WO2020133650A1 - 基于线面共轭的对构齿轮啮合副及其设计方法 - Google Patents

基于线面共轭的对构齿轮啮合副及其设计方法 Download PDF

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WO2020133650A1
WO2020133650A1 PCT/CN2019/075539 CN2019075539W WO2020133650A1 WO 2020133650 A1 WO2020133650 A1 WO 2020133650A1 CN 2019075539 W CN2019075539 W CN 2019075539W WO 2020133650 A1 WO2020133650 A1 WO 2020133650A1
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gear
curve
curved surface
line
contact
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PCT/CN2019/075539
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English (en)
French (fr)
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陈兵奎
张录合
彭帅
李朝阳
安来强
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重庆大学
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Priority to EP19903202.0A priority Critical patent/EP3882489A4/en
Publication of WO2020133650A1 publication Critical patent/WO2020133650A1/zh

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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
    • 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
    • F16H1/00—Toothed gearings for conveying rotary motion
    • 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

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  • the present application relates to the technical field of gear transmissions, in particular to a pair of meshing gear pairs based on line-plane conjugation and a design method thereof.
  • the gear determines the performance of the equipment to a large extent. Therefore, the design of high-performance gear transmission components also has very important significance and practical value in engineering.
  • the involute gear meshing pair is face-to-face conjugate.
  • the contact between the two faces forms a line contact, and the tooth shape is an involute.
  • the linear gear has the advantage of separability of the center distance, but it cannot cope with the deflection or pitch error of the gear axis, and the gear sliding rate of the linear contact is high, which has an adverse effect on the wear of the tooth surface, transmission efficiency and gear life.
  • a linear-conjugate involute gear transmission device is proposed.
  • This kind of gear has a small slip ratio and can achieve convex-concave type tooth surface contact.
  • the gear has a strong bearing capacity, but the linear-conjugate involute
  • the gear transmission is still based on the involute curve, the tooth profile still has certain limitations, and it cannot solve the problems of the axis deviation and pitch error of the gear.
  • the purpose of the present application is to provide a pair of contra-convex gear meshing pairs based on line-surface conjugation to solve the existing problems in the prior art.
  • the line-surface conjugate involute gear transmission device is based on involute, tooth shape It has certain limitations and cannot solve the technical problems of gear axis deviation and pitch error.
  • the present application also provides a design method of a pair of meshing gear pairs based on line-plane conjugation, to solve the existing problems in the prior art, the line-plane conjugate involute gear transmission device is based on involute,
  • the tooth shape has certain limitations, and cannot solve the technical problems of gear axis deviation and pitch error.
  • the present invention provides a pair of meshing gear pairs based on line-plane conjugate, which includes a first gear and a second gear, and the first gear and the second gear form a line-plane conjugate gear pair;
  • the line-plane conjugate gear pair includes a continuous tangent contact surface ⁇ 1 and a curve ⁇ 1 , the surface ⁇ 1 is located on the first gear, and only one curve on the surface ⁇ 1 is in contact with the curve ⁇ 1 ; At a predetermined time, curve ⁇ 1 maintains point contact with curved surface ⁇ 1 , and at a predetermined time, the contact point of curve ⁇ 1 with curved surface ⁇ 1 is unique;
  • the curve ⁇ 1 is a smooth curve, and the curved surface ⁇ 1 is a smooth curved surface.
  • a design method for a pair of meshing gear pairs based on line-plane conjugation includes the following steps:
  • u, v are surface parameters
  • ⁇ is a spatial parameter
  • step v also includes steps:
  • the fixed coordinate system S 1 and the dynamic coordinate system S 2 are respectively connected to the curve ⁇ 1 and the curved surface ⁇ 1 ;
  • step iv also includes:
  • ⁇ 1 is the angular velocity of the curve ⁇ 1 and ⁇ 2 is the angular velocity of the curved surface ⁇ 1 .
  • step vi also includes:
  • x(t) and z(t) are functions of the parameter t;
  • the method ⁇ c are located on a plane curve ⁇ 1, and curve ⁇ 1 intersect at a point, the curve passes through the point of intersection C [Gamma] [Gamma] C and the curve direction and a main surface normal vector at the point of intersection in the intersection point of ⁇ 1
  • the direction of the main normal is the same;
  • a pair of meshing gear pairs based on line-plane conjugate is provided.
  • the first gear and the second gear form a line-plane conjugate gear pair.
  • the profile of the first gear remains unchanged, and the curved surface ⁇ 1 is located the first gear, the curved only one [Sigma [Gamma] curves contact with the upper 11; at each time t, the curve ⁇ 1 ⁇ 1 is kept in contact with the surface point and the surface at each point ⁇ 1 unique Contact at time t, that is, the curve ⁇ 1 has a unique contact point with the curved surface ⁇ 1 ; the curved ⁇ 1 is a smooth curve, and the curved surface ⁇ 1 is a smooth curved surface, which realizes line-plane conjugation of the pair of meshing gears.
