Two second order instantaneous invariants of axodes

Abstract

It is well established that generalized spatial motion of a rigid body can be defined in terms of two axodes; a fixed axode and a moving axode. These axodes are ruled surfaces that uniquely roll and slide upon one another. Presented are two instantaneous invariants that define relative motion between two axodes. These invariants are based on the instantaneous screw axis (ISA) or line-tangency shared by two axodes in mesh. One invariant is the relative angular displacement (rolling) about the ISA and the other invariant is the relative axial displacement (sliding) along the ISA. Both invariants are ratios expressed in terms of the normal curvature and geodesic torsion for any point on the ISA. An induced curvature is introduced to quantify the combined rolling and sliding displacement. Two osculating hyperboloids are established in terms of induced curvature, which when combined, create a skew axis gear pair with uniform motion that matches the instantaneous invariants of the axodes. A skew axis gear set with elliptical motion is presented to demonstrate generalized axodes, instantaneous invariants, induced curvature, and osculating hyperboloids.