Nonlinear vibrations of a beam with time-varying rigidity and mass

We consider asymptotic solutions for nonlinear beams that can be described by a fourth order hyperbolic equation with an integral nonlinearity and some space and time dependent coefficients. These coefficients can describe varying mass and rigidity perturbations. A two-time scales perturbation method reduces this complicated equation to an infinite-dimensional Hamiltonian system for the Fourier modes. An analysis of this system shows that the corresponding dynamics is quasi-periodic and periodic in time if the coefficients are constant. For non-constant coefficients the dynamics changes significantly. For some special non-constant coefficients the Hamiltonian dynamics can be simplified. We obtain a simpler finite-dimensional system. Numerical simulations show existence of new interesting dynamical effects due to resonances between some Fourier modes. These resonances can lead to large oscillations, even for small nonlinearities. The phase portraits which correspond to these resonance cases will also be presented.