复合型悬臂梁压电俘能器理论与实验研究
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摘要
如何向微电子产品无线供能是当前研究的一个热点问题。由于电池供电本身的缺陷和使用环境的限制,使人们更加热衷于研究无源供能技术。由压电材料制成的压电发电装置具有不发热、无电磁干扰、无污染,结构简单,易于加工制作和实现机构的微小化、集成化等优点,因而备受关注。近年来,将压电材料产生的电能用作驱动微功率电器电源的研究才刚刚开始,但已经展示出压电发电与能量存储技术的光明前景。
在查阅大量文献的基础之上,本文针对复合型悬臂梁压电俘能器进行了理论和实验研究。为了达到初步调节俘能器振子振动频率特性的目的,本文分别设计了一套可实现长度调节的双晶悬臂梁压电振子以实现其固有频率的调节,并建立该压电振子固有频率的理论模型;设计了L型悬臂梁压电振子以调节其前两阶固有频率,并建立了其前两阶固有频率的理论模型;设计一个悬臂梁阵列以实现其在一定的频段内产生谐振或者近似谐振,并建立了其各阶固有频率的理论模型。
在理论分析的基础之上,对三种类型的悬臂梁压电振子进行有限元仿真。结果表明有限元仿真分析结果与数值计算基本吻合。由模态分析得出了悬臂梁压电振子固有频率随长度变化的关系,L型悬臂梁压电振子的结构参数及多臂型悬臂梁的各阶弯振模态的固有频率,并在此基础上进行了有限元谐响应分析,得出了三种悬臂梁压电俘能器的输出电压与激振频率的关系。
为了验证理论和有限元仿真分析结果的正确,本文制作三种类型的压电悬臂梁俘能器并建立了实验系统,分别采用PSV-400-m~2激光测振仪和HP4294A阻抗分析仪测量了各压电振子的振动模态和阻抗特性,测量结果与理论分析结果基本吻合。利用阻抗分析仪测量俘能器的集总等效参数并计算匹配阻抗,对各种压电俘能器进行了俘能实验研究,最后综合比较分析了各压电俘能器的频率调节性能和俘能能力。
It is a hotspot currently that how to supply the energy to microelectronic products. For the limitation of environment and self defect of battery supplying system, it is necessary to investigate new passive energy supplying technology. For the reason that piezoelectric power generator made of piezoelectric material has the predominance that low hear emission, none magnetic disturbance, none pollution, simple structure, workability, microminiaturize and integration etc, it is now highly paid attention to the research of supplying power produced by piezoelectric material to activate micro power electric appliance just starts these years, but a great future of technology of piezoelectric power generator and energy storage has been shown.
After consulting many literatures, this dissertation conducts theoretical and experimental study on compound piezoelectric cantilever energy harvester. Considering that cantilever’s resonant frequency could be changed by modifying its structure parameters, a length-adjusting device is created and the mathematical model of the resonant frequency is built. For the fact that L-shaped cantilever’s first two resonant frequencies could be tuned closer, an L-shaped piezoelectric cantilever is created and the mathematical model of the first two resonant frequencies is built. For the sake of creating various resonant frequencies in a wide band, an ensemble of cantilever beams is created and the mathematical model of the resonant frequencies is built.
Based on theory analysis, finite element analysis and simulations are conducted. The analysis and simulation results are in accordance with the theoretical prediction. Finite element modal analysis shows the relationship between resonant frequency and length of beam of the length-adjusting piezoelectric cantilever, the relationship between the first two frequencies of L-shaped piezoelectric cantilever, and that among the 8 orders resonant frequencies of multiple beams type piezoelectric cantilever. Finite element harmonic analysis shows the relationship between the voltages output and exciting frequencies of all the three kinds of piezoelectric cantilevers.
