Damped oscillations

In real systems an ever-present friction — air viscosity, internal friction of the spring, dissipation in the joints — removes energy from the oscillator: the amplitude does not stay constant, but decreases over time.

The standard model adds a friction force proportional to the velocity, $F_\text{att} = -b\,\dot x$. The equation of motion becomes

$m\ddot x = -K x - b\,\dot x \quad\Rightarrow\quad \ddot x + 2\zeta\omega_0\,\dot x + \omega_0^2\,x = 0$

with $\omega_0 = \sqrt{K/m}$ the natural angular frequency and $\zeta = \dfrac{b}{2\sqrt{mK}}$ the dimensionless damping coefficient. It is the value of $\zeta$ that decides the character of the motion.

Three damping regimes

• $\zeta < 1$ (underdamped): the body oscillates, but the amplitude decays exponentially, $x(t) = A_0\,e^{-\zeta\omega_0 t}\cos(\omega_d t + \varphi)$, with reduced angular frequency $\omega_d = \omega_0\sqrt{1-\zeta^2} < \omega_0$.
• $\zeta = 1$ (critical damping): the body returns to equilibrium in the shortest possible time, without oscillating. This is the regime of car shock absorbers.
• $\zeta > 1$ (overdamped): the body returns to equilibrium slowly, without ever overshooting it.

Underdamped regime (blue): oscillation with dashed exponential envelope. Critical regime (green): return to equilibrium without oscillating, in the minimum time.

The engineering optimum

Critical damping is the ideal compromise: it stops unwanted oscillations without “ringing” (bouncing) and without the slowness of overdamping. This is why a car’s shock absorbers are designed close to $\zeta = 1$.

Links

Topics: Oscillations and harmonic motion Concepts: Damped oscillations · Simple harmonic motion Objects: Spring

Related exercises: Worked exercise — Mass, spring and damping · Problem — Oscillation without damping · Worked exercise — Finding A and φ from initial conditions