| 摘要: | 本研究對變斷面懸臂梁形式之壓電調諧質量阻尼器(Piezoelectric-Tuned Mass Damper, Piezo-TMD)，推導完整力電耦合之運動方程式，並轉化為狀態空間後對其系統進行分析。首先將壓電本構方程式(Piezoelectric Constitutive Equation)以尤拉梁形式之懸臂梁結構，推導壓電懸臂梁力學與電路運動方程式，再利用有限元素概念將不同斷面之壓電懸臂梁分割成多個元素塊，並引入多項式形狀函數及疊加元素塊，推導變斷面壓電懸臂梁矩陣形式之運動方程式，以便分析壓電層與基底層尺寸不同的問題。最後加裝外接質量塊於懸臂梁末端形成完整力學方程式，並外加電阻於電路迴路形成完整電路方程式。壓電TMD之設計目的為：吸收結構振動能量至壓電TMD中，並轉換其振動能量為電能後進行擷取；亦即發電效率為其重點，因此本文以壓電阻尼比作為判斷其發電效率之指標。建立完成數值模型後，即可對壓電TMD進行分析，透過繪製頻率反應函數圖可了解元素數量多寡會影響可觀察的模態數量以及模態頻率的準確性，同時也可了解添加外接質量塊後壓電TMD反應受第一模態所主導。而為了解各項壓電材料層面參數影響壓電阻尼比之情況，接著對壓電材料參數進行敏感度分析。由分析發現，壓電材料的軟硬、力電耦合係數以及寄生電容的大小，皆會對最大壓電阻尼比產生不同的影響，從而瞭解挑選壓電材料時之依據。
透過先前團隊研究可知，單純增加壓電材料使用量無法提升最大壓電阻尼比，同時發現尋獲壓電TMD最大壓電阻尼比時，壓電TMD與結構之質量比數值無法再提升從而擁有更好的效果。本研究欲將多個壓電TMD組合成多元壓電調諧質量阻尼器(Piezoelectric - Mutiple Tuned Mass Dampers, Piezo-MTMDs)，藉由多元(Mutiple)形式加裝壓電TMD裝至氣彈模型上進行壓電MTMD最佳化設計，藉此打破質量比之上限。因此首先推導出氣彈模型加裝壓電MTMD之運動方程式，並運用最佳化設計方式尋找作為未知參數的外接質量與外接電阻，得以設計出壓電MTMD。設計時，先設計出作為基礎之壓電懸臂梁，其尺寸與參數套用至每組壓電TMD中。再結合氣彈模型，並使用直接搜尋法(Direct Search)進行最佳化設計，找出結構速度H2-norm值最小時各個單獨壓電TMD所匹配之懸臂梁外接質量以及外接電阻之組合，如此便完成壓電MTMD之設計。利用設計出來之壓電MTMD進行數值分析、繪製頻率反應函數圖，並以設計風力進行動力分析，可知壓電MTMD可在減振的同時具備良好的發電效率。最後製作出成品並對其進行系統識別及擬合分析，於系統識別中先使用靜態識別方法：透過靜力及直流電壓加載測試識別出壓電TMD之實際參數；再來分析動態擬合成果：透過自由振動試驗測量壓電TMD位移歷時與產生之電壓歷時。實驗結果與數值分析成果比較，發現藉由增添幾何勁度矩陣以及轉動慣量進入數值分析模型中，可使分析成果成功與壓電TMD動態實驗數據匹配，確認材料參數之正確性，同時完整數值分析模擬方法。
;This study focuses on the Piezoelectric-Tuned Mass Damper (Piezo-TMD) with variable cross-section cantilever beams. The complete electromechanical coupled equations are derived and transformed into state-space representation for system analysis. Firstly, the mechanical and electrical motion equations of the piezoelectric cantilever beam are derived by the piezoelectric constitutive equation with the Euler-Bernoulli beam structure. Then, the finite element concept is used to divide the piezoelectric cantilever beam with different cross-sections into multiple element blocks. The polynomial shape functions and superimposed element blocks are introduced to derive the matrix form of the motion equations for the variable cross-section piezoelectric cantilever beam, so that the problem of the size different between piezoelectric layer and the basic layer can be analysized easily. Finally, an external mass is attached to the end of the cantilever beam to form the complete mechanical equation, and an external resistance is added to the circuit loop to form the complete electrical circuit equation. The design purpose of the Piezo-TMD is to absorb structural vibration energy into the Piezo-TMD and convert its vibration energy into electrical energy for harvesting. Thus, the power generation efficiency of the Piezo-TMD is a key concern, and this study uses the piezoelectric damping ratio as an efficiency indicator. After establishing the numerical model, the Piezo-TMD can be analyzed. By drawing the frequency response function diagram, the number of the observable modes and the accuracy of the modal frequency affected by the number of elements can be understood. It is also possible to understand that the response of the Piezo-TMD is controlled by the first mode after adding an external mass block. In order to understand the influence of various piezoelectric material parameters on the piezoelectric damping ratio, sensitivity analysis of piezoelectric material parameters is conducted. The analysis shows that the softness and hardness of the piezoelectric material, the electromechanical coupled coefficient, and the size of the parasitic capacitance all have different effects on the maximum piezoelectric damping ratio, which provides a basis for selecting piezoelectric materials.
Based on the previous research by our team, it is known that simply increasing the amount of piezoelectric material cannot improve the maximum piezoelectric damping ratio. It was also found that when the Piezo-TMD exist the maximum piezoelectric damping ratio, the mass ratio between the Piezo-TMD and the structure cannot be further improved to achieve better results. Therefore, this study further aims to combine multiple Piezo-TMDs into Piezoelectric-Multiple Tuned Mass Dampers (Piezo-MTMDs) to break through the limitation of the mass ratio. The motion equations for the aeroelastic model structure with Piezo-MTMDs are derived, and the optimization design method is used to find the unknown parameters of the external mass and external resistance to design the Piezo-MTMDs. During the design process, priority is given to design the fundamental piezoelectric cantilever beam that the dimensions and the parameters are applied to each set of Piezo-TMDs before installed on the aeroelastic model structure. The study uses the Pattern-Search method for optimization design to find the combination of the external mass and external resistance of each individual Piezo-TMD; the structure velocity H2-norm value is the smallest, thus completing the design of the Piezo-MTMDs. The designed Piezo-MTMDs are used for numerical analysis, drawing frequency response function diagrams, and conducting dynamic analysis with design wind force, which show that the Piezo-MTMDs can achieve excellent power generation efficiency while effectively reducing vibrations. Finally, a designed Piezo-TMD which is manufactured is applied to system identification and model fitting analysis. During the system identification, the static identification method is proposed firstly: the actual parameters of the Piezo-TMD are identified through static loading tests with static force and DC voltage. Next, the dynamic model fitting results are analyzed: the displacement history and voltage history of the Piezo-TMD are measured through free vibration tests. By comparing the experimental results with the numerical analysis results, it is found that the analysis results can successfully match the dynamic experimental data of the Piezo-TMD by adding geometric stiffness matrices and rotational inertias into the numerical analysis model. The matched results can confirm the correctness of the material parameters, and complete the numerical analysis simulation method. |
