$12\mathrm{.}1$ Figure P$12.$I shows a shear frame $($i$.$e$.,$ rigid beams$)$ and its floor masses and story stiffnesses. This structure is subjected to harmonic horizontal force $p(t)=p_u$ sin ot at the top floor.

$($a$)$ Derive equations for the steady$-$state displacements of the structure by two methods:

$($i$)$ direct solution of coupled equations, and $($ii$)$ modal analysis.

$($b$)$ Show that both methods give equivalent results.

$($c$)$ Plot on the same graph the two displacement amplitudes $z_{10}$ and ${?}^{**}20$ as functions of the excitation frequency. Use appropriate normalizations of the displacement and frequency it scales. Nieglect damping.

Flgure P$12\mathrm{.}1$