Free vibration of a rotating non-uniform beam with arbitrary pretwist, an elastically restrained root and a tip mass

The governing differential equations for the coupled bending–bending vibration of a rotating beam with a tip mass, arbitrary pretwist, an elastically restrained root, and rotating at a constant angular velocity, are derived by using Hamiltonʹs principle. The frequency equation of the system is derived and expressed in terms of the transition matrix of the transformed vector characteristic governing equation. The influence of the tip mass, the rotary inertia of the tip mass, the rotating speed, the geometric parameter of the cross-section of the beam, the setting angle, and the pretwist parameters on the natural frequencies are investigated. The difference between the effects of the setting angle on the natural frequencies of pretwisted and unpretwisted beams is revealed.