Spatial tooth profile design of harmonic gears based on curve-surface conjugation theory

ZHANG Ruiliang; SU Yanrong; WANG Yuxin

doi:10.16578/j.issn.1004.2539.2026.06.006

Abstract：[Objective]The spatial angular displacement caused by the irregular spatial deformation of the flexspline cup in harmonic reducers makes it difficult for traditional planar conjugate tooth profile design methods to accurately describe the real contact law between the circular spline and flexspline.A novel spatial tooth profile design method for harmonic gears based on curve-surface conjugation mechanism was proposed.[Methods]Firstly,a spatial kinematic coordinate system and motion model for harmonic drive were established,considering spatial deformation factors such as radial displacement,tangential displacement and normal angular displacement of the flexspline cup,to provide an accurate kinematic basis for tooth profile design.Secondly,the curve-surface conjugation mechanism was introduced.A contact path was preselected on the given involute flexspline tooth surface,and the meshing equation was established to iteratively solve the corresponding conjugate contact curve of the circular spline.Then,the three-dimensional spatial tooth surface of the circular spline was generated according to the surface construction mechanism.Finally,the contact stress and meshing tooth number of the spatial tooth profile and the traditional planar tooth profile were compared and analyzed through numerical examples and finite element models to verify the effectiveness of the proposed method.[Results]The results show that the number of meshing tooth pairs of the harmonic gear constructed by the curve-surface conjugation design method increases by 5.9%compared with the traditional method,and the maximum contact stress on the tooth surface decreases by 7%.The proposed method effectively enlarges the conjugate contact area of gear teeth and improves the stress distribution state of the tooth surface,which provides a reference for the optimal design of spatial tooth profiles of high-precision harmonic gears.