Nonlinear Oscillations and Stability of Rotating Low-Aspect-Ratio Blades Containing Internal Resonance

Abstract

Rotating orthotropic blade equations based on the assumptions of the plate theory and the nonlinear von Kármán strain–displacement relations are obtained, and, furthermore, updated equations associated with prestressed configuration are provided. The extended Galerkin method is employed to determine the prestressed configuration due to the rotation and to obtain the linear frequencies and normal modes of updated equations. The multiple scales method, in combination with the Galerkin method, is used to study the nonlinear vibrations and to obtain analytical relations for the nonlinear frequencies and mode shapes. The role of different parameters on the backbone curves and effective nonlinearity coefficients is evaluated. In the nonlinear analysis, internal resonance is also considered. Moreover, the fourth-order Runge–Kutta method is applied to numerically analyze the nonlinear responses. A detailed investigation shows that in the torsion modes, the aspect ratio and the stiffness ratio changes influence the nonlinearity type and quality of the results. The quantities of effective nonlinearity coefficients depend on the rotational speed and the types of material and geometry. In the case of internal resonance, the stability of the nonlinear modes has been evaluated, and it is found that the rotating blades have different stable and unstable coupled modes.