The manoeuvring damping force concerns surge, sway and yaw degrees of freedom. Each component of the damping force is defined as the sum of non-linear terms depending in surge, sway and yaw velocities of the body.

$\begin{split}\mathbf{f}_{MD} = \begin{bmatrix} \sum_k C_X(k) |u|^{m_k} |v|^{n_k} |w|^{p_k} sgn(u) sgn(v) sgn(w) \\ \sum_k C_Y(k) |u|^{m_k} |v|^{n_k} |w|^{p_k} sgn(u) sgn(v) sgn(w) \\ \sum_k C_N(k) |u|^{m_k} |v|^{n_k} |w|^{p_k} sgn(u) sgn(v) sgn(w) \end{bmatrix}\end{split}$

where

• $$C_X(k)$$, $$C_Y(k)$$, $$C_N(k)$$ are the coefficients for the components respectively in surge, sway and yaw.
• $$u$$, $$v$$, $$w$$ are respectively the translational velocities in surge, sway and angular velocity in yaw.