A DQ based approach to simulate the vibrations of buckled beams

In this paper, the nonlinear planar response of a hinged–hinged buckled beam to a primary-resonance excitation of its first vibration mode is computed by a new numerical scheme. The beam is subjected to an axial force beyond the critical load of the first buckling mode and to a transverse harmonic excitation. The nonlinear dynamical problem is solved by deducing directly the discretized equations governing the problem thanks to a new approach, here called DQ based approach, since it is based on the application of the quadrature rules of the DQM. As it will be shown, for the problem here considered, the minimum number of degrees of freedom to be retained to limit the numerical errors is four. Computer simulations of the dynamic behaviour of the discretized system are conducted by means of the IDQ method, a method proposed and recently generalized by the author. A sequence of supercritical period-doubling bifurcations leading to chaos, snapthrough motions and quasi-periodic motions can be observed, similarly to some cases existing in literature.