# Damping force¶

## Linear damping¶

The linear damping force, applied on a body, can be expressed using generalized notations:

\[\mathbf{f}_{LD} = \mathbf{M}_{LD} \mathcal{V}\]

where \(\mathcal{V} = \begin{bmatrix} \mathbf{u} \\ \mathbf{\omega} \end{bmatrix}\) is the generalized velocity of the body and \(\mathbf{M}_{LD}\) is the damping matrix.

The velocity of the body can be taken relatively to a fluid flow velocity (air or water).

## Quadratic damping¶

The quadratic damping force is only applied on the translational velocity of a body, and therefor cannot be given using generalized notations:

\[\begin{split}\mathbf{f}_{QD} = -\frac{1}{2} \rho_{fluid} \begin{bmatrix} C_x S_x |u_x| u_x \\C_y S_y |u_y| u_y \\C_z S_z |u_z| u_z \\ \end{bmatrix}\end{split}\]

where

- \(\rho_{fluid}\) is the fluid density.
- \(C_i\) are the damping coefficients,
- \(S_i\) are the projected surfaces,
- \(\mathbf{u} = \begin{bmatrix}u_x & u_y & u_z \end{bmatrix}\) is the body velocity. It can also be taken relatively to a fluid flow velocity, but be careful not to use a current load which might be redundant.