Difference between revisions of "Category:Floating Elastic Plate"

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{{complete pages}}
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== Introduction ==
 
== Introduction ==
  
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We present here the theory of an elastic plate in a vacuum, concentrating of the two-dimensional problem.
 
We present here the theory of an elastic plate in a vacuum, concentrating of the two-dimensional problem.
  
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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
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{{equations for a uniform free beam}}
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by the following
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 +
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>\zeta(x,t) = e^{i\omega t} \bar{\zeta}(x) </math> from linearity. In this case the equations reduce to
 +
 
 
<center><math>
 
<center><math>
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D\frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2} = p
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\partial_x^2\left(\beta \partial_x^2 \bar{\zeta}\right)  -\omega^2 \rho_i h \bar{\zeta} = \bar{p}
 
</math></center>
 
</math></center>
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where <math>D</math> is the flexural rigidity, <math>\rho_i</math> is the density of the beam,
 
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<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.
 
  
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The edges of the plate can satisfy a range of boundary conditions. The natural boundary condition (i.e. free-edge boundary conditions).
+
== Linear Elastic Plate on Water ==
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<center><math>
 
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\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
+
We derive here the equations for a plate on a fluid, ignoring boundary conditions at the plate edge and assuming
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<math>\eta(x,t) = e^{i\omega t} \bar{\eta}(x) </math> from linearity. In this case the equations reduce to
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the plate occupies the entire fluid region.  
  
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<center><math>
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{{linear elastic plate on water time domain}}
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D\frac{\partial^4 \bar{\eta}}{\partial x^4} -\omega^2 \rho_i h \bar{\eta} = \bar{p}
 
−
</math></center>
 
  
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== Linear Elastic Thin Plate on Water ==
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== Frequency Domain Equations ==
  
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We begin with the [[Linear and Second-Order Wave Theory| linear equations]] for a fluid.
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{{frequency domain equations for a floating plate}}
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The kinematic condition is the same
 
−
<center><math>
 
−
\frac{\partial\zeta}{\partial t} = \frac{\partial\Phi}{\partial z}  , \ z=0;  </math></center>
 
−
but the dynamic condition needs to be modified to include the effect of the the plate
 
−
<center><math> \rho g\zeta  + \rho \frac{\partial\Phi}{\partial t}
 
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= D\frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2}
 
−
, \ 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>
 
−
where
 
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<math>\zeta</math> is the surface displacement, <math>\Phi</math> is the velocity potential,
 
−
and <math>\rho</math> is the fluid density.
 
  
 
== Nonlinear Elastic Thin Plate on Water ==
 
== Nonlinear Elastic Thin Plate on Water ==
  
 
The nonlinear plate equations can be found by include
 
The nonlinear plate equations can be found by include
−
We begin with the [[Linear and Second-Order Wave Theory| linear equations]] for a fluid.
+
We begin with the [[Conservation Laws and Boundary Conditions | nonlinear equations]] for a fluid.
−
The kinematic condition is the same
+
The kinematic condition is  
 
<center><math>  
 
<center><math>  
−
\frac{\partial\zeta}{\partial t} = \frac{\partial\Phi}{\partial z} , \ z=0;  </math></center>
+
\frac{\partial\zeta}{\partial t}+\frac{\partial\Phi}{\partial x} \frac{\partial\zeta}{\partial x}
−
but the dynamic condition needs to be modified to include the effect of the the plate
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=\frac{\partial\Phi}{\partial z},
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<center><math> \rho g\zeta  + \rho \frac{\partial\Phi}{\partial t}
+
\ z=\zeta(x,t) </math></center>
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= D\frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2}
+
the dynamic condition is
−
, \ z=0; </math></center>
 
−
 
 
 
<center><math>  
 
<center><math>  
−
D\frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2}   
+
\partial_x^2 \left( \frac{ D \partial_x^2 \zeta}{(1+(\partial_x\zeta^2))^{3/2}} \right)
−
= - \rho\frac{\partial\Phi}{\partial t}-\rho\frac{1}{2}\nabla\Phi \cdot \nabla \Phi - \rho g z  
+
+ \rho_i h \frac{\partial^2 \zeta}{\partial t^2}   
 +
= - \rho\frac{\partial\Phi}{\partial t}-\rho\frac{1}{2}\nabla\Phi \cdot \nabla \Phi - \rho g \zeta, \ z=\zeta(x,t)
 
