Electric load optimization of a nonlinear mono-stable duffing harvester excited by white noise

Abstract

This paper investigates the electric load optimization of a nonlinear mono-stable Duffing energy harvester under white noise excitations considering symmetric and asymmetric nonlinear restoring forces. Statistical linearization is utilized to obtain approximate analytical expressions for the statistical averages including the average output power, which is then optimized with respect to the electric load. It is shown that the optimal load is dependent on the nonlinearity unless the ratio between the period of the mechanical system and the time constant of the harvesting circuit is large. Furthermore, it is demonstrated that, under optimal electric loading, a mono-stable Duffing harvester with a symmetric nonlinear restoring force can never produce higher average power levels than an equivalent linear harvester regardless of the magnitude of the nonlinearity. On the other hand, asymmetries in the restoring force are shown to provide performance improvements over an equivalent linear harvester.

1 Introduction

A significant body of the current literature on vibratory energy harvesting is focused on the concept of purposeful inclusion of stiffness nonlinearities for broadband transduction [1–12]. When compared to their linear resonant counterparts, nonlinear vibratory energy harvesters (VEHs) have a wider steady-state frequency bandwidth, leading to the common belief that they can be utilized to improve performance especially in random and non-stationary vibratory environments. Uncertainty propagation analysis performed on linear and nonlinear energy harvesters served to further reinforce this conclusion by illustrating that, under harmonic excitations, a linear device is much more sensitive to uncertainties arising from imprecise characterization of the host environment and/or from manufacturing tolerances [13].

The basic concept of nonlinear VEHs lies in using external design means in order to purposefully introduce and control the magnitude and nature of the nonlinearity. The most common approach to the design of such systems introduces a nonlinear restoring force, using, for example, magnetic or mechanical forces [2–4]. There are two different classes of nonlinear VEHs. The first is concerned with designing energy harvesters that exhibit a nonlinear resonant behavior similar to that of a mono-stable Duffing oscillator with a hardening/softening nonlinearity [2–4]. The second class is designed to have a double-well potential energy function similar to that of a bi-stable Duffing oscillator.

While most environmental excitations under which VEHs are designed to operate have random or time-dependent characteristics, their design and optimization is currently, for the most part, based on steady-state analyses assuming harmonic excitations. While this constitutes an important first step, it does not provide the tools or insights necessary for improving their performance in an actual environment. Recently, few research studies have focused on presenting a clearer picture of how randomness and non-stationarities in the excitation influence the average power of nonlinear VEHs [14–23]. Such studies were mainly focused on analyzing the response of a mono-stable Duffing harvester with a symmetric restoring force to white noise excitations. It was determined that the ratio between the period of the mechanical system and the time constant of the harvesting circuit plays an important role in characterizing the influence of the nonlinearity on the average power.

Electric load optimization of mono-stable Duffing harvesters with under white noise was investigated by Green et al. [23]. They considered an electromagnetic energy harvester with hardening nonlinearities and used statistical linearization to show that the optimal load is not a function of the nonlinearity and is equal to that corresponding to the optimization of the linear problem. However, Green et al. [23] made two assumptions; first they neglected the inductance of the coil, and, second, they assumed a symmetric restoring force. In this paper, we alleviate these assumptions and delineate the influence of stiffness nonlinearities on the optimal electric load under white noise excitations. In particular, while there were several previous studies addressing the response of mono-stable energy harvesters to white noise excitations [23, 24], none to the authors’ knowledge have addressed obtaining general expressions for the optimal load without invoking the assumptions of Green et al. [23]. As such, this work investigates how the optimal load of mono-stable Duffing harvesters is influenced by nonlinearities and asymmetries in the restoring force. To achieve this goal, the rest of the manuscript is organized as follows: Section 2 presents a general electromechanical model of a nonlinear Duffing-type harvester. Section 3 formulates the FPK equation governing the probability distribution function (PDF) of the response under white noise. Section 4 presents an exact solution of the FPK equation for the linearized system, which is then used to find the optimal load. Section 5 uses statistical linearization techniques combined with an optimization algorithm to delineate the influence of the nonlinearity on the optimal electric load for both symmetric and asymmetric potentials. Finally, Sect. 6 presents the main conclusions.

2 Electromechanical model

To achieve the objectives of this work, we consider the basic physics of a nonlinear VEH which can be captured by considering a mechanical oscillator coupled to an electric circuit through an electromechanical coupling mechanism. This mechanism can either be piezoelectric, Fig. 1a; or electromagnetic, Fig. 1b. Assuming linear electromechanical coupling, the equations of motion can be written in the following general form:

$$m\ddot{\bar{x}}+c \dot{\bar{x}}+\frac{{\text {d}}{\bar{U}}({\bar{x}})}{{\text {d}}{\bar{x}}}+\theta {\bar{y}}=-m\ddot{{\bar{x}}}_b, $$

(1a)

$$C_p\dot{\bar{y}}+\frac{\bar{y}}{R}=\theta \dot{\bar{x}}, ({\text {piezoelectric}}),\quad L\dot{\bar{y}}+R{\bar{y}}=\theta \dot{\bar{x}}, ({\text {electromagnetic}})$$

