# Identification Procedure for Design Optimization of Gravitational Electromagnetic Energy Harvesters
## 3 Results and Discussions
### Design Optimization  
The validated model can be used for the numerical optimization of the harvester size and performance. The optimization phase is based on a design approach considering the application of the harvester and its constraints. The main design issue for a VEH is to tune its natural resonance frequency to the fundamental excitation frequency of the harvested vibrations. For this reason, the optimization of the generated power is not sufficient and should be performed along with the sizing and tuning process. This paper studies the influence of the magnet's dimensions on the GVEH asymmetric magnetic suspension characteristics of resonance frequency and equilibrium position. Sensitivity analyses on the diameter and height of both moving and fixed magnets allow us to evaluate the configuration of the suspension that has the target natural frequency of the destined application. Moreover, the study on the equilibrium position is linked to the tube length to give a proper stroke to the moving magnet in the asymmetric suspension. Once the target frequency and equilibrium position are matched, a study on the optimization of the generated power can be performed to maximize the efficiency of the device already complying with the design constraints of the application. As mentioned in Section 2.1, the asymmetric magnetic suspension consists of only one fixed magnet at the bottom end of the tube and a moving magnet that oscillates under the action of the magnetic repulsive and gravitational restoring forces. Ring-shaped magnets are used in this study due to the use of the internal guide in the tube. Consequently, the magnets parameters that are studied in the design stage are the external diameter, internal diameter, and the height along the axial direction. Table 8 reports the details of the configurations adopted in the numerical model for the sensitivity analyses. Figure 10: Experimental and model FRFs comparison for EH1, EH2, and EH3. The first sensitivity analysis is performed on the fixed magnet. The external diameter, radial thickness, and height variables are changed one at a time, keeping the others at a constant value. The moving magnet has the dimensions of EH2 configuration for all the sensitivity analyses of the fixed magnet, in order to have a comparison between the simulations and experimental results. Figure 11 reports the outcome of the study on the fixed magnet dimensions. The equilibrium position is referred to the air gap between the upper face of the fixed magnet and the lower face of the moving one. The curves show that increasing the volume of the moving magnet, whether via diameter or height, the resonance frequency decreases and the air gap increases. In particular, the most sensitive parameter is the external diameter as it generates the highest variation in terms of frequency and air gap. This result can be related to the formulation of the resonance frequency of the GVEH linearized system [32]:  
\[\omega_{n}=\sqrt{\frac{2g}{z_{0}}}=\sqrt{4g\sqrt{\frac{\pi mg}{\mu _{0}H_{moc}H_{fix}}}} \tag{17}\] \[H_{mocv}=H_{c,mocv}A_{mocv}\] (18) \[H_{fix}=H_{c,fix}A_{fix} \tag{19}\]  
where:  
* \(m\) is the seismic mass of the moving magnet;
* \(g\) is the gravitational acceleration;
* \(z_{0}\) is the equilibrium air gap;
* \(\mu_{0}\) is the vacuum permeability;
* \(H_{mov}\), \(H_{fix}\) are the magnetic field intensities of moving and fixed magnet, respectively;
* \(H_{c,mocv}\), \(H_{c,fix}\) are the coercive forces of moving and fixed magnet, respectively;
* \(A_{mov}\), \(A_{fix}\) are the pole surface areas of moving and fixed magnet, respectively;  
Since the field intensity depends on the polar surface area, the greater influence of the diameter parameter with respect to the height is justified. Moreover, increasing the fixed magnet height results in a low increase in the air gap, which is the relative distance between the magnet. However, the absolute position of the moving magnet increases strongly as