05 / CHAPTER 05
Resonance and bandwidth
Use a response curve to understand amplification and selectivity.
ANCHOR EQUATION
Natural frequency
In symbols: ω₀ = √(k/m)
The natural frequency sets the scale for resonance.
Resonance is selective energy absorption
The response peaks when the drive supplies energy at the phase needed to overcome damping. Resonance is not simply ‘matching frequencies’; phase and loss determine how much power enters the oscillator.
Read the response width
For weak damping, the amplitude curve peaks near ω₀ and its half-power bandwidth is approximately Δω = c/m. The quality factor Q ≈ mω₀/c measures how narrow and tall the response is. Larger Q means more cycles store energy before losses remove it.
For m = 0.5 kg, c = 0.20 N·s/m, k = 50 N/m, ω₀ = 10 rad/s and Q ≈ 25. The bandwidth estimate is 0.40 rad/s, so a drive 2 rad/s away is strongly attenuated compared with one near resonance.
A high Q is not always desirable
A pump mount may need to avoid resonance, while a sensor may exploit it. The same physics produces amplification or failure depending on whether the operating frequency is controlled.
Work it: A structure has f₀ = 30 Hz and Q = 12. Estimate its bandwidth in Hz using Δf ≈ f₀/Q, then decide whether a 1 Hz frequency drift matters.