(1b)

where the dot represents a derivative with respect to time, $\tau$. The variable ${\bar{x}}$ represents the relative displacement of the mass, m; c is a linear viscous damping coefficient; $\theta$ is a linear electromechanical coupling coefficient; $\ddot{\bar{x}}_b$ is the base acceleration; $C_p$ is the capacitance of the piezoelectric element; L is the inductance of the harvesting coil, and $\bar{y}$ is the electric quantity representing the induced voltage in capacitive harvesters and the induced current in inductive ones. These are measured across an equivalent resistive load, R. In piezoelectric energy harvesters, the load, R, is the parallel equivalent of the piezoelectric resistance, $R_p$, and the load resistance, $R_l$, i.e., $R=\frac{R_l R_p}{R_l+R_p}$. In inductive harvesters, it represents the series equivalent of the load and coil resistance, $R_c$, i.e., $R=R_l+R_c$. The function $\bar{U}(\bar{x})$ represents the potential energy of the mechanical oscillator and is given in the following general form:

$$\bar{U}({\bar{x}})=\frac{1}{2} k_1 {\bar{x}}^2+\frac{1}{3} k_2 {\bar{x}}^3+\frac{1}{4} k_3 {\bar{x}}^4,$$

(2)

where $k_1$, $k_2$ and $k_3$ are, respectively, the linear, quadratic, and cubic nonlinearity coefficients appearing in the restoring force. The equations of motion can be further non-dimensionalized by introducing the following dimensionless quantities:

$$ x=\frac{\bar{x}}{l_c}, \quad x_b =\frac{{\bar{x}}_b}{l_c}, \quad t=\tau \omega _n, \quad y=\frac{C_p}{\theta l_c} {\bar{y}} \, (\text {piezoelectric}), \quad y=\frac{L}{\theta l_c} {\bar{y}} \, (\text {electromagnetic}) $$

(3)

where $l_c$ is a length scale, and $\omega _n=\sqrt{k_1/m}$ is the natural frequency of the mechanical oscillator. With these transformations, the non-dimensional equations of motion can be expressed as

$$\ddot{x}+2\zeta \dot{x}+\frac{{\text {d}}U}{\text {d}x}+\kappa ^2 y=-\ddot{x}_b,$$

(4a)

$$\dot{y}+\alpha y=\dot{x},$$

(4b)

where

$$\frac{{\text {d}}U}{{\text {d}}x}=x + \lambda x^2+\delta x^3, $$

(5)

and

$$\begin{aligned}&\zeta =\frac{c}{2\sqrt{k_1 m}}, \quad \kappa ^2=\frac{\theta ^2}{k_1 C_p}\,(\text {capacitive}), \,\kappa ^2=\frac{\theta ^2}{k_1 L}\,(\text {inductive}), \\&\lambda = \frac{k_2 l_c}{k_1}, \quad \delta =\frac{k_3 l^2_c}{k_1}, \quad \alpha =\frac{1}{RC_p\omega _n} (\text {piezoelectric}),\,\alpha =\frac{R}{L\omega _n}\, (\text {electromagnetic}). \end{aligned}$$

Here, $\zeta$ is the mechanical damping ratio, $\kappa$ is a linear dimensionless electromechanical coupling coefficient, $\alpha$ is the ratio between the mechanical and electrical time constants of the harvester. Finally, $\lambda$ and $\delta$ are the coefficients of the quadratic and cubic nonlinearity.

Fig. 1

A simplified representation of a vibratory energy harvester

The form of Eq. (4) permits classifying energy harvesters, regardless of their coupling mechanism, into three major categories based on the shape of their potential energy function and the associated restoring force.

• 1. Linear when $\lambda =\delta =0$: In such a case, the restoring force is a linear function of the displacement. Most linear VEHs are only linear within a certain range of operation. Large deformations and electromechanical coupling mechanisms can introduce small nonlinearities that are usually neglected to avoid complexities in the analysis.

• 2. Nonlinear mono-stable when $\delta >0$ and $0\le \lambda < 2 \sqrt{\delta }$: When $\delta >0$ and $\lambda =0$, the potential shape is symmetric and has one minimum (mono-stable). The restoring force increases with the displacement and is said to be of the hardening type. When $\delta >0$ and $0<\lambda < 2 \sqrt{\delta }$, the potential function remains mono-stable but loses its symmetry around the equilibrium as shown in Fig. 2a.

• 3. Nonlinear bi-stable when $\delta >0$ and $\lambda \ge 2 \sqrt{\delta }$: Here, the potential function of the harvester has two potential wells separated by a potential barrier as depicted in Fig. 3.

In this paper, the analysis is restricted to mono-stable potentials as described in the first two cases.

Fig. 2

a Potential energy and b restoring force in nonlinear mono-stable VEHs

Fig. 3

a Potential energy and b restoring force in nonlinear bi-stable VEHs

3 Stochastic dynamics

The environmental excitation, $\ddot{x}_b$, to which the harvester is subjected is assumed to be a physical zero-mean Gaussian process with a very small correlation time which approaches zero. In such a case, $\ddot{x}_b$ can be approximated by a Gaussian white noise process such that

$$\langle \ddot{x}_b(t)\rangle =0, \quad \langle \ddot{x}_b(t)\ddot{x}_b(s)\rangle =2 \pi S_0 \delta _0 (s-t),$$

(6)

where $\langle \rangle$ denotes the expected value, $S_0$ is the spectral density of the process, and $\delta _0$ is the Dirac-delta function. While white noise of infinite bandwidth is only a theoretical construction, the assumption of white noise is not as restrictive as it may appear so long as the bandwidth of the excitation is sufficiently larger than that of the harvester’s. In the open literature it is a common practice to use the ratio between the correlation time of the noise and the time constant of the system’s response as a guideline to the validity of the white noise approximation [25]. If this ratio is <0.1, one can safely use the white noise approximation.

