Linear Wave-Body Interaction - Revision history

Meylan: /* Final System of Equations */

2011-04-26T12:58:18Z

Final System of Equations

← Older revision		Revision as of 12:58, 26 April 2011
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	Then the equations can be expressed as follows.		Then the equations can be expressed as follows.
	<center><math> \left[-\omega^2 \left(\mathbf{M} + \mathbf{A} \right) +		<center><math> \left[-\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{M}</math> is the mass matrix, <math>\mathbf{A}</math> is the added mass matrix,

Meylan: /* Final System of Equations */

2011-04-26T12:51:49Z

Final System of Equations

← Older revision		Revision as of 12:51, 26 April 2011
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	<center><math>		<center><math>
	-\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}} - \mathrm{i}\omega
	\sum_{\nu} \zeta_\nu \phi_{\nu}^{\mathrm{R}}\right) \mathbf{n}_{\mu}\, dS		\sum_{\nu} \zeta_\nu \phi_{\nu}^{\mathrm{R}}\right) \mathbf{n}_{\mu}\, dS
	- \sum_{\nu} C_{\mu\nu}\xi_{\nu},\quad \textrm{for} \qquad \mu=1,3,5,		- \sum_{\nu} C_{\mu\nu}\xi_{\nu},\quad \textrm{for} \qquad \mu=1,3,5,

Meylan: /* Equations for the Radiation Potential in Frequency Domain */

2011-04-26T12:51:05Z

Equations for the Radiation Potential in Frequency Domain

← Older revision		Revision as of 12:51, 26 April 2011
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	<math>\phi_\nu^{\mathrm{R}}</math> and the total potential is written as		<math>\phi_\nu^{\mathrm{R}}</math> and the total potential is written as
	<center><math>		<center><math>
	\phi = \phi^{\mathrm{I}} + \phi^{\mathrm{D}} +		\phi = \phi^{\mathrm{I}} + \phi^{\mathrm{D}} - \mathrm{i} \omega
	\sum_\nu \zeta_\nu \phi_\nu^{\mathrm{R}}		\sum_\nu \zeta_\nu \phi_\nu^{\mathrm{R}}
	</math>		</math>

Meylan: /* Final System of Equations */

2011-04-26T12:18:26Z

Final System of Equations

← Older revision		Revision as of 12:18, 26 April 2011
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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>
	-\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}} +
	\sum_{\nu} \zeta_\nu \phi_{\nu}^{\mathrm{R}}\right) \mathbf{n}_{\mu}\, dS		\sum_{\nu} \zeta_\nu \phi_{\nu}^{\mathrm{R}}\right) \mathbf{n}_{\mu}\, dS

Adi Kurniawan: /* Haskind relations of exciting forces */

2010-11-06T12:11:41Z

Haskind relations of exciting forces

← Older revision		Revision as of 12:11, 6 November 2010
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	Body-boundary conditions:		Body-boundary conditions:

	<center><math> \frac{\partial\phi_7}{\partial n} = -\frac{\partial\phi_I}{\partial n}, \quad \mbox{on} \quad S_B ~~</math></center>~~		<center><math>
			\begin{align}
	~~<center><math>~~ \phi_I = \frac{i g A}{\omega} e^{Kz-iKx\cos\beta-iKy\sin\beta+i\omega t} ~~</math></center>~~		\frac{\partial\phi_7}{\partial n} &= -\frac{\partial\phi_I}{\partial n}, \quad \mbox{on} \quad S_B \\
			\phi_I &= \frac{i g A}{\omega} e^{Kz-iKx\cos\beta-iKy\sin\beta+i\omega t} \\
	~~<center><math>~~ \frac{\partial\phi_j}{\partial n} = i\omega n_j, \quad j=1,\cdots,6 \, ~~</math></center>~~		\frac{\partial\phi_j}{\partial n} &= i\omega n_j, \quad j=1,\cdots,6 \, \\
			n_j &= \begin{cases}
	~~<center><math>~~ n_j = \begin{cases}
	n_j, & j=1,2,3 \\		n_j, & j=1,2,3 \\
	\left( \vec{x} \times \vec{n} \right), & j=4,5,6		\left( \vec{x} \times \vec{n} \right), & j=4,5,6
	\end{cases} </math></center>		\end{cases}
			\end{align}
			</math></center>

