$4\mathrm{.}89.$ Use Lagrange's equations to derive the equations of motion for the building model presented in Figure $4\mathrm{.}9$ of Example $4\mathrm{.}4\mathrm{.}3$ for the undamped case.

Figure $4\mathrm{.}9($a$)$ A simple model of the horizontal vibration of a four$-$story structure. Here each floor is modeled as a lumped mass, and the walls are modeled as providing horizontal stiffness. $($b$)$ The restoring forces acting on each mass $($floor$).$