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Revisit to the theoretical analysis of a classical piezoelectric vibration energy harvester

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Abstract

In this paper, we investigate the problem for a classical piezoelectric vibration energy harvester. Exact theoretical solution to the problem is derived and compared to the solutions proposed in the literature. Asymptotic expansions of the solution are explored in the hope of finding a plausible simpler approximation of the solution and corresponding output performance measures. Dependence of the output performance measures upon the electromechanical coupling factor is therefore studied. Some tips are then provided for the design of piezoelectric energy harvester.

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References

  1. Beeby, S.P., Tudor, M.J., White, N.: Energy harvesting vibration sources for microsystems applications. Meas. Sci. Technol. 17(12), R175 (2006)

    Article  Google Scholar 

  2. Anton, S.R., Sodano, H.A.: A review of power harvesting using piezoelectric materials (2003–2006). Smart Mater. Struct. 16(3), R1 (2007)

    Article  Google Scholar 

  3. Zhou, M., Al-Furjan, M.S.H., Zou, J., Liu, W.: A review on heat and mechanical energy harvesting from human-principles, prototypes and perspectives. Renew. Sustain. Energy Rev. 82, 3582–3609 (2018)

    Article  Google Scholar 

  4. Safaei, M., Sodano, H.A., Anton, S.R.: A review of energy harvesting using piezoelectric materials: state-of-the-art a decade later (2008–2018). Smart Mater. Struct. 28(11), 113001 (2019)

    Article  Google Scholar 

  5. Roundy, S., Wright, P.K., Rabaey, J.: A study of low level vibrations as a power source for wireless sensor nodes. Comput. Commun. 26(11), 1131–1144 (2003)

    Article  Google Scholar 

  6. Dutoit, N.E., Wardle, B.L., Kim, S.G.: Design considerations for MEMS-scale piezoelectric mechanical vibration energy harvesters. Integr. Ferroelectr. 71(1), 121–160 (2005)

    Article  Google Scholar 

  7. Stephen, N.G.: On energy harvesting from ambient vibration. J. Sound Vib. 293(1–2), 409–425 (2006)

    Article  Google Scholar 

  8. Cottone, F., Vocca, H., Gammaitoni, L.: Nonlinear energy harvesting. Phys. Rev. Lett. 102(8), 080601 (2009)

    Article  Google Scholar 

  9. Erturk, A., Inman, D.J.: On mechanical modeling of cantilevered piezoelectric vibration energy harvesters. J. Intell. Mater. Syst. Struct. 19(11), 1311–1325 (2008)

    Article  Google Scholar 

  10. Crandall, S.H.: Dynamics of Mechanical and Electromechanical Systems. McGraw-Hill, New York (1968)

    Google Scholar 

  11. Hagood, N.W., Chung, W.H., Von Flotow, A.: Modelling of piezoelectric actuator dynamics for active structural control. J. Intell. Mater. Syst. Struct. 1(3), 327–354 (1990)

    Article  Google Scholar 

  12. Sodano, H.A., Park, G., Inman, D.: Estimation of electric charge output for piezoelectric energy harvesting. Strain 40(2), 49–58 (2004)

    Article  Google Scholar 

  13. Lu, F., Lee, H., Lim, S.: Modeling and analysis of micro piezoelectric power generators for micro-electromechanical-systems applications. Smart Mater. Struct. 13(1), 57 (2003)

    Article  Google Scholar 

  14. Chen, S.N., Wang, G.J., Chien, M.C.: Analytical modeling of piezoelectric vibration-induced micro power generator. Mechatronics 16(7), 379–387 (2006)

    Article  Google Scholar 

  15. Ajitsaria, J., Choe, S.Y., Shen, D., Kim, D.: Modeling and analysis of a bimorph piezoelectric cantilever beam for voltage generation. Smart Mater. Struct. 16(2), 447 (2007)

    Article  Google Scholar 

  16. Kreyszig, E.: Introductory Functional Analysis with Applications, vol. 1. Wiley, New York (1978)

    MATH  Google Scholar 

  17. Erturk, A., Inman, D.J.: A distributed parameter electromechanical model for cantilevered piezoelectric energy harvesters. J. Vib. Acoust. 130(4), 041002 (2008)

    Article  Google Scholar 

  18. Erturk, A., Inman, D.J.: An experimentally validated bimorph cantilever model for piezoelectric energy harvesting from base excitations. Smart Mater. Struct. 18(2), 025009 (2009)

    Article  Google Scholar 

  19. Erturk, A., Tarazaga, P.A., Farmer, J.R., Inman, D.J.: Effect of strain nodes and electrode configuration on piezoelectric energy harvesting from cantilevered beams. J. Vib. Acoust. 131(1), 011010 (2009)

    Article  Google Scholar 

  20. Shu, Y., Lien, I.: Analysis of power output for piezoelectric energy harvesting systems. Smart Mater. Struct. 15(6), 1499 (2006)

