Second-Order Wave Effect On Offshore Platforms - Revision history

Adi Kurniawan: /* Spectral Density of Second-Order Effects */

2009-12-08T17:18:32Z

Spectral Density of Second-Order Effects

← Older revision		Revision as of 17:18, 8 December 2009
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	The spectral density of a sum-frequency force with complex quadratic transfer function <math> F^+ \left( \omega_k, \omega_l \right)\,</math> is given by the expression (stated here without proof):		The spectral density of a sum-frequency force with complex quadratic transfer function <math> F^+ \left( \omega_k, \omega_l \right)\,</math> is given by the expression (stated here without proof):

	<center><math> {S_F^{(2)}}^+ (\Omega) = 8 \int_0^{\Omega/2} S(\omega) S(\Omega-\omega) \left\|F^+ (\omega,\Omega-\omega)\right\|^2 d\omega </math></center>		<center><math> {S_F^{(2)}}^+ (\Omega) = 8 \int_0^{\Omega/2} S(\omega) S(\Omega-\omega) \left\|F^+ (\omega,\Omega-\omega)\right\|^2 \mathrm{d}\omega </math></center>

	where <math>S(\omega)\,</math> is the spectrum of the ambient seastate.		where <math>S(\omega)\,</math> is the spectrum of the ambient seastate.
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	The corresponding expression for the difference-frequency force is:		The corresponding expression for the difference-frequency force is:

	<center><math> {S_F^{(2)}}^- (\Omega) = 8 \int_0^\infty S(\omega) S(\Omega-\omega) \left\|F^- (\omega,\omega+\Omega)\right\|^2 d\omega </math></center>		<center><math> {S_F^{(2)}}^- (\Omega) = 8 \int_0^\infty S(\omega) S(\Omega-\omega) \left\|F^- (\omega,\omega+\Omega)\right\|^2 \mathrm{d}\omega </math></center>

	and by invoking Newman's approximation:		and by invoking Newman's approximation:

	<center><math> {S_F^{(2)}}^- (\Omega) \cong 8 \int_0^\infty S(\omega) S(\Omega-\omega) \mathbb{D} (\omega) \mathbb{D} (\omega+\Omega) d\omega </math></center>		<center><math> {S_F^{(2)}}^- (\Omega) \cong 8 \int_0^\infty S(\omega) S(\Omega-\omega) \mathbb{D} (\omega) \mathbb{D} (\omega+\Omega) \mathrm{d}\omega </math></center>

	Also it is useful to recall the mean value of the drift force in an ambient seastate with spectral density <math>S(\omega)\,</math>:		Also it is useful to recall the mean value of the drift force in an ambient seastate with spectral density <math>S(\omega)\,</math>:

	<center><math> \bar{D} = 2 \int_0^\infty S(\omega) \mathbb{D} (\omega) d\omega </math></center>		<center><math> \bar{D} = 2 \int_0^\infty S(\omega) \mathbb{D} (\omega) \mathrm{d}\omega </math></center>

	==Nonlinear Responses of Compliant Offshore Platforms Excited by Second-Order Effects==		==Nonlinear Responses of Compliant Offshore Platforms Excited by Second-Order Effects==

Adi Kurniawan: /* Spectral Density of Second-Order Effects */

2009-12-08T17:10:18Z

Spectral Density of Second-Order Effects

← Older revision		Revision as of 17:10, 8 December 2009
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	==Spectral Density of Second-Order Effects==		==Spectral Density of Second-Order Effects==

	* In practice it is necessary to be able to determine the statistical properties of linear and second-order forces and vessel responses.		In practice it is necessary to be able to determine the statistical properties of linear and second-order forces and vessel responses. This may be accomplished via direct simulation in the time domain when appropriate numerical methods are available or by spectral analysis in combination with approximate linear response models. In the case of second-order wave effects on offshore platforms, for example the slow-drift excitation and response of semi-submersibles, spars, and TLPs, moored or tethered to the ocean floor, linear response models exist. They are excited by a second-order force the spectral density of which is useful to know. The remainder of this section defines the spectral density of the sum and difference frequency force on any floating body.

	* This may be accomplished via direct simulation in the time domain when appropriate numerical methods are available or by spectral analysis in combination with approximate linear response models.		Let <math>S(\omega)\,</math> be the spectrum of the ambient seastate assumed unidirectional. The relation between <math>S(\omega)\,</math> and the amplitudes of the polychromatic ambient seastate <math>A_i\,</math> was given earlier and repeated here

	* In the case of second-order wave effects on offshore platforms, for example the slow-drift excitation and response of semi-submersibles, spar's and TLP's, moored or tethered to the ocean floor, linear response models exist. They are excited by a second-order force the spectral density of which is useful to know.

