Consider the torsional mass-spring system in Figure 5.63. The mass moments of inertia of the two disks are \(I_{1}\) and \(I_{2}\), respectively. The massless torsional springs represent the elasticity of the shafts.

a. Draw the necessary free-body diagrams and derive the differential equations of motion. Provide the equations in the second-order matrix form.

b. Determine the transfer functions \(\Theta_{1}(s) / T(s)\) and \(\Theta_{2}(s) / T(s)\). All the initial conditions are assumed to be zero.

c. Determine the state-space representation with the angular displacements \(\theta_{1}\) and \(\theta_{2}\) as the outputs.

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FIGURE 5.63 Problem 3. K 0000 1 K 02 0000 12