The bar shown in figure 8.5 is fixed at one end and free at the other end. Use two bar elements to estimate the natural frequencies and corresponding mode shapes. The length of the bar is \(2 \mathrm{~m}\), the axial rigidity \(E A=10^{6} \mathrm{~N}\), and the mass per unit length of the bar \(ho A=\) \(0.042 \mathrm{~kg} / \mathrm{m}\). Determine the natural frequencies of the bar using:

(a) consistent mass matrix; and

(b) lumped mass matrix.

 

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1 E1 2 E2 3 U1 1 m U2 1m Figure 8.5 Vibration of a clampedfree bar modeled using two elements