$12\mathrm{.}1$ Figure P$12\mathrm{.}1$ shows a shear frame $($i$.$e$.,$ rigid beams$)$ and its floor masses and story stiffnesses. This structure is subjected to harmonic horizontal force $p(t)=p_{o}sint$ 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 $u_{10}$ and $u_{20}$ as functions of the excitation frequency. Use appropriate normalizations of the displacement and frequency scales. Neglect damping.

gure P$12\mathrm{.}1$