For the purpose of verifying the results of theoretical and finite element analysis results, the piezoelectric cantilever energy harvester experimental system is built up. The laser vibrometer and impedance analyzer test the bending modes and the equivalent parameters separately, and all the results are in accordance with the theoretical prediction. The matching impedances are calculated by the use of lumped equivalent parameters tested by impedance analyzer and validated in the power scavenging experiments. The frequencies modify capabilities and powers output are also observed and analyzed.
在查阅大量文献的基础之上,本文针对复合型悬臂梁压电俘能器进行了理论和实验研究。为了达到初步调节俘能器振子振动频率特性的目的,本文分别设计了一套可实现长度调节的双晶悬臂梁压电振子以实现其固有频率的调节,并建立该压电振子固有频率的理论模型;设计了L型悬臂梁压电振子以调节其前两阶固有频率,并建立了其前两阶固有频率的理论模型;设计一个悬臂梁阵列以实现其在一定的频段内产生谐振或者近似谐振,并建立了其各阶固有频率的理论模型。
在理论分析的基础之上,对三种类型的悬臂梁压电振子进行有限元仿真。结果表明有限元仿真分析结果与数值计算基本吻合。由模态分析得出了悬臂梁压电振子固有频率随长度变化的关系,L型悬臂梁压电振子的结构参数及多臂型悬臂梁的各阶弯振模态的固有频率,并在此基础上进行了有限元谐响应分析,得出了三种悬臂梁压电俘能器的输出电压与激振频率的关系。
为了验证理论和有限元仿真分析结果的正确,本文制作三种类型的压电悬臂梁俘能器并建立了实验系统,分别采用PSV-400-m~2激光测振仪和HP4294A阻抗分析仪测量了各压电振子的振动模态和阻抗特性,测量结果与理论分析结果基本吻合。利用阻抗分析仪测量俘能器的集总等效参数并计算匹配阻抗,对各种压电俘能器进行了俘能实验研究,最后综合比较分析了各压电俘能器的频率调节性能和俘能能力。
It is a hotspot currently that how to supply the energy to microelectronic products. For the limitation of environment and self defect of battery supplying system, it is necessary to investigate new passive energy supplying technology. For the reason that piezoelectric power generator made of piezoelectric material has the predominance that low hear emission, none magnetic disturbance, none pollution, simple structure, workability, microminiaturize and integration etc, it is now highly paid attention to the research of supplying power produced by piezoelectric material to activate micro power electric appliance just starts these years, but a great future of technology of piezoelectric power generator and energy storage has been shown.
After consulting many literatures, this dissertation conducts theoretical and experimental study on compound piezoelectric cantilever energy harvester. Considering that cantilever’s resonant frequency could be changed by modifying its structure parameters, a length-adjusting device is created and the mathematical model of the resonant frequency is built. For the fact that L-shaped cantilever’s first two resonant frequencies could be tuned closer, an L-shaped piezoelectric cantilever is created and the mathematical model of the first two resonant frequencies is built. For the sake of creating various resonant frequencies in a wide band, an ensemble of cantilever beams is created and the mathematical model of the resonant frequencies is built.
Based on theory analysis, finite element analysis and simulations are conducted. The analysis and simulation results are in accordance with the theoretical prediction. Finite element modal analysis shows the relationship between resonant frequency and length of beam of the length-adjusting piezoelectric cantilever, the relationship between the first two frequencies of L-shaped piezoelectric cantilever, and that among the 8 orders resonant frequencies of multiple beams type piezoelectric cantilever. Finite element harmonic analysis shows the relationship between the voltages output and exciting frequencies of all the three kinds of piezoelectric cantilevers.
For the purpose of verifying the results of theoretical and finite element analysis results, the piezoelectric cantilever energy harvester experimental system is built up. The laser vibrometer and impedance analyzer test the bending modes and the equivalent parameters separately, and all the results are in accordance with the theoretical prediction. The matching impedances are calculated by the use of lumped equivalent parameters tested by impedance analyzer and validated in the power scavenging experiments. The frequencies modify capabilities and powers output are also observed and analyzed.
