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Results: Displacement RAOs |
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Results: Displacement RAOs |
These are reported on the displacement RAOs sheet of the results tables. Displacement RAOs are the amplitudes, $\xi_j$, of the radiation potentials in each degree of freedom $j$. They are the complex amplitudes of the body's oscillatory motion relative to its mean position (i.e. the position specified on the bodies page). The displacement RAOs are reported in body coordinates $\Bxyz$ hence, e.g., $\xi_1$ is the amplitude of surge in the $\mat{B}_\mathrm{x}$ direction.
| Note: | Almost all other results depend on the values of the displacement RAOs (the only exceptions are the added mass and damping, hydrostatics and load RAO results). If you change your model in a way that modifies the displacement RAOs, it will indirectly modify almost all other results. |
The displacement RAOs are obtained by solving the equation of motion for the body at first order in wave steepness. In its most basic form the equation of motion is given by \begin{equation} \label{eqEquationOfMotion} \left\{-\omega^2\left(M_{ij} + A_{ij} \right) + \textrm{i}\omega \left(B_{ij} + D_{ij} \right) + K_{ij} + \Kext_{ij}\right\}\xi_j = F_i \end{equation} where this matrix equation is written using the summation convention, and
$M_{ij}$ is the body inertia matrix
$A_{ij}$ and $B_{ij}$ are the added mass and damping matrices
$D_{ij}$ is the external damping matrix
$K_{ij}$ is the hydrostatic stiffness matrix
$\Kext_{ij}$ is the external stiffness matrix
$F_i$ is the load RAO, using the preferred load RAO calculation method.
$\omega$ is the wave (angular) frequency
| Note: | The equation of motion (\ref{eqEquationOfMotion}) pertains to oscillatory motion about the body's mean position. Therefore the body must be in static equilibrium when in its mean position in the absence of waves. If that equilibrium is not a simple hydrostatic (free-floating) equilibrium, the additional effects that contribute to the body's position must be captured in the constraints data. |
If a body degree of freedom is fixed, the equation of motion (\ref{eqEquationOfMotion}) is modified by adding a restraining load to ensure that the corresponding displacement RAO is zero.
If one body is rigidly connected to another body, the equation of motion (\ref{eqEquationOfMotion}) is modified by adding a connection load to ensure that the displacement RAOs of the parent and child together correspond to rigid-body motion.
If your model contains any Morison elements, the loads on those elements also contribute to the equation of motion (\ref{eqEquationOfMotion}). Drag forces must be linearised to contribute, which requires that a wave spectrum is specified to enable the linearisation to proceed. Inertia forces do not require linearisation.
Ultimately, all the loads are linear functions of the velocity of each sub-element, $\vobject$, and the fluid velocity at the centroid of each sub-element, $\vfluid$. The accelerations used for inertia forces are linear functions of these velocities since $\aobject = \textrm{i} \omega \vobject$ and $\afluid = \textrm{i} \omega \vfluid$. Finally, recall that fluid velocity is decomposed into contributions from the incident, scattered and radiation velocity potentials \begin{equation} \vfluid = \nabla\phi_I + \nabla\phi_S + \nabla\phi_R \end{equation} With this notation, the loads on Morison elements contribute to (\ref{eqEquationOfMotion}) as follows:
$\nabla\phi_I$ and $\nabla\phi_S$ are both constant (independent of $\xi_j$). Therefore these terms in $\vfluid$ generate loads that are added to the right-hand side of (\ref{eqEquationOfMotion})
$\vobject$ and $\nabla\phi_R$ are both linear functions of $\xi_j$. Coefficients for the loads generated by these terms are collected and added to the left-hand side of (\ref{eqEquationOfMotion})
| Note: | If the Morison fluid velocity is undisturbed incident wave, $\nabla\phi_I$ is the only contribution included in $\vfluid$. |