	* The remainder of this section defines the spectral density of the sum and difference frequency force on any floating body.

	* Let <math>S(\omega)\,</math> be the spectrum of the ambient seastate assumed unidirectional. The relation between <math>S(\omega)\,</math> and the amplitudes of the polychromatic ambient seastate <math>A_i\,</math> was given earlier and repeated here

	<center><math> S\left(\omega_i\right)\Delta\omega = \frac{1}{2} A_i^2 \, </math></center>		<center><math> S\left(\omega_i\right)\Delta\omega = \frac{1}{2} A_i^2 \, </math></center>
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	===Sum-frequency Problem===		===Sum-frequency Problem===

	* The spectral density of a sum-frequency force with complex quadratic transfer function <math> F^+ \left( \omega_k, \omega_l \right)\,</math> is given by the expression (stated here without proof):		The spectral density of a sum-frequency force with complex quadratic transfer function <math> F^+ \left( \omega_k, \omega_l \right)\,</math> is given by the expression (stated here without proof):

	<center><math> {S_F^{(2)}}^+ (\Omega) = 8 \int_0^{\Omega/2} S(\omega) S(\Omega-\omega) \left\|F^+ (\omega,\Omega-\omega)\right\|^2 d\omega </math></center>		<center><math> {S_F^{(2)}}^+ (\Omega) = 8 \int_0^{\Omega/2} S(\omega) S(\Omega-\omega) \left\|F^+ (\omega,\Omega-\omega)\right\|^2 d\omega </math></center>
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	===Difference-Frequency Problem===		===Difference-Frequency Problem===

	* The corresponding expression for the difference-frequency force is:		The corresponding expression for the difference-frequency force is:

	<center><math> {S_F^{(2)}}^- (\Omega) = 8 \int_0^\infty S(\omega) S(\Omega-\omega) \left\|F^- (\omega,\omega+\Omega)\right\|^2 d\omega </math></center>		<center><math> {S_F^{(2)}}^- (\Omega) = 8 \int_0^\infty S(\omega) S(\Omega-\omega) \left\|F^- (\omega,\omega+\Omega)\right\|^2 d\omega </math></center>

Adi Kurniawan: /* Second-Order Wave Forces */

2009-12-08T15:53:08Z

Second-Order Wave Forces

← Older revision		Revision as of 15:53, 8 December 2009
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	The second-order complex transfer functions <math>F_{kl}^{(2)\pm}\left(\omega_k,\omega_l\right)\,</math> for the sum and difference frequency problems are from the solution of the second-order problem.		The second-order complex transfer functions <math>F_{kl}^{(2)\pm}\left(\omega_k,\omega_l\right)\,</math> for the sum and difference frequency problems are from the solution of the second-order problem.

	For values of <math>\left(k,l\right)\,</math> that may be as large as 30-50 for realistic seastates and for complex three-dimensional structures, this represents a very significant computational task. It can be reduced by a factor of 2 by imposing the symmetry condition for ~~hermitian~~ forms:		For values of <math>\left(k,l\right)\,</math> that may be as large as 30-50 for realistic seastates and for complex three-dimensional structures, this represents a very significant computational task. It can be reduced by a factor of 2 by imposing the symmetry condition for Hermitian forms:

	<center><math> F_{kl}^\pm = F_{kl}^{\pm *}\,</math></center>		<center><math> F_{kl}^\pm = F_{kl}^{\pm *}\,</math></center>

Adi Kurniawan: /* Second-Order Wave Forces */

2009-12-08T15:51:23Z

Second-Order Wave Forces

← Older revision		Revision as of 15:51, 8 December 2009
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	In response to an ambient wave record consisting of the linear superposition of regular waves, the linear wave force on a floating structure, or any other linear effect, takes the form:		In response to an ambient wave record consisting of the linear superposition of regular waves, the linear wave force on a floating structure, or any other linear effect, takes the form:

	<center><math> F_1(t) = \mathrm{Re} \~~sum_K A_K F_K~~^{(1)} \left(\~~omega_K~~\right) e^{i\~~omega_K~~ t +i\~~phi_K~~} </math></center>		<center><math> F_1(t) = \mathrm{Re} \sum_k A_k F_k^{(1)} \left(\omega_k\right) e^{i\omega_k t +i\phi_k} </math></center>

	The complex force transfer function <math> ~~F_K~~^{(1)}\left(\~~omega_K~~\right) \,</math> is evaluated analytically or numerically as described earlier.		The complex force transfer function <math> F_k^{(1)}\left(\omega_k\right) \,</math> is evaluated analytically or numerically as described earlier.

