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3D buoy theory |
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3D buoy theory |
3D buoys are relatively simple objects. Their three degrees of freedom are the translational ones: they do not support rotation, so the loads described here are all pure forces with no moment contributions. Further, their local axes are always aligned with global axes, so the forces are defined with respect to global axis directions and are applied at the buoy origin.
All the fluid-related forces on a 3D buoy are scaled by the buoy's instantaneous proportion wet, $\PW$, calculated as \begin{equation} \PW = h\wet / h \end{equation} where
$h\wet=$ wetted height $= \frac12 h + (\text{depth of the buoy origin below the surface})$, taking depth as negative above the surface, so $0 \leq h\wet \leq h$
$h=$ buoy height.
The weight force, acting vertically downwards, is \begin{equation} \vec{f}_\mathrm{w} = -m\ g\ \vec{u}_\mathrm{Z} \end{equation} where
$m=$ buoy mass
$g=$ acceleration due to gravity
$\vec{u}_\mathrm{Z}$ is a unit vector vertically upwards.
The buoyancy force, acting vertically upwards, is \begin{equation} \vec{f}_\mathrm{b} = \rho\ g\ V\wet \vec{u}_\mathrm{Z} \end{equation} where
$\rho=$ water density
$V\wet=$ wetted volume $= \PW V$
$V=$ buoy volume.
The drag force applied in each axis direction is given by \begin{equation} \begin{aligned} f_\mathrm{DX} &= -\PW\ \tfrac12\ \rho\ \C{DX} A_\mathrm{X} v_\mathrm{X} \lvert \vec{v} \rvert \\ f_\mathrm{DY} &= -\PW\ \tfrac12\ \rho\ \C{DY} A_\mathrm{Y} v_\mathrm{Y} \lvert \vec{v} \rvert \\ f_\mathrm{DZ} &= -\PW\ \tfrac12\ \rho\ \C{DZ} A_\mathrm{Z} v_\mathrm{Z} \lvert \vec{v} \rvert \\ \end{aligned} \end{equation} where
$\CD=$ the given drag coefficient for each component
$A=$ the given drag area for each component
$\vec{v}=$ buoy velocity relative to the fluid velocity.
The added mass load $f_\mathrm{A}$ for each axial component direction follows the inertial part of Morison's equation. Writing $\ab$ for the buoy acceleration relative to earth in the component direction \begin{equation} \label{fA} f_\mathrm{A} = (1 + \Ca)\ \Delta\ \af - \Ca\Delta\ \ab \end{equation} where
$\Delta$ is the instantaneous reference mass, given by $\rho V\wet$
$\Ca$ is the added mass coefficient for this component
$\af$ is the component of fluid acceleration, relative to earth
$\ab=\af-\ar$ is the buoy acceleration relative to earth, where $\ar$ is the component of fluid acceleration relative to the buoy.
The $1$ in $(1{+}\Ca)$ in the first term of expression (\ref{fA}) represents what is known as the Froude-Krylov force; the $\Ca$ in the first term gives rise to the added mass force. The second term, $\Ca\Delta\ \ab$, represents the increase in buoy inertia due to the added mass.
Finally, 3D buoys are also subject to a reaction force $\vec{f}_\mathrm{R}$, from the seabed and any elastic solid with which they come into contact, of the form \begin{equation} \vec{f}_\mathrm{R} = r(d)\ a\ \vec{n} \end{equation} where
$d=$ depth of penetration of the buoy origin
$r(d)=$ reaction force per unit contact area at penetration $d$
$a$ is the 3D buoy contact area
$\vec{n}=$ unit vector in the reaction force direction for seabed or shapes.
If explicit integration is used in the dynamic analysis, 3D buoys may also experience a damping force. For details of both the reaction force and the damping force, see seabed theory and shape theory.
Frictional forces are also modelled when contact occurs, using the seabed friction coefficient and friction coefficient data, and following the model described in friction theory.