本研究主要針對壓電懸臂梁形式之壓電調諧質量阻尼器(Piezoelectric-Tuned Mass Damper, Piezo-TMD)，進行數值模擬分析與最佳化設計。本研究首先推導壓電懸臂梁力學與電路之運動方程式，並且將壓電懸臂梁之運動方程式引入多項式形狀函數以單自由度化其運動方程式，並且跟分布參數之形狀函數進行頻率反應函數異同的比較。對於壓電調諧質量阻尼器而言，在具備一定減振效果上，如何能有較佳的發電效率為其重點，而最大壓電阻尼比可以是判斷其發電效率的關鍵因素。因此本研究針對了壓電懸臂梁的寬度、長度、壓電層和基底層為相同厚度比例下之總厚度、以及外接質量塊質量，上述四項進行敏感度分析，可以知道它們的改變並不會影響最大壓電阻尼比的數值，也因此了解壓電調諧質量阻尼器之壓電阻尼比會有其極限，無法僅靠著增加壓電材料的使用量而輕易改變其最大壓電阻尼比，也就是不能單純藉由增加壓電材料而增加發電效率。故在壓電調諧質量阻尼器設計上，針對壓電懸臂梁的部分，其各項尺寸可預設一個初步的數值，靠著其初步數值便可以計算出其最大壓電阻尼比，而有了最大壓電阻尼比之後，便可以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，在此情況下就可以再藉著外接質量塊來調整整個壓電調諧質量阻尼器的總質量。壓電調諧質量阻尼器之自然頻率與電路頻率和結構頻率相調諧時，結構振動能轉為電能之效率為最佳，因此利用傳統調諧質量阻尼器最佳頻率比設計公式來找出其最佳頻率比，接著就可利用調整長度和寬度來調諧機械頻率，並調整電阻來匹配電路進而找到其最佳電阻。利用設計出來之壓電調諧質量阻尼器進行數值分析繪出其頻率反應函數圖，並做動力分析，可知在隨機風力作用下，若在氣彈模型上有加裝壓電調諧質量阻尼器時可有效減振，並同時具備不錯的發電效率。

This research focuses on numerical analysis and optimal design of the cantilever-type piezoelectric tuned mass damper (Piezo-TMD). At first, this research derives the mechanical and electrical of equations of motion of piezoelectric cantilever beam, and then the polynomial shape function is introduced to obtain the generalized single-degree-of-freedom equations of motion. The frequency response function is further derived to compare with the one derived by distributed parametric shape function. For the cantilever-type Piezo-TMD, in terms of having a certain vibration reduction effect, to have a better power generation efficiency is the key point, and the maximum piezoelectric damping ratio can be used to evaluate the power generation efficiency. Therefore, the research had tested four key parameters for sensitivity analysis, including the width and length of the piezoelectric cantilever beam, the total thickness of the piezoelectric layer and the base layer at the same thickness ratio, and the mass of the external proof mass. The results show that these four terms are not affecting the value of the maximum piezoelectric damping ratio, so just trying to add the amount of piezoelectric material will not lead to a higher power generation. In view of this, when designing the cantilever-type Piezo-TMD, a proper size of piezoelectric cantilever beam can be firstly chosen to realize the maximum piezoelectric damping ratio, then the optimal TMD mass ratio could be found by using the optimal damping ratio design formula of traditional tuned mass damper. Accordingly, the external proof mass is designed to match the optimal TMD mass ratio. For the best power generation and vibration reduction, the cantilever-type piezo-TMD have to tune to the main structure according to the design formula of optimal frequency ratio of traditional tuned mass damper. Thus, the length and width of piezoelectric beam can then be adjusted for the mechanical tuning, the resistance can also be adjusted for electrical matching. Use the aeroelastic model structure implemented with designed cantilever-type piezo-TMD to perform numerical analysis of frequency response function and time history analysis, the result shows that the cantilever-type piezo-TMD can effectively reduce the structural vibration and have great power generation simultaneously.

[1] 「建築物耐風設計規範與解說」，內政部營建署，中華民國95年9月22日台內營字第0950805664號。
[2] Manbachi A, Cobbold RSC., “Development and application of piezoelectric materials for ultrasound generation and detection”, Ultrasound,19 (4): 187-196 (2011).
