the expression of the conjugate surface , which greatly reduces the computational workload.
  • the elements in the fourth row of the B matrix are all zeros, and the 3 ⁇ 3 matrix on the upper left is an anti-symmetric matrix with fewer calculation elements; this simplifies the entire calculation process.
  • the present invention also discloses a double-stage compound wave movable tooth reducer, which includes a first-stage reduction assembly and a second-stage reduction assembly that is in transmission cooperation with the first-stage reduction assembly.
  • Both the deceleration assembly and the second-stage deceleration assembly are compound wave movable tooth reducers, and the internal tooth profile of the inner ring gear is obtained by using the above-mentioned design method; the first deceleration assembly and the second
  • the internal teeth of the second reduction assembly (the fixed internal teeth of the first-stage reduction assembly and the output internal teeth of the second-stage reduction assembly) are solved for the tooth profile, and the obtained tooth profile is shown in Figure 3 (only the internal teeth of the first-stage reduction assembly internal tooth profile of the ring gear as an example).
  • the elliptical cam 5 of the first-stage deceleration assembly is used to input power, the ring gear 1 of the first-stage deceleration assembly is fixed, and the movable tooth cage 7 of the first-stage deceleration assembly is connected with the second-stage deceleration assembly.
  • the movable tooth cage 8 is rigidly driven, and the elliptical cam 5 of the first-stage reduction assembly is rigidly transmitted with the elliptical cam 6 of the second-stage reduction assembly, and the ring gear 4 of the second-stage reduction assembly is used to output power;
  • the movable tooth holder 7 of the first deceleration assembly is slidably fitted with the movable tooth 2 of the first deceleration assembly in the radial direction
  • the movable tooth holder 8 of the second deceleration assembly is slidably fitted with the first
  • the movable tooth 3 of the deceleration assembly will not be described in detail here;
