Calculate the natural frequencies of a uniaxial bar clamped at both ends. Use:

(a) two,

(b) three, and

(c) four elements of equal length. Perform the analysis using both consistent mass matrix and lumped mass matrix. Plot the frequencies as a function of the number of elements. Comment on your observations. Note that the exact frequencies for a clamped clamped bar are: \(\omega_{n}=(n \pi / L) \sqrt{E / ho}\).

The properties of the bar are: length \(=0.6 \mathrm{~m}\); area of cross section \(=10^{-3} \mathrm{~m}^{2}\); Young's modulus \(=75 \mathrm{GPa} ;\) density \(=3,000 \mathrm{~kg} / \mathrm{m}^{3}\).