基于柔轮变形函数的谐波齿轮传动运动几何学及其啮合性能研究
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摘要
本论文在国家自然科学基金项目(No.59375178,50775016)和国家高技术研究发展计划项目(863项目)(No.2006AA04Z101)的资助下,结合谐波传动的设计和应用实际,对杯形柔轮的谐波齿轮传动运动几何学及其齿廓啮合性能进行了研究。
首先依据谐波齿轮传动输入、输出相对运动原理,提出了基于柔轮变形函数的谐波齿轮传动新模型,是传统的谐波齿轮传动摩擦模型、行星传动模型(等速曲线、节曲线)之外的更直接的数学模型。从而得到谐波齿轮传动的瞬时理论传动比、实际综合传动比的解析模型;并通过定义波发生器、刚轮、柔轮输出端和柔轮齿的坐标系,定义柔轮输出端与啮合端的弹性变形函数,推导了三构件的相对运动和轮齿的啮合运动的运动变换关系。
其次,确定了谐波齿轮传动中柔轮弹性变形函数求解方法,构造了基于ANSYS参数化设计语言APDL的有限元接触分析参数化模型;对柔轮在不同参数、不同载荷作用的弹性变形函数进行求解,并用双三次样条拟合得到柔轮壳体变形函数方程,实现了柔轮壳体弹性变形函数表述柔轮刚性输出端和啮合端的对应关系;以D120机型的谐波齿轮传动为例分析了柔轮变形函数特性;实验结果与柔轮弹性变形函数模型计算具有很高的一致性。
对于谐波传动中一系列啮合平面的齿廓运动进行了运动几何学研究:建立了谐波传动各截面的瞬心线解析方程,得到了柔轮齿相对刚轮齿运动的定、动瞬心线和柔轮齿的回转中心的解析表达式;讨论了瞬心线的特性,如瞬心线的形状与柔轮壳体变形函数相关性、瞬心线的周期性、不封闭性、渐近线存在性等;基于柔轮齿相对刚轮齿的运动几何学性质(定、动瞬心线的纯滚动),把传统的刚性齿轮齿廓共轭啮合定理推广到谐波齿轮传动的弹性共轭理论中,得到了实现谐波齿轮传动定传动比输入输出要求的轮齿弹性共轭求解方法。
建立了谐波传动弹性共轭齿廓综合方程;基于瞬心线的性状和共轭条件证明了谐波传动二次啮合(二次共轭点)的存在性:以D120机型的谐波齿轮传动中刚轮齿廓(渐开线)为已知齿廓,求解与其共轭的柔轮空间齿廓,这一空间齿廓的前后截面齿厚相差12.9%倍的模数;以此刚轮渐开线齿形为基准,分别研究了直线齿、单圆弧齿和双圆弧齿的共轭齿廓,提出以平均齿厚变化最小、重合度最大、共轭点连续为原则,综合与其共轭的柔轮齿廓,得到了新的齿廓特征:
(1)单圆弧作为刚轮齿廓曲线选择合适的齿形参数时,二次共轭啮合曲线能在柔轮齿形中应用,重合度增加了3.8倍以上;
(2)双圆弧齿在谐波传动中不仅重合度大,而且在一段区域、一个瞬时,齿廓上有两个点同时共轭,同时共轭段占全齿高的82.07%。
在谐波传动运动几何学基础上,提出基于瞬心线的谐波齿轮传动的理论重合度和实际重合度定义;给出了理论和实际重合度的计算方法;依据曲线的曲率的特征对刚轮齿廓曲线的性状分类;深入分析了刚轮齿廓曲线的性状、齿形参数与重合度的关系,得出在相同齿全高的条件下重合度大小按曲线的曲率排列为ε_(n|k_(th)＞0)＜ε_(n|k_(th)=0)＜ε_(n|k_(th)＜0);讨论了常用刚轮齿廓:渐开线、单圆弧和双圆弧曲线的重合度和二次啮合存在条件,给出了谐波传动齿形二次啮合条件。
同时,分析了负载作用下柔轮壳体畸变对瞬心线和共轭齿廓的影响,得到瞬心线在负载作用下沿轴向的非线性变化和随负载大小的变化特征;共轭齿廓在负载作用下出现偏转,偏转角沿轴线呈线性变化;随着负载的加大单圆弧和双圆弧齿形的共轭啮合重合度增大,由此说明在负载条件下单圆弧和双圆弧齿形仍然具有良好的啮合特性。
最后,对本文提出的谐波齿轮传动空间弹性共轭理论综合齿形进行了运动仿真,验证了本运动几何学模型的正确性与综合方法的有效可行性。由此完成的数学模型和求解方法,为谐波传动结构设计和齿廓综合提供了新途径。
A kinematic model and approach for vigorous analysis and conjugating tooth profile generation of harmonic drive is presented based on the basic principle of harmonic drive.This research supported by the National Natural Science Foundation of China under Grant No. 59375178,50775016 and National High Technology Research and Development Program of China(863 program) under Grant No.2006AA04Z101.
