Damped SHM

$x(t) = Ce^{-\gamma t} \cos(\omega_1 t + \phi)$

$x(t) = (C_1 + C_2 t) e^{-\gamma t}$

$x(t) = C_1 e^{-\left( \gamma - \sqrt{\gamma^2 - \omega_0^2} \right)t} + C_2 e^{-\left( \gamma + \sqrt{\gamma^2 - \omega_0^2} \right)t}$

$a(t) = -\frac{k}{m} x(t) - \frac{b}{m} v(t)$

$\gamma = \frac{b}{2m}$

$\omega_1 = \sqrt{\omega_0^2 - \gamma^2}$

$\gamma^2 - \omega_0^2 < 0$

$\gamma^2 - \omega_0^2 = 0$

$\gamma^2 - \omega_0^2 > 0$

$E_{\text{mech}}(t) = \frac{1}{2} k x(t)^2 = E_{\text{mech}}(0) e^{-\frac{t}{\tau}}$

Q = $\tau\omega_1$

$\tau = \frac{m}{b}$

$Q = 2\pi \frac{\text{initial energy stored in oscillating system}}{\text{energy dissipated in cycle}}$

Time taken for the energy of the damped oscillation to reduce to $e^{-1}$ or 36.8% of its initial value