Torsional vibration analysis of shaft with multi inertias

An analytical method is proposed to investigate the torsional vibration of the uniform circular shaft with multiple concentrated inertias. The governing equation is established based on the Hamiltonian principle and verified by the dynamical method. The theoretical solutions of frequencies and mode shapes under different boundary conditions are obtained using the separation variable method and integral transformation. The effectiveness of the proposed method is verified by comparison with existing literature. Considering the change of the magnitudes/positions/number of concentrated inertias, and different boundary conditions, the natural frequencies and mode shapes are discussed. Several general rules are obtained. Moreover, some interesting phenomena have been found and explained. The analytical method has applications in the design of shafting with multiple concentrated inertias and the reliability checking of the “approximate” solutions.