Dynamics of a delayed Duffing-type energy harvester under narrow-band random excitation

Abstract

In this paper, a Duffing-type energy harvester with time delay circuit under narrow-band random excitation is studied by using the method of multiple scales. The analytical expressions of the steady-state responses near the principal resonance and their stable conditions are derived. Results show that the phenomenon of stochastic jump is found, and the operational bandwidth of the nonlinear energy harvester can be extended. The effects of frequency detuning, noise intensity, time delay, piezoelectric coupling, and time constant ratio on the dynamical behaviors are discussed. From the viewpoint of improving system performance and optimizing harvester design, a proper choose of time delay and feedback gain is proposed by the power conversion efficiency and the RMS voltage. It indicates that the negative feedback gain is favorable for an energy harvester to provide more electrical output power and miniaturize design. Moreover, the Monte Carlo simulations are given to verify the validity of the theoretical method.

1 Introduction

Over the past decades, there has been an increasing amount of studies on vibrational energy harvesters (VEHs) to provide a continuous scalable energy generator for the low-power consumption and wireless electronics [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]. In general, the VEH is an electromechanical coupled device which can transform the ambient vibration into useful electricity. The VEH includes linear and nonlinear mechanical systems. Compared with the linear VEH, the nonlinear one can extend the bandwidth and harvest energy from ambient vibration with a wider spectrum. Thus, different types of the nonlinear VEH, such as nonlinear mono-stable [5, 6], bi-stable [7, 8] and multi-stable systems [9, 10] are designed to improve the harvesting performance. The common approaches to design nonlinear VEH are introducing a nonlinear restoring force [11] and matching the fundamental frequency by considering resonant behaviors [12, 13]. Meanwhile, most environmental vibrations have random characteristics, and the stochastic dynamical behaviors of the nonlinear VEH have been investigated [16,17,18,19,20,21,22,23,24,25]. For VEH under Gaussian white noise, many analytical methods have been applied to obtain the approximate solution, such as equivalent linearization technique [16], equivalent nonlinearization method [17], Fokker–Plank–Kolmogorov (FPK) equation method [18] and the stochastic averaging [19]. If the correlation time of the random excitation is large, the colored noise is chosen to describe the random process. Then, the stochastic averaging and its improved technique have been proved to be a better solving method [20,21,22,23,24,25]. For example, Bobryk et al. [21] considered a VEH system and found that colored noise has a significant effect on the enhancement of energy harvesting. Liu et al. [22] proposed a new quasi-conservative stochastic averaging method and analyzed the mono-stable response of nonlinear VEH system driven by colored noise. Zhang et al. [23,24,25] presented the improved stochastic averaging procedure based on energy-dependent frequency to deal with bi-stable and tri-stable systems. Based on the mechanism of parameter-optimized stochastic resonance, the appropriate choice of the system parameters is provided to enhance mean output power and power conversion efficiency greatly.

Time delays commonly exist in the feedback of controlled mechanical or electrical systems. Previous studies demonstrated that the time delays can be utilized to improve the control performance of dynamic systems. The time-delayed feedback control (TDFC) can be applied to stabilize strongly unstable orbits [26], reduce vibration [27, 28], and control bifurcation [29]. Therefore, the TDFC has been introduced to improve the efficiency of piezoelectric VEH. For example, Park et al. [30] observed that the power produced by the vibrations of piezoelectric devices is low and the equipment powered by the VEH is hard to operate effectively. To overcome this difficulty, they used a time delay circuit and reset IC chip to improve the efficiency of the electrical power generated. Belhaq et al. [31, 32] studied the periodic and the quasi-periodic responses in the delayed nonlinear VEH and explored the effects of time delay on the system performance. Alhazza et al. [33] presented a single-input and single-output multi-mode delayed feedback control to suppress the free vibrations of a flexible cantilever beam. They analyzed the effect of size and location of the piezoelectric patch and the location of the sensor on the stability of the response. Yang et al. [34] used the TDFC to enhance the stochastic resonance effect and output power in a novel hybrid energy harvester. Guo et al. [35] and Zhang et al. [36] used the TDFC to improve the harvesting performance of the multiple attractors wind-induced and bi-stable VEH systems under Gaussian white noise, respectively. To the best knowledge of the authors, no attention has been paid to the dynamics of the delayed nonlinear VEH under narrow-band random excitation.

The aim of this study is to reveal the dynamical behaviors and enhance efficiency of the output power in a Duffing-type VEH with time delay circuit, which is subjected to a narrow-band random excitation. The rest of the paper is organized as follows: In Sect. 2, the steady-state responses near the principal resonance and their stabilities of a delayed electromechanical VEH, driven by a narrow-band random excitation, are presented by using the method of multiple scales. The influences of the frequency detuning, the time delay, the feedback gain and coupling coefficient on the second-order moment of amplitude and the mean output power are discussed. In Sect. 3, the appropriate choice of the time delay and the feedback gain is proposed in order to miniaturize the device and enhance the harvesting performance. The output RMS voltage and the power conversion efficiency under the proper feedback parameters are computed to evaluate the system performance. Meanwhile, Monte Carlo simulations (MCS) results are given to verify the validity of the theoretical method. Finally, some specific conclusions are drawn in Sect. 4.

2 Stochastic response of a delayed Duffing-type VEH

The model of an electromechanically coupled VEH is given in Fig. 1, which consists of a mechanical oscillator coupled to an electrical circuit by a delayed piezoelectric device. The electromechanically coupled VEH under the narrow-band random excitation can be expressed as the following general form:

$$M\ddot{\bar{X}}(t) + \bar{C}\dot{\bar{X}}(t) + {{{\text{d}}\bar{U}(\bar{X})} \mathord{\left/ {\vphantom {{{\text{d}}\bar{U}(\bar{X})} {d\bar{X}}}} \right. \kern-0pt} {d\bar{X}}} - \bar{\chi }\bar{V}(t) = - M\ddot{\bar{X}}_{\text{b}} ,$$

(1.1)

$$C_{\text{p}} \dot{\bar{V}}(t) + {{\bar{V}(t)} \mathord{\left/ {\vphantom {{\bar{V}(t)} R}} \right. \kern-0pt} R} + \bar{\kappa }\dot{\bar{X}}(t) = \bar{\beta }\bar{V}(t - \tau )$$

