On the helical Cardan motion and related paradoxical chains

Article history: Received 2 February 2011 Received in revised form 6 January 2012 Accepted 20 January 2012 Available online 18 February 2012 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 motionwith 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 Sarrus' mechanism for the generation of a rectilinear translational motion is derived as well. © 2012 Elsevier Ltd. All rights reserved.