The response statistics associated with the stochastic dynamics of Eq. (4) can be generated by expressing the equations in the It$\hat{o}$ stochastic form as [26, 27]

$${\text {d}}{\mathbf {x}}(t)= {\mathbf {f}} ({\mathbf {x}} ,t){\text {d}}t+{\mathbf {G}} ({\mathbf {x}} ,t) {\text {d}}{\mathbf {B}} , $$

(7)

where ${\mathbf {x}} =(x_1, x_2, x_3)^T \equiv (x, \dot{x}, y)^T$, B is a Brownian motion process such that $\ddot{x}_b(t)={\text {d}}{\mathbf {B}} /{\text {d}}t$ and

$$\begin{aligned} {\mathbf {f}} ({\mathbf {x}} ,t)= \left\{ \begin{array}{c} x_2 \\ -2 \zeta x_2-\frac{{\text {d}}U}{{\text {d}}x_1}-\kappa ^2 x_3\\ -\alpha x_3-x_2\\ \end{array} \right\} , \quad {\mathbf {G}} ({\mathbf {x}} ,t) = \left\{ \begin{array}{c} 0 \\ -1\\ 0\\ \end{array} \right\} . \end{aligned}$$

(8)

The solution of Eq. (7) is determined by the evolution of the transition PDF, $P({\mathbf {x}} ,t)$, which, in turn, is governed by the following FPK equation:

$$\begin{aligned} \frac{\partial P({\mathbf {x}} ,t)}{\partial t}= & -\sum _{i=1}^3\frac{\partial }{\partial x_i}[P({\mathbf {x}} ,t){\mathbf {f}} _i({\mathbf {x}} ,t)] +\frac{1}{2} \sum _{i=1}^3 \sum _{j=1}^3 \frac{\partial ^2}{\partial x_i x_j}[P({\mathbf {x}} ,t)({\mathbf {Q}} {\mathbf {G}} {\mathbf {G}} ^T)_{ij}],\nonumber \\ \quad P(\infty ,t)= & \,P(-\infty ,t)=0, \end{aligned}$$

(9)

where

$$\begin{aligned} {\mathbf {Q}} =\left[ \begin{array}{ccc} 0& 0 & 0 \\ 0 & 2 \pi S_0 & 0 \\ 0 & 0 & 0\\ \end{array} \right] . \end{aligned}$$

With the knowledge of ${\mathbf {f}} ({\mathbf {x}} ,t)$ and ${\mathbf {G}} ({\mathbf {x}} ,t)$, the FPK equation reduces to

$$\begin{aligned} \frac{\partial P({\mathbf {x}} ,t)}{\partial t}= & -x_2\frac{\partial P({\mathbf {x}} ,t)}{\partial x_1}+2\zeta \frac{\partial (x_2P({\mathbf {x}} ,t))}{\partial x_2}+\left( \frac{{\text {d}}U}{{\text {d}}x_1}+\kappa ^2 x_3\right) \frac{\partial P({\mathbf {x}} ,t)}{\partial x_2}\nonumber \\&+\alpha \frac{\partial (x_3P({\mathbf {x}} ,t))}{\partial x_3}+x_2\frac{\partial P({\mathbf {x}} ,t)}{\partial x_3} +\pi S_0 \frac{\partial ^2 P({\mathbf {x}} ,t)}{\partial ^2 x_2},\nonumber \\&\quad P(\infty ,t)=P(-\infty ,t)=0. \end{aligned}$$

(10)

Upon solving Eq. (10) for $P({\mathbf {x}} ,t)$, the response statistics can then be obtained via

$$ \left\langle \prod _{i=1}^3 x^{k_i}_i \right\rangle = \int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty } \prod _{i=1}^3 x^{k_i}_i P ({\mathbf {x}} ,t) {\text {d}}x_1 {\text {d}}x_2 {\text {d}}x_3, $$

(11)

where $k_i=0,1,2,\ldots$.

4 Optimality of the linear system

For the purpose of performance comparison, we first investigate the response statistics and the optimality of the design parameters for the linear harvester with $\lambda =\delta =0$. This has been already discussed in Ref. [?] but is repeated consicely in this section for completeness. Since for linear systems, the response to a Gaussian input is also Gaussian, it is possible to obtain an exact stationary solution of Eq. (10) in the general Gaussian form

$$ P(x_1,x_2,x_3)=A \exp {\left( \sum _{i,j=1}^3 a_{ij} x_i x_j\right) }, $$

(12)

where A is a constant obtained via the following normalization scheme:

$$\int _{-\infty }^{\infty } \int _{-\infty }^{\infty } \int _{-\infty }^{\infty } P(x_1,x_2,x_3){\text {d}}x_1 {\text {d}}x_2 {\text {d}}x_3=1,$$