  • FIG. 1 is a schematic diagram of selecting a contact trace ⁇ 2 based on a design method of a counter-gear gear pair based on a line-surface conjugation provided by an embodiment of the present application;
  • Example 2 is a schematic diagram of selecting a contact trace ⁇ 2 based on a design method of a pair of counter-gear meshing pairs based on line-plane conjugation provided in Example 1 of the present application;
  • Example 3 is a schematic diagram of the simulation of the contact trace ⁇ 2 based on the design method of the contra-gear gear pair based on line-plane conjugation provided in Example 1 of the present application;
  • FIG. 4 is a schematic diagram of the normal direction of the contact trace ⁇ 2 based on the design method of the counter-gear gear pair based on the line-plane conjugation provided by the embodiment of the present application;
  • FIG. 5(a) is a schematic diagram 1 of the normal direction of the contact trace ⁇ 2 based on the design method of the counter-gear gear pair based on the line-plane conjugation provided by the embodiment of the present application;
  • FIG. 5(b) is a schematic diagram 1 of the normal direction of the contact trace ⁇ 2 based on the design method of the counter-gear gear pair based on the line-plane conjugation provided by the embodiment of the present application;
  • FIG. 6 is a spatial coordinate system of a method for designing a pair of contra-gear gears based on line-plane conjugation provided by an embodiment of the present application;
  • Example 8 is a schematic diagram of a conjugate curve of a method for designing a pair of meshing gears based on line-surface conjugation provided in Example 1 of the present application;
  • FIG. 9 is a schematic diagram of the meshing line of the design method of the pair of meshing gears based on the line-surface conjugation provided by the embodiment of the present application;
  • FIG. 10 is a schematic diagram of a simulation of a structure diagram of a tooth surface provided by a design method of a counter-gear gear pair based on line-plane conjugation provided in Example 1 of the present application;
  • FIG. 11(a) is a schematic diagram 1 of the simulation diagram of the structure of the tooth surface of the design method of the counter-gear gear pair based on line-plane conjugation provided in Example 1 of the present application;
  • 11(b) is a schematic diagram 2 of a simulation diagram of a tooth surface of a gear surface design method based on a line-plane conjugate contra-gear gear pair provided by Example 1 of the present application;
  • FIG. 11(c) is a schematic diagram 3 of the simulation of the tooth surface structure diagram of the design method of the counter-gear gear pair based on line-plane conjugation provided in Example 1 of the present application;
  • FIG. 12 is a schematic diagram of a pinion gear model based on a line-plane conjugate pair of counter-gear meshing pairs provided in Example 1 of this application;
  • FIG. 13 is a schematic structural diagram of an envelope method for solving a tooth shape equation based on a design method of a pair of contra-convex gear meshing pairs provided by Example 2 of the present application;
  • FIG. 14 is a schematic structural diagram of a tooth profile normal method for solving a tooth profile equation based on a design method of a pair of contra-convex gear meshing pairs provided by Example 3 of the present application;
  • Table 1 is the gear pair parameters.
  • connection should be understood in a broad sense, for example, it can be a fixed connection or a Detachable connection, or integral connection; it can be mechanical connection or electrical connection; it can be directly connected, or it can be indirectly connected through an intermediate medium, or it can be the connection between two components.
  • connection should be understood in specific situations.
  • the present application provides a pair of meshing gear pairs based on line-plane conjugate.
  • the pair of meshing gear pairs based on line-plane conjugate includes a first gear and a second gear.
  • the first gear is formed with the second gear
  • the linear-plane conjugate gear pair includes a continuous tangential contact surface ⁇ 1 and a curve ⁇ 1 , the surface ⁇ 1 is located in the first gear, said only one surface in contact with a curve on the graph 1 [Gamma] [Sigma; at a predetermined timing, the curve 1 and [Gamma] 1 [Sigma holding point contact with the curved surface, and the predetermined time, the contact point of the curve 1 and [Gamma] 1 [Sigma sole surface;
  • the curve ⁇ 1 is a smooth curve
  • the curved surface ⁇ 1 is a smooth curved surface.
  • the first gear and the second gear form a linear-plane conjugate gear pair
  • the curved surface ⁇ 1 is located on the first gear on the curved surface and the curve only one curve 1 on [Gamma] [Sigma contacting 1; at each time t, the curve 1 [Gamma] 1 [Sigma kept in contact with the surface point and the surface at each point [Sigma 1 only comes into contact time t, That is, the curve ⁇ 1 has a unique contact point with the curved surface ⁇ 1 ; the curved ⁇ 1 is a smooth curve, and the curved surface ⁇ 1 is a smooth curved surface, which realizes the line-plane conjugation of the pair of meshing gears.
  • the present application provides a method for designing a pair of meshing gear pairs based on line-plane conjugation.
  • This application takes an internal gear as an example.
  • the first gear is known as an internal gear
  • the tooth profile of the internal gear end is an involute end.
  • the equation can be expressed as:
  • r is the base circle radius
  • ⁇ involute parameter is the base circle radius
  • the helical gear tooth surface can be obtained by making the same spiral movement around the z-axis of the rotation axis at every point on the involute line, that is, on the one hand, it rotates at a constant speed around the z-axis, and at the same time makes a straight line motion along the z-axis at the same speed. Therefore, the internal gear tooth surface equation can be expressed as:
  • ⁇ is the angle of rotation around the z axis.
  • the contact trajectory of the line-surface meshing counter-gear can be selected according to need, that is, a smooth continuous curve is designated as the contact trace on the first gear.
  • a smooth continuous curve is designated as the contact trace on the first gear.
  • countless curves can be selected on the tooth surface, and there are various methods for selection.
  • a shortest curve can be determined, assuming that the surface parameter ⁇ has the simplest linear relationship with ⁇ :
  • B is the tooth width
  • p is the spiral parameter
  • the range of ⁇ is ⁇ 1 ⁇ 2.
  • ⁇ 1 and ⁇ 2 are the values of ⁇ at the start and end points on the curved surface.
  • the contact trace is simulated in MATLAB software. Unlike the general point meshing helical gear, the meshing trace is not a cylindrical spiral, but a smooth curve from the root to the top of the tooth.
  • the design method of a pair of gear pairs based on line-plane conjugation provided by this application.
  • the theoretical basis of the contact between the first gear and the second gear is line-plane meshing. It has strong adaptability to the deflection of the gear axis, pitch error and the center. Separable distance, and the actual contact state of the first gear and the second gear is point contact, the tooth surface is close to the theoretical pure rolling, and the sliding rate is small; and the type of tooth surface contact is not limited to convex and convex contact, but can also be convex and concave Contact, wide range of application, can improve the load-bearing capacity of the gear.
  • step v further includes the steps of: establishing a fixed coordinate system S 1 and a dynamic coordinate system S 2 , the fixed coordinate system S 1 and the dynamic coordinate system S 2 are respectively fixedly connected to the curve ⁇ 1 and the curved surface ⁇ 1 ; and the dynamic coordinate system is determined Conversion matrix between S 2 and fixed coordinate system S 1 : among them, Is the angle that curve ⁇ 1 turns, The angle through which the surface ⁇ 1 is turned.
