On tensegrity frameworks. (Russian. English summary) Zbl 1447.70002
Summary: Ideal designs, made of rigid bars (levers), inextensible cables and incompressible struts are considered. In English such constructions are called “tensegrity frameworks”. In the particular case of structures composed of only the levers, – this is a bar and joint frameworks. In recent times the tensegrity frameworks are increasingly used in architecture and construction, for example, the construction of bridges.In English mathematical literature geometric properties of such structures were studied since the seventies of the last century. This article is apparently the first in Russian mathematical literature devoted to this topic. It is a breath survey to the theory of tensegrity frameworks.
It introduces mathematical formalization of tensegrity frameworks in the spirit of the work of the author on hinge mechanisms. This formalization includes original Russian terminology, not reducible to the borrowing of English words. Only not pinned tensegrity frameworks are investigated. We call a tensegrity frameworks, allowing the internal stress, and not allowing a continuous deformation with a change of form, – a truss. A truss that cannot be assembled in a different way to be not congruent to the initial one is called globally rigid. If a tensegrity frameworks is globally rigid in \(R^n\) and also globally rigid in every \(R^N\) for \(N>n\) it is called universally rigid.
We focus on the problem – when a given tensegrity framework is globally rigid? We consider an effective method for solving this problem, based on investigation of a particular function – the potential energy of the structure. We search tensegrity frameworks for which this potential energy is minimal. The method is described in detail in this article. The main theorem, giving a sufficient condition of universal rigidity of tensegrity framework is proved in detail. The study of internal stress of a tensegrity framework and its stress matrix, by means of which the potential energy is written, is of fundamental importance. Examples of applications of this theorem to planar and spatial tensegrity frameworks are presented.
In general, this subject is not yet sufficiently developed, and is currently actively investigated. At the end of the article some open questions are formulated.
It introduces mathematical formalization of tensegrity frameworks in the spirit of the work of the author on hinge mechanisms. This formalization includes original Russian terminology, not reducible to the borrowing of English words. Only not pinned tensegrity frameworks are investigated. We call a tensegrity frameworks, allowing the internal stress, and not allowing a continuous deformation with a change of form, – a truss. A truss that cannot be assembled in a different way to be not congruent to the initial one is called globally rigid. If a tensegrity frameworks is globally rigid in \(R^n\) and also globally rigid in every \(R^N\) for \(N>n\) it is called universally rigid.
We focus on the problem – when a given tensegrity framework is globally rigid? We consider an effective method for solving this problem, based on investigation of a particular function – the potential energy of the structure. We search tensegrity frameworks for which this potential energy is minimal. The method is described in detail in this article. The main theorem, giving a sufficient condition of universal rigidity of tensegrity framework is proved in detail. The study of internal stress of a tensegrity framework and its stress matrix, by means of which the potential energy is written, is of fundamental importance. Examples of applications of this theorem to planar and spatial tensegrity frameworks are presented.
In general, this subject is not yet sufficiently developed, and is currently actively investigated. At the end of the article some open questions are formulated.
References:
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