On tensegrity frameworks

M. D. Kovalev

Čebyševskij sbornik, Tome 16 (2015) no. 3, pp. 306-321.

Résumé

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 can'not be assembled in a different way to be not congruent to 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 particular function — the potential energy of the structure. We search a tensegrity frameworks for which this potential energy is minimal. The method is described in detail in the article. The main theorem, giving a sufficient condition of Universal Rigidity of tensegrity framework is proved in details. The study of internal stresses 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. Bibliography: 15 titles.

Détail

Keywords: Tesegrity frameworks, global rigidity, potential function, stress matrix.

Bibliographie

[1] Kovalev M. D., “Geometric theory of linkages”, Russian Acad. Sci. Izv. Math., 44:1 (1995), 43–68 | MR | Zbl

[2] Kovalev M. D., “Questions geometry hinged devices and schemes”, Vestnik MSTU. Mechanical Engineering Series, 2001, no. 4, 33–51 (Russian)

[3] Kovalev M. D., “Straightenable hinged frameworks”, Sb. Math., 195:5–6 (2004), 833–858 | DOI | DOI | MR | Zbl

[4] Connelly R., “Rigidity and Energy”, Invent. Math., 66:1 (1982), 11–33 | DOI | MR | Zbl

[5] Connelly R., “Rigidity”, Handbook of Convex Geometry, Chapter 1.7, v. A, eds. P. M. Gruber, J. M. Wills, Elsevier, 1993 | MR

[6] Asimov L., Roth B., “The rigidity of Graphs, II”, Journal of Math. analysis and appl., 68:1 (1979), 171–190 | DOI | MR

[7] Crapo H., Whiteley W., “Statics of Frameworks and Motions of Panel Structures, a Projective Geometric Introduction”, Structural Topology, 1982, no. 6, 43–82 | MR

[8] Connelly R., “The Rigidity of Certain Cabled Frameworks and the Second-Order Rigidity of Arbitrarily Triangulated Convex Surfaces”, Advances in Math., 37:3 (1980), 272–299 | DOI | MR | Zbl

[9] Roth B., Whiteley W., “Tensegrity Frameworks”, Trans. Amer. Math. Soc., 265:2 (1981), 419–446 | DOI | MR | Zbl

[10] Grunbaum B., Shepard G., “Rigidity of Polyhedra, Frameworks and Cabled Frameworks”, Notices Amer. Math. Soc., 25 (1978), Abstract 760-D3, A-642

[11] Connelly R., Terrell M., “Globally rigid symmetric tensegrities”, Dual French-English text, Structural Topology, 21 (1995), 59–78 | MR | Zbl

[12] Connelly R., “Generic global rigidity”, Discrete Comput. Geom., 33:4 (2005), 549–563 | DOI | MR | Zbl

[13] Connelly R., Whiteley W., “Global rigidity. The effect of coning”, Discrete Comput. Geom., 43 (2010), 717–735 | DOI | MR | Zbl

[14] Connelly R., What is ... a tensegrity?, Notices Amer. Math. Soc., 60:1 (2013), 78–80 | DOI | MR | Zbl

[15] Connelly R., Gortler S., Iterative Universal Rigidity, 28 Jan 2015, arXiv: 1401.7029v2[math.MG] | MR