View source for Category: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.

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 [http://en.wikipedia.org/wiki/Euler_Bernoulli_beam_equation Bernoulli-Euler Beam] on the surface of the water, the equation of motion is given
by the following
<center><math>
D\frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2} = p
</math></center>
where <math>D</math> is the flexural rigidity, <math>\rho_i</math> is the density of the beam,
<math>h</math> is the thickness of the beam (assumed constant), <math> p</math> is the pressure
and <math>\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). 
<center><math>
\frac{\partial^2 \eta}{\partial x^2} = 0, \,\,\frac{\partial^3 \eta}{\partial x^3} = 0
</math></center>
at the edges of the plate.

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

<center><math>
D\frac{\partial^4 \bar{\eta}}{\partial x^4} -\omega^2 \rho_i h \bar{\eta} = \bar{p}
</math></center>

== Linear Elastic Thin Plate on Water ==

We begin with the [[Linear and Second-Order Wave Theory| linear equations]] for a fluid.
The dynamic condition is the same
<center><math> \zeta = -\frac{1}{g} \frac{\partial\Phi}{\partial t}, \ z=0; </math></center>
but the kinematic condition needs to be modified to include the effect of the the plate
<center><math> 
\frac{\partial\zeta}{\partial t} - \frac{\partial\Phi}{\partial z} = , \ z=0;  </math></center>
We also have Laplace's equation
<center><math>
\Delta \Phi = 0,\,\,-h<z,0
</math></center>
and the usual non-flow condition at the bottom surface
<center><math>
\partial_z \Phi = 0,\,\,z=-h
</math></center>
<math>\zeta</math> is the surface displacement and <math>\Phi</math> is the velocity potential.
<center>
<math>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>
</center>
plus the equations within  the fluid 
<center>
<math>\nabla^2\phi =0 </math>
</center>
<center>
<math>\frac{\partial \phi}{\partial z} = 0, \, z=-h</math></center>
where <math> \,g </math> is the acceleration due to gravity,  <math> \,\rho_i </math> and <math> \,\rho </math> 
are the densities of the plate and the water respectively, <math> \,d </math> and <math> \,D </math> are
the thickness and flexural rigidity of the plate.


<center><math> \frac{\partial\zeta_1}{\partial t} = \frac{\partial\Phi_1}{\partial z} , \ z=0;  </math></center>
which follows from the Kinematic equation and 
<center><math> \zeta_1 = -\frac{1}{g} \frac{\partial\Phi_1}{\partial t}, \ z=0; </math></center>
which follows from the Dynamic equation. These are the linear free surface conditions. 

[[Category:Linear Hydroelasticity]]