[[../02 Areas/Physics/Damped Harmonic Oscillators#SHM with driving forces (Resonance)]] Example: Let there be some damped spring system. ![[../00 Excalidraw/202509291340 .excalidraw]] $\huge \begin{align} k &= 1 \pu{ N/m }\\ m &= 200 \pu{ {kg} }\\ b &= 700 \pu{ \frac{N\cdot s}m } \\ \omega &= 1 \pu{ rad/s } \\ \omega_{0}^2 &= \frac{k}{m} = 5\cdot 10^{-3} \\ \omega_{0} = \sqrt{ 5\cdot 10^{-3} } &= 2\cdot 10^{-1} \end{align}$ $ \huge \begin{align} \omega_{0}^2 - \omega^2 &= 5 \cdot 10^{-3} - 1 = -0.995 \\ A_{0} &= \frac{F_{0}}{200}\pa{ (-0.995)^2 + \frac{700^2 1^2}{200^2} }^{-1/2}\\ A_{0} &= \frac{F_{0}}{200} \cdot \frac{1}{\sqrt{ 13.25 }}\\ \phi_{0} &= \arctan\pa{ \frac{-0.995}{ 1 \cdot \frac{700}{200} } } \\ &\approx -16\degree\\ \end{align}$ So the full solution for $\vec x(t)$ is $\huge \vec x(t) = \frac{F_{0}}{200} \cdot \frac{1}{\sqrt{ 13.25 }} \sin(t - 16\degree) $