The paper presents the first elastic meshing law and spatial elastic conjugation tooth profile for harmonic gear drives. The kinematics and geometry of harmonic drives that have a cup-type flexible spline with an oval wave generator is investigated. The deformation function of the neutral layer of a flexspline is investigated and calculation example is taken by non-linear finite element software. The mathematic model of kinematics of harmonic drive is set up, in which the spatial deformation of a flexspline is separated into a set of deformed curves on the cross sections vertical to the axis. Thus, the spatial engagement in harmonic drive can be visualized as a set of planar engagement. The properties of the instant center and the centrodes of the flexspline tooth relative to the circular spline are studied. The phenomenon of twice engagement at one tooth point of circular spline is found for the first time. The planar gearing engagement and the instant center equation are analyzed and a vigorous spatial elastic conjugation theory is provided for harmonic gear drive.

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