A detailed study of the parametric excitation of a vertical heavy rod using the method of multiple scales

An analytical solution for the problem of an immersed flexible and vertical heavy rod subjected to a vertical top motion is developed using the multiple scales method directly applied to the partial differential equations of motion. The obtained results show good agreement with a numerical solution obtained using the finite element method for a study case. The analytical solution is then used to carry out some sensitivity studies. The effects of the structural nonlinearities, hydrodynamic and structural damping terms are investigated. It is shown that the nonlinearities play a role in defining the frequency of the top motion that causes the maximum amplitude of response, but not the value of the amplitude itself. In turn, the major role played by the hydrodynamic damping in defining the response amplitude is addressed. It is also shown that the structural damping have an important effect on the response amplitude even in the case of small damping ratio. This occurs due to the combined effect of the structural with the hydrodynamic damping. Finally, it is pointed out that small differences in the structural damping ratio can lead to significant differences in the response amplitude.