Derivative Seakeeping Quantities - Revision history

Meylan at 09:12, 16 October 2009

2009-10-16T09:12:36Z

← Older revision		Revision as of 09:12, 16 October 2009
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	{{Ocean Wave Interaction with Ships and Offshore Structures		{{Ocean Wave Interaction with Ships and Offshore Structures
	\| chapter title = Derivative Seakeeping Quantities		\| chapter title = Derivative Seakeeping Quantities
	\| next chapter = [[]]		\| next chapter = [[Seakeeping In Random Waves]]
	\| previous chapter = [[Ship Roll-Sway-Yaw Motions]]		\| previous chapter = [[Ship Roll-Sway-Yaw Motions]]
	}}		}}

Meylan at 09:12, 16 October 2009

2009-10-16T09:12:17Z

← Older revision		Revision as of 09:12, 16 October 2009
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			{{Ocean Wave Interaction with Ships and Offshore Structures
			\| chapter title = Derivative Seakeeping Quantities
			\| next chapter = [[]]
			\| previous chapter = [[Ship Roll-Sway-Yaw Motions]]
			}}

			{{incomplete pages}}


	The principal seakeeping quantity from a seakeeping analysis of a floating body at zero or forward speed is the Response Amplitude Operator (or RAO)		The principal seakeeping quantity from a seakeeping analysis of a floating body at zero or forward speed is the Response Amplitude Operator (or RAO)

Adi Kurniawan at 04:03, 27 May 2009

2009-05-27T04:03:11Z

Adi Kurniawan at 03:17, 23 September 2008

2008-09-23T03:17:28Z

← Older revision		Revision as of 03:17, 23 September 2008
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	<center><math> Z(t) = \mathbb{R}\mathbf{e} \left\{ \mathbb{Z}(\omega) e^{i\omega t} \right\}, \quad RAO=\frac{\mathbb{Z}(\omega)}{A} </math></center>		<center><math> Z(t) = \mathbb{R}\mathbf{e} \left\{ \mathbb{Z}(\omega) e^{i\omega t} \right\}, \quad RAO=\frac{\mathbb{Z}(\omega)}{A} </math></center>

	==Example 1==		==Example 1 - Acceleration RAO at the bow of a ship==

	* Acceleration RAO at the bow of a ship

	The vertical displacement of point <math>A\,</math> due to the vessel heave & pitch motions is		The vertical displacement of point <math>A\,</math> due to the vessel heave & pitch motions is
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	So the RAO of the vertical acceleration at the bow is a linear combination of the heave and pitch RAO's.		So the RAO of the vertical acceleration at the bow is a linear combination of the heave and pitch RAO's.

	==Example 2==		==Example 2 - Hydrodynamic pressure disturbance at a fixed point on a ship hull oscillating in heave & pitch in waves==

	* Hydrodynamic pressure disturbance at a fixed point on a ship hull oscillating in heave & pitch in waves

	The linear hydrodynamic pressures at a point <math>A\,</math> located at <math> \vec{X}_A\,</math> relative to the ship frame is:		The linear hydrodynamic pressures at a point <math>A\,</math> located at <math> \vec{X}_A\,</math> relative to the ship frame is:

Adi Kurniawan at 03:16, 23 September 2008

2008-09-23T03:16:38Z

← Older revision		Revision as of 03:16, 23 September 2008
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	~~<u>Derivative seakeeping quantities</u>~~

	The principal seakeeping quantity from a seakeeping analysis of a floating body at zero or forward speed is the Response Amplitude Operator (or RAO)		The principal seakeeping quantity from a seakeeping analysis of a floating body at zero or forward speed is the Response Amplitude Operator (or RAO)

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	<center><math> Z(t) = \mathbb{R}\mathbf{e} \left\{ \mathbb{Z}(\omega) e^{i\omega t} \right\}, \quad RAO=\frac{\mathbb{Z}(\omega)}{A} </math></center>		<center><math> Z(t) = \mathbb{R}\mathbf{e} \left\{ \mathbb{Z}(\omega) e^{i\omega t} \right\}, \quad RAO=\frac{\mathbb{Z}(\omega)}{A} </math></center>

	~~<u>~~Example 1~~</u>~~		==Example 1==

	* Acceleration RAO at the bow of a ship		* Acceleration RAO at the bow of a ship
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	So the RAO of the vertical acceleration at the bow is a linear combination of the heave and pitch RAO's.		So the RAO of the vertical acceleration at the bow is a linear combination of the heave and pitch RAO's.

