$11-4.$ A rigid rectangular slab is supported by three columns rigidly attached to the

slab and at the base $($as shown in Fig. P$11-2).$

FIGURE P$11-2$

$($a$)$ Evaluate the mass and stiffness matrices for this system $($in terms of

$m,EI,$ and $L,$ considering the three displacement coordinates shown.

$($Hint: Apply a unit displacement or acceleration corresponding to each

coordinate and evaluate the forces acting in each coordinate required for

equilibrium.$)$

$($b$)$ Compute the frequencies and mode shapes of this sytem, normalizing the

mode shapes so that either $v_2$ or $v_3$ is unity. Explain the steps in detail