  • step v it is necessary to first establish a fixed coordinate system S 1 and a dynamic coordinate system S 2 , where the fixed coordinate system S 1 and the dynamic coordinate system S 2 are respectively fixed to the curve ⁇ 1 and the curved surface ⁇ 1 , as shown in FIG. 6 Shows, where S o (O o -x o , y o , z o ) and S p (O p -x p , y p , z p ) are two coordinate systems fixed in space, the z axis and the curve ⁇ 1 the pivot axis coincides with the axis of pivot axis Z p ⁇ 1 coincides with the curved surface.
  • the x axis coincides with the x p axis, which is the direction of the shortest distance between the two axes, that is, the center distance ⁇ .
  • the fixed coordinate system S 1 (O 1 -x 1 , y 1 , z 1 ) and the dynamic coordinate system S 2 (O 2 -x 2 , y 2 , z 2 ) are respectively different from S o (O o- x o , y o , z o ) and S p (O p -x p , y p , z p ) coincide.
  • the curve ⁇ 1 rotates around the Z 0 axis at an angular velocity ⁇ 1
  • the curved surface ⁇ 1 rotates around the z p axis at an angular velocity ⁇ 2 .
  • the curve ⁇ 1 turns Angle, surface ⁇ 1 turned angle.
  • step iv further includes: determining the relative motion speed v (12) of the dynamic coordinate system S 2 relative to the fixed coordinate system S 1 ; Where ⁇ 1 is the angular velocity of the curve ⁇ 1 and ⁇ 2 is the angular velocity of the curved surface ⁇ 1 .
  • i 2 , j 2 , and k 2 be the unit vectors of the coordinate axes x 2 , y 2 , and z 2
  • the point p t is any contact point between the curve and the surface in space.
  • the coordinate values in the coordinate system S 2 are (x 2 , y 2 , z 2 ); the angular velocities of the curve ⁇ 1 and the curved surface ⁇ 1 in the coordinate system S 2 are expressed by vectors as
  • ⁇ 1 is the modulus of the angular velocity of the curve ⁇ 1
  • ⁇ 2 is the modulus of the angular velocity of the curved surface ⁇ 1 .
  • i 12 is the transmission ratio
  • step vi also includes: selecting the normal section curve as ⁇ c ; Wherein, X (t) and z (t) as a function of the parameter t; ⁇ c positioned on the normal plane curve ⁇ 1, and curve ⁇ 1 intersect at a point, the curve passes through the point of intersection ⁇ c and the curve
  • the main normal vector direction of ⁇ c at the intersection point is the same as the main normal direction of the curved surface ⁇ 1 at the intersection point; determine the tooth surface ⁇ 2 equation: among them, Is the main normal vector of the surface ⁇ 2 at the intersection point, Is the tangent vector of the contact locus ⁇ 2 at the contact point on the surface ⁇ 2 , by get.
  • the normal section method is used to construct the second gear tooth surface
  • the tooth surface of the second gear may be constructed by continuously changing the curved surface of the normal section along the conjugate curve.
  • the normal section can be composed of any smooth curve, according to the formula Available, the normal profile tooth profile equation can be expressed as:
  • k 1 is the arc radius of the tooth surface.
  • the cut vector ⁇ can be expressed as
  • the normal section is a circular arc or a hyperbolic tooth profile.
  • a circular arc or hyperbolic tooth profile is selected for the normal section.
  • the arc tooth profile is taken as the research object.
  • step vii according to the gear tooth surface equation, write a program to solve the tooth surface equation.
  • Fig. 11(a) is the starting point of meshing
  • Fig. 11(b) is any position during meshing
  • Fig. 11(c ) Is the end point of meshing. It can be seen that the curve and the curved surface always maintain point contact and mesh along the meshing line respectively. The meshing point is clearly visible, and the trajectory of the meshing point on the curved surface is consistent with the desired meshing trace.
  • step viii calculating the coordinates of the points on the tooth surface according to the solved tooth surface equations, importing the coordinates of the points on the tooth surface into the 3D software, and the 3D software generates the gear tooth solid model.
  • step vi uses the envelope method to determine the gear tooth profile equation.
  • the tooth profile 1 forms a family of curves on the plane of the gear 2. Since the tooth shape 2 and the tooth shape 1 are in tangential contact at every instant, mathematically speaking, the tooth shape 2 should be the envelope of the curve family formed by the tooth shape 1. Using this principle, the tooth profile 2 can be obtained from the motion law of the gear pair and the tooth profile 1, this method is called the envelope method.
  • the tooth shape of another gear is enveloped by the tooth shape of another gear. This phenomenon can be clearly observed when the gear is processed by the generative method.
  • the tooth shape of the gear shaping knife (including the side edge) And top edge) as the tooth shape 1, it gradually envelops the tooth shape 2 of the workpiece during the forming process.
  • step vi uses the tooth profile normal method to determine the gear profile equation.
  • M is the contact point of the two tooth shapes at this instant.
  • the design method of the mating gear pair based on line-plane conjugation is not limited to involute gears, and the theoretical basis of the contact between the first gear and the second gear is line-plane meshing, which has a skew to the gear axis, It has the characteristics of strong adaptability of pitch error and separability of center distance, and the actual contact state of the first gear and the second gear is point contact, the tooth surface is close to theoretical pure rolling, and the sliding rate is small; and the type of tooth surface contact is not limited to
  • the convex-convex contact can also be a convex-concave contact, which has a wide range of applications and can improve the load-bearing capacity of the gear.