(1.2)

where $\bar{X}$ denotes the displacement of the mass M, $\dot{\bar{X}}$ denotes the velocity of the mass M, $\bar{X}_{\text{b}}$ denotes the base acceleration, and $\bar{C}$ denotes the linear viscous damping coefficient. The potential function $\bar{U}(\bar{X}) = {{k_{1} \bar{X}^{2} } \mathord{\left/ {\vphantom {{k_{1} \bar{X}^{2} } 2}} \right. \kern-0pt} 2} + {{k_{2} \bar{X}^{4} } \mathord{\left/ {\vphantom {{k_{2} \bar{X}^{4} } 4}} \right. \kern-0pt} 4}$, $\bar{\chi }$ represents the linear electromechanical coupling coefficients, C_p represents the effective capacitance of the piezoelectric element, $\bar{V}$ represents the inductive voltage, and $R$ denotes the equivalent resistance load, i.e., $R = R_{\text{l}} R_{\text{p}} /(R_{\text{l}} + R_{\text{p}} )$, in which R_l is the load resistance and R_p is the piezoelectric resistance. $\bar{\kappa }$ is the piezoelectric coupling term in the electrical circuit, $\bar{\beta }$ and $\tau$ are the feedback gain and the time delay in the electrical circuit, respectively.

Fig. 1

a Schematic diagram of an electromechanical VEH coupled with b a delayed piezoelectric mechanism

The following transformations are introduced to make Eq. (1) dimensionless, i.e.,

$$\begin{aligned} & x = \frac{{\bar{X}}}{l},\,\,x_{b} = \frac{{\bar{X}_{b} }}{l},\,\,V = \frac{{C_{p} }}{{\bar{\chi }l}}\bar{V},\,\,\omega_{n} = \sqrt {\frac{{k_{1} }}{M}},\,\, C = \frac{{\bar{C}}}{M}, \\ &\hat{\alpha } = \frac{{k_{2} l^{2} }}{M},\,\,\hat{\chi } = \frac{{\bar{\chi }^{2} l}}{{MC_{p} }},\,\,\lambda = \frac{1}{{RC_{p} }},\,\,\kappa = \frac{{\bar{\kappa }}}{{\bar{\chi }}},\,\,\beta = \frac{{\bar{\beta }}}{{C_{p} }}. \\ \end{aligned}$$

Then, the dimensionless coupled VEH under the narrow-band random excitation can be expressed as

$$\ddot{x}(t) + C\dot{x}(t) + {{\text{d} U(x)} \mathord{\left/ {\vphantom {{\text{d} U(x)} {\text{d} x}}} \right. \kern-0pt} {\text{d} x}} - \hat{\chi }V(t) = - \ddot{x}_{b} ,$$

(2.1)

$$\dot{V}(t) + \lambda V(t) + \kappa \dot{x}(t) = \beta V(t - \tau ).$$

(2.2)

where $x(t)$ denotes the dimensionless displacement of the mass $M$, $V(t)$ is the dimensionless voltage across the load resistance, the dimensionless linear viscous damping coefficient $C = 2\hat{\mu }\omega_{n},$ and the dimensionless potential function $U(x) = {{\omega_{n}^{2} x^{2} } \mathord{\left/ {\vphantom {{\omega_{n}^{2} x^{2} } 2}} \right. \kern-0pt} 2} + {{\hat{\alpha }x^{4} } \mathord{\left/ {\vphantom {{\hat{\alpha }x^{4} } 4}} \right. \kern-0pt} 4}$. $\hat{\mu }$ and $\hat{\alpha }$ denote the dimensionless damping and stiffness coefficients, respectively, $\omega_{n}$ is the natural frequency of system, $l$ in the transformations denotes the length scale, $\hat{\chi }$ denotes the dimensionless electromechanical coupling coefficient, $\lambda$ denotes the dimensionless time constant ratio, $\kappa$ is the dimensionless piezoelectric coupling term in the electrical circuit, and $\beta$ and $\tau$ are the dimensionless feedback gain and the time delay in the electrical circuit, respectively.

The base acceleration $- \ddot{x}_{b}$ in Eq. (2.1) is considered as the ambient noise $\xi (t)$ which describes as a narrow-band random process $\xi (t) = \hat{F}\cos (\varOmega t + \gamma W(t))$ [37]. Here the standard Wiener process $W(t)$ is the diffusion process with a null drift coefficient and a unit diffusion coefficient, $\hat{F}$ is a deterministic amplitude, $\varOmega$ is the constant rotation speed, and $\gamma$ is the intensity of Gaussian white noise $\dot{W}(t)$. The power spectrum $S(\omega )$ of $\xi (t)$ is given as [38]

$$S(\omega ) = \frac{1}{2}\frac{{\hat{F}^{2} \bar{\gamma }^{2} (\varOmega^{2} + \omega^{2} + {{\bar{\gamma }^{4} } \mathord{\left/ {\vphantom {{\bar{\gamma }^{4} } 4}} \right. \kern-0pt} 4})}}{{(\varOmega^{2} - \omega^{2} + {{\bar{\gamma }^{4} } \mathord{\left/ {\vphantom {{\bar{\gamma }^{4} } 4}} \right. \kern-0pt} 4})^{2} + \omega^{2} \bar{\gamma }^{4} }} .$$

(3)

From Eq. (3), the variances of $S(\omega )$ with the noise intensity $\bar{\gamma }$ are presented in Fig. 2. As the intensity $\bar{\gamma }$ increases, the bandwidth of the peak increases, too. When the limiting procedure $\bar{\gamma } \to \infty$, Eq. (3) tends to the uniformly distributed power spectrum of white noise. When $\bar{\gamma } \to 0$, $S(\omega )$ vanishes in the entire frequency range except at the singular frequency $\omega = \pm \varOmega$, where $S( \pm \varOmega ) \to \infty$. This is a typical spectrum of the narrow-band noise.