Firstly,based on the relative motion principle of harmonic drive,the five frames and the elastic deformation function of a flexspline at meshing part relative to the output part are defined,which can express the three components relative motion and flexspline tooth relative flexspline output part,and the motion transformation between the three components and the flexspline tooth part are established.Thus a new mathematic model that is different from traditional friction model of harmonic drive is established,which an analytical model of instant and synthesis transmission radio is derived.A finite element contact analysis parametric model for a flexspline elastic deformation investigation in harmonic drive is built by ANSYS Parametric Design Language(APDL).Using this model,a spatial elastic deformation of a cup shaped flexspline with any parameters can be solved,and then flexspline elastic deformation function is fit by double cubic spline function.Thereby the relationship between the rigid output part and meshing part of a flexspline is clearly expessed by the elastic deformation function of a flexspline.D120 harmonic drive is taken as an example calculating and testing the deformation of flexspline shell.It proves that the model and the results are correct.
For a set of planar tooth profile motion of harmonic drive,the kinematics and geometry of harmonic drive is investigated.The mathematical equation of the centrode is established, from which the analytical expressions between flexspline tooth and circular spline such as absolute centrode,fixed centrode,moving centrode and relative centrode are derived by means of differential geometry methode.The characteristic of the centrodes is discussed, which includes the relationship between the centrode shape and deformation function of flexspline,the periodicity of the centrode and the existence of the asymptote and so on.Based on the kinematics characteristic of the flexspline tooth relative to the circular spline tooth,the elastic conjugation method of harmonic drive is proposed,which satisfies the desired transmission ratio at harmonic drive output part.
The conjugating tooth profile equation is set up.The twice engagement in harmonic drive is first discovered and demonstrated by means of gear meshing principle.D120 type harmonic drive is as example in which the circular spline tooth profile is given,such as the involute tooth profile,straight line tooth profile,the single arc tooth profile and the double arc tooth profile,to find the conjugating tooth profile.There are two first discoveries,the one first discovery is that the twice engagement can be used in the circle arc tooth profile,which can make contact radio increase 348%;the second first discovery is that the double circle arc tooth profile in the part of 82.07%whole tooth has two point meshing at the same time.
The definition of the contact ratio in harmonic drive is proposed.The calculating formula of the contact radio is given.The engagement properties of the tooth profiles of the circular spline with the curvature characteristic k_(th)＞0,k_(th)=0,k_(th)＜0 are analyzed,the common used tooth profiles such as involute tooth profile,single circle arc tooth profile and double circle arcs tooth profile are discussed,leads out the theorem that:for any kind of profiles of circular spline,there are always be twice engagement as long as the contact ratio is more than zero, and indicated that the circular-arc profile of circular spline with the curvature k_(th)＜0 has better engagement property for it with larger contact ratio.In addition,theorem of twice engagement in harmonic drive is presented.
Meanwhile,the flexspline deformation function under the load influencing to the centrodes and conjugation profiles are analyzed.The result is that under the load condition the centrodes along the axis of rotating shaft is nonlinearity distribution,the deflection of the conjugating tooth profile yields and along the axis of rotating shaft is nonlinearity variation, the contact radio is larger than no-load condition and the tooth profiles with single arc curve and double arc curve have better conjugating property.
