Find the first two natural frequencies of vibration in the vertical direction of the floor of the suspension bridge shown in Fig. 8.27 under the following assumptions:

1. The floor can be considered as a uniform beam with simple supports at both ends \(C\) and \(D\).

2. The floor has a width \((w) 12 \mathrm{~m}\), thickness \((t) 0.75\mathrm{~m}\) and weight, including the supporting girders, \(3000 \mathrm{~N} / \mathrm{m}\).

3. The Young's modulus of the floor is \(175 \mathrm{GPa}\).

Figure 8.27:-

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A' C' Tower Cables B h = 50 m Floor 1 = span 500 m FIGURE 8.27 A suspension bridge. B' Tower D'