雙自由度折返樑於旋轉式壓電能量採集之分析

Abstract

本研究建立一幾何非線性的旋轉雙自由度折返懸臂樑於鉛直平面旋轉振動的系統，探究其非線性行為對頻率響應以及穩定度的影響，並建立理論模型，以驗證實驗結果。首先，利用牛頓第二定律搭配非線性的邊界條件，建立單旋轉樑於鉛直平面旋轉的統御方程式。而後使用Galerkin’s discretization 將空間及時間函數分離，找到其正規化條件以推導出其運動方程式，並將非線性取至三次項。可觀察出此方程式為一Mathieu’s 方程式搭配著幾何非線性項。接下來採用微擾法對其共振頻附近進行分析，得到一近似解析解，此解可用來研究各參數對整體系統振幅的影響；第二，延伸上述單自由度旋轉樑的研究步驟，建立雙自由度旋轉折返樑於鉛直平面下振動的模型，比較離心力對於不同擺放方向之樑的影響及其造成共振頻的改變，並觀察幾何非線性於旋轉環境下所造成的硬化或軟化非線性；第三，在貼附單壓電片於雙自由度折返樑後，分別對其進行鉛直激振以及旋轉環境的掃頻試驗，觀察其電壓頻率響應。最後，配合實驗以及前面推出的模型來擬合對照，對其定性討論現象，並觀察定量上與實驗的差距，探討其誤差的可能性。

We model a geometrically nonlinear rotating two-degree-of-freedom cantilever cutout beam in a vertical plane and exploit the influence of nonlinear behavior on frequency responses and stability of the beam. A theoretical model is developed and verified with the experimental results. First, Newton's second law along with nonlinear boundary conditions is used to establish the governing equation for a single rotating beam in the vertical plane. The Galerkin’s discretization scheme is used to separate the spatial and temporal variables. The normalization conditions are determined and used to derive the equation of motion. The nonlinearity is taken up to the cubic term. It can be observed that the equation of motion is a Mathieu’s equation with the geometric nonlinear terms. Next, the perturbation method is used to obtain an approximate analytical solution near the resonance frequency. This solution can be used to study the influence of each parameter on the amplitude of the overall system. Second, continued with the above-mentioned single-degree-of-freedom rotating beam, we further established and the model of the vibration of the two-degree-of-freedom rotating cutout beam in a vertical plane. The centrifugal force is examined to see its influence on thenatural frequency of the beam installed in different orientations. The hardening or softening phenomenon due to geometric nonlinearity from rotating environment are then discussed. Third, a piezoelectric layer is attached on the two-degree-of-freedom cutout beam. The base-excitation and the sweeping tests of the rotating environment are carried out respectively to observe the voltage responses., The waveforms at specific driving frequencies are also investigated. Finally, we compare the model with the experiment results and discuss the possible causes of the errors.