压电俘能器俘能性能数学建模及仿真分析
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摘要
随着MEMS、微电子等低耗能电子产品的广泛应用,对便携式、移动式电源的需求越来越强烈,而以传统的化学电池为这些微电子器件供能存在许多明显的弊端,如体积大、供能寿命有限、需要定期更换等。为解决这个问题,人们开始致力于研究从环境中吸取能量并转化为电能的方法。基于压电能量转换的压电俘能器具有能量密度大、结构简单、能耗低、易微型化等优点,因此具有非常广阔的应用前景。以往对压电俘能器的研究,要么集中对压电俘能器中压电振子结构进行优化研究,要么集中对压电储能电路进行研究,而都没有将压电振子结构与储能电路耦合起来分析,同时压电俘能器现主要有两种典型的结构形式:悬臂梁式和圆盘式,因此本文将以典型的悬臂梁和圆盘式压电俘能器为研究对象,将压电振子结构与储能电路耦合起来分析,针对其发电性能的数学建模和仿真分析展开研究。
本文首先针对悬臂梁式压电俘能器进行研究,利用弹性力学与压电理论建立压电振子结构与能量存储电路耦合作用的发电性能数学模型,并在该模型的基础上分析压电振子主要结构参数、压电材料参数和负载对压电俘能器输出性能的影响关系,为悬臂梁式压电俘能器的结构设计与优化提供理论研究。
在理论分析的基础上,建立悬臂梁式压电俘能器的机电耦合作用有限元仿真分析模型,分析压电振子主要结构参数、压电材料参数和负载对压电俘能器输出性能的影响关系,验证理论模型的正确性。
最后,本文对带中心质量的圆盘式压电俘能器进行研究,利用理论弹性力学与压电理论建立圆盘式压电俘能器输出性能的数学模型,分析压电俘能器在串联与并联连接两种情况下,压电振子振动频率和负载对压电俘能器低频时输出性能的影响关系,同时利用有限元方法建立圆盘式压电俘能器机电耦合作用分析模型,仿真验证理论模型的正确性。
With using MEMS, micro-electronics and other low power electronic products more convenience, the need for portable, mobile power is more strongly. Many drawbacks of the traditional chemical battery which can not be ignored, such as big size, large quality, limited lifespan, periodic replacement. People begin to study the method that can harvest energy from environment and convert to electricity. The energy harvester based on piezoelectric has many advantages, such as big energy density, simple structure, low energy consumption, easy miniaturization, so it is very broad application prospects. Previous research on piezoelectric energy harvester, either focus on the piezoelectric structure optimization, or focus on energy storage circuit, they are ignored the coupling of the piezoelectric structure and storage circuit. There are two main forms of piezoelectric structures: cantilever and disc pattern, in this article two kinds of energy harvester will be studied, the coupling of the piezoelectric structure and storage circuit will be considered. Their mathematical model and simulation for power generation performance will be studied.
In this paper, cantilever energy harvester is firstly studied. Using theory of elasticity and piezoelectric established piezoelectric structure and energy storage circuit coupled mathematical model of power generation performance, and based on the model analyze the impact of main structural parameters, piezoelectric material parameters and load on energy harvester’s output performance, for the cantilever piezoelectric energy harvester structure design and optimization provide a theoretical study.
Based on theoretical analysis, the electro-mechanical coupling finite element simulation model of cantilever piezoelectric energy harvester is established. Use the model analyze the impact of main structural parameters of the energy harvester, piezoelectric material parameters and load on energy harvester’s output performance, Verify the correctness of theoretical model.
Finally, disc piezoelectric energy harvester with a central-attached mass is studied, Using theory of elasticity and piezoelectric established the coupled mathematical model of power generation performance, Use the model analyze the impact of frequency and load on energy harvester’s output performance in both series connection and parallel connection, at the same time, the electro-mechanical coupling finite element simulation model of disc piezoelectric energy harvester is established, simulation verify the correctness of theoretical model.
