Perturbative computation of nonlinear harvesting through a path integral approach

Statistical field theories provide powerful tools to study complex dynamical systems. In this work those tools are used to analyze the dynamics of a kinetic energy harvester, which is modeled by a system of coupled stochastic nonlinear differential equations and driven by colored noise. Using the Martin-Siggia-Rose response fields we analytically approach the problem through path integrals in the phase space and represent the moments that correspond to physical observables through Feynman diagrams. This analysis method is tested by comparing the solution to the linear case with previous analytical results. Through a perturbative expansion it is calculated how the nonlinearity affects, to the first order, the energy harvest supporting the results through numerical simulations.

Figure 1

Bandwidth for the linear oscillator with $k=1$ and different $\zeta$ values: $\zeta =0$ [Eq. (29), dotted line], $\zeta =0.01$ (dashed line), $\zeta =0.1$ (solid line), and $\zeta =1$ (dash–dotted line).

Figure 2

Bandwidth for the linear oscillator with $\zeta =0.1$ and different $k$ values: $k=0$ (dotted line), $k=0.5$ (dashed line), $k=1$ (solid line), and $k=2$ (dash–dotted line).

Figure 3

Factor $\lambda A$ from Eq. (52) vs $\zeta$ and $k=\lambda =a=1$ (solid line), vs $k$ and $\zeta =\lambda =a=1$ (dot–dashed line), vs $\lambda$ and $\zeta =k=a=1$ (dashed line), and vs $a$ and $\zeta =k=\lambda =1$. All curves intersect at ${A\lambda |}_{\zeta =k=\lambda =a=1}=0.2$.

Figure 4

Mean electrical power harvested on a load resistor $R$. The lines correspond to Eq. (56) and the points to the numerical results, for $\beta =0$ (solid with circles), $\beta =-0.1$ (dash with squares), and $\beta =0.1$ (dot-dash with diamonds). The remaining parameters are $\alpha =0.5$ and $k_c=k_V=m={\gamma }_m=C=R=D={\lambda }_I=1$.

Figure 5

Mean input power from the forcing $\eta$. Same parameters and indicators as in Fig. 4.

Figure 6

Mean efficiency $\rho$. Same parameters and indicators as in Fig. 4.