Stability analysis of inhomogeneous equilibrium for axially and transversely excited nonlinear beam

In this work we consider the dynamical response of a non-linear beam with viscous damping, perturbed in both the transverse and axial direc- tions. The system is modeled using coupled non-linear momentum equations for the axial and transverse displacements. In particular we show that for a class of boundary conditions (beam clamped at the extremes) and uniformly distributed load, there exists a non-uniform equilibrium state. Different mod- els of damping are considered: first, third and fifth order dissipation terms. We show that in all cases in the presence of the damping forces, the excited beam is stable near the equilibrium for any perturbation. An energy estimate approach is used in order to identify the space in which the solution of the perturbed system is stable.