Fig. 2

The power spectrum $S(\omega )$ of random excitation $\xi (t)$ for fixed $\hat{F}{ = 0} . 2 5$, $\varOmega = 1.0$

In order to apply the method of multiple scales, we introduce a small parameter $\varepsilon$ to obtain the nondimensional parameters: $\hat{\mu } = \varepsilon \mu$, $\hat{\alpha } = \varepsilon \alpha$, $\hat{\chi } = \varepsilon \chi$, $\hat{F} = \varepsilon F$. Equation (2) can be rewritten in the following nondimensional form as

$$\ddot{x}(t) + 2\varepsilon \mu \omega_{n} \dot{x}(t) + \omega_{n}^{2} x(t) + \varepsilon \alpha x^{3} (t) - \varepsilon \chi V(t) = \varepsilon F\cos (\varOmega t + \gamma W(t)),$$

(4.1)

$$\dot{V}(t) + \lambda V(t) + \kappa \dot{x}(t) = \beta V(t - \tau ).$$

(4.2)

By using the method of multiple scales, the solutions of Eq. (4) can be assumed in the following forms:

$$x(t) = x_{0} (T_{0} ,T_{1} ) + \varepsilon x_{1} (T_{0} ,T_{1} ) + O(\varepsilon^{2} ),$$

(5.1)

$$V\left( t \right) = V_{0} \left( {T_{0} ,T_{1} } \right) + \varepsilon V_{1} \left( {T_{0} ,T_{1} } \right) + O\left( {\varepsilon^{2} } \right),$$

(5.2)

where $T_{0} = t$, $T_{1} = \varepsilon t$. Introducing the differential operators $D_{0} = {\partial \mathord{\left/ {\vphantom {\partial {\partial T_{0} }}} \right. \kern-0pt} {\partial T_{0} }}$ and $D_{1} = {\partial \mathord{\left/ {\vphantom {\partial {\partial T_{1} }}} \right. \kern-0pt} {\partial T_{1} }}$, one finds

$$\left\{ {\begin{array}{*{20}c} {\frac{\text{d} }{{\text{d} t}} = D_{0} + \varepsilon D_{1} + O(\varepsilon^{2} ),\;\quad \;} \\ {\frac{{\text{d}^{2} }}{{\text{d} t^{2} }} = D_{0}^{2} + 2\varepsilon D_{0} D_{1} + O(\varepsilon^{2} ).} \\ \end{array} } \right.$$

(6)

Substituting Eqs. (5) and (6) into Eq. (4) and equating the same power of $\varepsilon$, one obtains the equations at the first order as

$$\left\{ {\begin{array}{*{20}c} {D_{0}^{2} x_{0} (T_{0} ,T_{1} ) + \omega_{n}^{2} x_{0} (T_{0} ,T_{1} ) = 0,\quad \quad \quad \quad \quad \quad \;\quad \quad \quad \quad } \\ {D_{0}^{{}} V_{0} (T_{0} ,T_{1} ) + \lambda V_{0} (T_{0} ,T_{1} ) + \kappa D_{0} x_{0} (T_{0} ,T_{1} ) = \beta V_{0} (T_{0} - \tau ,T_{1} )}, \\ \end{array} } \right.$$

(7)

and at the second order as

$$\left\{ {\begin{array}{*{20}l} {D_{0}^{2} x_{1} (T_{0} ,T_{1} ) + \omega_{n}^{2} x_{1} (T_{0} ,T_{1} ) = - 2D_{0} D_{1} x_{0} (T_{0} ,T_{1} ) - 2\mu \omega_{n} D_{0} x_{0} (T_{0} ,T_{1} )}\hfill \\ { - \alpha x_{0}^{3} (T_{0} ,T_{1} ) + \chi V_{0} (T_{0} ,T_{1} ) + F\cos (\varOmega T_{0} + \gamma W(T_{1} )), }\hfill \\ {D_{0} V_{1} (T_{0} ,T_{1} ) + \lambda V_{1} (T_{0} ,T_{1} ) = - D_{1} V_{0} (T_{0} ,T_{1} ) - kD_{0} x_{0} (T_{0} ,T_{1} ) + \beta V_{1} (T_{0} - \tau ,T_{1} ).} \hfill \\ \end{array} } \right.$$

(8)

Then, the general solutions of Eqs. (7) and (8) read

$$ {x_{0} (T_{0} ,T_{1} ) = A_{1} (T_{1} )\exp (\text{i} \omega_{n} T_{0} ) + \text{cc} ,} $$

(9.1)

$$ {V_{0} (T_{0} ,T_{1} ) = B_{1} (T_{1} )\exp (\text{i} \omega_{n} T_{0} ) + \text{cc}}$$

(9.2)

where $\text{cc}$ denotes the conjugate term. Substituting Eq. (9.2) into Eq. (7) one finds

$$B_{1} \cdot \text{i} \omega_{n} + \lambda B_{1} + A_{1} \cdot \text{i} \omega_{n} \kappa = \beta B_{1} \cdot \exp ( - \text{i} \omega_{n} \tau ).$$

(10)

From Eq. (10), the amplitude $B_{1} (T_{1} )$ can be expressed as

$$B_{1} (T_{1} ) = \frac{{ - \text{i} \omega_{n} \kappa }}{{\text{i} \omega_{n} + \lambda - \beta \exp ( - \text{i} \omega_{n} \tau )}}A_{1} (T_{1} ),$$

(11)

where $A_{1} (T_{1} )$ is the slowly varying amplitude of the solution.

Substituting Eq. (9.1) into Eq. (8) one arrives at

$$\begin{aligned} D_{0}^{2} x_{1} + \omega_{n}^{2} x_{1} & = \left[ { - 2\text{i} \omega_{n} A_{1}^{{^{\prime } }} - 2\text{i} \omega_{n}^{2} \mu A_{1} - 3\alpha \bar{A}_{1} A_{1}^{2} } \right]\exp (\text{i} \omega_{n} T_{0} ) \\ & \quad - \frac{{\text{i} \omega_{n} \kappa \chi A_{1} }}{{\text{i} \omega_{n} + \lambda - \beta \exp ( - \text{i} \omega_{n} \tau )}}\exp (\text{i} \omega_{n} T_{0} ) - \alpha A_{1}^{3} \exp (3\text{i} \omega_{n} T_{0} ) \\ & \quad + \frac{F}{2}\exp [\text{i} (\varOmega T_{0} + \gamma W(T_{1} ))] + \text{cc} , \\ \end{aligned}$$

