Analysis of Structural Vibrations of Vertical Axis Wind Turbine Blades via Hamilton’s Principle — Part 2: Exact and Approximate Solutions

Abstract

In this part 2, we provide a detailed solution to the beam equations derived in Part 1 for a VAWT blade. The main results have been outlined in the abstract of Part 1 which includes the effects of various geometrical and physical parameters such as blade length [Formula: see text], chord length [Formula: see text], radius [Formula: see text] and angular speed [Formula: see text] as well as material damping [Formula: see text]. It is shown that among the four dimensions of deformation lateral bending is the dominant factor in determining the natural frequencies of the blade. In case the blade is rotating with a constant angular speed, the dispersion relation of the 1 degree-of-freedom (DOF) motion for lateral bending can be exactly derived that an implicit function of the natural, i.e. resonant frequencies has to be solved by a root-finding method. The mode shape function is then explicitly obtained. In particular, the natural frequencies of the 1-DOF motion for the rotating blade are shown to be conveniently approximated by simple analytical formulas in terms of those for the stationary blade and the angular speed of rotation. The dependence of the natural frequencies on the chord length can also be approximated analytically by a ratio formula for given blade length to chord ratio. In addition, resonance maps of natural frequency plotted versus various parameters are provided to fully exploit the usefulness of the main results. Through a series of analysis of 2-DOF systems, we show the respective importance of the centrifugal force, coupling deformation and the Coriolis force in modifying the natural frequencies of the 1-DOF model rather than simply ignoring any of these effects.