引文
1 Bor-Shun Lee, Jyun-Jhang He, Wen-Jong Wu, et al. MEMS generator of power harvesting by vibrations using piezoelectric cantilever beam with digitate electrode. William, W. Clark, Mehdi, AhmadianArnold, Lumsdaine, 2006, SPIE:61690B
2 S. R. Anton, H. A. Sodano. A review of power harvesting using piezoelectric materials (2003-2006). Smart Materials & Structures. 2007, 16(3):R1-R21
3 A. J. Schoenecker, T. Daue, B. Brueckner, et al. Overview on macro fiber composite applications - art. no. 61701K. Smart Structures and Materials 2006: Active Materials: Behavior and Mechanics. 2006, 6170:K1701-K1701
4 H. A. Sodano, D. J. Inman, G. Park. Comparison of piezoelectric energy harvesting devices for recharging batteries. Journal of Intelligent Material Systems and Structures. 2005, 16(10):799-807
5 J. Cho, M. Anderson, R. Richards, et al. Optimization of electromechanical coupling for a thin-film PZT membrane: I. Modeling. Journal of Micromechanics and Microengineering. 2005, 15(10):1797-1803
6 Y Ammar, A Buhrig, M Marzencki, et al. Wireless sensor network node with asynchronous architecture and vibration harvesting micro power generator, 2005, ACM New York, NY, USA:287-292
7方科,李欣欣,杨志刚.压电式能量获取装置的研究现状[J]. 2006
8 M. Ferrari, V. Ferrari, M. Guizzetti, et al. Piezoelectric multifrequency energy converter for power harvesting in autonomous microsystems. Sensors and Actuators a-Physical. 2008, 142(1):329-335
9 S Kim, TJ Johnson, WW Clark. Harvesting energy from a cantilever piezoelectric beam, 2004, SPIE:259-268
10 R. Want, K. I. Farkas, C. Narayanaswami. Energy harvesting and conservation. Ieee Pervasive Computing. 2005, 4(1):14-17
11 T. Starner. Human-powered wearable computing. Ibm Systems Journal. 1996, 35(3-4):618-629
12 S. W. Arms, C. P. Townsend, D. L. Churchill, et al. Power management for energy harvesting wireless sensors. Smart Structures and Materials 2005: Smart Electronics, Mems, BioMems, and Nanotechnology. 2005, 5763:267-275
13 T. A. Anderson, D. W. Sexton. A vibration energy harvesting sensor platform for increased industrial efficiency - art. no. 61741Y. Smart Structures and Materials 2006: Sensors and Smart Structures Technologies for Civil, Mechanical , and Aerospace Systems, Pts 1 and 2. 2006, 6174:Y1741-Y1741
14 J Baker, S Roundy, P Wright. Alternative Geometries for Increasing Power Density in Vibration Energy Scavenging for Wireless Sensor Networks
15 Shad Roundy, Paul K. Wright, Jan Rabaey. A study of low level vibrations as a power source for wireless sensor nodes. Computer Communications. 2003, 26(11):1131-1144
16 J. S. Yang, H. G. Zhou, Y. T. Hu, et al. Performance of a piezoelectric harvester in thickness-stretch mode of a plate. Ieee Transactions on Ultrasonics Ferroelectrics and Frequency Control. 2005, 52(10):1872-1876
17 C. D. Richards, M. J. Anderson, D. F. Bahr, et al. Efficiency of energy conversion for devices containing a piezoelectric component. Journal of Micromechanics and Microengineering. 2004, 14(5):717-721
18 J. Rastegar, C. Pereira, H. L. Nguyen. Piezoelectric-based power sources for harvesting energy from platforms with low frequency vibration - art. no. 617101. Smart Structures and Materials 2006: Industrial and Commercial Applications of Smart Structures Technologies. 2006, 6171:17101-17101
