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Seabed theory |
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Seabed theory |
The seabed reaction force is the sum of a penetration resistance force in the seabed normal direction and a friction force in the direction tangential to the seabed plane and towards the friction target position. If explicit integration is used in the dynamic analysis then, in addition, seabed damping forces are applied in these normal and tangential directions.
The penetration resistance force depends on the choice of seabed model used – for details, see the theory for either the elastic seabed model or the nonlinear soil model. The friction force is described under the friction theory topic.
Lines, 3D buoys and 6D buoys, and drag chains are the only objects which interact with the seabed. Other objects are not affected by it: they are simply unaware of its presence.
Line interaction occurs when one of the line nodes penetrates the seabed. The seabed reaction forces are calculated using the penetration of the lower outer surface of the line (based on the line type outer contact diameter), and the seabed forces are applied at that point. The seabed lateral friction force is calculated using the line type seabed friction coefficient.
A 3D buoy interacts when the buoy origin penetrates the seabed. The seabed reaction forces are calculated using the penetration of the buoy origin, and are applied at the buoy origin. The seabed friction force is calculated using the buoy seabed friction coefficient.
A 6D buoy interacts when any of its vertices penetrates the seabed. Each penetrating vertex experiences its own seabed normal reaction and lateral friction forces, based on the penetration of the vertex and displacement of the vertex from its friction target position, and the forces are applied at that vertex. This gives a model where each vertex behaves rather like a pad, such as the landing pad on a lunar module.
The nature of drag chain interaction with the seabed is somewhat different, and is described separately.
Under the elastic seabed model, the seabed behaves as an elastic spring in the normal direction, with spring strength determined by the normal seabed stiffness data item.
The normal stiffness reaction force $\vec{f}_\mathrm{R}$ is given by \begin{equation} \vec{f}_\mathrm{R} = r(d)\ a\ \vec{n} \end{equation} where
$d=$ depth of penetration
$r(d)=$ reaction force per unit contact area at penetration $d$
$a=$ contact area for the 3D buoy, 6D buoy or line
$\vec{n}=$ unit vector in the outward normal direction to the seabed.
For simple linear stiffness, $r(d)$ is given by $k_\mathrm{n}d$ where $k_\mathrm{n}$ is the normal seabed stiffness. For nonlinear stiffness, $r(d)$ may be defined by a variable data table.
If explicit integration is used in the dynamic analysis, then seabed damping forces are applied in the normal and tangential directions, as follows.
The normal seabed damping force is only applied when the penetrating object is travelling into the seabed, not when it is coming out of the seabed, and is given by \begin{equation} \vec{f}_\mathrm{Dn} = \begin{cases} 2 \lambda (m\ k_\textrm{nom} a)^{1/2}\ v_\textrm{n} \vec{n} & \text{for $v_\mathrm{n}\gt0$} \\ 0 & \text{for $v_\mathrm{n}\leq0$} \end{cases} \end{equation} where
$\lambda=$ seabed percent critical damping / 100
$m=$ mass of the object (e.g. the mass of a node of a line)
$k_\textrm{nom}=$ nominal normal stiffness
$v_\mathrm{n}=$ component of velocity normal to the seabed, positive when travelling into the seabed and negative when coming out.
For linear stiffness, $k_\textrm{nom} = k_\mathrm{n}$. For nonlinear stiffness, $k_\textrm{nom}$ is taken to be the normal stiffness corresponding to a penetration of zero.
The tangential seabed damping force, $\vec{f}_\mathrm{Dt}$, is applied so as to oppose the tangential component of the velocity of the penetrator and is given by \begin{equation} \vec{f}_\mathrm{Dt} = -2 \lambda (m\ k_\mathrm{t} a)^{1/2}\ v_\mathrm{t} \vec{t} \end{equation} where
$k_\mathrm{t}=$ seabed shear stiffness
$v_\mathrm{t}=$ component of penetrator velocity tangential to the seabed
$\vec{t}=$ unit vector in the tangential velocity direction.