Overconstrained Mechanisms Based on Trapezohedra

We start with a special polyhedron, the so-called ‘trapezohedron’ (see E. WEISSTEIN [11]). It is dual to an antiprism and can be generated as the intersection of two suitable congruent regular pyramids with regular n-gons as directing polygons (n > 2). Its faces are 2n congruent trapezoids (kites). We use this polyhedron to construct some generalisations of the Fulleroid-like mechanisms described by G. KIPER [2]-[3], K. WOHLHART [12]-[13] and the author [8]. These generalisations consist of 2n congruent special parallel four-bars in the faces of this trapezohedron which are interlinked by spherical 2R-joints (at some fixed angles). This mechanism consists of 8n rigid bodies interlinked via 8n 1R-joints and 4n 2R-joints. As the classical Grübler-Kutzbach-Chebyshev-formula gives a theoretical degree of freedom F = 6 8n at first sight this mechanism is supposed to be rigid. But owing to our special geometric dimensions physical models (for some values of n > 2) seem to admit at least a one-parametric self-motion. The particular case n=3 can also be viewed as some generalization of the Fulleroid-like-mechanism presented in [3]. The aim of this paper is to work out and elucidate the existence of highly symmetric one-parametric self-motions of the mechanisms for general values of n>2. All these different self–motions share two higher order singular positions.