[3] Gautschi, G., “Piezoelectric Sensorics: Force, Strain, Pressure, Acceleration and Acoustic Emission Sensors, Materials and Amplifiers”, Springer (2002).
[4] IEEE Std 176-1987., “Standard on Piezoelectricity’’, ANSI/IEEE, New York, 1988.
[5] Park G, Cudney HH, and Inman DJ., “Impedance-based health monitoring of civil structural components’’, ASCE Journal of Infrastructure Systems, 6(4): 153-160 (2000).
[6] Zhao X, Gao H, Zhang G, Ayhan B, Yan F, Kwan C and Rose JL., “Active health monitoring of an aircraft wing with embedded piezoelectric sensor/actuator network: I. defect detection, localization and growth monitoring”, Smart Materials and Structures, 16(4):1218-1225 (2007).
[7] Moura JDRV, and Steffen V., “Impedance-based health monitoring for aeronautic structures using statistical meta-modeling”, Journal of Intelligent Material Systems and Structures, 17(11):1023-1036 (2006).
[8] Annamdas VGM, Yang Y, and Soh CK., “Influence of loading on the electromechanical admittance of piezoceramic transducers”, Smart Materials and Structures, 16(5):1888-1897 (2007).
[9] Mall S., “Integrity of Graphite/Epoxy Laminate Embedded with Piezoelectric Sensor/ Actuator under Monotonic and Fatigue Loads”, Smart Materials and Structures, 11(4):527-533 (2002).
[10] Abé M, Park G, and Inman, DJ., “Impedance-based monitoring of stress in thin structural members”, Proceedings of 11th International Conference on Adaptive Structures and Technologies, 285-292 (2000).
[11] Sun FP, Chaudry Z, Rogers CA, Majmundar M, and Liang C., “Automated real-time structure health monitoring via signature pattern recognition”, Proceedings of the Smart Structures Materials Conference, Proceedings of SPIE, 2443:236-247 (1995).
[12] Ong CW, Yang YW, Naidu ASK, Lu Y, and Soh CK., “Application of the electro-mechanical impedance method for the identification of in-situ stress in structures”, Smart Structures, Devices, and Systems, Proceedings of SPIE, 4935:503-514 (2002).
[13] Song G, Gu H, and Mo YL., “Concrete structural health monitoring using embedded piezoceramic transducers”, Smart Materials and Structures, 16(4):959-968 (2007).
[14] Song G, Gu H, and Mo YL., “Smart aggregates: multi-functional sensors for concrete structures - a tutorial and a review”, Smart Materials and Structures, 17(3):1-17 (2008).
[15] Yan S, Sun W, and Song G., “Health monitoring of reinforced concrete shear walls using smart aggregates”, Smart Materials and Structures, 18(4):047001 (2009).
[16] Liao WI, Lin CH, Huang JS and Song G., “Seismic health monitoring of RC frame structures using smart aggregates”, Earthquake Engineering and Engineering Vibration, 12(1):25-32 (2013).
[17] Kamada T, Fujita T, Hatayama T, Arikabe T, Murai N, Aizawa S, and Tohyama K., “Active vibration control of flexural-shear type frame structures with smart structures using piezoelectric actuators”, Smart Materials and Structures, 7:479-488 (1998).
[18] Sethi V and Song G., “Multimode vibration control of a smart model frame structure”, Smart Materials and Structures, 15:473-479 (2006).
[19] Shimazaki M and Fujita T., “Experimental study of piezoelectric actuators and magnetostrictive actuators for large-scale smart structures”, The 14th World Conference on Earthquake Engineering (2008).
[20] Garrett GT and Chen G., “Experimental characterization of piezoelectric friction dampers”, Smart Structures and Materials 2001: Smart Systems for Bridges, Structures, and Highways, Proceedings of SPIE, 4330:405-415(2001).
[21] Lu LY, Lin GL and Lin CY., “Experimental verification of a piezoelectric smart isolation system”, Structural Control and Health Monitoring, 18(8):869-889 (2011).
