Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Improving the off-resonance energy harvesting performance using dynamic magnetic preloading


Piezoelectric stack transducers in d33 mode have a much higher mechanical-to-electric energy conversion efficiency compared with d31 mode piezoelectric harvesters. However, multilayered piezoelectric stacks usually operate in off-resonance due to the higher stiffness and thereby have a lower power output under low-frequency excitations. This paper proposes to apply the dynamic magnetic pre-loading to a piezoelectric stack transducer to significantly increase the power output. The energy harvesting system consists of a multilayered piezoelectric stack with a compliant force amplification frame, a proof mass, and two magnets configured in attraction. The static force–displacement relationship of the magnets is identified from experiments and extended to a dynamic model capable of characterizing the dynamic magnetic interaction. An electromechanical model is developed based on the theoretical derivation and the experimentally identified parameters to predict the voltage outputs under different resistive loads. Approximate analytical solutions are derived by using the harmonic balance method and show good agreements with the numerical and experimental results. The performance of the system is examined and compared with that of the harvester without magnetic pre-loading. The influences of the distance between the two magnets and the electrical resistive loads on the power output are investigated. Results indicate the energy harvesting system with magnetic pre-loading can produce over thousand times more power than the system without magnetic pre-loading at the base excitation of 3 Hz and 0.5 m/s2, far below the resonance at 243 Hz

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13


  1. 1.

    Wang, W., Cao, J., Mallick, D., et al.: Comparison of harmonic balance and multi-scale method in characterizing the response of monostable energy harvesters. Mech. Syst. Signal Process. 108, 252–261 (2018)

  2. 2.

    Fang, F., Xia, G., Wang, J.: Nonlinear dynamic analysis of cantilevered piezoelectric energy harvesters under simultaneous parametric and external excitations. Acta. Mech. Sin. 34(3), 561–577 (2018)

  3. 3.

    Cao, D., Gao, Y., Hu, W.: Modeling and power performance improvement of a piezoelectric energy harvester for low-frequency vibration environments. Acta. Mech. Sin. 35(4), 894–911 (2019)

  4. 4.

    Yuan, T.C., Yang, J., Chen, L.Q.: Nonlinear vibration analysis of a circular composite plate harvester via harmonic balance. Acta. Mech. Sin. 35(4), 912–925 (2019)

  5. 5.

    Yang, B., Lee, C., Xiang, W., et al.: Electromagnetic energy harvesting from vibrations of multiple frequencies. J. Micromech. Microeng. 19(3), 035001 (2009)

  6. 6.

    Donelan, J.M., Li, Q., Naing, V., et al.: Biomechanical energy harvesting: generating electricity during walking with minimal user effort. Science 319, 807–810 (2008)

  7. 7.

    Rome, L.C., Flynn, L., Goldman, E.M., et al.: Generating electricity while walking with loads. Science 309(5741), 1725–1728 (2005)

  8. 8.

    Pan, Y., Lin, T., Qian, F., et al.: Modeling and field-test of a compact electromagnetic energy harvester for railroad transportation. Appl. Energy 247, 309–321 (2019)

  9. 9.

    Mitcheson, P., Miao, P., Start, B., et al.: MEMS electrostatic micro-power generator for low frequency operation. Sens. Actuators A. Phys. 115(2–3), 523–529 (2004)

  10. 10.

    Eun, Y., Kwon, D.S., Kim, M.O., et al.: A flexible hybrid strain energy harvester using piezoelectric and electrostatic conversion. Smart Mater. Struct. 23(4), 045040 (2014)

  11. 11.

    Luo, L., Liu, D., Zhu, M., et al.: Metamodel-assisted design optimization of piezoelectric flex transducer for maximal bio-kinetic energy conversion. J. Intell. Mater. Syst. Struct. 28(18), 2528–2538 (2017)

  12. 12.

    Zhu, J., Niu, X., Hou, X., et al.: Highly reliable real-time self-powered vibration Sensor based on a piezoelectric nanogenerator. Energy Technol. 6(4), 781–789 (2018)

  13. 13.

    Qian, F., Zhou, W., Kaluvan, S., et al.: Theoretical modeling and experimental validation of a torsional piezoelectric vibration energy harvesting system. Smart Mater. Struct. 27(4), 045018 (2018)

  14. 14.

    Malakooti, M.H., Sodano, H.A.: Piezoelectric energy harvesting through shear mode operation. Smart Mater. Struct. 24(5), 055005 (2015)

  15. 15.

    Wen, S., Xu, Q., Zi, B.: Design of a new piezoelectric energy harvester based on compound two-stage force amplification frame. IEEE Sens. J. 18(10), 3989–4000 (2018)

  16. 16.

    Xu, T.B., Siochi, E.J., Kang, J.H., et al.: Energy harvesting using a PZT ceramic multilayer stack. Smart Mater. Struct. 22(6), 065015 (2013)

  17. 17.

    Li, Z., Xu, Q., Tam, L.M.: Design of a new piezoelectric energy harvesting handrail with vibration and force excitations. IEEE Access. 7, 151449–151458 (2019)

  18. 18.

    Qian, F., Xu, T.B., Zuo, L.: Design, optimization, modeling and testing of a piezoelectric footwear energy harvester. Energy Convers. Manag. 171, 1352–1364 (2018)

  19. 19.

