Problem $2$

$($a$)$ Determine the natural vibration frequencies and modes; express the frequencies in terms of $\backslash ($ m$,$ E I $\backslash ),$ and $\backslash ($ h $\backslash ).$ Sketch the modes and identify the associated natural frequencies.

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

$($c$)$ Normalize each mode so that the modal mass $\backslash ($ M$\_\{$n$\}\backslash )$ has unit value. Sketch these normalized modes. Compare these modes with those obtained in part $($a$)$ and comment on the differences.