Elastic Plate on Shallow Water - Revision history

Meylan at 09:37, 20 October 2009

2009-10-20T09:37:42Z

← Older revision		Revision as of 09:37, 20 October 2009
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			{{incomplete pages}}

	We derive the equations for an elastic plate on shallow water.		We derive the equations for an elastic plate on shallow water.
	These equations can be found in [[Stoker 1957]]. The plate is		These equations can be found in [[Stoker 1957]]. The plate is

Meylan at 12:34, 13 June 2007

2007-06-13T12:34:42Z

← Older revision		Revision as of 12:34, 13 June 2007
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	may be approximated as [[Shallow Depth\|shallow]]. The amplitudes are assumed		may be approximated as [[Shallow Depth\|shallow]]. The amplitudes are assumed
	small enough that the linear theory is appropriate, and the plate is		small enough that the linear theory is appropriate, and the plate is
	sufficiently thin that the [[~~shallow draft~~] approximation may be made		sufficiently thin that the [[Shallow Draft]] approximation may be made

	The kinematic condition is		The kinematic condition is

Meylan at 01:20, 28 August 2006

2006-08-28T01:20:20Z

← Older revision		Revision as of 01:20, 28 August 2006
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	We ~~first begin by deriving~~ the equations for an ~~[[Elastic Plate~~ on ~~Shallow Water]]~~		We derive the equations for an elastic plate on shallow water.
	These equations can be found in [[Stoker 1957]]. The plate is		These equations can be found in [[Stoker 1957]]. The plate is
	infinite in the <math>y</math> direction, so that only the <math>x</math> and <math>z</math> directions are		infinite in the <math>y</math> direction, so that only the <math>x</math> and <math>z</math> directions are

Meylan at 01:19, 28 August 2006

2006-08-28T01:19:46Z

New page

We first begin by deriving the equations for an [[Elastic Plate on Shallow Water]]
These equations can be found in [[Stoker 1957]]. The plate is
infinite in the <math>y</math> direction, so that only the <math>x</math> and <math>z</math> directions are
considered. The <math>x</math> direction is horizontal, the positive <math>z</math> axis points
vertically up, and the plate covers the region <math>-b\leq x\leq b.</math>
The water is of uniform depth <math>h</math> which is small enough that the water
may be approximated as [[Shallow Depth|shallow]]. The amplitudes are assumed
small enough that the linear theory is appropriate, and the plate is
sufficiently thin that the [[shallow draft] approximation may be made

The kinematic condition is
<center><math>
\partial _{t}\zeta =-h\partial _{x}^{2}\phi , \,\,\,(1)
</math></center>
where <math>\phi </math> is the velocity potential of the water (averaged over the
depth) and <math>\zeta </math> is the displacement of the water surface or the plate
(from the shallow draft approximation). The equation derived by equating the
pressure at the free surface is
<center><math>
-\rho g\zeta -\rho \partial _{t}\phi =\left\{
\begin{matrix}
0,\;\;x\notin (-b,b), \\
D\partial _{x}^{4}\zeta +\rho ^{\prime }d\partial _{t}^{2}\zeta ,\;\;x\in
(-b,b),
\end{matrix}
\right. \,\,\, (2)
</math></center>
where <math>D</math> is the bending rigidity of the plate per unit length, <math>\rho </math> is
the density of water, <math>\rho ^{\prime }</math> is the average density of the plate,
<math>g</math> is the acceleration due to gravity, and <math>d</math> is the thickness of the plate
<math>.</math> At the ends of the plate the free edge boundary conditions
<center><math>
\lim_{x\downarrow -b}\partial _{x}^{2}\zeta =\lim_{x\uparrow b}\partial
_{x}^{2}\zeta =\lim_{x\downarrow -b}\partial _{x}^{3}\zeta =\lim_{x\uparrow
b}\partial _{x}^{3}\zeta =0 \,\,\,(3)
</math></center>
are applied, however the theory which will be developed applies
equally to any of the energy-conserving edge conditions such as clamped or
pinned and there is no need for the boundary conditions to be symmetric.
Equation (3) gives the following implied boundary
conditions for <math>\phi </math>
<center><math>
\lim_{x\downarrow -b}\partial _{x}^{4}\phi =\lim_{x\uparrow b}\partial
_{x}^{4}\phi =\lim_{x\downarrow -b}\partial _{x}^{5}\phi =\lim_{x\uparrow
b}\partial _{x}^{5}\phi =0 \,\,\,(4)
</math></center>

[[Category:Floating Elastic Plate]]