Two-Dimensional Floating Elastic Plate - Revision history

Meylan: /* Equations of Motion */

2009-04-04T07:15:58Z

Equations of Motion

← Older revision		Revision as of 07:15, 4 April 2009
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	== Equations of Motion ==		== Equations of Motion ==

	{{equations for a ~~uniform free~~ beam}}		{{equations for a beam}}

	We derive here the equations for a plate on a fluid, ignoring boundary conditions at the plate edge and assuming		We derive here the equations for a plate on a fluid, ignoring boundary conditions at the plate edge and assuming
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	{{frequency domain equations for a floating plate}}		{{frequency domain equations for a floating plate}}


	== Computational Methods ==		== Computational Methods ==

Meylan at 01:35, 3 April 2009

2009-04-03T01:35:44Z

← Older revision		Revision as of 01:35, 3 April 2009
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	[[Category:Floating Elastic Plate]]		[[Category:Floating Elastic Plate]]
			[[Category:Complete Pages]]

Meylan at 01:35, 3 April 2009

2009-04-03T01:35:06Z

@@ Line 8: / Line 8: @@
 {{linear elastic plate on water time domain}}
-The pressure is given by the linearised Bernoulli equation at the wetted surface (assuming zero
+=== Frequency Domain Equations ===
-=Energy Balance=
+== Computational Methods ==
-= Solution Methods =
+This wiki contains a number of methods to solve for a floating elastic plate in the frequency domain.
 There are many different methods to solve the corresponding equations ranging from highly analytic such

Meylan: /* Eigenfunction Matching Method */

2009-03-30T09:50:02Z

Eigenfunction Matching Method

← Older revision		Revision as of 09:50, 30 March 2009
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	Other authors have extended mode-matching approach of [[Fox and Squire 1990]]. The eigenfunction matching solution was extended		Other authors have extended mode-matching approach of [[Fox and Squire 1990]]. The eigenfunction matching solution was extended
	to plates of different properties by [[Barrett and Squire 1996]]. [[Sahoo et al. 2001]] defined an inner product enabling the solution to be found by using <math>N</math> eigenfunctions and inverting an <math>N\times N</math> matrix, Earlier [[Chakrabarti 2000]] had used an infinite depth mode-matching scheme to set up a singular integral equation, the equivalent problem to a residue calculus problem when the eigenvalue spectrum is continuous. This singular integral equation was solved by transforming it into a Hilbert problem ([[Roos 1969]]), a generalization of the [[Wiener-Hopf]] problem (although in practice both are solved in exactly the same way).		to plates of different properties by [[Barrett and Squire 1996]]. [[Sahoo et al. 2001]] defined an inner product enabling the solution to be found by using <math>N</math> eigenfunctions and inverting an <math>N\times N</math> matrix, Earlier [[Chakrabarti 2000]] had used an infinite depth mode-matching scheme to set up a singular integral equation, the equivalent problem to a residue calculus problem when the eigenvalue spectrum is continuous. This singular integral equation was solved by transforming it into a Hilbert problem ([[Roos 1969]]), a generalization of the [[:Category:Wiener-Hopf\|Wiener-Hopf]] problem (although in practice both are solved in exactly the same way).

	===[[:Category:Residue Calculus\|Residue Calculus]]===		===[[:Category:Residue Calculus\|Residue Calculus]]===

Meylan at 10:07, 27 March 2009

2009-03-27T10:07:02Z

← Older revision		Revision as of 10:07, 27 March 2009
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	= Equations of Motion =		== Equations of Motion ==

	{{equations for a uniform free beam}}		{{equations for a uniform free beam}}

			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.

