Analysis for the free vibration of rotating variable cross-section beam

This paper studies the free vibration characteristics of a rotating variable cross-section flexible beam mounted on a rigid hub system, especially foucses on the coupling effects of the dynamic of the hub on the rotating beam. Firstly, we employ the Hamilton principle to derive the governing equation of motion of the Euler-Bernoulli beam and the equation for the hub's dynamics, in which the coupling between the hub and the beam is taken into account in the modeling. The coupling inculdes the effects from the centrifugal force and the geometrical nonlinearity of the beam's deformation. Secondly, the Fourier transformation is used to convert the partial differential equation into an ordinary differential equation, which can then be efficiently solved by using the difference method for obtaining the natural frequencies and mode shapes of the rotating beam. Furthermore, we validate our model by comparing the results from the reduced model with those found in the existing literatures, and then investigate how the natural frequencies and mode shapes of the rotating beam are affected by the geometric features of the beam's cross-section, the rotating speed and the inertial ratio between the hub and the beam. We conclude that the steady-state response of the coupled nonlinear hub-beam system not only depends on the inherent vibration characteristics of the flexible beam in structural dynamics, but also it is closely related to the external dynamic environment.