A new exact approach for the analysis of torsional vibration of a non-uniform shaft carrying an arbitrary number of rigid disks is presented. The exact solutions for non-uniform shafts with arbitrary distribution of torsional stiffness or mass polar moment of inertia are obtained for several important cases. The fundamental solutions developed in this paper lead to a single frequency equation for a non-uniform shaft with classical or non-classical boundary conditions and carrying an arbitrary number of rigid disks.

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