Abstract

The parametric response of a cantilever thick beam with a tip mass subjected to harmonic axial support motion is investigated. The governing non-linear equation of motion is derived for an arbitrary axial support motion. To formulate a simple, physically correct dynamic model for the stability and periodicity analyses, the governing equation is truncated to the first characteristic mode of vibration. Using Green’s function and Schauder’s fixed point theorem, the necessary and sufficient conditions for the existence of periodic oscillatory behavior of the beam are established. The influence of the harmonic base excitation parameters, i.e., the excitation amplitude and frequency, on the steady-state amplitude of vibration are determined. Depending on the values of the excitation amplitude and frequency in the stable and unstable regions, the solution exhibits many shapes besides the transition periodic shapes. Numerical results indicate that for a given beam under a known excitation, increasing the tip mass would usually reduce the stable periodic region. To show the effect of rotary inertia and shear deformation, the beam model is reduced to the Euler-Bernoulli and purely flexural beam theories, respectively. The results show that using purely flexural or even Euler-Bernoulli model rather than Timoshenko, would produce an incorrect periodic region.

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