  • the present invention uses an elliptical cam to replace the traditional eccentric wheel to realize the axisymmetric meshing area, so that the forces acting on the cam can cancel each other and reduce vibration, noise and impact; at the same time, combined with the planetary transmission configuration method with less tooth difference, through the complex wave two
  • the multi-stage transmission realizes the large-speed-ratio compound-wave movable tooth transmission with simple structure, which reduces the difficulty of parts processing.
  • the internal teeth in the first stage are fixed internal teeth
  • the internal teeth in the second stage are output internal teeth.
  • the internal teeth are fixed, the elliptical cam rotates, and the cam profile curve generates a radial thrust, forcing the cylindrical movable tooth to roll along the working tooth profile of the internal tooth in the opposite direction, and the rolling of the movable tooth drives the rotation of the cage. Because the cages of the two stages are integrated, the movement is transmitted to the second stage, and under the action of the movable teeth and the cam in the second stage, the output internal teeth rotate, thereby realizing the speed conversion function of the movable tooth transmission.

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Abstract

本发明提供的一种复波式活齿减速器内齿廓设计方法及两级减速器，针对椭圆凸轮轮廓的时变节曲线啮合形式，采用B矩阵运动学法求解分别求解两级传动的内齿齿廓，由于啮合矩阵具有不随共轭曲面形式的改变而变动的唯一性，因此，B矩阵的计算与共轭曲面的几何参数无关，无论共轭曲面的表达式如何，都可以采用同一个B矩阵进行计算分析，相比传统基于包络理论的活齿齿廓设计方法，简化了计算过程，提高了设计效率与设计精度，降低后续加工的复杂程度，从而减少加工成本。

Description

复波式活齿减速器内齿廓设计方法及两级减速器 技术领域
本发明涉及变速器领域，尤其涉及一种复波式活齿减速器内齿廓设计方法及两级减速器。
背景技术
谐波和RV减速器通过多齿啮合的误差均化效应实现高精度、并增加啮合齿数提高承载能力，广泛应用于工业机器人关节。谐波减速器通过薄壁柔轮变形使齿轮分度圆变成椭圆实现双波多齿啮合，刚度差，并且时变节曲线啮合副的精确设计与修形困难，齿面易产生磨损，精度保持性差，另外柔轮加工工艺性差，在周期交变应力左右下容易发生疲劳断裂。RV减速器通过接触弹性变形实现多齿啮合，因此对加工精度要求极高,微米级的误差就会使多齿啮合转变为多齿干涉，产生磨损、振动、噪声等可靠性问题，另一方面，为实现大速比和多摆线轮的动平衡，结构非常复杂，装配尺寸链加长后进一步提高了保证加工精度的难度。
活齿传动通过释放齿轮与齿圈的约束实现时变节曲线的多齿啮合，并且避免了谐波与RV减速器实现多齿啮合时的局限性，因此结构简单、工艺性好。但其内齿圈的齿廓的设计则需要复杂的计算才能获得，并且精度并不理想，使得后续的加工过程也相对复杂，增加加工成本。
因此，需要一种设计方法，通过相对于现有技术简单的计算即能够获得内齿圈的内齿廓形，降低后续加工的复杂程度，从而减少加工成本。
发明内容
有鉴于此，本发明提供一种复波式活齿减速器内齿廓设计方法及两级减速器，通过相对于现有技术简单的计算即能够获得内齿圈的内齿廓形，降低后续加工的复杂程度，从而减少加工成本。
本发明提供的复波式活齿减速器内齿廓设计方法，活齿减速器包括内齿圈、 活齿组件和椭圆凸轮，所述活齿组件包括活齿和活齿保持架，所述活齿在椭圆凸轮的作用下与内齿圈啮合滚动或或脱离啮合；其中所述内齿圈的内齿廓的设计方法包括下列步骤：
a.建立坐标系，包括
以椭圆转动中心为原点O，凸轮短轴为X轴，长轴为Y轴建立平面直角坐标系OXY；以活齿中心为原点O 1，竖直方向为Y 1轴，水平方向为X 1轴建立活齿平面直角坐标系S 1{O 1X 1Y 1}；以椭圆转动中心为原点O 2，竖直方向为Y 2轴，水平方向为X 2轴建立内齿平面直角坐标系S 2{O 2X 2Y 2}，其中原点O 2与原点O重合，转动前，Y 2轴与Y轴重合；
