Chapter 4.P1, Problem 1P

Denote by $y(t)$ the displacement of the platform from it equilibrium position relative to a fixed frame of reference. Relative to the same fixed reference frame, let $u(t)$ represent the displacement of the plant floor from its equilibrium position. Show that the differential equation for the motion of the platform is

$y^{″}+2\xi {\omega }_0{y}^{\prime }+{\omega }_0{}^{2}y=2\xi {\omega }_0{u}^{\prime }+{\omega }_0{}^{2}u$, (i)

where, ${\omega }_0=\sqrt{k/m}$ and $\xi =\left(\gamma /2\sqrt{mk}\right)$ is a dimensionless parameter known as the viscous damping ratio.