Please show hand calculations. For the system below:

a$.$ Determine the natural vibration frequencies and modes; express the frequencies

in terms of $m,EI,$ and $h.$

b$.$ Verify that the modes satisfy the orthogonality properties.

c$.$ Normalize each mode so that the roof displacement is unity. Sketch the modes

and identify the associated natural frequencies.

d$.$ Normalize each mode so that the modal mass $M_n$ has unit value. Compare these

modes with those obtained in part $($c$)$ and comment on the differences.

The structure of the previous example is modified so that the columns are hinged at the

base. Determine the natural vibration frequencies and modes of the modified system

and compare them with the vibration properties of the original structure. Comment on the

effect of the column support condition on the vibration properties.