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

Equilibrium and stability of tensegrity structures: A convex analysis approach

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

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.
Citation: Franco Maceri, Michele Marino, Giuseppe Vairo. Equilibrium and stability of tensegrity structures: A convex analysis approach. Discrete and 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), 1329-1350.

[2]

C. R. Calladine and S. Pellegrino, First-order infinitesimal mechanisms, International Journal of Solids and Structures, 27 (1991), 505-515.

[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), 5223-5252. 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), 419-446. doi: 10.1090/S0002-9947-1981-0610958-6.

[6]

R. Connelly, Rigidity and energy, Inventiones Mathematichae, 66 (1982), 11-33. 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), 453-491. doi: 10.1137/S0895480192229236.

[8]

R. Connelly and A. Back, Mathematics and tensegrity, American Scientists, 86 (1998), 142-151. doi: 10.1511/1998.2.142.

[9]

F. Maceri, M. Marino and G. Vairo, Convex analysis and ideal tensegrities, Comptes Rendus Mecanique, 339 (2011), 683-691.

[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), 55-68.

[12]

J. J. Moreau, Fonctionnelles convexes, Editions of Department of Civil Engineering, University of Rome Tor Vergata, ISBN 9788862960014, Roma, 2003.

[13]

S. Guest, The stiffness of prestressed frameworks: A unifying approach, International Journal of Solids and Structures, 43 (2006), 842-854.

[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), 857-882. 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), 3875-3886.

show all references

References:
[1]

S. Pellegrino, Analysis of prestressed mechanisms, International Journal of Solids and Structures, 26 (1990), 1329-1350.

[2]

C. R. Calladine and S. Pellegrino, First-order infinitesimal mechanisms, International Journal of Solids and Structures, 27 (1991), 505-515.

[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), 5223-5252. 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), 419-446. doi: 10.1090/S0002-9947-1981-0610958-6.

[6]

R. Connelly, Rigidity and energy, Inventiones Mathematichae, 66 (1982), 11-33. 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), 453-491. doi: 10.1137/S0895480192229236.

[8]

R. Connelly and A. Back, Mathematics and tensegrity, American Scientists, 86 (1998), 142-151. doi: 10.1511/1998.2.142.

[9]

F. Maceri, M. Marino and G. Vairo, Convex analysis and ideal tensegrities, Comptes Rendus Mecanique, 339 (2011), 683-691.

[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), 55-68.

[12]

J. J. Moreau, Fonctionnelles convexes, Editions of Department of Civil Engineering, University of Rome Tor Vergata, ISBN 9788862960014, Roma, 2003.

[13]

S. Guest, The stiffness of prestressed frameworks: A unifying approach, International Journal of Solids and Structures, 43 (2006), 842-854.

[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), 857-882. 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), 3875-3886.

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