19 JW Sohn, SB Choi, DY Lee. An investigation on piezoelectric energy harvesting for MEMS power sources. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science. 2005, 219(4):429-436
20 Tsz-Ho Ng, Wei-Hsin Liao. Feasibility study of a self-powered piezoelectric sensor. Vijay, K. Varadan, 2004, SPIE:377-388
21 K. Mossi, C. Green, Z. Ounaies, et al. Harvesting energy using a thin unimorph prestressed bender: Geometrical effects. Journal of Intelligent Material Systems and Structures. 2005, 16(3):249-261
22 Mingjie Guan, Wei-Hsin Liao. On the energy storage devices in piezoelectric energy harvesting. William, W. Clark, Mehdi, AhmadianArnold, Lumsdaine, 2006, SPIE:61690C
23 D. Guyomar, A. Badel, E. Lefeuvre, et al. Toward energy harvesting using active materials and conversion improvement by nonlinear processing. Ieee Transactions on Ultrasonics Ferroelectrics and Frequency Control. 2005, 52(4):584-595
24 S. N. Jiang, X. F. Li, S. H. Guo, et al. Performance of a piezoelectric bimorph for scavenging vibration energy. Smart Materials & Structures. 2005, 14(4):769-774
25 S. R. Platt, S. Farritor, H. Haider. On low-frequency electric power generation with PZT ceramics. Ieee-Asme Transactions on Mechatronics. 2005, 10(2):240-252
26 M. Ericka, D. Vasic, F. Costa, et al. Energy harvesting from vibration using a piezoelectric membrane. Journal De Physique Iv. 2005, 128:187-193
27 AD Danak, HS Yoon, GN Washington. Optimization of electrical output in response to mechanical input in piezoceramic laminated shells. Proc. ASME Int. Mech. Eng. Congr., Washington, DC. 2003
28程光明,庞建志,唐可洪.压电陶瓷发电能力测试系统的研制.吉林大学学报(工学版). 2007(02)
29陈子光,胡元太,杨嘉实.基于扭转模态的角振动压电俘能器研究.应用数学和力学. 2007(06)
30胡洪平,高发荣,薛欢.低频螺旋状压电俘能器结构性能分析.固体力学学报. 2007(01)
31 Yabin Liao, Henry A. Sodano. Structural Effects and Energy Conversion Efficiency of Power Harvesting. Journal of Intelligent Material Systems and Structures. 2009, 20(5):505-514
32 H. A. Xue, Y. T. Hu, Q. M. Wang. Broadband piezoelectric energy harvesting devices using multiple bimorphs with different operating frequencies. Ieee Transactions on Ultrasonics Ferroelectrics and Frequency Control. 2008, 55(9):2104-2108
33 P. J. Cornwell, J. Goethal, J. Kowko, et al. Enhancing power harvesting using a tuned auxiliary structure. Journal of Intelligent Material Systems and Structures. 2005, 16(10):825-834
34 S Roundy, Y Zhang. Toward self-tuning adaptive vibration-based microgenerators, 2005:373
35 C. B. Williams, R. B. Yates. Analysis of a micro-electric generator for microsystems. Sensors and Actuators a-Physical. 1996, 52(1-3):8-11
36 HW Kim, A Batra, S Priya, et al. Energy harvesting using a piezoelectric" cymbal" transducer in dynamic environment. Japanese Journal of Applied Physics. 2004, 43(9 A):6178-6183
37 J Han, A Von Jouanne, T Le, et al. Novel power conditioning circuits for piezoelectric micropower generators, 2004
38 AJ du Plessis, MJ Huigsloot, FD Discenzo. Resonant packaged piezoelectric power harvester for machinery health monitoring, 2005:224
39 G. A. Lesieutre, G. K. Ottman, H. F. Hofmann. Damping as a result of piezoelectric energy harvesting. Journal of Sound and Vibration. 2004, 269(3-5):991-1001
