The multi$-$store building shown in Fig. $2-($a$)$ has an elastic foundation that consists of many heavy$-$duty structural springs and dashpots to limit the vertical vibrations. Suppose that an earthquake hits and causes the ground to move by the sine wave: $y(t)=0\mathrm{.}05sin(20t)$ as shown in the equivalent model of Fig. $4-($b$).$ The total mass of the building is $m_{eq}=500,000kg,$ and the elastic behavior of the building foundation is modeled by an equivalent spring stiffness of $k_{eq}=32{10}^6\frac{N}{m},$ and equivalent damping coefficient of: $c_{eq}=4{10}^{5}N\frac{s}{m}.$

Fig. $2-($a$)$

Fig. $2-($b$)$

a$)$ Determine the amplitude $(x)$ of the steady$-$state vibrations of the building. $($Show your work$).$

b$)$ Determine the earthquake magnitude $(Y)$ for which the building amplitude reaches: $x=0\mathrm{.}05m,$ which may cause the building to collapse $($Suppose the rest of the parameters remain the same as in part $($a$)).$

NOTE: The numerical values of this problem are arbitrarily chosen to provide a simple case study. They may not reflect the values of a real building under a real earthquake.