# 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.