本文首先针对悬臂梁式压电俘能器进行研究,利用弹性力学与压电理论建立压电振子结构与能量存储电路耦合作用的发电性能数学模型,并在该模型的基础上分析压电振子主要结构参数、压电材料参数和负载对压电俘能器输出性能的影响关系,为悬臂梁式压电俘能器的结构设计与优化提供理论研究。
在理论分析的基础上,建立悬臂梁式压电俘能器的机电耦合作用有限元仿真分析模型,分析压电振子主要结构参数、压电材料参数和负载对压电俘能器输出性能的影响关系,验证理论模型的正确性。
最后,本文对带中心质量的圆盘式压电俘能器进行研究,利用理论弹性力学与压电理论建立圆盘式压电俘能器输出性能的数学模型,分析压电俘能器在串联与并联连接两种情况下,压电振子振动频率和负载对压电俘能器低频时输出性能的影响关系,同时利用有限元方法建立圆盘式压电俘能器机电耦合作用分析模型,仿真验证理论模型的正确性。
With using MEMS, micro-electronics and other low power electronic products more convenience, the need for portable, mobile power is more strongly. Many drawbacks of the traditional chemical battery which can not be ignored, such as big size, large quality, limited lifespan, periodic replacement. People begin to study the method that can harvest energy from environment and convert to electricity. The energy harvester based on piezoelectric has many advantages, such as big energy density, simple structure, low energy consumption, easy miniaturization, so it is very broad application prospects. Previous research on piezoelectric energy harvester, either focus on the piezoelectric structure optimization, or focus on energy storage circuit, they are ignored the coupling of the piezoelectric structure and storage circuit. There are two main forms of piezoelectric structures: cantilever and disc pattern, in this article two kinds of energy harvester will be studied, the coupling of the piezoelectric structure and storage circuit will be considered. Their mathematical model and simulation for power generation performance will be studied.
In this paper, cantilever energy harvester is firstly studied. Using theory of elasticity and piezoelectric established piezoelectric structure and energy storage circuit coupled mathematical model of power generation performance, and based on the model analyze the impact of main structural parameters, piezoelectric material parameters and load on energy harvester’s output performance, for the cantilever piezoelectric energy harvester structure design and optimization provide a theoretical study.
Based on theoretical analysis, the electro-mechanical coupling finite element simulation model of cantilever piezoelectric energy harvester is established. Use the model analyze the impact of main structural parameters of the energy harvester, piezoelectric material parameters and load on energy harvester’s output performance, Verify the correctness of theoretical model.
Finally, disc piezoelectric energy harvester with a central-attached mass is studied, Using theory of elasticity and piezoelectric established the coupled mathematical model of power generation performance, Use the model analyze the impact of frequency and load on energy harvester’s output performance in both series connection and parallel connection, at the same time, the electro-mechanical coupling finite element simulation model of disc piezoelectric energy harvester is established, simulation verify the correctness of theoretical model.
引文
1方科,李欣欣,杨志刚,程光明,阚君武.压电式能量获取装置的研究现状.传感器与微系统. 2006, 2(l0):7~9
2杜小振,褚金奎,朴相镐,张海军.基于微型悬臂梁的发电机制探索.中国机械工程.2005,16:41~43
3 S. Roundy, P. K. Wrisht, J. Rabaey. A study of low level vibrations as apower source for wireless sensor nodes. Computer Communications.2003,26:1131~1144
4 G. Poulin, E. Sarraute, F. Costa. Generation of electrical energy for portable devices comparative study of an electromagnetic and a piezoelectric system. Sensors and Actuators A. 2004,114:461~471
5胡洪平,高发荣,薛欢,胡元太.低频螺旋状压电俘能器结构性能分析.固体力学学报. 2007,28(1):87
6 Y. Amnar, A. Buhrig, M. Marzencki. Wireless sensor network node with asynchronous architecture and vibration harvesting micro power generator. Joint soc-EUSAI conference.2005,1~6
7 Y. B. Jeon, R. Sood, J. H. Jeong. MEMS power generator with transversemode thin film PZT. Sensors and Actuators A. 2005, 122:16~22
