$2\mathrm{.}4$ A $2-$DOF mechanical system with a damper and spring in series is shown in Fig. P$2\mathrm{.}4.$ The independent displacement of the mass $m$ is $z_1$ and $z_2$ is the independent displacement of the node point between the damper $b$ and spring $k.$ The spring is undeflected when $z_1=z_2=0.$ An external force $f_a(t)$ acts on mass $m.$ Derive the mathematical model of the mechanical system $[$Hint: draw FBDs for the node and mass $m$ and neglect the inertia of the "massless node"$].$

rigure r$<mn>.</mn><mn>4</mn>$