	The definition of the corresponding second-order force (to be added to the linear one defined above) may be cast in the form:		The definition of the corresponding second-order force (to be added to the linear one defined above) may be cast in the form:

Adi Kurniawan: /* Second-Order Wave Forces */

2009-12-08T15:50:40Z

Second-Order Wave Forces

← Older revision		Revision as of 15:50, 8 December 2009
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	In response to an ambient wave record consisting of the linear superposition of regular waves, the linear wave force on a floating structure, or any other linear effect, takes the form:		In response to an ambient wave record consisting of the linear superposition of regular waves, the linear wave force on a floating structure, or any other linear effect, takes the form:

	<center><math> F_1(t) = \~~mathbb~~{~~R}\mathbf{e~~} \sum_K A_K F_K^{(1)} \left(\omega_K\right) e^{i\omega_K t +i\phi_K} </math></center>		<center><math> F_1(t) = \mathrm{Re} \sum_K A_K F_K^{(1)} \left(\omega_K\right) e^{i\omega_K t +i\phi_K} </math></center>

	The complex force transfer function <math> F_K^{(1)}\left(\omega_K\right) \,</math> is evaluated analytically or numerically as described earlier.		The complex force transfer function <math> F_K^{(1)}\left(\omega_K\right) \,</math> is evaluated analytically or numerically as described earlier.
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	The definition of the corresponding second-order force (to be added to the linear one defined above) may be cast in the form:		The definition of the corresponding second-order force (to be added to the linear one defined above) may be cast in the form:

	<center><math> f_2(t) = \~~mathbb~~{~~R}\mathbf{e~~} \sum_k \sum_l A_k A_l F_{kl}^{(2)^+} e^{i\left(\omega_k+\omega_l\right)t + i\left(\phi_k+\phi_l\right)} </math></center>		<center><math> f_2(t) = \mathrm{Re} \sum_k \sum_l A_k A_l F_{kl}^{(2)^+} e^{i\left(\omega_k+\omega_l\right)t + i\left(\phi_k+\phi_l\right)} </math></center>
	<center><math> \~~mathbb{R}\mathbf~~{e} \sum_k \sum_l A_k A_l^* F_{kl}^{(2)^-} e^{i\left(\omega_k-\omega_l\right)t + i\left(\phi_k-\phi_l\right)} </math></center>		<center><math> \mathrm{Re} \sum_k \sum_l A_k A_l^* F_{kl}^{(2)^-} e^{i\left(\omega_k-\omega_l\right)t + i\left(\phi_k-\phi_l\right)} </math></center>

	The second-order complex transfer functions <math>F_{kl}^{(2)\pm}\left(\omega_k,\omega_l\right)\,</math> for the sum and difference frequency problems are from the solution of the second-order problem.		The second-order complex transfer functions <math>F_{kl}^{(2)\pm}\left(\omega_k,\omega_l\right)\,</math> for the sum and difference frequency problems are from the solution of the second-order problem.

Adi Kurniawan: /* Second-Order Wave Forces */

2009-12-08T15:49:43Z

Second-Order Wave Forces

Adi Kurniawan at 15:48, 8 December 2009

2009-12-08T15:48:51Z

← Older revision		Revision as of 15:48, 8 December 2009
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	\| previous chapter = [[Second-Order Wave Theory]]		\| previous chapter = [[Second-Order Wave Theory]]
	}}		}}

	{{incomplete pages}}		{{incomplete pages}}

	==Second-Order Wave Forces==		==Second-Order Wave Forces==

Adi Kurniawan at 15:48, 8 December 2009

2009-12-08T15:48:03Z

← Older revision		Revision as of 15:48, 8 December 2009
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	{{incomplete pages}}		{{incomplete pages}}

	~~<u>~~Second-Order Wave Forces~~</u>~~		==Second-Order Wave Forces==

	* In response to an ambient wave record consisting of the linear superposition of regular waves, the linear wave force on a floating structure, or any other linear effect takes the form:		* In response to an ambient wave record consisting of the linear superposition of regular waves, the linear wave force on a floating structure, or any other linear effect takes the form:
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	* The evaluation of the resulting second-order force record <math>f_2^-(t)\,</math> may be carried out using FFT routings.		* The evaluation of the resulting second-order force record <math>f_2^-(t)\,</math> may be carried out using FFT routings.