8 J. Baker, S. Roundy, P. Wright. Alternative geometries for increasing power density in vibration energy scavenging for wireless sensor networks Proc. 3rd Int. Energy Conversion Engineering Conf. 2005,8:959~970
9 T. H. Ng, W. H. Liao. Feasibility study of a self-powered piezoelectric sensor. Proc. Smart Structures and Materials Conf. 2004, 5389:377~388
10 S. R. Platt, S. Farritor, K. Garvin, H. Haider. The use of piezoelectric ceramics for electric power generation. IEEE/ASME transactions on mechatronics. 2005,(10):455~461
11 H. W. Kim, A. Batra, S. Priya. Energy harvesting using a piezoelectric‘cymbal’transducer in dynamic environment. Appl Phys.2004, 6:178~183
12 H. W. Kim, S. Priya, K. Uchino. Piezoelectric energy harvesting under high pre-stressed cyclic vibrations. Journal of Electroceramics.2005,15:27~34
13 C. L. Sun, K. H. Lam. A novel drum piezoelectric-actuator. Appl. Phys.2006,6:385~389
14 S. Wang, K. H. Lam, C. L. Sun, K. W. Kwok, H. L. Chan, M. S. Guo,X. Z. Zhao. Energy harvesting with piezoelectric drum transducer. Appliedphysics letters. 2007, 90:113506
15 S. N Jiang, Y. T. Hu. Analysis of a Piezoelectric Bimorph Plate with a Central-Attached Mass as an Energy Harvester, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control. 2007, 54(7):1463~1469
16 C. W, R. Xia, J. Brewer. Piezoelectric Micro Power Generator(PMPG):A MEMS based Portable Power Device. http://mit.edu/micronanosystems/www/MTL%20AR2004%20PMPG%20300words,2005
17 G. W. Taylor, J. R. Burns, S. M. Kaman, W. B. Powers, T. R. Welsh. The Energy Harvesting Eel: A Small Subsurface Ocean/River Power Generator. IEEE Journal of oceanic engineering.2001, 26(4):539~547
18 S. W. Arms, C. P. Townsend, D. L. Churchill. Power management for energy harvesting wireless sensors Proc. Smart Structures and Materials Conf. 2005,5763:267~75
19 S. Priya. Modeling of electric energy harvesting using piezoelectric windmill. Appl. Phys. Lett. 2005,87: 184101
20 M. Kimura. Piezoelectric generation device. US Patent, 5801475
21 N. Ching, G. M. Chan, W. J. Li. PCB integrated micro-generator for wireless systems. International Symposium on Smart Structures and Microsystems, 2000,6:19~21
22 J. Rastegar, D. Haarhoff, C. Pereira. Piezoelectric-based energy-harvesting power sources for gun-fired munitions. Conference on Industrial and Commercial Applications of Smart Structures Technologies, 2006,6174:61740
23 N. G. Elvin, A. A. Elvin, M. Spector. A self-powered mechanical strain energy sensor. Smart Materials and Structures, 2001,10(2): 293~299
24 R. J. Duggirala, H. Li, A. Lal. An autonomous self-powered acoustic transmitter using radioactive thin films. IEEE ultrasonics symposium,2004,2:1328~1321
25朱永华.压电供电式红外遥控器的研究.吉林大学工学硕士学位论文. 2007:49~67
26刘艳涛.压电自供电型无线发射装置研究.吉林大学工学硕士学位论文.2006:59~70
27董璐.基于MEMS的压电型微能量采集器的研究.上海交通大学工学硕士学位论文.2007:48~56
28 S. Roundy, J. Rabaey. A study of low lever vibrations as a power sourcefor wireless sensor nodes. Computer Communications.2003,26:1131~1144
29 Y. S. Cho, Y. E. Pak. Five-port equivalent electric circuit of piezoelectricbimorph beam.Sensors Actuators.2000,84:140~148
30 R. Aoyagi, H. Tanaka. Equivalent circuit analysis of piezoelectric bendingvibrators. Japanese Journal of Applied Physics.1995,33:3010~3014
31 G. K. Ottman, H. F. Hofmann, A. C. Bhatt. Adaptive piezoelectric energyharvesting circuit for wireless remote power supply. IEEE Transactions on Power Electronics. 2002,17(5):669~676
32 C. Keawboonchuay, T. G. Engel. Maximum power generation in a piezoelectric pulse generator. IEEE Transations on Plasma Science. 2003,31:123~128
33 S. Roundy, P. K. Wright. A piezoelectric vibration based generator for wireless electronics. Smart Mater.Struct.2004,13:1131~1142
