A simple structure is idealized as a single$-$degree$-$of$-$freedom $[$SDF$]$ system and is shown in Figure $3$ below. Its whole mass $12,000Kg$ is assumed to be concentrated at top which is supported by two steel columns with hollow cross$-$sections, The columns are assumed massless, rigid in vertical direction and fimly fixed to the rigid ground. Important structural dimensions and the column's cross$-$section are shown in Figur $3.$ The modulus of elasticity of steel is $2{10}^{11}\frac{N}{m^2}$

Figure a: A simple structure supportef by two columns and a lateral spring

Task $1)$ Determine the natural fraquency $($f$),$ natural circular frequency $($omega$)$ and the natural time period $($ T $)$ of this simple structure.

Task $3$: Determine and plot the undamped free vibration response of the system for the following three initial conditions.

a$)$ u$(0)=0\mathrm{.}03m1st$ derivative $ofu(0)=0$ m$/$s

b$)u(0)=0\mathrm{.}03m,1st$ derivative $ofu(0)=0\mathrm{.}2\frac{m}{s}$

c$)u(0)=0\mathrm{.}03m,u(0\mathrm{.}5sec)=-0\mathrm{.}02\frac{m}{s}$

Overlay the plots of dislacement responses for each case in the same flgure,

Tack $4.$ Consider the case when the simple structue shown in Flgure $3$ is modeled with an energy dissipation mechanism in the form of viscous damping. Suppose that the critical damping ratio $($E$)$ of this structure is $0\mathrm{.}50.$ Determine and plot the damped displacement response of structure for all three sets of intial condions given in Task $3.$ Compare the displacement$-$vs$-$time plots for undamped and damped cases. Overlay the plots of displscement responses for both cases in the same figure.