## 2 Materials and Methods
### Magnetic Levitation Force  
As far as magnetic levitation is concerned, the main characteristic of harvesters is magnetic levitation suspension. Therefore, in this section, the interaction between two moving magnets and two fixed magnets is described. The magnet's orientation determines attractive or repulsive forces. The forces vary depending on the magnet's separation distance, shape, diameter, and magnetization direction. The magnetic repulsive forces were induced in the experiment using a specially designed device (Figure 2a) and in the numerical simulation by FEM using the Comsol Multiphysics software ([[https://www.comsol.com/](https://www.comsol.com/)]([https://www.comsol.com/](https://www.comsol.com/)), accessed on 16 May 2023) with progressively finer meshes to reach a higher precision (Figure 2b). There are many methods for computing magnetic forces in Comsol. The magnetic field was modeled using the magnetic field interface in the AC/DC module. The force is based on the surface integration of the Maxwell Stress Tensor, which is an integral computed over the exterior surfaces that are an arbitrary closed path. The geometry domain of magnets is selected for calculation. The movement of the magnet was modeled using the moving mesh interface. The material for the magnet used in the FEM simulation was neodymium N38, which was obtained from the materials module and solid mechanics interfaces selected in the coupling feature. The measuring system (Figure 2a) was composed of a nonmagnetic tube with a compact sensor FX 292X-100A-0025-L for axial force measurement. Two cylindrical magnets were installed in the tube, and the distance between them could be adjusted with a vertical screw. A small sensor for magnetic force measurement was mounted inside the tube. Magnetic forces between the moving magnets (\(A_{1}\) or \(A_{2}\)) and the fixed magnets \(B_{1}\) and \(B_{2}\) are presented in Figure 3a. To estimate the magnetic force characteristics, each moving magnet was analyzed separately (without the second moving magnet). Then, the position of the magnet was changed, and the force was measured. Figure 3b shows the repulsive force between two moving magnets (between \(A_{1}\) and \(A_{2}\)). The blue dots mark the experimental results, while the black line marks the FEM results. The experimental  
Figure 2: Specially patented device for measuring magnetic forces (**a**), and an FEM model for magnetic force calculation (**b**). and numerical magnetic forces (\(A_{1}\) and \(A_{2}\)) show high agreement. The magnetic force becomes too strong when the two moving magnets are too close to each other. The smaller the distance between the magnets is, the greater the difference between the FEM and experimental results becomes. The proposed mathematical polynomial model (Equation (1)) is marked with a blue line. The red points and dashed vertical line mark the equilibrium position of both magnets. The equilibrium points (static position) are shifted away from the center point (point 0 in Figure 2b). It should be noted that the force between the moving magnets and two end magnets is different from that in the 1 DOF harvester. For the classical 1 DOF harvester, the magnetic restoring force resembles the Duffing characteristic [18; 20]. However, in this case, the characteristic is different due to the orientation of the magnets (the moving magnet is attracted by one of the fixed magnets). The magnet \(A_{1}\) is repelled by the fixed bottom magnet \(B_{1}\) and attracted to the fixed top magnet (see Figure 2b). The magneto-elastic forces as well as the magnetic field depend on the distance between the magnets. The relationship can be visualized in the form of magnetic field lines passing through the magnet along its direction of magnetization. The field strength relates to the density of the field lines in a certain area. The fixed magnet produces a field in its core and in its external surroundings. The magnetic field direction \"flows\" from one pole to the other.