(13)

and the $a_{ij}$ are attained by substituting Eq. (12) into Eq. (10), then forcing the solvability conditions. This yields

$$ a_{ij}=-\frac{1}{2}\frac{|R|_{ij}}{|R|}, $$

(14)

where

$$\begin{aligned} R=\frac{2\pi S_0}{2\zeta (1+\alpha ^2+2\alpha \zeta )+\kappa ^2(\alpha +2\zeta )} \left[ \begin{array}{ccc} 1+\alpha ^2+2\alpha \zeta & 0 & \frac{1}{2}\\ 0 &\frac{1+\alpha ^2+2\alpha \zeta +\kappa ^2}{2} & \frac{\alpha }{2} \\ \frac{1}{2} & \frac{\alpha }{2} & \frac{1}{2}\\ \end{array} \right] . \end{aligned}$$

Here, |R| and $|R|_{ij}$ are, respectively, the determinant and co-factors of R.

With the knowledge of the exact stationary probability function, the required response statistics can now be obtained using Eq.  (11). Of special importance are the mean square values of the displacement, velocity, and electric quantity which can be expressed as

$$\left\langle x^2_1\right\rangle= \pi S_0\frac{1+\alpha ^2+2\alpha \zeta }{2\zeta (1+\alpha ^2+2\alpha \zeta )+\kappa ^2(\alpha +2\zeta )}, $$

(15a)

$$\left\langle x^2_2\right\rangle= \pi S_0\frac{1+\alpha ^2+2\alpha \zeta + \kappa ^2}{2\zeta (1+\alpha ^2+2\alpha \zeta )+\kappa ^2(\alpha +2\zeta )}, $$

(15b)

$$ \left\langle x^2_3\right\rangle= \pi S_0\frac{1}{2\zeta (1+\alpha ^2+2\alpha \zeta )+\kappa ^2(\alpha +2\zeta )}. $$

(15c)

Using Eq. (15c), the dimensional average power can also be expressed in the simple form

$$\langle {\mathrm {P}} \rangle =k_1 l^2_c \omega _n\alpha \kappa ^2 \langle x^2_3 \rangle , $$

(16)

Equations (16) can be utilized to investigate the optimal time constant ratio, $\alpha$, of the linear VEH. This can be achieved by differentiating Eq. (16) with respect to $\alpha$, then solving for $\alpha _{opt}$ to obtain

$$\alpha _{opt}=\sqrt{1+\kappa ^2}. $$

(17)

The preceding expression reveals that the optimal time constant ratio of the linear harvester is only dependent on the electromechanical coupling, $\kappa$. Using the definition of the time constant ratio, we conclude that the optimal load occurs at $R_{opt}=1/(C_p \omega _n\sqrt{1+\kappa ^2})$ for a piezoelectric VEH, and at $R_{opt}=L \omega _n \sqrt{1+\kappa ^2}$ for an inductive one.

5 Optimality of the nonlinear system

In this section, we discuss electric load optimization for the nonlinear harvester. Due to its simplicity, the case of a large time constant ratio, $\alpha$, is first discussed in Sect. 5.1. This serves to represent the behavior when either L or $C_p$ are very small. In fact, it is a common practice in the literature to neglect the inductance of the coils in electromagnetic harvesters, and, sometimes the capacitance of the piezoelectric element for simplicity, [3, 19, 23]. Subsequently, in Sect. 5.2, the general case of any time constant ratio is investigated. The case of symmetric potential function is treated first, then the influence of potential function asymmetries is studied.

5.1 The special case of a large time constant ratio

From a mathematical point of view, the circuit dynamics, Eq. (4), represents a first-order low-pass filter with the velocity being its input, the electric quantity, y, representing its output, and $\alpha$ characterizing the inverse of its time constant. When $\alpha$ is large, the bandwidth of the filter is large, and the circuit dynamics can be approximated via $\alpha y=\dot{x}$. This allows the dynamics of the coupled system, Eq. (4a), to be reduced to the following form:

$$\ddot{x}+\zeta _{eff} \dot{x}+\frac{{\text {d}}U}{{\text {d}}x}=-\ddot{x}_b $$

(18)

where $\zeta _{eff}= 2\zeta +\kappa ^2/\alpha$. For the reduced system, the PDF of the response can be obtained by solving a reduced FPK equation of the form

$$\begin{aligned} \frac{\partial P({\mathbf {x}} ,t)}{\partial t}= & -x_2\frac{\partial P({\mathbf {x}} ,t)}{\partial x_1}+2\zeta _{eff}\frac{\partial (x_2P({\mathbf {x}} ,t))}{\partial x_2}+\left( \frac{{\text {d}}U}{{\text {d}}x_1}\right) \frac{\partial P({\mathbf {x}} ,t)}{\partial x_2}+\pi S_0 \frac{\partial ^2 P({\mathbf {x}} ,t)}{\partial ^2 x_2}, \nonumber \\&\quad P(\infty ,t)=P(-\infty ,t)=0, \end{aligned}$$

(19)

where $( x_1, x_2)^{T} \equiv (x, \dot{x})^T$. In the stationary sense, the transition probability function is time invariant, i.e. $\partial P({\mathbf {x}} ,t)/\partial t=0$ or $P({\mathbf {x}} ,t)=P({\mathbf {x}} )$, and Eq. (19) admits the following stationary solution:

$$P(x_1,x_2)=A_1 \exp \left\{ \frac{-\zeta _{eff}}{\pi S_0}U(x_1)\right\} \times A_2 \exp \left\{ \frac{-\zeta _{eff}}{\pi S_0}\frac{x^2_2}{2}\right\} , $$