	~~<u>~~Spectral Density of Second-Order Effects~~</u>~~		==Spectral Density of Second-Order Effects==

	* In practice it is necessary to be able to determine the statistical properties of linear and second-order forces and vessel responses.		* In practice it is necessary to be able to determine the statistical properties of linear and second-order forces and vessel responses.
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	For a small frequency band <math>\Delta\omega\,</math> centered around <math>\omega_i\,</math>.		For a small frequency band <math>\Delta\omega\,</math> centered around <math>\omega_i\,</math>.

	* <u>Linear System Theory~~</u>~~		===Linear System Theory===

	Given <math>S(\omega)\,</math> and the complex transfer function of a linear force (or other effect) <math>F(\omega)\,</math>, the spectral density of <math>F\,</math> is given by		Given <math>S(\omega)\,</math> and the complex transfer function of a linear force (or other effect) <math>F(\omega)\,</math>, the spectral density of <math>F\,</math> is given by
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	This result is extended below to second-order forces (and moments).		This result is extended below to second-order forces (and moments).

	~~<u>~~Sum-frequency Problem~~</u>~~		===Sum-frequency Problem===

	* The spectral density of a sum-frequency force with complex quadratic transfer function <math> F^+ \left( \omega_k, \omega_l \right)\,</math> is given by the expression (stated here without proof):		* The spectral density of a sum-frequency force with complex quadratic transfer function <math> F^+ \left( \omega_k, \omega_l \right)\,</math> is given by the expression (stated here without proof):
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	where <math>S(\omega)\,</math> is the spectrum of the ambient seastate.		where <math>S(\omega)\,</math> is the spectrum of the ambient seastate.

	~~<u>~~Difference-Frequency Problem~~</u>~~		===Difference-Frequency Problem===

	* The corresponding expression for the difference-frequency force is:		* The corresponding expression for the difference-frequency force is:
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	<center><math> \bar{D} = 2 \int_0^\infty S(\omega) \mathbb{D} (\omega) d\omega </math></center>		<center><math> \bar{D} = 2 \int_0^\infty S(\omega) \mathbb{D} (\omega) d\omega </math></center>

	~~<u>~~Nonlinear Responses of Compliant Offshore Platforms Excited by Second-Order Effects~~</u>~~		==Nonlinear Responses of Compliant Offshore Platforms Excited by Second-Order Effects==

	The above two figures illustrate two typical nonlinear responses of compliant offshore platforms designed to operate in waters of large depth for the extraction of oil and gas from sub-sea reservoirs.		The above two figures illustrate two typical nonlinear responses of compliant offshore platforms designed to operate in waters of large depth for the extraction of oil and gas from sub-sea reservoirs.

Meylan at 23:40, 16 October 2009

2009-10-16T23:40:01Z

← Older revision		Revision as of 23:40, 16 October 2009
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			{{Ocean Wave Interaction with Ships and Offshore Structures
			\| chapter title = Second-Order Wave Effect On Offshore Platforms
			\| next chapter =
			\| previous chapter = [[Second-Order Wave Theory]]
			}}

			{{incomplete pages}}

	<u>Second-Order Wave Forces</u>		<u>Second-Order Wave Forces</u>

Adi Kurniawan: typo

2009-01-12T05:18:18Z

typo

← Older revision		Revision as of 05:18, 12 January 2009
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	* Slow-Drift Responses <math> X(t)\,</math>		* Slow-Drift Responses <math> X(t)\,</math>

	These are typically large amplitude low-frequency responses of the order of minutes excited by second-order difference-frequency effects, as ~~will~~ as time dependent wind & currents. The latter are of viscous nature.		These are typically large amplitude low-frequency responses of the order of minutes excited by second-order difference-frequency effects, as well as time dependent wind & currents. The latter are of viscous nature.

	An effective model for <math>X(t)\,</math> is:		An effective model for <math>X(t)\,</math> is:

← Older revision		Revision as of 15:49, 8 December 2009
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	==Second-Order Wave Forces==		==Second-Order Wave Forces==

	* In response to an ambient wave record consisting of the linear superposition of regular waves, the linear wave force on a floating structure, or any other linear effect takes the form:		In response to an ambient wave record consisting of the linear superposition of regular waves, the linear wave force on a floating structure, or any other linear effect, takes the form:

	<center><math> F_1(t) = \mathbb{R}\mathbf{e} \sum_K A_K F_K^{(1)} \left(\omega_K\right) e^{i\omega_K t +i\phi_K} </math></center>		<center><math> F_1(t) = \mathbb{R}\mathbf{e} \sum_K A_K F_K^{(1)} \left(\omega_K\right) e^{i\omega_K t +i\phi_K} </math></center>
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	The complex force transfer function <math> F_K^{(1)}\left(\omega_K\right) \,</math> is evaluated analytically or numerically as described earlier.		The complex force transfer function <math> F_K^{(1)}\left(\omega_K\right) \,</math> is evaluated analytically or numerically as described earlier.

