"The dynamics of a wind turbine blade under bend-bend-twist coupled vibrations is investigated. The potential and kinetic energy expressions for a straight nonuniform blade are written in terms of beam parameters. Then, the energies are expressed in terms of modal coordinates by using the assumed modes method, and the equations of motion are found by applying Lagrange's formula. The bend-bend-twist equations are coupled with each other, and have stiffness variations due to centrifugal effects and gravitational parametric terms which vary cyclicly with the hub angle. To determine the natural frequencies and mode shapes of the system, a modal analysis is applied on the linearized coupled equations of constant angle snapshots of a blade with effects of constant speed rotation. Lower modes of the coupled bend-bend-twist model are dominantly in-plane or out-of-plane modes. To investigate the parametric effects, several blade models are analyzed at different angular positions. The stiffness terms involving centrifugal and gravitational effects can be significant for long blades. To further see the effect of blade length on relative parametric stiffness change, the blade models are scaled in size, and analyzed at constant rotational speeds, at horizontal and vertical orientations. Blade-hub dynamics of a horizontal-axis wind turbine is also studied. Blade equations are coupled through the hub equation, and have parametric terms due to cyclic aerodynamic forces, centrifugal effects and gravitational forces. The modal inertia of a single blade is defined by the linear mass density times the square of transverse displacements from blade's undeflected axis. For reasonable transverse displacements, the modal inertia of a blade is usually small compared to the rotor inertia which is the combined inertia of the hub plus all three blades about the shaft. This 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. By casting the rotor angle as the independent variable, and assuming small variations in rotor speed, the leading order blade equations are decoupled from the rotor equation. The interdependent blade equations constitute a three-degree-of-freedom system with periodic parametric and direct excitation. The response is analyzed by using the method of multiple scales. The system has superharmonic and subharmonic resonances due to direct and parametric effects introduced by gravity. Amplitude-frequency relations and stabilities of these resonances are studied. The Mathieu equation represents the transient dynamics of a single-mode blade model. Approximate solutions to the linear unforced Mathieu equation, and their stabilities, are investigated. Floquet theory shows that the solution can be written as a product between an exponential part and a periodic part at the same frequency or half the frequency of excitation. An approach combining Floquet theory with the harmonic balance method is investigated. A Floquet solution having an exponential part with an unknown exponential argument and a periodic part consisting of a truncated series of harmonics is assumed. Then, performing harmonic balance, the Floquet exponents and and harmonic coefficients are found. From this frequencies of the response and stability of the solution are determined. The truncated solution is consistent with an existing infinite series solution for the undamped case. The truncated solution is then applied to the damped Mathieu equation and to parametric excitation with two harmonics. Solutions and stability of multi-degree-of-freedom Mathieu-type systems are also investigated. A procedure similar to the one applied for the Mathieu equation is used to find the initial conditions response, frequency content, and stability characteristics. The approach is applied to two- and three-degrees-of-freedom examples. For a few parameter sets, the results obtained from this method are compared to the numerical solutions. This study provides a framework for a transient analysis of three-blade turbine equations."--Pages ii-iii.
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