Nonlinear parametric instability of wind turbine wings

Nonlinear rotor dynamic is characterized by parametric excitation of both linear and nonlinear terms caused by centrifugal and Coriolis forces when formulated in a moving frame of reference. Assuming harmonically varying support point motions from the tower, the nonlinear parametric instability of a wind turbine wing has been analysed based on a two-degrees-of-freedom model with one modal coordinate representing the vibrations in the blade direction and the other vibrations in edgewise direction. The functional basis for the eigenmode expansion has been taken as the linear undamped fixed-base eigenmodes. It turns out that the system becomes unstable at certain excitation amplitudes and frequencies. If the ratio between the support point motion and the rotational frequency of the rotor is rational, the response becomes periodic, and Floquet theory may be used to determine instability. In reality the indicated frequency ratio may be irrational in which case the response is shown to be quasi-periodic, rendering the Floquet theory useless. Moreover, as the excitation frequency exceeds the eigenfrequency in the edgewise direction, the response may become chaotic. For this reason stability of the system has in all cases been evaluated based on a Lyapunov exponent approach. Stability boundaries are determined as a function of the amplitude and frequency of the support point motion, the rotational speed, damping ratios and eigenfrequencies in the blade and edgewise directions.