Mathematical Modelling of Piezoelectric Generators on the Base of the Kantorovich Method

  • Arkadiy N. SolovievEmail author
  • Valerii A. Chebanenko
  • Ivan A. Parinov
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 81)


In this chapter, applied semi-analytical theories were constructed, allowing preliminary estimations of the output characteristics of piezoelectric generators (PEG) of various configurations. The developed theories are based on the Hamiltonian principle, extended to the theory of electroelasticity. In the first part of the work, within the framework of the Euler-Bernoulli hypotheses, a model for a cantilever PEG was developed. The main model’s peculiarity is the consideration of the structural features of cantilever PEGs. In the second part, a model was developed for multilayer stacked PEGs, where the energy generation process was considered as forced oscillations of an electroelastic rod. Solutions for both cases were carried out using the Kantorovich method. The adequacy of the theories obtained in both cases was verified by comparison with finite-element calculations.


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This work has been supported by the Government Contract (project part) 9.1001.2017/PCh, by the Russian Foundation for the Basic Research Grant 16-58-52013 MNT-a and by Russian State Mission 007-01114-16 PR (project 0256-2015-0074).


  1. Adhikari S, Friswell MI, Inman DJ (2009) Piezoelectric energy harvesting from broadband random vibrations. Smart Materials and Structures 18(11):115,005Google Scholar
  2. Baker J, Roundy S, Wright P (2005) Alternative geometries for increasing power density in vibration energy scavenging for wireless sensor networks. In: 3rd International Energy Conversion Engineering Conference, American Institute of Aeronautics and AstronauticsGoogle Scholar
  3. Cavallier B, Berthelot P, Nouira H, Foltete E, Hirsinger L, Ballandras S (2005) Energy harvesting using vibrating structures excited by shock. In: IEEE Ultrasonics Symposium, IEEE, vol 2, pp 943–945Google Scholar
  4. Chebanenko VA, Akopyan VA, Parinov IA (2015) Piezoelectric generators and energy harvesters: Modern state of the art. In: Parinov IA (ed) Piezoelectrics and Nanomaterials: Fundamentals, Developments and Applications, Nova Science Publishers, New York, pp 243–277Google Scholar
  5. Deng Q, Kammoun M, Erturk A, Sharma P (2014) Nanoscale flexoelectric energy harvesting. International Journal of Solids and Structures 51(18):3218–3225Google Scholar
  6. Dutoit NE, Wardle BL (2007) Experimental verification of models for microfabricated piezoelectric vibration energy harvesters. AIAA journal 45(5):1126–1137Google Scholar
  7. Dutoit NE, Wardle BL, Kim SG (2005) Design considerations for MEMS-scale piezoelectric mechanical vibration energy harvesters. Integrated Ferroelectrics 71(1):121–160Google Scholar
  8. Elvin N, Erturk A (2013) Advances in Energy Harvesting Methods. Springer, HeidelbergGoogle Scholar
  9. Erturk A, Inman DJ (2008) On mechanical modeling of cantilevered piezoelectric vibration energy harvesters. Journal of Intelligent Material Systems and Structures 19(11):1311–1325Google Scholar
  10. Erturk A, Inman DJ (2011) Piezoelectric Energy Harvesting. John Wiley and Sons, Ltd., New YorkGoogle Scholar
  11. Feenstra J, Granstrom J, Sodano H (2008) Energy harvesting through a backpack employing a mechanically amplified piezoelectric stack. Mechanical Systems and Signal Processing 22(3):721–734Google Scholar
  12. Goldfarb M, Jones LD (1999) On the efficiency of electric power generation with piezoelectric ceramic. Trans ASME J Of Dyn Syst Measurement and Control 121:566–571Google Scholar
  13. Han B, Vassilaras S, Papadias CB, Soman R, Kyriakides MA, Onoufriou T, Nielsen RH, Prasad R (2013) Harvesting energy from vibrations of the underlying structure. Journal of Vibration and Control 19(15):2255–2269Google Scholar
  14. Kerr AD, Alexander H (1968) An application of the extended kantorovich method to the stress analysis of a clamped rectangular plate. Acta Mechanica 6(2-3):180–196Google Scholar
