# Identification Procedure for Design Optimization of Gravitational Electromagnetic Energy Harvesters
## 2 Materials and Methods
### Numerical Model  
Ansys Maxwell software allows us to perform magnetostatic and transient electromagnetic simulations to evaluate the force of the magnetic springs and the magnetic flux linkage through the coils. Models for the three harvesters are built in 2D cylindrical coordinates given the axisymmetry of the configurations. Figure 4a,b show the magnetic field resulting from the magnetostatic analysis (a) and the flux linkage from the transient electromagnetic analysis (b) for EH3 configuration with the moving magnet in its equilibrium position. The magnitude of the magnetic repulsive force and of the magnetic flux linkage both depend on the position of the moving magnet inside the tube. This dependence introduces non-linearities in both stiffness and damping characteristics, as the moving magnet position varies in time under external excitation. A numerical approach is necessary to derive the system characteristics which determine the dynamics behavior of the energy harvesters. The main mechanical characteristics of the harvesters are the stiffness of the magnetic spring \(k\) and the viscous damping \(c_{vis}\). Magnetostatic analyses in Ansys Maxwell need to be performed for the definition of the stiffness characteristic. The moving magnet position  
Figure 3: Simulink block scheme. is swept from the bottom of the tube, ideally touching the fixed magnet, to the top. The simulations enable the evaluation of the distribution of magnetic field vectors created by the two opposing magnets for each step of the moving magnet position, as Figure 4 shows. For every position of the moving magnet, Ansys Maxwell computes the corresponding repulsive force between the two magnets of the system \(F_{mag}(z)\). As stated in [30], the magnetic force acting between two magnetic dipoles having the same magnetic moment and their axes aligned is proportional to \(1/z^{4}\). Even though the magnets in the three configurations have significantly different magnetic moments, the proposed function fits the simulation data quite precisely. The expression of the function is the following:  
\[F_{mag,fit}(z)=\frac{p_{1}}{z^{4}+q_{1}z^{3}+q_{2}z^{2}+q_{3}z+q_{4}} \tag{8}\]  
The coefficients are determined in the MATLAB environment for each harvester configuration. Figure 5 shows the plots of the magnetic force fitting curves for EH1, EH2, and EH3 compared to the simulation data. The stiffness characteristic can be easily obtained by deriving again the magnetic force fitting function with respect to \(z\). \[k(z)=\frac{dF_{mag,fit}(z)}{dz} \tag{9}\]  
The computation of the magnet equilibrium position can be done by finding the coordinate \(z_{0}\) at which the magnetic force function equals the moving magnet weight. \[F_{mag,fit}(z_{0})=F_{0}=mg \tag{10}\]  
The system can be linearized by evaluating the stiffness in the equilibrium position and the corresponding linear resonance frequency. \[f_{n}=\frac{1}{2\pi}\sqrt{\frac{k(z_{0})}{m}}=\frac{1}{2\pi}\sqrt{\frac{2g}{z _{0}}} \tag{11}\]  
Table 4 summarizes the main results of static simulations on the three harvesters. Figure 4: (**a**) Magnetic field vectors for EH3. (**b**) Magnetic flux linkage for EH3. Another mechanical characteristic of the system is the viscous damping due to air action and surface friction of the moving magnet. The damping coefficient is considered a constant and is defined through experimental tests on the devices. Free-fall tests of the moving magnet are carried out on the experimental apparatus (described in Section 2.3). The harvester time response of output voltage signal is processed to extract the damping value. As can be seen in Figure 6 the peaks of the output signal are interpolated with an exponential function of time:  
\[y=ae^{-bt} \tag{12}\]  
where \(b\) is the exponential decay constant that allows us to evaluate the viscous damping [31].