2.1 The Three Damping Regimes

A mass on a spring, slowed by friction and pushed by nothing, obeys

$x^{″} + 2 a x^{'} + (a^{2} + b^{2}) x = 0 .$

The coefficients are written in this particular way because of what happens next. Guessing $x = e^{r t}$ and substituting turns the differential equation into an ordinary algebraic one — the characteristic equation

$r^{2} + 2 a r + (a^{2} + b^{2}) = 0,$

whose roots are $r = - a \pm i b$ . The guess pays off because differentiating an exponential returns the same exponential times a constant, so the whole equation collapses to a polynomial. The roots then tell you everything: the general solution is built from $e^{- a t} \cos b t$ and $e^{- a t} \sin b t$ , a decaying amplitude $e^{- a t}$ multiplying an oscillation of frequency $b$ . The real part $- a$ is the decay rate; the imaginary part $b$ is the ringing.

One particular solution, with $x open parenthesis 0 close parenthesis equals 1$ and a specific starting velocity, is

$x (t) = e^{- a t} (\cos b t + \frac{1}{2} \sin b t),$

and that is what the figure below plots. Both parameters are live.

A red oscillating curve that starts at height 1 and wiggles along the horizontal time axis inside a shrinking envelope. The slider a controls the damping: at a = 0 the wiggles keep full size forever, and as a grows they die out faster until barely one swing survives. The slider b sets how many oscillations fit in the window. — Damped free vibration $x (t) = e^{- a t} (\cos b t + \frac{1}{2} \sin b t)$ . The slider $a$ is the damping and the slider $b$ the oscillation frequency; together they are the real and imaginary parts of the characteristic roots.

Explore

Observe. Set $a equals 0$ , $b equals 3$ . With no damping the motion repeats forever at constant amplitude. Count how many complete oscillations fit in the window, and predict what setting $b equals 1.5$ will do to that count before you try it.
Predict. Keeping $b equals 3$ , what changes as $a$ grows from $0$ to $0.5$ — the height of the wiggles, the spacing of the wiggles, or both? Commit to an answer.
Verify. Drag $a$ to $0.5$ and check. The curve stays inside an invisible envelope $\pm C e^{- a t}$ ; the wiggle spacing is set by $b$ alone. Confirm the two sliders act independently.
Observe. Now push $a$ to $2$ with $b$ still at $3$ . How many times does the curve cross zero before it becomes indistinguishable from the axis? Compare with the count at $a equals 0.3$ .
Predict, then verify. Set $a equals 2$ and drag $b$ down to $0.5$ . The curve barely completes a single swing. What would $b equals 0$ mean for the characteristic roots, and what shape would the motion have?

Connect

Written in the standard form $x^{″} + p x^{'} + q x = 0$ , the roots are $r = (- p \pm \sqrt{p^{2} - 4 q}) / 2$ , and the sign of the discriminant $p squared minus 4 q$ splits behavior into three cases. When $p squared less than 4 q$ the roots are complex, $- a \pm i b$ with $b greater than 0$ : underdamped motion, oscillating inside a decaying envelope, everything the figure shows with $b$ positive. When $p squared greater than 4 q$ both roots are real and negative: overdamped motion, a sum of two decaying exponentials that crosses zero at most once and never oscillates. The boundary case $p squared equals 4 q$ gives a repeated real root and is called critically damped; its solutions are $(c_{1} + c_{2} t) e^{- a t}$ , and among all damping choices it returns the mass to rest fastest without overshoot.

That last fact is engineering, not trivia. A screen door closer, a car's shock absorber, and an instrument needle are all deliberately tuned near critical damping — enough resistance to kill the ringing, not so much that the return crawls. In the figure, dragging $a$ upward with $b$ fixed walks you toward that tuning: the oscillations you have to eliminate are visible right up to the point where the last one disappears.

An original work of XYZ Homework, built around interactive XYZ 3D figures, following the standard first-course sequence (first-order equations through systems and the phase plane). Distinct from the LibreTexts-sourced Differential Equations for Engineers (Lebl) edition, which has its own attribution. License: CC-BY-NC-SA-4.0.