Free damped vibration: three cases

Equation of motion

For mass $m$, spring $k$ and viscous damper $c$, with displacement $x$ from equilibrium:

$$m\ddot{x} + c\dot{x} + kx = 0$$

Assume $x = Ae^{st}$:

$$ms^2 + cs + k = 0 \;\Rightarrow\; s_{1,2} = -\frac{c}{2m} \pm \sqrt{\left(\frac{c}{2m}\right)^2 - \frac{k}{m}}$$

The critical damping coefficient is the value of $c$ that makes the term under the square root zero: $c_c = 2\sqrt{km} = 2m\omega_n$. The damping ratio is $\zeta = c/c_c$. Then

$$s_{1,2} = \omega_n\left(-\zeta \pm \sqrt{\zeta^2 - 1}\right)$$

Case 1: under-damped ($\zeta < 1$)

The roots are complex: $s = -\zeta\omega_n \pm i\omega_d$, with damped frequency $\omega_d = \omega_n\sqrt{1-\zeta^2}$.

$$x(t) = Xe^{-\zeta\omega_n t}\sin(\omega_d t + \phi)$$

The motion is oscillatory with an exponentially decaying amplitude.

Case 2: critically damped ($\zeta = 1$)

Equal real roots $s = -\omega_n$:

$$x(t) = (A + Bt)\,e^{-\omega_n t}$$

There is no oscillation; the mass returns to equilibrium in the shortest possible time.

Case 3: over-damped ($\zeta > 1$)

Two distinct negative real roots:

$$x(t) = Ae^{(-\zeta + \sqrt{\zeta^2-1})\omega_n t} + Be^{(-\zeta - \sqrt{\zeta^2-1})\omega_n t}$$

There is no oscillation; the return to equilibrium is slower than in the critical case.

 x
 x0*
 | \\ \
 |  \\  '.__
 |   \ '.    '----.______________  over-damped (slow return)
 |    \  '-.___
 |     \       '------.________    critically damped (fastest, no overshoot)
 +------\-----------.------.----------> t
 |       \        /   \__/         under-damped (oscillates, decays)
 |        '.____.'