Classical Mechanics
Driven Oscillator and Resonance
mẍ + bẋ + kx = F₀cos(ωt) — amplitude peaks when driving frequency ω matches natural ω₀
A driven damped oscillator obeys mẍ + bẋ + kx = F₀ cos(ωt). The steady-state response is x(t) = A(ω) cos(ωt − φ) with amplitude A(ω) = F₀ / √((k − mω²)² + (bω)²). When the driving frequency ω equals the natural frequency ω₀ = √(k/m), the amplitude peaks — phenomenon known as resonance. The peak amplitude is A_max = F₀ / (bω₀) = (F₀/k) · Q (gain at resonance is the Q factor). Sharper peak = higher Q. Famous catastrophes from unaccounted resonance: Tacoma Narrows Bridge collapse 1940 (vortex-induced resonance), Millennium Bridge wobble 2000 (lateral pedestrian resonance, fixed with TMDs). Critical for opera glass shattering, MRI tuning (Larmor frequency), and laser cavity Q-switching.
- Equationmẍ + bẋ + kx = F₀cos(ωt)
- Steady stateA(ω)cos(ωt − φ)
- Resonanceω = ω₀ = √(k/m)
- Peak amplitudeF₀/(bω₀) = (F₀/k)Q
- Phase shift0 to π
- Tacoma Narrows1940 collapse
Interactive visualization
Press play, or step through manually. The visualization is yours to drive — try it before reading on.
Watch the 60-second explainer
A condensed visual walkthrough — narrated, captioned, under a minute.
Why resonance matters
- Musical instruments. The body of a violin, the air column of a flute, and the soundboard of a piano are all driven resonators selecting and amplifying the harmonics of the strings or reed.
- MRI. Hydrogen nuclei precess at the Larmor frequency ω = γB; an RF pulse at exactly that frequency tips spins out of equilibrium. Spatial gradients shift the Larmor frequency to encode position.
- Mechanical engineering. Every rotor, shaft, blade, and hull has natural modes; designs avoid running near them. Engine mounts, hard drives, and turbine blades fail at resonance.
- Atomic clocks. Cesium-133 atoms are driven at 9.192 631 770 GHz, the energy gap between two hyperfine states. The narrow atomic linewidth gives Q ≈ 10¹⁰, defining the second.
- Radio tuning. A tuned LC circuit is a high-Q driven oscillator selecting one carrier frequency from a forest of broadcast signals.
- Laser cavities. Optical resonators amplify photons at modes nλ = 2L. Q-switching dumps stored energy in a giant pulse by suddenly raising cavity Q.
- Earthquakes. Building heights are tuned away from common ground-motion frequencies; tuned mass dampers add backup.
Common misconceptions
- "Resonance is always destructive." Most resonance is benign or useful — every musical instrument, every radio receiver, every MRI scan, every quartz watch is resonance working as intended.
- "You need exact equality of frequencies." The peak has finite width Δω = ω₀/Q. Within that band the response is still strong; only outside it does the response decay.
- "Linear theory always applies." Real systems become nonlinear at large amplitude — Duffing oscillators, parametric resonance, mode coupling, and limit cycles all kick in. Tacoma Narrows was nonlinear flutter, not pure linear resonance.
- "Damping kills resonance." Damping just lowers and broadens the peak. The peak survives down to Q = 1/√2 (where it merges with low-frequency response).
- "Resonance only happens at the natural frequency." Subharmonic, superharmonic, and parametric resonances exist at ω = ω₀/n, nω₀, and 2ω₀ respectively in nonlinear systems.
- "Soldiers must break step on bridges." Modern bridges are designed not to resonate at marching frequencies; the rule is conservative tradition more than active engineering risk for new construction.
Frequently asked questions
Why does amplitude peak at the natural frequency?
The steady-state amplitude is A(ω) = F₀ / √((k − mω²)² + (bω)²). The first term in the denominator vanishes exactly at ω = ω₀ = √(k/m), where the elastic force −kx and the inertial force −mẍ cancel each other for a sinusoidal motion, leaving only the damping term bẋ to balance the drive. With the largest of the two quadratic terms killed, the denominator hits a minimum and amplitude is maximal. Formally for light damping the peak is at ω_R ≈ ω₀ √(1 − 1/(2Q²)), nudged slightly below ω₀, but for high-Q systems the difference is negligible.
What is the phase shift between driver and response?
tan(φ) = bω / (k − mω²). At low driving frequency (ω ≪ ω₀) the spring dominates, response is in phase with the drive (φ ≈ 0). At resonance (ω = ω₀) the spring and inertia cancel, response lags the drive by exactly π/2 (90°). At high frequency (ω ≫ ω₀) the inertia dominates, response is exactly out of phase (φ ≈ π). The π/2 lag at resonance is what makes pushing a swing efficient — you push at the bottom of the arc, in phase with velocity, putting maximum power into the system.
Why was the Tacoma Narrows resonance disaster?
On 7 November 1940 the Tacoma Narrows suspension bridge twisted itself apart in a 40 mph wind. The mechanism was not a simple linear resonance but aeroelastic flutter — wind-induced vortex shedding fed energy into the bridge's torsional mode at exactly its natural frequency, with positive feedback because the deck's lift coefficient changed with twist angle. Amplitudes grew from inches to feet to over five feet of twist before the deck failed. Modern suspension bridges use slotted decks, stiffening trusses, and aerodynamic fairings to suppress this coupling, plus tuned mass dampers as a backstop.
How do tuned mass dampers prevent resonance?
A tuned mass damper (TMD) is a secondary mass-spring-dashpot tuned so that its natural frequency matches the dominant mode you want to suppress. When the building (or bridge) tries to oscillate at that mode, the TMD oscillates 180° out of phase, draining energy into its dashpot. Taipei 101's TMD is a 660-tonne steel sphere on cables, visible to tourists. The Millennium Bridge's wobble was fixed in 2002 by adding 37 viscous dampers and 52 TMDs after pedestrians' lateral foot fall locked into a positive-feedback synchronization.
What is the half-power bandwidth (FWHM)?
The full width at half maximum (FWHM) Δω of the resonance peak in power (amplitude squared) is the range of driving frequencies over which the response stays within a factor of √2 of the peak amplitude. For light damping Δω ≈ ω₀ / Q. Equivalently Q = ω₀ / Δω. So a high-Q resonance is sharp — only a narrow band of frequencies excite it strongly — while a low-Q system has a broad, gentle peak. Used to characterize radio filters, laser cavities, MRI receive coils, and atomic absorption lines.
Why does pushing a swing work — anharmonic resonance?
A pumping rider on a swing is not a linear driven oscillator — they raise and lower their center of mass twice per swing cycle, parametrically modulating the effective length. This is parametric resonance: the drive is at 2ω₀, not ω₀. A toddler pushed by a parent is closer to the linear case — a small force per cycle, applied in phase with velocity (at the bottom of the arc) builds amplitude quickly thanks to the high Q of an underdamped pendulum. Both work, but they exploit different resonance physics.