	* The definition of the corresponding second-order force (to be added to the linear one defined above) may be cast in the form:		The definition of the corresponding second-order force (to be added to the linear one defined above) may be cast in the form:

	<center><math> f_2(t) = \mathbb{R}\mathbf{e} \sum_k \sum_l A_k A_l F_{kl}^{(2)^+} e^{i\left(\omega_k+\omega_l\right)t + i\left(\phi_k+\phi_l\right)} </math></center>		<center><math> f_2(t) = \mathbb{R}\mathbf{e} \sum_k \sum_l A_k A_l F_{kl}^{(2)^+} e^{i\left(\omega_k+\omega_l\right)t + i\left(\phi_k+\phi_l\right)} </math></center>
	<center><math> \mathbb{R}\mathbf{e} \sum_k \sum_l A_k A_l^* F_{kl}^{(2)^-} e^{i\left(\omega_k-\omega_l\right)t + i\left(\phi_k-\phi_l\right)} </math></center>		<center><math> \mathbb{R}\mathbf{e} \sum_k \sum_l A_k A_l^* F_{kl}^{(2)^-} e^{i\left(\omega_k-\omega_l\right)t + i\left(\phi_k-\phi_l\right)} </math></center>

	* The second-order complex transfer functions <math>F_{kl}^{(2)\pm}\left(\omega_k,\omega_l\right)\,</math> for the sum and difference frequency problems are from the solution of the second-order problem.		The second-order complex transfer functions <math>F_{kl}^{(2)\pm}\left(\omega_k,\omega_l\right)\,</math> for the sum and difference frequency problems are from the solution of the second-order problem.

	* For values of <math>\left(k,l\right)\,</math> that may be as large as 30-50 for realistic seastates and for complex three-dimensional structures, this represents a very significant computational task. It can be reduced by a factor of 2 by imposing the symmetry condition for hermitian forms:		For values of <math>\left(k,l\right)\,</math> that may be as large as 30-50 for realistic seastates and for complex three-dimensional structures, this represents a very significant computational task. It can be reduced by a factor of 2 by imposing the symmetry condition for hermitian forms:

	<center><math> F_{kl}^\pm = F_{kl}^{\pm *}\,</math></center>		<center><math> F_{kl}^\pm = F_{kl}^{\pm *}\,</math></center>
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	Yet this leaves a significant computational effort which in its generality is carried out by three-dimensional panel methods solving the sum and difference frequency radiation and diffraction problems around floating structures.		Yet this leaves a significant computational effort which in its generality is carried out by three-dimensional panel methods solving the sum and difference frequency radiation and diffraction problems around floating structures.

	* In the difference-frequency problem a useful and accurate approximation for <math>f_2(t)\,</math> is available known as Newman's approximation.		In the difference-frequency problem a useful and accurate approximation for <math>f_2(t)\,</math> is available known as Newman's approximation.

	In unidirectional waves it approximates the transfer function as follows:		In unidirectional waves it approximates the transfer function as follows:
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	where <math>\mathbb{D}_k\left(\omega_k\right)\,</math> is the mean drift force on the floating platform at frequency <math>\omega_k\,</math> which is evaluated with the methods discussed earlier.		where <math>\mathbb{D}_k\left(\omega_k\right)\,</math> is the mean drift force on the floating platform at frequency <math>\omega_k\,</math> which is evaluated with the methods discussed earlier.

	* The result of this approximation is that the solution of the second-order problem is circumvented entirely! This is in practice very desirable due to the complexity and effort in solving second-order problems.		The result of this approximation is that the solution of the second-order problem is circumvented entirely! This is in practice very desirable due to the complexity and effort in solving second-order problems.

	* The evaluation of the resulting second-order force record <math>f_2^-(t)\,</math> may be carried out using FFT routings.		The evaluation of the resulting second-order force record <math>f_2^-(t)\,</math> may be carried out using FFT routings.

	==Spectral Density of Second-Order Effects==		==Spectral Density of Second-Order Effects==