I'm modelling particles in a system using a spring and damper force.
$$F= kx -cv$$
$x=x_i -x_f$, where $x_i$=centre of spring and $x_f$= displaced position.
Above $x$ is the displacement and $v$ is the velocity in the previous timestep. I don't want the particles to vibrate too much upon displacement, thus I increase $c$ to very high. But I notice they then start moving rapidly along the centre of displacement. Does this make sense or am going wrong somewhere when coding it?
Best Answer
What matters here is how the value of $c$ compares to the value of $k$.
Let us choose a $\zeta = \frac{c}{2\sqrt{mk}}$
One can show that when $\zeta =1 $ the system is critically damped, and will not exhibited any oscillations and will return to the origin in the shortest possible time interval.
When $\zeta > 1 $ the system is over damped and will take longer than reach the equilibrium position, but will still not exhibit oscillatory motion.
When $\zeta < 1 $ the system will exhibit oscillatory motion at the natural frequency, and the amplitude will gradually decrease over time.
Therefore if you want the system to exhibit no oscillatory motion and move to the origin in the shortest time frame, choose $\zeta = \frac{c}{2\sqrt{mk}} = 1$and thus set $c = 2\sqrt{mk}$.
The diagram below shows the oscillatory behavior for various values of $\zeta$: