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On the Improved Modeling of the Magnetoelastic Force in a Vibrational Energy Harvesting System

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A Correction to this article was published on 17 December 2019

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Abstract

Objective

The efficiency of vibrational energy harvesting systems that consist of a cantilever beam with attached piezoceramic layers can be increased by intentionally introducing nonlinearities. These nonlinearities are often implemented in the form of permanent magnets near the beam’s free end. The influence of these magnets is typically assumed to be a single transverse force that depends cubically on the displacement of the beam tip. The coefficients of a corresponding single degree of freedom model are often found heuristically, without an explicit modeling of the magnetoelastic force.

Methods

In this paper, we present and assess the validity of a procedure to determine the magnetoelastic forces acting on the beam from physical laws of the magnetic field with corresponding parameters. The paper outlines the method itself, describing initially how the magnetic field is computed by a Finite Element simulation. In the second step, the total transverse force on the beam is determined from the magnetic field by means of a numerical evaluation of the Maxwell stress tensor. The required minimum degree of a suitable polynomial force approximation of the numeric values is discussed briefly. The validity of this model is then considered by investigating its bifurcation behavior with respect to mono-, bi-, and tristability for different distances between the magnets and comparing the findings to results found by experiments.

Results

While the model’s predictions of the number of equilibrium positions for any magnet distance are generally in good agreement with results of experiments, there are deviations when it comes to the exact positions of the equilibria. With respect to these findings, the limitations of a two-dimensional magnetic field modeling with only linear material models are addressed.

Conclusion

The paper concludes that the method outlined here is a step toward the deduction of a detailed model based on physical laws of the magnetic field with corresponding parameters replacing simple heuristic polynomial magnetic force laws.

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Change history

  • 17 December 2019

    The original version of this article contains��an error. In Table��1 ���Magnetic material parameters��� the value for the ���relative permeability of steel��� is��1000 (instead of 300).

Notes

  1. D. C. Meeker, Finite Element Method Magnetics, Version 4.0.1, http://www.femm.info.

References

  1. Kim SK, Kim JH, Kim J (2011) A review of piezoelectric energy harvesting based on vibration. Int J Precis Eng Manuf 12(6):1129–1141. https://doi.org/10.1007/s12541-011-0151-3

    Article  Google Scholar 

  2. Priya SJ (2007) Advances in energy harvesting using low profile piezoelectric transducers. J Electroceram 19:165–182. https://doi.org/10.1007/s10832-007-9043-4

    Article  Google Scholar 

  3. Wei C, Jing X (2017) A comprehensive review on vibration energy harvesting: modelling and realization. Renew Sustain Energy Rev 74:1–18. https://doi.org/10.1016/j.rser.2017.01.073

    Article  MathSciNet  Google Scholar 

  4. Ajitsaria J, Choe SY, Shen D, Kim DJ (2007) Modeling and analysis of a bimorph piezoelectric cantilever beam for voltage generation. Smart Mater Struct 16(2):447–454. https://doi.org/10.1088/0964-1726/16/2/024

    Article  Google Scholar 

  5. Erturk A, Inman DJ (2011) Piezoelectric energy harvesting. Wiley, New York. https://doi.org/10.1002/9781119991151

    Book  Google Scholar 

  6. Sodano HA, Park G, Leo DJ, Inman DJ (2003) Model of piezoelectric power harvesting beam. In: ASME international mechanical engineering congress and expo, 15–21 Nov 2003, Washington DC, vol 40, no 2. http://dx.doi.org/10.1115/IMECE2003-43250

  7. Yu S, He S, Li W (2010) Theoretical and experimental studies of beam bimorph piezoelectric power harvesters. J Mech Mater Struct 5(3):427–445. https://doi.org/10.2140/jomms.2010.5.427

    Article  Google Scholar 

  8. Erturk A, Hoffmann J, Inman DJ (2009) A piezomagnetoelastic structure for broadband vibration energy harvesting. Appl Phys Lett 94:11–14. https://doi.org/10.1063/1.3159815

    Article  Google Scholar 

  9. Gammaitoni L, Neri I, Vocca H (2009) Nonlinear oscillators for vibration energy harvesting. Appl Phys Lett 94:164102. https://doi.org/10.1063/1.3120279

