Q$2.$ A five bay$-$three story shear frame has rigid beams and lumped masses at each floor level. The masses are equal to $m$ at each floor level. Assume that all the members are axially riqid.

$E=30000$MPa;$h=3m$;$m=20$ tons.

Determine

a$)(7$ p$.)$ the equation of motion for undamped free vibration,

b$)(7$ p$.)$ the natural frequencies.

c$)(7$ p$.)$ mode shapes.

d$)(7$ p$.)$ normalized mode shapes and show that the mode shapes are orthogonal with respect to the mass matrix.

e$)(7$ p$.)$ the modal displacement coordinates, if the $u_o=\left[\begin{array}{c}1\mathrm{.}0\\ 3\mathrm{.}0\end{array}\right].$