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. The plate is assumed to be isotropic while the water motion is irrotational and inviscid.
Two Dimensional Problem
Equations of Motion
When considering a two dimensional problem, the [math]\displaystyle{ y }[/math] variable is dropped and the plate is regarded as a beam. There are various beam theories that can be used to describe the motion of the beam. The simplest theory is the Bernoulli-Euler beam which is commonly used in the two dimensional hydroelastic analysis. Other beam theories include the Timoshenko beam theory and Reddy-Bickford beam theory where shear deformation of higher order is considered.
For a Bernoulli-Euler beam on the surface of the water, the equation of motion 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 beam, [math]\displaystyle{ h }[/math] is the thickness of the beam (assumed constant), [math]\displaystyle{ p }[/math] is the pressure and [math]\displaystyle{ \eta }[/math] is the beam vertical displacement.
The edges of the plate satisfy the natural boundary condition (i.e. free-edge boundary conditions).
[math]\displaystyle{ \frac{\partial^2 \eta}{\partial x^2} = 0, \,\,\frac{\partial^3 \eta}{\partial x^3} = 0 }[/math]
at the edges of the plate.
The pressure is given by the linearised Bernoulli equation at the wetted surface (assuming zero pressure at the surface), i.e.
[math]\displaystyle{ p = \rho g \frac{\partial \phi}{\partial z} + \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 beam-covered (or possible multiple beams covered) region of the fluid 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] for water of Finite Depth are the following. At the surface we have the dynamic condition
[math]\displaystyle{ D\frac{\partial^4 \eta}{\partial x^4} +\left(\rho g- \omega^2 \rho_i h \right)\eta = i\omega \rho \phi, \, z=0, \, x\in P }[/math]
[math]\displaystyle{ 0= \rho g \frac{\partial \phi}{\partial z} + i\omega \rho \phi, \, x\in F }[/math]
and the kinematic condition
[math]\displaystyle{ \frac{\partial\phi}{\partial z} = i\omega\eta }[/math]
The equation within the fluid is governed by Laplace's Equation
[math]\displaystyle{ \nabla^2\phi =0 }[/math]
and we have the no-flow condition through the bottom boundary
[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]). [math]\displaystyle{ g }[/math] is the acceleration due to gravity, [math]\displaystyle{ \rho_i }[/math] and [math]\displaystyle{ \rho }[/math] are the densities of the plate and the water respectively, [math]\displaystyle{ h }[/math] and [math]\displaystyle{ D }[/math] the thickness and flexural rigidity of the plate.
Solution Method
There are many different methods to solve the corresponding equations ranging from highly analytic such as the Wiener-Hopf to very numerical based on Eigenfunction Matching Method which are applicable and have advantages in different situations.
Three Dimensional Problem
Equations of Motion
For a classical thin plate, the equation of motion is given by
[math]\displaystyle{ D\nabla ^4 W + \rho _i hW = p }[/math]
