本研究對壓電懸臂梁形式之壓電調諧質量阻尼器(Piezoelectric-Tuned Mass Damper, Piezo-TMD)，進行運動方程式推導及系統分析，再結合氣彈模型，設計壓電TMD之參數使其做為氣彈模型之調諧質量阻尼器使用，並進行數值模擬。首先將壓電本構方程式結合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壓電懸臂梁分割成元素塊，並帶入多項式形狀函數推導，疊加形成矩陣形式之壓電懸臂梁運動方程式，最後於外加電路串聯電阻及電感，形成完整電路迴路。為了檢驗矩陣形式之推導結果是否符合真實情況，繪製其頻率反應函數圖，並與文獻提供之分布參數形式之頻率反應函數圖比較。壓電TMD之設計目的為：在吸收結構能量的同時，從中擷取能量—亦即發電效率為其重點。本文定義壓電阻尼比參數為判斷其發電效率之指標。接著對壓電TMD進行敏感度分析，了解哪些尺寸參數影響壓電阻尼比。由分析發現，僅固定總厚度調整壓電層與基底層比例將有最佳的厚度比達到最大壓電阻尼比，其餘尺寸參數不影響最大壓電阻尼比。由此可知單純增加壓電材料使用量無法提升最大壓電阻尼比，亦代表壓電TMD之發電效率有其上限。故為求最大減振效果，壓電TMD之設計流程與傳統質量阻尼器不同，應先決定阻尼比再求質量比。設計時，壓電懸臂梁之尺寸可預設一組初步的數值來計算其可達到之最大壓電阻尼比。再結合氣彈模型，利用傳統調諧質量阻尼器最佳阻尼比設計公式推算出其質量比，並使用直接搜尋法(Direct Search)做最佳化設計，找出結構速度H2-norm值最小時之懸臂梁長度、電阻及電感之組合，如此便完成壓電TMD之設計。利用設計出來之壓電TMD進行數值分析、繪製頻率反應函數圖，和以設計風力進行動力分析，可知壓電TMD可在減振的同時具備不錯的發電效率。最後依據分析壓電材料在懸臂梁上之發電特性，當壓電層以最佳的長度比例極化時，壓電層有最佳發電效率。以及在高模態時，不同的彎曲方向將降低發電效率。

關鍵字: 壓電懸臂梁、調諧質量阻尼器、有限元素模型、壓電材料、能量擷取、RLC電路、最佳化設計、H2-norm最佳化

In this study, the equation of motion and system analysis of a Piezoelectric-Tuned Mass Damper (Piezo-TMD) in the form of a piezoelectric cantilever beam are derived. The piezoelectric TMD is firstly derived from the piezoelectric constitutive equation combined with the cantilever beam according to the Euler–Bernoulli beam theory. Then, the piezoelectric cantilever beam is divided into element blocks with the finite element concept to superimpose in the form of a matrix equations. Afterward, the resistance and inductance are connected in series within the circuit to form a complete circuit loop. To verify the correctness of the derived matrix equations, the frequency response function is plotted to confirm the consistency with reference which is derived according to distributed parameters. The Piezo-TMD is not only designed to reduce the vibration of the structure, but also to harvest the vibration energy of the main structure. Therefore, the power generation efficiency is also considered as a second priority in design process. The piezoelectric damping ratio defined in this paper can be used as an indicator to represent the power generation efficiency. In order to understand which parameters will affect the maximum piezoelectric damping ratio, the sensitivity analysis of parameters of the Piezo-TMD is conducted. Accordioning to the sensitivity analysis, only the thickness ratio of the piezoelectric layer to the base layer will affect the maximum piezoelectric damping ratio. The best thickness layer ratio can be found to achieve the maximum piezoelectric damping ratio. The analysis results also shown that simply increasing the amount of piezoelectric materials cannot increase the piezoelectric damping ratio so that the piezoelectric damping ratio has its upper limit, that is, the power generation efficiency has its upper bound. Because of the limitation of the maximum piezoelectric damping ratio, the proposed design method of the Piezo-TMD is different from the traditional TMD, the damping ratio is determined before the mass ratio. In the design, each dimension of the piezoelectric cantilever beam can be preset as a preliminary value, and the maximum piezoelectric damping ratio can be calculated accordingly. The mass ratio is therefore calculated by the optimum damping ratio design formula of the traditional tuned mass damper. To optimize the other parameters of the Piezo-TMD, the Direct Search method is used to find the optimum beam length, resistance, and inductance when the structural velocity H2-norm is minimum. The designed Piezo-TMD is analyzed numerically, the frequency response function and time history analysis of subjected to wind force shows that the Piezo-TMD performs well in both structural vibration reduction and power generation. Finally, according to the analysis of polarized range of the piezoelectric material on the cantilever beam, the best polarized length ratio can be found to achieve maximum power generation efficiency in the first mode. However, the power generation efficiency will be reduced in high modes due to the different bending directions.

Keywords: piezoelectric cantilever beam, tuned mass dampers, finite element model, piezoelectric materials, energy harvesting, RLC circuit, optimal design, H2-norm optimization

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