American Institute of Mathematical Sciences

Advanced
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.   Google Scholar

[2]

C. R. Calladine and S. Pellegrino, First-order infinitesimal mechanisms,, International Journal of Solids and Structures, 27 (1991), 505.   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.  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.  doi: 10.1090/S0002-9947-1981-0610958-6.  Google Scholar

[6]

R. Connelly, Rigidity and energy,, Inventiones Mathematichae, 66 (1982), 11.  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.  doi: 10.1137/S0895480192229236.  Google Scholar

[8]

R. Connelly and A. Back, Mathematics and tensegrity,, American Scientists, 86 (1998), 142.  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.   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.   Google Scholar

[12]

J. J. Moreau, Fonctionnelles convexes,, Editions of Department of Civil Engineering, (9788).   Google Scholar

[13]

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

show all references

References:
[1]

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

[2]

C. R. Calladine and S. Pellegrino, First-order infinitesimal mechanisms,, International Journal of Solids and Structures, 27 (1991), 505.   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.  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.  doi: 10.1090/S0002-9947-1981-0610958-6.  Google Scholar

[6]

R. Connelly, Rigidity and energy,, Inventiones Mathematichae, 66 (1982), 11.  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.  doi: 10.1137/S0895480192229236.  Google Scholar

[8]

R. Connelly and A. Back, Mathematics and tensegrity,, American Scientists, 86 (1998), 142.  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.   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.   Google Scholar

[12]

J. J. Moreau, Fonctionnelles convexes,, Editions of Department of Civil Engineering, (9788).   Google Scholar

[13]

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

[1]

Andrea Braides, Valeria Chiadò Piat. Non convex homogenization problems for singular structures. Networks & Heterogeneous Media, 2008, 3 (3) : 489-508. doi: 10.3934/nhm.2008.3.489
[2]

S. E. Pastukhova. Asymptotic analysis in elasticity problems on thin periodic structures. Networks & Heterogeneous Media, 2009, 4 (3) : 577-604. doi: 10.3934/nhm.2009.4.577
[3]

Henry O. Jacobs, Hiroaki Yoshimura. Tensor products of Dirac structures and interconnection in Lagrangian mechanics. Journal of Geometric Mechanics, 2014, 6 (1) : 67-98. doi: 10.3934/jgm.2014.6.67
[4]

Samir Adly, Daniel Goeleven, Dumitru Motreanu. Periodic and homoclinic solutions for a class of unilateral problems. Discrete & Continuous Dynamical Systems - A, 1997, 3 (4) : 579-590. doi: 10.3934/dcds.1997.3.579
[5]

Yazheng Dang, Fanwen Meng, Jie Sun. Convergence analysis of a parallel projection algorithm for solving convex feasibility problems. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 505-519. doi: 10.3934/naco.2016023
[6]

Melvin Leok, Diana Sosa. Dirac structures and Hamilton-Jacobi theory for Lagrangian mechanics on Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 421-442. doi: 10.3934/jgm.2012.4.421
[7]

Wenguo Shen. Unilateral global interval bifurcation for Kirchhoff type problems and its applications. Communications on Pure & Applied Analysis, 2018, 17 (1) : 21-37. doi: 10.3934/cpaa.2018002
[8]

Nelly Point, Silvano Erlicher. Convex analysis and thermodynamics. Kinetic & Related Models, 2013, 6 (4) : 945-954. doi: 10.3934/krm.2013.6.945
[9]

Sanming Liu, Zhijie Wang, Chongyang Liu. On convergence analysis of dual proximal-gradient methods with approximate gradient for a class of nonsmooth convex minimization problems. Journal of Industrial & Management Optimization, 2016, 12 (1) : 389-402. doi: 10.3934/jimo.2016.12.389
[10]

Piotr Gwiazda, Piotr Minakowski, Agnieszka Świerczewska-Gwiazda. On the anisotropic Orlicz spaces applied in the problems of continuum mechanics. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1291-1306. doi: 10.3934/dcdss.2013.6.1291
[11]

Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159
[12]

Florent Berthelin, Thierry Goudon, Bastien Polizzi, Magali Ribot. Asymptotic problems and numerical schemes for traffic flows with unilateral constraints describing the formation of jams. Networks & Heterogeneous Media, 2017, 12 (4) : 591-617. doi: 10.3934/nhm.2017024
[13]

Junjiang Lai, Jianguo Huang. A finite element method for vibration analysis of elastic plate-plate structures. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 387-419. doi: 10.3934/dcdsb.2009.11.387
[14]

Gang Bao, Bin Hu, Peijun Li, Jue Wang. Analysis of time-domain Maxwell's equations in biperiodic structures. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 259-286. doi: 10.3934/dcdsb.2019181
[15]

Tomáš Roubíček. On certain convex compactifications for relaxation in evolution problems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 467-482. doi: 10.3934/dcdss.2011.4.467
[16]

Anulekha Dhara, Aparna Mehra. Conjugate duality for generalized convex optimization problems. Journal of Industrial & Management Optimization, 2007, 3 (3) : 415-427. doi: 10.3934/jimo.2007.3.415
[17]

Abdelmalek Aboussoror, Samir Adly, Vincent Jalby. Weak nonlinear bilevel problems: Existence of solutions via reverse convex and convex maximization problems. Journal of Industrial & Management Optimization, 2011, 7 (3) : 559-571. doi: 10.3934/jimo.2011.7.559
[18]

Hui Zhang, Jian-Feng Cai, Lizhi Cheng, Jubo Zhu. Strongly convex programming for exact matrix completion and robust principal component analysis. Inverse Problems & Imaging, 2012, 6 (2) : 357-372. doi: 10.3934/ipi.2012.6.357
[19]

Jan J. Sławianowski, Vasyl Kovalchuk, Agnieszka Martens, Barbara Gołubowska, Ewa E. Rożko. Essential nonlinearity implied by symmetry group. Problems of affine invariance in mechanics and physics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 699-733. doi: 10.3934/dcdsb.2012.17.699
[20]

Kim S. Bey, Peter Z. Daffer, Hideaki Kaneko, Puntip Toghaw. Error analysis of the p-version discontinuous Galerkin method for heat transfer in built-up structures. Communications on Pure & Applied Analysis, 2007, 6 (3) : 719-740. doi: 10.3934/cpaa.2007.6.719

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences