On Spatial Euler-Savary Equations for Envelopes

摘要

Presented are three equations that are believed to be original and new to the kinematics community. These three equations are extensions of the planar Euler-Savary relations (for envelopes) to spatial relations. All three spatial forms parallel the existing well established planar Euler-Savary equations. The genesis of this work is rooted in a system of cylindroidal coordinates specifically developed to parameterize the kinematic geometry of generalized spatial gearing and consequently a brief discussion of such coordinates is provided. Hyperboloids of osculation are introduced by considering an instantaneously equivalent gear pair. These analog equations establish a relation between the kinematic geometry of hyperboloids of osculation in mesh (viz., second-order approximation to the axode motion) to the relative curvature of conjugate surfaces in direct contact (gear teeth). Planar Euler-Savary equations are presented first along with a discussion on the terms in each equation. This presentation provides the basis for the proposed spatial Euler-Savary analog equations. A lot of effort has been directed to establishing generalized spatial Euler-Savary equations resulting in many different expressions depending on the interpretation of the planar Euler-Savary equation. This work deals with the interpretation where contacting surfaces are taken as the spatial analog to the contacting planar curves.