## 2 Materials and Methods
### Modeling  
The proposed theoretical nonlinear electromagnetic harvester can be simplified to a 2 DOF lumped-parameter model connected by magnetic springs, as shown in Figure 5. Both levitating magnets move axially on the tube when the harvester is excited by the shaker. The mass of the suspended moving magnets is identical: \(m_{1}=m_{2}=m\). In the harvester, there are three damping sources: friction from contact of the magnet with the tube, air damping, and electromagnetic damping. Because both magnets are identical, the mechanical damping describing the total damping of both magnets is also the same and denoted by \(c\). The model consists of three subsystems, two of which are mechanical and one electromagnetic. Based on [PERSON]'s second law, the system dynamics can be described by time-dependent coupled nonlinear differential equations, two of which describe the dynamics of the magnets, and one that represents the electrical circuit output connected to the external resistive load: \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline m & 0.09 (kg) & P\({}_{1}\) & \(1.098\times 10^{9}\) \\ c & 0.07 (Ns/m) & P\({}_{2}\) & \(1.106\times 10^{4}\) \\ k\({}_{11}\) & \(-\)0.000039577 (N/m) & P\({}_{3}\) & \(-7.762\times 10^{5}\) \\ k\({}_{22}\) & \(-\)99,042,000 (N/m) & P\({}_{4}\) & \(-38.85\) \\ L & 1.46 (H) & P\({}_{5}\) & 2170 \\ R & 1000 (O) & P\({}_{6}\) & 0.03027 \\ \(\alpha_{1}\) & 30 (N/A) & P\({}_{7}\) & \(-0.1523\) \\ \(\alpha_{2}\) & 30 (N/A) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of the 2 DOF harvester and the nonlinear coefficients of the magnetic spring. \[m\ddot{x}_{1}+c\dot{x}_{1}+F_{B_{1}A_{1}B_{2}}(x_{1})-F_{A_{1}A_{2}}(x_{1},x_{2})+ \alpha_{1}\dot{i}=m(\ddot{x}_{s}+g), \tag{3}\]  
\[m\ddot{x}_{2}+c\dot{x}_{2}-F_{B_{1}A_{2}B_{2}}(x_{2})+F_{A_{1}A_{2}}(x_{1},x_{2})+ \alpha_{2}\dot{i}=m(\ddot{x}_{s}+g), \tag{4}\]  
\[Li+Ri=\alpha_{1}\dot{x}_{1}+\alpha_{2}\dot{x}_{2}. \tag{5}\]  
The coordinates \(x_{1}=X_{1}-x_{s}\) and \(x_{2}=X_{2}-x_{s}\) denote the relative magnet's displacements, where \(x_{s}\) is the shaker's displacement. The \(i\) is the induced current in the coil, from which the recovered energy can be easily estimated (\(P=\dot{I}^{2}R\)). The magnet's positions are measured from the equilibrium points of both magnets relative to the coil's center. Parameters \(\alpha_{1}\) and \(\alpha_{2}\) are the electromechanical coupling coefficients that characterize interaction between mechanical and electrical subsystems. These parameters can be estimated based on a geometry magnet-coil system or by simply testing a magnet moving through a coil with known speed [18]. The coil with a resistance \(R_{C}\) and an inductance \(L\) is connected to the resistor with a load resistance \(R_{L}\) (the total resistance is \(R=R_{L}+R_{C}\)). The magneto-elastic forces described in Section 2.2 are \(F_{B_{1}A_{1}B_{2}}(x_{1})\), \(F_{B_{1}A_{2}B_{2}}(x_{2})\), and \(F_{A_{1}A_{2}}(x_{1},x_{2})\). The moving magnets generate a complex current-time signal when the system is excited. Harmonic excitation is generated by the shaker and LMS software ([[https://eleaminofustry.com/directory/software-categories/learning-management-systems](https://eleaminofustry.com/directory/software-categories/learning-management-systems)]([https://eleaminofustry.com/directory/software-categories/learning-management-systems](https://eleaminofustry.com/directory/software-categories/learning-management-systems)), accessed on 16 May 2023), by a harmonic force \(\ddot{x}_{s}=A\text{cos}\omega t\), where \(\omega\) and \(A\) are the frequency and amplitude of excitation, respectively. A detailed analysis of Equation (5) shows that the induced voltage (and current) for a given electrical load depends on the levitating magnets' velocities and the electromechanical coupling coefficients; hence, the mechanical and electromagnetic subsystems are strongly coupled.