Difference between revisions of "Derivative Seakeeping Quantities"
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− | + | {{Ocean Wave Interaction with Ships and Offshore Structures | |
+ | | chapter title = Derivative Seakeeping Quantities | ||
+ | | next chapter = [[Seakeeping In Random Waves]] | ||
+ | | previous chapter = [[Ship Roll-Sway-Yaw Motions]] | ||
+ | }} | ||
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+ | {{incomplete pages}} | ||
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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) = \ | + | <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 = \ | + | <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) = \ | + | <center><math> Z(t) = \mathrm{Re} \left\{ \mathbb{Z}(\omega) e^{i\omega t} \right\}, \quad RAO=\frac{\mathbb{Z}(\omega)}{A} </math></center> |
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− | + | ==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 | The vertical displacement of point <math>A\,</math> due to the vessel heave & pitch motions is | ||
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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) = \ | + | <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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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 - Hydrodynamic pressure disturbance at a fixed point on a ship hull oscillating in heave & pitch in waves== | |
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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 = \ | + | <center><math> P_A = \mathrm{Re} \left\{ \mathbb{P}_A e^{i\omega t} \right\} \, </math></center> |
where | where | ||
− | <center><math> \mathbb{P}_A = - \rho \left\{ \left( i\omega - U \frac{\partial}{\partial X} \right) \left( \phi_3 + \phi_5 \right) + | + | <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> |
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<center><math> RAO = \frac{\mathbb{P}_A}{A} \, </math></center>. | <center><math> RAO = \frac{\mathbb{P}_A}{A} \, </math></center>. |
Latest revision as of 09:12, 16 October 2009
Wave and Wave Body Interactions | |
---|---|
Current Chapter | Derivative Seakeeping Quantities |
Next Chapter | Seakeeping In Random Waves |
Previous Chapter | Ship Roll-Sway-Yaw Motions |
The principal seakeeping quantity from a seakeeping analysis of a floating body at zero or forward speed is the Response Amplitude Operator (or RAO)
where [math]\displaystyle{ 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
It follows that the only seakeeping quantity with [math]\displaystyle{ RAO\equiv 1 \, }[/math] is [math]\displaystyle{ \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
Example 1 - Acceleration RAO at the bow of a ship
The vertical displacement of point [math]\displaystyle{ A\, }[/math] due to the vessel heave & pitch motions is
So the corresponding RAO in waves of amplitude [math]\displaystyle{ A\, }[/math] is:
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]\displaystyle{ A\, }[/math] located at [math]\displaystyle{ \vec{X}_A\, }[/math] relative to the ship frame is:
where
.
Ocean Wave Interaction with Ships and Offshore Energy Systems