	~~<u>~~Example 2~~</u>~~		==Example 2==

	* Hydrodynamic pressure disturbance at a fixed point on a ship hull oscillating in heave & pitch in waves		* Hydrodynamic pressure disturbance at a fixed point on a ship hull oscillating in heave & pitch in waves

Syan077 at 08:03, 13 March 2007

2007-03-13T08:03:14Z

← Older revision		Revision as of 08:03, 13 March 2007
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	where		where

	<center><math> \mathbb{P}_A = - \rho \left\{ \left( i\omega - U \frac{\partial}{\partial X} \right) \left( \phi_3 + \phi_5 \right) + ~~\right. \,</math></center>~~		<center><math> \mathbb{P}_A = - \rho \left\{ \left( i\omega - U \frac{\partial}{\partial X} \right) \left( \phi_3 + \phi_5 \right) + \left( i\omega - U \frac{\partial}{\partial x} \right) \left( \phi_I + \phi_D \right) + g \left( \Pi_3 - X \Pi_5 \right) \right\} _{\vec{X}_A} \, </math></center>
	~~<center><math> \left.~~ \left( i\omega - U \frac{\partial}{\partial x} \right) \left( \phi_I + \phi_D \right) + g \left( \Pi_3 - X \Pi_5 \right) \right\} _{\vec{X}_A} \, </math></center>

	<center><math> RAO = \frac{\mathbb{P}_A}{A} \, </math></center>.		<center><math> RAO = \frac{\mathbb{P}_A}{A} \, </math></center>.

Syan077 at 08:02, 13 March 2007

2007-03-13T08:02:20Z

← Older revision		Revision as of 08:02, 13 March 2007
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	where		where

	<center><math> \mathbb{P}_A = - \rho \left\{ \left( i\omega - U \frac{\partial}{\partial X} \right) \left( \phi_3 + \phi_5 \right) + \,</math></center>		<center><math> \mathbb{P}_A = - \rho \left\{ \left( i\omega - U \frac{\partial}{\partial X} \right) \left( \phi_3 + \phi_5 \right) + \right. \,</math></center>
	<center><math> \left( i\omega - U \frac{\partial}{\partial x} \right) \left( \phi_I + \phi_D \right) + g \left( \Pi_3 - X \Pi_5 \right) \right\} _{\vec{X}_A} \, </math></center>		<center><math> \left. \left( i\omega - U \frac{\partial}{\partial x} \right) \left( \phi_I + \phi_D \right) + g \left( \Pi_3 - X \Pi_5 \right) \right\} _{\vec{X}_A} \, </math></center>

	<center><math> RAO = \frac{\mathbb{P}_A}{A} \, </math></center>.		<center><math> RAO = \frac{\mathbb{P}_A}{A} \, </math></center>.

Syan077 at 08:00, 13 March 2007

2007-03-13T08:00:46Z

New page

Derivative seakeeping quantities

The principal seakeeping quantity from a seakeeping analysis of a floating body at zero or forward speed is the Response Amplitude Operator (or RAO)

<center><math> \xi_j(t) = \mathbb{R}\mathbf{e} \left\{ \Pi_j (\omega) e^{i\omega t} \right\} \, </math></center>

<center><math> RAO = \frac{\Pi_j(\omega)}{A}, \quad j=1,2,3 \, </math></center>

<center><math> = \frac{\Pi_j(\omega)}{A/L}, \quad j=4,5,6 \, </math></center>

where <math>L\,</math> is a characteristic length. The RAO is a complex quantity with phase defined relative to the ambient wave elevation at the origin of the coordinate system

<center><math> \zeta_I = \mathbb{R}\mathbf{e} \left\{ A e^{i\omega t} \right\} \, </math></center>

It follows that the only seakeeping quantity with <math>RAO\equiv 1 \,</math> is <math> \zeta_I(t)\,</math>.

A partial list of derivative seakeeping quantities of interest in practice is:

* Free-surface elevation. Needed to estimate the clearance under the deck of offshore platforms.

* Vessel kinematics at specified points, e.g. needed to estimate the motion properties of containerized cargo.

* Relative wave elevation and velocity near the bow of a ship. Needed to estimate the occurrence and severity of slamming.

* Local and global structural loads needed for the vessel structural design.

According to linear theory, all derivative quantities which are linear superpositions of other quantities, take the form

<center><math> Z(t) = \mathbb{R}\mathbf{e} \left\{ \mathbb{Z}(\omega) e^{i\omega t} \right\}, \quad RAO=\frac{\mathbb{Z}(\omega)}{A} </math></center>

Example 1

* Acceleration RAO at the bow of a ship

The vertical displacement of point <math>A\,</math> due to the vessel heave & pitch motions is

<center><math> \xi_A (t) = \xi_3 (t) - X_A \xi_5 (t) \, </math></center>

<center><math> \frac{d^2\xi_A(t)}{dt^2} = \ddot{\xi}_3(t) - X_A \ddot{\xi}_5(t) = \mathbb{R}\mathbf{e} \left\{ -\omega^2 \left[ \Pi_3 - X_A \Pi_5 \right] e^{i\omega t} \right\} </math></center>

So the corresponding RAO in waves of amplitude <math>A\,</math> is:

<center><math> RAO = \frac{-\omega^2 \left(\Pi_3 -X_A \Pi_5 \right)}{A} = -\omega^2 \left(RAO_3 - X_A RAO_5 \right) </math></center>

So the RAO of the vertical acceleration at the bow is a linear combination of the heave and pitch RAO's.

