## 2 Experimental Methods
### Concept Theory  
As described in [25], The VEH will have a stable equilibrium state, resulting from the attractive force in the air gap and the restoring force from the elastic components. The total restoring force is the difference between these two forces. The attractive magnetic force has a non-linear square dependency [56] on the flux in the air gap while the mechanical spring force from the elastic components is linear (for small displacement). For ease of description, we have added Figure 2, which shows the relevant features of linear curves intersecting a square curve. The three linear curves in Figure 2 represent the restoring mechanical spring force for three, decreasing, initial air gap distances, i.e., the air gap size when the beam is horizontal. Using Figure 2 as a simplified example, stable equilibrium states are obtained at the points marked by dots. Unstable equilibria are marked by arrows. The equilibrium state can be adjusted by modifying the initial distance of the air gap. As this initial distance is decreased, the linear curve in Figure 2 moves to the left and the point of stable equilibria is found at a smaller air gap distance. As the system experiences a harmonic external force excitation, the beam will oscillate around its stable equilibrium. Figure 1: Cross-section of the proposed energy harvester design, with iron core (gray), neodymium magnets (blue/red), aluminum substrate (green) and galfenol magnetostrictive layer (purple). Iron core components are numbered, 1 to 6, and are referred to in the main text. Figure 2: Hypothetical linear and square-dependent forces representing an ideal version of the system described in this work. Dashed lines correspond to the mechanical spring force at three different initial air gaps. The solid line corresponds to the magnetic force. Dots represent stable equilibria and arrows represent unstable equilibria. The total restoring force experienced by the proof mass, while oscillating, will differ depending on the equilibrium state, leading to differences in frequency response. The closer the linear spring force tangents of the square magnetic force, the smaller the restoring force becomes, thus leading to a lower resonance frequency. This comes at the cost of a smaller oscillation allowed before the magnetic force dominates (i.e., the point indicated by an arrow is passed) and the air gap is permanently closed. In this work, we include the effects from magnetostriction, which lead to altered magnetization [57] due to strain (Villari effect), and mechanical strain due to magnetization (Joule effect). The Villari effect can in this case make it possible to achieve larger time variations in the magnetic field (in the region of the magnetostrictive material) as the beam oscillates, while the Joule effect can have a beneficial effect on the beam displacement. We choose to use galfenol as magnetostrictive material, as its favorable characteristics [34; 39] have made it a common choice for use in vibration energy harvesters making use of magnetostriction. Galfenol has the benefit of having a significant magnetostriction (\(\lambda_{\text{s}}\)-200 ppm) and being easy to machine (as it is ductile). A layer of galfenol is in this case glued to the elastic beam. The magnetostriction constant for galfenol is positive [57; 58], which results in an increase in magnetization with positive stress (tension) and an increase in strain with an applied magnetic field. For this reason, the magnetostrictive layer of the beam is on the bottom side, which experiences tension when the air gap decreases. As mentioned, the magnets provide a magnetic field bias for the system. Both a magnetic field and stress bias are required for an optimal performance of magnetostrictive components. As can be seen from the magnetization and strain vs. applied magnetic field curves in [57; 59], for a certain pre-stress, there is a region of the applied magnetic field that has the largest variation in strain and magnetization.