40 G. K. Ottman, H. F. Hofmann, G. A. Lesieutre. Optimized piezoelectric energy harvesting circuit using step-down converter in discontinuous conduction mode. Ieee Transactions on Power Electronics. 2003, 18(2):696-703
41 E Lefeuvre, A Badel, C Richard, et al. A comparison between several vibration-powered piezoelectric generators for standalone systems. Sensors & Actuators: A. Physical. 2006, 126(2):405-416
42 N. S. Shenck, J. A. Paradiso. Energy scavenging with shoe-mounted piezoelectrics. Ieee Micro. 2001, 21(3):30-42
43 S. Priya. Modeling of electric energy harvesting using piezoelectric windmill. Applied Physics Letters. 2005, 87(18):-
44张福学.现代压电学,上册.科学出版社, 2001
45あおぐと光るうちわ.日新电机, 2004
46 I.A. Karnovsky, O.L. Lebed. Free Vibration of Beams and Frames. McGraw-Hill, 2004
47 R. W. Clough, J. Penzin. Dynamics of Structures. Second. McGraw-Hill, 1993
48 HA Sodano, DJ Inman, G Park. A Review of Power Harvesting from Vibration Using Piezoelectric Materials. The Shock and Vibration Digest. 2004, 36(3):197
49周祝林,王亚熊.多层复合材料弹性模量理论估算.玻璃钢. 2002:6-11
50 AG Haddow, ADS Barr, DT Mook. Theoretical and experimental study of modal interaction in a two-degree-of-freedom structure. Journal of Sound and Vibration. 1984, 97:451-473
51 A. Erturk, J. M. Renno, D. J. Inman. Modeling of Piezoelectric Energy Harvesting from an L-shaped Beam-mass Structure with an Application to UAVs. Journal of Intelligent Material Systems and Structures. 2009, 20(5):529-544
52 S. M. Shahruz. Design of mechanical band-pass filters for energy scavenging. Journal of Sound and Vibration. 2006, 292(3-5):987-998
53盛美萍,王敏庆,孙进才.噪声与振动控制技术基础.科学出版社, 2001
54 A. Nayfeh, D. Mook. Nonlinear Oscillations. Wiley, 1979
2 S. R. Anton, H. A. Sodano. A review of power harvesting using piezoelectric materials (2003-2006). Smart Materials & Structures. 2007, 16(3):R1-R21
3 A. J. Schoenecker, T. Daue, B. Brueckner, et al. Overview on macro fiber composite applications - art. no. 61701K. Smart Structures and Materials 2006: Active Materials: Behavior and Mechanics. 2006, 6170:K1701-K1701
4 H. A. Sodano, D. J. Inman, G. Park. Comparison of piezoelectric energy harvesting devices for recharging batteries. Journal of Intelligent Material Systems and Structures. 2005, 16(10):799-807
5 J. Cho, M. Anderson, R. Richards, et al. Optimization of electromechanical coupling for a thin-film PZT membrane: I. Modeling. Journal of Micromechanics and Microengineering. 2005, 15(10):1797-1803
6 Y Ammar, A Buhrig, M Marzencki, et al. Wireless sensor network node with asynchronous architecture and vibration harvesting micro power generator, 2005, ACM New York, NY, USA:287-292
7方科,李欣欣,杨志刚.压电式能量获取装置的研究现状[J]. 2006
8 M. Ferrari, V. Ferrari, M. Guizzetti, et al. Piezoelectric multifrequency energy converter for power harvesting in autonomous microsystems. Sensors and Actuators a-Physical. 2008, 142(1):329-335
9 S Kim, TJ Johnson, WW Clark. Harvesting energy from a cantilever piezoelectric beam, 2004, SPIE:259-268
10 R. Want, K. I. Farkas, C. Narayanaswami. Energy harvesting and conservation. Ieee Pervasive Computing. 2005, 4(1):14-17
11 T. Starner. Human-powered wearable computing. Ibm Systems Journal. 1996, 35(3-4):618-629