			{{linear elastic plate on water time domain}}

	The pressure is given by the linearised Bernoulli equation at the wetted surface (assuming zero		The pressure is given by the linearised Bernoulli equation at the wetted surface (assuming zero

Meylan: /* Equations of Motion */

2009-03-27T10:03:20Z

Equations of Motion

← Older revision		Revision as of 10:03, 27 March 2009
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	{{equations for a uniform free beam}}		{{equations for a uniform free beam}}

	~~For a 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 satisfy 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.~~

	The pressure is given by the linearised Bernoulli equation at the wetted surface (assuming zero		The pressure is given by the linearised Bernoulli equation at the wetted surface (assuming zero

Meylan: /* Equations of Motion */

2009-03-27T10:00:50Z

Equations of Motion

← Older revision		Revision as of 10:00, 27 March 2009
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	= Equations of Motion =		= Equations of Motion =

	When considering a two dimensional problem, the <math>y</math> variable is dropped and the plate is regarded as a beam. There are various beam theories that can be used to describe the motion of the beam. The simplest theory is the [http://en.wikipedia.org/wiki/Euler_Bernoulli_beam_equation Bernoulli-Euler Beam] which is commonly used in the two dimensional hydroelastic analysis. Other beam theories include the [[Timoshenko Beam]] theory and [[Reddy-Bickford Beam]] theory where shear deformation of higher order is considered.

	{{equations for a uniform free beam}}		{{equations for a uniform free beam}}

Meylan: /* Equations of Motion */

2009-03-27T09:59:40Z

Equations of Motion

← Older revision		Revision as of 09:59, 27 March 2009
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	When considering a two dimensional problem, the <math>y</math> variable is dropped and the plate is regarded as a beam. There are various beam theories that can be used to describe the motion of the beam. The simplest theory is the [http://en.wikipedia.org/wiki/Euler_Bernoulli_beam_equation Bernoulli-Euler Beam] which is commonly used in the two dimensional hydroelastic analysis. Other beam theories include the [[Timoshenko Beam]] theory and [[Reddy-Bickford Beam]] theory where shear deformation of higher order is considered.		When considering a two dimensional problem, the <math>y</math> variable is dropped and the plate is regarded as a beam. There are various beam theories that can be used to describe the motion of the beam. The simplest theory is the [http://en.wikipedia.org/wiki/Euler_Bernoulli_beam_equation Bernoulli-Euler Beam] which is commonly used in the two dimensional hydroelastic analysis. Other beam theories include the [[Timoshenko Beam]] theory and [[Reddy-Bickford Beam]] theory where shear deformation of higher order is considered.

			{{equations for a uniform free beam}}

	For a Bernoulli-Euler beam on the surface of the water, the equation of motion is given		For a Bernoulli-Euler beam on the surface of the water, the equation of motion is given

Meylan: /* Energy Balance */

2006-12-13T08:53:47Z

Energy Balance

← Older revision		Revision as of 08:53, 13 December 2006
Line 70:		Line 70:
	energy balance condition becomes the following		energy balance condition becomes the following
	<center><math>		<center><math>
	D\|T)\|^2 + \|R\|^2 = \|I\|^2,		D\|T\|^2 + \|R\|^2 = \|I\|^2,
	</math></center>		</math></center>
	where <math>D</math> is found by applying Green's theorem to <math>\phi</math> and its conjugate,		where <math>D</math> is found by applying Green's theorem to <math>\phi</math> and its conjugate,

Meylan: /* Energy Balance */

2006-12-13T08:53:29Z

Energy Balance

← Older revision		Revision as of 08:53, 13 December 2006
Line 70:		Line 70:
	energy balance condition becomes the following		energy balance condition becomes the following
	<center><math>		<center><math>
	D\|~~T_{\Lambda}(0~~)\|^2 + \|~~R_{1}(0)~~\|^2 = \|I\|^2,		D\|T)\|^2 + \|R\|^2 = \|I\|^2,
	</math></center>		</math></center>
	where <math>D</math> is found by applying Green's theorem to <math>\phi</math> and its conjugate,		where <math>D</math> is found by applying Green's theorem to <math>\phi</math> and its conjugate,