    Article  Google Scholar 

  21. Shu, Y., Lien, I., Wu, W.: An improved analysis of the SSHI interface in piezoelectric energy harvesting. Smart Mater. Struct. 16(6), 2253 (2007)

    Article  Google Scholar 

  22. Qiu, J., Jiang, H., Ji, H., Zhu, K.: Comparison between four piezoelectric energy harvesting circuits. Front. Mech. Eng. China 4(2), 153–159 (2009)

    Article  Google Scholar 

  23. Landau, L.D., Lifshitz, E.: Course of Theoretical Physics, Theory of Elasticity, vol. 7. Pergamon Press, Oxford (1986)

    Google Scholar 

  24. Jackson, J.D.: Classical Electrodynamics. Wiley, New York (2007)

    MATH  Google Scholar 

  25. Weaver Jr., W., Timoshenko, S.P., Young, D.H.: Vibration Problems in Engineering. Wiley, New York (1990)

    Google Scholar 

  26. Timoshenko, S., Young, D.H.: Elements of Strength of Materials. Van Nostrand, New York (1968)

    Google Scholar 

  27. Erturk, A., Inman, D.J.: Piezoelectric Energy Harvesting. Wiley, New York (2011)

    Book  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the financial support from the National Natural Science Foundation of China (NSFC) under Contract Number 51705112. The research presented in this paper is also supported by the Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ18E050007.

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Appendix

Appendix

$$\begin{aligned} \left\{ \begin{aligned}&A_{k+1} + C_{k+1} = 0, \\&B_{k+1} + D_{k+1} = 0, \\&\left( - A_{k+1} \cos {\sqrt{\sigma }} - B_{k+1} \sin {\sqrt{\sigma }} + C_{k+1} \cosh {\sqrt{\sigma }} + D_{k+1} \sinh {\sqrt{\sigma }} \right) + \\&\frac{j \beta \sqrt{\sigma }}{ j\sigma \beta + 1 } \left( - A_{k} \sin {\sqrt{\sigma }} + B_{k} \cos {\sqrt{\sigma }} + C_{k} \sinh {\sqrt{\sigma }} + D_{k} \cosh {\sqrt{\sigma }} \right) = 0, \\&A_{k+1} \sin {\sqrt{\sigma }} - B_{k+1} \cos {\sqrt{\sigma }} + C_{k+1} \sinh {\sqrt{\sigma }} + D_{k+1} \cosh {\sqrt{\sigma }} = 0, \end{aligned}\right. \end{aligned}$$
(38)

whose solution is expressed by

$$\begin{aligned} \left\{ \begin{aligned} A_{k+1}&= \left( \frac{j \beta \sqrt{\sigma }}{1+j \beta \sigma } \right) \left( \frac{\cos \sqrt{\sigma }+\cosh \sqrt{\sigma }}{2 \cos \sqrt{\sigma }\cosh \sqrt{\sigma }+2} \right) \left( Q_k \right) , \\ B_{k+1}&= \left( \frac{j \beta \sqrt{\sigma }}{1+j \beta \sigma } \right) \left( \frac{-\sinh \sqrt{\sigma }+\sin \sqrt{\sigma }}{2 \cos \sqrt{\sigma }\cosh \sqrt{\sigma }+2} \right) \left( Q_k \right) , \\ C_{k+1}&= \left( \frac{j \beta \sqrt{\sigma }}{1+j \beta \sigma } \right) \left( -\frac{\cos \sqrt{\sigma }+\cosh \sqrt{\sigma }}{2 \cos \sqrt{\sigma } \cosh \sqrt{\sigma }+2} \right) \left( Q_k \right) , \\ D_{k+1}&= \left( \frac{j \beta \sqrt{\sigma }}{1+j \beta \sigma } \right) \left( \frac{-\sin \sqrt{\sigma }+\sinh \sqrt{\sigma }}{2 \cos \sqrt{\sigma }\cosh \sqrt{\sigma }+2} \right) \left( Q_k \right) , \end{aligned}\right. \end{aligned}$$
(39)

in which

$$\begin{aligned} Q_k = - A_{k} \sin {\sqrt{\sigma }} + B_{k} \cos {\sqrt{\sigma }} + C_{k} \sinh {\sqrt{\sigma }} + D_{k} \cosh {\sqrt{\sigma }}. \end{aligned}$$
(40)

In terms of \(Q_k\) (\(k \ge 0\)), we have the following iterative relation

$$\begin{aligned} Q_{k+1} = - \left( \frac{ \sin \sqrt{\sigma } \cosh \sqrt{\sigma } + \cos \sqrt{\sigma } \sinh \sqrt{\sigma } }{ \cos \sqrt{\sigma }\cosh \sqrt{\sigma }+1 } \right) \left( \frac{j \beta \sqrt{\sigma }}{1+j \beta \sigma } \right) Q_k, \end{aligned}$$
(41)

and the initial two values \(Q_0\) and \(Q_1\):

$$\begin{aligned} \left\{ \begin{aligned} Q_0&= \frac{\sinh \sqrt{\sigma }-\sin \sqrt{\sigma }}{\cos \sqrt{\sigma } \cosh \sqrt{\sigma }+1}, \\ Q_{1}&= \frac{j \beta \sqrt{\sigma }}{1+j \beta \sigma } \left( \frac{\sin \sqrt{\sigma } -\sinh \sqrt{\sigma }}{\cos \sqrt{\sigma } \cosh \sqrt{\sigma }+1} \right) \left( \frac{\cos \sqrt{\sigma } \sinh \sqrt{\sigma }+\sin \sqrt{\sigma } \cosh \sqrt{\sigma }}{\cos \sqrt{\sigma } \cosh \sqrt{\sigma }+1} \right) . \end{aligned}\right. \end{aligned}$$
(42)

