Internal, primary and combination resonances of a wind turbine blade with coupled flapwise and edgewise motions

Abstract

In recent decades blades of wind turbines have been designed longer and longer for better performance of energy harvesting. The growth in structural flexibility has strengthened the effect of geometric nonlinearity and led to complicated nonlinear dynamic behavior of vibration response. In this paper, an investigation on internal, primary and multi-mode combination resonances of a flexible wind turbine blade with coupled flapwise and edgewise motions is studied. Through the Hamilton's principle, the dynamical model is established using the Euler-Bernoulli beam considering the actions of gravitational and unsteady aerodynamic force. The displacement of the blade is expanded into four basic modes based on the Galerkin's method, three in flapwise direction and one in edgewise direction. The governing equations of the four modal coordinates are solved using the method of multiple scales in transient and steady-state. For the transient responses, the internal and multi-mode combination resonances are revealed by the transfer of energy between resonant modes. Unlike the case of primary and internal resonances, the quasi-periodic under the multi-mode combination resonance is discovered through phase diagram of the first edgewise mode. For the steady-state responses, amplitude-frequency curves and phase diagrams of resonance modes under the primary and multi-mode combination resonance are derived. Influence of the combination resonance on the primary resonance is discussed for the steady-state responses with different aerodynamic force and geometric nonlinearity. Finally, parametric analyses are presented to show how steady-state motion of flapwise and edgewise modes vary with detuning parameters and design parameters.