Three-Dimensional Floating Elastic Plate - Revision history

Meylan at 05:48, 30 October 2012

2012-10-30T05:48:46Z

← Older revision		Revision as of 05:48, 30 October 2012
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			{{incomplete pages}}

			We develop here a theory to solve for a three-dimensional floating elastic plate.

	= Equations of Motion =		= Equations of Motion =

	For a classical thin plate, the equation of motion is given by		For a classical thin plate, the equation of motion is given by
	<center><math>		<center><math>
	D\nabla ^4 w - \rho _i h \frac{\partial^2 w}{\partial t^2} = p ~~\qquad (1)~~		D\nabla ^4 w - \rho _i h \frac{\partial^2 w}{\partial t^2} = p
	</math></center>		</math></center>
	Equation (1) is subject to the free edge boundary		Equation (1) is subject to the free edge boundary
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	\left[ \nabla^2 - (1-\nu)		\left[ \nabla^2 - (1-\nu)
	\left(\frac{\partial^2}{\partial s^2} + \kappa(s)		\left(\frac{\partial^2}{\partial s^2} + \kappa(s)
	\frac{\partial}{\partial n} \right) \right] w = 0, ~~\qquad(1.1)~~		\frac{\partial}{\partial n} \right) \right] w = 0,
	</math></center>		</math></center>
	<center><math>		<center><math>
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	\frac{\partial}{\partial s}		\frac{\partial}{\partial s}
	\left( \frac{\partial}{\partial n} \frac{\partial}{\partial s}		\left( \frac{\partial}{\partial n} \frac{\partial}{\partial s}
	-\kappa(s) \frac{\partial}{\partial s} \right) \right] w = 0, ~~\qquad(1.2)~~		-\kappa(s) \frac{\partial}{\partial s} \right) \right] w = 0,
	</math></center>		</math></center>
	where <math>\nu</math> is Poisson's ratio and		where <math>\nu</math> is Poisson's ratio and

Meylan: Reverted edits by Shagman (Shagman); changed back to last version by Meylan

2006-12-09T09:44:39Z

Reverted edits by Shagman (Shagman); changed back to last version by Meylan

← Older revision		Revision as of 09:44, 9 December 2006
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	[[Category:Floating Elastic Plate]]		[[Category:Floating Elastic Plate]]



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Shagman at 21:43, 7 December 2006

2006-12-07T21:43:29Z

← Older revision		Revision as of 21:43, 7 December 2006
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	[[Category:Floating Elastic Plate]]		[[Category:Floating Elastic Plate]]



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Meylan: /* Equations of Motion */

2006-09-11T09:34:47Z

Equations of Motion

← Older revision		Revision as of 09:34, 11 September 2006
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	as a function of arclength <math>s</math> along <math>\partial \Delta</math>;		as a function of arclength <math>s</math> along <math>\partial \Delta</math>;
	<math>\partial/\partial s</math> and <math>\partial/\partial n</math> represent derivatives		<math>\partial/\partial s</math> and <math>\partial/\partial n</math> represent derivatives
	tangential and normal to the boundary <math>\partial \Delta</math>, respectively.		tangential and normal to the boundary <math>\partial \Delta</math>, respectively
	[[Porter ~~paper?~~]] where <math>n</math> and <math>s</math> denote the normal and tangential		([[Porter and Porter 2004]]) where <math>n</math> and <math>s</math> denote the normal and tangential
	directions respectively.		directions respectively.

Meylan: /* Solution of the Equations of Motion */

2006-09-11T09:11:32Z

Solution of the Equations of Motion

← Older revision		Revision as of 09:11, 11 September 2006
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	There are a number of methods to solve this problem. We will describe a		There are a number of methods to solve this problem. We will describe a
	method which generalises the [[Linear Wave Scattering for Floating Rigid Body]] to a plate which		method which generalises the [[Linear Wave Scattering for a Floating Rigid Body]] to a plate which
	has an infinite number of degrees of freedom. Many other methods of solution		has an infinite number of degrees of freedom. Many other methods of solution
	have been presented, most of which consider some kind of regular plate shape		have been presented, most of which consider some kind of regular plate shape

Mdmgreen: /* Equations of Motion for the Water */

2006-09-11T08:31:47Z

Equations of Motion for the Water

← Older revision		Revision as of 08:31, 11 September 2006
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	x}\notin \Delta ,		x}\notin \Delta ,
	\end{matrix}		\end{matrix}
	\right\} \qquad(5~~)%(bvp_nond~~)		\right\} \qquad(5)
	</math></center>		</math></center>
	The vector <math>\mathbf{x=(}x,y)</math> is a		The vector <math>\mathbf{x=(}x,y)</math> is a
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	occupied by the plate.		occupied by the plate.