    Article  Google Scholar 

  10. Harne RL, Wang KW (2013) A review of the recent research on vibration energy harvesting via bistable systems. Smart Mater Struct 22:023001. https://doi.org/10.1088/0964-1726/22/2/023001

    Article  Google Scholar 

  11. Tang L, Yang Y, Soh CK (2010) Toward broadband vibration-based energy harvesting. J Intell Mater Syst Struct 21:1867–1897. https://doi.org/10.1177/1045389X10390249

    Article  Google Scholar 

  12. Daqaq MF, Masana R, Erturk A, Quinn DD (2014) On the role of nonlinearities in vibratory energy harvesting: a critical review and discussion. Appl Mech Rev 66(4):040801–040801-23. https://doi.org/10.1115/1.4026278

    Article  Google Scholar 

  13. Moon FC, Holmes P (1979) A magnetoelastic strange attractor. J Sound Vib 65:275–296. https://doi.org/10.1016/0022-460x(79)90520-0

    Article  MATH  Google Scholar 

  14. Litak G, Friswell MI, Adhikari S (2010) Magnetopiezoelastic energy harvesting driven by random excitations. Appl Phys Lett 96:214103-1. https://doi.org/10.1063/1.3436553

    Article  Google Scholar 

  15. Tam JI (2013) Numerical and experimental investigations into the nonlinear dynamics of a magneto-elastic system. Final report, Department of Mechanical and Aerospace Engineering, Princeton University

  16. Noll M-U, Lentz L, von Wagner U (2019) On the discretization of a bistable cantilever beam with application to energy harvesting. Facta Univ Ser Mech Eng (submitted)

  17. Yan B, Zhou SX, Litak G (2018) Nonlinear analysis of the tristable energy harvester with a resonant circuit for performance enhancement. Int J Bifurc Chaos. https://doi.org/10.1142/S021812741850092X

    Article  MATH  Google Scholar 

  18. Zhou S, Cao J, Inman DJ, Lin J, Li D (2016) Harmonic balance analysis of nonlinear tristable energy harvesters for performance enhancement. J Sound Vib 373:223–235. https://doi.org/10.1016/j.jsv.2016.03.017

    Article  Google Scholar 

  19. Zhou Z, Qin W, Zhu P (2017) A broadband quad-stable energy harvester and its advantages over bi-stable harvester: simulation and experiment verification. Mech Syst Signal Process 84:158–168. https://doi.org/10.1016/j.ymssp.2016.07.001

    Article  Google Scholar 

  20. Erturk A, Inman DJ (2011) Broadband piezoelectric power generation on high-energy orbits of the bistable Duffing oscillator with electromechanical coupling. J Sound Vib 330:2339–2353. https://doi.org/10.1016/j.jsv.2010.11.018

    Article  Google Scholar 

  21. Masana R, Daqaq MF (2011) Relative performance of a vibratory energy harvester in mono- and bi-stable potentials. J Sound Vib 330:6036–6052. https://doi.org/10.1016/j.jsv.2011.07.031

    Article  Google Scholar 

  22. Friswell MI, Ali FS, Bilgen O, Adhikari S, Lees AW, Litak G (2012) Non-linear piezoelectric vibration energy harvesting from a vertical cantilever beam with tip mass. J Intell Mater Syst Struct 23(13):1501–1521. https://doi.org/10.1177/1045389X12455722

    Article  Google Scholar 

  23. Lan C, Qin W (2017) Enhancing ability of harvesting energy from random vibration by decreasing the potential barrier of bistable harvester. Mech Syst Signal Process 85:71–81. https://doi.org/10.1016/j.ymssp.2016.07.047

    Article  Google Scholar 

  24. Stanton SC, McGehee CC, Mann PB (2010) Nonlinear dynamics for broadband energy harvesting: investigation of a bistable piezoelectric inertial generator. Physica D 239:640–653. https://doi.org/10.1016/j.physd.2010.01.019

    Article  MATH  Google Scholar 

  25. Wang H, Tang L (2017) Modeling and experiment of bistable two-degree-of-freedom energy harvester with magnetic coupling. J Mech Syst Signal Process 86:29–39. https://doi.org/10.1016/j.ymssp.2016.10.001