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Abstract

一种线面共轭齿轮副，包括第一齿轮和第二齿轮，第一齿轮与第二齿轮形成线面共轭齿轮副，线面共轭齿轮副包括连续相切接触的曲面Σ1和曲线Г1，曲面Σ1位于第一齿轮上，曲面Σ1上仅有一条曲线与曲线Г1接触，在预定时刻，曲线Г1与曲面Σ1保持点接触，且曲线Г1与曲面Σ1的接触点唯一；曲线Г1为光滑曲线，曲面Σ1为光滑曲面。一种线面共轭齿轮副的设计方法也被公开。

Description

基于线面共轭的对构齿轮啮合副及其设计方法
相关申请的交叉引用
本申请要求于2018年12月27日提交中国专利局的申请号为CN201811616387.8、名称为“基于线面共轭的对构齿轮啮合副及其设计方法”的中国专利申请的优先权，其全部内容通过引用结合在本申请中。
技术领域
本申请涉及齿轮传动装置技术领域，尤其是涉及一种基于线面共轭的对构齿轮啮合副及其设计方法。
背景技术
齿轮作为一种典型的机械基础件，在很大程度上决定着装备的性能，因而针对高性能齿轮传动元件的设计也具有十分重要的意义和工程实用价值。
现有圆柱齿轮啮合副中应用最广的是渐开线齿轮，渐开线齿轮啮合副是面和面共轭，两个面之间的接触形成线接触，齿形为渐开线，渐开线齿轮具有中心距可分性的优点，但是不能应对齿轮轴线的偏斜或者俯仰误差，而且线接触的齿轮滑动率较高，对齿面的磨损、传动效率及齿轮寿命有不利的影响。
基于上述问题，提出了线面共轭的渐开线齿轮传动装置，这种齿轮的滑动率小，可以实现凸凹类型的齿面接触，齿轮的承载能力强，但是线面共轭的渐开线齿轮传动装置仍然是基于渐开线建立的，齿形仍具有一定的局限性，而且也无法解决齿轮的轴线偏差和俯仰误差的问题。
发明内容
本申请的目的在于提供一种基于线面共轭的对构齿轮啮合副，以解决现有技术中存在的，线面共轭的渐开线齿轮传动装置是基于渐开线建立的，齿形具有一定的局限性，无法解决齿轮的轴线偏差和俯仰误差的技术问题。
本申请还提供了一种基于线面共轭的对构齿轮啮合副的设计方法，以解决现有技术中存在的，线面共轭的渐开线齿轮传动装置是基于渐开线建立的，齿形具有一定的局限性，无法解决齿轮的轴线偏差和俯仰误差的技术问题。
本申请提供的一种基于线面共轭的对构齿轮啮合副，包括第一齿轮和第二齿轮，所述第一齿轮与第二齿轮形成线面共轭齿轮副；
所述线面共轭齿轮副包括连续相切接触的曲面Σ 1和曲线Г 1，所述曲面Σ 1位于所述第一齿轮上，所述曲面Σ 1上仅有一条曲线与曲线Г 1接触；在预定时刻，曲线Г 1与曲面Σ 1保持点接触，且预定时刻，曲线Г 1与曲面Σ 1的接触点唯一；
所述曲线Г 1为光滑曲线，所述曲面Σ 1为光滑曲面。
本申请提供的一种基于线面共轭的对构齿轮啮合副的设计方法，包括如下步骤：
i)确定第一齿轮的曲面Σ 1方程为：
Figure PCTCN2019075539-appb-000001
其中，u，v为曲面参数；
ii)确定第一齿轮的齿面上的接触迹线Г 2方程为：
Figure PCTCN2019075539-appb-000002
其中，φ为空间参数；
iii)确定第一齿轮的齿面上的接触迹线的法向量
Figure PCTCN2019075539-appb-000003
为：
Figure PCTCN2019075539-appb-000004
iv)由公式n·v (12)＝0，定曲线Г 1曲面Σ 1的啮合方程；
v)根据接触迹线Г 2在固定坐标系S 1下的方程与啮合方程，确定共轭曲线；
Figure PCTCN2019075539-appb-000005
vi)根据共轭曲线，确定第二齿轮齿面Σ 2方程。
进一步的，步骤v中还包括步骤：
建立固定坐标系S 1与动坐标系S 2，固定坐标系S 1与动坐标系S 2分别与曲线Г 1与曲面Σ 1固联；
确定动坐标系S 2与固定坐标系S 1之间的转化矩阵：
Figure PCTCN2019075539-appb-000006
其中，
Figure PCTCN2019075539-appb-000007
为曲线Г 1转过的角度，
Figure PCTCN2019075539-appb-000008
为曲面Σ 1转过的角度。
进一步的，步骤iv中还包括：
确定所述动坐标系S 2相对于固定坐标系S 1的相对运动速度v (12)；
Figure PCTCN2019075539-appb-000009
其中，ω 1为曲线Г 1转动的角速度，ω 2为曲面Σ 1转动的角速度。
进一步的，步骤vi中还包括：
选取法截面曲线为Г c：
Figure PCTCN2019075539-appb-000010
其中，x(t)和z(t)为参数t的函数；
所述Г c位于曲线Г 1的法平面上，且与曲线Г 1相交于一点，所述曲线Г c穿过相交点且曲线Г c在相交点的主法矢方向和曲面Σ 1在相交点的主法线方向一致；
确定齿面Σ 2方程：
Figure PCTCN2019075539-appb-000011
Figure PCTCN2019075539-appb-000012
其中，
Figure PCTCN2019075539-appb-000013
为曲面Σ 2在相交点的主法线向量，
Figure PCTCN2019075539-appb-000014
为曲面Σ 2上接触轨迹Г 2接触点处的切矢量，
Figure PCTCN2019075539-appb-000015
由
Figure PCTCN2019075539-appb-000016
得到。
本申请提供的基于线面共轭的对构齿轮啮合副，所述第一齿轮与第二齿轮形成线面共轭齿轮副，所述第一齿轮齿廓保持不变，所述曲面Σ 1位于所述第一齿轮上，所述曲面Σ 1上仅有一条曲线与曲线Г 1接触；在每一时刻t，曲线Г 1与曲面Σ 1保持点接触，且曲面Σ 1每一点都在唯一的时刻t进入接触，即曲线Г 1与曲面Σ 1存在唯一的接触点；所述曲线Г 1为光滑曲线，所述曲面Σ 1为光滑曲面，实现对构齿轮啮合副的线面共轭。
