Use two elements of equal length to determine the natural frequencies and mode shapes of a clampedclamped beam shown in figure 8.6. The length of the beam is \(2 \mathrm{~m}\), \(E=100 \mathrm{GPa}, A=10^{-5} \mathrm{~m}^{2}, I=10^{-8} \mathrm{~m}^{4}\), and \(ho=4000 \mathrm{~kg} / \mathrm{m}^{3}\). Determine the natural frequencies of the beam using:

(a) consistent mass matrix; and

(b) lumped mass matrix.

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1 X E1 1m V2 E2 1m 2 Figure 8.6 Free vibration of a clampedclamped beam using two beam elements 3 3