34 H. Y. Kim, S. H. Priya, H. A. Stephanou, K. E. Uchino. Consideration of impedance matching techniques for efficient piezoelectric energy harvesting. IEEE transactions on ultrasonic, ferroelectrics, and frequency control. 2007,54:1851~1859
35 Y. B. Liao, H. A. Sodano. Model of a single mode energy harvester and properties for optimal power generation. Smart Mater. Struct.2008,17:1~13
36 E. Lefeuvre, D. Audigier, C. Richard, D. Guyomar. Buck-Boost Converterfor Sensor less Power Optimization of Piezoelectric Energy Harvester. IEEE transactions on power electronics. 2007,22:2018~2025
37辛雪花.压电振子发电的基本特性及试验研究.吉林大学工学硕士学位论文.2006:43~48
38 J. Ajitsaria1, S. Y. Choe1, D. Shen, D. J. Kim. Modeling and analysis ofa bimorph piezoelectric cantilever beam for voltage generation. Smart Mater. Struct. 2007,16:447~454
39 H. Y. Kim, S. H. Priya, K. J. Uchino. Modeling of Piezoelectric Energy Harvesting Using Cymbal Transducers. Japanese Journal of Applied Physics.2006,45:5836~5840
40 F. Lu, H. P. Lee1, S. P. Lim. Modeling and analysis of micro piezoelectric power generators for micro-electromechanical-systems applications. Smart Mater. Struct. 2004, 13:57~63
41 Y. T. Hu, X. S. Zhang, J. S. Yang, Q. Jiang. Transmitting Electric Energy Through a Metal. Acoustic Waves Using Piezoelectric Transducers. 2003,7(50):773~780
42徐芝伦.弹性力学(下册).第四版.高等教育出版社,2006:1~9
43栾桂冬,张金铎等.压电换能器和换能器阵.北京大学出版社, 2005:40~49
44 Yoon, Washington, Danak. Modeling, Optimization and design of efficientinitially curved piezoceramic unimorphs for energy harvesting applications. Journal of intelligent material systems and structures. 2005,16:877~888
45 K. H. Sung. Admittance matrix of asymmetric piezoelectric bimorph with two separate electrical ports under general distributed loads. IEEE transactions on ultrasonics, ferroelectrics, and frequency control. 2001,48:976~983
46 S. N. Jiang, X. F. Li. Performance of a piezoelectric bimorph for scavenging vibration energy. Smart materials and structures. 2005,14:769~774
47莫喜平. ANSYS软件在模拟分析声学换能器中的应用.声学技术. 2007,26:1279~1290
48 S. H. Chang, J. F. Lin. Analysis and optimization of trimorph ring transducers. Journal of Sound and Vibration. 2003,263:831~851
49叶会英,张德辉.压电元件机电耦合系数定义研究.电子元件与材料, 2004,23:1~6
50 R. C. Lawrence, W. W. Clark. Comparison of low-frequency piezoelectricswitching shunt techniques for structural damping. Smart Mater. Struct. 2002,11:370~376
51 N. W. Hagood, A von flotow damping of structural vibrations with piezoelectric materials and passive electrical networks. Journal of sound and vibration. 1991,146(2):243~268
52 C. M. Junior, A. Erturk, D. Inman. An electromechanical finite element model for piezoelectric energy harvester plates. Journal of Sound and Vibration. 2009,327:9~25
53 A. Erturk, D. J. Inman. A distributed parameter electromechanical model for cantilevered piezoelectric energy harvesters. Journal of vibration and Acoustics. 2008,130:1~14
54 A. Erturk, D. J. Inman. An experimentally validated bimorph cantilever model for piezoelectric energy harvesting from base excitations. Smart Mater.Struct. 2009,18:1~17
55刘品宽,孙立宁,祝宇虹,赵研峰.双压电复合薄圆板驱动器的理论分析.压电与声光. 2002,4:111~115
56 S. N. Jiang, Y. T. Hu. Analysis of a piezoelectric bimorph plate with a central-attached mass as an energy harvester. IEEE transactions on ultrasonics, ferroelectrics, and frequency control. 2007,54(7):1463~1469
2杜小振,褚金奎,朴相镐,张海军.基于微型悬臂梁的发电机制探索.中国机械工程.2005,16:41~43
3 S. Roundy, P. K. Wrisht, J. Rabaey. A study of low level vibrations as apower source for wireless sensor nodes. Computer Communications.2003,26:1131~1144
4 G. Poulin, E. Sarraute, F. Costa. Generation of electrical energy for portable devices comparative study of an electromagnetic and a piezoelectric system. Sensors and Actuators A. 2004,114:461~471
