Single degree of freedom system $m\ddot{q}(t) + c\dot{q}(t)+kq(t)=f(t)$ With initial conditions defined as $q(0)=q_{0}$ $\dot{q}(0)=\dot{q}_{0}$ ##### Free Vibration Natural harmonic oscillation from some initial displacement or velocity Governing equation: $m\ddot{q}(t) + c\dot{q}(t)+kq(t)=f(t)$ or $\ddot{q}+2\zeta\omega\dot{q} + \omega^{2}q = k\omega^{2}f(t)$ **Undamped Case** $q(t) = q_{0}\cos\omega t+\frac{\dot{q_{0}}}{\omega}\sin\omega t$ or $q(t) = Q\cos(\omega t-\theta)$ where $Q = \sqrt{ q_{0}^{2}+\left( \frac{\dot{q}}{\omega} \right)^{2} }$ and $\theta = \arctan\left( \frac{\dot{q_{0}}/(\omega Q)}{q_{0}/Q} \right)$ $\omega=\sqrt{ \frac{k}{m} }$ is the undamped natural frequency $2\zeta\omega = \frac{c}{m}$ $\frac{\zeta}{\sqrt{ 1-\zeta^{2} }}= \frac{1}{2\pi n}\ln \frac{Q_{i}}{Q_{i+n}} \approx \zeta$ for small $\zeta$