	The boundary value problem (~~(bvp)~~) is subject to an incident wave which		The boundary value problem (5) is subject to an incident wave which
	is imposed through the		is imposed through the
	[[Sommerfeld Radiation Condition]]		[[Sommerfeld Radiation Condition]]
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	\lim_{\left\| \mathbf{x}\right\| \rightarrow \infty }\sqrt{\|\mathbf{x}\|}\left(		\lim_{\left\| \mathbf{x}\right\| \rightarrow \infty }\sqrt{\|\mathbf{x}\|}\left(
	\frac{\partial }{\partial \|\mathbf{x}\|}-i\alpha \right) (\phi -\phi ^{		\frac{\partial }{\partial \|\mathbf{x}\|}-i\alpha \right) (\phi -\phi ^{
	\mathrm{In}})=0, \qquad(6~~) %(sommerfield~~)		\mathrm{In}})=0, \qquad(6)
	</math></center>		</math></center>
	where the incident potential <math>\phi ^{\mathrm{In}}</math> is		where the incident potential <math>\phi ^{\mathrm{In}}</math> is

Mdmgreen: /* Solution of the Equations of Motion */

2006-09-11T08:31:01Z

Solution of the Equations of Motion

← Older revision		Revision as of 08:31, 11 September 2006
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	boundary value problem into an integral equation using a Green function		boundary value problem into an integral equation using a Green function
	Performing such a transformation, the boundary		Performing such a transformation, the boundary
	value problem (~~(bvp)~~) and (~~(summerfield)~~) become		value problem (5) and (6) become
	<center><math>		<center><math>
	\phi (\mathbf{x})=\phi ^{i}(\mathbf{x})+\iint_{\Delta }G_{\alpha }(\mathbf{x}		\phi (\mathbf{x})=\phi ^{i}(\mathbf{x})+\iint_{\Delta }G_{\alpha }(\mathbf{x}
	;\mathbf{y})\left( \alpha \phi (\mathbf{x})+i\sqrt{\alpha }w(\mathbf{x}		;\mathbf{y})\left( \alpha \phi (\mathbf{x})+i\sqrt{\alpha }w(\mathbf{x}
	)\right) dS_{\mathbf{y}}. (water)		)\right) dS_{\mathbf{y}}. \qquad(8)(water)
	</math></center>		</math></center>
	where <math>G_{\alpha }</math> is the [[Free-Surface Green Function]]		where <math>G_{\alpha }</math> is the [[Free-Surface Green Function]]

Mdmgreen: /* Equations of Motion for the Water */

2006-09-11T08:30:07Z

Equations of Motion for the Water

← Older revision		Revision as of 08:30, 11 September 2006
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	x}\notin \Delta ,		x}\notin \Delta ,
	\end{matrix}		\end{matrix}
	\right\} (bvp_nond)		\right\} \qquad(5)%(bvp_nond)
	</math></center>		</math></center>
	The vector <math>\mathbf{x=(}x,y)</math> is a		The vector <math>\mathbf{x=(}x,y)</math> is a
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	\lim_{\left\| \mathbf{x}\right\| \rightarrow \infty }\sqrt{\|\mathbf{x}\|}\left(		\lim_{\left\| \mathbf{x}\right\| \rightarrow \infty }\sqrt{\|\mathbf{x}\|}\left(
	\frac{\partial }{\partial \|\mathbf{x}\|}-i\alpha \right) (\phi -\phi ^{		\frac{\partial }{\partial \|\mathbf{x}\|}-i\alpha \right) (\phi -\phi ^{
	\mathrm{In}})=0, (~~summerfield~~)		\mathrm{In}})=0, \qquad(6) %(sommerfield)
	</math></center>		</math></center>
	where the incident potential <math>\phi ^{\mathrm{In}}</math> is		where the incident potential <math>\phi ^{\mathrm{In}}</math> is
	<center><math>		<center><math>
	\phi ^{\mathrm{In}}(x,y,z)=\frac{A}{{\omega }}e^{i\alpha (x\cos \theta		\phi ^{\mathrm{In}}(x,y,z)=\frac{A}{{\omega }}e^{i\alpha (x\cos \theta
	+y\sin \theta )}e^{\alpha z}, (input)		+y\sin \theta )}e^{\alpha z}, \qquad (7)(input)
	</math></center>		</math></center>
	where <math>A</math> is the non-dimensional wave amplitude.		where <math>A</math> is the non-dimensional wave amplitude.

