Understanding vibration of wind turbine blades is of fundamental importance. This study focuses on the effect of blade mistuning on the coupled blade-hub dynamics. Unavoidably, the set of blades are not precisely identical due to inhomogeneous materials, manufacturer tolerances, etc. This work is focused on the blade-hub dynamics of horizontal-axis wind turbines with mistuned blades. The reduced-order equations of motion are derived for the wind turbine blades and hub exposed to centrifugal effects and gravitational and cyclic aerodynamic forces. Although the blades and hub equations are coupled, they can be decoupled from the hub by changing the independent variable from time to rotor angle and by using a small parameter approximation. The resulting blade equations include parametric and direct excitation terms. The method of multiple scales is applied to examine response of the linearized system. This analysis shows that superharmonic and primary resonances exist and are influenced by the mistuning. Resonance cases and the relations between response amplitude and frequency are studied. Besides illustrating the effects of damping and forcing level, the first-order perturbation solutions are verified withcomparisons to numerical simulations at superharmonic resonance of order two. The simulation point to speed-locking phenomenon, in which the superharmonic speed is locked in for an interval of applied mean loads. Additionally, the effect of rotor loading onthe rotor speed and blade amplitudes is investigated for different initial conditions and mistuning cases.Next, a second-order method of multiple scales is applied in the rotor-angle domain to analyze in-plane blade-hub dynamics. A superharmonic resonance case at one third the natural frequency is revealed. This resonance case is not captured by a first-order perturbation expansion. The relationship between response amplitude and frequency is studied. Resonances under constant loading are also analyzed. The effect of blade mistuning on the coupled blade-hub dynamics is taken into account. To better understand parametrically excited multi-degree-of-freedom behavior, approximate solutions to tuned and mistuned four-degree-of-freedom systems with parametric stiffness are studied. The solution and stability of a four-degree-of-freedom Mathieu-type system is investigated with and without broken symmetry. The analysis is done using Floquet theory with harmonic balance. A Floquet-type solution is composed of a periodic and an exponential part. The harmonic balance is applied when the Floquet solution is inserted into the original differential equation of motion. The analysis brings about an eigenvalue problem. By solving this, the Floquet characteristic exponents and the corresponding eigenvectors that give the Fourier coefficients are found in terms of the system parameters. The stability transition curve can be found by analyzing the real parts of the characteristic exponents. The frequency content can be determined by analyzing imaginary parts at the exponents. A response that involves a single Floquet exponent (and its complex conjugate) can be generated with a specific set of initial conditions, and can be regarded as a "modal response''. The method is applied to both tuned and detuned four-degree-of-freedom examples.
Read
