Vessel theory: Other damping

$\newcommand{\vx}{v_\mathrm{x}}$ $\newcommand{\vy}{v_\mathrm{y}}$ $\newcommand{\vz}{v_\mathrm{z}}$ $\newcommand{\vh}{v_\mathrm{h}}$ $\newcommand{\wx}{\omega_\mathrm{x}}$ $\newcommand{\wy}{\omega_\mathrm{y}}$ $\newcommand{\wz}{\omega_\mathrm{z}}$ $\newcommand{\wh}{\omega_\mathrm{h}}$

The other damping load on the vessel is the sum of linear and quadratic terms that are calculated using the other damping coefficients specified on the vessel type data form. It is only calculated if other damping is included on the vessel data form.

The velocity vectors used in the calculation, the translational velocity, $\vec{v}$, and angular velocity, $\vec{\omega}$, depend on the value specified for other damping calculated from primary, as follows:

The damping load is applied at the reference origin.

The damping model used depends on the vessel type symmetry, as follows.

If the symmetry is not circular then the damping model treats each degree of freedom independently, and the damping force $(\fx,\fy,\fz)$ and moment $(\mx,\my,\mz)$ are given by \begin{equation} \begin{aligned} \fx &= -L\urm{surge} \vx - Q\urm{surge} \vx\lvert \vx\rvert \\ \fy &= -L\urm{sway} \vy - Q\urm{sway} \vy\lvert \vy\rvert \\ \fz &= -L\urm{heave} \vz - Q\urm{heave} \vz\lvert \vz\rvert \\ \mx &= -L\urm{roll} \wx - Q\urm{roll} \wx \lvert \wx \rvert \\ \my &= -L\urm{pitch} \wy - Q\urm{pitch} \wy \lvert \wy \rvert \\ \mz &= -L\urm{yaw} \wz - Q\urm{yaw} \wz \lvert \wz \rvert \end{aligned} \end{equation} where

$L$ and $Q$ are the linear and quadratic other damping coefficients, respectively

subscripts $x, y, z$ denote the axes of the other damping frame of reference.

If the vessel type symmetry is circular, then the quadratic term is instead applied as a cross-flow drag model, with a vertical axis, so \begin{equation} \begin{aligned} \fx &= -L\urm{surge} \vx - Q\urm{surge} \vx\lvert \vh\rvert \\ \fy &= -L\urm{sway} \vy - Q\urm{sway} \vy\lvert \vh\rvert \\ \fz &= -L\urm{heave} \vz - Q\urm{heave} \vz\lvert \vz\rvert \\ \mx &= -L\urm{roll} \wx - Q\urm{roll} \wx \lvert \wh \rvert \\ \my &= -L\urm{pitch} \wy - Q\urm{pitch} \wy \lvert \wh \rvert \\ \mz &= -L\urm{yaw} \wz - Q\urm{yaw} \wz \lvert \wz \rvert \end{aligned} \end{equation} where \begin{align} \vh &= (\vx, \vy, 0)\ \text{ is the horizontal vector component of }\vec{v} \\ \wh &= (\wx, \wy, 0)\ \text{ is the horizontal vector component of }\vec{\omega} \end{align} The axes used when the other damping loads are applied are those of the other damping frame of reference.