Nonlinear vibration analysis of rotating beams undergoing parametric instability: Lagging-axial motion

The nonlinear free vibration and principal parametric resonance of rotating beams are investigated taking into account the lagging-axial coupling motion due to Coriolis force. This work tackles analytically the problem of parametric resonances induced by periodic modulation of the angular speed. The nonlinear equations of motion are obtained via a direct Lagrangian formulation. The method of multiple scales is employed to perform a perturbation analysis of the nondimensional equations of motion to deliver the effective nonlinearity of the lagging and axial modes and the critical conditions for the onset of parametric resonances. A comprehensive study on the effect of the rotational speed and the damping ratio on the modes nonlinearity and on the instability regions is presented. Comparisons in terms of effective nonlinearity coefficient and principal parametric resonance response were carried out so as to illustrate the importance of the exact geometrical formulation against ad hoc beam theories such as the Euler-Bernoulli beam model