b.将活齿齿廓坐标的径矢r 1与法向量n 1变换到内齿坐标系S 2中，
则
Figure PCTCN2021139627-appb-000001
其中，M 21为由活齿坐标系S 1到内齿坐标系S 2的坐标变换矩阵，W 21为由活齿坐标系S 1到内齿坐标系S 2的底矢变换矩阵，r 2为活齿齿廓坐标在内齿坐标系S 2中的径矢，n 2为活齿齿廓坐标在内齿坐标系S 2中的法向量；
c.获取啮合矩阵B
表达式为：
Figure PCTCN2021139627-appb-000002
d.获取共轭转角
通过公式
Figure PCTCN2021139627-appb-000003
求解不同弧长参数s对应的共轭转角
Figure PCTCN2021139627-appb-000004
其中v 2 (12)为两共轭曲面活齿曲面与内齿曲面在接触点O f处的相对速度；
e.将步骤d中的共轭转角
Figure PCTCN2021139627-appb-000005
代入公式(1)，获取内齿廓。
进一步，步骤c中，
Figure PCTCN2021139627-appb-000006
其中，δ为接触点O f法线与Y轴夹角；ω为椭圆凸轮的径向变形量；
Figure PCTCN2021139627-appb-000007
为活齿相对于凸轮长轴的转角；上述参数带有上标“′”为求导数。
进一步，步骤b中，
设在坐标系S 1中，活齿齿廓坐标的径矢r 1与法向量n 1分别为
Figure PCTCN2021139627-appb-000008
其中s为活齿的弧长参量；n x为法向量n 1在X轴的分量；n y为法向量n 1在Y轴的分量；φ为接触点O f矢径与Y轴夹角；
坐标系S 2中的共轭方程为，
Figure PCTCN2021139627-appb-000009
且
Figure PCTCN2021139627-appb-000010
本发明还公开了一种双级复波式活齿减速器，包括第一级减速组件和与第一级减速组件传动配合的第二级减速组件，所述第一级减速组件和第二级减速组件均为复波式活齿减速器，且均采用所述的设计方法获得内齿圈的内齿廓。
进一步，所述第一级减速组件的椭圆凸轮用于输入动力，第一级减速组件的内齿圈固定，第一级减速组件的活齿保持架与第二级减速组件的活齿保持架刚性传动，以及，第一级减速组件的椭圆凸轮与第二级减速组件的椭圆凸轮刚性传动，所述第二级减速组件的内齿圈用于输出动力。
本发明的有益效果：本发明的复波式活齿减速器内齿廓设计方法及两级减速器，针对椭圆凸轮轮廓的时变节曲线啮合形式，采用B矩阵运动学法求解分别求解两级传动的内齿齿廓，由于啮合矩阵具有不随共轭曲面形式的改变而变动的唯一性，因此，B矩阵的计算与共轭曲面的几何参数无关，无论共轭曲面的表达式如何，都可以采用同一个B矩阵进行计算分析，相比传统基于包络理论的活齿齿廓设计方法，简化了计算过程，提高了设计效率与设计精度，降低后续加工的复杂程度，从而减少加工成本。
附图说明
下面结合附图和实施例对本发明作进一步描述：
图1为复波式活齿传动运动原理示意图；
图2为二级复波式活齿减速器机构简图；
图3为一级减速组件横截面示意图(二级减速组件结构与其相同，参数可能不同)。
具体实施方式
要确定本发明提供的复波式活齿减速器内齿廓设计方法，首先要行以下基础工作：
以椭圆凸轮回转中心O为原点，以凸轮长轴为Y轴，则椭圆凸轮和活齿的相对运动几何关系如图1所示，其中各参数意义如下表：
Figure PCTCN2021139627-appb-000011
活齿一般为圆柱或者圆珠形，中心指的是横截面的圆心，同理，椭圆凸轮的中心也指的是横截面的椭圆中心，在此不再赘述；凸轮椭圆采用的是标准椭圆，根据标准椭圆凸轮的参数方程，此时椭圆凸轮曲线
Figure PCTCN2021139627-appb-000012
的矢径ρ为
Figure PCTCN2021139627-appb-000013
椭圆凸轮的径向变形量为
ω＝ρ-r b     (9)
式中，r b为凸轮基圆半径
椭圆凸轮转动后后，活齿与其接触点矢径与法线的夹角为：
Figure PCTCN2021139627-appb-000014
活齿中心与椭圆凸轮中心的连线相对于接触点O f与椭圆凸轮中心的连线的夹角：
Figure PCTCN2021139627-appb-000015
根据几何三角关系，由图1可得：
Figure PCTCN2021139627-appb-000016
其中，r为圆柱活齿半径，i为单级活齿传动比
基于上述内容，本实施例的复波式活齿减速器内齿廓设计方法包括下列步骤：
首先，确定活齿及内齿齿数：
复波式活齿行星传动，两级的活齿齿数与内齿(固定内齿或输出内齿)齿数的确定需要根据NN型传动机构的原理来确定；比如两级减速器，如图2所示，第一级减速组件的活齿1，内齿轮2为固定齿轮；第二级减速组件的活齿2，内齿轮4为输出内齿。根据机构转化法，可以求得机构传动比i H4为
Figure PCTCN2021139627-appb-000017
式中，n H、n 4分别为椭圆凸轮和输出内齿的转速，Z 1、Z 3分别为第一级减速组件和第二级减速组件的活齿齿数，Z 2、Z 4分别为第一级减速组件和第二级减速组件的固定内齿和输出内齿齿数。
