垂直軸風機葉片撓曲與扭轉非線性振動分析

Abstract

垂直軸風力發電機具有無風向限制以及噪音較小等等的優點，近年來有越來越多的人投入心力研究，此論文以 NACA0015 型號作為分析模型，利用非線性分析風機葉片 在運轉的共振響應以及受力與位移的穩定狀況。系統的非線性來源為撓曲所造成的變 形引發軸力產生，軸力又因葉片幾何的關係對其他方向位移的交互作用產生非線性項， 本研究從基本的材料力學理論出發，建構出各方向的應變位能，並保留到二階非線性 項，再以垂直風機真實運轉的運動狀況考慮其動能，之後利用漢彌爾頓定理推算出控 制方程式，將方程式無因次化後，考慮真實葉片模型對方程式作適量的省略，再利用 Galerkin Method 簡化方程式之後對其做響應分析得到不同外力頻率下，葉片振幅與頻 率的關係曲線，最後利用微小擾動法 (Perturbation Methods) 分界其穩定與不穩定區間 來判斷葉片可能的運動行為，在此因在線性系統時葉片較硬的方向為獨立系統，而葉 片較軟的方向與扭轉耦合，所以我們先將獨立的方向捨去只考慮一撓曲一扭轉的方程 組，在各章的最後小節在考慮兩撓曲做一個平行比較。同時也對所推導的控制方程式 用數值的方法求解，我們採用四階倫基 -庫達法當求解器求解，將結果做初步的呈現後 再對其做快速傅立葉分析以求更加了解其響應的組成，再以相位圖的方式呈現數值結 果，可以看出非線性系統所具有的結果，同時與分析的結果做一個比較驗證。

Vertical axis wind turbine has the benefit of operation that is independent of the direction, and less noise. In recent years there are more and more people involved in research of VAWTs. In this present paper, we adopt NACA0015 to be analysis model. Using nonlinear analysis to analyze resonance and stability of amplitude – force frequency relation of wind blade . The nonlinear term is cause by the axial force that owing to bending deformation. We start form basic theory of mechanics of materials. To find the strain potential energy and retained to the second order nonlinearity. Consider the kinetic energy of operating VAWTs. Using the Hamilton’s principle to derive the governing equation and then dimensionless the equation. Consider real condition to simplify the governing equation. Appling Galerkin’s method to equation and get the amplitude force frequency relation for different frequency. Using perturbation methods to designate the stable and unstable zones. In linear system the leading edge direction is uncouple, and the other bending direction is couple with torsion. So we first consider the bending and torsion coupling equation. In the last part of section we consider the two bending equation to compared with above. Simultaneously applied Runge-Kutta fourth order method to solve equation. And using FFT to analyze the result. Then use different method to present the result. Finally make a verification by compare the analysis and numerical result.