Derivative Seakeeping Quantities

Wave and Wave Body Interactions
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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)

[math]\displaystyle{ \xi_j(t) = \mathrm{Re} \left\{ \Pi_j (\omega) e^{i\omega t} \right\} \, }[/math] [math]\displaystyle{ RAO = \frac{\Pi_j(\omega)}{A}, \quad j=1,2,3 \, }[/math] [math]\displaystyle{ = \frac{\Pi_j(\omega)}{A/L}, \quad j=4,5,6 \, }[/math]

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

[math]\displaystyle{ \zeta_I = \mathrm{Re} \left\{ A e^{i\omega t} \right\} \, }[/math]

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

[math]\displaystyle{ Z(t) = \mathrm{Re} \left\{ \mathbb{Z}(\omega) e^{i\omega t} \right\}, \quad RAO=\frac{\mathbb{Z}(\omega)}{A} }[/math]

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

[math]\displaystyle{ \xi_A (t) = \xi_3 (t) - X_A \xi_5 (t) \, }[/math] [math]\displaystyle{ \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]

So the corresponding RAO in waves of amplitude [math]\displaystyle{ A\, }[/math] is:

[math]\displaystyle{ RAO = \frac{-\omega^2 \left(\Pi_3 -X_A \Pi_5 \right)}{A} = -\omega^2 \left(RAO_3 - X_A RAO_5 \right) }[/math]

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:

[math]\displaystyle{ P_A = \mathrm{Re} \left\{ \mathbb{P}_A e^{i\omega t} \right\} \, }[/math]

where

[math]\displaystyle{ \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] [math]\displaystyle{ RAO = \frac{\mathbb{P}_A}{A} \, }[/math]

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Ocean Wave Interaction with Ships and Offshore Energy Systems