On the helical Cardan motion and related paradoxical chains

A novel special motion called helical Cardan motion is introduced. The well known Cardanic motion is a special case with a zero pitch. A Cardan motion is a planar cycloidal motion, which can be produced by the internal rolling of a circle on a fixed circle whose diameter is twice the moving circle diameter. The newly studied motion is a combination of a Cardan motion with a helical motion. Any point lying on the surface of a cylinder having the smaller rolling circle as its cross section traces out a planar sinusoidal curve. Frame-free vector calculations are used to establish the essential properties of the special motion. Its corresponding characteristics are further utilized to make new derivations of related paradoxical chains, namely the new isosceles triangle HHHP chain, the HHHH rhomboid of Delassus and the two-loop Sarrus-type 6H-1P chain. A generalization of Sarrusʹ mechanism for the generation of a rectilinear translational motion is derived as well.