The shear deformation effect on the nonlinear dynamic of a simple rotor blade

Abstract

The nonlinear planar response of a cantilever rotating slender beam to a principal parametric resonance of its first bending mode is analyzed considering the effect of shear deformation. The equation of motion is obtained in the form of an integro-partial differential equation, taking into account mid-plane stretching, a rotation speed and modal damping. A composite linear elastic material is considered and the cross-section properties are assumed to be constant given the assumption of small strains. The beam is subjected to a harmonic transverse load in the presence of internal resonance. The internal resonance can be activated for a range of the beam rotating speed, where the second natural frequency is approximately three times the first natural frequency. The method of multiple scales method is used to derive four-first ordinary differential equations that govern the evolution of the amplitude and phase of the response. These equations are used to determine the steady state responses and their stability. Nonlinear normal modes are obtained for the two models, considering and neglecting the effect of shear deformation. The results of the analysis show that the equilibrium solutions are influenced by the transverse shear effect. When this effect is ignored the amplitude of vibration is reduced significantly, thus altering the dynamic response of the beam. This alteration can lead to an incorrect stability prediction of the periodic solutions.