Flexible polyhedra - Exploring finite mechanisms of triangulated polyhedra

Change log

In a quest to design novel deployable structures, flexible polyhedra provide interesting insights. This work follows the discovery of flexible polyhedra and aims to make flexible polyhedra more useful.

The dissertation describes how flexible polyhedra can be made. The flexible polyhedra first considered in this dissertation have a rotational degree of freedom. The range of this rotational movement is measured and maximised in this work by numerical maximisation. All polyhedra are established computationally: an iterative solution method is used to find vertex coordinates; several clash detecting methods are described to define whether each rotational position of a flexible polyhedron is physically possible; then a range of motion is defined between occurrences of clashes at the two ends; finally, an optimisation tool is used to maximise the range of motion.

By using these tools, the range of motion of two types of simplest flexible polyhedra are maximised. The first type is a series of flexible polyhedra generalised from the Steffen flexible polyhedron. The range of motion of this type is improved to double that of Steffen’s original, from 27° to 59°. Another type of flexible polyhedron is expanded from a model provided by Tachi. Based on the understanding of Steffen’s flexible polyhedron, optimisation parameters are carefully given. This new type has achieved a wider range of motion, so now the range of motion of flexible polyhedron is tripled to 80°.

After enlarging the range of motion of the degree of freedom in the 1-dof systems, the dissertation found multiple degrees of freedom in one polyhedron. The multiple mechanisms can be even repetitive, so that an n-dof polyhedron is found. A polyhedron of two degrees of freedom is first presented. Then, a unit cell for any number of mechanisms is found. As a repetitive structure, a 3-dof polyhedron is presented. Finally, this work presents the possibility of configuring a flexible polyhedral torus and a closed polyhedral surface that is able to flex without the need to stop.

Guest, Simon
rigidity, flexible, flexibility, deployable structures, finite mechanisms, triangles, polyhedron, tetrahedral, Bricard octahedron, torus, Maxwell's rule, structural engineering, optimisation, simulated annealing, polygon, Matlab, clash detection in computation, Steffen flexible polyhedron, novel spatial structures, rigidity theorem, Bellows theorem, geometriy, Euler, Cauchy, convex polyhedron, crinkle, origami structures, triangulated, Pareto optimal, symmetry, continuous motion, range of motion, constant volume
Doctor of Philosophy (PhD)
Awarding Institution
University of Cambridge