Asymptotic finite-dimensional approximations for a class of extensible elastic systems

We consider the equation

$$u_{tt}+\delta u_t+A^{2}u+{\Vert A^{\theta /2}u\Vert }^2{A}^{\theta }u=g$$

where $A^2$ is a diagonal, self-adjoint and positive-definite operator and $\theta \in [0,1]$ and we study some finite-dimensional approximations of the problem. First, we analyze the dynamics in the case when the forcing term $g$ is a combination of a finite number of modes. Next, we estimate the error we commit by neglecting the modes larger than a given $N$. We then prove, for a particular class of forcing terms, a theoretical result allowing to study the distribution of the energy among the modes and, with this background, we refine the results. Some generalizations and applications to the study of the stability of suspension bridges are given.