(12)

where the prime represents the derivative with respect to $T_{1}$. The elimination of the secular term in Eq. (12) requires

$$- 2\text{i} \omega_{n} A_{1}^{{^{\prime } }} - 2\text{i} \omega_{n}^{2} \mu A_{1} - 3\alpha \bar{A}_{1} A_{1}^{2} - \frac{{\text{i} \omega_{n} \kappa \chi A_{1} }}{{\text{i} \omega_{n} + \lambda - \beta \exp ( - \text{i} \omega_{n} \tau )}} + \frac{F}{2}\exp [\text{i} (\varOmega T_{0} - \omega_{n} T_{0} + \gamma W(T_{1} ))] = 0.$$

(13)

2.1 Principal resonance

In order to investigate the principal resonance of Eq. (4), one can introduce the detuning frequency $\sigma$ to express the principal resonance as

$$\varOmega = \omega_{n} + \varepsilon \sigma .$$

(14)

From Eq. (14), one can rewrite Eq. (13) as follows:

$$- 2\text{i} \omega_{n} A_{1}^{{^{\prime } }} - 2\text{i} \omega_{n}^{2} \mu A_{1} - 3\alpha \bar{A}_{1} A_{1}^{2} - \frac{{\text{i} \omega_{n} \kappa \chi A_{1} }}{{\text{i} \omega_{n} + \lambda - \beta \exp ( - \text{i} \omega_{n} \tau )}} + \frac{F}{2}\exp [\text{i} (\sigma T_{1} + \gamma W(T_{1} ))] = 0.$$

(15)

To simplify Eq. (15), it is natural to assume

$$A_{1} (T_{1} ) = \frac{{a(T_{1} )}}{2}\exp (\text{i} \varphi (T_{1} )) .$$

(16)

Substituting Eq. (16) into Eq. (15) and separating the real and imaginary parts of Eq. (15), one obtains

$$\left\{ {\begin{array}{*{20}l} {a^{{^{\prime } }} = - \mu_{1} a + \frac{F}{{2\omega_{n} }}\sin \eta ,} \hfill \\ {a\eta^{{^{\prime } }} = \sigma_{1} a - \frac{{3\alpha a^{3} }}{{8\omega_{n} }} + \frac{F}{{2\omega_{n} }}\cos \eta + \gamma aW^{\prime } (T_{1} ),} \hfill \\ \end{array} } \right. $$

(17)

where $\eta (T_{1} ) = \sigma T_{1} + \gamma W(T_{1} ) - \varphi (T_{1} )$, $\mu_{1} = \mu \omega_{n} + {{\chi \kappa (\lambda - \beta \cos \omega_{n} \tau )} \mathord{\left/ {\vphantom {{\chi \kappa (\lambda - \beta \cos \omega_{n} \tau )} H}} \right. \kern-0pt} H}$, $\sigma_{1} = \sigma - {{\chi \kappa (\omega_{n} + \beta \sin \omega_{n} \tau )} \mathord{\left/ {\vphantom {{\chi \kappa (\omega_{n} + \beta \sin \omega_{n} \tau )} H}} \right. \kern-0pt} H}$, $H = 2\left[ {(\omega_{n} + \beta \sin \omega_{n} \tau )^{2} + (\lambda - \beta \cos \omega_{n} \tau )^{2} } \right]$.

Solving Eq. (17) for the amplitude $a$ and the phase $\eta$, one gets the first-order uniform expansion of the solution of Eq. (3) as follows:

$$x(t) = a(\varepsilon t)\cos (\varOmega t - \eta (\varepsilon t)) + O(\varepsilon ) .$$

(18)

According to Eqs. (11) and (18), the first-order uniform expansion of the solution of Eq. (4.2) is given as follows:

$$V(t) = \frac{{\omega_{n} \kappa }}{{\sqrt {{H \mathord{\left/ {\vphantom {H 2}} \right. \kern-0pt} 2}} }}a(\varepsilon t)\cos (\varOmega t - \eta (\varepsilon t) + \theta ) + O(\varepsilon ) ,$$

(19)

where $\theta = \arctan [{{(\lambda - \beta \cos \omega_{n} \tau )} \mathord{\left/ {\vphantom {{(\lambda - \beta \cos \omega_{n} \tau )} {(\omega_{n} + \beta \sin \omega_{n} \tau )}}} \right. \kern-0pt} {(\omega_{n} + \beta \sin \omega_{n} \tau )}}],\;(\omega_{n} + \beta \sin \omega_{n} \tau \ne 0)$.

2.2 The steady-state solution and its stability

From Eq. (17), the it $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{o}$ stochastic differential equation (SDE) of the steady-state solution is written as

$$\left\{ {\begin{array}{*{20}c} {\text{d} a = \left[ { - \mu_{1} a + \frac{F}{{2\omega_{n} }}\sin \eta } \right]\text{d} T_{1} ,\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {} & {} \\ \end{array} } & {} \\ \end{array} } & {} & {} & {} \\ \end{array} } \\ {\text{d} \eta = \left[ {\sigma_{1} - \frac{{3\alpha a^{2} }}{{8\omega_{n} }} + \frac{F}{{2a\omega_{n} }}\cos \eta } \right]\text{d} T_{1} + \gamma \text{d} W(T_{1} ).} \\ \end{array} } \right.$$

(20)

When $\gamma = 0$, Eq. (20) turns to the case without random excitation. Then, the steady-state solution $a = a_{0}$ and $\eta = \eta_{0}$ can be found by assuming conditions $a^{\prime} = 0$, $\eta^{\prime} = 0$ [32],

$$\left\{ {\begin{array}{*{20}l} {\mu_{1} a_{0} = \frac{F}{{2\omega_{n} }}\sin \eta_{0} ,} \hfill \\ {\sigma_{1} a_{0} - \frac{{3\alpha a_{0}^{3} }}{{8\omega_{n} }} = - \frac{F}{{2\omega_{n} }}\cos \eta_{0} .} \hfill \\ \end{array} } \right.$$

(21)

