Flapwise dynamic response of a wind turbine blade in super-harmonic resonance

The flapwise dynamic response of a rotating wind turbine blade in super-harmonic resonance is studied in this paper, while the blade is subjected to unsteady aerodynamic loads. Coupled extensional–bending vibrations of the blade are considered; the governing equations which are coupled through linear and quadratic terms arising from rotating and geometric effects respectively are obtained by applying the Hamiltonian principle. The lth flapwise linear frequency and the rotational frequency are assumed to be in an almost 3:1 ratio, so super-harmonic resonance occurs when this linear frequency is close to the associated nonlinear frequency. By using the first n, no less than l, linear undamped modal functions as a functional basis and applying the Galerkin procedure, a 2n-degree-of-freedom discrete model with quadratic and cubic terms owing to geometric effect is derived. The generalized displacements corresponding to the discrete system are disintegrated into static and dynamic displacements. Perturbation method is adopted to get analytical solutions of the discrete dynamic system for positive aerodynamic dampings. The coning angle and the inflow ratio are chosen as two control parameters to analyze aeroelastic behaviors of the blade. By assuming that the static and dynamic displacements are of the same order in resonance region, and there is no other resonance except the super-harmonic resonance, the multiple-scales method is employed to obtain a set of amplitude modulation equations whose coefficients depend on two control parameters. The frequency-response equation is derived from the amplitude modulation equations. A method to estimate the functional dependence of the detuning parameter on two control parameters is introduced. The amplitude of the harmonic response is derived from the frequency-response equation after knowing the detuning parameter. Then the stability of the steady motion with respect to control parameters can be determined. The evolution of the dynamic response of the resonance mode with decreasing aerodynamic damping is discussed by means of both perturbation and numerical methods.