Five steps, in order: one mass on one spring, the same motion seen as a trajectory, what happens when you shake it at every frequency, why a square wave can hit resonance at a frequency you never drove, and a real design job where you have to make the shaking stop.
Progress0 / 5
✓
Step 1
The Spring–Mass–Damper
One mass, one spring, one damper: the equation underneath almost every wobble in engineering. Start by letting it move on its own, with no outside force at all.
Try thisStay in Free (pluck) mode and pick the Underdamped scenario.
Now noticeThe mass overshoots and rings down inside a shrinking envelope. It oscillates at very nearly the natural frequency set by k and m. Light damping barely shifts that frequency; mostly it decides how fast the ringing dies.
Break itSwitch to Overdamped, then to Critical.
Why it changesOverdamped, it crawls home without ever oscillating. Critical is the sweet spot: back to rest as fast as possible with no overshoot, which is exactly what a door closer or a car suspension is tuned for. One number, the damping ratio ζ, decides which regime you're in.
✓
Step 2
The Same Motion as a Phase Portrait
Plot position against velocity instead of position against time, and the whole history of the oscillator becomes a single curve. Every vibration problem has a picture like this hiding inside it.
Try thisChoose the Stable spiral preset and click a few places to drop trajectories.
Now noticeEvery trajectory spirals into the origin. That spiral is the ring-down from Step 1: each loop is one oscillation, and the shrinking is the damping eating energy.
Break itSwitch to the Center preset.
Why it changesWith zero damping, nothing removes energy, so trajectories close into loops and circle forever. That's the frictionless oscillator, and any trace of damping at all turns those loops into spirals.
✓
Step 3
Frequency Response & Resonance
Now shake it. Drive the oscillator with a pure sine wave, sweep the frequency, and read off how much it amplifies and how far it lags behind: its Bode plot.
Try thisLeave damping near ζ = 0.10 and sweep the drive ratio r slowly from well below 1 to well above it.
Now noticeAt low r the output follows the input in step. Near r = 1 it peaks and lags by a quarter cycle (90°). Far above, it shrinks and moves almost exactly opposite to the input (180°).
Break itPark r at 1 and drag ζ down toward its minimum.
Why it failsAt resonance, only damping limits the motion: the peak grows roughly like 1/(2ζ). Halve the damping and the worst-case shaking doubles. That's why every rotating machine is designed to stay away from its natural frequency.
✓
Step 4
The Auto-Tuner: Resonance You Didn't Ask For
Real forcing is rarely a pure sine wave. A square or sawtooth wave is a stack of harmonics, and each one passes through the frequency response from Step 3 on its own.
Try thisChoose the Square wave, keep damping low, and set the fundamental drive ω₀/ωₙ to about 0.33.
Now noticeThe output rings at three times the drive frequency. The square wave's 3rd harmonic landed right on resonance and swallowed the whole response, even though it's only a third the size of the fundamental.
Break itMove ω₀/ωₙ to 0.50, then switch the waveform to Sawtooth.
Why it changesA square wave contains only odd harmonics, so at 0.50 there's no 2nd harmonic to hit resonance and the output goes quiet. A sawtooth contains every harmonic, so its 2nd harmonic finds the resonance and the ringing comes straight back.
✓
Step 5 · Capstone
Putting It Together: Vibration Isolator Tuning
A design brief that uses everything above. A rooftop pump is shaking the office below, and you choose the mounts it sits on. Three specs pull in different directions: isolation, the startup peak, and static sag.
Try thisSoften the mounts until the operating point slides past r = √2 into the green isolation zone.
Now noticeIsolation only starts past √2. Below it, mounts make the shaking worse. Once you're isolating, low damping looks great at operating speed but fails the startup peak, because the pump passes through resonance on the way up.
Break itSwitch to Variable 900–1,800 rpm with the 15 mm sag limit.
Why it failsThe green region on the design map disappears: no mount meets all three specs. That's the correct engineering answer, not a bug. Knowing which spec to renegotiate is part of the design.
🎉 Path complete!
You've followed one oscillator from a single pluck to a real isolation design, and you've seen resonance turn up three different ways. Ready for another path, or want to revisit a step above?
EngineeringCandy · Learning Paths · your progress is saved in this browser only