When $\gamma \ne 0$, the nontrivial solution of Eq. (20) is assumed in the following form:

$$a = a_{0} + a_{1} ,\,\,\eta = \eta_{0} + \eta_{1} ,$$

(22)

where $a_{0}$, $\eta_{0}$ are satisfying Eq. (21), and $a_{1}$, $\eta_{1}$ are the perturbation terms. Substituting Eq. (22) into Eq. (20), one reaches the linearization of Eq. (20) at $(a_{0} ,\eta_{0} ),$

$$\left\{ {\begin{array}{*{20}l} {a_{1}^{\prime } = - \mu_{1} a_{1} + \frac{F}{{2\omega_{n} }}\cos \eta_{0} \cdot \eta_{1} ,} \hfill \\ {\eta_{1}^{\prime } = \left( {\frac{{\sigma_{1} }}{{a_{0} }} - \frac{{9\alpha a_{0} }}{{8\omega_{n} }}} \right)a_{1} - \frac{F}{{2a_{0} \omega_{n} }}\sin \eta_{0} \cdot \eta_{1} + \gamma W^{\prime } (T_{1} ),} \hfill \\ \end{array} } \right.$$

(23)

According to Eqs. (21), Eqs. (23) can be rewritten as follows:

$$\left\{ {\begin{array}{*{20}l} {a_{1}^{\prime } = - \mu_{1} a_{1} - M_{1} \eta_{1} ,} \hfill \\ {\eta_{1}^{\prime } = M_{2} a_{1} - \mu_{1} \eta_{1} + \gamma W^{\prime } (T_{1} ),} \end{array} } \right.$$

(24)

where $M_{1} = \sigma_{1} a_{0} - {{3\alpha a_{0}^{3} } \mathord{\left/ {\vphantom {{3\alpha a_{0}^{3} } {8\omega_{n} }}} \right. \kern-0pt} {8\omega_{n} }}$, $M_{2} = {{\sigma_{1} } \mathord{\left/ {\vphantom {{\sigma_{1} } {a_{0} }}} \right. \kern-0pt} {a_{0} }} - {{9\alpha a_{0}^{{}} } \mathord{\left/ {\vphantom {{9\alpha a_{0}^{{}} } {8\omega_{n} }}} \right. \kern-0pt} {8\omega_{n} }}$.

The second-order moments of Eq. (23) are satisfying the following equalities:

$$\left\{ {\begin{array}{*{20}l} {\frac{{{\text{d}}Ea_{1}^{2} }}{{{\text{d}}T_{1} }} = - 2\mu_{1} Ea_{1}^{2} - 2M_{1} Ea_{1} \eta_{1} ,} \hfill \\ {\frac{{{\text{d}}Ea_{1} \eta_{1} }}{{{\text{d}}T_{1} }} = M_{2} Ea_{1}^{2} - M_{1} E\eta_{1}^{2} - 2\mu_{1} Ea_{1} \eta_{1} ,} \hfill \\ {\frac{{{\text{d}}E\eta_{1}^{2} }}{{{\text{d}}T_{1} }} = 2M_{2} Ea_{1} \eta_{1} - 2\mu_{1} E\eta_{1}^{2} + \gamma^{2} ,} \hfill \\ \end{array} } \right.$$

(25)

where $\text{E} [ \cdot ]$ denotes the mathematical expectation.

Let ${{\text{dE} a_{1}^{2} } \mathord{\left/ {\vphantom {{\text{dE} a_{1}^{2} } {\text{d} T_{1} }}} \right. \kern-0pt} {\text{d} T_{1} }} = {{\text{dE} a_{1}^{{}} \eta_{1} } \mathord{\left/ {\vphantom {{\text{dE} a_{1}^{{}} \eta_{1} } {\text{d} T_{1} }}} \right. \kern-0pt} {\text{d} T_{1} }} = {{\text{dE} \eta_{1}^{2} } \mathord{\left/ {\vphantom {{\text{dE} \eta_{1}^{2} } {\text{d} T_{1} }}} \right. \kern-0pt} {\text{d} T_{1} }} = 0$, then one can get the second-order steady-state moments by solving Eqs. (25),

$$\left\{ {\begin{array}{*{20}c} {\text{E} a_{1}^{2} = \frac{{M_{1}^{2} \gamma^{2} }}{{4\mu_{1} (\mu_{1}^{2} + M_{1} M_{2} )}},} \\ {\text{E} a_{1} \eta_{1} = \frac{{ - M_{1} \gamma^{2} }}{{4(\mu_{1}^{2} + M_{1} M_{2} )}},} \\ {\text{E} \eta_{1}^{2} = \frac{{(2\mu_{1}^{2} + M_{1} M_{2} )\gamma^{2} }}{{4\mu_{1} (\mu_{1}^{2} + M_{1} M_{2} )}}.} \\ \end{array} } \right.$$

(26)

Taking the expectation on both sides of Eq. (22), one obtains the first-order and the second-order steady-state moments of the solution,

$$\left\{ {\begin{array}{*{20}c} {\text{E} a = a_{0} ,\text{E} a^{2} = a_{0}^{2} + \text{E} a_{1}^{2} .} \\ {\text{E} \eta = \eta_{0} ,\text{E} \eta^{2} = \eta_{0}^{2} + \text{E} \eta_{1}^{2} .} \\ \end{array} } \right.$$

(27)

Similarly, the first-order and the second-order steady-state moments of the output voltage are given as follows:

$$\text{E} V = \frac{{\omega_{n} \kappa }}{{\sqrt {{H \mathord{\left/ {\vphantom {H 2}} \right. \kern-0pt} 2}} }}a_{0} ,\quad \text{E} V^{2} = \frac{{2\omega_{n}^{2} \kappa^{2} }}{H}(a_{0}^{2} + \text{E} a_{1}^{2} ).$$

(28)

The mean output power is obtained as

$$\text{E} P = \lambda \chi \text{E} V^{2} .$$

(29)

To guarantee the existence of second-order steady-state moments (26), the necessary conditions should be satisfied as $Ea_{1}^{2} > 0$ and $E\eta_{1}^{2} > 0$. That is,

$$\mu_{1} > 0,\quad \mu_{1}^{2} + M_{1} M_{2} > 0.$$

(30)

Based on the Routh–Hurwitz criterion, the second-order steady-state moments (26) are asymptotically stable if and only if the following conditions can be satisfied:

$$\mu_{1} > 0,\quad \mu_{1}^{2} + {{ 2M_{1} M_{2} } \mathord{\left/ {\vphantom {{ 2M_{1} M_{2} } 9}} \right. \kern-0pt} 9} > 0.$$

