NapkinCalc

Differential Equations B — Oscillation & Damping

Damping — oscillation meets decay

continues from lesson 1 — values defined earlier in the course stay live here

ELI5: add friction and the two stories from this track combine: the exponential decay of part A becomes an envelope that slowly strangles the oscillation. The result, e^(−ct)·cos(ωt), is the heartbeat of every real vibration — nothing rings forever.

exp(-0.4 * x) * cos(3 * x)
-0.500.510246810

damped oscillation — a decaying envelope squeezing the cosine

✓ pass abs(exp(−0.4⋅2)⋅cos(3⋅2))≤exp(−0.4⋅2)\mathrm{abs}\left(\mathrm{exp}\left(-0.4 \cdot 2\right) \cdot \mathrm{cos}\left(3 \cdot 2\right)\right) \le \mathrm{exp}\left(-0.4 \cdot 2\right) the decaying envelope e^(−0.4t) bounds the wiggle

Real-world hook: damping is shock absorbers smoothing a bump, a plucked guitar string fading, and the engineered decay that keeps a bridge from shaking itself apart.

Try it yourself: the amplitude envelope is e^(−0.4t). When has it decayed to half (e^(−0.4t) = 0.5)?

tenv=log(2)0.4t_{env} = \frac{\mathrm{log}\left(2\right)}{0.4} = 1.73291.7329 ✏️ Your turn: find when the envelope e^(−0.4t) reaches 0.5. The check evaluates the envelope at your t. (Hint: t = ln 2 / 0.4.)
✓ pass abs(exp(−0.4⋅tenv)−0.5)<0.001\mathrm{abs}\left(\mathrm{exp}\left(-0.4 \cdot t_{env}\right) - 0.5\right) < 0.001 green when the envelope has halved

You now hold the full Mathematics spine: algebra to manipulate, trig for waves, calculus to differentiate and accumulate, matrices for systems, and differential equations to model anything that moves. This is the math engineering is written in.