Rigidity of frameworks with coordinated constraint relaxations

Serocold, Hattie and Schulze, Bernd (2019) Rigidity of frameworks with coordinated constraint relaxations. PhD thesis, Lancaster University.

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Abstract

This thesis is concerned with the rigidity of coordinated frameworks. These are considered to be bar-joint frameworks for which the requirement that the lengths of bars be kept fixed is relaxed on some collection of bars, with the caveat that all bars within a coordination class must change length by the same amount. We begin by formulating the conditions for a framework to be continuously coordinated rigid, infinitesimally coordinated rigid, and statically coordinated rigid. We prove that static and infinitesimal rigidity are equivalent for coordinated frameworks, and that for regular coordinated frameworks, continuous rigidity and infinitesimal rigidity are equivalent. We give a characterisation of the rigidity of frameworks in d-dimensional Euclidean space with k coordination classes, based on the rigidity of the structure graph of such a framework. Since minimal infinitesimal rigidity of bar-joint frameworks is characterised in 1- and 2-dimensions, we extend the standard characterisations to a combinatorial characterisation of minimally infinitesimally rigid frameworks with one class of coordinated bars, and with two classes of coordinated bars, in both dimension 1 and dimension 2. We also obtain an inductive characterisation of such minimally infinitesimally rigid frameworks using coordinated analogues to standard inductive constructions. We conclude by considering coordinated frameworks with symmetric realisations, and give some initial results in this area.