Long-Term Dynamics of the Coupled Suspension Bridge System

Abstract

<jats:p>In this paper we study the long-term dynamics of a nonlinear suspension bridge system. The road bed and the main cable are modeled as a nonlinear beam and a vibrating string, respectively, and their coupling is carried out by one-sided springs. First, we scrutinize the set of stationary solutions, which turns out to be nontrivial when the axial load exceeds some critical value. Then, we prove the existence of a bounded global attractor of optimal regularity and we give its characterization in terms of the steady states of the problem.</jats:p>

Mathematical Models for Suspension Bridges

“…More recently, Bochicchio et al [52][53][54] connected the beam, as described by Eq. (2.106), with a moving cable through hangers, thereby generalising (2.103).…”

Buckling and nonlinear dynamics of elastically coupled double-beam systems

“…The proof is omitted in that it can be obtained by paralleling Proposition 1 in [9], and relies on a standard Faedo-Galerkin approximation scheme (see [2,3]) together with a slight generalization of the usual Gronwall lemma. In particular, the uniform-intime estimates needed to prove the global existence are exactly the same as in Lemma 1.…”

“…This result is achieved by two steps: first the boundedness of these orbits in a more regular space, 2 , is proved in Lemma 8, then the compactness of R is ensured by the compact embedding ⋐ 2 (see [9,10]). …”

Asymptotic dynamics of nonlinear coupled suspension bridge equations

“…When h = 0, it is apparent that functions F L and F OS , respectively, provide simple models of Linear and One-Sided springs with stiffness k 2 (see [6,8]). On the other hand, the dissipative terms appearing when h > 0 model some damping effect inside the springs.…”

“…Apart from a lot of works which deal with approximations and numerical simulations, some of them are devoted to scrutinize the periodic or longtime global dynamics by analytical methods [6,8,9,22,26].…”