Figure P$12.$I

$12\mathrm{.}3$ Consider the system of Fig. P$12\mathrm{.}1$ with modal damping ratios $6$ s subjected to the same excitation. Derive equations for the steady$-$state displacement amplitudes of the system.

$12\mathrm{.}4$ The undamped system of Fig. P$12.$I is subjected to an impulsive force at the first$-$floor mass: $P_1(t)={}_0(t).$ Derive equations for the lateral floor displacements as functions of time.

$12\mathrm{.}5$ The undamped system of Fig. P$12\mathrm{.}1$ is subjected to a suldenly applied force at the first$-$floor mass: $p_1(t)=p_{0}t0.$ Derive equations for $($a$)$ the lateral floor displacements as functions of time, and $($b$)$ the story drift $($or deformation$)$ in the second story as a function of time.

$12\mathrm{.}6$ The undamped system of Fig. P$12\mathrm{.}1$ is subjected to a rectangular pulse force at the first floor. The palse has an amplitude $p_e$ and duration $f_d=\frac{T_1}{2},$ where $T_1$ is the fundamental vibration period of the system. Derive equations for the floor displacements as functions of time.