5胡洪平,高发荣,薛欢,胡元太.低频螺旋状压电俘能器结构性能分析.固体力学学报. 2007,28(1):87
6 Y. Amnar, A. Buhrig, M. Marzencki. Wireless sensor network node with asynchronous architecture and vibration harvesting micro power generator. Joint soc-EUSAI conference.2005,1~6
7 Y. B. Jeon, R. Sood, J. H. Jeong. MEMS power generator with transversemode thin film PZT. Sensors and Actuators A. 2005, 122:16~22
8 J. Baker, S. Roundy, P. Wright. Alternative geometries for increasing power density in vibration energy scavenging for wireless sensor networks Proc. 3rd Int. Energy Conversion Engineering Conf. 2005,8:959~970
9 T. H. Ng, W. H. Liao. Feasibility study of a self-powered piezoelectric sensor. Proc. Smart Structures and Materials Conf. 2004, 5389:377~388
10 S. R. Platt, S. Farritor, K. Garvin, H. Haider. The use of piezoelectric ceramics for electric power generation. IEEE/ASME transactions on mechatronics. 2005,(10):455~461
11 H. W. Kim, A. Batra, S. Priya. Energy harvesting using a piezoelectric‘cymbal’transducer in dynamic environment. Appl Phys.2004, 6:178~183
12 H. W. Kim, S. Priya, K. Uchino. Piezoelectric energy harvesting under high pre-stressed cyclic vibrations. Journal of Electroceramics.2005,15:27~34
13 C. L. Sun, K. H. Lam. A novel drum piezoelectric-actuator. Appl. Phys.2006,6:385~389
14 S. Wang, K. H. Lam, C. L. Sun, K. W. Kwok, H. L. Chan, M. S. Guo,X. Z. Zhao. Energy harvesting with piezoelectric drum transducer. Appliedphysics letters. 2007, 90:113506
15 S. N Jiang, Y. T. Hu. Analysis of a Piezoelectric Bimorph Plate with a Central-Attached Mass as an Energy Harvester, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control. 2007, 54(7):1463~1469
16 C. W, R. Xia, J. Brewer. Piezoelectric Micro Power Generator(PMPG):A MEMS based Portable Power Device. http://mit.edu/micronanosystems/www/MTL%20AR2004%20PMPG%20300words,2005
17 G. W. Taylor, J. R. Burns, S. M. Kaman, W. B. Powers, T. R. Welsh. The Energy Harvesting Eel: A Small Subsurface Ocean/River Power Generator. IEEE Journal of oceanic engineering.2001, 26(4):539~547
18 S. W. Arms, C. P. Townsend, D. L. Churchill. Power management for energy harvesting wireless sensors Proc. Smart Structures and Materials Conf. 2005,5763:267~75
19 S. Priya. Modeling of electric energy harvesting using piezoelectric windmill. Appl. Phys. Lett. 2005,87: 184101
20 M. Kimura. Piezoelectric generation device. US Patent, 5801475
21 N. Ching, G. M. Chan, W. J. Li. PCB integrated micro-generator for wireless systems. International Symposium on Smart Structures and Microsystems, 2000,6:19~21
22 J. Rastegar, D. Haarhoff, C. Pereira. Piezoelectric-based energy-harvesting power sources for gun-fired munitions. Conference on Industrial and Commercial Applications of Smart Structures Technologies, 2006,6174:61740
23 N. G. Elvin, A. A. Elvin, M. Spector. A self-powered mechanical strain energy sensor. Smart Materials and Structures, 2001,10(2): 293~299
24 R. J. Duggirala, H. Li, A. Lal. An autonomous self-powered acoustic transmitter using radioactive thin films. IEEE ultrasonics symposium,2004,2:1328~1321
25朱永华.压电供电式红外遥控器的研究.吉林大学工学硕士学位论文. 2007:49~67
26刘艳涛.压电自供电型无线发射装置研究.吉林大学工学硕士学位论文.2006:59~70
27董璐.基于MEMS的压电型微能量采集器的研究.上海交通大学工学硕士学位论文.2007:48~56
28 S. Roundy, J. Rabaey. A study of low lever vibrations as a power sourcefor wireless sensor nodes. Computer Communications.2003,26:1131~1144
29 Y. S. Cho, Y. E. Pak. Five-port equivalent electric circuit of piezoelectricbimorph beam.Sensors Actuators.2000,84:140~148
30 R. Aoyagi, H. Tanaka. Equivalent circuit analysis of piezoelectric bendingvibrators. Japanese Journal of Applied Physics.1995,33:3010~3014