    Article  Google Scholar 

  26. Henrotte F, Deliege G, Hameyer K (2002) The eggshell method for the computation of electromagnetic forces on rigid bodies in 2D and 3D. CEFC 2002, Perugia, Italy, 16–18 Apr 2002. https://doi.org/10.1108/03321640410553427

  27. Martens W, von Wagner U, Litak G (2013) Stationary response of nonlinear magneto-piezoelectric energy harvester systems under stochastic excitation. Eur Phys J Spec Top 222:1665–1673. https://doi.org/10.1140/epjst/e2013-01953-5

    Article  Google Scholar 

  28. Noll M-U, Lentz L (2016) On the refined modeling of the force distribution in a bistable magnetoelastic energy harvesting system due to a magnetic field. Proc Appl Math Mech 16(1):289–290. https://doi.org/10.1002/pamm.201610133

    Article  Google Scholar 

  29. Kim P, Seok J (2014) A multi-stable energy harvester: dynamic modeling and bifurcation analysis. J Sound Vib 333(21):5525–5547. https://doi.org/10.1016/j.jsv.2014.05.054

    Article  Google Scholar 

  30. Lentz L (2018) Zur Modellbildung und Analyse von bistabilen Energy-Harvesting-Systemen. Doctoral thesis, TU Berlin. https://doi.org/10.14279/depositonce-7525

  31. Noll M-U, Lentz L (2017) On the modeling of the distributed force in a bistable magnetoelastic energy harvesting system. In: 4th Workshop in devices, materials and structures for energy harvesting and storage in Oulu, Finland

  32. Moon FC (1997) Magneto-solid mechanics. Wiley, New York

    Google Scholar 

  33. Furlani EP (2001) Permanent magnet and electromechanical devices. Academic Press, New York

    Google Scholar 

  34. Reich F (2017) Coupling of continuum mechanics and electrodynamics—an investigation of electromagnetic force models by means of experiments and selected problems. Doctoral thesis, TU Berlin. http://dx.doi.org/10.14279/depositonce-6518

  35. Derby N, Olbert S (2010) Cylindrical magnets and ideal solenoids. Am J Phys 78(3):229–235. https://doi.org/10.1119/1.3256157

    Article  Google Scholar 

  36. de Medeiros LH, Reyne G, Meunier G (1999) About the distribution of forces in permanent magnets. IEEE Trans Magn 34:3560–3563. https://doi.org/10.1109/20.767168

    Article  Google Scholar 

  37. de Medeiros LH, Reyne G, Meunier G (1998) Comparison of global force calculations on permanent magnets. IEEE Trans Magn 34:3560–3563. https://doi.org/10.1109/20.717840

    Article  Google Scholar 

  38. McFee S, Webb JP, Lowther DA (1988) A tunable volume integration formulation for force calculation in finite-element based computational magnetostatics. IEEE Trans Magn 24(1):439–442. https://doi.org/10.1109/20.43951

    Article  Google Scholar 

  39. Noll M-U (2018) Energy harvesting system. https://doi.org/10.6084/m9.figshare.7492208

  40. Noll M-U (2018) Magnetic field. https://doi.org/10.6084/m9.figshare.7492469

  41. Noll M-U (2019) Experimental setup of an energy harvesting system III. https://doi.org/10.6084/m9.figshare.7993115

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Funding

This study was funded by Deutsche Forschungsgemeinschaft (Grant No. WA 1427/23-1,2).

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Authors and Affiliations

  1. Chair of Mechatronics and Machine Dynamics, Technische Universität Berlin, Einsteinufer 5, 10587, Berlin, Germany

    Max-Uwe Noll, Lukas Lentz & Utz von Wagner

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  1. Max-Uwe Noll
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  2. Lukas Lentz
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Correspondence to Max-Uwe Noll.

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Noll, MU., Lentz, L. & von Wagner, U. On the Improved Modeling of the Magnetoelastic Force in a Vibrational Energy Harvesting System. J. Vib. Eng. Technol. 8, 285–295 (2020). https://doi.org/10.1007/s42417-019-00159-4

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