Please answer all sections of the question quick as possible: There is a $5$ storey building model, as shown

in figure, with $k=7\mathrm{.}005{10}^6\frac{N}{c}m,m=$

$1\mathrm{.}0{10}^{5}kg,$ and the building is subjected to

seismic excitation $x_i,$ the corresponding

power spectral density of $x_g$ is

$S_{x_8}()=\{\frac{{}_0^4+4{}_0^2{}_0^2{}^2}{({}^2-{}_0^2{)}^2+4{}_0^2{}_0^2{}^2}\}{S}_0$

in which os$,$ so are respectively nature

frequency and damping ratio of the soil of

the building foundation.

Here ${}_0^2=242\mathrm{.}0,\frac{1}{s^2},{}_0=0\mathrm{.}6398$ and the structural

damping ratio ${}_i=0\mathrm{.}05(i=1,2,$dotsdots,$5),$ and $S_0=5\mathrm{.}75c\frac{m^2}{s^3}$

The questions are:

$(1)$Please derive the formula of Kanai$--$Tajimi

model.

$(2)$ calculate the natural frequencies and mode

shape matrix $($or mode matrix$).$

$(3)$ calculate the power spectral density of the top

storey respectively for displacement response and

acceleration response, and please draw their figures.

$(4)$Calculate the root mean squares of the top storey

of the building respectively for displacement

response and acceleration response.

$(5)$ Please use the Matlab to design the programs

and, do the calculation above.