A damped spring–mass system has m=10 kg, stiffness 1000 N/m and damping coefficient 20 N s/m. Find the ratio of the first positive displacement peak to the fifth positive peak (one decimal place).
$\omega_n=\sqrt{k/m}=10$ rad/s and $\zeta=c/(2m\omega_n)=0.1$. The logarithmic decrement per cycle is $\delta=2\pi\zeta/\sqrt{1-\zeta^2}=0.6315$. Four complete cycles separate the first and fifth like-signed peaks, so $x_1/x_5=e^{4\delta}\simeq12.5$.
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