Difference between revisions of "Category:Floating Elastic Plate"

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by the following
 
by the following
 
<center><math>
 
<center><math>
−
\partial_x^2\left(D\partial_x^2 \eta\right) + \rho_i h \frac{\partial^2 \eta}{\partial t^2} = p
+
\partial_x^2\left(D\partial_x^2 \zeta\right) + \rho_i h \frac{\partial^2 \zeta}{\partial t^2} = p
 
</math></center>
 
</math></center>
 
where <math>D</math> is the flexural rigidity, <math>\rho_i</math> is the density of the plate,
 
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, <math> p</math> is the pressure
 
<math>h</math> is the thickness of the plate, <math> p</math> is the pressure
−
and <math>\eta</math> is the plate vertical displacement. Note that this equations simplifies if the plate has constant
+
and <math>\zeta</math> is the plate vertical displacement. Note that this equations simplifies if the plate has constant
 
properties.   
 
properties.   
  
 
The edges of the plate can satisfy a range of boundary conditions. The natural boundary condition (i.e. free-edge boundary conditions).  
 
The edges of the plate can satisfy a range of boundary conditions. The natural boundary condition (i.e. free-edge boundary conditions).  
 
<center><math>
 
<center><math>
−
\frac{\partial^2 \eta}{\partial x^2} = 0, \,\,\frac{\partial^3 \eta}{\partial x^3} = 0
+
\frac{\partial^2 \zeta}{\partial x^2} = 0, \,\,\frac{\partial^3 \zeta}{\partial x^3} = 0
 
</math></center>
 
</math></center>
 
at the edges of the plate.
 
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
 
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
+
<math>\zeta(x,t) = e^{i\omega t} \bar{\zeta}(x) </math> from linearity. In this case the equations reduce to
  
 
<center><math>
 
<center><math>
−
\partial_x^2\left(D\partial_x^2 \bar{\eta}\right)  -\omega^2 \rho_i h \bar{\eta} = \bar{p}
+
\partial_x^2\left(D\partial_x^2 \bar{\zeta}\right)  -\omega^2 \rho_i h \bar{\zeta} = \bar{p}
 
</math></center>
 
</math></center>
  

Revision as of 21:42, 6 November 2008

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{ \partial_x^2\left(D\partial_x^2 \zeta\right) + \rho_i h \frac{\partial^2 \zeta}{\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, [math]\displaystyle{ p }[/math] is the pressure and [math]\displaystyle{ \zeta }[/math] is the plate vertical displacement. Note that this equations simplifies if the plate has constant properties.

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 \zeta}{\partial x^2} = 0, \,\,\frac{\partial^3 \zeta}{\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{ \zeta(x,t) = e^{i\omega t} \bar{\zeta}(x) }[/math] from linearity. In this case the equations reduce to

[math]\displaystyle{ \partial_x^2\left(D\partial_x^2 \bar{\zeta}\right) -\omega^2 \rho_i h \bar{\zeta} = \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 dynamic 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]

where [math]\displaystyle{ \zeta }[/math] is the surface displacement, [math]\displaystyle{ \Phi }[/math] is the velocity potential, and [math]\displaystyle{ \rho }[/math] is the fluid density.

Nonlinear Elastic Thin Plate on Water

The nonlinear plate equations can be found by include We begin with the nonlinear equations for a fluid. The kinematic condition is

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

the dynamic condition is

[math]\displaystyle{ \partial_x^2 \left( \frac{ D \partial_x^2 \zeta}{(1+(\partial_x\zeta^2))^{3/2}} \right) + \rho_i h \frac{\partial^2 \eta}{\partial t^2} = - \rho\frac{\partial\Phi}{\partial t}-\rho\frac{1}{2}\nabla\Phi \cdot \nabla \Phi - \rho g z, \ z=\zeta }[/math]

There are different versions of the nonlinear boundary condition for a plate and this one is based on Parau and Dias 2002. We also have Laplace's equation

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

and the usual non-flow condition at the bottom surface

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