Problem $8$

Consider the two$-$story shear building model shown in the figure, in which $x_i(t),f_i(t),c_i,$ and $k_i,$ respectively, denote the displacement, applied lateral force, mass, viscous damping coefficient, and lateral stiffness of the $i-$th floor, $i=1,2.$ The second floor is attached to the strong wall by a viscous damper with damping coefficient $c_w$ The masses are given by $m_1=m_2=0\mathrm{.}5176$ kip$*\frac{s^2}{i}n.$ From a test, the undamped natural frequency and the mode shape of the first mode were identified as ${}_1=33\mathrm{.}3ra\frac{d}{s},{}_1=[0\mathrm{.}5267,0\mathrm{.}8507],$ where the first and second element represents the first and second floors, respectively.

a$.$ Find the floor stiffnesses $k_1$ and $k_2.$

b$.$ Find the undamped natural frequency and mode shape of the second mode. Sketch both mode shapes, scaling them so that ${}_i^T{}_i=1,i=1,2.$

c$.$ Assuming proportional $($Rayleigh$)$ damping, determine the damping matrix of the given system that will result in $2\%$ critical damping in each mode. Determine the viscous damping coefficients $c_1,c_2$ and $c_w.$

d$.$ Obtain the response of the first floor when an impulse load is applied at the second floor, i$.$e$.f_1(t)=0$ and $f_2(t)=3(t),$ where $(t)$ the a Dirac delta function.