Nonlinear Dynamics of a Beam on Elastic Foundation

The nonlinear dynamics of a simply supported beam resting on a nonlinear spring bed with cubic stiffness is analyzed. The continuous differential operator describing the mathematical model of the system is discretized through the classical Galerkin procedure and its nonlinear dynamic behavior is investigated using the method of Normal Forms. This model can be regarded as a simple system describing the oscillations of flexural structures vibrating on nonlinear supports and then it can be considered as a simple investigation for the analysis of more complex systems of the same type. Indeed, the possibility of the model to exhibit actually interesting nonlinear phenomena (primary, superharmonic, subharmonic and internal resonances) has been shown in a range of feasibility of the physical parameters. The singular perturbation approach is used to study both the free and the forced oscillations; specifically two parameter families of stationary solutions are obtained for the forced oscillations.