	<center><math> j=1: \ \mbox{Surge} \qquad j=2: \ \mbox{Sway} \qquad j=3: \ \mbox{Heave} </math></center>		<center><math> j=1: \ \mbox{Surge} \qquad j=2: \ \mbox{Sway} \qquad j=3: \ \mbox{Heave} </math></center>

Adi Kurniawan: /* Haskind relations of exciting forces */

2010-11-06T12:06:11Z

Haskind relations of exciting forces

← Older revision		Revision as of 12:06, 6 November 2010
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	== Haskind relations of exciting forces ==		== Haskind relations of exciting forces ==

	<center><math> \mathbf{X}_i(\omega) = - i\omega\rho\iint_{S_B} (\phi_I + \phi_7) n_i \mathrm{d}S ~~</math></center>~~		<center><math>
			\begin{align}
	~~<center><math>~~ = - \rho \iint_{S_B} (\phi_I + \phi_7) \frac{\partial \phi_i}{\partial n} \mathrm{d}S </math></center>		\mathbf{X}_i(\omega) &= - i\omega\rho\iint_{S_B} (\phi_I + \phi_7) n_i \mathrm{d}S \\
			&= - \rho \iint_{S_B} (\phi_I + \phi_7) \frac{\partial \phi_i}{\partial n} \mathrm{d}S
			\end{align}
			</math></center>

	where the radiation velocity potential <math> \phi_i \,</math> is known to satisfy:		where the radiation velocity potential <math> \phi_i \,</math> is known to satisfy:
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	where		where

	<center><math> M_{ij}~~(\omega)~~: \mbox{Body inertia matrix including moments of inertia for rotational modes. For details refer to MH} \, </math></center>		<center><math> M_{ij}: \mbox{Body inertia matrix including moments of inertia for rotational modes. For details refer to MH} \, </math></center>

	<center><math> A_{ij}(\omega): \mbox{Added mass matrix} \,</math></center>		<center><math> A_{ij}(\omega): \mbox{Added mass matrix} \,</math></center>
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	<center><math> \frac{\partial\phi_7}{\partial n} = -\frac{\partial\phi_I}{\partial n}, \quad \mbox{on} \quad S_B </math></center>		<center><math> \frac{\partial\phi_7}{\partial n} = -\frac{\partial\phi_I}{\partial n}, \quad \mbox{on} \quad S_B </math></center>

	<center><math> \phi_I = \frac{i g A}{\omega} e^{KZ-~~iKX~~\cos\beta-~~iKY~~\sin\beta+i\omega t} </math></center>		<center><math> \phi_I = \frac{i g A}{\omega} e^{Kz-iKx\cos\beta-iKy\sin\beta+i\omega t} </math></center>

	<center><math> \frac{\partial\phi_j}{\partial n} = i\omega n_j, \quad j=1,\cdots,6 \, </math></center>		<center><math> \frac{\partial\phi_j}{\partial n} = i\omega n_j, \quad j=1,\cdots,6 \, </math></center>

	<center><math> n_j = \begin{~~Bmatrix~~}		<center><math> n_j = \begin{cases}
	& n_j, ~~\qquad~~ & j=1,2,3 \\		n_j, & j=1,2,3 \\
	& \left( \vec{X} \times \vec{n} \right)~~_{j+3}~~, ~~\quad~~ j=4,5,6		\left( \vec{x} \times \vec{n} \right), & j=4,5,6
	\end{~~Bmatrix~~} </math></center>		\end{cases} </math></center>

