Nonlinear Interaction of Free Vibrations of Cylinder in Water Near a Plane Bottom

Abstract

Free undamped vibrations in a fluid of an elastically supported cylinder, placed in the vicinity of a horizontal, plane and rigid bottom are considered. It is assumed that the fluid is incompressible and inviscid, and the boundaries are perfect. The two-dimensional problem is posed, that is the cylinder can oscillate in both the horizontal and vertical directions. The Lagrangian formulation is used, in which the kinetic energy of the cylinder and the fluid is calculated, as well as the potential energy of linear elastic vertical and horizontal springs. In the paper explicit expressions are established for the coefficients in the series describing the complex potential of the fluid. Also, explicit expressions are given for the kinetic energy of the mechanical system and the lift forces acting on the cylinder. The Euler equations of the variational formulation yield two nonlinear coupled equations for the unknown horizontal and vertical displacements of the centre of the cylinder. By substitution of finite differences in the variational formulation and differentiation with respect to the displacements at the assumed time steps, a stable numerical solution is obtained. The calculated displacements are irregular functions in time, but the trajectories of motion of the centre of the cylinder lie in a well-defined area delimited by an envelope. This envelope for small vertical displacements may be approximated by two parabolas and two vertical lines.