Category:Floating Elastic Plate

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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.

The solution methods are divided up into those for Two-Dimensional Floating Elastic Plate and those for a Three-Dimensional Floating Elastic Plate.

Linear Thin Elastic Plate Theory

We present here the theory of an elastic plate in a vacuum, concentrating of the two-dimensional problem.

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 can satisfy a range of boundary conditions. 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.

If we assume that the pressure is of the form [math]\displaystyle{ p(x,t) = e^{i\omega t} \bar{p}(x) }[/math] then it follows that [math]\displaystyle{ \eta(x,t) = e^{i\omega t} \bar{\eta}(x) }[/math] from linearity. In this case the equations reduce to

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

Linear Elastic Thin Plate on Water

We begin with the linear equations for a fluid. The kinematic condition is the same

[math]\displaystyle{ \frac{\partial\zeta}{\partial t} = \frac{\partial\Phi}{\partial z} , \ z=0; }[/math]

but the kinematic condition needs to be modified to include the effect of the the plate

[math]\displaystyle{ \rho g\zeta + \rho \frac{\partial\Phi}{\partial t} = D\frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2} , \ z=0; }[/math]

We also have Laplace's equation

[math]\displaystyle{ \Delta \Phi = 0,\,\,-h\lt z\lt 0 }[/math]

and the usual non-flow condition at the bottom surface

[math]\displaystyle{ \partial_z \Phi = 0,\,\,z=-h }[/math]

[math]\displaystyle{ \zeta }[/math] is the surface displacement and [math]\displaystyle{ \Phi }[/math] is the velocity potential.

[math]\displaystyle{ D\frac{\partial^4}{\partial x^4} \frac{\partial \phi}{\partial z} + \left(\rho g - \omega^2 \rho_i d \right) \frac{\partial \phi}{\partial z} = - \rho \omega^2 \phi, \, z=0 }[/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]

where [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{ \,d }[/math] and [math]\displaystyle{ \,D }[/math] are the thickness and flexural rigidity of the plate.


[math]\displaystyle{ \frac{\partial\zeta_1}{\partial t} = \frac{\partial\Phi_1}{\partial z} , \ z=0; }[/math]

which follows from the Kinematic equation and

[math]\displaystyle{ \zeta_1 = -\frac{1}{g} \frac{\partial\Phi_1}{\partial t}, \ z=0; }[/math]

which follows from the Dynamic equation. These are the linear free surface conditions.

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