Nonlinear super-harmonic and sub-harmonic resonance analysis of the horizontal-axis wind turbine blades using multiple scale method including the pre-twist effects

Abstract

The focus of this paper is on the nonlinear secondary resonance of HAWT blades under the super-harmonic and sub-harmonic excitations. To pursue this goal, a 3-DOF dynamic model of the blade is presented including the pre-twist effects and then the method of multiple scale (MS) method is employed to investigate the nonlinear secondary resonance. Nine different cases of secondary resonances including 3 (2) super-harmonic resonance cases of order 2 (3) as well as 2 (2) sub-harmonic resonance cases of order 1/2 (1/3) are detected. For each case, the amplitude-frequency and phase-frequency responses are derived analytically. The stability of each branch of the solution is evaluated through the Hartman-Grobman theorem and several saddle-node bifurcation points are revealed, considering the NREL 5 MW HAWT blade as the case study. Moreover, a parametric study is carried out to study the effects of damping, pre-twist, angular velocity, and excitation force amplitude. It is found that these parameters have a great influence on amplitude-frequency response (AFR) curves. Eventually, some comparisons are made to demonstrate the validity of the formulation and simulation results.