Results: Added mass and damping

These quantities are reported on the added mass and damping sheets of the results tables. Both are reported in body coordinates $\Bxyz$. The added mass ($A_{ij}$) and damping ($B_{ij}$) are real coefficients given in terms of the radiation potentials, $\phi_j$, by \begin{equation} A_{ij} - \frac{\textrm{i}}{\omega} B_{ij} = \rho \int_{\SB}(n_{\textrm{vel}})_i \phi_j \ud S \end{equation} A modified formula for added mass and damping holds in the presence of dipole panels.

The units of the $3{\times}3$ blocks of an added mass matrix for a single body are \begin{equation} \nonumber \left[ \begin{matrix} M & ML \\ ML & ML^2 \end{matrix} \right] \end{equation} where $M$ and $L$ denote the units of mass and length, respectively.

The units of the $3{\times}3$ blocks of a damping matrix for a single body are \begin{equation} \nonumber \left[ \begin{matrix} \dfrac{F}{L/T} & \dfrac{F}{\rad/T} \\ \dfrac{FL}{L/T} & \dfrac{FL}{\rad/T} \end{matrix} \right] \end{equation} where $F$, $L$ and $T$ denote the units of force, length and time, respectively.

In a multibody model with $N$ bodies the added mass and damping matrices are $6N{\times}6N$, with the units of each $6{\times}6$ block given by the expressions above.