On Spatial Euler–Savary Equations for Envelopes

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.