Study Of Kinematics And Meshing Characteristic Of Harmonic Gear Drives Based On The Deformation Function Of The Flexspline

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.