本申请提供的基于线面共轭的对构齿轮啮合副的设计方法，先根据已知的第一齿轮确定第一齿轮的曲面Σ 1的方程；之后确定第一齿轮的齿面上的接触迹线Г 2的方程；根据接 触迹线Г 2的方程确定接触迹线的法向量
Figure PCTCN2019075539-appb-000017
之后，由公式n·v (12)＝0，确定曲线Г 1与曲面Σ 1的啮合方程；根据接触迹线Г 2在固定坐标系S 1下的方程与啮合方程，确定共轭曲线；根据共轭曲线，确定第二齿轮齿面Σ 2方程，实现对第二齿轮齿面方程的确定，第一齿轮与第二齿轮的接触理论基础是线面啮合，具有对齿轮轴线偏斜、俯仰误差的适应能力强及中心距可分性的特点，且第一齿轮与第二齿轮的实际接触状态为点接触，齿面间接近理论纯滚动，滑动率小；而且齿面接触类型不限于凸凸接触，也可以是凸凹接触，适用范围广，可以提高齿轮的承载能力。
附图说明
为了更清楚地说明本申请具体实施方式或现有技术中的技术方案，下面将对具体实施方式或现有技术描述中所需要使用的附图作简单地介绍，显而易见地，下面描述中的附图是本申请的一些实施方式，对于本领域普通技术人员来讲，在不付出创造性劳动的前提下，还可以根据这些附图获得其他的附图。
图1为本申请实施例提供的基于线面共轭的对构齿轮啮合副的设计方法的接触迹线Г 2的选取示意图；
图2为本申请实施例一提供的基于线面共轭的对构齿轮啮合副的设计方法的接触迹线Г 2的选取示意图；
图3为本申请实施例一提供的基于线面共轭的对构齿轮啮合副的设计方法的接触迹线Г 2的模拟示意图；
图4为本申请实施例提供的基于线面共轭的对构齿轮啮合副的设计方法的接触迹线Г 2的法线方向的示意图；
图5(a)为本申请实施例提供的基于线面共轭的对构齿轮啮合副的设计方法的接触迹线Г 2的法线方向的示意图一；
图5(b)为本申请实施例提供的基于线面共轭的对构齿轮啮合副的设计方法的接触迹线Г 2的法线方向的示意图一；
图6为本申请实施例提供的基于线面共轭的对构齿轮啮合副的设计方法的空间坐标系；
图7为本申请实施例提供的基于线面共轭的对构齿轮啮合副的设计方法的相对运动速度的空间坐标系；
图8为本申请实施例一提供的基于线面共轭的对构齿轮啮合副的设计方法的共轭曲线的示意图；
图9为本申请实施例提供的基于线面共轭的对构齿轮啮合副的设计方法的啮合线的示意图；
图10为本申请实施例一提供的基于线面共轭的对构齿轮啮合副的设计方法的齿面的结构图的模拟示意图；
图11(a)为本申请实施例一提供的基于线面共轭的对构齿轮啮合副的设计方法的齿面的结构图的仿真示意图一；
图11(b)为本申请实施例一提供的基于线面共轭的对构齿轮啮合副的设计方法的齿面的结构图的仿真示意图二；
图11(c)为本申请实施例一提供的基于线面共轭的对构齿轮啮合副的设计方法的齿面的结构图的仿真示意图三；
图12为本申请实施例一提供的基于线面共轭的对构齿轮啮合副的小齿轮的模型示意图；
图13为本申请实施例二提供的基于线面共轭的对构齿轮啮合副的设计方法的包络法求解齿形方程的结构示意图；
图14为本申请实施例三提供的基于线面共轭的对构齿轮啮合副的设计方法的齿形法线法求解齿形方程的结构示意图；
表1为齿轮副参数。
具体实施方式
下面将结合附图对本申请的技术方案进行清楚且完整地描述，显然，所描述的实施例是本申请一部分实施例，而不是全部的实施例。基于本申请中的实施例，本领域普通技术人员在没有做出创造性劳动前提下所获得的所有其他实施例，都属于本申请保护的范围。
在本申请的描述中，需要说明的是，如出现术语“中心”、“上”、“下”、“左”、“右”、“竖直”、“水平”、“内”或“外”等指示的方位或位置关系为基于附图所示的方位或位置关系，仅是为了便于描述本申请和简化描述，而不是指示或暗示所指的装置或元件必须具有特定的方位、以特定的方位构造和操作，因此不能理解为对本申请的限制。此外，如出现术语“第一”、“第二”或“第三”仅用于描述目的，而不能理解为指示或暗示相对重要性。
在本申请的描述中，需要说明的是，除非另有明确的规定和限定，如出现术语“安装”、“相连”或“连接”应做广义理解，例如，可以是固定连接，也可以是可拆卸连接，或一体地连接；可以是机械连接，也可以是电连接；可以是直接相连，也可以通过中间媒介间接相连，可以是两个元件内部的连通。对于本领域的普通技术人员而言，可以具体情况理解上述术语在本申请中的具体含义。
本申请提供了一种基于线面共轭的对构齿轮啮合副，所述基于线面共轭的对构齿轮啮 合副包括第一齿轮和第二齿轮，所述第一齿轮与第二齿轮形成线面共轭齿轮副，所述第一齿轮齿廓保持不变；所述线面共轭齿轮副包括连续相切接触的曲面Σ 1和曲线Г 1，所述曲面Σ 1位于所述第一齿轮上，所述曲面Σ 1上仅有一条曲线与曲线Г 1接触；在预定时刻，曲线Г 1与曲面Σ 1保持点接触，且预定时刻，曲线Г 1与曲面Σ 1的接触点唯一；所述曲线Г 1为光滑曲线，所述曲面Σ 1为光滑曲面。
本申请提供的基于线面共轭的对构齿轮啮合副，如图1所示，所述第一齿轮与第二齿轮形成线面共轭齿轮副，所述曲面Σ 1位于所述第一齿轮上，所述曲面Σ 1上仅有一条曲线与曲线Г 1接触；在每一时刻t，曲线Г 1与曲面Σ 1保持点接触，且曲面Σ 1每一点都在唯一的时刻t进入接触，即曲线Г 1与曲面Σ 1存在唯一的接触点；所述曲线Г 1为光滑曲线，所述曲面Σ 1为光滑曲面，实现对构齿轮啮合副的线面共轭。
实施例一
本申请提供了一种基于线面共轭的对构齿轮啮合副的设计方法，所述基于线面共轭的对构齿轮啮合副的设计方法包括如下步骤：i)确定第一齿轮的曲面Σ 1方程为：
Figure PCTCN2019075539-appb-000018
其中，u，v为曲面参数；ii)确定第一齿轮的齿面上的接触迹线Г 2方程为：
