## 2 Experimental Methods
### FEM Model  
In this work, we use a finite element modeling tool (COMSOL Multiphysics 6.2), with a geometry equivalent to Figure 1 and suitably chosen material properties (Table 1). COMSOL includes the necessary physics modules to account for stress, strain, magnetic fields and resulting forces as well as the effect of magnetostriction (both Joule and Villari effects). To reduce computation time, we use a 2D model (2D modeling of a magnetostrictive beam has been shown to be sufficiently accurate, compared to 3D modeling [35]). Magnetic flux is in this case only in-plane and induced currents (eddy currents) are only out-of-plane. A global out-of-plane thickness is specified in the software, thus allowing cross-sectional flux and accompanying forces to be calculated.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material Parameter** & **Fe** & **Neodymium** & **FeGa** & **Al** \\ \hline E [GPa] & 200 & 70 & 60 & 70 \\ \hline M\({}_{\text{S}}\) [MA/m] & 1.91 & N/A & 1.5 & N/A \\ \hline \(\lambda_{\text{S}}\) [ppm] & N/A & N/A & 200 & N/A \\ \hline \(\rho\) [kg/m\({}^{3}\)] & 7850 & 7520 & 7870 & 2700 \\ \hline \(\sigma\) [MS/m] & 10.15 & N/A & 5.96 & N/A \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material properties used in COMSOL. Definitions: E = Young’s modulus, M\({}_{\text{S}}\) = magnetization saturation, \(\sigma\) = electrical conductivity, \(\rho\) = density, \(\lambda_{\text{S}}\) = saturation magnetostriction, and N/A = not applicable with this specific COMSOL model.  
Material-dependent B-H curves are used when modeling the magnetic fields within the core components. The magnets are modeled with a remanent flux density of 1.18 T. The magnetic field in the magnetostrictive component is modeled non-linearly using the Langevin function [60] of an effective magnetic field, which is the sum of the applied magnetic field and the magnetic field due to stress in the magnetostrictive material. The material properties used in the model are based on soft magnetic steel for the core components, neodymium for the magnets, aluminum for the substrate and galfenol for the magnetostrictive component. Mechanical damping is included as Rayleigh damping, with mass damping assumed to be negligible [50] and a stiffness damping of \(10^{-5}\), corresponding to a damping ratio of 0.0081. The electrical conductivity of the core corresponds to that of iron, allowing for eddy currents. These currents reduce performance by creating a counteracting magnetic field and damping of the system [60]. These effects are small for low frequencies and can typically be significantly mitigated using laminated cores. Numerical damping, resulting from the FEM time-stepping algorithm, is minimized by choosing the generalized \(\alpha\) method and setting \"\(\alpha\)\" to 0.97.  
The key output from the FEM simulations is the frequency response of the energy harvester, with regard to open-circuit coil voltage. Although the simulation is carried out in 2D, the expected coil voltage can be estimated from the derivative of the flux. The flux used for the voltage calculation is the average of flux in a 10 mm \(\times\) 10 mm \(\times\) 10 mm region of the core, close to the air gap. As the output is assumed to be non-linear, a frequency domain simulation cannot be used. Instead, time domain simulations are performed for a number of frequencies in the region of interest. Each simulation corresponds to 60 oscillations of the system. To exclude any initial transient behavior, we use only the second half of the time domain data, from which the RMS voltage for each frequency is determined.