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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

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), 1329-1350. Google Scholar

[2]

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

[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.  Google Scholar

[4]

R. Motro, "Tensegrity: Structural Systems for the Future," Kogan Page Science, 2003. doi: 10.1016/B978-190399637-9/50035-4.  Google Scholar

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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.  Google Scholar

[6]

R. Connelly, Rigidity and energy, Inventiones Mathematichae, 66 (1982), 11-33. doi: 10.1007/BF01404753.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[10]

M. Frémond, "Non-Smooth Thermomechanics," Springer-Verlag Berlin, 2001.  Google Scholar

[11]

P. D. Panagiotopoulos, Convex analysis and unilateral static problems, Archive of Applied Mechanics, 45 (1976), 55-68. Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

[15]

W. O. Williams, A primer on the mechanics of tensegrity structures, preprint, (2007). Google Scholar

[16]

J. Y. Zhang and M. Ohsaki, Stability conditions for tensegrity structures, International Journal of Solids and Structures, 44 (2007), 3875-3886.  Google Scholar

show all references

References:
[1]

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

[2]

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

[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.  Google Scholar

[4]

R. Motro, "Tensegrity: Structural Systems for the Future," Kogan Page Science, 2003. doi: 10.1016/B978-190399637-9/50035-4.  Google Scholar

[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.  Google Scholar

[6]

R. Connelly, Rigidity and energy, Inventiones Mathematichae, 66 (1982), 11-33. doi: 10.1007/BF01404753.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[10]

M. Frémond, "Non-Smooth Thermomechanics," Springer-Verlag Berlin, 2001.  Google Scholar

[11]

P. D. Panagiotopoulos, Convex analysis and unilateral static problems, Archive of Applied Mechanics, 45 (1976), 55-68. Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

[15]

W. O. Williams, A primer on the mechanics of tensegrity structures, preprint, (2007). Google Scholar

[16]

J. Y. Zhang and M. Ohsaki, Stability conditions for tensegrity structures, International Journal of Solids and Structures, 44 (2007), 3875-3886.  Google Scholar

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