Category:Linear Hydroelasticity - Revision history

Meylan: /* Fluid Structure Equations */

2010-04-28T09:42:59Z

Fluid Structure Equations

← Older revision		Revision as of 09:42, 28 April 2010
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	<center><math> \left[\mathbf{K} -\mathrm{i}\mathbf{S} -\omega^2 \left(\mathbf{M} + \mathbf{A} \right) +		<center><math> \left[\mathbf{K} -\mathrm{i}\mathbf{S} -\omega^2 \left(\mathbf{M} + \mathbf{A} \right) +
	\mathrm{i}\omega \mathbf{B} + \mathbf{C} \right] \vec{\zeta} = \mathbf{f} </math></center>		\mathrm{i}\omega \mathbf{B} + \mathbf{C} \right] \vec{\zeta} = \mathbf{f} </math></center>
	where <math>\mathbf{M}</math> is the mass matrix, <math>\mathbf{A}</math> is the added mass matrix,		where
			<math>\mathbf{K}</math> is the stiffnes matrix, <math>\mathbf{S}</math> is the structural damping matrix,
			<math>\mathbf{M}</math> is the mass matrix, <math>\mathbf{A}</math> is the added mass matrix,
	<math>\mathbf{B}</math> is the damping matrix, <math>\mathbf{C}</math> is the hydrostatic matrix,		<math>\mathbf{B}</math> is the damping matrix, <math>\mathbf{C}</math> is the hydrostatic matrix,
	<math>\vec{\zeta}</math> is the vector of body displacements and <math>\mathbf{f}</math> is the force.		<math>\vec{\zeta}</math> is the vector of body displacements and <math>\mathbf{f}</math> is the force.

Meylan: /* Frequency Domain Problem */

2010-04-28T09:41:53Z

Frequency Domain Problem

← Older revision		Revision as of 09:41, 28 April 2010
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	The eigenvalue problem for the "dry" natural vibrations yields:		The eigenvalue problem for the "dry" natural vibrations yields:
	<center><math>		<center><math>
	\left( \begin{bmatrix}K\end{bmatrix}-\~~omega^2~~ \begin{bmatrix}M\end{bmatrix} \right )\begin{bmatrix}w\end{bmatrix}=\begin{bmatrix}0\end{bmatrix}		\left( \begin{bmatrix}K\end{bmatrix}-\lambda \begin{bmatrix}M\end{bmatrix} \right )\begin{bmatrix}w\end{bmatrix}=\begin{bmatrix}0\end{bmatrix}
	</math></center>		</math></center>
	As a solution of the eigenvalue problem for each natural mode one obtains <math>\omega_n</math>, the n-th dry natural frequency and <math>\begin{bmatrix}w_n\end{bmatrix}</math>, the corresponding dry natural mode. Note that only the first modes		As a solution of the eigenvalue problem for each natural mode one obtains <math>\omega_n</math>, the n-th dry natural frequency and <math>\begin{bmatrix}w_n\end{bmatrix}</math>, the corresponding dry natural mode. Note that only the first modes
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	</math></center>		</math></center>
	Associated with each of these modes is a normal motion on the body surface given by		Associated with each of these modes is a normal motion on the body surface given by
	<math>\mathbf{n}_{\nu}</math>.		<math>\mathbf{n}_{\nu}</math>. The first modes are the standard rigid modes.

	== Equations for the Fluid ==		== Equations for the Fluid ==

Meylan: /* Fluid Structure Equations */

2010-04-28T09:37:57Z

Fluid Structure Equations

← Older revision		Revision as of 09:37, 28 April 2010
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	- \sum_{\nu} C_{\mu\nu}\xi_{\nu},		- \sum_{\nu} C_{\mu\nu}\xi_{\nu},
	</math></center>		</math></center>

			{{added mass damping and force matrices definition}}

			Then the equations can be expressed as follows.
			<center><math> \left[\mathbf{K} -\mathrm{i}\mathbf{S} -\omega^2 \left(\mathbf{M} + \mathbf{A} \right) +
			\mathrm{i}\omega \mathbf{B} + \mathbf{C} \right] \vec{\zeta} = \mathbf{f} </math></center>
			where <math>\mathbf{M}</math> is the mass matrix, <math>\mathbf{A}</math> is the added mass matrix,
			<math>\mathbf{B}</math> is the damping matrix, <math>\mathbf{C}</math> is the hydrostatic matrix,
			<math>\vec{\zeta}</math> is the vector of body displacements and <math>\mathbf{f}</math> is the force.