Hence it is shown that for \(k \ge 0\),

$$\begin{aligned} Q_{k} = \left[ - \left( \frac{j \beta \sqrt{\sigma }}{1+j \beta \sigma } \right) \left( \frac{ \sin \sqrt{\sigma } \cosh \sqrt{\sigma } + \cos \sqrt{\sigma } \sinh \sqrt{\sigma } }{ \cos \sqrt{\sigma }\cosh \sqrt{\sigma }+1 } \right) \right] ^k \left( \frac{\sinh \sqrt{\sigma }-\sin \sqrt{\sigma }}{\cos \sqrt{\sigma } \cosh \sqrt{\sigma }+1} \right) . \end{aligned}$$
(43)

As a result, we obtain that for \(k \ge 1\),

$$\begin{aligned} \left\{ \begin{aligned} A_{k}&= \left( \frac{j \beta \sqrt{\sigma }}{1+j \beta \sigma } \right) ^{k} \left( \frac{ -\sin \sqrt{\sigma } \cosh \sqrt{\sigma } - \cos \sqrt{\sigma } \sinh \sqrt{\sigma } }{ \cos \sqrt{\sigma }\cosh \sqrt{\sigma }+1 } \right) ^{k-1} \left( \frac{\sinh \sqrt{\sigma }-\sin \sqrt{\sigma }}{\cos \sqrt{\sigma } \cosh \sqrt{\sigma }+1} \right) \left( \frac{\cos \sqrt{\sigma }+\cosh \sqrt{\sigma }}{2 \cos \sqrt{\sigma }\cosh \sqrt{\sigma }+2} \right) , \\ B_{k}&= \left( \frac{j \beta \sqrt{\sigma }}{1+j \beta \sigma } \right) ^{k} \left( \frac{ -\sin \sqrt{\sigma } \cosh \sqrt{\sigma } - \cos \sqrt{\sigma } \sinh \sqrt{\sigma } }{ \cos \sqrt{\sigma }\cosh \sqrt{\sigma }+1 } \right) ^{k-1} \left( \frac{\sinh \sqrt{\sigma }-\sin \sqrt{\sigma }}{\cos \sqrt{\sigma } \cosh \sqrt{\sigma }+1} \right) \left( \frac{-\sinh \sqrt{\sigma }+\sin \sqrt{\sigma }}{2 \cos \sqrt{\sigma }\cosh \sqrt{\sigma }+2} \right) , \\ C_{k}&= \left( \frac{j \beta \sqrt{\sigma }}{1+j \beta \sigma } \right) ^{k} \left( \frac{ -\sin \sqrt{\sigma } \cosh \sqrt{\sigma } - \cos \sqrt{\sigma } \sinh \sqrt{\sigma } }{ \cos \sqrt{\sigma }\cosh \sqrt{\sigma }+1 } \right) ^{k-1} \left( \frac{\sinh \sqrt{\sigma }-\sin \sqrt{\sigma }}{\cos \sqrt{\sigma } \cosh \sqrt{\sigma }+1} \right) \left( \frac{-\cos \sqrt{\sigma }-\cosh \sqrt{\sigma }}{2 \cos \sqrt{\sigma } \cosh \sqrt{\sigma }+2} \right) , \\ D_{k}&= \left( \frac{j \beta \sqrt{\sigma }}{1+j \beta \sigma } \right) ^{k} \left( \frac{ -\sin \sqrt{\sigma } \cosh \sqrt{\sigma } - \cos \sqrt{\sigma } \sinh \sqrt{\sigma } }{ \cos \sqrt{\sigma }\cosh \sqrt{\sigma }+1 } \right) ^{k-1} \left( \frac{\sinh \sqrt{\sigma }-\sin \sqrt{\sigma }}{\cos \sqrt{\sigma } \cosh \sqrt{\sigma }+1} \right) \left( \frac{-\sin \sqrt{\sigma }+\sinh \sqrt{\sigma }}{2 \cos \sqrt{\sigma }\cosh \sqrt{\sigma }+2} \right) . \end{aligned}\right. \end{aligned}$$

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Zhou, M., Zhao, H. Revisit to the theoretical analysis of a classical piezoelectric vibration energy harvester . Arch Appl Mech 90, 2379–2395 (2020). https://doi.org/10.1007/s00419-020-01727-x

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