Analysis of Structural Vibrations of Vertical Axis Wind Turbine Blades via Hamilton’s Principle — Part 1: General Formulation

Abstract

Energy harvesting by wind turbines is of great concern in many countries/areas, yet its safety is inevitably related to the structural vibration of the turbine system. In this study, we present a complete linear analysis of structural vibrations for vertical axis wind turbines (VAWTs) based on Euler’s beam theory by Lagrangian mechanics. The un-deformed blade is assumed to be vertically straight. There are several findings from solving the resultant equations which represent four dimensions of deformation, involving motion: lateral bending-chordwise bending-axial torsion-axial extension (BBTE) (1) There is no deformation coupling between axial tension and axial torsion. (2) The natural frequencies of the blade are mainly determined by lateral bending, and [Formula: see text] ([Formula: see text]) denote the natural frequencies determined solely by lateral bending. (3) The centrifugal force credited to blade deformation is the primary factor that modifies the natural frequencies. (4) The Coriolis force can exist only in the coupled system, but in any case, the Coriolis force will not be generated by coupling lateral bending and axial tension. (5) The Coriolis force, when lateral bending is coupled with chord bending or axial torsion, can only slightly modify the natural frequencies. (6) In the case of fixed speed of rotation, [Formula: see text], where [Formula: see text] is angular speed and [Formula: see text] is the distance from the rotation axis to the elastic center of the blade: given [Formula: see text]-the blade length to chord ratio, it is found that the natural frequencies [Formula: see text] of the blade are, in close approximations, inversely proportional to [Formula: see text], i.e. [Formula: see text], where [Formula: see text] is the base chord length and [Formula: see text] is the base blade length. (7) In the general case of rotating blade ([Formula: see text], we let [Formula: see text] denote the [Formula: see text]th natural frequency when [Formula: see text]. It is found that the natural frequencies [Formula: see text] are closely approximated by [Formula: see text] (8) The material damping yields the imaginary part of the modified system frequency [Formula: see text], which deteriorates the energy absorption rate of the blade. Perturbation analysis with a solvability condition is performed to determine the imaginary part of [Formula: see text]. Given the same material, [Formula: see text] is inversely proportional to [Formula: see text], i.e. [Formula: see text].