On spatial relations to the Euler-Savary formula

Abstract

The Euler-Savary formula is a well-established relation with multiple interpretations. This formula is revisited and cast as a geared three-link spatial mechanism with uniform motion and gear rotation axes without axial pitch. Two versions of the spatial Euler-Savary formula emerge; the classical version that matches motion parallel to the ISA and a new version that matches motion perpendicular to the ISA akin to the planar scenario. These two versions are combined resulting in a circle line surface of moving axes together with a conjugate center line surface of stationary axes. These two surfaces define axis pairs used to match second-order axode motion. Also included are inflection surfaces as the spatial analog to the planar inflection circle. One of the inflection surfaces is contrasted with an invariant relative curvature surface. Subsequently, a ruled surface is presented to emulate the planar Bresse circle by combining two axes on the inflection surface to establish a third axis on a Bressesque surface. Lastly, an RSSR mechanism is presented by combining aspects of the invariant curvature surface and an R-R interpretation of the Euler-Savary formula.