Figure PCTCN2019075539-appb-000019
其中，φ为空间参数；iii)确定第一齿轮的齿面上的接触迹线的法向量
Figure PCTCN2019075539-appb-000020
为：
Figure PCTCN2019075539-appb-000021
iv)由公式n·v (12)＝0，确定曲线Г 1与曲面Σ 1的啮合方程；v)根据接触迹线Г 2在固定坐标系S 1下的方程与啮合方程，确定共轭曲线；
Figure PCTCN2019075539-appb-000022
vi)根据共轭曲线，确定第二齿轮齿面Σ 2方程。
本申请以内啮合齿轮为例，如图2所示，第一齿轮已知为内啮合齿轮，内啮合端面齿廓为一端渐开线，其方程可以表示为：
Figure PCTCN2019075539-appb-000023
其中，r为基圆半径，θ渐开线参数。
斜齿轮齿面可以由渐开线上的每一点绕旋转轴z轴做相同的螺旋运动得到，即一方面绕z轴等速转动，同时沿z轴做等速直线运动。因此，内齿轮齿面方程可表示为：
Figure PCTCN2019075539-appb-000024
两式合并，得曲面Σ 1方程：
Figure PCTCN2019075539-appb-000025
其中，α为绕z轴旋转的角度。
之后，在第一齿轮上选取接触迹线Г 2，确定Г 2的方程：
具体地，线面啮合对构齿轮的接触轨迹可根据需要进行选择，即在已经第一齿轮上指定一条光滑连续曲线作为接触迹线。对于内齿轮，在其齿面上可以选取无数条曲线，选取的方法多种多样。在第一齿轮的曲面Σ 1上确定起始点和终止点后，可以确定一条最短的曲线，假定曲面参数θ与α存在最简单的线性关系：
α＝kθ+b
其中，
Figure PCTCN2019075539-appb-000026
B为齿宽，p为螺旋参数，θ的范围为θ 1≤θ≤θ 2，θ 1，θ 2为曲面上的起始点与终止点处θ的值。
因此，内齿轮上的接触迹线Г 2可以表示为：
Figure PCTCN2019075539-appb-000027
如图3所示，在MATLAB软件中模拟接触迹线，与一般点啮合斜齿轮不同的是，啮合迹线不是圆柱螺旋线，而是一条从齿根到齿顶的光滑曲线。
之后，对
Figure PCTCN2019075539-appb-000028
按照公式
Figure PCTCN2019075539-appb-000029
进行运算，
如图4所示，确定内齿轮曲面Σ 1上接触迹线Г 2的法线方向
Figure PCTCN2019075539-appb-000030
Figure PCTCN2019075539-appb-000031
需要说明的是，对于所得到的法向量
Figure PCTCN2019075539-appb-000032
不同方向代表不同的啮合形式。如图5(a)所示，当法向量
Figure PCTCN2019075539-appb-000033
指向曲面凸的方向时，构建出来的齿轮副为外啮合方式，如图5(b)所 示，当法向量
Figure PCTCN2019075539-appb-000034
指向曲面凹向时，所构建出来的齿轮副为内啮合方式。
之后根据公式n·v (12)＝0，确定曲线Г 1与曲面Σ 1的啮合方程；
之后根据公式
Figure PCTCN2019075539-appb-000035
确定共轭曲线；
最后，根据共轭曲线，确定第二齿轮齿面Σ 2方程。
本申请提供的基于线面共轭的对构齿轮啮合副的设计方法，第一齿轮与第二齿轮的接触理论基础是线面啮合，具有对齿轮轴线偏斜、俯仰误差的适应能力强及中心距可分性的特点，且第一齿轮与第二齿轮的实际接触状态为点接触，齿面间接近理论纯滚动，滑动率小；而且齿面接触类型不限于凸凸接触，也可以是凸凹接触，适用范围广，可以提高齿轮的承载能力。
进一步地，步骤v中还包括步骤：建立固定坐标系S 1与动坐标系S 2，固定坐标系S 1与动坐标系S 2分别与曲线Г 1与曲面Σ 1固联；确定动坐标系S 2与固定坐标系S 1之间的转化矩阵：
Figure PCTCN2019075539-appb-000036
其中，
Figure PCTCN2019075539-appb-000037
为曲线Г 1转过的角度，
Figure PCTCN2019075539-appb-000038
为曲面Σ 1转过的角度。
具体地，在步骤v中需要首先建立固定坐标系S 1与动坐标系S 2，其中固定坐标系S 1与动坐标系S 2分别与曲线Г 1与曲面Σ 1固联，如图6所示，其中S o(O o-x o，y o，z o)及S p(O p-x p，y p，z p)是两个在空间固定的坐标系，z轴与曲线Г 1的回转轴线重合，Z p轴与曲面Σ 1的回转轴线重合。x轴与x p轴重合，该方向为两轴线最短距离方向，即中心距α。固定坐标系S 1(O 1-x 1，y 1，ｚ 1）及动坐标系S 2(O 2-x 2，y 2，z 2)在起始位置时分别与S o(O o-x o，y o，z o)及S p(O p-x p，y p，z p)重合。曲线Г 1以角速度ω 1绕Z 0轴转动，曲面Σ 1以角速度ω 2绕z p轴转动。从起始位置经过一段时间后，曲线Г 1转过
Figure PCTCN2019075539-appb-000039
角，曲面Σ 1转过
Figure PCTCN2019075539-appb-000040
角。
根据图6所示，动坐标系S 2与固定坐标系S 1之间的转换关系为：
Figure PCTCN2019075539-appb-000041
由此根据公式
Figure PCTCN2019075539-appb-000042
可以求得接触迹线Г 2在固定坐标系S 1下的方程。
进一步地，步骤iv中还包括：确定所述动坐标系S 2相对于固定坐标系S 1的相对运动速度v (12)；
Figure PCTCN2019075539-appb-000043