(31)

Therefore, Eqs. (30) and (31) give the stability condition for the second-order steady-state moments of the system solution. From Eqs. (21), (26), and (27)–(29), the effects of first-order and second-order steady-state moments of system (4) are analyzed by choosing parameters $\chi = 1. 0$, $\mu = 1. 0$, $\omega_{n} = 1. 0$, $f = 2.5$, $\varepsilon = 0.1$. In Fig. 3, the first-order moments of the amplitude $Ea$ and the output voltage $EV$ versus detuning frequency $\sigma$ are plotted near the principal resonance. With the increase of the nonlinear stiffness, both $Ea$ and $EV$ show the right-bended resonance curves. That is, the operational bandwidth of VEH is extended by considering the stiffness nonlinearity. Figure 4 shows the curves of the second-order moments $Ea^{2}$, $EV^{2}$ and the mean output power $EP$ as functions of $\sigma$. It is seen that there are three branches and stochastic jump phenomena in the steady-state response [39]. According to Eqs. (30) and (31), we know that the top and the bottom ones are stable high-energy and low-energy branches, respectively, while the middle one is an unstable branch. When the noise intensity $\gamma$ is small enough, the noise is not able to change the stability of these branches.

Fig. 3

The first-order steady-state moments of the amplitude $Ea$ and the voltage $EV$ as functions of the detuning frequency $\sigma$ with different nonlinear stiffnesses $\alpha$ ($\kappa = 0. 5$, $\beta = 0. 5$, $\tau = 0.5$, $\lambda = 0.5$)

Fig. 4

The second-order steady-state moments of system $Ea^{2}$, $EV^{2}$ and mean output power $EP$ as functions of the detuning frequency $\sigma$ ($\kappa = 0. 5$, $\beta = 0. 5$, $\tau = 0.5$, $\lambda = 0.5$, $\alpha = 5. 0$, $\gamma = 0.1$)

To check the stability of the second-order moments $Ea^{2}$ and $EV^{2}$, conditions (30) and (31) are plotted for fixed $\sigma = 2. 0$ in Fig. 5. When the amplitude $a_{0}$ is chosen as the top or the bottom branches of the solution, the expressions $\mu_{1}^{2} + M_{1} M_{2}$ and $\mu_{1}^{2} + {{ 2M_{1} M_{2} } \mathord{\left/ {\vphantom {{ 2M_{1} M_{2} } 9}} \right. \kern-0pt} 9}$ are marked with dash and dot lines, respectively. Obviously, conditions (30) and (31) are satisfied; then, the top and the bottom branches in Fig. 4 are stable. If $a_{0}$ is chosen as the middle branch, the expressions $\mu_{1}^{2} + M_{1} M_{2}$ and $\mu_{1}^{2} + {{ 2M_{1} M_{2} } \mathord{\left/ {\vphantom {{ 2M_{1} M_{2} } 9}} \right. \kern-0pt} 9}$ are marked with a solid line. From Fig. 5a, it is found that $\mu_{1}^{2} + M_{1} M_{2} < 0$ and the stability condition (30) is unsatisfied. It is proven that the middle one shown in Fig. 4 is unstable.

Fig. 5

Stability conditions (30) and (31) for the second-order steady-state moments of system ($\kappa = 0. 5$, $\beta = 0. 5$, $\tau = 0.5$, $\lambda = 0.5$, $\alpha = 5. 0$, $\sigma = 2. 0$, $\gamma = 0.1$). The curves correspond to the top branch of the amplitude (dash line) and the bottom ones (dot line), the third branch is between them (solid line)

The second-order moments of amplitude $\text{E} a^{2}$ and the mean output power $\text{E} P$ are important for miniaturization of the device and enhancement of energy harvesting. In Figs. 6, 7, and 8, the effects of the system parameters on the harvesting performances are explored based on the above theoretical results (27) and (29). Figure 6 shows the curves of the $\text{E} a^{2}$ and the $\text{E} P$ versus the detuning frequency $\sigma$ for different feedback gains $\beta$ and fixed $\tau = 0.5$. On the one hand, the peak value of $\text{E} a^{2}$ at principal resonance increases with the increase of $\beta$. On the other hand, the peak value of $\text{E} P$ for $\beta { = 0}$ is the largest one among the three cases. In other words, for fixed time delay $\tau = 0.5$, the consideration of the time-delayed feedback in the circuit decreases the mean output power of VEH.

Fig. 6

The second-order steady-state moments of the amplitude $Ea^{2}$ and the mean output power $EP$ as functions of the detuning frequency $\sigma$ with different feedback gains $\beta$ ($\kappa = 0. 5$, $\tau = 0.5$, $\lambda = 0.5$, $\alpha = 5. 0$, $\gamma = 0.1$)

Fig. 7

The second-order steady-state moments of the amplitude $Ea^{2}$ and the mean output power $EP$ as functions of the feedback gain $\beta$ with different time delays $\tau$ ($\kappa = 0. 5$, $\sigma = 2. 0$, $\lambda = 0.5$, $\alpha = 5. 0$, $\gamma = 0.1$)

Fig. 8

The second-order steady-state moments of the amplitude $Ea^{2}$ and the mean output power $EP$ as functions of the piezoelectric coupling in the circuit $\kappa$ with different time constant ratios $\lambda$ ($\beta = 0. 5$,$\sigma = 2. 0$, $\tau = 0.5$, $\alpha = 5. 0$, $\gamma = 0.1$)

To better understand the influences of the time-delayed feedback term, the variances of $\text{E} a^{2}$ and $\text{E} P$ versus $\beta$ are plotted for different time delays in Fig. 7. From Fig. 7a, there is an optimal $\beta$ where the value of $\text{E} a^{2}$ arrives at a minimum. In Fig. 7b, the value of $\text{E} P$ reaches a maximum at a proper $\beta$. That is, the $\text{E} a^{2}$ and the $\text{E} P$ can reach a minimum and a maximum, respectively, by choosing the optimal value of $\beta$, which can be used to improve the energy harvesting performance and optimize the design of VEH. However, the optimal value of $\beta$ depends on the time delay $\tau$. For example, for fixed $\tau { = 1}$, the optimal value of $\beta$ can be found as $\beta { = - 1} . 0$ to minimize $\text{E} a^{2}$ and maximize $\text{E} P$ in Fig. 7. Thus, it is possible to improve the harvesting performance of VEH by choosing appropriate feedback gain and time delay.