  15. Liao Y, Sodano HA (2009) Structural effects and energy conversion efficiency of power harvesting. Journal of Intelligent Material Systems and Structures 20(5):505–514Google Scholar
  16. Liu Y, Tian G, Wang Y, Lin J, Zhang Q, Hofmann HF (2009) Active piezoelectric energy harvesting: General principle and experimental demonstration. Journal of Intelligent Material Systems and Structures 20(5):575–585Google Scholar
  17. Nechibvute A, Chawanda A, Luhanga P (2012) Finite element modeling of a piezoelectric composite beam and comparative performance study of piezoelectric materials for voltage generation. ISRN Mater Science ID 921361:11 pagesGoogle Scholar
  18. Roundy S, Wright PK (2004) A piezoelectric vibration based generator for wireless electronics. Smart Materials and Structures 13(5):1131Google Scholar
  19. Shevtsov S, Akopyan V, Rozhkov E, Chebanenko V, Yang CC, Jenny Lee CY, Kuo CX (2016) Optimization of the electric power harvesting system based on the piezoelectric stack transducer. In: Parinov IA, Chang SH, Topolov VY (eds) Advanced Materials: Manufacturing, Physics, Mechanics and Applications, Springer International Publishing, Cham, pp 639–650Google Scholar
  20. Soloviev AN, Vatulyan AO (2011) Non-classical biem in electroelasticity and inverse coefficient problem. In: Parinov IA (ed) Piezoceramic Materials and Devices, Nova Science Publishers, New York, pp 1–51Google Scholar
  21. Soloviev AN, Parinov IA, Duong LV, Yang CC, Chang SH, Lee JCY (2013) Analysis of finite element models for piezoelectric devices of energy harvestingm. In: Parinov IA, Chang SH (eds) Physics and Mechanics of New Materials and their Applications, Nova Science Publishers, New York, pp 335–352Google Scholar
  22. Soloviev AN, Chebanenko VA, Zakharov YN, Rozhkov EV, Parinov IA, Gupta VK (2017) Study of the output characteristics of ferroelectric ceramic beam made from non-polarized ceramics pzt-19: Experiment and modeling. In: Parinov IA, Chang SH, Jani MA (eds) Advanced Materials: Techniques, Physics, Mechanics and Applications, Springer International Publishing, Cham, pp 485–499Google Scholar
  23. Solovyev AN, Duong LV (2016) Optimization for the harvesting structure of the piezoelectric bimorph energy harvesters circular plate by reduced order finite element analysis. International Journal of Applied Mechanics 8(3):1650,029Google Scholar
  24. Solovyev AN, Duong LV, Akopyan VA, Rozhkov EV, Chebanenko VA (2016) Numerical simulation of the experiment on pulsed excitation of stack type piezoelectric generator. Vestnik DSTU 1(84):19–26Google Scholar
  25. Vatulyan AO, Soloviev AN (2009) Direct and inverse problems for homogeneous and inhomogeneous elastic and electroelastic bodies (in Russ.). SFEDU Publishers, Rostov-on-DonGoogle Scholar
  26. Wang J, Shi Z, Han Z (2013) Analytical solution of piezoelectric composite stack transducers. Journal of Intelligent Material Systems and Structures 24(13):1626–1636Google Scholar
  27. Yu S, He S, Li W (2010) Theoretical and experimental studies of beam bimorph piezoelectric power harvesters. J of Mechanics of Mater and Structures 5(3):427–445Google Scholar
  28. Zhao S, Erturk A (2014) Deterministic and band-limited stochastic energy harvesting from uniaxial excitation of a multilayer piezoelectric stack. Sensors and Actuators A: Physical 214:58–65Google Scholar

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© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Arkadiy N. Soloviev
    • 1
    Email author
  • Valerii A. Chebanenko
    • 2
  • Ivan A. Parinov
    • 3
  1. 1.Don State Technical University, Gagarin sq., 1 & I. I. Vorovich Institute of Mathematics, Mechanics and Computer SciencesSouthern Federal UniversityRostov-on-DonRussia
  2. 2.Southern Scientific Center of Russian Academy of ScienceRostov-on-DonRussia
  3. 3.I. I. Vorovich Institute of Mathematics, Mechanics and Computer SciencesSouthern Federal UniversityRostov-on-DonRussia

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