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].

中文翻译：

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通过汉密尔顿原理分析垂直轴风力涡轮机叶片的结构振动 - 第 1 部分：一般公式

风力涡轮机的能量收集是许多国家/地区非常关注的问题，但其安全性不可避免地与涡轮机系统的结构振动有关。在这项研究中，我们基于拉格朗日力学的欧拉梁理论，对垂直轴风力涡轮机 (VAWT) 的结构振动进行了完整的线性分析。假设未变形的叶片是垂直笔直的。求解代表变形的四个维度的合成方程有几个发现，包括运动：横向弯曲 - 弦向弯曲 - 轴向扭转 - 轴向延伸 (BBTE) (1) 轴向张力和轴向扭转之间没有变形耦合。(2)叶片的固有频率主要由横向弯曲决定，[公式：见正文]（[公式：见正文]) 表示仅由横向弯曲确定的固有频率。(3) 归因于叶片变形的离心力是改变固有频率的主要因素。(4)科里奥利力只能存在于耦合系统中，但无论如何，横向弯曲和轴向拉力耦合都不会产生科里奥利力。(5) 当横向弯曲与弦弯曲或轴向扭转耦合时，科里奥利力只能略微改变固有频率。(6)在固定转速的情况下，[公式：见文]，其中[公式：见文]为角速度，[公式：见文]为旋转轴到叶片弹性中心的距离：给定[公式：见正文]-叶片长度与弦比，发现叶片的固有频率[公式：见正文]为，近似地，与[公式：见文本]成反比，即[公式：见文本]，其中[公式：见文本]是基本弦长，[公式：见文本]是基本叶片长度。(7) 在旋转叶片的一般情况下（[公式：见文]，我们让[公式：见文]表示[公式：见文]时的[公式：见文]次固有频率。发现固有频率[公式：见正文]与[公式：见正文]非常近似 (8) 材料阻尼产生修正系统频率的虚部[公式：见正文]，这会降低叶片的能量吸收率。进行具有可解性条件的微扰分析以确定[公式：见文本]的虚部。给定相同的材料，[公式：见文本]与[公式：见文本]成反比，即。e. [公式：见正文]。