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研究生： 劉德燻
Liou, De-Shiun
論文名稱： 部份嵌入壓電材料懸臂樑具焦電效應之動態響應分析
Study of Timoshenko Beam Embedded With Piezoelectric material Including Pyroelectric Effects
指導教授： 王榮泰
Wang, Rung-tai
學位類別： 碩士
Master
系所名稱： 工學院 - 工程科學系
Department of Engineering Science
論文出版年： 2008
畢業學年度： 96
語文別： 中文
論文頁數： 91
中文關鍵詞： 壓電材料 、振動 、有限元素法 、頻率
外文關鍵詞： Vibration, Piezoelectric, Finite element method, Frequency
相關次數： 點閱：73  下載：2 
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    • 本文壓電材料除了一般考慮的壓電效應，為求更接近實際情況，還考慮了焦電效應，部份嵌入式壓電材料懸臂樑，不同於以往貼附式壓電材料懸臂樑，可以避免突起的壓電材料對原始結構的外在影響，有更接近完整的樑結構，計算方法使用了有限元素法對此結構做動態響應分析，並用模態法來驗證有限元素分析的準確性，再以有限元素分析所得結果做回授控制，再分別探討壓電材料不考慮溫差影響和考慮溫差影響的制振效果。

    數學模型的假設建立皆為Timoshenko理論的三明治樑所組成。再此結構中將分成三個跨距，每ㄧ跨距皆分三層，第二跨距上下層為壓電材料，可視為此樑之制動器及感測器，第一、三跨距都為鋁材，但仍分成三層，再利用連續位移條件整合，使得邊界條件的代入更為方便。

    有限元素法的計算是依應力場和應變場推導出動能和應變能，並利用Hamilton’s Principle 對動能和應變能做變分得出運動方程式，再用去除時間項的靜態平衡方程式計算出各跨距的勁度矩陣和質量矩陣，建立出有限元素模型後以堆疊方式經Lagrange’s equation 計算出系統的模態頻率。

    模態法則是將運動方程式中的單變數轉換為時間和距離雙變數，再配合邊界條件計算出力函數，進而解出模態頻率，並將其結果與有限法相比較，確定有限元素法的可行性。

    回授控制分析是利用電位移計算出感測器之電流，經Gain值轉換給予制動器逆向電壓，並利用有限元素法搭配動態阻尼帶入Lagrange’s equation ，再配合Newmark’s數值積分法進行回授控制及抑制振動的模擬分析。

    Finite element modeling of the cantilever beams with piezoelectric sensor and actuator layers is considered in this paper. The piezoelectric beam element is based on Timoshenko beam theory. The mathematical model is based on a displacement field, linear temperature field, electrical potential field, piezoelectric field and pyroelectric field. The natural frequencies obtained by the finite element method will be campared with those analytic results.

    In vibration control, constant-gain negative velocity feedback control has been used in a closed control loop. Newmark method is taken to analyze the influence of the gain, displace, length and depth on the dynamic responses. The influence of temperature also be investigated.

    摘要 I Abstract II 誌謝 III 目錄 IV 表目錄 VII 圖目錄 VIII 圖目錄 VIII 符號說明 XV 符號說明 XV 第一章 緒論 1 §1-1 研究動機 1 §1-2 研究目的 2 §1-3 文獻探討 3 第二章 研究架構 6 §2-1 研究架構流程 6 §2-2 本文基本假設 7 第三章 研究方法及內容 8 §3-1 運動方程式推導 8 §3-1-1 初始設定 8 §3-1-2 應力、應變 9 §3-1-3 應變能、動能 10 §3-1-4 利用Hamilton’s Principle得出運動方程式和邊界條件 12 §3-1-5 結構之運動方程式和邊界條件 14 §3-2 有限元素法分析 17 §3-2-1 靜態平衡方程式 17 §3-2-2 解聯立方程式可得各位移函數通解 18 §3-2-3 位移場用單位元素兩端點表示之表示式 19 §3-2-4 代入應變能及動能計算單位質量矩陣、勁度矩陣 20 §3-2-5 堆疊完整結構，得完整結構之運動方程式 20 §3-3 模態法分析 22 §3-3-1 將原運動方程式改寫，將雙變數函數拆成兩個單變數函數 22 §3-3-2 解聯立方程式得出各位移函數通解 23 §3-3-3 找出力場相關函數 25 §3-3-4 用兩端點來表示位移場和力場組合( ) 25 §3-3-5 給予適當邊界條件計算自然振動頻率 26 §3-4 回授控制分析 27 §3-4-1 計算感測器輸出之電流 27 §3-4-2 制動器與感測器之作用力 28 §3-4-3 動態回授阻尼矩陣 29 §3-4-4 Newmark’s Scheme 29 第四章 研究結果與模擬數據分析 31 §4-1 案例探討 31 §4-1-1 各項參數值 31 §4-1-2 有限元素法和模態法之自然頻率比較 32 §4-1-3 各方向位移的模態圖 36 §4-2 探討壓電材料位置及長度對整體結構的影響 39 §4-2-1 改變 長度對整體結構W方向位移變化的影響 39 §4-2-2 改變 長度對整體結構W方向位移變化的影響 40 §4-2-3 改變 長度對整體結構W方向位移變化的影響 41 §4-3 探討回授控制影響 42 §4-3-1 各效應回授影響推論 42 §4-3-2 壓電感測器上之應變 44 §4-3-3 Gain值效應 46 §4-3-4 長度效應（嵌入壓電材料位置） 55 §4-3-5 長度效應（嵌入壓電材料長度） 64 §4-3-6 長度效應（嵌入壓電材料厚度） 73 第五章 結論與建議 82 §5-1 結論 82 §5-2 建議 84 參考文獻 85 附錄A 88 附錄B 89 附錄C 90 自述 91

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