$2\mathrm{.}1$ A vibration isolation system is shown in Fig. P$2\mathrm{.}1.$ Damper $b,$ connects mass $m$ to the fixed overhead

horizontal surface. The vibration mounts that support the mass on the moving base are modeled by lumped

stiffness $k$ and viscous friction $b_2.$ The displacement of the base $z_{in}(t)$ is the input to the system and $z$ is the

vertical displacement of mass $m$ as measured from the static equilibrium position. Derive the mathematical

model of this mechanical system.

Figure P$2\mathrm{.}1$