Real oscillators lose energy due to friction, drag, internal material losses, or other dissipative effects. A common linear model for a damped
mass–spring oscillator is
\[
x'' + 2\gamma x' + \omega_0^2 x = 0,
\]
where \(\omega_0\) is the natural (undamped) angular frequency and \(\gamma\) is the damping coefficient (with units of \(1/\text{s}\)).
The damping term \(2\gamma x'\) represents a force proportional to velocity, a good approximation for many systems at moderate speeds.
The behavior depends on the relative size of \(\gamma\) and \(\omega_0\). Define the discriminant-like quantity
\(\omega_0^2-\gamma^2\). When \(\gamma<\omega_0\), the system is underdamped and still oscillates, but with a decaying amplitude:
\[
x(t)=A e^{-\gamma t}\cos(\omega' t + \phi),
\]
where the damped angular frequency is
\[
\omega'=\sqrt{\omega_0^2-\gamma^2}.
\]
The exponential factor \(e^{-\gamma t}\) is the amplitude envelope: each peak is smaller than the previous one by a factor determined by \(\gamma\).
When \(\gamma=\omega_0\), the system is critically damped. It returns to equilibrium as quickly as possible without oscillating.
When \(\gamma>\omega_0\), it is overdamped, returning to equilibrium more slowly without oscillations. In both of these non-oscillatory cases,
\(\omega'\) is not real, which is why the underdamped cosine formula is not the appropriate general solution. Instead, the solution is a sum of
decaying exponentials. This analyzer still visualizes the decay trend and clearly labels the regime.
A useful decay metric is the time constant \(\tau\), defined (for \(\gamma>0\)) by
\[
\tau=\frac{1}{\gamma}.
\]
Over each time interval of length \(\tau\), the envelope amplitude \(A e^{-\gamma t}\) is multiplied by \(e^{-1}\approx 0.368\).
Another common measure is the half-life of the envelope, the time for the amplitude to drop to half:
\[
t_{1/2}=\frac{\ln 2}{\gamma}.
\]
These are purely envelope measures; the oscillation frequency (when underdamped) is set by \(\omega'\).
Damping appears in engineering everywhere: car suspensions (shock absorbers), instrument needles, door closers, and vibration isolation systems.
Underdamping gives oscillatory “ringing,” while critical damping yields fast settling with minimal overshoot. Overdamping reduces overshoot too,
but can make the return sluggish. Advanced models include nonlinear damping (e.g., quadratic drag) and forced oscillations where a periodic drive
injects energy, leading to resonance and steady-state amplitudes.
