Damped oscillator

A damped oscillator has some velocity-dependent friction force $F = -\Gamma \dot x$ . In general, its equation of motion is:

\ddot x + \Gamma \dot x + \omega_0^2 x = 0.

We solve this by guessing $x=\mathrm{Re}[z]$ where $z = A e^{i \alpha t}$ .

Then

\begin{align*} (-\alpha^2 + i \alpha \Gamma + \omega_0^2) A e^{i \alpha t} &= 0 \\ \alpha^2 - i \alpha \Gamma - \omega_0^2 &= 0. \end{align*}

Then solve for $\alpha$ using the quadratic formula.

\alpha = \frac{i \Gamma}{2} \pm \sqrt{-\frac{\Gamma^2}{4} + \omega_0^2}.

Define $\omega = -\frac{\Gamma^2}{4} + \omega_0^2$ . Then the solution for $z$ is:

z(t) = Ae^{-\frac{\Gamma}{2} t} e^{\pm i \omega t}.

The solution for $x$ will depend on the specific damping regime.

Damping regimes

There are three damping regimes depending on the sign of $\omega$ . In other words, if the roots of $\alpha$ are real or complex.

Underdamped

When $\omega_0^2 > \Gamma^2/4$ , the oscillations are stronger than the damping. In this case the discriminant is positive and $\alpha$ has real roots. An underdamped oscillator may oscillate for some time before damping out.

The solution for an underdamped oscillator is:

x(t) = \mathrm{Re}[z(t)] = e^{-\frac{\Gamma}{2} t} (A \cos(\omega t) + B \sin(\omega t)).

Or alternatively:

x(t) = A e^{-\frac{\Gamma}{2} t} \cos(\omega t + \phi).

Overdamped

When $\omega_0^2 < \Gamma^2/4$ , the damping is stronger than the oscillations. In this case the discriminant is negative and $\alpha$ has complex roots.

We rewrite $\alpha = i\left(\frac{\Gamma}{2} \pm \sqrt{\frac{\Gamma^2}{4} - \omega_0^2}\right)$ . Define $\Gamma_\pm = \frac{\Gamma}{2} \pm \sqrt{\frac{\Gamma^2}{4} - \omega_0^2}$ .

Then our solution is:

x(t) = \mathrm{Re}\left[ A e^{i \cdot i \Gamma_+} + e^{i \cdot i \Gamma_-} \right] = Ae^{-\Gamma_+} + Be^{-\Gamma_-}.

Note that there is no oscillation.

Critically damped

Critically damped means $\omega_0^2 = \Gamma^2/4$ . In this case the descriminant is $0$ , so $x(t) = Ae^{-\frac{\Gamma}{2}t}$ . However, since this has only one free variable it cannot be the full solution to the second-order equation. The full solution is:

x(t) = (A+Bt)e^{-\frac{\Gamma}{2}t}.

A critically damped oscillator will return to equilibrium the fastest.