Electromechanical Modeling of Cantilevered Piezoelectric Energy Harvesters for Persistent Base Motions

  • Alper Erturk
  • Daniel J. Inman


This chapter investigates electromechanical modeling of cantilevered piezoelectric energy harvesters excited by persistent base motions. The modeling approaches are divided here into two sections as lumped parameter modeling and distributed parameter modeling. The first section discusses the amplitude-wise correction of the existing lumped parameter piezoelectric energy harvester model for base excitation. For cantilevers operating in the transverse and longitudinal vibration modes, it is shown that the conventional base excitation expression used in the existing lumped parameter models may yield highly inaccurate results in predicting the vibration response of the structure. Dimensionless correction factors are derived to improve the predictions of the coupled lumped parameter piezoelectric energy harvester model. The second section of this chapter presents coupled distributed parameter modeling of unimorph and bimorph cantilevers under persistent base excitations for piezoelectric energy harvesting. Closed-form solutions are obtained by considering all vibration modes and the formal representation of the direct and converse piezoelectric effects. Steady state electrical and mechanical response expressions are derived for arbitrary frequency excitations. These multi-mode solutions are then reduced to single-mode solutions for excitations around the modal frequencies. Finally, the analytical expressions derived here are validated experimentally for a cantilevered bimorph with a proof mass.


