Parametric resonances of a three-blade-rotor system with reference to wind turbines

Coupled blade-hub dynamics of a coupled three-blade-rotor system with parametric stiffness, which is similar to a horizontal-axis wind turbine, is studied. Blade equations have parametric and direct excitation terms due to gravity and are coupled through the hub equation. For a single degree-of-freedom blade model with only in-plane transverse vibrations, the reduced-order model shows parametric resonances. A small parameter is established for large blades, which enables us to treat the effect of blade motion as a perturbation on the rotor motion. The rotor speed is not constant, and the cyclic variations cannot be expressed as explicit functions of time. Therefore, it is more convenient to use the rotor angle as the independent variable. By expressing the system dynamics in the rotor angle domain and assuming small variations in rotor speed, the blade equations are decoupled from the rotor equation. The interdependent blade equations constitute a three-degree-offreedom system with periodic parametric and direct excitation. The response is analyzed by using a first-order method of multiple scales (MMS). The system has a superharmonic and a subharmonic resonances due to direct and parametric effects introduced by gravity. Amplitude-frequency relations and stabilities of these resonances are studied. The MMS solutions are compared with numerical simulations for verification.