Resonance: Driven Damped Oscillator

Resonance

One, two, or three spring-mass systems hang from a shared driver that oscillates at frequency ω_d. Each has independent mass, stiffness, and damping — so each has a different natural frequency ω₀ = √(k/m). When ω_d matches one system's ω₀, that one resonates wildly while the others barely move. This is the core insight of resonance: same force, selective response.

Sample Learning Goals

• Explain the conditions required for resonance.
• Identify the variables that affect the natural frequency of a mass-spring system.
• Explain the distinction between driving frequency and natural frequency.
• Explain the distinction between transient and steady-state behavior.
• Identify which variables affect the duration of transient behavior.
• Recognize the phase relationship between driver and oscillator, especially how phase differs above and below resonance.
• Give examples of real-world resonance and explain why understanding it matters.

Key Equations

• ODE: m·x'' + b·x' + k·x = F₀·cos(ω_d·t)
• Natural frequency: ω₀ = √(k/m)
• Q factor: Q = √(km) / b — higher Q = sharper resonance peak
• Steady-state amplitude: A(ω_d) = (F₀/m) / √((ω₀²−ω_d²)² + (2γω_d)²)
• Phase lag: φ = −arctan(2γω_d / (ω₀²−ω_d²))

The Phase Flip

• Below resonance (ω_d < ω₀): Mass moves nearly in phase with driver (φ ≈ 0°)
• At resonance (ω_d = ω₀): Mass lags driver by exactly 90°
• Above resonance (ω_d > ω₀): Mass is nearly anti-phase (φ ≈ −180°)

Try These Experiments

1. Find resonance: With 1 spring, slowly drag ω_d near ω₀. Watch the amplitude grow.
2. Two springs, one resonates: Switch to "2 springs". Give them different masses. Drive at one's ω₀ — it goes wild while the other barely moves. Same force, selective response.
3. Radio Tuner (3 springs): Use the "Radio Tuner" preset. Three springs with ω₀ = 2, 3.16, and 5. Sweep the driver — each lights up in turn as ω_d passes through its resonance. This is exactly how a radio tunes stations.
4. Same ω₀, Different Q: Use "Same ω₀, Different Q" preset. Three springs with identical natural frequency but different damping. At resonance, the sharp one (low damping) has huge amplitude while the damped one barely moves. This teaches why Q factor matters.
5. Sweep mode: Click "Sweep" to auto-ramp ω_d. With 3 springs you'll see each peak up sequentially.
6. Effect of damping: Compare "No Damping" (amplitude grows without bound!) vs "Heavy Damping" (barely resonates).
7. Transient behavior: Change ω_d suddenly. The time graph shows messy transient → smooth steady-state. More damping = faster settling.
8. Energy at resonance: Switch to Energy tab. The green total line shows energy accumulating at resonance until input = dissipation.

Real-World Resonance

• Tacoma Narrows Bridge (1940): Wind vortices drove the bridge at its natural frequency. Amplitude grew until the bridge collapsed.
• Wine glass shatter: High-Q resonance. A singer matching the glass's natural frequency pumps energy in faster than damping removes it.
• Car suspension: Heavily damped (low Q) on purpose — you don't want resonance when driving over bumps.
• Radio tuning: An LC circuit has a resonance peak. Tuning the frequency selects one station from the spectrum.
• MRI machines: Hydrogen nuclei resonate at a specific radio frequency in a magnetic field. The resonance signal creates the image.
• Earthquake engineering: Buildings are designed so their natural frequency doesn't match typical seismic frequencies.