Research On The Meshing Force Performance Of Harmonic Gear Based On Backlash And Meshing Stiffness Matrix

Harmonic drive can achieve movement or power transmission by controllable elastic deformation of flexspline squeezed by the wave generator continuously,and its transmission principle is essentially different from that of rigid gear transmission.Therefore,harmonic drive has many advantages,such as large transmission ratio,many meshing teeth,high transmission efficiency,high transmission precision and little return difference.It has been widely used in aeronautics and astronautics,industrial robots,machine tools,medical instruments and various electronic devices.The reasonable tooth profile parameters are the precondition to ensure the transmission precision,and the index of tooth profile parameters is mainly the distribution of backlash and meshing force.In this paper,the distribution of the backlash and the meshing force between tooth profile of flexspline and tooth profile of circular spline are calculated and studied.First,the calculation methods of backlash of harmonic gear are improved.The tooth profile equation under coordinate transformation is established,and the backlash algorithm based on the orientation of the circumferential displacement is proposed.An accurate algorithm is used to iteratively calculate the position of the gear root after the assembly deformation,and a backlash algorithm based on the arc length positioning is proposed.These backlash methods are more consistent with the backlash results of the finite element model.Based on the linear generatrix assumption of the flexspline deformation,the spatial taper deformation of the flexspline cylinder are solved,and the theoretical calculation method of backlash of the spatial meshing tooth profile is given.Then,the meshing stiffness matrix of flexspline is solved by the theory and the finite element model.According to the torsion theory of thin-walled cylinder and disc,beam bending energy method,the torsional deformation of cylinder and the displacement of meshing point on the tooth body are calculated respectively when the meshing point of flexspline is subjected to the uint circumferntial force,respectively.In order to reflect the trend of flexspline meshing flexibility,a unit circumferntial force is applied to the flexspline meshing point of the solid element,and its circumferential displacement is extracted.And the flexibility matrix of the meshing point is constructed.Finally the linear meshing stiffness matrix is obtained from the inverse of flexibility matrix.According to the geometric relationship between the circumferential displacement of the meshing point on circular spline,the circumferential displacement of the flexspline meshing point and the no-load backlash between the tooth profiles,a theoretical iterative algorithm for meshing force based on no-load backlash and linear meshing stiffness matrix is put forward.Through the gradual increase of the rotational displacement on circular spline,the load backlash and the meshing force are calculated iteratively.The linear meshing stiffness matrix that is adopted in the theoretical algorithm does not consider the.nonlinear influence of mesh stiffness,and especially the nonlinear variation of meshing stiffness caused by change of the wrapping angle that different meshing force are subject on the double disc wave generator.In order to verify the theoretical algorithm,a plane tooth profile and 3D solid finite element model are established respectively.The load backlash and meshing force are numerically solved by finite element nonlinear contact analysis.The research shows that:(1)The calculated results from the improved backlash algorithm are more consistent with the results of the finite element model.(2)Under lower load condition,the meshing force on a single tooth is linearly proportional relationship with the rotation angle of the circular spline.Under large load condition,the meshing force on single tooth has a nonlinear relationship with the rotation angle of the circular spline.(3)Under the rated torque condition,the distribution of the meshing force given by the theoretical iterative algorithm is basically consistent with the finite element results.When the load is much larger,the number of meshing teeth is less than that of the theoretical algorithm,and the amplitude of the meshing force is larger.As the load reaches a certain extent,the meshing stiffness of flexspline presents evidently nonlinear property.