其中，ω 1为曲线Г 1转动的角速度，ω 2为曲面Σ 1转动的角速度。
如图7所述，为使矢量运算方便，令i 2，j 2，k 2为坐标轴x 2，y 2，z 2的单位矢量，点p t为空间中曲线与曲面任意接触点，其在坐标系S 2坐标值为(x 2，y 2，z 2)；曲线Г 1和曲面Σ 1分别在坐标系S 2中的角速度分别用矢量表示为
Figure PCTCN2019075539-appb-000044
其中，ω 1为曲线Г1角速度的模，ω 2为曲面Σ 1角速度的模。
如图7所示，
Figure PCTCN2019075539-appb-000045
r 2＝O 2P t＝x 2i 2+y 2j 2+z 2k 2
P t点随曲线Г1和曲面Σ1运动时在坐标系S o下的速度分别为
Figure PCTCN2019075539-appb-000046
Figure PCTCN2019075539-appb-000047
同时，坐标系S 2中曲线Г 1和曲面Σ 1在点P t的相对运动速度可以表示为
Figure PCTCN2019075539-appb-000048
如果写成
Figure PCTCN2019075539-appb-000049
综上所述，可得
Figure PCTCN2019075539-appb-000050
由公式n·v (12)＝0，进而得出啮合方程：
Figure PCTCN2019075539-appb-000051
令A＝i 12an x2，B＝i 12an y2，M＝(i 12-1)(x 2n y2-y 2n x2)
i 12为传动比，且
Figure PCTCN2019075539-appb-000052
可得啮合方程的简化形式：
Figure PCTCN2019075539-appb-000053
之后，根据接触迹线Г 2在固定坐标系S 1下的方程与啮合方程，确定共轭曲线：
Figure PCTCN2019075539-appb-000054
之后，运用MATLAB，按照上述的步骤，所求得的共轭曲线如图8所示。
进一步地，步骤vi中还包括：选取法截面曲线为Γ c；
Figure PCTCN2019075539-appb-000055
其中，x(t)和z(t)为参数t的函数；所述Г c位于曲线Г 1的法平面上，且与曲线Г 1相交于一点，所述曲线Г c穿过相交点且曲线Г c在相交点的主法矢方向和曲面Σ 1在相交点的主法线方向一致；确定齿面Σ 2方程：
Figure PCTCN2019075539-appb-000056
Figure PCTCN2019075539-appb-000057
其中，
Figure PCTCN2019075539-appb-000058
为曲面Σ 2在相交点的主法线向量，
Figure PCTCN2019075539-appb-000059
为曲面Σ 2上接触轨迹Г 2在接触点处的切矢量，
Figure PCTCN2019075539-appb-000060
由
Figure PCTCN2019075539-appb-000061
得到。
实施例一中采用法截面法来进行第二齿轮齿面的构建；
具体地，如图9所示，第二齿轮齿面可由法截面曲面沿着共轭曲线连续变化而构建。法截面可由任意光滑曲线构成，根据公式
Figure PCTCN2019075539-appb-000062
可得，法截面齿廓方程可以表示为：
Figure PCTCN2019075539-appb-000063
其中，k 1为齿面圆弧半径。
根据矢量间关系及必要的矩阵变换，可以计算出法向量β的方程表达式：
Figure PCTCN2019075539-appb-000064
切矢α可表示为
Figure PCTCN2019075539-appb-000065
副法矢γ可以由表达式γ＝α×β确定，得到
Figure PCTCN2019075539-appb-000066
求得其单位向量为：
Figure PCTCN2019075539-appb-000067
Figure PCTCN2019075539-appb-000068
Figure PCTCN2019075539-appb-000069
由此，得到第二齿轮的齿面方程：
Figure PCTCN2019075539-appb-000070
构建的曲面如图10所示。
进一步地，所述法截面为圆弧或者双曲线齿廓。
优选地，综合考虑齿面承载能力及加工方便性，法截面选取圆弧或者双曲线齿廓，本实施例以圆弧齿廓作为研究对象。
进一步地，还包括步骤vii：根据齿轮齿面方程，编写程序，求解齿面方程。
齿面方程得出后，运用MATLAB编写程序，得到第二齿轮齿面的结构图，如图10所示。
构建曲线曲面啮合副后，在MATLAB中进行啮合仿真，如图11所示，具体参数见表1，图11(a)为啮合起点，图11(b)为啮合中任意位置，图11(c)为啮合终点，可以看出，曲线与曲面始终保持点接触，分别沿着啮合线啮合，啮合点清晰可见，啮合点在曲面上的轨迹与所求啮合迹线一致。
进一步地，还包括步骤viii：根据求解的齿面方程，计算齿面上点的坐标，将齿面上点的坐标导入三维软件中，由三维软件生成轮齿实体模型。
之后依据表1中齿轮设计参数，将齿面数据点采集并导入UG三维设计软件中，建立函数关系式约束齿顶圆、齿根圆和分度圆等尺寸，通过曲面造型等功能建立完整的齿轮模型，如图12所示。
实施例二
本实施例与实施例一不同的是，步骤vi采用包络法确定齿轮齿形方程。
具体地，如图13所示，设想齿轮2固定不动，把瞬心线I在瞬心线П上纯滚动，齿形1就在齿轮2的平面上形成曲线族。由于齿形2与齿形1在每个瞬时都是相切接触的，从数学上讲，齿形2就应是齿形1形成的曲线族的包络。用这个原理，可以由齿轮副的运动规律及齿形1求得齿形2，这种方法称为包络法。