    Liu, X., Wang, J., Li, W.: Dynamic analytical solution of a piezoelectric stack utilized in an actuator and a generator. Appl. Sci. 8(10), 1779 (2018)

  20. 20.

    Zhou, Z., Qin, W., Zhu, P.: Harvesting performance of quad-stable piezoelectric energy harvester: modeling and experiment. Mech. Syst. Signal Process. 110, 260–272 (2018)

  21. 21.

    Morris, D.J., Youngsman, J.M., Anderson, M.J., et al.: A resonant frequency tunable, extensional mode piezoelectric vibration harvesting mechanism. Smart Mater. Struct. 17(6), 065021 (2008)

  22. 22.

    Xie, Z., Xiong, J., Zhang, D., et al.: Design and experimental investigation of a piezoelectric rotation energy harvester using bistable and frequency up-conversion mechanisms. Appl. Sci. 8(9), 1418 (2018)

  23. 23.

    Gu, L., Livermore, C.: Compact passively self-tuning energy harvesting for rotating applications. Smart Mater. Struct. 21(1), 015002 (2011)

  24. 24.

    Foisal, A.R.M., Hong, C., Chung, G.S.: Multi-frequency electromagnetic energy harvester using a magnetic spring cantilever. Sens. Actuators, A 182, 106–113 (2012)

  25. 25.

    Li, M., Zhou, J., Jing, X.: Improving low-frequency piezoelectric energy harvesting performance with novel X-structured harvesters. Nonlinear Dyn. 94(2), 1409–1428 (2018)

  26. 26.

    Xue, H., Hu, Y., Wang, Q.M.: Broadband piezoelectric energy harvesting devices using multiple bimorphs with different operating frequencies. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 55(9), 2104–2108 (2008)

  27. 27.

    Yang, Z., Zu, J.: High-efficiency compressive-mode energy harvester enhanced by a multi-stage force amplification mechanism. Energy Convers. Manag. 88, 829–833 (2014)

  28. 28.

    Yang, Z., Zhu, Y., Zu, J.: Theoretical and experimental investigation of a nonlinear compressive-mode energy harvester with high power output under weak excitations. Smart Mater. Struct. 24(2), 025028 (2015)

  29. 29.

    Kulah, H., Najafi, K.: Energy scavenging from low-frequency vibrations by using frequency up-conversion for wireless sensor applications. IEEE Sens. J. 8(3), 261–268 (2008)

  30. 30.

    Gu, L., Livermore, C.: Impact-driven, frequency up-converting coupled vibration energy harvesting device for low frequency operation. Smart Mater. Struct. 20(4), 045004 (2011)

  31. 31.

    Challa, V.R., Prasad, M.G., Shi, Y., et al.: A vibration energy harvesting device with bidirectional resonance frequency tenability. Smart Mater. Struct. 17(1), 015035 (2008)

  32. 32.

    Challa, V.R., Prasad, M.G., Fisher, F.T.: Towards an autonomous self-tuning vibration energy harvesting device for wireless sensor network applications. Smart Mater. Struct. 20(2), 025004 (2011)

  33. 33.

    Qian, F., Zhou, S., Zuo, L.: Approximate solutions and their stability of a broadband piezoelectric energy harvester with a tunable potential function. Commun. Nonlinear Sci. Numer. Simul. 80, 104984 (2020)

  34. 34.

    Lan, C., Qin, W.: Enhancing ability of harvesting energy from random vibration by decreasing the potential barrier of bistable harvester. Mech. Syst. Signal Process. 85, 71–81 (2017)

  35. 35.

    Wang, G., Liao, W.H., Yang, B., et al.: Dynamic and energetic characteristics of a bistable piezoelectric vibration energy harvester with an elastic magnifier. Mech. Syst. Signal Process. 105, 427–446 (2018)

  36. 36.

    Huang, D., Zhou, S., Litak, G.: Analytical analysis of the vibrational tristable energy harvester with a RL resonant circuit. Nonlinear Dyn. 97(1), 663–677 (2019)

  37. 37.

    Li, H., Qin, W., Lan, C., et al.: Dynamics and coherence resonance of tri-stable energy harvesting system. Smart Mater. Struct. 25(1), 015001 (2015)

  38. 38.

    Leinonen, M., Juuti, J., Jantunen, H., et al.: Energy harvesting with a bimorph type piezoelectric diaphragm multilayer structure and mechanically induced pre-stress. Energy Technol. 4(5), 620–624 (2016)

  39. 39.

    Wang, L., Chen, S., Zhou, W., et al.: Piezoelectric vibration energy harvester with two-stage force amplification. J. Intell. Mater. Syst. Struct. 28(9), 1175–1187 (2017)

Download references


The authors gratefully acknowledge the support of Commonwealth Research Commercialization Fund (CRCF) from the Center for Innovative Technology (CIT) of Virginia. The work was completed while S. Zhou was a postdoc at Virginia Tech.

Author information

Correspondence to Lei Zuo.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Qian, F., Zhou, S. & Zuo, L. Improving the off-resonance energy harvesting performance using dynamic magnetic preloading. Acta Mech. Sin. (2020).

Download citation


  • Energy harvesting
  • Piezoelectric
  • Off-resonance
  • Harmonic balance analysis
  • Magnetic interaction