Resonance Phenomenon - Interactive Visualization

Interactive visualization of resonance phenomenon in driven harmonic oscillators - explore amplitude-frequency response curves, phase relationships, damping effects, and real-world applications

Spring-Mass System

Displacement: 0.00 m
Driving Force: 0.00 N
Natural Frequency f₀: 1.00 Hz

Amplitude-Frequency Response Curve

Current Frequency: 1.00 Hz

System Parameters

Driving Force Parameters

Quick Presets

Phase Difference Between Driving Force and Response

Phase Difference φ: 0°
φ = 0°: Force in phase with displacement (low frequency)
φ = 90°: Force in phase with velocity (at resonance)
φ = 180°: Force opposite to displacement (high frequency)

Real-Time Statistics

Amplitude
0.00 m
Max Amplitude
0.00 m
Resonance Frequency
0.00 Hz
Q Factor
0.00

Physical Formulas

Natural Frequency

f₀ = (1/2π)√(k/m)
Natural frequency of spring-mass system

Driving Force

F = F₀·cos(ωt)
Periodic external driving force

Amplitude Response

A(ω) = F₀/m / √((ω₀² - ω²)² + (γω)²)
Amplitude as function of driving frequency

Phase Difference

φ = arctan(γω/(ω₀² - ω²))
Phase lag of response relative to driving force

Real-World Applications

Swing Pushing

Pushing a swing at its natural frequency increases amplitude dramatically - the classic resonance example everyone experiences.

Bridge Resonance (Tacoma Narrows)

The famous 1940 bridge collapse caused by wind-induced vortex shedding at the bridge's natural frequency.

Musical Instruments

Resonance in instrument bodies amplifies specific frequencies, creating rich tones and acoustic feedback.

MRI Radio Frequency

MRI machines use radio frequency pulses at the Larmor frequency to resonate with hydrogen nuclei in body tissues.

Radio Tuning

Radio receivers use LC circuits tuned to resonate at specific frequencies to select stations.

Microwave Ovens

Microwaves at 2.45 GHz resonate with water molecules, heating food through molecular friction.

Understanding Resonance

What is Resonance?

Resonance occurs when a system is driven at its natural frequency, causing the amplitude of oscillation to reach its maximum. At resonance, energy transfer from the driving force to the system is most efficient.

Resonance Conditions

  • Driving Frequency = Natural Frequency: fdrive = f₀ is the primary resonance condition
  • Low Damping: Lower damping leads to sharper and higher resonance peaks
  • Phase Relationship: At resonance, the driving force is in phase with velocity (90° phase lag)

Damping Effects

Damping reduces the amplitude at resonance and broadens the resonance curve. The quality factor Q = ω₀/γ quantifies resonance sharpness - higher Q means sharper resonance. Try adjusting the damping slider to see how it affects the resonance peak!

Frequency Sweep

Use the frequency sweep preset to watch how the amplitude changes as the driving frequency sweeps through the resonance. You'll see the amplitude peak dramatically at fdrive = f₀.