Abstract
In this paper we discuss the well-posedness and the asymptotic behavior of a doubly nonlinear problem describing the vibrations of a Kelvin–Voigt string–beam system which models a suspension bridge. For this model we obtain the existence and uniqueness of solutions and the exponential stability of the homogeneous system, provided that the constant axial force \(p\) is smaller than a critical value. For a general \(p,\) the existence of the regular global attractor is proved when the external loads are independent of time.
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References
Abdel-Ghaffar AM, Rubin LI (1983) Non linear free vibrations of suspension bridges: theory. ASCE J Eng Mech 109:313–329
Abdel-Ghaffar AM, Rubin LI (1983) Non linear free vibrations of suspension bridges: application. ASCE J Eng Mech 109:330–345
Ahmed NU, Harbi H (1998) Mathematical analysis of dynamic models of suspension bridges. SIAM J Appl Math 109:853–874
Babin AV, Vishik MI (1992) Attractors of evolution equations. North-Holland, Amsterdam
Bochicchio I, Giorgi C, Vuk E (2010) Long-term damped dynamics of the extensible suspension bridge. Int J Differ Equ. doi:10.1155/2010/3834202010
Bochicchio I, Giorgi C, Vuk E (2010) On some nonlinear models for suspension bridges. In: Andreucci D, Carillo S, Fabrizio M, Loreti P, Sforza D (eds) Proceedings of the conference “Evolution Equations and Materials with Memory Proceedings”, “La Sapienza” Università di Roma 2011, Rome, 12–14 July 2010. ISBN 978-88-95814-51-3
Bochicchio I, Giorgi C, Vuk E (2012) Long-term dynamics of the coupled suspension bridge system. Math Models Methods Appl Sci 22:1250021. doi:10.1142/S02182025125002122012
Bochicchio I, Giorgi C, Vuk E (2013) Asymptotic dynamics of nonlinear coupled suspension bridge equations. J Math Anal Appl 402:319–333. doi:10.1016/j.jmaa.2013.01.036
Choi QH, Jung T (1999) A nonlinear suspension bridge equation with nonconstant load. Nonlinear Anal 35:649–668
Conti M, Pata V (2005) Weakly dissipative semilinear equations of viscoelasticity. Commun Pure Appl Anal 4:705–720
Conti M, Pata V, Squassina M (2006) Singular limit of differential systems with memory. Indiana Univ Math J 55:169–216
Drábek P, Holubová G, Matas A, Nečesal P (2003) Nonlinear models of suspension bridges: discussion of the results. Appl Math 48:497–514
Giorgi C, Pata V, Vuk E (2008) On the extensible viscoelastic beam. Nonlinearity 21:713–733
Hale JK (1988) Asymptotic behavior of dissipative systems. American Mathematical Society, Providence
McKenna PJ, Walter W (1987) Nonlinear oscillations in a suspension bridge. Arch Ration Mech Anal 98:167–177
Pata V, Prouse G, Vishik MI (1998) Traveling waves of dissipative nonautonomous hyperbolic equations in a strip. Adv Methods Differ Equ 3:249–270
Temam R (1997) Infinite-dimensional dynamical systems in mechanics and physics. Springer, New York
Zhong C, Ma Q, Sun C (2007) Existence of strong solutions and global attractors for the suspension bridge equations. Nonlinear Anal 67:442–454
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Bochicchio, I., Giorgi, C. & Vuk, E. Well-posedness and longtime behaviour of a coupled nonlinear system modeling a suspension bridge. Meccanica 50, 665–673 (2015). https://doi.org/10.1007/s11012-014-9996-8
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DOI: https://doi.org/10.1007/s11012-014-9996-8