由一个齿轮的齿形包络出另一个齿轮的齿形，这种现象在用展成法加工齿轮时可以观察地很清楚，例如，在插齿时把插齿刀的齿形(包括侧刃及顶刃)作为齿形1，在展成过程中它就逐步地包络出工件的齿形2。
实施例三
本实施例与实施例一不同的是，步骤vi采用齿形法线法确定齿轮齿形方程。
具体地，令齿轮1及2都作转动，设在某一瞬时，齿形1在图14中实线所示的位置，根据n·v＝0的条件，可以在齿轮1上求得一点M，它的法线通过瞬心点P，则M就是这个瞬时齿形1和2的接触点。齿轮1转过一个角度，齿形转到图13中虚线所示的位置时，同样可以求得法线通过点P的点M',则M'又是这个瞬时两个齿形的接触点。用这种方法求得齿轮1转动过程中的一系列接触点位置，它在固定平面中的轨迹就称为啮合线(因为两个齿形是沿着这条线连续地发生啮合的)。接触点在转动着的齿轮2平面上的轨迹就是要求的齿形2。
本申请提供的基于线面共轭的对构齿轮啮合副的设计方法不止局限于渐开线齿轮，而且第一齿轮与第二齿轮的接触理论基础是线面啮合，具有对齿轮轴线偏斜、俯仰误差的适应能力强及中心距可分性的特点，且第一齿轮与第二齿轮的实际接触状态为点接触，齿面间接近理论纯滚动，滑动率小；而且齿面接触类型不限于凸凸接触，也可以是凸凹接触，适用范围广，可以提高齿轮的承载能力。
最后应说明的是：以上各实施例仅用以说明本申请的技术方案，而非对其限制；尽管参照前述各实施例对本申请进行了详细的说明，本领域的普通技术人员应当理解：其依然可以对前述各实施例所记载的技术方案进行修改，或者对其中部分或者全部技术特征进行等同替换；而这些修改或者替换，并不使相应技术方案的本质脱离本申请各实施例技术方案的范围。

Claims (5)

  1. 一种基于线面共轭的对构齿轮啮合副，其特征在于，包括第一齿轮和第二齿轮，所述第一齿轮与第二齿轮形成线面共轭齿轮副；
    所述线面共轭齿轮副包括连续相切接触的曲面Σ 1和曲线Г 1，所述曲面Σ 1位于所述第一齿轮上，所述曲面Σ 1上仅有一条曲线与所述曲线Г 1接触；在预定时刻，所述曲线Г 1与所述曲面Σ 1保持点接触，且预定时刻，所述曲线Г 1与所述曲面Σ 1的接触点唯一；
    所述曲线Г 1为光滑曲线，所述曲面Σ 1为光滑曲面。
  2. 一种基于线面共轭的对构齿轮啮合副的设计方法，其特征在于，包括如下步骤：
    i)确定第一齿轮的曲面Σ 1方程为：
    Figure PCTCN2019075539-appb-100001
    其中，u，v为曲面参数；
    ii)确定所述第一齿轮的齿面上的接触迹线Г 2方程为：
    Figure PCTCN2019075539-appb-100002
    其中，φ为空间参数；
    iii)确定所述第一齿轮的齿面上的接触迹线的法向量
    Figure PCTCN2019075539-appb-100003
    为：
    Figure PCTCN2019075539-appb-100004
    iv)由公式n·ν (12)＝0，确定曲线Г 1与所述曲面Σ 1的啮合方程；
    v)根据所述接触迹线Г 2在固定坐标系S 1下的方程与啮合方程，确定共轭曲线；
    Figure PCTCN2019075539-appb-100005
    vi)根据共轭曲线，确定第二齿轮齿面Σ 2方程。
  3. 根据权利要求2所述的基于线面共轭的对构齿轮啮合副的设计方法，其特征在于，所述步骤v中还包括步骤：
    建立所述固定坐标系S 1与动坐标系S 2，所述固定坐标系S 1与动坐标系S 2分别与所述曲线Г 1与所述曲面Σ 1固联；
    确定所述动坐标系S 2与所述固定坐标系S 1之间的转化矩阵：
    Figure PCTCN2019075539-appb-100006
    其中，
    Figure PCTCN2019075539-appb-100007
    为所述曲线Г 1转过的角度，
    Figure PCTCN2019075539-appb-100008
    为所述曲面Σ 1转过的角度。
  4. 根据权利要求3所述的基于线面共轭的对构齿轮啮合副的设计方法，其特征在于，所述步骤iv中还包括：
    确定所述动坐标系S 2相对于所述固定坐标系S 1的相对运动速度ν (12)；
    Figure PCTCN2019075539-appb-100009
    其中，ω 1为所述曲线Г 1转动的角速度，ω 2为所述曲面Σ 1转动的角速度。
  5. 根据权利要求4所述的基于线面共轭的对构齿轮啮合副的设计方法，其特征在于，所述步骤vi中还包括：
    选取法截面曲线为Г c：
    Г (t)＝(x(t) 0 z(t) 1) Tr，
    其中，x(t)和z(t)为参数t的函数；
    所述Г c位于所述曲线Г 1的法平面上，且与所述曲线Г 1相交于一点，所述曲线Г c穿过相交点且所述曲线Г c在相交点的主法矢方向和所述曲面Σ 1在相交点的主法线方向一致；
    确定曲面Σ 2方程：
    Figure PCTCN2019075539-appb-100010
    Figure PCTCN2019075539-appb-100011
    其中，
    Figure PCTCN2019075539-appb-100012
    为所述曲面Σ 2在相交点的主法线向量，
    Figure PCTCN2019075539-appb-100013
    为所述曲面Σ 2上接触轨迹Г 2在接触点处的切矢量，
    Figure PCTCN2019075539-appb-100014
    由
    Figure PCTCN2019075539-appb-100015
    得到。
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