(20)

where $A^{-1}_1=\int _{-\infty }^{\infty } \exp \left\{ \frac{- \zeta _{eff}}{\pi S_0}U(x_1)\right\} {\text {d}}x_1$ and $A^{-1}_2= \int _{-\infty }^{\infty }\exp \left\{ \frac{- \zeta _{eff}}{\pi S_0}\frac{x^2_2}{2}\right\} {\text {d}}x_2$. Note that, the resulting PDF can be factored into a function of the displacement, $x_1$, and a function of the velocity, $x_2$. This implies that the displacement and velocity can be treated as two independent random variables. In such a case, the expected mean square value of the velocity, $\langle x^2_2\rangle$, is independent of the displacement, nonlinearity, and the potential function altogether; and is given by

$$ \langle \dot{x}^2\rangle \equiv \langle x^2_2 \rangle = A_2 \int _{-\infty }^{\infty }x^2_2\exp \left\{ \frac{-\zeta _{eff}}{\sigma ^2}x^2_2\right\} {\text {d}}x_2=\frac{\pi S_0}{\zeta _{eff}}. $$

(21)

Using the relation $y=\dot{x}/\alpha$ in conjunction with Eq. (21), the expected mean square value of the electric quantity can then be written as

$$ \langle y^2 \rangle \equiv \langle x^2_3 \rangle = \frac{\pi S_0}{\alpha ^2 \zeta _{eff}}.$$

(22)

Equation (22) reveals that the expected value of the electric quantity, voltage in the case of piezoelectric harvesters, and current in the case of electromagnetic ones, is independent of the shape of the potential function leading to the conclusion that for large values of the time constant ratio $\alpha$, no matter how the potential function of the harvester is altered, it has no influence on the average output power. This conclusion holds for all harvesters with nonlinearities appearing in the restoring force.

Referring back to Eq. (15c), it can be noted that when $\alpha \gg \zeta$ , i.e., when L or $C_p$ approaches a very small number, the mean square value of the electric output reduces to

$$ \langle x^2_3 \rangle = \frac{\pi S_0}{\alpha ^2 \zeta _{eff}}, $$

(23)

which is equivalent to Eq. (22).This illustrates that the linear and nonlinear system have similar expressions for the electric quantity when $\alpha$ is large. Thus, the optimal load is the same for both cases as given by Eq. (17). In dimensional terms, since $C_p$ and L approach zero when $\alpha$ is large, $R_{opt}$ approaches short circuit in electromagnetic VEHs, and approaches open circuit in piezoelectric ones.

It is worth noting that, while the variance of the electric quantity is independent of the nonlinearity as previously described, the variance of the displacement still decreases with the nonlinearity as discussed earlier in Ref. [23]. This implies that the nonlinearity helps produce the same average power but for a smaller variance in the displacement. Based on this result, the authors of Ref. [23] concluded that the nonlinearity can help produce a more compact device. However, this conclusion should be approached with caution since a reduction in variance does not necessarily prevent the instantaneous displacement from being large at some instants in time.

5.2 The general case of any time constant ratio

When the time constant ratio is not necessarily large and the restoring force has a nonlinear dependence on the displacement, an exact solution of the FPK equation, Eq. (10), is not easily attainable even in the stationary sense. To approximate the response statistics in such a scenario, it is common to seek approximate response statistics. One approach is based on statistically linearizing the governing equation of motion, Eq. (4a). For the mono-stable nonlinear system at hand, methods of statistical linearization are capable of providing accurate response statistics for weakly nonlinear systems subjected to Gaussian excitations of moderate intensity. For a better understanding of the results, the case of a symmetric mono-stable potential is first considered.

5.2.1 Symmetric potential

When the potential function is symmetric, the nonlinear equation of motion can be replaced by an equivalent linear system in the form

$$\begin{aligned}&\ddot{x} + 2\zeta \dot{x}+ \omega ^2_e x+\kappa ^2 y=-\ddot{x}_b,\nonumber \\&\dot{y}+\alpha y=\dot{x}, \end{aligned}$$

(24)

where $\omega ^2_e x$ is an equivalent linear restoring force that best approximates the nonlinear restoring force of Eq. (4a). To obtain the unknown coefficient, $\omega ^2_e$, we minimize the main square error, E, between the actual restoring force and its linear equivalent, i.e., we let

$$ \frac{\partial {\langle E^2\rangle }}{\partial \omega ^2_e}=0, \quad \frac{\partial ^2{\langle E^2\rangle }}{\partial (\omega ^2_e)^2}>0, $$

(25)

where $\langle E^2\rangle =\langle (\omega ^2_e x- x - \delta x^3)^2\rangle$. This yields

$$(\omega ^2_e -1) \langle x^2\rangle -\delta \langle x^4\rangle =0, \quad \langle x^2 \rangle > 0, $$

(26)

To find an approximate analytical expression for $\omega ^2_e$, the fourth order statistical moment of the displacement is approximated as $\langle x^4 \rangle \approx 3 \langle x^2 \rangle ^2$, which yields

$$ \omega ^2_e=1+3\delta \langle x^2 \rangle . $$

(27)

