Near-resonant mechanical susceptibility ($R\simeq [2m{\omega }_0(-\delta +i\gamma ){]}^{-1}$)

For the convention $x=\mathrm{Im}(RF{e}^{i\omega t})$, viscous damping $b$ gives $R=[m({\omega }_0^2-{\omega }^2)+ib\omega {]}^{-1}$. Put $\delta =\omega -{\omega }_0$ and $\gamma =b/(2m)$. If both are small relative to ${\omega }_0$, the displayed approximation follows. Its squared modulus has half-width $\gamma$, and the displacement lags the force by $\pi /2$ at $\omega ={\omega }_0$.