Asymptotic dynamics of nonlinear coupled suspension bridge equations

Abstract

a b s t r a c tIn this paper we study the long-term dynamics of a doubly nonlinear abstract system which involves a single differential operator to different powers. For a special choice of the nonlinear terms, the system describes the motion of a suspension bridge where 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 nonlinear springs. The set of stationary solutions turns out to be nonempty and bounded. As the external loads vanish, the null solution of the system is proved to be exponentially stable provided that the axial load does not exceed some critical value. Finally, we prove the existence of a bounded global attractor of optimal regularity in connection with an arbitrary axial load and quite general nonlinear terms.

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).…”

“…Since this models the case where the sustaining cable is fixed, we skipped directly to the more reliable model (2.107); here the coupling term is just k.u v/ C and we refer to [54] for more general coupling terms involving also the derivatives u t and v t . The proof of Proposition 2.16 is somehow standard and may be obtained with the Galerkin method, see [ …”

Global attractors for the coupled suspension bridge system with temperature

“…If this is the case, the previous system takes the form When , we recover at once . Assuming F to represent the action of one‐sided elastic springs and viscous dampers, the corresponding solution semigroup is known to possess a regular global attractor . On the other hand, there are some cases in which F can be reasonably assumed to be linear (e.g., ).…”

“…In particular, the asymptotic analysis is highly nontrivial and requires the exploitation of certain dissipation integrals and sharp energy‐type estimates. In addition, unlike some previous papers on cable–beam systems involving the coupling between parabolic and hyperbolic equations , we are able here to obtain existence and optimal regularity of the global attractor without resorting to a bootstrap argument.…”

Buckling and nonlinear dynamics of elastically coupled double-beam systems

“…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]). …”