Finally,the meshing motion of four kind of conjugating tooth pairs is simulated.It proves that theory proposed is correct and the synthesis method is available.The mathematic model and the method achieved in this dissertation provide a new approach for structure design and tooth profile synthesis of harmonic gear drives.
首先依据谐波齿轮传动输入、输出相对运动原理,提出了基于柔轮变形函数的谐波齿轮传动新模型,是传统的谐波齿轮传动摩擦模型、行星传动模型(等速曲线、节曲线)之外的更直接的数学模型。从而得到谐波齿轮传动的瞬时理论传动比、实际综合传动比的解析模型;并通过定义波发生器、刚轮、柔轮输出端和柔轮齿的坐标系,定义柔轮输出端与啮合端的弹性变形函数,推导了三构件的相对运动和轮齿的啮合运动的运动变换关系。
其次,确定了谐波齿轮传动中柔轮弹性变形函数求解方法,构造了基于ANSYS参数化设计语言APDL的有限元接触分析参数化模型;对柔轮在不同参数、不同载荷作用的弹性变形函数进行求解,并用双三次样条拟合得到柔轮壳体变形函数方程,实现了柔轮壳体弹性变形函数表述柔轮刚性输出端和啮合端的对应关系;以D120机型的谐波齿轮传动为例分析了柔轮变形函数特性;实验结果与柔轮弹性变形函数模型计算具有很高的一致性。
对于谐波传动中一系列啮合平面的齿廓运动进行了运动几何学研究:建立了谐波传动各截面的瞬心线解析方程,得到了柔轮齿相对刚轮齿运动的定、动瞬心线和柔轮齿的回转中心的解析表达式;讨论了瞬心线的特性,如瞬心线的形状与柔轮壳体变形函数相关性、瞬心线的周期性、不封闭性、渐近线存在性等;基于柔轮齿相对刚轮齿的运动几何学性质(定、动瞬心线的纯滚动),把传统的刚性齿轮齿廓共轭啮合定理推广到谐波齿轮传动的弹性共轭理论中,得到了实现谐波齿轮传动定传动比输入输出要求的轮齿弹性共轭求解方法。
建立了谐波传动弹性共轭齿廓综合方程;基于瞬心线的性状和共轭条件证明了谐波传动二次啮合(二次共轭点)的存在性:以D120机型的谐波齿轮传动中刚轮齿廓(渐开线)为已知齿廓,求解与其共轭的柔轮空间齿廓,这一空间齿廓的前后截面齿厚相差12.9%倍的模数;以此刚轮渐开线齿形为基准,分别研究了直线齿、单圆弧齿和双圆弧齿的共轭齿廓,提出以平均齿厚变化最小、重合度最大、共轭点连续为原则,综合与其共轭的柔轮齿廓,得到了新的齿廓特征:
(1)单圆弧作为刚轮齿廓曲线选择合适的齿形参数时,二次共轭啮合曲线能在柔轮齿形中应用,重合度增加了3.8倍以上;
(2)双圆弧齿在谐波传动中不仅重合度大,而且在一段区域、一个瞬时,齿廓上有两个点同时共轭,同时共轭段占全齿高的82.07%。
在谐波传动运动几何学基础上,提出基于瞬心线的谐波齿轮传动的理论重合度和实际重合度定义;给出了理论和实际重合度的计算方法;依据曲线的曲率的特征对刚轮齿廓曲线的性状分类;深入分析了刚轮齿廓曲线的性状、齿形参数与重合度的关系,得出在相同齿全高的条件下重合度大小按曲线的曲率排列为ε_(n|k_(th)＞0)＜ε_(n|k_(th)=0)＜ε_(n|k_(th)＜0);讨论了常用刚轮齿廓:渐开线、单圆弧和双圆弧曲线的重合度和二次啮合存在条件,给出了谐波传动齿形二次啮合条件。
同时,分析了负载作用下柔轮壳体畸变对瞬心线和共轭齿廓的影响,得到瞬心线在负载作用下沿轴向的非线性变化和随负载大小的变化特征;共轭齿廓在负载作用下出现偏转,偏转角沿轴线呈线性变化;随着负载的加大单圆弧和双圆弧齿形的共轭啮合重合度增大,由此说明在负载条件下单圆弧和双圆弧齿形仍然具有良好的啮合特性。
最后,对本文提出的谐波齿轮传动空间弹性共轭理论综合齿形进行了运动仿真,验证了本运动几何学模型的正确性与综合方法的有效可行性。由此完成的数学模型和求解方法,为谐波传动结构设计和齿廓综合提供了新途径。
A kinematic model and approach for vigorous analysis and conjugating tooth profile generation of harmonic drive is presented based on the basic principle of harmonic drive.This research supported by the National Natural Science Foundation of China under Grant No. 59375178,50775016 and National High Technology Research and Development Program of China(863 program) under Grant No.2006AA04Z101.
