Triangle Pseudocongruence in Constraint Singularity of Constant-Velocity Couplings

Abstract

Congruent triangles establish that a class of intersecting-shaft couplings is constant velocity. These mechanisms employ a pair of linkages in parallel: a spherical joint at the intersection of the shafts and the intersection of straight-line tracks away from the shaft center to transmit rotation. A proof of constant velocity follows from the congruence of an initial pair of triangles with two matching sides and one excluded angle. This side-side-angle (SSA) condition is a pseudocongruence because it allows two different values for the included angle, indicating that such shaft couplings have symmetric and skewed assembly configurations. If the other excluded angle happens to be 90 deg, the SSA condition guarantees congruence because there is a single solution for the included angle. The 90 deg condition, however, occurs at a posture with a constraint singularity, where the shaft coupling is unable to transmit torque. Motion screw analysis establishes the same geometric condition for a coupling based on a revolute-spherical-revolute Clemens linkage. An upper bound on shaft deflection imposed by hyperextension of that linkage, along with a bound on deflection where constraint singularity occurs, identifies couplings where the constraint singularity can occur within the physical limits.