	<center><math> i=1: \ \mbox{Surge} \qquad i=2: \ \mbox{Sway} \qquad i=3: \ \mbox{Heave} </math></center>		<center><math> j=1: \ \mbox{Surge} \qquad j=2: \ \mbox{Sway} \qquad j=3: \ \mbox{Heave} </math></center>
	<center><math> i=4: \ \mbox{Roll} \qquad i=5: \ \mbox{Pitch} \qquad i=6: \ \mbox{Yaw} </math></center>		<center><math> j=4: \ \mbox{Roll} \qquad j=5: \ \mbox{Pitch} \qquad j=6: \ \mbox{Yaw} </math></center>

	At large distances from the body the velocity potentials satisfy the radiation condition:		At large distances from the body the velocity potentials satisfy the radiation condition:

	<center><math> \phi_j (R,\theta) \sim \frac{A_j(\theta)}{\sqrt{R}} e^{KZ-iKR} + O \left( \frac{1}{R^{3/2}} \right) </math></center>		<center><math> \phi_j (R,\theta) \sim \frac{A_j(\theta)}{\sqrt{R}} e^{Kz-iKR} + O \left( \frac{1}{R^{3/2}} \right) </math></center>

	with		with

Adi Kurniawan: /* Equations for a Fixed Body in Frequency Domain */

2010-11-06T11:41:59Z

Equations for a Fixed Body in Frequency Domain

← Older revision		Revision as of 11:41, 6 November 2010
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	We decompose the potential as		We decompose the potential as
	~~<center>~~
	<math>		<math>
	\phi = \phi^{\mathrm{I}} + \phi^{\mathrm{D}}		\phi = \phi^{\mathrm{I}} + \phi^{\mathrm{D}} \,,
	</math>		</math>
	~~</center>~~
	where <math>\phi^{\mathrm{I}}</math> is the incident potential and <math>\phi^{\mathrm{D}}</math>		where <math>\phi^{\mathrm{I}}</math> is the incident potential and <math>\phi^{\mathrm{D}}</math>
	is the diffracted potential. The boundary condition for the diffracted potential is		is the diffracted potential. The boundary condition for the diffracted potential is
	<center><math>		<center><math>
	\Delta\phi^{\mathrm{D}}=0, \, -h<z<0,\,\,\,\mathbf{x} \in \Omega		\begin{align}
	~~</math></center>~~		\Delta\phi^{\mathrm{D}}&=0, &-h<z<0,\,\,\mathbf{x} \in \Omega \\
	~~<center><math>~~		\partial_n\phi^{\mathrm{D}} &= 0, &z=-h, \\
	\partial_n\phi^{\mathrm{D}} = 0, \, z=-h,		\partial_n \phi^{\mathrm{D}} &= \alpha \phi, &z=0,\,\,\mathbf{x} \in \partial\Omega_{F},
	~~</math></center>~~		\end{align}
	~~<center><math>~~
	\partial_n \phi^{\mathrm{D}} = \alpha \phi,\,z=0,\,\,\mathbf{x} \in \partial\Omega_{F},
	</math></center>		</math></center>
	plus		plus

Adi Kurniawan: /* Haskind relations of exciting forces */

2010-11-06T11:39:02Z

Haskind relations of exciting forces

← Older revision		Revision as of 11:39, 6 November 2010
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	<center><math> \sum_{j=1}^6 \left[ - \omega^2 \left( M_{ij} + A_{ij} \right) + i \omega B_{ij} + C_{ij} \right] \Pi_j = \mathbf{X}_j, \quad i=1,\cdots,6 </math></center>		<center><math> \sum_{j=1}^6 \left[ - \omega^2 \left( M_{ij} + A_{ij} \right) + i \omega B_{ij} + C_{ij} \right] \Pi_j = \mathbf{X}_j, \quad i=1,\cdots,6 </math></center>