With the knowledge of $\omega ^2_e$, the response statistics associated with the equivalent linear system, Eq. (24), can now be obtained in a manner similar to that described in Sect. 4, resulting in the following expression:

$$ \left\langle x_1^2\right\rangle= \frac{\pi S_0}{1+3 \delta \langle x^2 \rangle }\frac{1+3 \delta \langle x^2 \rangle +\alpha ^2+2\alpha \zeta }{2\zeta (1+3 \delta \langle x^2 \rangle +\alpha ^2 +2\alpha \zeta ) +\kappa ^2(\alpha +2\zeta )}, $$

(28a)

$$ \left\langle x_2^2\right\rangle= \pi S_0\frac{1+3 \delta \langle x^2 \rangle +\alpha ^2+2\alpha \zeta + \kappa ^2}{2\zeta (1+3 \delta \langle x^2 \rangle +\alpha ^2 +2\alpha \zeta ) +\kappa ^2(\alpha +2\zeta )}, $$

(28b)

$$\left\langle x_3^2\right\rangle= \pi S_0\frac{1}{2\zeta (1+3 \delta \langle x^2 \rangle +\alpha ^2+2\alpha \zeta )+\kappa ^2(\alpha +2\zeta )}.$$

(28c)

and the dimensionless average power, $\langle P \rangle =\alpha \langle x_3^2\rangle$. Since the coefficient $\omega ^2_e$ still depends on the unknown variance of the displacement of the original nonlinear system, it is still necessary to approximate $\langle x_1^2 \rangle$. To achieve this goal, two approaches are commonly adopted in the literature. In the first approach, $\langle x_1^2 \rangle$ is approximated using the variance of the linear system, in other words, using Eq. (15a). In the second approach, Eq. (28a) is solved for $\langle x_1^2 \rangle$ and then substituted into Eq. (28c). Naturally, the second approach is more accurate but involves the solution of a six order polynomial in $\langle x_1^2 \rangle$.

To investigate the accuracy of both approaches, variation of $\langle x_1^2 \rangle$ with the nonlinearity is compared to a numerical integration of the stochastic differential equations, Eq. (7), as depicted in Fig. 4. It is evident that using the approximate variance based on the linear system, Eq. (15a), to approximate $\langle x_1^2 \rangle$, yields results that significantly underestimates the numerical simulations especially for larger values of $\delta$. Thus, the exact variance as obtained by solving Eq.  (28a) is used in this work to approximate the electric quantity and average output power.

Fig. 4

Variation of $\langle x^2 \rangle$ with the nonlinearity, $\delta$, obtained for $\kappa =0.65$, $\alpha =0.5$, $\zeta =0.01$, and $S_0=0.04$. Asterisks represent solutions obtained via numerical integration

To investigate the optimal electric load embedded within the time constant ratio, $\alpha$, the equation $\frac{\partial \langle P \rangle }{\partial \alpha _{opt}}=0$ is solved for $\alpha _{opt}$ subject to the condition $\frac{\partial ^2 \langle P \rangle }{\partial \alpha ^2_{opt}}<0$. Results are shown in Fig. 5 illustrating that the optimal power always decreases with the nonlinearity regardless of the excitation spectral density. The optimal time constant ratio, on the other hand, increases with the magnitude of the nonlinearity.

Fig. 5

Variation of a optimal $\langle P \rangle$ and b optimal $\alpha$ with the nonlinearity, $\delta$ for different values of $S_0$. Results are obtained for $\kappa =0.65$ and $\zeta =0.01$

To put these non-dimensional quantities in a better perspective, we use the parameter values of the electromagnetic generator studied by Green et al. [23]. In their paper, the authors neglected the influence of the inductance on the power and found that the optimal load and power do not depend on the nonlinearity. This is the same conclusion we arrived at in Sect. 5.1. Upon dimensionalizing Eq. (18), we arrive at the same power and optimal load expressions presented in Ref. [23]. These are given by

$$ \langle P \rangle _{opt}= \frac{\pi S_0 \theta ^2 R_L}{2 \zeta \omega _n (R_L+R_c)^2+\frac{\theta ^2(R_L+R_c)}{m}},$$

(29)

$$ R_{l_{opt}}= \frac{\sqrt{(2 \zeta \omega _n R_c)^2+2\zeta \omega _n R_c \theta ^2}}{2 \zeta \omega _n m}. $$

(30)

Clearly, the above expressions are independent of the nonlinearity. However, as shown in Fig. 6, if the inductance of the coil is not sufficiently small, the optimal power and associated electric load are actually a function of the nonlinearity. When L is as small as 0.01 H, there is a clear dependence of the optimal power on both the inductance and the nonlinearity. Thus, by neglecting the inductance, the harvester operates away from its optimal conditions which reduces the average output power. It is worth noting that an inductance value as large as 0.08 H was reported in Ref. [28] for an actual electromagnetic energy harvester.

