Free undamped vibrations in a fluid of an elastically supported cylind
er, placed in the vicinity of a horizontal, plane and rigid bottom are
considered. It is assumed that the fluid is incompressible and invisc
id, and the boundaries are perfect. The two-dimensional problem is pos
ed, that is the cylinder can oscillate in both the horizontal and vert
ical directions. The Lagrangian formulation is used, in which the kine
tic energy of the cylinder and the fluid is calculated, as well as the
potential energy of linear elastic vertical and horizontal springs. I
n the paper explicit expressions are established for the coefficients
in the series describing the complex potential of the fluid. Also, exp
licit expressions are given for the kinetic energy of the mechanical s
ystem and the lift forces acting on the cylinder. The Euler equations
of the variational formulation yield two nonlinear coupled equations f
or 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 c
alculated displacements are irregular functions in time, but the traje
ctories of motion of the centre of the cylinder lie in a well-defined
area delimited by an envelope. This envelope for small vertical displa
cements may be approximated by two parabolas and two vertical lines. (
C) 1995 Academic Press Limited