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We study the nonlinear free undamped motions of a hinged-hinged beam exhibiting geometric stretching-induced nonlinearity and arbitrary initial conditions. We treat the governing integral-partial-differential equation of motion as an infinite dimensional Hamiltonian system. We analytically obtain a quantitative Birkhoff Normal Form via a nonlinear coordinate transformation that yields the reduced (modulation) equations describing the free oscillations to within a certain nonlinear order with an estimate of the reminder. The obtained solutions provide a very precise description of small amplitude oscillations over large time scales. The analytical optimization of the involved estimates yields time stability results obtained for plausible values of the physical quantities and of the perturbation parameter. The role played by internal resonances in determining the time stability of the solution is highlighted and discussed. We show that initial conditions with a finite number of eigenfunctions yield bounded solutions living on invariant subspaces of the involved modes at all times. Conversely, initial conditions comprising the full (infinite) spectrum of eigenfunctions provide solutions for which time stability for all times cannot be stated.
Gregorio et al. (Mon,) studied this question.