A nonlinear mathematical model for dynamic analyses of a cantilevered beam with a tip-mass under support excitation

Abstract

The dynamic response of a cantilevered beam subjected to horizontal, vertical and angular support excitation is numerically investigated. An extensible Bernoulli–Euler beam model is assumed, and the governing equations of motion are derived using the extended Hamilton’s principle and the Galerkin’s method. Modal-amplitude time histories, as well as maps of post-critical steady-state vibration amplitudes, are obtained and discussed. It is pointed out the existence of similarities with the principal Mathieu’s instability observed in a one degree-of-freedom system, i.e., the condition in which the frequency of stiffness variation is twice the structural natural frequency. Particularly, the angular support excitation may lead to a principal region of parametric instability and the presence of simultaneous vertical and angular support excitation increased a second region of important responses. Another finding is that the horizontal support excitation leads to an increase in the maximum lateral response near the frequencies associated with the parametric resonance.