Abstract A new exact general approach to study the free and forced torsional vibrations of a system with N-stepped changes in its properties, governed by the one-dimensional wave equation, and supported by N + 1 springs and dampers, and with concentrated masses has been developed. This approach is computationally efficient as it leads to a single equation for any N . The closed form frequency equations are presented for N = 1-4. The modifications required to apply this approach to the transverse vibrations of strings, longitudinal vibrations of stepped bars, longitudinal vibrations of partially embedded rods, elastically attached masses, repeated sections and torsional vibrations of coupled system of shafts are presented. The theoretical predictions were checked using simple and complex torsional vibrating systems, and they agree with the previous theoretical and experimental results and with the additional results obtained using a finite element method and a lumped parameter method. Results are also presented where a multi-stepped system was used as an approximation to a continuously changing system, and they agree with the previous results.