## 3 Results and Discussion
### Frequency Response Curve  
In the linear harvester, the magnitude of excitation does not affect the resonance bandwidth. However, in the nonlinear harvester, excitation has a great impact on response. Therefore, frequency response analysis is one of the most important analyses in nonlinear systems, because it shows the resonance frequencies of a system and the relative peak amplitude. The main aim of the frequency response is to obtain information about the magnet oscillation level and to obtain the frequencies where the recovered current (and power) is highest. In this subsection, the 2 DOF nonlinear harvester is investigated using different excitation accelerations, e.g., from 0.2\(\div\)1 \(g\). The maximum (MAX(x\({}_{1}\)), MAX(x\({}_{2}\))) and minimum (MIN(x\({}_{1}\)), MIN(x\({}_{2}\))) responses of both magnets and the output maximum  
Figure 6: Comparison and validation of the bottom magnet resonance curve (**a**) and induced current (**b**), for an excitation amplitude of \(A=0.6\,g\) and \(R=1\,\mathrm{k}\Omega\). current (MAX(i)) are analyzed. The displacements of magnets are measured according to the coordinate system from Figure 5.  
In Figure 7, the maximum and minimum displacements of the top (Figure 7a) and bottom (Figure 7b) magnets, as well as the maximum recovered current (Figure 7c), are shown. These diagrams were plotted by sweeping the frequency \(\omega\) from 20 rad/s to 140 rad/s, for different amplitudes: 0.2 \(g\) (black line), 0.6 \(g\) (blue line), and 1 \(g\) (green line). The dashed line denotes the static position (equilibrium) of both magnets, while the continuous line indicates the stable solution (stable fixed point). The red line between the fold (saddle-node) bifurcation points \(FB\) denotes the unstable solution (unstable fixed point). Fold bifurcations occur when two equilibria come together and disappear [32]. It can be noted that the main (first) resonance is observed for \(\omega\approx\) 60 rad/s, while the second one is for \(\omega\approx\) 120 rad/s. It should be noted that the maximal and minimal displacements are not symmetric. An increase in the amplitude of excitation causes an increase in the magnet oscillations, and the hardening effect can be observed. In the first resonance, the hardening effect characterizes two stable and one unstable solution, occurring for \(A=1\)\(g\) (green line). The multistability is caused by a fold bifurcation (\(FB\)) where the stable and unstable branches meet. The peak-to-peak top and bottom magnets' output responses are quite similar and amount to 0.022 m (for \(A=1\)\(g\)). Interestingly, the resonance of the recovered current shows two peaks close to the frequencies of \(\omega\approx\) 60 rad/s and 70 rad/s (Figure 7c). These peaks are the result of the dynamic vibration absorption effect. The bottom magnet, for some parameters, causes vibration reduction of the top magnet (region between both peaks). This phenomenon exists in a 2 DOF system. Between the two similar peaks, the recovered  
Figure 7: Simulated resonance curves of the top magnet (**a**), bottom magnet (**b**), induced current (**c**), and unstable tongues close to the first and second resonances (**d**) for different amplitudes of excitation \(A\). The solid lines mark the numerically stable solutions, and the dashed curves mark the unstable ones. The orange region indicates the multistability region, and the pink region denotes the unstable region caused by the \(NS\) bifurcation. The top line means \(MAX\), while the bottom line means \(MIN\) values of the magnet displacements. current decreases from 28 mA (for \(\omega=58\) rad/s) to 14 mA (for \(\omega=60\) rad/s). This means that the instantaneous power changes from 0.78 W to 0.2 W.  
The second resonance exhibits a multistability behavior, which could already be observed for 0.6 \(g\). This resonance is characteristic of lower magnet oscillations, and the recovered current is about 0.01 mA (0.1 W). However, the peak-to-peak top and bottom magnet response is about 0.02 m (for \(A=1\,g\)).