Question $2-$ Forced vibration

a$)$ Imagine you have a SDOF system as shown in Figure $2-$a$.$ You are asked to imagine an equivalent SDOF with one story similar to Figure $1-$b$.$ Please label the mass and the stiffnesses, the external force i$.$e$.$ fill in the five blanks depicted in Figure $1-$b$!$ Hint: Assume that the columns have the same lateral stiffness.

b$)$ Write down the equation of motion for the SDOF system depicted in Part a$.$

c$)$ Write down the solution for the eauation of motion in Part b$,$ i$.$e$.$ find $x(t)!$ Assume the system is initially at rest, that is $x_0$ and $x_0^{\text{'}}$ is equal to zero!

d$)$ Assume the following system characteristics for the SDOF under undamped harmonic loading $($shown in Figire $2$c with a mass of $20$ tons, a critical damping ratio of $0,$ column dimensions of $25cm40cm(E=30,000,000k\frac{N}{m}2),$ external time$-$dependent force of $P(t)=1000sin(5t)kN.$ The system is initially at rest Part c$.$ Calculate the system's lateral stiffness in $k\frac{N}{m},$ system's angular frequency in rad$/$sec$,$ maximum displacement in meters, maximum shear force in one column! Please do the calculations by hand and show them on your assignment.

e$)$ Use NONLIN software to repeat Part d$,$ compare your hand calculations with NONLIN results! You can take $3$sec as the duration of the force and $4$ sec as the length of signal.

f$)$ What happens to the displacement amplitudes if you would change the period of the forcing function so that it equals the structural period? Plot the displacement time history and explain the resulting resonance phenomenon using dynamic magnifier concept? Hint: $=\frac{{}_{\text{f}\text{o}\text{r}\text{c}\text{e}}}{{}_{str}}$ and $\frac{1}{1-{}^2}$ is named as dynamic magnifier.

g$)$ Now add damping to the system