Refer to Figure P4.77, which shows a turbine driving an

Chapter , Problem 4.77

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Refer to Figure P4.77, which shows a turbine driving an electrical generator through a gear pair. The diameter ratio of the gears is \(D_{2} / D_{1}=1.5\). The inertias in \(\mathrm{kg} \cdot \mathrm{m}^{2}\) of gear 1 and gear 2 are 100 and 500, respectively. The turbine and generator inertias are 2000 and 1000, respectively. The torsion stiffness of shaft 1 is \(3 \times 105 \mathrm{~N} \cdot \mathrm{m} / \mathrm{rad}\), and that of shaft 2 is \(8 \times 104 \mathrm{~N} \cdot \mathrm{m} / \mathrm{rad}\). Disregard the shaft inertias and obtain the natural frequencies of the system.