Problem $2$: For the same frame asked in Problem $1,$ consider the system when the beam is rigid $)=($;

a$)$ Find the damped period $(T_D)$ if the structure is assumed to have a damping ratio of $=10\%($Write your result with three digits after the decimal$).$ Comment on your result.

b$)$ If this frame is given an initial displacement and velocity of $u(0)=u_0=5cm,u^{}(0)=u_0^{}=200c\frac{m}{s},$ respectively; compute the undamped displacement response and damped displacement responses for damping ratios $=5\%$ and ${}^{}=20\%.$ Then, plot these responses on the same $u-t$ graph for a total duration of $2$ seconds $($One $u-t$ graph having three curves$).$ You are recommended to use a time increment of $t=0\mathrm{.}01sec$ for your calculations. Comment on your plot with one sentence only.

Notes:

Undamped free vibration response: $u(t)=$Acos${}_{n}t+$Bsin${}_{n}t$ where $A=u_0$ and $B=\frac{u_0^{}}{{}_n}$

Damped $($underdamped$)$ free vibration response: $u(t)=e^{-{}_{n}t}(Acos{}_{D}t+$Bsin${}_{D}t)$

where $A=u_0$ and $B=\frac{u_0^{}+{}_n{u}_0}{{}_D}$

c$)$ If this frame is subjected to a harmonic lateral load of $p(t)=p_{0}sint($at the DOF level$)$ when it is initially at rest $)=0,u_0^{}=(0$ and if $p_0=2000-kN$ and $=0\mathrm{.}25{}_n,$ compute the undamped displacement response and damped displacement response for a damping ratio of ${}^{}=10\%.$ Then, for both undamped and damped cases, plot the transient, steady$-$state, and total $($complete$)$ responses on the same $u-t$ graph for a total duration of $2$ seconds $($Two $u-t$ graph, each graph has three curves$).$ You are recommended to use a time increment of $t=0\mathrm{.}01sec$ for your calculations. Comment on your plot with one sentence only.

Notes:

Undamped harmonic response: $u(t)=$Acos${}_{n}t+$Bsin${}_{n}t+\frac{p_0}{k}\frac{1}{1-{}^2}sint$

where $A=u_0$ and $B=\frac{u_0^{}}{{}_n}-\frac{p_0}{k}\frac{}{1-{}^2}$

Damped harmonic response: $u(t)=e^{-{}_{n}t}(Acos{}_{D}t+$Bsin${}_{D}t)+$Csin$t+$Dcos$t$

transient steady$-$state

where $C=\frac{p_0}{k}\frac{1-{}^2}{(1-{}^2{)}^2+(2{)}^2}$ and $D=\frac{p_0}{k}\frac{-2}{(1-{}^2{)}^2+(2{)}^2}$

The constants A and $B$ in the above equation can be determined by satisfying the initial conditions in the complete solution.

Remember that $=$ Damping ratio, $=\frac{}{{}_n}=$ Frequency ratio