Generalized forms of an overconstrained sliding mechanism consisting of two congruent tetrahedra

TL;DR

This paper explores the finite motions of a bar structure made of two congruent tetrahedra with specific edge constraints, generalizing previous work from cubes to more general parallelepipeds and proposing further generalizations.

Contribution

It extends the analysis of overconstrained tetrahedral mechanisms from cubes to general parallelepipeds and introduces new potential generalizations with examples of finite motions.

Findings

All finite motions of the structure are determined.

Generalization from cubes to rectangular parallelepipeds is achieved.

Examples of finite motions for new generalizations are provided.

Abstract

We investigate the motions of a bar structure consisting of two congruent tetrahedra, whose edges in their basic position form the face diagonals of a rectangular parallelepiped. The constraint of the motion is that the originally intersecting edges should remain coplanar. We determine all finite motions of our bar structure. This generalizes our earlier work, where we did the same for the case when the rectangular parallelepiped was a cube. At the end of the paper we point out three further possibilities to generalize the question about the cube, and give for them examples of finite motions.