Abstract
This work is focused on the doubly nonlinear equation , whose solutions represent the bending motion of an extensible, elastic bridge suspended by continuously distributed cables which are flexible and elastic with stiffness k2. When the ends are pinned, long‐term dynamics is scrutinized for arbitrary values of axial load p and stiffness k2. For a general external source f, we prove the existence of bounded absorbing sets. When f is time‐independent, the related semigroup of solutions is shown to possess the global attractor of optimal regularity and its characterization is given in terms of the steady states of the problem.
Highlights
We scrutinize the longtime behavior of a nonlinear evolution problem describing the damped oscillations of an extensible elastic bridge of unitary natural length suspended by means of flexible and elastic cables
The model equation ruling its dynamics can be derived from the standard modeling procedure, which relies on the basic assumptions of continuous distribution of the stays’ stiffness along the girder and of the dominant truss behavior of the bridge see, e.g., 1
In the pioneer papers by McKenna and coworkers see 2–4, the dynamics of a suspension bridge is given by the well-known damped equation
Summary
We scrutinize the longtime behavior of a nonlinear evolution problem describing the damped oscillations of an extensible elastic bridge of unitary natural length suspended by means of flexible and elastic cables. ∂ttu ∂xxxxu ∂tu k2u f, International Journal of Differential Equations where u u x, t : 0, 1 × R → R accounts for the downward deflection of the bridge in the vertical plane, and u stands for its positive part, namely,. A geometric nonlinearity appears into the bending equation This is achieved by combining the pioneering ideas of Woinowsky-Krieger on the extensible elastic beam 5 with 1.1. D ∂xxxx w ∈ H4 0, 1 : w 0 w 1 ∂xxw 0 ∂xxw 1 0 This operator is strictly positive selfadjoint with compact inverse, and its discrete spectrum is given by λn n4π4, n ∈ N. If pinned ends are considered, the initial-boundary value problem 1.3 – 1.5 can be described by means of a single operator A ∂xxxx, which enters the equation at the powers 1 and 1/2. As we shall show throughout the paper, this model leads to exact results which are rather simple to prove and, are capable of capturing the main behavioral dynamic characteristics of the bridge
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