The propeller of a ship, of weight \(10^{5} \mathrm{~N}\) and polar mass moment of inertia \(10,000 \mathrm{~kg}-\mathrm{m}^{2}\), is connected to the engine through a hollow stepped steel propeller shaft, as shown in Fig. 3.47. Assuming that water provides a viscous damping ratio of 0.1, determine the torsional vibratory response of the propeller when the engine induces a harmonic angular displacement of \(0.05 \sin 314.16 t \mathrm{rad}\) at the base (point \(A\) ) of the propeller shaft.

Figure 3.47:-

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(a) Water (provides damping) 20 m 30 m 0.4m A 0.4 m Engine 0.6 m 0.2 m Propeller (105 N) Hollow stepped propeller shaft Engine vibratory disturbance (acos wt) (b) FIGURE 3.47 Propeller of a ship.