12 S. W. Arms, C. P. Townsend, D. L. Churchill, et al. Power management for energy harvesting wireless sensors. Smart Structures and Materials 2005: Smart Electronics, Mems, BioMems, and Nanotechnology. 2005, 5763:267-275
13 T. A. Anderson, D. W. Sexton. A vibration energy harvesting sensor platform for increased industrial efficiency - art. no. 61741Y. Smart Structures and Materials 2006: Sensors and Smart Structures Technologies for Civil, Mechanical , and Aerospace Systems, Pts 1 and 2. 2006, 6174:Y1741-Y1741
14 J Baker, S Roundy, P Wright. Alternative Geometries for Increasing Power Density in Vibration Energy Scavenging for Wireless Sensor Networks
15 Shad Roundy, Paul K. Wright, Jan Rabaey. A study of low level vibrations as a power source for wireless sensor nodes. Computer Communications. 2003, 26(11):1131-1144
16 J. S. Yang, H. G. Zhou, Y. T. Hu, et al. Performance of a piezoelectric harvester in thickness-stretch mode of a plate. Ieee Transactions on Ultrasonics Ferroelectrics and Frequency Control. 2005, 52(10):1872-1876
17 C. D. Richards, M. J. Anderson, D. F. Bahr, et al. Efficiency of energy conversion for devices containing a piezoelectric component. Journal of Micromechanics and Microengineering. 2004, 14(5):717-721
18 J. Rastegar, C. Pereira, H. L. Nguyen. Piezoelectric-based power sources for harvesting energy from platforms with low frequency vibration - art. no. 617101. Smart Structures and Materials 2006: Industrial and Commercial Applications of Smart Structures Technologies. 2006, 6171:17101-17101
19 JW Sohn, SB Choi, DY Lee. An investigation on piezoelectric energy harvesting for MEMS power sources. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science. 2005, 219(4):429-436
20 Tsz-Ho Ng, Wei-Hsin Liao. Feasibility study of a self-powered piezoelectric sensor. Vijay, K. Varadan, 2004, SPIE:377-388
21 K. Mossi, C. Green, Z. Ounaies, et al. Harvesting energy using a thin unimorph prestressed bender: Geometrical effects. Journal of Intelligent Material Systems and Structures. 2005, 16(3):249-261
22 Mingjie Guan, Wei-Hsin Liao. On the energy storage devices in piezoelectric energy harvesting. William, W. Clark, Mehdi, AhmadianArnold, Lumsdaine, 2006, SPIE:61690C
23 D. Guyomar, A. Badel, E. Lefeuvre, et al. Toward energy harvesting using active materials and conversion improvement by nonlinear processing. Ieee Transactions on Ultrasonics Ferroelectrics and Frequency Control. 2005, 52(4):584-595
24 S. N. Jiang, X. F. Li, S. H. Guo, et al. Performance of a piezoelectric bimorph for scavenging vibration energy. Smart Materials & Structures. 2005, 14(4):769-774
25 S. R. Platt, S. Farritor, H. Haider. On low-frequency electric power generation with PZT ceramics. Ieee-Asme Transactions on Mechatronics. 2005, 10(2):240-252
26 M. Ericka, D. Vasic, F. Costa, et al. Energy harvesting from vibration using a piezoelectric membrane. Journal De Physique Iv. 2005, 128:187-193
27 AD Danak, HS Yoon, GN Washington. Optimization of electrical output in response to mechanical input in piezoceramic laminated shells. Proc. ASME Int. Mech. Eng. Congr., Washington, DC. 2003
28程光明,庞建志,唐可洪.压电陶瓷发电能力测试系统的研制.吉林大学学报(工学版). 2007(02)
