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

A Correction to this article was published on 17 December 2019

This article has been updated



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.


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.


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.


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).


  1. D. C. Meeker, Finite Element Method Magnetics, Version 4.0.1,


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

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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).

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  • Energy harvesting
  • Bistable oscillator
  • Magnetoelastic force
  • Maxwell stress tensor