The preceding discussion clearly illustrates that the mono-stable Duffing harvester with a symmetric potential always produces lower average power than its linear counterpart with equivalent linear stiffness. Thus, even the unintentionally introduced hardening nonlinearities commonly seen in the first mode dynamics of beams, will inadvertently reduce the average output power of the harvester when operated in a white noise environment. Furthermore, it is shown that the optimal load has a clear dependence on the nonlinearity when L and $C_p$ are not sufficiently small. As such, extreme caution should be practiced before neglecting the inductance of the coil or the capacitance of the piezoelectric element while performing optimization analysis as this may yield suboptimal results.

Fig. 6

Variation of a optimal $\langle P \rangle$ and b optimal $R_L$ with the nonlinearity $\delta$ for $\omega _n=40$ rad/s, $\theta =0.2$ V$\cdot$ s/m, $m=0.02$ kg, $\zeta =0.05$, $S_0=1$ Watt/Hz and $R_c=0.5\,\Omega$

5.2.2 Asymmetric potential

In many cases, the potential energy function of nonlinear mono-stable harvesters is asymmetric due to asymmetries in the nonlinear restoring force. For instance, structural imperfections, initial curvature, and added masses produce a quadratic nonlinearity in beam-type harvesters. Additionally, in the process of intentionally introducing nonlinearities to the harvester’s through external design mechanisms, it is often difficult to create a perfectly symmetric restoring force.

Since the potential energy is no longer symmetric, harvesters with both quadratic and cubic nonlinearity do not necessarily have a zero mean value of displacement $\langle x\rangle \ne 0$, [29]. To account for this, we introduce the following transformation:

$$x(t) = x_0(t)+\langle x\rangle , $$

(31)

where $x_0(t)$ is the dynamics measured with respect to the mean of x(t). Next, to statistically linearize the asymmetric system, we seek a linear restoring force that best approximates the nonlinear one in the form

$$\frac{{\text {d}}U}{{\text {d}}x}=\omega ^2_{eq}x_0+b,$$

(32)

where b is introduced to account for a possible shift in the approximate linear force due to asymmetries in the original nonlinear restoring force. With that, the equivalent linear system can be written as

$$\ddot{x}_0+2\zeta \dot{x}_0 + \omega ^2_{eq}x_0 + b +\kappa ^2 y=-\ddot{x}_b, $$

(33a)

$$\begin{aligned}&\dot{y}+\alpha y=\dot{x}. \end{aligned}$$

(33b)

To obtain the unknown coefficient, $\omega ^2_{eq}$, we minimize the main square error, E, between the actual restoring force and its linear equivalent, i.e., we let

$$\begin{aligned} \frac{\partial {\langle E^2\rangle }}{\partial \omega ^2_{eq}}=0, \quad \frac{\partial {\langle E^2\rangle }}{\partial b}=0, \quad \frac{\partial ^2{\langle E^2\rangle }}{\partial (\omega ^2_{eq})^2}>0, \quad \frac{\partial ^2{\langle E^2\rangle }}{\partial b^2}>0. \end{aligned}$$

(34)

where $\langle E^2\rangle =\langle ((x+\lambda x^2+\delta x^3)-(\omega ^2_{eq}(x-\langle x \rangle )+b))^2\rangle$.

Once the candidate solutions of $\omega ^2_{eq}$ and b are found, they should also satisfy the following condition.

A function $T(\omega ^2_{eq},b)$ is defined such that

$$\begin{aligned} T(\omega ^2_{eq},b)=\frac{\partial ^2{\langle E^2\rangle }}{\partial (\omega ^2_{eq})^2}\frac{\partial ^2{\langle E^2\rangle }}{\partial b^2}-\left( \frac{\partial ^2{\langle E^2\rangle }}{\partial \omega ^2_{eq}\partial b}\right) ^2, \end{aligned}$$

(35)

Substitute the optimal $\omega ^2_{eq}$ and b into Eq. (35). If $T>0$, then it can generate the minimum value of $E^2$.

By taking the statistical average of both sides of Eqs. (33a) and (33b), it can be shown that $b=\langle -\ddot{x}_b \rangle = 0$, and $\langle y \rangle = 0$. Furthermore, by assuming that the response PDF follows a Gaussian process, the higher-order moments can be approximated via

$$\langle x^3 \rangle \approx 3 \langle x \rangle \langle x^2 \rangle - 2 \langle x \rangle ^3,$$

(36)

$$ \langle x^4 \rangle \approx 3 \langle x^2 \rangle ^2 - 2\langle x \rangle ^4.$$

(37)

Substituting Eqs. (36) and (37) into Eq. (34), and solving for $\omega ^2_{eq}$ and b yields

$${\omega }^2_{eq}= 3\delta \langle x^2 \rangle + 2 \lambda \langle x \rangle + 1, $$

(38)

$$ {b}= 3 \delta \langle x \rangle \langle x^2 \rangle - 2\delta \langle x \rangle ^3 + \lambda \langle x^2 \rangle + \langle x \rangle =0.$$

(39)

It is worth noting that, in the case of the asymmetric potential, $\omega ^2_{eq}$ depends on the nonzero mean value of x which can be obtained using Eq. (39).