[22] Ottman G K, Hofmann H F, Bhatt A C and Lesieutre G A., “Adaptive piezoelectric energy harvesting circuit for wireless remote power supply”, IEEE Transactions on Power Electronics, 17(5):669–676(2002).
[23] Guan M J and Liao W H., “On the efficiencies of piezoelectric energy harvesting circuits towards storage device voltages”, Smart Mater. Struct, 16(2):498–505(2007).
[24] Guyomar D, Badel A and Lefeuvre E., “Toward energy harvesting using active materials and conversion improvement by nonlinear processingIEEE Trans”, Ultrason. Ferroelectr. Freq. Control, 52584–95(2005).
[25] Shu YC and Lien IC., “Analysis of power output for piezoelectric energy harvesting system”, Smart Materials and Structures, 15:1499-1512(2006).
[26] Shu YC and Lien IC., “Efficiency of energy conversion for a piezoelectric power harvesting system”, Journal of Micromechanics and Microengineering, (2006).
[27] Shu YC and Lien IC., “Array of piezoelectric energy harvesting by the equivalent impedance approach”, Structures,21: 082001(2012).
[28] Ajitsaria J, Choe SY, Shen D and Kim DJ., “Modeling and analysis of a bimorph piezoelectric cantilever beam for voltage generation”, Smart Materials and Structures, 16(2):447-454 (2007).
[29] Erturk A and Inman DJ., “Mechanical considerations for modeling of vibration-based energy harvesters”, Proceedings of the ASME IDETC 21st Biennial Conference on Mechanical Vibration and Noise, Las Vegas, NV, Sep.4–7 (2007).
[30] Erturk A and Inman DJ., “A distributed parameter electomechanical model for cantilevered piezoelectric energy harvesters”, ASME J. Vib, Acoust.13004102 (2008).
[31] Erturk A and Inman DJ., “An experimentally validated bimorph cantilever model for piezoelectric energy harvesting from base excitations”, Smart Mater.Struct, 18:025009 (2009).
[32] Mateu L and Moll F., “Optimum piezoelectric bending beam structures for energy harvesting using shoe inserts”, Journal of Intelligent Material Systems and Structures, 16(10):835-845 (2005).
[33] Paquin S and St-Amant Y., “Improving the performance of a piezoelectric energy harvester using a variable thickness beam”, Smart Material and Structures, 19:105020 (2010).
[34] Mo C, Radziemski LJ and Clark WW., “Analysis of piezoelectric circular diaphragm energy harvesters for use in a pressure fluctuating system”, Smart Material and Structure, 19:02016 (2011).
[35] Cho J, Anderson M, Richards R, Bahr D and Richards C., “Optimization of electromechanical coupling for a thin-film PZT membrane: I. Modeling”, Journal of Micromechanics and Microengineering, 15:1797-180 (2005).
[36] Park JH, Kang J, Ahn H, Kim SB, Liu D, Kim DJ., “Analysis of Stress Distribution in Piezoelectric MEMS Energy Harvester Using Shaped Cantilever Structure”, Ferroelectrics, 409(1):55-61 (2010).
[37] Allik H and Hughes TJR., “Finite Element Method for Piezoelectric vibration. International Journal for Numerical Methods in Engineering”, International Journal for Numerical Methods In Engineering, 2:151-157 (1970).
[38] Lee AJ, Wang Y and Inman DJ., “Energy harvesting of piezoelectric stack actuator from a shock event”, Journal of Vibration and Acoustics, 136:01116 (2014).
[39] Shevtsova S and Flekc M., “Random vibration energy harvesting by piezoelectric stack charging the battery”, Procedia Engineering, 144:645-652 (2016).
[40] Han D and Yun KS., “Piezoelectric energy harvester using mechanical frequency up-conversion for operation at low-level accelerations and low-frequency vibration”, Microsystem Technologies, 21(8) 1669-1676 (2015).
[41] Jeon YB, Sood R, Jeong JH and Kim SG., “MEMS power generator with transverse mode thin film PZT”, Sensors and Actuators A: Physical, 122(1):16‐22 (2005).
[42] Feenstra J, Granstrom J and Sodano H., “Energy harvesting through a backpack employing a mechanically amplified piezoelectric stack”, Mechanical Systems and Signal Processing, 22(3):721-734 (2008).
