What does the damping ratio do, and why does a door slam or crawl?

Start at a slam, then turn up the damping until the door just glides shut. Watch the two poles meet.

Challenge: Close the door in under 3 seconds without slamming. The box under the picture turns green when you get it.

Stuck? Pick one of the examples from the “Try an example” menu, or press “New example.”

The door obeys a differential equation: the spring's pull and the damper's drag balance the door's inertia.

$$\theta'' + 2\zeta\omega\,\theta' + \omega^2\theta = 0$$

The Laplace transform turns each derivative into a power of $s$, so calculus becomes algebra:

$$\Theta(s) = \frac{\theta_0\,(s + 2\zeta\omega)}{s^2 + 2\zeta\omega s + \omega^2}$$

The poles are where the bottom is zero, and they describe the whole motion:

• Complex pair ($\zeta < 1$): the door would swing past shut and back. The frame stops it: a slam.
• One double pole ($\zeta = 1$): the fastest close with no overshoot.
• Two real poles ($\zeta > 1$): no overshoot, but the slower pole drags it out.

The small box plots the poles. Slide the damping and watch them move together, meet, then split along the axis.

Set the damping and spring strength. Open "More settings" to change how far the door starts open.

• "Closed" means within 1° of the frame for a door that doesn't slam, which is when the latch catches.
• A real closer also has a separate "latch speed" valve for the last few degrees. This model treats the damping as constant.