Example 2

* Hydrodynamic pressure disturbance at a fixed point on a ship hull oscillating in heave & pitch in waves

The linear hydrodynamic pressures at a point <math>A\,</math> located at <math> \vec{X}_A\,</math> relative to the ship frame is:

<center><math> P_A = \mathbb{R}\mathbf{e} \left\{ \mathbb{P}_A e^{i\omega t} \right\} \, </math></center>

where

<center><math> \mathbb{P}_A = - \rho \left\{ \left( i\omega - U \frac{\partial}{\partial X} \right) \left( \phi_3 + \phi_5 \right) + \,</math></center>
<center><math> \left( i\omega - U \frac{\partial}{\partial x} \right) \left( \phi_I + \phi_D \right) + g \left( \Pi_3 - X \Pi_5 \right) \right\} _{\vec{X}_A} \, </math></center>

<center><math> RAO = \frac{\mathbb{P}_A}{A} \, </math></center>.

[[Ocean Wave Interaction with Ships and Offshore Energy Systems]]

← Older revision		Revision as of 04:03, 27 May 2009
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	The principal seakeeping quantity from a seakeeping analysis of a floating body at zero or forward speed is the Response Amplitude Operator (or RAO)		The principal seakeeping quantity from a seakeeping analysis of a floating body at zero or forward speed is the Response Amplitude Operator (or RAO)

	<center><math> \xi_j(t) = \~~mathbb~~{~~R}\mathbf{e~~} \left\{ \Pi_j (\omega) e^{i\omega t} \right\} \, </math></center>		<center><math> \xi_j(t) = \mathrm{Re} \left\{ \Pi_j (\omega) e^{i\omega t} \right\} \, </math></center>

	<center><math> RAO = \frac{\Pi_j(\omega)}{A}, \quad j=1,2,3 \, </math></center>		<center><math> RAO = \frac{\Pi_j(\omega)}{A}, \quad j=1,2,3 \, </math></center>
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	where <math>L\,</math> is a characteristic length. The RAO is a complex quantity with phase defined relative to the ambient wave elevation at the origin of the coordinate system		where <math>L\,</math> is a characteristic length. The RAO is a complex quantity with phase defined relative to the ambient wave elevation at the origin of the coordinate system

	<center><math> \zeta_I = \~~mathbb~~{~~R}\mathbf{e~~} \left\{ A e^{i\omega t} \right\} \, </math></center>		<center><math> \zeta_I = \mathrm{Re} \left\{ A e^{i\omega t} \right\} \, </math></center>

	It follows that the only seakeeping quantity with <math>RAO\equiv 1 \,</math> is <math> \zeta_I(t)\,</math>.		It follows that the only seakeeping quantity with <math>RAO\equiv 1 \,</math> is <math> \zeta_I(t)\,</math>.
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	According to linear theory, all derivative quantities which are linear superpositions of other quantities, take the form		According to linear theory, all derivative quantities which are linear superpositions of other quantities, take the form

	<center><math> Z(t) = \~~mathbb~~{~~R}\mathbf{e~~} \left\{ \mathbb{Z}(\omega) e^{i\omega t} \right\}, \quad RAO=\frac{\mathbb{Z}(\omega)}{A} </math></center>		<center><math> Z(t) = \mathrm{Re} \left\{ \mathbb{Z}(\omega) e^{i\omega t} \right\}, \quad RAO=\frac{\mathbb{Z}(\omega)}{A} </math></center>

	==Example 1 - Acceleration RAO at the bow of a ship==		==Example 1 - Acceleration RAO at the bow of a ship==
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	<center><math> \xi_A (t) = \xi_3 (t) - X_A \xi_5 (t) \, </math></center>		<center><math> \xi_A (t) = \xi_3 (t) - X_A \xi_5 (t) \, </math></center>

	<center><math> \frac{d^2\xi_A(t)}{dt^2} = \ddot{\xi}_3(t) - X_A \ddot{\xi}_5(t) = \~~mathbb~~{~~R}\mathbf{e~~} \left\{ -\omega^2 \left[ \Pi_3 - X_A \Pi_5 \right] e^{i\omega t} \right\} </math></center>		<center><math> \frac{d^2\xi_A(t)}{dt^2} = \ddot{\xi}_3(t) - X_A \ddot{\xi}_5(t) = \mathrm{Re} \left\{ -\omega^2 \left[ \Pi_3 - X_A \Pi_5 \right] e^{i\omega t} \right\} </math></center>

	So the corresponding RAO in waves of amplitude <math>A\,</math> is:		So the corresponding RAO in waves of amplitude <math>A\,</math> is:
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	The linear hydrodynamic pressures at a point <math>A\,</math> located at <math> \vec{X}_A\,</math> relative to the ship frame is:		The linear hydrodynamic pressures at a point <math>A\,</math> located at <math> \vec{X}_A\,</math> relative to the ship frame is:

	<center><math> P_A = \~~mathbb~~{~~R}\mathbf{e~~} \left\{ \mathbb{P}_A e^{i\omega t} \right\} \, </math></center>		<center><math> P_A = \mathrm{Re} \left\{ \mathbb{P}_A e^{i\omega t} \right\} \, </math></center>

	where		where