Bend‐bend‐twist vibrations of a wind turbine blade

Abstract

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 mode 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. These studies show that the parametric stiffness effects should be taken into account when designing long blades. KEYWORDSblade vibrations, modal analysis, parametric excitation INTRODUCTIONVibrational analysis of a wind turbine blade plays an important role in turbine design. In horizontal-axis wind turbines, failure often takes place in the hub and gearbox, because of the cyclic loads applied by the blades. To prevent failure and make improvements in turbine design, dynamics of the blades must be investigated.Blades are under cyclic loading due to turbine rotation. A steady wind speed usually varies with altitude, so as the blade rotates with the hub, it is affected by a cyclicly varying amount of wind force. Also the tangential and radial components of gravity force vary cyclicly, changing the effective stiffness of the blade with hub angle. These effects introduce parametric terms into the equations of motion.For practical reasons, the blade vibrations are usually investigated in flap-and edge-wise directions separately. 1-3 These 2 directions are uncoupled only when the product moment of inertia is zero. Yet, for a general airfoil cross section, it is not zero, which introduces coupling between 2 bending directions. Dawson 4 studied bend-bend coupled vibrations in pretwisted beams and formulated natural frequencies using energy methods. Torsional vibrations are also coupled with bending vibrations for most blades. 5-7 Dokumaci 5 derived the torsion-bending coupled equations analytically, and Bishop et al 6 improved the theory by introducing warping. Cooley and Parker also worked on bend-twist coupled vibration of spinning beams, taking the centrifugal effects into account. 8 Hodges and Dowell found the equations of motion for a blade goi...

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