Figure 8 presents the effects of piezoelectric coupling in the circuit $\kappa$ and time constant ratio $\lambda$ on $\text{E} a^{2}$ and $\text{E} P$. It is seen that the value of $\text{E} a^{2}$ decreases with increasing $\kappa$, while the $\text{E} P$ increases with increasing $\kappa$. From the viewpoint of harvesting performance, a large value of $\kappa$ should be considered in the design of VEH. Meanwhile, for fixed $\kappa > 1. 5$, both $\text{E} a^{2}$ and $\text{E} P$ decrease with the increase of $\lambda$.

3 Energy harvesting performance

In this Section, the numerical solutions of Eq. (4) are obtained by using the fourth-order Runge–Kutta algorithm. The random excitation $\xi (t)$ in Eq. (4) can be recast into the following form:

$$\begin{aligned} & \xi (t) = F\cos (\phi (t)), \\ & \dot{\phi }\left( t \right) = \varOmega + \gamma \zeta \left( t \right),\,\,\zeta \left( t \right) = \dot{W}\left( t \right). \\ \end{aligned}$$

(32)

The formal derivation $\zeta (t)$ of the unit Wiener process is a Gaussian white noise, which has the power spectrum of a constant and is unrealized. For the numerical simulations related to stochastic dynamics, it is more convenient to use the pseudorandom signal to model $\zeta (t)$ [40],

$$\zeta (t) = \sqrt {\frac{4\omega }{N}} \sum\limits_{k = 1}^{N} {\cos \left[ {\frac{2k - 1}{N}\omega t + \phi_{k} } \right]} ,$$

(33)

where $\varphi_{k}$ are independent and uniformly distributed in $(0,2 \pi ]$, $N$ is a large positive integer.

In the following analysis, the main system parameters in Eqs. (4) and (33) are set as $\chi = 1. 0$, $\mu = 1. 0$, $\omega_{n} = 1. 0$, $f = 2.5$, $\varepsilon = 0.1$, $\kappa = 0. 5$, $\lambda = 0.5$, $\alpha = 5.0$, $N = 1000$, $\omega = 1.0$, unless otherwise mentioned. The initial values are chosen as $x(t) = 0$, $\dot{x}(t) = 0$, $V(t) = 0$, $t \in \left[ { - \tau ,0} \right]$.

3.1 Response analysis of the system

First, the numerical results are presented in Fig. 9 to verify the theoretical results (26), (27) obtained by the method of multiple scales in the first-order and the second-order steady-state moments of the amplitude. The solid lines denote the theoretical solutions of Eq. (27), and the circles represent the numerical solutions of the original system (4). Obviously, they coincide with each other very well.

Fig. 9

The first-order and the second-order steady-state moments of the amplitude as functions of the detuning frequency $\sigma$. ($\beta = 0. 5$, $\tau = 0.5$, $\gamma = 0.1$). Solid lines: theoretical solutions (27); circles: numerical solutions of the original system (4)

Then, the stochastic response of the system is further explored. In the following numerical experiment, the time history and phase trajectory of system (4), starting from initial values $x(t) = 0$, $\dot{x}(t) = 0$, $V(t) = 0$, $t \in \left[ { - \tau ,0} \right]$, with different noise intensity and detuning frequency are shown in Figs. 10 and 11, respectively. By comparing these subplots in Fig. 10, we find that the output responses of displacement, velocity, and the instantaneous output power fluctuate and the phase trajectory changes from a limit cycle to a diffuse limit cycle with the increase of noise intensity. Surprisingly, all the responses including the instantaneous output power will be weakened with a large dose of noise intensity in the energy harvester under a narrow-band random excitation. Figure 11 presents the stochastic response and the phase trajectory in velocity and voltage with different detuning frequencies. One can see that the increase of detuning frequency $\sigma$ causes all responses to heighten slowly and then sharply weakened in $\sigma \in \left( {2,2.5} \right)$. Subsequently, all responses are weakened in a mild manner. This numerical result just verifies the stochastic jump phenomenon of the theoretical results shown in Fig. 3.

Fig. 10

Time history and phase trajectory of Eq. (4) with different noise intensities, where a $\gamma = 0$ and b $\gamma = 0.5$. ($\beta = 0. 5$, $\tau = 0.5$, $\sigma = 3$)

Fig. 11

The stochastic response and phase trajectory with different detuning frequencies $\sigma$($\beta = 0. 5$, $\tau = 0.5$, $\gamma = 0.2$)

3.2 Effects of the time-delayed feedback on the harvesting performance

In order to evaluate the harvesting performance of VEH, the effects of TDFC on the second-order moment of amplitude $\text{E} a^{2}$, mean output power $\text{E} P$, RMS voltage $V_{rms},$ and power conversion efficiency $\rho\%$ of the energy harvester are discussed in this Subsection.

The dimensionless coupled VEH under a narrow-band random excitation in Eq. (4) can be rearranged as

$$\frac{\text{d} }{{\text{d} t}}\left( {\frac{1}{2}\dot{x}^{2} + \frac{1}{2}\omega_{n}^{2} x^{2} (t) + \frac{1}{4}\varepsilon \alpha x^{4} (t)} \right) + 2\varepsilon \mu \omega_{n} \dot{x}^{2} (t) - \varepsilon \chi \dot{x}V(t) = \varepsilon \dot{x}F\cos (\varOmega t + \gamma W(t)),$$

(34.1)

$$\frac{\text{d} }{{\text{d} t}}V^{2} (t) = 2\beta V(t)V(t - \tau ) - 2\lambda V^{2} (t) - 2\kappa V(t)\dot{x}(t).$$

(34.2)

Due to ${{\text{d} V^{2} (t)} \mathord{\left/ {\vphantom {{\text{d} V^{2} (t)} {\text{d} t}}} \right. \kern-0pt} {\text{d} t}} = 0$ at the steady state, Eqs. (34.1) and (34.2)  can be rewritten as

$$\frac{\text{d} }{{\text{d} t}}E(x,\dot{x}) + 2\varepsilon \mu \omega_{n} \dot{x}^{2} (t) + \frac{\varepsilon \chi }{\kappa }\left( {\lambda V^{2} (t) - \beta V(t)V(t - \tau )} \right) = \varepsilon \dot{x}F\cos (\varOmega t + \gamma W(t)),$$

