On the vertical darboux motion

Abstract

A vertical Darboux motion (VDM) is a 1-DoF motion. It is a special case of a general Darboux motion in which all the trajectories in the moving body are planar ellipses. An axis is globally invariant in a VDM, which is a special cylindrical motion. Beyond the transformation of point coordinates, which is available in the literature, some mechanical generators of a VDM are revealed. These VDM generators are systematically synthesized by the parallel arrangement of one cylindrical (C) pair and one generator of a 5-dimensional (5D) manifold of rigid-body displacements, in which one point is compelled to move in a fixed plane. For instance, the center of a spherical S pair in a serial PPS chain moves in a plane that is parallel to the prismatic P pairs. Replacing the PPS chain with other generators of planar-spherical motion yields isoconstrained generators of VDM. In addition, overconstrained realizations of VDM generators with exceptional mobility are also unveiled.