A turbine rotor is mounted on a stepped shaft that is fixed at both ends as shown in Fig. 3.51. The torsional stiffnesses of the two segments of the shaft are given by \(k_{t 1}=3000 \mathrm{~N}-\mathrm{m} / \mathrm{rad}\) and \(k_{t 2}=4000 \mathrm{~N}-\mathrm{m} / \mathrm{rad}\). The turbine generates a harmonic torque given by \(M(t)=M_{0} \cos \omega t\) about the shaft axis with \(M_{0}=200 \mathrm{~N}-\mathrm{m}\) and \(\omega=500 \mathrm{rad} / \mathrm{s}\). The mass moment of inertia of the rotor about the shaft axis is \(J_{0}=0.05 \mathrm{~kg}-\mathrm{m}^{2}\). Assuming the equivalent torsional damping constant of the system as \(c_{t}=2.5 \mathrm{~N}-\mathrm{m}-\mathrm{s} / \mathrm{rad}\), determine the steady-state response of the rotor, \(\theta(t)\).

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Turbine rotor, Jo 8(t) k M(t) = Mo COS of FIGURE 3.51 Turbine rotor subjected to harmonic torque.