A quantitative Birkhoff Normal Form for the hinged-hinged beam equation with geometric nonlinearity

We consider an undamped nonlinear hinged-hinged beam with stretching nonlinearity as an infinite dimensional hamiltonian system. We obtain analytically a quantitative Birkhoff Normal Form, via a nonlinear coordinate transformation that allows us to integrate the system up to a small reminder, providing a very precise description of small amplitude solutions over large time scales. The optimization of the involved estimates yields results obtained for realistic values of the physical quantities and of the perturbation parameter.