Q$.2($a$)$ A two$-$storey building is idealized as shown in Fig. Q$\mathrm{.}2.$ Determine the following dynamic properties of the building:

$($i$)$ stiffness matrix K;

$($ii$)$ mass matrix $M$;

$($iii$)$ natural cyclic frequencies of the building; and

$($iv$)$ the corresponding mode shapes of vibration.

$(5$ marks$)$

$(2$ marks$)$

$(5$ marks$)$

$(4$ marks$)$

$($b$)$ If the structure in part $($a$)$ has a damping ratio of $10\%,$ determine the damped natural cyclic frequencies and damped natural periods of vibration.$[$Hint: ${}_D=(1-{}^2{)}^{\frac{1}{2}}]$

$(4$ marks$)$

Fig. Q$.2$