Energy Harvester Proof Mass Lump Parameter Model Base Excitation Piezoelectric Energy Harvester 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ajitsaria J, Choe S Y, Shen D, and Kim D J 2007 Modeling and analysis of a bimorph piezoelectric cantilever beam for voltage generation Smart Materials and Structures 16:447–454CrossRefGoogle Scholar
  2. Anton S R and Sodano H A 2007 A review of power harvesting using piezoelectric materials (2003–2006) Smart Materials and Structures 16:R1–R21CrossRefGoogle Scholar
  3. Arnold D 2007 Review of microscale magnetic power generation IEEE Transactions on Magnetics 43:3940–3951CrossRefGoogle Scholar
  4. Banks T L and Inman D J 1991 On damping mechanisms in beams ASME Journal of Applied Mechanics 58:716–723MATHGoogle Scholar
  5. Beeby S P, Tudor M J, and White N M 2006 Energy harvesting vibration sources for microsystems applications, Measurement Science and Technology 13:175–195CrossRefGoogle Scholar
  6. Chen S-N, Wang G–J, and Chien M-C 2006 Analytical modeling of piezoelectric vibration-induced micro power generator, Mechatronics 16:387–397CrossRefGoogle Scholar
  7. Clough R W and Penzien J 1975 Dynamics of Structures John Wiley and Sons, New YorkMATHGoogle Scholar
  8. Cook-Chennault K A, Thambi N, and Sastry A M 2008 Powering MEMS portable devices – a review of non-regenerative and regenerative power supply systems with emphasis on piezoelectric energy harvesting systems, Smart Materials and Structures 17:043001:1–33CrossRefGoogle Scholar
  9. Crandall S H, Karnopp D C, Kurtz Jr E F, and Pridmore-Brown D C 1968 Dynamics of Mechanical and Electromechanical Systems McGraw-Hill, New YorkGoogle Scholar
  10. Daqaq M, Renno J M, Farmer J R, and Inman D J 2007 Effects of system parameters and damping on an optimal vibration-based energy harvester Proceedings of the 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference Google Scholar
  11. duToit N E, Wardle B L, and Kim S-G 2005 Design considerations for MEMS-scale piezoelectric mechanical vibration energy harvesters, Integrated Ferroelectrics 71:121–160CrossRefGoogle Scholar
  12. duToit N E and Wardle B L 2007 Experimental verification of models for microfabricated piezoelectric vibration energy harvesters, AIAA Journal 45:1126–1137CrossRefGoogle Scholar
  13. Elvin N and Elvin A 2008 A general equivalent circuit model for piezoelectric generators, Journal of Intelligent Material Systems and Structures 19 in press (DOI: 10.1177/1045389X08089957)Google Scholar
  14. Erturk A and Inman D J 2008a On mechanical modeling of cantilevered piezoelectric vibration energy harvesters, Journal of Intelligent Material Systems and Structures 19:1311–1325CrossRefGoogle Scholar
  15. Erturk A and Inman D J 2008b Issues in mathematical modeling of piezoelectric energy harvesters, Smart Materials and Structures in pressGoogle Scholar
  16. Erturk A and Inman D J 2008c A distributed parameter electromechanical model for cantilevered piezoelectric energy harvesters, ASME Journal of Vibration and Acoustics 130:041002-1-15CrossRefGoogle Scholar
  17. Erturk A and Inman D J 2008d An experimentally validated bimorph cantilever model for piezoelectric energy harvesting from base excitations, Smart Materials and Structures acceptedGoogle Scholar
  18. Erturk A and Tarazaga P A, Farmer J R, and Inman D J 2008e Effect of strain nodes and electrode configuration on piezoelectric energy harvesting from cantilevered beams, ASME Journal of Vibration and Acoustics in press (DOI: 10.1115/1.2981094)Google Scholar
  19. Fang H-B, Liu J-Q, Xu Z-Y, Dong L, Chen D, Cai B-C, and Liu Y 2006 A MEMS-based piezoelectric power generator for low frequency vibration energy harvesting, Chinese Physics Letters 23:732–734CrossRefGoogle Scholar
  20. Glynne-Jones P, Tudor M J, Beeby S P, and White N M 2004 An electromagnetic, vibration-powered generator for intelligent sensor systems, Sensors and Actuators A 110:344–349CrossRefGoogle Scholar
  21. Hagood N W, Chung W H, and Von Flotow A 1990 Modelling of piezoelectric actuator dynamics for active structural control, Journal of Intelligent Material Systems and Structures 1:327–354CrossRefGoogle Scholar
  22. IEEE Standard on Piezoelectricity 1987 IEEE, New York.Google Scholar
  23. Jeon Y B, Sood R, Jeong J H, and Kim S 2005 MEMS power generator with transverse mode thin film PZT, Sensors & Actuators A 122:16–22CrossRefGoogle Scholar
  24. Lesieutre G A, Ottman G K, and Hofmann H F 2004 Damping as a result of piezoelectric energy harvesting, Journal of Sound and Vibration 269:991–1001CrossRefGoogle Scholar
  25. Lin J H, Wu X M, Ren T L, and Liu L T 2007 Modeling and simulation of piezoelectric MEMS energy harvesting device, Integrated Ferroelectrics 95:128–141.CrossRefGoogle Scholar
  26. Lu F, Lee H, and Lim S 2004 Modeling and analysis of micro piezoelectric power generators for micro-electromechanical-systems applications, Smart Materials and Structures 13:57–63CrossRefGoogle Scholar
  27. Mitcheson P, Miao P, Start B, Yeatman E, Holmes A, and Green T 2004 MEMS electrostatic micro-power generator for low frequency operation, Sensors and Actuators A 115:523–529CrossRefGoogle Scholar
  28. Ottman G K, Hofmann H F, Bhatt A C, and Lesieutre G A 2002 Adaptive piezoelectric energy harvesting circuit for wireless remote power supply, IEEE Transactions on Power Electronics 17:669–676.CrossRefGoogle Scholar
  29. Priya S 2007 Advances in energy harvesting using low profile piezoelectric transducers, Journal of Electroceramics 19:167–184CrossRefGoogle Scholar
  30. Roundy S, Wright P K, and Rabaey J 2002 Micro-electrostatic vibration-to-electricity converters Proceedings of the ASME 2002 International Mechanical Engineering Congress and Exposition Google Scholar
  31. Roundy S, Wright P K, and Rabaey J 2003 A study of low level vibrations as a power source for wireless sensor nodes, Computer Communications 26:1131–1144CrossRefGoogle Scholar
  32. Sodano H A, Inman D J, and Park G 2004a A review of power harvesting from vibration using piezoelectric materials, The Shock and Vibration Digest 36:197–205CrossRefGoogle Scholar
  33. Sodano H A, Park G, and Inman D J 2004b Estimation of electric charge output for piezoelectric energy harvesting, Strain 40:49–58CrossRefGoogle Scholar
  34. Sodano H, Inman D, and Park G 2005 Generation and storage of electricity from power harvesting devices, Journal of Intelligent Material Systems and Structures 16:67–75CrossRefGoogle Scholar
  35. Stephen N G 2006 On energy harvesting from ambient vibration, Journal of Sound and Vibration 293:409–425CrossRefGoogle Scholar
  36. Strutt J W (Lord Rayleigh) 1894 The Theory of Sound MacMillan Company, LondonMATHGoogle Scholar
  37. Timoshenko S, Young D H, and Weaver W 1974 Vibration Problems in Engineering John Wiley and Sons, New YorkGoogle Scholar
  38. Williams C B and Yates R B 1996 Analysis of a micro-electric generator for microsystems, Sensors and Actuators A 52:8–11CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Alper Erturk
    • 1
  • Daniel J. Inman
  1. 1.Center for Intelligent Material Systems and Structures Department of Engineering Science and MechanicsVirginia Polytechnic Institute and State UniversityBlacksburgUSA

Personalised recommendations