31 G. K. Ottman, H. F. Hofmann, A. C. Bhatt. Adaptive piezoelectric energyharvesting circuit for wireless remote power supply. IEEE Transactions on Power Electronics. 2002,17(5):669~676
32 C. Keawboonchuay, T. G. Engel. Maximum power generation in a piezoelectric pulse generator. IEEE Transations on Plasma Science. 2003,31:123~128
33 S. Roundy, P. K. Wright. A piezoelectric vibration based generator for wireless electronics. Smart Mater.Struct.2004,13:1131~1142
34 H. Y. Kim, S. H. Priya, H. A. Stephanou, K. E. Uchino. Consideration of impedance matching techniques for efficient piezoelectric energy harvesting. IEEE transactions on ultrasonic, ferroelectrics, and frequency control. 2007,54:1851~1859
35 Y. B. Liao, H. A. Sodano. Model of a single mode energy harvester and properties for optimal power generation. Smart Mater. Struct.2008,17:1~13
36 E. Lefeuvre, D. Audigier, C. Richard, D. Guyomar. Buck-Boost Converterfor Sensor less Power Optimization of Piezoelectric Energy Harvester. IEEE transactions on power electronics. 2007,22:2018~2025
37辛雪花.压电振子发电的基本特性及试验研究.吉林大学工学硕士学位论文.2006:43~48
38 J. Ajitsaria1, S. Y. Choe1, D. Shen, D. J. Kim. Modeling and analysis ofa bimorph piezoelectric cantilever beam for voltage generation. Smart Mater. Struct. 2007,16:447~454
39 H. Y. Kim, S. H. Priya, K. J. Uchino. Modeling of Piezoelectric Energy Harvesting Using Cymbal Transducers. Japanese Journal of Applied Physics.2006,45:5836~5840
40 F. Lu, H. P. Lee1, S. P. Lim. Modeling and analysis of micro piezoelectric power generators for micro-electromechanical-systems applications. Smart Mater. Struct. 2004, 13:57~63
41 Y. T. Hu, X. S. Zhang, J. S. Yang, Q. Jiang. Transmitting Electric Energy Through a Metal. Acoustic Waves Using Piezoelectric Transducers. 2003,7(50):773~780
42徐芝伦.弹性力学(下册).第四版.高等教育出版社,2006:1~9
43栾桂冬,张金铎等.压电换能器和换能器阵.北京大学出版社, 2005:40~49
44 Yoon, Washington, Danak. Modeling, Optimization and design of efficientinitially curved piezoceramic unimorphs for energy harvesting applications. Journal of intelligent material systems and structures. 2005,16:877~888
45 K. H. Sung. Admittance matrix of asymmetric piezoelectric bimorph with two separate electrical ports under general distributed loads. IEEE transactions on ultrasonics, ferroelectrics, and frequency control. 2001,48:976~983
46 S. N. Jiang, X. F. Li. Performance of a piezoelectric bimorph for scavenging vibration energy. Smart materials and structures. 2005,14:769~774
47莫喜平. ANSYS软件在模拟分析声学换能器中的应用.声学技术. 2007,26:1279~1290
48 S. H. Chang, J. F. Lin. Analysis and optimization of trimorph ring transducers. Journal of Sound and Vibration. 2003,263:831~851
49叶会英,张德辉.压电元件机电耦合系数定义研究.电子元件与材料, 2004,23:1~6
50 R. C. Lawrence, W. W. Clark. Comparison of low-frequency piezoelectricswitching shunt techniques for structural damping. Smart Mater. Struct. 2002,11:370~376
51 N. W. Hagood, A von flotow damping of structural vibrations with piezoelectric materials and passive electrical networks. Journal of sound and vibration. 1991,146(2):243~268
52 C. M. Junior, A. Erturk, D. Inman. An electromechanical finite element model for piezoelectric energy harvester plates. Journal of Sound and Vibration. 2009,327:9~25
53 A. Erturk, D. J. Inman. A distributed parameter electromechanical model for cantilevered piezoelectric energy harvesters. Journal of vibration and Acoustics. 2008,130:1~14
54 A. Erturk, D. J. Inman. An experimentally validated bimorph cantilever model for piezoelectric energy harvesting from base excitations. Smart Mater.Struct. 2009,18:1~17
55刘品宽,孙立宁,祝宇虹,赵研峰.双压电复合薄圆板驱动器的理论分析.压电与声光. 2002,4:111~115
56 S. N. Jiang, Y. T. Hu. Analysis of a piezoelectric bimorph plate with a central-attached mass as an energy harvester. IEEE transactions on ultrasonics, ferroelectrics, and frequency control. 2007,54(7):1463~1469