Resonance in Forced Oscillations (Maximum amplitude when driving frequency ω_d = natural)
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Maximum amplitude when driving frequency ω_d = natural frequency ω₀ = √(k/m)
What each symbol means
| Symbol | What it stands for |
|---|---|
| b | Damping constant — smaller b gives sharper and taller resonance peak (kg/s) |
| ω_d | Angular frequency of external driving force (rad/s) |
| A_max | Maximum amplitude at resonance (m) |
| ω₀ | Natural angular frequency of the system (rad/s) |
When to use this
In forced oscillations, a system driven by F = F₀cos(ω_d·t) oscillates at ω_d (not ω₀) in steady state. Amplitude is maximum when ω_d ≈ ω₀ (resonance). Sharpness of resonance increases with decreasing damping. Examples: soldiers breaking step on bridge, tuning radio, swing pushed at natural frequency.