Solution
= Solution
The amplitude envelope falls by $e^{-1/2}$ when $e^{-\gamma t/2}=e^{-1/2}$, so $t=1/\gamma$. Dividing by the period $2\pi/\omega_d$ gives
$$
\boxed{N=\frac{\omega_d}{2\pi\gamma}\simeq\frac{\omega_0}{2\pi\gamma}.}
$$
This is a cycle-count estimate in the small-damping regime, not necessarily an integer. With zero damping there is no amplitude decay and no finite such time.