Floating Elastic Plate

Introduction

The floating elastic plate is one of the best studied problems in hydroelasticity. It can be used to model a range of physical structures such as a floating break water, an ice floe or a VLFS). The equations of motion were formulated more than 100 years ago and a discussion of the problem appears in Stoker 1957. The problem can be divided into the two and three dimensional formulations which are closely related.

Equations of Motion

The equation for a elastic plate which is governed by Kirkoffs equation is given by the following

[math]\displaystyle{ D\frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2} = p }[/math]

where [math]\displaystyle{ D }[/math] is the flexural rigidity, [math]\displaystyle{ \rho_i }[/math] is the density of the plate, [math]\displaystyle{ h }[/math] is the thickness of the plate (assumed constant), [math]\displaystyle{ p }[/math] is the pressure and [math]\displaystyle{ \eta }[/math] is the plate displacement.

The pressure is given by the linearised Bernouilli equation at the wetted surface (assuming zero pressure at the surface), i.e.

[math]\displaystyle{ p = \rho g \phi + \rho \frac{\partial \phi}{\partial t} }[/math]

where [math]\displaystyle{ \rho }[/math] is the water density and [math]\displaystyle{ g }[/math] is gravity, and [math]\displaystyle{ \phi }[/math] is the velocity potential. The velocity potential is governed by Laplace's equation through out the fluid domain subject to the free surface condition and the condition of no flow through the bottom surface. If we denote the region of the fluid surface covered in the plate (or possible multiple plates) by [math]\displaystyle{ P }[/math] and the free surface by [math]\displaystyle{ F }[/math] the equations of motion for the Frequency Domain Problem with frequency [math]\displaystyle{ \omega }[/math] is

[math]\displaystyle{ D\frac{\partial^4 \eta}{\partial x^4} - \omega^2 \rho_i h \eta = \rho g \phi + i\omega \rho \phi x\in P }[/math]

[math]\displaystyle{ 0= \rho g \phi + i\omega \rho \phi \, x\in F }[/math]

plus the equations within the fluid

[math]\displaystyle{ \nabla^2\phi =0 }[/math]

[math]\displaystyle{ \frac{\partial \phi}{\partial z} = 0, \, z=-h }[/math]

(so we have a fluid of constant depth with the bottom surface at [math]\displaystyle{ z=-h }[/math] and the free surface or plate covered surface are at [math]\displaystyle{ z=0 }[/math].