Analysis of influence of nonlinearities and noise correlation time in a single-DOF energy-harvesting system via power balance description

Abstract

We discuss the role played by the time correlation properties of stochastic sources and by model nonlinearities in single-degree of freedom energy-harvesting systems. After transforming the state equations into energy-angle coordinates, we apply a stochastic projection operator technique to obtain the system power balance equation. The latter allows to evaluate both the magnitude of the power injected by noise into the system and the harvested power, thus providing a tool instrumental for designers to optimize the harvester. We show that for systems with modulated (multiplicative) noise, nonlinear energy harvesters can outperform their linear counterparts, highlighting the physical mechanism that explains their better performance.

Appendices

Appendix A: Solution of the nonlinear oscillator

In the absence of noise and for the potential $V(x) = a x^2/2 + b x^4/4$ Eq. (2) becomes

$$\begin{aligned} \dot{x} =&y \\ \dot{y} =&- a x - b x^3 \end{aligned}$$

Dividing one equation by the other, separating variables and integrating one finds

$$\begin{aligned} \dfrac{y^2}{2} + a \dfrac{x^2}{2} + b \dfrac{x^4}{4} = H_0 \end{aligned}$$

where $H_0$ is a constant representing the energy. Solving with respect to y and considering only the positive determination yields

$$\begin{aligned} y = \sqrt{4H_0 - 2a x^2 - b x^4} \end{aligned}$$

Substituting in $\dot{x} = y$ and separating variables gives the first-order differential equation

$$\begin{aligned} \dfrac{\text {d}x}{\sqrt{4H_0 - 2a x^2 - b x^4}} = \text {d}t \end{aligned}$$

Integrating both sides, it turns out that x(t) is in fact given by [43]

$$\begin{aligned} x(t) = \left( \dfrac{4E^2}{a^2+ 4 b E}\right) ^{1/4} \text {sd}[(a^2+4bE)^{1/4}t,k] \end{aligned}$$

where $E=4H_0$ represents the energy level. Differentiating with respect to time

$$\begin{aligned} y(t)= & {} \sqrt{2E} \, \text {cd}[(a^2+4bE)^{1/4}t,k] \, \\&\text {nd}[(a^2+4bE)^{1/4}t,k] \end{aligned}$$

Appendix B: Integral functions $I_n$

The well-known relationships between the squares of Jacobi elliptic functions

$$\begin{aligned} k^2 \, \text {sd}^2 u =&\text {nd}^2 u - 1\\ k^2 \, \text {cd}^2 u =&1 - k'^2 \text {nd}^2 u \end{aligned}$$

imply that (42) can be reduced to the solution of the integrals

$$\begin{aligned} I_n = \int _0^{4{\mathcal {K}}(E)} \text {nd}^n u \, \text {d}u \end{aligned}$$

For $n=2,4$ the integrals are easily computed with the help of the Fourier series for $\text {nd}^2 u$ and $\text {nd}^4 u$ [44], obtaining

$$\begin{aligned} I_2 =&\int _0^{4 {\mathcal {K}}(E)} \text {nd}^2 u \, \text {d}u = \dfrac{4{\mathcal {E}}(k)}{k'^2}\\ I_4 =&\int _0^{4 {\mathcal {K}}(E)} \text {nd}^4 u \, \text {d}u = \dfrac{8(2-k^2){\mathcal {E}} - 4 (1-k^2){\mathcal {K}}}{3k'^4} \end{aligned}$$

where ${\mathcal {E}}(k)$ is the complete elliptic integral of the second kind.

For larger values of n, integrals $I_n$ can be computed using a recursive relation. Following the procedure described in [45] we find

$$\begin{aligned}&\dfrac{\text {d}}{\text {d}u} \left( \text {nd}^l u \, \text {sd}u \, \text {cd}u \right) \\&\quad = -(l+2) \dfrac{k'^2}{k^2} \, \text {nd}^{l+3} u \\&\qquad + (l+1) \dfrac{k'^2+1}{k^2} \, \text {nd}^{l+1} u - l \dfrac{1}{k^2} \, \text {nd}^{l-1} u \end{aligned}$$

For $l=n-3$

$$\begin{aligned}&\dfrac{\text {d}}{\text {d}u} \left( \text {nd}^{n-3} u \, \text {sd}u \, \text {cd}u \right) \\&\quad = (1-n) \dfrac{k'^2}{k^2} \, \text {nd}^n u \\&\qquad + (n-2) \dfrac{k'^2+1}{k^2} \, \text {nd}^{n-2} u - (3-n) \dfrac{1}{k^2} \, \text {nd}^{n-4} u \end{aligned}$$

Integrating and rearranging terms

$$\begin{aligned} I_n =&\dfrac{1}{(1-n)k'^2} \big [ (2-n) (k'^2 + 1) I_{n-2} + (n-3) I_{n-4} \big ] \end{aligned}$$