Mdmgreen: /* Equations of Motion */

2006-09-11T08:27:31Z

Equations of Motion

← Older revision		Revision as of 08:27, 11 September 2006
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	For a classical thin plate, the equation of motion is given by		For a classical thin plate, the equation of motion is given by
	<center><math>		<center><math>
	D\nabla ^4 w - \rho _i h \frac{\partial^2 w}{\partial t^2} = p\~~,\,\,~~(1)		D\nabla ^4 w - \rho _i h \frac{\partial^2 w}{\partial t^2} = p \qquad (1)
	</math></center>		</math></center>
	Equation (1) is subject to the free edge boundary		Equation (1) is subject to the free edge boundary
	conditions for a thin plate		conditions for a thin plate
	<center><math> ~~(boundary1)~~		<center><math>
	\left[ \nabla^2 - (1-\nu)		\left[ \nabla^2 - (1-\nu)
	\left(\frac{\partial^2}{\partial s^2} + \kappa(s)		\left(\frac{\partial^2}{\partial s^2} + \kappa(s)
	\frac{\partial}{\partial n} \right) \right] w = 0,		\frac{\partial}{\partial n} \right) \right] w = 0, \qquad(1.1)
	</math></center>		</math></center>
	<center><math> ~~(boundary2)~~		<center><math>
	\left[ \frac{\partial}{\partial n} \nabla^2 +(1-\nu)		\left[ \frac{\partial}{\partial n} \nabla^2 +(1-\nu)
	\frac{\partial}{\partial s}		\frac{\partial}{\partial s}
	\left( \frac{\partial}{\partial n} \frac{\partial}{\partial s}		\left( \frac{\partial}{\partial n} \frac{\partial}{\partial s}
	-\kappa(s) \frac{\partial}{\partial s} \right) \right] w = 0,		-\kappa(s) \frac{\partial}{\partial s} \right) \right] w = 0, \qquad(1.2)
	</math></center>		</math></center>
	where <math>\nu</math> is Poisson's ratio and		where <math>\nu</math> is Poisson's ratio and
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	become		become
	<center><math>		<center><math>
	\beta \nabla^{4}{w}+\gamma \frac{\partial^2 w}{\partial t^2}=\frac{\partial {\Phi}}{\partial {t}}-{w}, (n-d_ice)		\beta \nabla^{4}{w}+\gamma \frac{\partial^2 w}{\partial t^2}=\frac{\partial {\Phi}}{\partial {t}}-{w}, \qquad(3)% (n-d_ice)
	</math></center>		</math></center>
	where		where
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	This leads to the following equation		This leads to the following equation
	<center><math>		<center><math>
	\beta \nabla ^{4}w+\alpha \gamma w=-i\omega\phi -w. (plate2)		\beta \nabla ^{4}w+\alpha \gamma w=-i\omega\phi -w. \qquad(4)%(plate2)
	</math></center>		</math></center>

Meylan: /* Solution of the Equations of Motion */

2006-06-16T00:08:27Z

Solution of the Equations of Motion

← Older revision		Revision as of 00:08, 16 June 2006
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	There are a number of methods to solve this problem. We will describe a		There are a number of methods to solve this problem. We will describe a
	method which generalises the [[Floating Body ~~Solution~~]] to a plate which		method which generalises the [[Linear Wave Scattering for Floating Rigid Body]] to a plate which
	has an infinite number of degrees of freedom. Many other methods of solution		has an infinite number of degrees of freedom. Many other methods of solution
	have been presented, most of which consider some kind of regular plate shape		have been presented, most of which consider some kind of regular plate shape