A motor shaft of diameter \(50 \mathrm{~mm}\) drives a pump shaft of diameter \(100 \mathrm{~mm}\) through a spur gear pair, as shown in Fig. 17.58 . The motor rotor has a moment of inertia of \(500 \mathrm{~kg} \cdot \mathrm{m}^{2}\) and the pump rotor has \(1500 \mathrm{~kg} \cdot \mathrm{m}^{2}\). The speed ratio is \(3: 1 . G=84 \mathrm{GPa}\). Calculate the natural frequency of the gear system.

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Motor 50 mm 500 kg-m Pinion -100 mm Pump Gear 1500 kg.m 250 500 mm- mm Fig.17.58 Geared system