Dampened Spring Systems

A damping element resists relative velocity across it. In other words, it exerts a damping force:

$$f=cv=c\dot{x}$$

where $c$ is the damping coefficient.

In the context of a dampened mass-spring system, we then have:

$$m\ddot{x}=f-kx-c\dot{x}$$

or

$$f=m\ddot{x}+c\dot{x}+kx$$