由于新型活齿传动是对称多齿啮合，因此活齿和内齿齿数以及两级啮合齿 数差Z 2-Z 1、Z 4-Z 3尽量选取偶数以避免凸轮高速运转时不平衡力的出现。作为少齿差活齿传动，两级内啮合齿数差Z 2-Z 1＝Z 4-Z 3＝k≤2、同时为了保证输出内齿的转动方向与凸轮保持一致，应该满足Z 1≥Z 3；将上述齿数关系带入传动比公式可得
Figure PCTCN2021139627-appb-000018
根据设计所需的传动比以及选取适当的k，通过凑整法即可得到两级活齿齿数Z 1、Z 3。
其次，设计内齿廓的廓形，包括下列步骤：
a.建立坐标系，包括：
以椭圆转动中心为原点O，凸轮短轴为X轴，长轴为Y轴建立平面直角坐标系OXY；以活齿中心为原点O 1，竖直方向为Y 1轴，水平方向为X 1轴建立活齿平面直角坐标系S 1{O 1X 1Y 1}；以椭圆转动中心为原点O 2，竖直方向为Y 2轴，水平方向为X 2轴建立内齿平面直角坐标系S 2{O 2X 2Y 2}，其中原点O 2与原点O重合，转动前，Y 2轴与Y轴重合；
如图1所示，以第一级减速组件为例(虽然第二级减速组件内齿输出动力，但运动关系原理一致，所以求解齿廓的方法相同，在此不再赘述)，椭圆凸轮为输入装置，内齿固定，保持架输出；凸轮顺时针转动，活齿在凸轮的作用下逆时针转动，转动过程中始终与凸轮相切，而固定不动的内齿始终和活齿接触，基于相对运动关系，利用下述的B矩阵运动学法对内齿齿廓进行求解；
运用相对角速度计算单级活齿传动比，假设内齿齿数为 m，活齿齿数为 n，凸轮转动角度为
Figure PCTCN2021139627-appb-000019
保持架的转动角度为
Figure PCTCN2021139627-appb-000020
故单级活齿传动的传动比为：
Figure PCTCN2021139627-appb-000021
两个相对运动的活齿曲面与内齿曲面要实现共轭运动，必须满足基本的共轭方程：
Figure PCTCN2021139627-appb-000022
式中，
Figure PCTCN2021139627-appb-000023
和
Figure PCTCN2021139627-appb-000024
分别为两共轭曲面(活齿曲面与内齿曲面)在接触点O f处 的相对速度矢和公法矢；
b.将活齿齿廓坐标的径矢r 1与法向量n 1变换到内齿坐标系S 2中，
则
Figure PCTCN2021139627-appb-000025
其中，M 21为由活齿坐标系S 1到内齿坐标系S 2的坐标变换矩阵，W 21为由活齿坐标系S 1到内齿坐标系S 2的底矢变换矩阵，r 2为活齿齿廓坐标在内齿坐标系S 2中的径矢，n 2为活齿齿廓坐标在内齿坐标系S 2中的法向量；
在坐标系S 1中，活齿齿廓坐标的径矢r 1与法向量n 1分别为
Figure PCTCN2021139627-appb-000026
其中s为活齿的弧长参量；n x为法向量n 1在X轴的分量；n y为法向量n 1在Y轴的分量；
在坐标系S 2中计算啮合方程，则式(16)的矩阵形式为：
Figure PCTCN2021139627-appb-000027
c.获取啮合矩阵B
表达式为：
Figure PCTCN2021139627-appb-000028
即啮合矩阵，根据如图的相对运动关系可得：
Figure PCTCN2021139627-appb-000029
其中，δ为接触点O f法线与Y轴夹角；ω为椭圆凸轮的径向变形量；
Figure PCTCN2021139627-appb-000030
为活齿相对于凸轮长轴的转角；上述参数带有上标“′”为求导数。
d.获取共轭转角
由于相对速度
Figure PCTCN2021139627-appb-000031
是矢径对时间的微分，因此相对速度：
Figure PCTCN2021139627-appb-000032
将式(1)与式(7)带入上述B矩阵运动学法的啮合方程式(6)，可得下 述公式，
Figure PCTCN2021139627-appb-000033
通过公式(3)结合公式(4)和(5)求解不同弧长参数s对应的共轭转角
Figure PCTCN2021139627-appb-000034
共轭转角的集合表示为：
Figure PCTCN2021139627-appb-000035
其中v 2 (12)为两共轭曲面活齿曲面和内齿曲面在接触点O f处的相对速度；
e.将步骤d中的共轭转角集
Figure PCTCN2021139627-appb-000036
代入公式(1)，并结合公式(4)、(5)获取内齿廓的廓形；
由于啮合矩阵具有不随共轭曲面形式的改变而变动的唯一性，因此，B矩阵的计算与共轭曲面的几何参数无关，无论共轭曲面的表达式如何，都可以采用同一个B矩阵进行计算分析，这大大减小了计算工作量。并且，B矩阵的第四行元素全为零，左上3×3的矩阵为反对称矩阵，计算元素较少；简化整个计算过程。
如图2所示，本发明还公开了一种双级复波式活齿减速器，包括第一级减速组件和与第一级减速组件传动配合的第二级减速组件，所述第一级减速组件和第二级减速组件均为复波式活齿减速器，且均采用所述的设计方法获得内齿圈的内齿廓；采用本实施例的上述方法分别对第一减速组件和第二减速组件内齿(地一级减速组件的固定内齿和第二级减速组件的输出内齿)进行齿廓求解，求解所得齿廓如图3所示(仅以地一级减速组件的内齿圈的内齿廓为例)。
