## 3 Results and Discussion
### Influence of Electromechanical Coupling  
The electromechanical coupling coefficient (a) measures the efficiency of converting input mechanical energy into electrical energy. It can be changed by modifying the magnet-coil design, using specially designed magnets with separators, adding additional magnets, or modifying the resonance behavior [33]. An increase in the coupling coefficient also causes greater damping in the harvester system, which makes this parameter crucial for energy harvesting. In strong nonlinear systems, especially those with many degrees of freedom and many electromechanical couplings, their impact is difficult to predict. A comparison of the resonance curves for different electromechanical couplings is given in Figures 10 and 11. The resonance curves were obtained by path following of the periodic solutions, starting from \(\omega=20\) rad/s. The circles denote the bifurcation fold points. Two strategies for electromechanical coupling analysis were applied. First, the electromechanical coupling between the top magnet and the coil (\(\alpha_{2}\)) was modified, and the fixed value of the bottom magnet's electromechanical coupling was assumed (\(\alpha_{1}=30\) N/A). For \(\alpha_{2}=0\) N/A (no coupling between the mechanical and electrical subsystems \(-\) black line), both magnets exhibit the highest oscillation, and the main nonlinear resonance (two peaks with a hardening behavior) is observed close to the frequency \(\omega=60\div 80\) rad/s. For this case, the recovered current is the highest and amounts to about \(i_{MAX}=0.038\) A (Figure 10c). An increase in \(\alpha_{2}\) to 10 N/A causes a reduction in the oscillation of both magnets in the first resonance (red line). When both electromechanical couplings are identical (green line), then the hardening effect occurs in the second resonance. For \(\alpha_{2}=60\) N/A (blue line), the hardening effect is reduced in both resonances, and no bifurcation points occur. The obtained resonance curves are similar to those of the linear system [34] (the bending effect is not observed). Interestingly, the peak in the second resonance is the lowest for \(\alpha_{2}=0\) N/A, but higher for \(\alpha_{2}=30\) N/A and \(\alpha_{2}=60\) N/A. This means that the higher electromechanical coupling between the bottom magnet and the coil improved recovered current near the second resonance. Figure 10: Influence of \(\alpha_{2}\) on the resonance curves of a bottom magnet (**a**), top magnet (**b**) and induced current (**c**), for \(\alpha_{1}=30\) N/A and \(A=0.6\)\(g\). The second strategy is similar: a fixed electromechanical coupling between the top magnet and the coil was assumed (\(\alpha_{2}=30\) N/A), and the parameter \(\alpha_{1}\) was changed. Figure 11 shows the influence of \(\alpha_{1}\) on the harvester response and induced current. It can be observed that the \(\alpha_{1}\) parameter plays a similar role to \(\alpha_{2}\). An increase in \(\alpha_{1}\) causes the magnet amplitude to decrease and the main nonlinear resonance to be reduced. However, the in the second resonance, the recovered current is higher for an \(\alpha_{1}\) of 30 N/A and 60 N/A. An analysis of the obtained curves in Figures 10 and 11 reveals that the electromechanical coupling influences the foldover effect that is responsible for the bending of the resonance curve in an amplitude versus frequency plot. The effect is characterized by two stable and one unstable solution. In Figure 12a, a two-parameter continuation was used to obtain the foldover region in the main resonance. First, it was assumed that \(\alpha_{1}=0\) N/A and that \(\alpha_{2}\) and \(\omega\) were changed (dark orange color). Second, \(\alpha_{2}=0\) N/A, and \(\alpha_{1}\) and \(\omega\) were changed (light region). The continuation started at the \(FB\) point (black point close to \(\omega\approx 77\) rad/s) and followed the path to its subsequent positions (black borderline). The direction of path-following continuation was marked with arrows.