Discussion
The amplitude of free vibration of a damped structure decays
Answer: D
Correct answer: exponentially with time
The free vibration solution contains the factor \( e^{-\xi\omega_n t} \), so the envelope decays exponentially.
The consequence is that the ratio of successive peaks is constant, which is exactly what makes the logarithmic decrement a well-defined quantity: taking the logarithm of a constant ratio gives the same value for every pair of peaks, so several cycles can be averaged. Linear decay would give a constant difference instead, and no such measurement would be possible.
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