Firstly,based on the relative motion principle of harmonic drive,the five frames and the elastic deformation function of a flexspline at meshing part relative to the output part are defined,which can express the three components relative motion and flexspline tooth relative flexspline output part,and the motion transformation between the three components and the flexspline tooth part are established.Thus a new mathematic model that is different from traditional friction model of harmonic drive is established,which an analytical model of instant and synthesis transmission radio is derived.A finite element contact analysis parametric model for a flexspline elastic deformation investigation in harmonic drive is built by ANSYS Parametric Design Language(APDL).Using this model,a spatial elastic deformation of a cup shaped flexspline with any parameters can be solved,and then flexspline elastic deformation function is fit by double cubic spline function.Thereby the relationship between the rigid output part and meshing part of a flexspline is clearly expessed by the elastic deformation function of a flexspline.D120 harmonic drive is taken as an example calculating and testing the deformation of flexspline shell.It proves that the model and the results are correct.
For a set of planar tooth profile motion of harmonic drive,the kinematics and geometry of harmonic drive is investigated.The mathematical equation of the centrode is established, from which the analytical expressions between flexspline tooth and circular spline such as absolute centrode,fixed centrode,moving centrode and relative centrode are derived by means of differential geometry methode.The characteristic of the centrodes is discussed, which includes the relationship between the centrode shape and deformation function of flexspline,the periodicity of the centrode and the existence of the asymptote and so on.Based on the kinematics characteristic of the flexspline tooth relative to the circular spline tooth,the elastic conjugation method of harmonic drive is proposed,which satisfies the desired transmission ratio at harmonic drive output part.
The conjugating tooth profile equation is set up.The twice engagement in harmonic drive is first discovered and demonstrated by means of gear meshing principle.D120 type harmonic drive is as example in which the circular spline tooth profile is given,such as the involute tooth profile,straight line tooth profile,the single arc tooth profile and the double arc tooth profile,to find the conjugating tooth profile.There are two first discoveries,the one first discovery is that the twice engagement can be used in the circle arc tooth profile,which can make contact radio increase 348%;the second first discovery is that the double circle arc tooth profile in the part of 82.07%whole tooth has two point meshing at the same time.
The definition of the contact ratio in harmonic drive is proposed.The calculating formula of the contact radio is given.The engagement properties of the tooth profiles of the circular spline with the curvature characteristic k_(th)＞0,k_(th)=0,k_(th)＜0 are analyzed,the common used tooth profiles such as involute tooth profile,single circle arc tooth profile and double circle arcs tooth profile are discussed,leads out the theorem that:for any kind of profiles of circular spline,there are always be twice engagement as long as the contact ratio is more than zero, and indicated that the circular-arc profile of circular spline with the curvature k_(th)＜0 has better engagement property for it with larger contact ratio.In addition,theorem of twice engagement in harmonic drive is presented.
Meanwhile,the flexspline deformation function under the load influencing to the centrodes and conjugation profiles are analyzed.The result is that under the load condition the centrodes along the axis of rotating shaft is nonlinearity distribution,the deflection of the conjugating tooth profile yields and along the axis of rotating shaft is nonlinearity variation, the contact radio is larger than no-load condition and the tooth profiles with single arc curve and double arc curve have better conjugating property.
Finally,the meshing motion of four kind of conjugating tooth pairs is simulated.It proves that theory proposed is correct and the synthesis method is available.The mathematic model and the method achieved in this dissertation provide a new approach for structure design and tooth profile synthesis of harmonic gear drives.
引文
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