Meylan: /* Fluid Structure Equations */

2010-04-28T09:28:54Z

Fluid Structure Equations

← Older revision		Revision as of 09:28, 28 April 2010
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	- \sum_{\nu} C_{\mu\nu}\xi_{\nu},		- \sum_{\nu} C_{\mu\nu}\xi_{\nu},
	</math></center>		</math></center>


	<center><math>\begin{bmatrix}W\end{bmatrix}</math>=<math>\begin{bmatrix}\mathbf{w_1}\,\mathbf{w_2}\,\ldots \end{bmatrix}</math> is matrix of dry natural modes, with modes being sorted column-wise,</center>

	~~<center><math>\begin{bmatrix}\xi\end{bmatrix}</math>is natural modes coefficients vector (modal amplitudes).</center>~~


	~~<center><math>\begin{bmatrix}W\end{bmatrix}^T \begin{bmatrix}K\end{bmatrix} \begin{bmatrix}W\end{bmatrix}~~
	~~\begin{bmatrix}\xi\end{bmatrix}+\begin{bmatrix}W\end{bmatrix}^T \begin{bmatrix}S\end{bmatrix} \begin{bmatrix}W\end{bmatrix}~~
	~~\begin{bmatrix}\dot\xi\end{bmatrix}~~
	~~+\begin{bmatrix}W\end{bmatrix}^T \begin{bmatrix}M\end{bmatrix} \begin{bmatrix}W\end{bmatrix}~~
	~~\begin{bmatrix}\ddot\xi\end{bmatrix}</math>=<math>\begin{bmatrix}W\end{bmatrix}^T\begin{bmatrix}F(t)\end{bmatrix}</math></center>~~


	<center><math>\begin{bmatrix}k\end{bmatrix} \begin{bmatrix}\xi\end{bmatrix}+\begin{bmatrix}s\end{bmatrix} \begin{bmatrix}\dot\xi\end{bmatrix}+\begin{bmatrix}m\end{bmatrix} \begin{bmatrix}\ddot\xi\end{bmatrix}</math>=<math>\begin{bmatrix}f(t)\end{bmatrix}</math></center>


	~~<center><math>\begin{bmatrix}k\end{bmatrix}=\begin{bmatrix}W^T\end{bmatrix}\begin{bmatrix}K\end{bmatrix}\begin{bmatrix}W\end{bmatrix}</math>is the modal stiffness matrix,~~
	~~<math>\begin{bmatrix}m\end{bmatrix}=\begin{bmatrix}W^T\end{bmatrix}\begin{bmatrix}M\end{bmatrix}\begin{bmatrix}W\end{bmatrix}</math>is the modal mass matrix.</center>~~


	~~Hydroelastic analysis of the general 3D structure is thus preformed using the modal superposition method.~~


	~~Let us assume time-harmonic motion. Then the following is valid:~~

	~~<center><math>\begin{bmatrix}\xi(t)\end{bmatrix}=\begin{bmatrix}\tilde{\xi}(\omega)\end{bmatrix}\cdot e^{i \omega t},~~
	~~\; \begin{bmatrix}f(t)\end{bmatrix}=\begin{bmatrix}\tilde{f}(\omega)\end{bmatrix}\cdot e^{i \omega t}</math></center>~~


	<center><math>\left ( \begin{bmatrix}k\end{bmatrix}+i\omega\begin{bmatrix}s\end{bmatrix}-\omega^2\begin{bmatrix}m\end{bmatrix} \right ) \begin{bmatrix}\tilde{\xi}\end{bmatrix}=\begin{bmatrix}\tilde{f}\end{bmatrix}</math></center>


	~~Modal hydrodynamic forces are calculated by pressure work integration over the wetted surface:~~


	~~<center><math>\tilde{f}^{hd}_i(t)=-i\omega\rho\iint_{S}\tilde{\phi}\,\mathbf{h_i}\mathbf{n}\,\mbox{d}S</math></center>~~


	~~Total velocity potential can be decomposed as:~~


	~~<center><math>\tilde{\phi}=\tilde{\phi}^I+\tilde{\phi}^D-i\omega\sum_{j=1}^N\tilde{\xi}_j\,\tilde{\phi}_j^R</math></center>~~


	~~to be continued............~~

Meylan: /* Fluid Structure Equations */

2010-04-28T09:28:17Z

Fluid Structure Equations

← Older revision		Revision as of 09:28, 28 April 2010
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	We substitute the expansion for the potential into the equations in the frequency domain and we obtain		We substitute the expansion for the potential into the equations in the frequency domain and we obtain
	<center><math>		<center><math>
	\sum_{\nu} M_{\mu\nu}\zeta_{\nu}		\sum_{\nu} K_{\mu\nu}\zeta_{\nu}
	-\mathrm{i}\omega\sum_{\nu} M_{\mu\nu}\zeta_{\nu}		-\mathrm{i}\omega\sum_{\nu} S_{\mu\nu}\zeta_{\nu}
	-\omega^2 \sum_{\nu} M_{\mu\nu}\zeta_{\nu}=-\mathrm{i}\omega\rho\iint_{\partial\Omega_{B}}		-\omega^2 \sum_{\nu} M_{\mu\nu}\zeta_{\nu}=-\mathrm{i}\omega\rho\iint_{\partial\Omega_{B}}
	\left(\phi^{\mathrm{I}} + \phi^{\mathrm{D}} +		\left(\phi^{\mathrm{I}} + \phi^{\mathrm{D}} +

