Motion equations
\[ m\ddot{x} = -k\bigl(x - y(t)\bigr) - c\bigl(\dot{x} - \dot{y}(t)\bigr),\qquad y(t) = Y_0\cos(\Omega t) \]
\[ m\ddot{x} + c\dot{x} + kx = Y_0\sqrt{k^2+(c\Omega)^2}\,\cos(\Omega t - \alpha),\quad \alpha = \operatorname{atan}\frac{-c\Omega}{k} \]
\[ x(t) = A\,e^{-2\zeta\omega_n t}\cos(\omega_s t + \phi) + X\cos(\Omega t - \psi) \]
\[ \omega_n = \sqrt{\dfrac{k}{m}},\quad \zeta = \dfrac{c}{2\sqrt{km}},\quad \omega_s = \omega_n\sqrt{1-\zeta^2} \]
Dynamic amplification factor
\[ \frac{X}{Y_0} = \sqrt{\dfrac{k^2+(c\Omega)^2}{(k-m\Omega^2)^2+(c\Omega)^2}} = \sqrt{\dfrac{1+\left(2\zeta\dfrac{\Omega}{\omega_n}\right)^2}{\left(1-\left(\dfrac{\Omega}{\omega_n}\right)^2\right)^2+\left(2\zeta\dfrac{\Omega}{\omega_n}\right)^2}} \]
Phase (absolute response)
\[ \psi = \operatorname{atan}\frac{-c\Omega}{k} - \operatorname{atan}\frac{-c\Omega}{k-m\Omega^2} \;=\; \operatorname{atan}\left(-2\zeta\dfrac{\Omega}{\omega_n}\right) - \operatorname{atan}\frac{-2\zeta\dfrac{\Omega}{\omega_n}}{1-\dfrac{\Omega^2}{\omega_n^2}} \]
Relative motion
\[ s(t) = x(t) - y(t) \]
\[ m\ddot{s} + c\dot{s} + k s = m\Omega^2 Y_0\cos(\Omega t) \]
\[ s(t) = B\,e^{-2\zeta\omega_n t}\cos(\omega_s t - \delta) + S_0\cos(\Omega t - \gamma) \]
Dynamic amplification factor (relative)
\[ \frac{S_0}{Y_0} = \frac{m\Omega^2}{\sqrt{(k-m\Omega^2)^2+(c\Omega)^2}} = \frac{\left(\dfrac{\Omega}{\omega_n}\right)^2}{\sqrt{\left(1-\left(\dfrac{\Omega}{\omega_n}\right)^2\right)^2+\left(2\zeta\dfrac{\Omega}{\omega_n}\right)^2}} \]
Phase (relative response)
\[ \gamma = \operatorname{atan}\frac{c\Omega}{k-m\Omega^2} \;=\; \operatorname{atan}\frac{2\zeta\dfrac{\Omega}{\omega_n}}{1-\dfrac{\Omega^2}{\omega_n^2}} \]