Problem $1$

$($a$)$ Determine the natural vibration frequencies and mode shapes $($normalized such that the roof displacement is $1)$; express the frequencies in terms of $m,EI,$ and $h.$ Draw these modes.

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

$($c$)$ Normalize each mode so that the modal mass $M_n$ has unit value. Compare these modes with those obtained in part $($a$)$ and comment on the differences.

HINT: IT MAY BE HELPFUL TO SOLVE FOR THE MASS AND STIFFNESS MATRICES FIRST