Repeating similar steps to those described in Sect. 5.2.1, the important response statistics can be expressed as

$$ \left\langle x^2\right\rangle = \left\langle x_1^2\right\rangle =\frac{\pi S_0}{\omega _{eq}^2}\frac{\omega _{eq}^2+\alpha ^2+2\alpha \zeta }{2\zeta (\omega _{eq}^2 +\alpha ^2 +2\alpha \zeta ) +\kappa ^2(\alpha +2\zeta )}+ \langle x_1 \rangle ^2, $$

(40a)

$$ \left\langle \dot{x}^2\right\rangle= \left\langle x_2^2\right\rangle =\pi S_0\frac{\omega _{eq}^2+\alpha ^2+2\alpha \zeta + \kappa ^2}{2\zeta (\omega _{eq}^2 +\alpha ^2 +2\alpha \zeta ) + \kappa ^2(\alpha +2\zeta )}, $$

(40b)

$$ \left\langle y^2\right\rangle= \left\langle x_3^2\right\rangle =\pi S_0\frac{1}{2\zeta (\omega _{eq}^2+\alpha ^2+2\alpha \zeta )+\kappa ^2(\alpha +2\zeta )}. $$

(40c)

and the dimensionless average power is given by, $\langle P \rangle = \alpha \langle x_3^2 \rangle$.

Figure 7 depicts variation of average power with $\delta$ for different values of $\lambda$ as obtained via statistical linearization and long-time integration of the original equations of motion. Good qualitative agreement between the statistical linearization results and the numerical integration is observed further validating the approximate results. It is also evident that the average power increases as $\lambda$ is increased, i.e., as the restoring force becomes more asymmetric. The maximum value of the average power takes place at the extreme case of mono-stability where $\lambda = 2 \sqrt{\delta }$. Beyond this value, the potential function becomes bi-stable. Most importantly, it can also be seen that, the average power of the mono-stable VEH with asymmetric potential can be larger than the linear VEH with $\delta =0$ when $\lambda$ is sufficiently large. This indicates that the asymmetry in the restoring force can help improve the average power of nonlinear mono-stable VEHs under white noise.

Fig. 7

Variation of $\langle P \rangle$ with the nonlinearity $\delta$ obtained for $\alpha =0.5$, $\kappa =0.65$, $\zeta =0.01$ and $S_0=0.05$. The markers represent solutions obtained via numerical integration

To better understand the influence of the time constant ratio on the output power in the case of an asymmetric potential, we study variation of the mean square displacement and mean power with $\delta$ in the extreme case of $\lambda = 2\sqrt{\delta }$ for two different values of $\alpha$ as depicted in Figs. 8 and 9. It is clear that, for both values of $\alpha$, the average power exhibits a maximum at some $\delta$ value. Furthermore, variation of the average power follows similar trends as the mean square displacement and is more pronounced for smaller values of $\alpha$ corresponding to large values of L or $C_p$. This trend can be explained by inspecting Eq. (4) and noting that, when $\alpha$ is small, the electric quantity (voltage of current) becomes directly proportional to the displacement.

Fig. 8

Variation of a $\langle P \rangle$ and b $\langle x^2 \rangle$ with the non-linearity $\delta$ for $\lambda =2\sqrt{\delta }$, $\alpha =0.1$, $\kappa = 0.65$ and $\zeta = 0.01$. The markers represent the solutions obtained via numerical integration

Fig. 9

Variation of a $\langle P \rangle$ and b $\langle x^2 \rangle$ with the nonlinearity $\delta$ for $\lambda =2\sqrt{\delta }$, $\alpha =0.5$, $\kappa = 0.65$ and $\zeta = 0.01$. The markers represent the solutions obtained via numerical integration

Figures 8 and 9 also reveal that the average power is larger for $\alpha =0.5$ than it is for $\alpha =0.1$ indicating the presence of an optimal value. Figure 10a investigates how this optimal value varies with $\delta$ for different values of the noise’s spectral density. Results indicate that the optimal load decreases initially with $\delta$, exhibits a minimum value then increases again as $\delta$ in increased further. The minimum value of $\alpha _{opt}$ corresponds to the maximum value of $\langle P \rangle _{opt}$ as shown in Fig. 10b, c and d. When compared to the symmetric potential ($\lambda =0$) or the linear system ($\lambda =\delta =0$), it is evident that the asymmetric potential produces higher optimal average power levels for all values of noise’s spectral density.

Fig. 10

Variation of a $\alpha _{opt}$ and $\langle P \rangle _{opt}$ with the nonlinearity $\delta$ for b $S_0=0.01$, c $S_0=0.03$ and d $S_0=0.05$. Results are obtained for $\kappa =0.65$, $\zeta =0.01$ and $\lambda = 2\sqrt{\delta }$

6 Conclusion

This paper investigated electric load optimization of nonlinear mono-stable Duffing VEHs subjected to white noise excitations. Both symmetric and asymmetric nonlinear restoring forces were considered. It was clearly shown that, in both scenarios, the optimal load exhibits a clear dependence on the nonlinearity especially when the ratio between the period of the mechanical system and the time constant of the harvesting circuit is not large. Regardless of the magnitude of the nonlinearity or the spectral density of the excitation, mono-stable Duffing harvesters with a symmetric potential (restoring force) were shown to produce lower optimal average power levels as compared to a linear VEH with an equivalent linear stiffness. On the other hand, under optimal electric loading, VEHs with asymmetric restoring forces are shown to provide performance enhancement over a linear device or a nonlinear device with a symmetric potential. Finally, while our best approximation to many environmental excitations is white noise, some realistic excitation sources can also deviate from this assumption necessitating additional experimental investigations [30].