Why a Door Closer Never Slams

A door closer has a spring that pulls the door shut and an oil damper that resists motion. With enough damping the door glides closed without ever swinging past. Its motion is a mix of two decaying exponentials.

How it works

Second-order equations with real, distinct roots

In one sentence

If the characteristic equation $ar^2 + br + c = 0$ of $ay'' + by' + cy = 0$ has distinct real roots $r_1 \ne r_2$ , the general solution is $y = c_1e^{r_1t} + c_2e^{r_2t}$ .

General formula

{\htmlData{sym=0}{y}}'' + {\htmlData{sym=1}{c}}\,{\htmlData{sym=0}{y}}' + {\htmlData{sym=2}{k}}\,{\htmlData{sym=0}{y}} = 0, \quad {\htmlData{sym=1}{c}}^2 > 4{\htmlData{sym=2}{k}} \quad\Rightarrow\quad {\htmlData{sym=0}{y}} = A\,e^{{\htmlData{sym=3}{r_1}} t} + B\,e^{{\htmlData{sym=4}{r_2}} t}, \quad r_{1,2} = \frac{-{\htmlData{sym=1}{c}} \pm \sqrt{{\htmlData{sym=1}{c}}^2 - 4{\htmlData{sym=2}{k}}}}{2}

Each root is a pure exponential motion that solves the equation by itself; the starting position and speed fix how much of each the door uses.

What the symbols mean

$y$
Door angle
How far open the door is, as a fraction of fully open.
$c$
Damping
How strongly the oil damper resists motion.
$k$
Spring stiffness
How strongly the spring pulls the door shut.
$r_1$
Slow decay rate
The root closer to zero; it sets how long closing takes.
$r_2$
Fast decay rate
The more negative root; its part dies out almost immediately.

The closer's spring pushes the door shut ( $ky$ ) and its damper resists motion ( $cy'$ ).
Try $y = e^{rt}$ : the characteristic equation $r^2 + cr + k = 0$ has two real roots when $c^2 > 4k$ .
Both roots are negative, so both exponentials decay; the general solution is any combination of them.
The slower root $r_1$ sets how long the door takes to close; the faster one dies out almost at once.

Learn the math

Second Order Linear EquationsDiff Eq · Lesson 13, Second Order Differential Equations Real, Distinct RootsDiff Eq · Lesson 14, Second Order Differential Equations Fundamental Sets and the WronskianDiff Eq · Lesson 18, Second Order Differential Equations