本实施例中，所述第一级减速组件的椭圆凸轮5用于输入动力，第一级减速组件的内齿圈1固定，第一级减速组件的活齿保持架7与第二级减速组件的活齿保持架8刚性传动，以及，第一级减速组件的椭圆凸轮5与第二级减速组件的椭圆凸轮6刚性传动，所述第二级减速组件的内齿圈4用于输出动力；如图所示，第一减速组件的活齿保持架7沿径向滑动配合设有第一减速组件的活齿2，第二减速组件的活齿保持架8沿径向滑动配合设有第一减速组件的活齿3，在此不再赘述；
本发明利用椭圆凸轮取代传统偏心轮，实现啮合区域轴对称，使作用在凸 轮上的力相互抵消，减小振动、噪声与冲击；同时结合少齿差行星传动构型方法，通过复波式二级传动实现了结构简单的大速比复波式活齿传动，降低了零件加工的难度。双级传动中，第一级中的内齿为固定内齿，第二级中的内齿为输出内齿。在第一级中，内齿固定，椭圆凸轮转动，凸轮的轮廓曲线产生径向推力，迫使圆柱活齿反方向沿着内齿工作齿廓滚动，活齿的滚动带动保持架的转动。由于两级的保持架一体，所以运动传到第二级，在第二级中的活齿和凸轮的作用下，输出内齿转动，从而实现活齿传动的转速变换作用。
最后说明的是，以上实施例仅用以说明本发明的技术方案而非限制，尽管参照较佳实施例对本发明进行了详细说明，本领域的普通技术人员应当理解，可以对本发明的技术方案进行修改或者等同替换，而不脱离本发明技术方案的宗旨和范围，其均应涵盖在本发明的权利要求范围当中。

Claims (5)

  1. 一种复波式活齿减速器内齿廓设计方法，其特征在于：活齿减速器包括内齿圈、活齿组件和椭圆凸轮，所述活齿组件包括活齿和活齿保持架，所述活齿在椭圆凸轮的作用下与内齿圈啮合滚动或或脱离啮合；其中所述内齿圈的内齿廓的设计方法包括下列步骤：
    a.建立坐标系，包括
    以椭圆转动中心为原点O，凸轮短轴为X轴，长轴为Y轴建立平面直角坐标系OXY；以活齿中心为原点O 1，竖直方向为Y 1轴，水平方向为X 1轴建立活齿平面直角坐标系S 1{O 1X 1Y 1}；以椭圆转动中心为原点O 2，竖直方向为Y 2轴，水平方向为X 2轴建立内齿平面直角坐标系S 2{O 2X 2Y 2}，其中原点O 2与原点O重合，转动前，Y 2轴与Y轴重合；
    b.将活齿齿廓坐标的径矢r 1与法向量n 1变换到内齿坐标系S 2中，
    则
    Figure PCTCN2021139627-appb-100001
    其中，M 21为由活齿坐标系S 1到内齿坐标系S 2的坐标变换矩阵，W 21为由活齿坐标系S 1到内齿坐标系S 2的底矢变换矩阵，r 2为活齿齿廓坐标在内齿坐标系S 2中的径矢，n 2为活齿齿廓坐标在内齿坐标系S 2中的法向量；
    c.获取啮合矩阵B
    表达式为：
    Figure PCTCN2021139627-appb-100002
    d.获取共轭转角
    通过公式
    Figure PCTCN2021139627-appb-100003
    求解不同弧长参数s对应的共轭转角
    Figure PCTCN2021139627-appb-100004
    其中v 2 (12)为两共轭曲面活齿曲面与内齿曲面在接触点O f处的相对速度；
    e.将步骤d中的共轭转角
    Figure PCTCN2021139627-appb-100005
    代入公式(1)，获取内齿廓。
  2. 根据权利要求1所述的复波式活齿减速器内齿廓设计方法，其特征在 于：步骤c中，
    Figure PCTCN2021139627-appb-100006
    其中，δ为接触点O f法线与Y轴夹角；ω为椭圆凸轮的径向变形量；
    Figure PCTCN2021139627-appb-100007
    为活齿相对于凸轮长轴的转角；上述参数带有上标“′”为求导数。
  3. 根据权利要求2所述的复波式活齿减速器内齿廓设计方法，其特征在于：步骤b中，
    设在坐标系S 1中，活齿齿廓坐标的径矢r 1与法向量n 1分别为
    Figure PCTCN2021139627-appb-100008
    其中s为活齿的弧长参量；n x为法向量n 1在X轴的分量；n y为法向量n 1在Y轴的分量；φ为接触点O f矢径与Y轴夹角；
    坐标系S 2中的共轭方程为，
    Figure PCTCN2021139627-appb-100009
    且
    Figure PCTCN2021139627-appb-100010
  4. 一种双级复波式活齿减速器，其特征在于：包括第一级减速组件和与第一级减速组件传动配合的第二级减速组件，所述第一级减速组件和第二级减速组件均为复波式活齿减速器，且均采用权利要求1、2或3的设计方法获得内齿圈的内齿廓。
  5. 根据权利要求4所述的复波式活齿减速器内齿廓设计方法，其特征在于：所述第一级减速组件的椭圆凸轮用于输入动力，第一级减速组件的内齿圈固定，第一级减速组件的活齿保持架与第二级减速组件的活齿保持架刚性传动，以及，第一级减速组件的椭圆凸轮与第二级减速组件的椭圆凸轮刚性传动，所述第二级减速组件的内齿圈用于输出动力。
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