 Damped harmonic oscillator

A damped harmonic oscillator has displacement $x$ governed byHere $m$ is mass, $k$ the spring constant, and $-b\dot{x}$ a viscous damping force. Its mechanical energy $E=m{\dot{x}}^2/2+k{x}^2/2$ obeys $\dot{E}=-b{\dot{x}}^2$, so the model makes the loss of stored energy explicit.

The roots $r=(-b\pm \sqrt{b^2-4mk})/(2m)$ separate underdamping, critical damping and overdamping. When $b^2<4mk$, $x=e^{-bt/(2m)}(A\cos {\omega }_{d}t+B\sin {\omega }_{d}t)$ with ${\omega }_d^2=k/m-b^2/(4m^2)$. A measured ringdown tests the exponential envelope and damped frequency separately.^[ref]