[43] Pan P, Zhang DB, Nie X, Chen HW., “Development of piezoelectric energy-harvesting tuned mass damper”, Science China Technological Sciences, 60(3):467-478 (2016).
[44] Lai YA, Kim JY, Yang CSW, Chung LL., “A low-cost and efficient d33-mode piezoelectric tuned mass damper with simultaneously optimized electrical and mechanical tuning”, Journal of Intelligent Material Systems and Structures, 1045389 (2020).
[45] Kim SB, Park H, Kim SH, Wikle HC, Park JH, Kim DJ., “Comparison of MEMS PZT Cantilevers Based on d31 and d33 Modes for Vibration Energy Harvesting”, Journal of Microelectromechanical Systems, 22(1):26-33 (2013).
[46] Den Hartog JP., “Mechanical Vibrations’’, 4th edn, McGraw-Hill, New York (1956).
[47] Ioi T and Ikeda K., “On the dynamic vibration damped absorber of the vibration system”, Bulletin of the Japanese Society of Mechanical Engineering, 21:64-71 (1978).
[48] Warburton GB and Ayorinde EO., “Optimum absorber parameters for simple systems. Earthquake Engineering and Structural Dynamics”, 8:197-217 (1980).
[49] Ayorinde EO and Warburton GB., “Minimizing structural vibrations with absorbers. Earthquake Engineering and Structural Dynamics”, 8:219-236 (1980).
[50] Warburton GB., “Optimum absorber parameters for various combinations of response and excitation parameters”, Earthquake Engineering and Structural Dynamics, 10:381-401 (1982).
[51] Bakre SV and Jangid RS., “Optimum parameters of tuned mass damper for damped main system. Structural Control and Health Monitoring”, 14:448-470 (2007).
[52] Lin CC, Hu CM, Wang JF and Hu RY., “Vibration Control Effectiveness of Passive Tuned Mass Dampers, Journal of the Chinese Institute of Engineers”, 17:367-376 (1994).
[53] Wang JF, Lin CC and Chen BL., “Vibration Suppression for High Speed Railway Bridges Using Tuned Mass Dampers”, International Journal of Solids and Structures, 40:465-491 (2003).
[54] Lee CL, Chen YT, Chung LL, Wang YP., “Optimal design theories and applications of tuned mass dampers”, Engineering Structures, 28:43-53 (2006).
[55] Ghosh A and Basu B., “A closed-form optimal tuning criterion for TMD in damped structures”, Structural Control and Health Monitoring, 14:681-692 (2005).
[56] Chang CC., “Mass dampers and their optimal designs for building vibration control”, Engineering Structures, 22:454-463 (1999).
[57] Fujino Y and Abe M., “Design formulas for tuned mass dampers based on a perturbation technique”, Engineering and Structural Dynamics, 22:833-854 (1993).
[58] Lallart M, Yan L, Wu YC, Guyomar D., “Electromechanical semi-passive nonlinear tuned mass damper for efficient vibration damping”, Journal of Sound and Vibration, 332:5696–5709 (2013).
[59] Bonell P, Rafique S, Shuttleworth R., “A theoretical study of a smart electromechanical tuned mass damper beam device”, Smart Mater. Struct, 21:125004 (2012).
[60] Xie XD, Wu N, Yuen KV, Wang Q., “Energy harvesting from high-rise buildings by a piezoelectric coupled cantilever with a proof mass”, International Journal of Engineering Science, 72:98–106 (2013).
[61] Friswell MI, Ali SF, Bilgen O, Adhikari S, Lees RW, Litak G., “Non-linear piezoelectric vibration energy harvesting from a vertical cantilever beam with tip mass”, Journal of Intelligent Material Systems and Structures, 23(13):1505–1521 (2012).
[62] Dent ACE, Bowen CR, Stevens R, Cain MG, Stewart M., “Tensile Strength of Active Fibre Composites-Prediction and Measurement”, Ferroelectric, 368:209-215 (2008).
[63] Chopra AK. “Dynamics of Structure”, Theory and applications to earthquake engineering Fourth edition (2012).