Equilibrium and stability of tensegrity structures: A convex analysis approach

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April 2013, 6(2): 461-478. doi: 10.3934/dcdss.2013.6.461

Equilibrium and stability of tensegrity structures: A convex analysis approach

Franco Maceri ^1, , Michele Marino ^1, and Giuseppe Vairo ^1,

1.	Department of Civil Engineering, University of Rome "Tor Vergata", Via del Politecnico, 1 - 00133 Rome, Italy, Italy, Italy

Received August 2011 Revised March 2012 Published November 2012

Abstract
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In this paper, tensegrity structures are modeled by introducing suitable energy convex functions. These allow to enforce both ideal and non-ideal constraints, gathering compatibility, equilibrium, and stability problems, as well as their duality relationships, in the same functional framework. Arguments of convex analysis allow to recover consistently a number of basic results, as well as to formulate new interpretations and analysis criterions.

Keywords: unilateral problems, Mechanics of tensegrity structures, convex analysis..

Mathematics Subject Classification: 74G65, 74G99, 70E50, 52A41, 52A4.

Citation: Franco Maceri, Michele Marino, Giuseppe Vairo. Equilibrium and stability of tensegrity structures: A convex analysis approach. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 461-478. doi: 10.3934/dcdss.2013.6.461

References:

[1]	S. Pellegrino, Analysis of prestressed mechanisms,, International Journal of Solids and Structures, 26 (1990), 1329.
[2]	C. R. Calladine and S. Pellegrino, First-order infinitesimal mechanisms,, International Journal of Solids and Structures, 27 (1991), 505.
[3]	C. Sultan, M. Corless and R. E. Skelton, The prestressability problem of tensegrity structures: some analytical solutions,, International Journal of Solids and Structures, 38 (2001), 5223. doi: 10.1016/S0020-7683(00)00401-7.
[4]	R. Motro, "Tensegrity: Structural Systems for the Future,", Kogan Page Science, (2003). doi: 10.1016/B978-190399637-9/50035-4.
[5]	B. Roth and W. Whiteley, Tensegrity frameworks,, Transactions of the American Mathematical Society, 265 (1981), 419. doi: 10.1090/S0002-9947-1981-0610958-6.
[6]	R. Connelly, Rigidity and energy,, Inventiones Mathematichae, 66 (1982), 11. doi: 10.1007/BF01404753.
[7]	R. Connelly and W. Whiteley, Second-order rigidity and prestress stability for tensegrity frameworks,, Journal on Discrete Mathematics, 9 (1996), 453. doi: 10.1137/S0895480192229236.
[8]	R. Connelly and A. Back, Mathematics and tensegrity,, American Scientists, 86 (1998), 142. doi: 10.1511/1998.2.142.
[9]	F. Maceri, M. Marino and G. Vairo, Convex analysis and ideal tensegrities,, Comptes Rendus Mecanique, 339 (2011), 683.
[10]	M. Frémond, "Non-Smooth Thermomechanics,", Springer-Verlag Berlin, (2001).
[11]	P. D. Panagiotopoulos, Convex analysis and unilateral static problems,, Archive of Applied Mechanics, 45 (1976), 55.
[12]	J. J. Moreau, Fonctionnelles convexes,, Editions of Department of Civil Engineering, (9788).
[13]	S. Guest, The stiffness of prestressed frameworks: A unifying approach,, International Journal of Solids and Structures, 43 (2006), 842.
[14]	A. Micheletti and W. O. Williams, A marching procedure for form-finding for tensegrity structures,, Journal of Mechanics of Materials and Structures, 2 (2007), 857. doi: 10.2140/jomms.2007.2.857.
[15]	W. O. Williams, A primer on the mechanics of tensegrity structures,, preprint, (2007).
[16]	J. Y. Zhang and M. Ohsaki, Stability conditions for tensegrity structures,, International Journal of Solids and Structures, 44 (2007), 3875.

show all references

References:

[1]	S. Pellegrino, Analysis of prestressed mechanisms,, International Journal of Solids and Structures, 26 (1990), 1329.
[2]	C. R. Calladine and S. Pellegrino, First-order infinitesimal mechanisms,, International Journal of Solids and Structures, 27 (1991), 505.
[3]	C. Sultan, M. Corless and R. E. Skelton, The prestressability problem of tensegrity structures: some analytical solutions,, International Journal of Solids and Structures, 38 (2001), 5223. doi: 10.1016/S0020-7683(00)00401-7.
[4]	R. Motro, "Tensegrity: Structural Systems for the Future,", Kogan Page Science, (2003). doi: 10.1016/B978-190399637-9/50035-4.
[5]	B. Roth and W. Whiteley, Tensegrity frameworks,, Transactions of the American Mathematical Society, 265 (1981), 419. doi: 10.1090/S0002-9947-1981-0610958-6.
[6]	R. Connelly, Rigidity and energy,, Inventiones Mathematichae, 66 (1982), 11. doi: 10.1007/BF01404753.
[7]	R. Connelly and W. Whiteley, Second-order rigidity and prestress stability for tensegrity frameworks,, Journal on Discrete Mathematics, 9 (1996), 453. doi: 10.1137/S0895480192229236.
[8]	R. Connelly and A. Back, Mathematics and tensegrity,, American Scientists, 86 (1998), 142. doi: 10.1511/1998.2.142.
[9]	F. Maceri, M. Marino and G. Vairo, Convex analysis and ideal tensegrities,, Comptes Rendus Mecanique, 339 (2011), 683.
[10]	M. Frémond, "Non-Smooth Thermomechanics,", Springer-Verlag Berlin, (2001).
[11]	P. D. Panagiotopoulos, Convex analysis and unilateral static problems,, Archive of Applied Mechanics, 45 (1976), 55.
[12]	J. J. Moreau, Fonctionnelles convexes,, Editions of Department of Civil Engineering, (9788).
[13]	S. Guest, The stiffness of prestressed frameworks: A unifying approach,, International Journal of Solids and Structures, 43 (2006), 842.
[14]	A. Micheletti and W. O. Williams, A marching procedure for form-finding for tensegrity structures,, Journal of Mechanics of Materials and Structures, 2 (2007), 857. doi: 10.2140/jomms.2007.2.857.
[15]	W. O. Williams, A primer on the mechanics of tensegrity structures,, preprint, (2007).
[16]	J. Y. Zhang and M. Ohsaki, Stability conditions for tensegrity structures,, International Journal of Solids and Structures, 44 (2007), 3875.

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