</math></center>
 
</math></center>
−
 
+
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
 
We also have Laplace's equation
 
<center><math>
 
<center><math>
−
\Delta \Phi = 0,\,\,-h<z<0
+
\Delta \Phi = 0,\,\,-h<z<\zeta,
 
</math></center>
 
</math></center>
 
and the usual non-flow condition at the bottom surface
 
and the usual non-flow condition at the bottom surface
 
<center><math>
 
<center><math>
−
\partial_z \Phi = 0,\,\,z=-h,
+
\partial_z \Phi = 0,\,\,z=-h.
 
</math></center>
 
</math></center>
−
where
+
 
−
<math>\zeta</math> is the surface displacement, <math>\Phi</math> is the velocity potential,
 
−
and <math>\rho</math> is the fluid density.
 
 
[[Category:Linear Hydroelasticity]]
 
[[Category:Linear Hydroelasticity]]

Latest revision as of 09:39, 20 October 2009


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.

There are various beam theories that can be used to describe the motion of the beam. The simplest theory is the Bernoulli-Euler Beam theory (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, the equation of motion is given by the following

[math]\displaystyle{ \partial_x^2\left(\beta(x)\partial_x^2 \zeta\right) + \gamma(x) \partial_t^2 \zeta = p }[/math]

where [math]\displaystyle{ \beta(x) }[/math] is the non dimensionalised flexural rigidity, and [math]\displaystyle{ \gamma }[/math] is non-dimensionalised linear mass density function. Note that this equations simplifies if the plate has constant properties (and that [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) .

The edges of the plate can satisfy a range of boundary conditions. The natural boundary condition (i.e. free-edge boundary conditions).

[math]\displaystyle{ \partial_x^2 \zeta = 0, \,\,\partial_x^3 \zeta = 0 }[/math]

at the edges of the plate.

The problem is subject to the initial conditions

[math]\displaystyle{ \zeta(x,0)=f(x) \,\! }[/math]
[math]\displaystyle{ \partial_t \zeta(x,0)=g(x) }[/math]

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(\beta \partial_x^2 \bar{\zeta}\right) -\omega^2 \rho_i h \bar{\zeta} = \bar{p} }[/math]

Linear Elastic Plate on Water

We derive here the equations for a plate on a fluid, ignoring boundary conditions at the plate edge and assuming the plate occupies the entire fluid region.

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.

Frequency Domain Equations

If we make the assumption of Frequency Domain Problem that everything is proportional to [math]\displaystyle{ \exp (-\mathrm{i}\omega t)\, }[/math] the equations become

[math]\displaystyle{ \begin{align} -\mathrm{i}\omega\zeta &= \partial_z\phi , &z=0 \\ \rho g\zeta - \mathrm{i}\omega\rho \phi &= D \partial_x^4 \eta -\omega^2 \rho_i h \zeta, &z=0 \\ \Delta \phi &= 0, &-h\lt z\lt 0 \\ \partial_z \phi &= 0, &z=-h, \end{align} }[/math]

where [math]\displaystyle{ \zeta }[/math] is the surface displacement and [math]\displaystyle{ \phi }[/math] is the velocity potential in the frequency domain.

These equations can be simplified by defining [math]\displaystyle{ \alpha = \omega^2/g }[/math], [math]\displaystyle{ \beta = D/\rho g }[/math] and [math]\displaystyle{ \gamma = \rho_i h/\rho }[/math] to obtain

[math]\displaystyle{ \begin{align} \Delta \phi &= 0, &-h \lt z \leq 0 \\ \partial_z \phi &= 0, &z = - h \\ \beta \partial_x^4 \zeta + \left( 1 - \gamma\alpha \right) \zeta &= -\mathrm{i} \sqrt{\alpha}\phi, &z = 0 \\ -\mathrm{i}\omega\zeta &= \partial_z\phi , &z=0 . \end{align} }[/math]

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(x,t) }[/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 \zeta}{\partial t^2} = - \rho\frac{\partial\Phi}{\partial t}-\rho\frac{1}{2}\nabla\Phi \cdot \nabla \Phi - \rho g \zeta, \ z=\zeta(x,t) }[/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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