Category:Floating Elastic Plate - Revision history

Meylan at 09:39, 20 October 2009

2009-10-20T09:39:44Z

2009-07-03T01:16:44Z

Linear Thin Elastic Plate Theory

← Older revision		Revision as of 01:16, 3 July 2009
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	<center><math>		<center><math>
	\partial_x^2\left(D\partial_x^2 \bar{\zeta}\right) -\omega^2 \rho_i h \bar{\zeta} = \bar{p}		\partial_x^2\left(\beta \partial_x^2 \bar{\zeta}\right) -\omega^2 \rho_i h \bar{\zeta} = \bar{p}
	</math></center>		</math></center>

2009-04-03T01:28:12Z

Frequency Domain Equations

← Older revision		Revision as of 01:28, 3 April 2009
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	== Frequency Domain Equations ==		== Frequency Domain Equations ==

	{{~~frequecy~~ domain equations for a floating plate}}		{{frequency domain equations for a floating plate}}

	== Nonlinear Elastic Thin Plate on Water ==		== Nonlinear Elastic Thin Plate on Water ==

2009-04-03T01:28:03Z

Frequency Domain Equations

@@ Line 31: / Line 31: @@
 == Frequency Domain Equations ==
-If we make the assumption of [[Frequency Domain Problem]] that everything is proportional to
+{{frequecy domain equations for a floating plate}}
 == Nonlinear Elastic Thin Plate on Water ==

2009-03-24T21:22:51Z

Linear Thin Elastic Plate Theory

← Older revision		Revision as of 21:22, 24 March 2009
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	{{equations for a uniform free beam}}		{{equations for a uniform free beam}}

			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>
			\partial_x^2\left(D\partial_x^2 \bar{\zeta}\right) -\omega^2 \rho_i h \bar{\zeta} = \bar{p}
			</math></center>

	== Linear Elastic Plate on Water ==		== Linear Elastic Plate on Water ==

2009-03-23T02:40:59Z

Linear Elastic Plate on Water

← Older revision		Revision as of 02:40, 23 March 2009
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	== Linear Elastic Plate on Water ==		== Linear Elastic Plate on Water ==

	We derive here the equations for a plate on a fluid, ignoring boundary conditions at the plate edge.		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}}		{{linear elastic plate on water time domain}}

2009-03-23T02:40:38Z

Linear Elastic Plate on Water

← Older revision		Revision as of 02:40, 23 March 2009
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	== Linear Elastic Plate on Water ==		== Linear Elastic Plate on Water ==

			We derive here the equations for a plate on a fluid, ignoring boundary conditions at the plate edge.

	{{linear elastic plate on water time domain}}		{{linear elastic plate on water time domain}}

2009-03-23T02:39:51Z

Frequency Domain Equations

← Older revision		Revision as of 02:39, 23 March 2009
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	z = 0.		z = 0.
	</math></center>		</math></center>


	~~{{finite plate frequency domain}}~~

	== Nonlinear Elastic Thin Plate on Water ==		== Nonlinear Elastic Thin Plate on Water ==

2009-03-23T02:39:15Z

Frequency Domain Equations

← Older revision		Revision as of 02:39, 23 March 2009
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	If we make the assumption of [[Frequency Domain Problem]] that everything is proportional to		If we make the assumption of [[Frequency Domain Problem]] that everything is proportional to
	~~<center>~~<math>\exp (i\omega t)\,</math~~></center~~> the equations become		<math>\exp (i\omega t)\,</math> the equations become
	<center><math>		<center><math>
	i\omega\zeta = \partial_z\phi , \ z=0; </math></center>		i\omega\zeta = \partial_z\phi , \ z=0; </math></center>
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	\beta \partial_x^4\partial_z \phi		\beta \partial_x^4\partial_z \phi
	- \left( \gamma\alpha - 1 \right) \partial_z \phi + \alpha\phi = 0, \;\;		- \left( \gamma\alpha - 1 \right) \partial_z \phi + \alpha\phi = 0, \;\;
	z = 0~~, \;\;\; -L \leq x \leq L,~~		z = 0.
	</math></center>		</math></center>

2009-03-23T02:38:48Z

Frequency Domain Equations

← Older revision		Revision as of 02:38, 23 March 2009
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	These equations can be simplified by defining <math>\alpha = \omega^2/g</math>,		These equations can be simplified by defining <math>\alpha = \omega^2/g</math>,
	<math>\beta = D/\rho g</math> and <math>\gamma = \rho_i h/\rho</math>		<math>\beta = D/\rho g</math> and <math>\gamma = \rho_i h/\rho</math> to obtain
			<center><math>
			\Delta \phi = 0, \;\;\; -h < z \leq 0,
			</math></center>
			<center><math>
			\partial_z \phi = 0, \;\;\; z = - h,
			</math></center>
			<center><math>
			\beta \partial_x^4\partial_z \phi
			- \left( \gamma\alpha - 1 \right) \partial_z \phi + \alpha\phi = 0, \;\;
			z = 0, \;\;\; -L \leq x \leq L,
			</math></center>