Seakeeping In Random Waves - Revision history

Adi Kurniawan at 15:26, 6 August 2010

2010-08-06T15:26:38Z

← Older revision		Revision as of 15:26, 6 August 2010
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	{{incomplete pages}}		{{incomplete pages}}

	~~<u>Seakeeping in random waves</u>~~		Assume known the ambient wave spectral density <math>S_{\zeta}(\omega_0)\,</math> assumed unidirectional for simplicity

	* Assume known the ambient wave spectral density <math>S_{\zeta}(\omega_0)\,</math> assumed unidirectional for simplicity

	<center><math> \frac{1}{2}A_i^2 = S(\omega_i)\Delta\omega \, </math></center>		<center><math> \frac{1}{2}A_i^2 = S(\omega_i)\Delta\omega \, </math></center>

Meylan at 23:34, 16 October 2009

2009-10-16T23:34:44Z

← Older revision		Revision as of 23:34, 16 October 2009
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			{{Ocean Wave Interaction with Ships and Offshore Structures
			\| chapter title = Seakeeping In Random Waves
			\| next chapter = [[Solution of Wave-Body Flows, Green's Theorem]]
			\| previous chapter = [[Derivative Seakeeping Quantities]]
			}}

			{{incomplete pages}}

	<u>Seakeeping in random waves</u>		<u>Seakeeping in random waves</u>

Adi Kurniawan at 07:20, 24 February 2009

2009-02-24T07:20:39Z

← Older revision		Revision as of 07:20, 24 February 2009
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	<center><math> \frac{1}{2}A_i^2 = S(\omega_i)\Delta\omega \, </math></center>		<center><math> \frac{1}{2}A_i^2 = S(\omega_i)\Delta\omega \, </math></center>

	* <math> \int_0^\infty S_\zeta(\omega) d\omega = \sigma_\zeta^2 \equiv \,</math> Variance of the wave elevation of ambient random seastate, assumed Gaussian with zero mean		* <math> \int_0^\infty S_\zeta(\omega) \mathrm{d}\omega = \sigma_\zeta^2 \equiv \,</math> Variance of the wave elevation of ambient random seastate, assumed Gaussian with zero mean

	* Assuming that the <math>RAO(\omega)\,</math> of a seakeeping quantity <math> X(t) \,</math> has been determined from a frequency domain analysis;		* Assuming that the <math>RAO(\omega)\,</math> of a seakeeping quantity <math> X(t) \,</math> has been determined from a frequency domain analysis;
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	The standard deviation of heave follows by simple integration over <math> \omega_0\,</math>:		The standard deviation of heave follows by simple integration over <math> \omega_0\,</math>:

	<center><math> \sigma_3^2 = \int_0^\infty d\omega_0 S_\zeta \left(\omega_0\right) \left\|RAO_3^*\left(\omega_0\right)\right\|^2 </math></center>		<center><math> \sigma_3^2 = \int_0^\infty \mathrm{d}\omega_0 S_\zeta \left(\omega_0\right) \left\|RAO_3^*\left(\omega_0\right)\right\|^2 </math></center>

	* The opposite choice of parameterizing the above integral w.r.t. <math>\omega\,</math> ends up with a lot of unnecessary algebra.		* The opposite choice of parameterizing the above integral w.r.t. <math>\omega\,</math> ends up with a lot of unnecessary algebra.

Syan077 at 22:36, 14 March 2007

2007-03-14T22:36:18Z

← Older revision		Revision as of 22:36, 14 March 2007
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	<center><math> \left\|RAO_3\right\|(\omega) = \left\|RAO_3\right\| \left( \omega_0 - U \frac{\omega_0^2}{g} \cos\beta \right) \equiv \left\|RAO_3 \right\| (\omega_0) \,</math></center>		<center><math> \left\|RAO_3\right\|(\omega) = \left\|RAO_3\right\| \left( \omega_0 - U \frac{\omega_0^2}{g} \cos\beta \right) \equiv \left\|RAO_3 \right\| (\omega_0) \,</math></center>

	New function of <math>\omega_0\,</math> by virtue of the <math> \omega \leftrightarrow\,</math> relation.		New function of <math>\omega_0\,</math> by virtue of the <math> \omega \leftrightarrow\, \omega_0 </math> relation.

