## 3 Results and Discussion  
### Comparing Measurements with Simulations  
Figure 4 shows the simulated and measured frequency dependencies of the open-circuit output voltage from a coil around the long core component (part 1 in Figure 1) of the energy harvester. The simulated data also contain a set where the magnetostrictive effect is removed by setting the saturation magnetostriction, \(\lambda_{\mathrm{S}}\), to 0. Both measured and simulated data show resonance at approximately 195 Hz, at \(\beta\) = 3 mm. As \(\beta\) increases, the resonance frequency decreases following the description in Section 2.2. For both measurements and simulations, \(\beta\) is increased from 3.0 mm in steps of 0.5 mm. Using \(\beta\) = 4.5 mm results in a permanently closed air gap at equilibrium for both the lab prototype and simulation model. The last data set therefore corresponds to the maximum \(\beta\), which can be used before the air gap is permanently closed. We found this maximum \(\beta\) to be 4.15 mm for the simulation model and 4.4 for the lab prototype. A likely explanation for this difference is that the total system reluctance of the lab prototype is larger than that of the simulation model (due to glue gaps). This would also explain the smaller resonance frequency tuning in the measured data, as compared to simulated frequency (see Figure 4a,b). Both sets of data show a non-linear relationship between resonance frequency and \(\beta\), with the highest sensitivity at large \(\beta\). Comparing the simulated frequency response, with \(\lambda_{\mathrm{S}}\) = 0 and \(\lambda_{\mathrm{S}}\) = 200 ppm (Figure 4a), magnetostriction does not contribute to the voltage amplitude, but increases the sensitivity of resonance frequency shift when decreasing the air gap. According to the simulated data of Figure 5b, most of the magnetostrictive material is probably close to magnetic saturation (see \(\mathrm{M_{S}}\) for galfenol in Table 1). Comparing the magnetic field data with the von Mises stress data of Figure 5a, only small coupling can be seen. Thus, we conclude that the effect of magnetostriction is here primarily an additional strain bias, which increases beam elasticity, an effect also known as the \(\Delta\)E effect [61]. The additional strain also reduces the equilibrium air gap distance, which in turn further reduces the resonance frequency (as described in Section 2.2). Figure 4: RMS of the open-circuit coil voltage. Curves from right to left result from using an increasingly thicker spacer, i.e., increasing \(\beta\). (**a**) Simulated values. Circles and crosses are data points. Solid lines correspond to using \(\lambda_{\mathrm{S}}\) = 200 and dashed lines for \(\lambda_{\mathrm{S}}\) = 0. (**b**) Measured on prototype. Dots are data points. Lines are for visual aid. For large air gap distances (\(\beta=3\) mm and \(\beta=3.5\) mm), the measured and simulated frequency responses fit well (see Figure 4a,b). As the air gap diminishes, the discrepancy between measured and simulated data, with regard to resonance frequency, increases. We assume the beams' mechanical properties to be correctly modeled, i.e., that the natural frequencies are essentially equal for both the modeled and measured system. As we do not measure the material properties of the prototype, there is a possibility that the material parameter values used in the COMSOL model differ from the physical prototype, leading to different output characteristics. However, as the magnetostrictive material is magnetically saturated in this case (see Figure 5b), the effect from errors in modeled magnetostrictive properties should be small. Thus, the discrepancy is likely explained by a larger total reluctance of the system (e.g., due to thick layers of glue), leading to smaller coupling between the air gap distance and flux (and equivalently force). The main point of this paper is however to determine if magnetostriction can have a synergetic effect with the dynamics of the energy harvester.