Meylan: /* Equations for the Fluid */

2010-04-28T09:27:33Z

Equations for the Fluid

@@ Line 56: / Line 56: @@
 {{standard linear wave scattering equations without body condition}}
 {{frequency domain equations for the radiation modes}}
 <center><math>\begin{bmatrix}W\end{bmatrix}</math>=<math>\begin{bmatrix}\mathbf{w_1}\,\mathbf{w_2}\,\ldots \end{bmatrix}</math> is matrix of dry natural modes, with modes being sorted column-wise,</center>

Meylan: /* Frequency Domain Problem */

2010-04-28T09:23:37Z

Frequency Domain Problem

← Older revision		Revision as of 09:23, 28 April 2010
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	We expand the displacement in <math>N</math> modes,		We expand the displacement in <math>N</math> modes,
	<center><math>		<center><math>
	w = \sum_{nu} \xi_n \mathbf{w}_{\nu}		w = \sum_{\nu} \xi_n \mathbf{w}_{\nu}
	</math></center>		</math></center>
	Associated with each of these modes is a normal motion on the body surface given by		Associated with each of these modes is a normal motion on the body surface given by

Meylan: /* Equations for the Fluid */

2010-04-28T09:22:07Z

Equations for the Fluid

← Older revision		Revision as of 09:22, 28 April 2010
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	== Equations for the Fluid ==		== Equations for the Fluid ==

			The equation for the fluid are as follows
	{{standard linear wave scattering equations without body condition}}		{{standard linear wave scattering equations without body condition}}
	{{frequency domain equations for the radiation modes}}		{{frequency domain equations for the radiation modes}}

Meylan: /* Frequency Domain Problem */

2010-04-28T09:21:34Z

Frequency Domain Problem

← Older revision		Revision as of 09:21, 28 April 2010
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	We expand the displacement in <math>N</math> modes,		We expand the displacement in <math>N</math> modes,
	<center><math> w = \sum_{n} \~~xi_n~~		<center><math>
			w = \sum_{nu} \xi_n \mathbf{w}_{\nu}
			</math></center>
			Associated with each of these modes is a normal motion on the body surface given by
			<math>\mathbf{n}_{\nu}</math>.

	~~</math></center>~~		== Equations for the Fluid ==

			{{standard linear wave scattering equations without body condition}}
			{{frequency domain equations for the radiation modes}}

	<center><math>\begin{bmatrix}W\end{bmatrix}</math>=<math>\begin{bmatrix}\mathbf{w_1}\,\mathbf{w_2}\,\ldots \end{bmatrix}</math> is matrix of dry natural modes, with modes being sorted column-wise,</center>		<center><math>\begin{bmatrix}W\end{bmatrix}</math>=<math>\begin{bmatrix}\mathbf{w_1}\,\mathbf{w_2}\,\ldots \end{bmatrix}</math> is matrix of dry natural modes, with modes being sorted column-wise,</center>

Meylan: /* Frequency Domain Problem */

2010-04-28T09:16:19Z

Frequency Domain Problem

← Older revision		Revision as of 09:16, 28 April 2010
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	</math></center>		</math></center>
	As a solution of the eigenvalue problem for each natural mode one obtains <math>\omega_n</math>, the n-th dry natural frequency and <math>\begin{bmatrix}w_n\end{bmatrix}</math>, the corresponding dry natural mode. Note that only the first modes		As a solution of the eigenvalue problem for each natural mode one obtains <math>\omega_n</math>, the n-th dry natural frequency and <math>\begin{bmatrix}w_n\end{bmatrix}</math>, the corresponding dry natural mode. Note that only the first modes
	are accurate approximations .		are accurate approximations.

			We expand the displacement in <math>N</math> modes,
			<center><math> w = \sum_{n} \xi_n

	~~Generalized nodal displacements vector can be expressed using calculated "dry" structure natural modes. This~~		</math></center>
	~~greatly reduces the dimension of the system since there are far fewer natural modes than there are generalized nodal~~
	~~displacements.~~


	~~<center><math>\begin{bmatrix}D\end{bmatrix}</math>=<math>\begin{bmatrix}W\end{bmatrix}\cdot\begin{bmatrix}\xi\end{bmatrix}~~</math></center>