The arrangement of the compressor, turbine, and generator in a thermal power plant is shown in Fig. 6.11. This arrangement can be considered as a torsional system where J; denote the mass moments of inertia of the three components (compressor, turbine, and generator), M; indicate the external moments acting on the components, and k; represent the torsional spring constants of the shaft between the components, as indicated in Fig. 6.11. Derive the equations of motion of the system using Lagrange's equations by treating the angular displacements of the components 0; as generalized coordinates.