Discussion
A structure with zero damping subjected to a harmonic force at its natural frequency would
Answer: C
Correct answer: build up amplitude without limit
At resonance the magnification is \( 1/2\xi \), which grows without bound as \( \xi \to 0 \). With no damping the forcing adds energy every cycle and nothing removes it, so the amplitude grows linearly with time.
Real structures always have some damping, so the response reaches a finite steady state - but the mathematics shows why damping is what limits resonant response, and why a small change in it matters so much near a natural frequency.
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