| 摘要: | 本研究針對雙層壓電懸臂梁結合剛性桿件之多元壓電調諧質量阻尼器(Piezoelectric - Multiple Tuned Mass Dampers, Piezo-MTMDs)，推導運動方程式，經最佳化設計參數後，進行實驗驗證。首先將壓電本構方程式(Piezoelectric Constitutive Equation)以尤拉梁之形式，推導雙層壓電層串聯情況之力學與電路運動方程式，再利用有限元素之概念，將壓電懸臂梁與一般桿件各視為一元素塊，推導不同斷面組合而成之矩陣形式的多自由度運動方程式，並加入外接質量塊與電阻形成完整力學與電路運動方程式，在此稱多自由度矩陣形式之數學模型為完整模型。而後另以剛性桿件末端位移自由度之多項式形狀函數，代入壓電運動方程式中得到二自由度之簡化模型。透過兩種數學模型之運動方程式改寫為狀態空間，可分別對其系統進行分析，繪製頻率反應函數，確認簡化模型可取代完整模型作為後續模擬之數學模型。其中，以壓電阻尼比作為判斷發電效率之指標。因此對外接質量塊、剛性桿件之長度、壓電懸臂梁總厚度與厚度比例，以及厚度比值與電阻進行敏感度分析，由分析結果知，外接質量塊之大小、壓電懸臂梁與剛性桿件之尺寸以及厚度比例皆會對壓電阻尼比造成不同影響，需視實際需求以選擇材料尺寸。透過本團隊先前研究可知，壓電阻尼比會受到幾何勁度之軸向荷載影響，為提升其數值，將壓電TMD以倒擺懸臂梁之形式架設，可將軸向荷載對壓電阻尼比之影響轉變為正效果。而為了增加質量比以達更好之減振效果與能量擷取，在此配置多個壓電TMD組合成壓電MTMDs，即可提升質量比，並降低最佳阻尼比需求，以利壓電材料使用。而後推導氣彈模型加裝壓電MTMDs之運動方程式，固定單一壓電TMD之模型基礎，利用直接搜尋法(Direct Search)進行最佳化設計，找出結構速度頻率反應函數H2-norm值最小與平均功率數值最大化時，各顆壓電TMD相對應之外接質量塊與電阻，經分析後，以平均功率最大化設計之最佳化參數在相近的減振效果下，能夠擷取更多的功率，因此後續之分析使用此參數。在繪製頻率反應函數，與以設計風力進行動力分析後，可知壓電MTMDs在減振同時兼具良好的發電效果。最後製作出6顆壓電TMD試體並對其各別進行系統識別，找出符合實際情形之壓電材料參數，再重新最佳化設計各顆之外接質量塊與電阻後，進行氣彈模型加裝壓電MTMDs風洞實驗。由實驗結果可知，壓電MTMDs對於氣彈模型有實質的減振作用，並可同時產生電能，以達到能源永續利用之目標。;In this study, the motion equations of Piezoelectric - Multiple Tuned Mass Dampers (Piezo-MTMDs) combined with double-layered piezoelectric cantilever beams and rigid rods were derived, and experimental verification was carried out after optimizing the design parameters. First, the piezoelectric constitutive equation was used in the form of an Euler–Bernoulli beam to derive the mechanics and circuit motion equations of the double-layer piezoelectric layer in series. Then, the finite elements concept was used to combine the piezoelectric cantilever beam with the regular rod. Each was regarded as an element, and a multi-degree-of-freedom motion equation in the form of a matrix composed of different sections was deduced. An external mass block and a resistor were added to obtain a complete mechanical and circuit motion equation. Here, the mathematical model in the form of a multi-degree-of-freedom matrix was called the Full model. Then, the polynomial shape function of the rigid rod end displacement degree of freedom was substituted into the piezoelectric motion equation to obtain the Simplify model with a single degree of freedom. By rewriting the motion equations of the two mathematical models into state spaces, their systems could be analyzed separately, frequency response functions could be drawn, and it was confirmed that the Simplify model could replace the Full model as the mathematical model for subsequent simulations. Among them, the piezo-damping ratio was used as an indicator to judge the power generation efficiency. Therefore, a sensitivity analysis was performed on the size of the external mass, the length of the rigid rod, the total thickness of the piezoelectric cantilever beam and the thickness ratio, and the thickness ratio and resistance. From the analysis results, it was seen that the size of the external mass block, the size and thickness ratio of the piezoelectric cantilever beam, and the rigid rod had different effects on the piezo-damping ratio. The material size needed to be selected based on actual requirements. Through previous research conducted by our team, we knew that the piezo-damping ratio was affected by the axial load of geometric stiffness. In order to improve its value, the Piezo-TMD was erected in the form of an inverted cantilever beam, which could transform the influence of the axial load on the piezo-damping ratio into a positive effect. In order to increase the mass ratio to achieve a better vibration reduction effect and energy harvest, multiple Piezo-TMDs were configured to form Piezo-MTMDs, which could increase the mass ratio and reduce the demand for the optimal damping ratio to facilitate piezoelectric materials used. Then the motion equations of adding Piezo-MTMDs to the aeroelastic model were deduced, the model basis of a single Piezo-TMD was fixed, the Direct Search method was used for optimal design, and the Piezo-TMD external mass block and resistor corresponding to the two optimal design methods were found when the structure velocity frequency response function H2-norm value was minimum and the mean power value was maximum. After analysis, the optimal parameters designed to maximize the mean power could capture more energy under similar vibration reduction effects, so subsequent analysis used this parameter. After plotting the frequency response function and conducting dynamic analysis based on the design wind force, it was seen that Piezo-MTMDs had good power generation effects while reducing vibration. Finally, 6 Piezo-TMD specimens were produced, and each of them was systematically identified to find out the piezoelectric material parameters that fit the actual situation. After re-optimizing the design of the external mass block and resistor of each Piezo-TMD specimen, the wind tunnel experiments were conducted on an aeroelastic model equipped with Piezo-MTMDs. It was evident from the experiment results that Piezo-MTMDs had a significant impact on reducing vibrations in the aeroelastic model, while simultaneously producing electrical energy to promote sustainable energy utilization. |