	~~Where:~~		where

	<center><math> \mbox{Body inertia matrix including moments of inertia for rotational modes. For details refer to MH} \, </math></center>		<center><math> M_{ij}(\omega): \mbox{Body inertia matrix including moments of inertia for rotational modes. For details refer to MH} \, </math></center>

	<center><math> A_{ij}(\omega): \mbox{Added mass matrix} \,</math></center>		<center><math> A_{ij}(\omega): \mbox{Added mass matrix} \,</math></center>

Adi Kurniawan: /* Haskind relations of exciting forces */

2010-11-05T15:10:30Z

Haskind relations of exciting forces

← Older revision		Revision as of 15:10, 5 November 2010
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	These take the form:		These take the form:

	<u>2D</u>: <math> ~~B_ii~~ = \frac{\left\| \mathbf{X}_i \right\|^2}{2\rho g V_g}, \quad V_g = \frac{g}{2\omega}, </math> Deep water		<u>2D</u>: <math> B_{ii} = \frac{\left\| \mathbf{X}_i \right\|^2}{2\rho g V_g}, \quad V_g = \frac{g}{2\omega}, </math> Deep water

	<u>3D</u>: <math> ~~B_33~~ = \frac{K}{4\rho g V_g} \left\| \mathbf{X}_3 \right\|^2 \,</math> --- Heave		<u>3D</u>: <math> B_{33} = \frac{K}{4\rho g V_g} \left\| \mathbf{X}_3 \right\|^2 \,</math> --- Heave

	(Axisymmetric bodies) <math> ~~B_22~~ = \frac{K}{8\rho g V_g} \left\| \mathbf{X}_2 \right\|^2 \,</math> --- Sway		(Axisymmetric bodies) <math> B_{22} = \frac{K}{8\rho g V_g} \left\| \mathbf{X}_2 \right\|^2 \,</math> --- Sway

	So knowledge of <math> \mathbf{X}_i(\omega)\,</math> allows the direct evaluation of the diagonal damping coefficients. These expressions are useful in deriving theoretical results in wave-body interactions to be discussed later.		So knowledge of <math> \mathbf{X}_i(\omega)\,</math> allows the direct evaluation of the diagonal damping coefficients. These expressions are useful in deriving theoretical results in wave-body interactions to be discussed later.

Adi Kurniawan: /* Haskind relations of exciting forces */

2010-11-05T15:09:48Z

Haskind relations of exciting forces

← Older revision		Revision as of 15:09, 5 November 2010
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	<center><math> \mathbf{X}_i(\omega) = \rho \iint_{S_B} \left[ \phi_I \frac{\partial\phi_i}{\partial n} - \phi_i \frac{\partial\phi_I}{\partial n} \right] \mathrm{d}S </math></center>		<center><math> \mathbf{X}_i(\omega) = \rho \iint_{S_B} \left[ \phi_I \frac{\partial\phi_i}{\partial n} - \phi_i \frac{\partial\phi_I}{\partial n} \right] \mathrm{d}S </math></center>

	The symmetry of the <math> ~~A_ij~~(\omega), ~~B_ij~~(\omega) \,</math> matrices applies in 2D and 3D. The application of Green's Theorem in 3D is very similar using the far-field representation for the potential <math> \phi_j\,</math>		The symmetry of the <math> A_{ij}(\omega), B_{ij}(\omega) \,</math> matrices applies in 2D and 3D. The application of Green's Theorem in 3D is very similar using the far-field representation for the potential <math> \phi_j\,</math>

	<center><math> \phi_j \sim \frac{A_j(\theta)}{\sqrt{R}} e^{KZ-iKR} + O\left(\frac{1}{R^{3/2}}\right) </math></center>		<center><math> \phi_j \sim \frac{A_j(\theta)}{\sqrt{R}} e^{KZ-iKR} + O\left(\frac{1}{R^{3/2}}\right) </math></center>