- 學位論文
受強激振下懸臂樑式壓電振動子之非線性研究
Investigation of Nonlinear Response of a Cantilevered Piezoelectric Oscillator Under Amplified Excitation
摘要
本研究主要探討懸臂樑式壓電振子所具有的不同非線性因素，在強激振的環境下，對於系統整體動態響應所造成的影響。本研究除了探討壓電材料中之非線性組成律外，亦考慮了懸臂樑在大幅度變形時，所出現的幾何非線性與慣性非線性。在尤拉－柏努利與不可伸縮樑的兩種假設下，結合瑞利-里茲近似法(Raleigh-Ritz approximation)，最後透過漢米爾頓原理(Hamilton’s principle)推導出壓電懸臂樑的控制方程式。 為進一步了解控制方程式中，各係數對於系統整體所造成的影響，本研究使用了多尺度分析法求解出系統之近似解析解，並定義非線性的等效參數Neff，發現此參數對於懸臂樑振子的動態響應特性上，扮演極為重要之角色。並以實際振子參數，分別使用三種不同形狀函數進行模擬，根據比較結果發現，以懸臂樑於點外力作用下所產生的靜態變形曲線為形狀函數，可作為本研究系統中，控制方程式的係數來源之標準。 接著藉由實驗數據得到的頻率響應曲線，與近似解析解中的頻率響應關係式進行曲線擬合，即可推測出系統的機械阻尼係數以及材料非線性等效係數，並可連同其它等效係數一併進行數值模擬。最後，根據比較結果顯示，本研究所建立的壓電振子數學模型無論是在系統的共振頻率，或是輸出端的電壓峰值預測上，與實驗所得的頻率響應曲線皆具有優異的一致性。
並列摘要
A study of nonlinear vibration of MEMs cantilevered piezoelectric oscillator subjected to intense excitations is presented. Several nonlinear sources within the oscillator will be considered and discussed for the effect to the dynamic response of the system. First, the material nonlinearity in the PZT layer of cantilever beam is considered, the geometrical nonlinearity and inertia nonlinearity which caused by the large deformation of cantilever beam are taken into account in this thesis as well. The derivation of governing equations is based on Hamilton variational principle, together with several assumptions including Euler-Bernoulli beam theory and inextensible beam condition. Reduced-order models by Rayleigh-Ritz approximation are also developed to focus on the first vibration mode of the system. For further understanding of the system, the approximation analytic solution of the system can be obtained by the method of multiple scale analysis. The effective nonlinear parameter Neff defined in the frequency-response equation is found to be a key parameter for the dynamic response of the system. By substituting the real system dimensions into the simulation, the shape function with point load exerted on the end tip of the beam is found to be a choice to become a standard for the effective coefficients of the governing equations. The damping and material nonlinearity coefficients in the governing equations are estimated by the curve fitting via experimental data. A good qualitative agreement is obtained between experimental and numerical results.