29陈子光,胡元太,杨嘉实.基于扭转模态的角振动压电俘能器研究.应用数学和力学. 2007(06)
30胡洪平,高发荣,薛欢.低频螺旋状压电俘能器结构性能分析.固体力学学报. 2007(01)
31 Yabin Liao, Henry A. Sodano. Structural Effects and Energy Conversion Efficiency of Power Harvesting. Journal of Intelligent Material Systems and Structures. 2009, 20(5):505-514
32 H. A. Xue, Y. T. Hu, Q. M. Wang. Broadband piezoelectric energy harvesting devices using multiple bimorphs with different operating frequencies. Ieee Transactions on Ultrasonics Ferroelectrics and Frequency Control. 2008, 55(9):2104-2108
33 P. J. Cornwell, J. Goethal, J. Kowko, et al. Enhancing power harvesting using a tuned auxiliary structure. Journal of Intelligent Material Systems and Structures. 2005, 16(10):825-834
34 S Roundy, Y Zhang. Toward self-tuning adaptive vibration-based microgenerators, 2005:373
35 C. B. Williams, R. B. Yates. Analysis of a micro-electric generator for microsystems. Sensors and Actuators a-Physical. 1996, 52(1-3):8-11
36 HW Kim, A Batra, S Priya, et al. Energy harvesting using a piezoelectric" cymbal" transducer in dynamic environment. Japanese Journal of Applied Physics. 2004, 43(9 A):6178-6183
37 J Han, A Von Jouanne, T Le, et al. Novel power conditioning circuits for piezoelectric micropower generators, 2004
38 AJ du Plessis, MJ Huigsloot, FD Discenzo. Resonant packaged piezoelectric power harvester for machinery health monitoring, 2005:224
39 G. A. Lesieutre, G. K. Ottman, H. F. Hofmann. Damping as a result of piezoelectric energy harvesting. Journal of Sound and Vibration. 2004, 269(3-5):991-1001
40 G. K. Ottman, H. F. Hofmann, G. A. Lesieutre. Optimized piezoelectric energy harvesting circuit using step-down converter in discontinuous conduction mode. Ieee Transactions on Power Electronics. 2003, 18(2):696-703
41 E Lefeuvre, A Badel, C Richard, et al. A comparison between several vibration-powered piezoelectric generators for standalone systems. Sensors & Actuators: A. Physical. 2006, 126(2):405-416
42 N. S. Shenck, J. A. Paradiso. Energy scavenging with shoe-mounted piezoelectrics. Ieee Micro. 2001, 21(3):30-42
43 S. Priya. Modeling of electric energy harvesting using piezoelectric windmill. Applied Physics Letters. 2005, 87(18):-
44张福学.现代压电学,上册.科学出版社, 2001
45あおぐと光るうちわ.日新电机, 2004
46 I.A. Karnovsky, O.L. Lebed. Free Vibration of Beams and Frames. McGraw-Hill, 2004
47 R. W. Clough, J. Penzin. Dynamics of Structures. Second. McGraw-Hill, 1993
48 HA Sodano, DJ Inman, G Park. A Review of Power Harvesting from Vibration Using Piezoelectric Materials. The Shock and Vibration Digest. 2004, 36(3):197
49周祝林,王亚熊.多层复合材料弹性模量理论估算.玻璃钢. 2002:6-11
50 AG Haddow, ADS Barr, DT Mook. Theoretical and experimental study of modal interaction in a two-degree-of-freedom structure. Journal of Sound and Vibration. 1984, 97:451-473
51 A. Erturk, J. M. Renno, D. J. Inman. Modeling of Piezoelectric Energy Harvesting from an L-shaped Beam-mass Structure with an Application to UAVs. Journal of Intelligent Material Systems and Structures. 2009, 20(5):529-544
52 S. M. Shahruz. Design of mechanical band-pass filters for energy scavenging. Journal of Sound and Vibration. 2006, 292(3-5):987-998
53盛美萍,王敏庆,孙进才.噪声与振动控制技术基础.科学出版社, 2001
54 A. Nayfeh, D. Mook. Nonlinear Oscillations. Wiley, 1979