II$.$ Damped Free Vibration. The structural frame $($shear building$)$ shown has a rigid beam and is

rigid jointed at both ends of the three columns. The mass of the structure of $5000kg$ is

concentrated at the top. The columns are $3\mathrm{.}0m$ long and each has an $EI=4\mathrm{.}5{10}^{6}N-m^2.$ The

structure has a viscous damping $($h$)$ of $4\%.$ Model the structure as an SDOF shear building.

a$.$ Calculate the undamped dynamic properties $(f,T,)$

b$.$ Calculate the damped dynamic properties

c$.$ The top was displaced to the right by $25mm$ and then released. Compute the displacement,

velocity and acceleration after one period. Plot the displacement as a function of time for at

least three periods.

d$.$ Using the plot in $($c$)$ determine the damping ratio $($h$)$ using the log decrement method.

Compare with the given $h.$