Driven Oscillations & Mechanical Resonance — 3D Lab Simulation
Resonant amplitude peaking when driving frequency matches natural frequency
Theoretical Foundation & Historical Context
When an oscillator with damping is driven by a periodic external force, it settles into steady-state motion. When the driving frequency matches the natural frequency ω₀ = √(k/m), the vibration amplitude surges to a dramatic peak.
Law: Damped Driven Harmonic Motion & Resonance Curve | Scientist: Robert Hooke & Lord Rayleigh | Year: 1678
Key Learning Objectives & Competencies
- ✓ Analyze the resonant peak condition where excitation driving frequency equals the system's natural frequency (ω = ω₀).
- ✓ Demonstrate how damping ratio ζ and quality factor Q dictate the sharpness and amplitude of the resonance peak.
- ✓ Evaluate the phase shift transition from 0° to -180°, crossing precisely -90° at resonance.
Mathematical Formulations & Differential Equations
m · ẍ + γ · ẋ + k · x = F₀ · cos(ω · t)
Inertial, damping, restoring, and driving forces
ω₀ = √(k / m)
Undamped natural frequency of mass-spring system
A(ω) = F₀ / √( m²·(ω² - ω₀²)² + γ²·ω² )
Resonance peak shape described by Lorentzian curve
Physical Parameters & Units of Measurement
| Symbol | Physical Quantity | SI Unit | Default |
|---|---|---|---|
| m | Oscillator mass (kg) | — | — |
| k | Spring constant stiffness (N/m) | — | — |
| γ | Viscous damping coefficient (N·s/m) | — | — |
| ω | External driving frequency (rad/s) | — | — |
Real-World & Industrial Engineering Applications
Tuned Mass Dampers installed in skyscrapers (Taipei 101, Shanghai Tower) to counteract earthquake and typhoon sway.
Nuclear Magnetic Resonance (NMR) and Medical Magnetic Resonance Imaging (MRI) body scanning.
Radio receiver RF antenna LC tank circuits selecting a single desired frequency from atmospheric broadcasts.
3-Step Controlled Experimental Protocol
- Step 1: Baseline Setup — Initialize the simulation in standard conditions and observe undisturbed system dynamics.
- Step 2: Parameter Variation — Gradually modulate physical parameters using the control panel sliders.
- Step 3: Verification & Phase Space Analysis — Compare phase space trajectories and conservation metrics against analytical ODE solutions.
Frequently Asked Questions (FAQ)
What caused the catastrophic collapse of the Tacoma Narrows Bridge in 1940?
While historically cited as simple forced resonance, modern aeroelastic analysis shows it collapsed due to aeroelastic flutter—a self-excited torsional instability where vortex shedding fed energy directly into the bridge's natural torsional mode.
What happens to resonant amplitude when damping approaches zero?
In a theoretical system with zero damping (ζ = 0), the amplitude at resonance approaches infinity over time. In real physical structures, the amplitude grows until material failure, elastic fracture, or nonlinear limits occur.
Why are soldiers ordered to break step when marching across suspension bridges?
Rhythmic marching imparts a periodic external force. If the cadence matches any natural vibrational frequency of the bridge span, mechanical resonance could amplify sway to dangerous levels, as occurred on the Broughton Suspension Bridge in 1831.
Related Simulations & Experiments
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