Spring-base displacement response with a grounded damper ($R(\omega )=k/(k-m{\omega }^2+ib\omega )$)

A spring's support displacement supplies force $k{x}_1$, while a damper attached to a fixed wall supplies $-b\dot{x}$. Measuring from static equilibrium gives $m\ddot{x}+b\dot{x}+kx=k{x}_1$. The sinusoidal displacement transfer function is the displayed ratio, with low-frequency limit1 and high-frequency limit $-k/(m{\omega }^2)$. Moving both the spring and damper supports would introduce a different numerator. A forced particular response must be supplemented by a transient unless initial conditions already select steady motion.