Natural vibration of a flexible beam-water coupled system with a concentrated mass attached at the free end of the beam

Abstract

The dynamical behaviour of a flexible beam-water interaction system having a concentrated mass (with accompanying moment of inertia) at the free end of the beam is examined. In the water domain, the coupled system is subject to an undisturbed boundary condition at infinity and a zero surface wave or linear surface disturbance condition on the free surface. The governing equations describing the behaviour of the system are analysed using the separation of variables method and their solutions presented. The eigenvalue equation of the natural vibration of the beam-water system is derived and exact solutions for each combination of boundary conditions are obtained. Calculations show that, for the undisturbed condition at infinity in the water domain, the natural frequencies of the coupled beam-concentrated mass dynamical system are lower than those of the beam alone. With constant ratio of concentrated mass to mass of the beam, as the beam changes from thick to thin, the concentrated mass and moment of inertia become less influential on the natural vibration behaviour of the coupled system.