	The standard deviation of heave follows by simple integration over <math> \omega_0\,</math>:		The standard deviation of heave follows by simple integration over <math> \omega_0\,</math>:

Syan077 at 22:34, 14 March 2007

2007-03-14T22:34:56Z

← Older revision		Revision as of 22:34, 14 March 2007
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	<u>Spectral analysis with forward-speed</u>		<u>Spectral analysis with forward-speed</u>

	<center><math> \omega = \left\| \omega_0 - U \frac{\omega_0^2}{g} \cos \~~beat~~ \right\| <\math></center>		<center><math> \omega = \left\| \omega_0 - U \frac{\omega_0^2}{g} \cos \beta \right\| </math></center>

	* Ambient wave spectral density <math> S_\zeta(\omega_0)\,</math> is defined relative to the absolute wave frequency <math> \omega_0\,</math>.		* Ambient wave spectral density <math> S_\zeta(\omega_0)\,</math> is defined relative to the absolute wave frequency <math> \omega_0\,</math>.
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	* The <math> RAO_X(\omega) \,</math> is usually defined relative to the encounter frequency <math> \omega\,</math>.		* The <math> RAO_X(\omega) \,</math> is usually defined relative to the encounter frequency <math> \omega\,</math>.

	* The relation of <math> \omega \~~longleftrightarrow~~ \omega_0 \, </math> is not single valued. The question thus arises of what is the <math>\sigma_X^2\,</math>?		* The relation of <math> \omega \leftrightarrow \omega_0 \, </math> is not single valued. The question thus arises of what is the <math>\sigma_X^2\,</math>?

	<u>Answer</u>		<u>Answer</u>

Syan077 at 22:33, 14 March 2007

2007-03-14T22:33:13Z

New page

Seakeeping in random waves

* Assume known the ambient wave spectral density <math>S_{\zeta}(\omega_0)\,</math> assumed unidirectional for simplicity

<center><math> \frac{1}{2}A_i^2 = S(\omega_i)\Delta\omega \, </math></center>

* <math> \int_0^\infty S_\zeta(\omega) d\omega = \sigma_\zeta^2 \equiv \,</math> Variance of the wave elevation of ambient random seastate, assumed Gaussian with zero mean

* Assuming that the <math>RAO(\omega)\,</math> of a seakeeping quantity <math> X(t) \,</math> has been determined from a frequency domain analysis;

Spectral analysis with forward-speed

<center><math> \omega = \left| \omega_0 - U \frac{\omega_0^2}{g} \cos \beat \right| <\math></center>

* Ambient wave spectral density <math> S_\zeta(\omega_0)\,</math> is defined relative to the absolute wave frequency <math> \omega_0\,</math>.

* The <math> RAO_X(\omega) \,</math> is usually defined relative to the encounter frequency <math> \omega\,</math>.

* The relation of <math> \omega \longleftrightarrow \omega_0 \, </math> is not single valued. The question thus arises of what is the <math>\sigma_X^2\,</math>?

Answer

* Given <math>\omega_0 \,</math>, a single value of <math>\omega\,</math> always follows.

* The opposite is not always true. Given <math>\omega\,</math> there may exist multiple <math>\omega_0\,</math>'s satisfying the encounter frequency relation.

* Therefore it is much simpler to parameterize with respect to <math>\omega_0\,</math>, even when the <math>RAO(\omega)\,</math> is evaluated as a function of <math>\omega\,</math>.

Proceed as follows:

Simply redefine the <math>RAO(\omega)\,</math> as follows:

<center><math> \left|RAO_3\right|(\omega) = \left|RAO_3\right| \left( \omega_0 - U \frac{\omega_0^2}{g} \cos\beta \right) \equiv \left|RAO_3 \right| (\omega_0) \,</math></center>

New function of <math>\omega_0\,</math> by virtue of the <math> \omega \leftrightarrow\,</math> relation.

The standard deviation of heave follows by simple integration over <math> \omega_0\,</math>:

<center><math> \sigma_3^2 = \int_0^\infty d\omega_0 S_\zeta \left(\omega_0\right) \left|RAO_3^*\left(\omega_0\right)\right|^2 </math></center>

* The opposite choice of parameterizing the above integral w.r.t. <math>\omega\,</math> ends up with a lot of unnecessary algebra.

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