Synthesize new 5-bar paradoxical chains via the oblique circular cylinder

Based on the geometric properties of the oblique circular or elliptic cylinder and an invariant subset of translations, a family of related chains having a paradoxical mobility is systematically synthesized. First of all, vector calculation is used to describe the plane symmetry operation. The plane symmetry property of an oblique circular (OC) cylinder, which has circular directrices, is verified. Next, we introduce a mechanical generator of 2-DoF translation along the surface of an OC cylinder and derive a novel PaPPa paradoxical chain, where Pa denotes the composite joint of a 4-R hinged parallelogram with parallel axes. Then, we establish its corresponding image chain. The implementation of pseudo-planar and Delassus parallelograms leads to more general chains of type PHPPH, where PH designates the composite joint of a 4-H parallelogram with four parallel axes. The related image chain having two couples of H pairs with parallel axes and equal pitches separately is a new 1-DoF paradoxical linkage.