Non-linear dynamic response of a wind turbine blade

Abstract

The non-linear equations of motion for an isolated three-degree flap-lag-feather rotor blade under the action of a gravity field are derived, by using Lagrange's equations, for arbitrarily large angular deflections. Quasi-steady airfoil theory is used to obtain aerodynamic forces for sheared wind. A consistent set of non-linear equations are obtained by using non-linear terms up to the third order. The limit cycle analysis for forced oscillations as well as the determination of the principal parametric resonance of the blade due to periodic forces (from gravity and sheared wind) are performed by using the harmonic balance method and solving the resulting non-linear algebraic equations numerically by the Newton-Raphson iterative technique. The stability check of the non-linear solutions is carried out by using the perturbation technique and with the assumption of slowly changing functions. The results are first obtained for a two-degree flap-lag blade and then the effect of the third degree of freedom (i.e., feather) is investigated. The effects of several parameters on the forced response of the blade are examined, including coning angle, structural damping, Lock number, inflow ratio and wind velocity gradient. The lead-lag response is primarily influenced by the gravity field whereas the wind gradient has a more prominent effect on flap motion. The self-excited flutter solutions are also obtained by using the harmonic balance method, for a blade in a uniform wind and with gravity field forces neglected. The flutter stability boundaries are obtained for several configurations. The limit cycle flutter solution of a typical configuration shows a non-linear softening spring behaviour. This reveals the possibility of sustained limit cycle flutter oscillation occuring below the linear instability speed if large enough disturbances are given to the blade.