# Moment-Based Stochastic Analysis of a Bistable Energy Harvester with Matching Network
## 6 Nonlinear System Analysis  
When the inertial magnetic mass and the tip magnet are close enough, magnetic repulsion induces bi-stability, forcing the beam to the left or to the right of the vertical position. The magnetic force makes two vertically tilted positions stable equilibrium points, associated to the local minima of the potential energy, while the vertical position becomes an unstable equilibrium associated to a local potential energy maximum. For ambient mechanical vibrations of small amplitude, the beam is expected to oscillate around one of the two equilibria, remaining confined within the corresponding potential well. If the vibration intensity is large enough, however, fluctuations of the beam around the resting position are combined with random jumps from one well to the other. As the noise intensity becomes large enough, the beam will exhibit more frequent excursions between the two wells, with a motion ultimately resembling a random wandering around the vertical position. To account for such bi-stability, we shall assume a potential function of the form \(U(x)=-k_{1}\,x^{2}/2+k_{3}\,x^{4}/4+U_{0}\), where \(k_{1}\) and \(k_{3}\) are the linear and the nonlinear elastic constants, respectively, and \(U_{0}=k_{1}^{2}/(4k_{3})\). Using the diagonal matrix (17), the following SDEs system is obtained:  
\[d\mathbf{X}_{t}=(\mathbf{A}_{0}\,\mathbf{X}_{t}+\mathbf{n}(\mathbf{X}_{t}))dt+ \varepsilon\mathbf{B}\,dW_{t} \tag{25}\]  
where  
\[\mathbf{A}_{0}=\left[ \begin{array}{cccc}0&1&0&0&0\\ 1&-\rho&-\beta&0&0\\ 0&\alpha&0&-1&0\\ 0&0&\mu&0&-\mu\\ 0&0&0&\eta&-\delta\end{array} \right],\qquad\mathbf{B}=[0,\sigma,0,0,0]^{T} \tag{26}\]  
and \(\mathbf{n}(x)=[0,-\kappa x^{3},0,0,0]^{T}\), with \(\kappa=k_{3}/k_{1}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Configuration** & **Voltage (rms)** & **Maximum Power** & **Efficiency** \\ \hline Resistive load & 4.43 V & 19.61 \(\mu\)W & 3.9\% \\ Matched load & 20.59 V & 424.1 \(\mu\)W & 84.8\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Root mean square value of the output voltage, output average power and power efficiency for the energy harvester with the two different load setups. Figure 3: Root mean square of the output voltage as a function of the matching network parameters \(L_{S}\) and \(C_{P}\). Proceeding in full analogy with the linear case, the evaluation of the output average power corresponds to the calculation of the second-order moments. However, differently from the linear case, for a generic nonlinear systems, it is impossible to derive closed equations for the moments. In fact, because of the nonlinear term in (25), the ODEs system for \(\mathrm{E}[\mathbf{X}_{t}^{n}]\) will include terms such as \(\mathrm{E}[\mathbf{X}_{t}^{n+2}]\) for all \(n=1,2,\ldots\), thus leading to an infinite hierarchy of coupled ODEs. A possible workaround is to make some a priori assumption about the probability density function for \(\mathbf{X}_{t}\). Under a suitable hypothesis, it is possible to express higher-order moments as functions of the lower-order ones, a methodology termed moment closure technique [47]. A quite common choice amounts to assume that the stochastic process obeys a multivariate Gaussian distribution. In general, it is neither possible to justify such an assumption rigorously nor to prove that the approximated results converge to the exact values. However, there is a large amount of evidence that the method provides accurate enough results, especially if the system is subject to weak white Gaussian noise and is characterized by underlying deterministic dynamics exhibiting a stable solution [47]. We proceed as follows:  
* We assume \(\mathbf{X}_{t}\sim\mathcal{N}(\mathbf{n},\sigma)\), and we derive the ODEs system for the first two order moments. All higher order moments in the ODE system are expressed as functions of the first and second order moments.