Problem $27$

The mass matrix, stiffness matrix, and modal properties of a seismically excited three$-$storey structure are given as below.

$M=m\left[\begin{array}{ccccccc}1& 0& 0& 1& 0& 0& 0\mathrm{.}5\end{array}\right]$

$0$

$0$

$0$

$K=k\left[\begin{array}{ccccccc}4& -2& 0& 3& -1& -1& 1\end{array}\right]$

$0$

$-2$

$=\left[\begin{array}{ccccccc}0\mathrm{.}427& -0\mathrm{.}551& ???& -0\mathrm{.}355& ???& 1\mathrm{.}000& ???\end{array}\right]$

$1\mathrm{.}000$

$0\mathrm{.}745$

$x_g^{}$

a$.$ From eigenvalue analysis, the natural periods of the structure are found to be $0\mathrm{.}879,0\mathrm{.}382,$ and $0\mathrm{.}261sec$ for the $1$st$,2$nd$,$ and $3$rd modes, respectively. Based on the given information find the mode shape of the $3$rd mode $($normalize the $3$rd mode shape such that the roof displacement is $1).$

b$.$ Assuming $2\%$ damping in the first and third modes of this structure, determine the damping matrix using the Rayleigh damping approximation.

c$.$ Determine the $2nd$ modal damping ratio for the structure.

d$.$ Write the equations of motion in state space form, i$.$e$.$

$z^{}=Az+B{x}_g^{}(t)$

$y=Cz+D{x}_g^{}(t)$

where the output $y={\left[\begin{array}{cc}{u}_2^{}& u_1^{}\end{array}\right]}^T.$