Principal Parametric Instability Analysis of Rotating Beams with Varying Rotational Speed

Parametric resonance is one of the key topics in nonlinear dynamic analysis of structures. In this paper nonlinear dynamic analysis of rotating beams with varying rotational speed in the presence of principal parametric resonance is investigated. The equations of motion are based on the von Karman strain-displacement relationship. The beam is isotropic and has a rectangular cross section. The flapping and axial motions are considered along the thickness and length of the beam. The multiple scales method is applied directly to the ordered equations of motion to nonlinear dynamic instability analysis. The multiple scales method delivers a closed form equation for the instability region of principal parametric resonance in terms of the adimensional rotational speed, axial mode frequency and damping coefficient. The Galerkin discretization approach is implemented to determine the natural frequencies. The results cover the effects of damping coefficient and mode number on the critical parametric excitation amplitude and also the parametric excitation frequency. Also, damping coefficient, mode number and parametric excitation amplitude influences on the instability boundaries are examined