Difference between revisions of "Floating Elastic Plate"

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= Equations of Motion =
 
= Equations of Motion =
  
The [[Frequency Domain Problem]] for a floating elastic plate which is governed by Kirkoffs equation is given
+
The equation for a elastic plate which is governed by Kirkoffs equation is given
 
by the following
 
by the following
 +
 +
<math>D\frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2} = P</math>
 +
 +
where <math>D</math> is the flexural rigidity, <math>\rho_i</math> is the density of the plate,
 +
<math>h</math> is the thickness of the plate (assumed constant), <math> P</math> is the pressure
 +
and <math>\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>P = \rho g \phi + \rho \frac{\partial \phi}{\partial t}</math>
 +
 +
where <math>\rho</math> is the water density and <math>g</math> is gravity, and <math>\phi</math>
 +
is the velocity potential. The velocity potential is governed by Laplace's equation through out
 +
the fluid domain subject to the
 +
 +
[[Frequency Domain Problem]]

Revision as of 10:32, 1 May 2006

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 1960. The problem can be divided into the two and three dimensional formulations which are closely related.

Two Dimensional Problem

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

Frequency Domain Problem