Question $2$:

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 firmly fixed to the rigid ground. Important structural dimensions and the column's cross$-$section are shown in Figure $3.$ The modulus of elasticity of steel is $2{10}^{11}\frac{N}{m^2}.$

$G=6$

Figure $3$: A simple structure supported by two columns and a lateral spring

Task $1$: Determine and plot the undamped free vibration response of the system for the following three initial conditions.

a$)u(0)=0\mathrm{.}03m,u^{}(0)=0\frac{m}{s}$

b$)u(0)=0\mathrm{.}03m,u^{}(0)=0\mathrm{.}2\frac{m}{s}$

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

Overlay the plots of displacement responses for each case in the same figure.

Task $2$: Consider the case when the simple structure shown in Figure $3$ is modelled with an energy dissipation mechanism in the form of viscous damping. Suppose that the critical damping ratio $()$ of this structure is $0\mathrm{.}05.$ Determine and plot the damped displacement response of structure for all three sets of initial conditions given in Task $1.$ Compare the displacement$-$vs$-$time plots for undamped and damped cases. Overlay the plots of displacement responses for both cases in the same figure.