(35)

where $E(x,\dot{x}) = {{\dot{x}^{2} } \mathord{\left/ {\vphantom {{\dot{x}^{2} } 2}} \right. \kern-0pt} 2} + {{\omega_{n}^{2} x^{2} } \mathord{\left/ {\vphantom {{\omega_{n}^{2} x^{2} } 2}} \right. \kern-0pt} 2} + {{\varepsilon \alpha x^{4} } \mathord{\left/ {\vphantom {{\varepsilon \alpha x^{4} } 4}} \right. \kern-0pt} 4}$ denotes the sum of kinetic and potential energy, $2\varepsilon \mu \omega_{n} \dot{x}^{2} (t)$ is the power dissipated by friction, $\varepsilon \dot{x}F\cos (\varOmega t + \gamma W(t))$ denotes the input power provided by the ambient noise, and $\varepsilon \chi {{\left( {\lambda V^{2} (t) - \beta V(t)V(t - \tau )} \right)} \mathord{\left/ {\vphantom {{\left( {\lambda V^{2} (t) - \beta V(t)V(t - \tau )} \right)} \kappa }} \right. \kern-0pt} \kappa }$ represents the instantaneous power transduced into electrical power, which only considers the electrical energy harvested from the ambient noise.

The index $\rho\%$ is defined to assess the total efficiency in the power conversion from the provided mechanical power to the final net electrical power. That is,

$$\rho \% = P_{e} /P_{m} [100\% ],$$

(36)

where $P_{e} = \left\langle {{{\chi \left( {\lambda V^{2} (t) - \beta V(t)V(t - \tau )} \right)} \mathord{\left/ {\vphantom {{\chi \left( {\lambda V^{2} (t) - \beta V(t)V(t - \tau )} \right)} \kappa }} \right. \kern-0pt} \kappa }} \right\rangle$, $P_{m} = \left\langle {\dot{x}F\cos (\varOmega t + \gamma W(t))} \right\rangle = \left\langle {\dot{x}S_{\xi } (\omega )} \right\rangle,$ and $\langle\cdot\rangle$ implies time average and ensemble average.

For small noise intensity $\gamma = 0.01$, the effects of time-delayed feedback on the second-order steady-state moments of the amplitude $\text{E} a^{2}$ and the mean output power $\text{E} P$ are shown in Fig. 12. Both $\text{E} a^{2}$ and $\text{E} P$ are nonmonotonic functions with $\beta$ and $\tau$. For any $\tau$, with the increase of $\beta$, $\text{E} a^{2}$ first decreases to a minimum and then increases in the overall trend, whereas the mean output power $\text{E} P$ first increases to a maximum and then decreases. At the red points marked in Fig. 12, $\text{E} P$ reaches the maximum when $\text{E} a^{2}$ reaches the minimum for any $\tau$. It indicates that there is an optimal feedback gain $\beta_{opt}$ that maximizes the mean output power and at which the second-order moment reaches the minimum. Besides, with the increase in $\tau$, the maximum of $\text{E} P$ marked with red points does not change much, while the corresponding minimum of $\text{E} a^{2}$ decreases gradually. Obviously, these findings are beneficial to the minimization design of an energy harvester without losing the output power.

Fig. 12

The second-order steady-state moment of the amplitude $\text{E} a^{2}$ and the mean output power $\text{E} P$ as functions of the feedback gain $\beta$ and time delay $\tau$ for fixed $\gamma = 0.01$ and $\sigma = 1$

Nevertheless, it should be noted that the optimal $\beta_{opt}$ corresponding to the minimum of $\text{E} a^{2}$ and the maximum of $\text{E} P$ are not necessarily the same especially when the noise intensity is too large, e.g., $\gamma = 0.6$, as shown in Fig. 13. With an increase in the noise intensity, the minimum of $\text{E} a^{2}$ decreases and shifts to the left, while the maximum of $\text{E} P$ decreases and shifts to the right. This indicates that the increase in noise intensity reduces the mean output power, for the reason that large noise intensity causes the limit cycle to diffuse to the origin as shown in Fig. 10. It can also be observed from Figs. 12 and 13 that the optimal value $\beta_{opt}$ usually occurs at a value less than zero. It indicates that negative feedback gain is favorable for an energy harvester to provide more electrical power compared with the positive feedback gain.

Fig. 13

The second-order steady-state moments of the amplitude $\text{E} a^{2}$ and the mean output power $\text{E} P$ as a function of feedback gain $\beta$ with different noise intensity $\gamma$ for fixed $\sigma = 1$ and $\tau = 1$

Figure 14 shows the effects of time constant ratio $\lambda$ on $\text{E} a^{2}$ and the $\text{E} P$. In this numerical experiment, for each $\beta$, $\text{E} a^{2}$ displays nonmonotonous changes versus the time constant ratio $\lambda$, whereas $\text{E} P$ increases monotonously as $\lambda$ increases. From Fig. 14a, one also can find that a negative feedback gain $\beta$ is more beneficial to minimization of the energy harvester for $\lambda$ in [0.2,1]. On the other hand, when $\lambda < 0.6$, the mean output power of the energy harvester under a negative feedback gain is superior to that under a positive feedback gain.

Fig. 14

The second-order steady-state moments of the amplitude $\text{E} a^{2}$ and the mean output power $\text{E} P$ as functions of the time constant ratio $\lambda$ with different feedback gains $\beta$ for fixed $\sigma = 1$, $\gamma = 0.01,$ and $\tau = 0.5$

Figure 15 presents the effects of the time-delayed feedback on the power conversion efficiency and the RMS voltage. From Fig. 15a, one can see that for any $\tau$ there exists an optimal $\beta_{opt}$ maximizing the power conversion efficiency. Moreover, compared with the nondelayed feedback, negative feedback gain can enhance the power conversion efficiency, while the positive feedback gain will weaken the power conversion efficiency, as shown in Fig. 15b. Furthermore, in Fig. 15c, an optimal $\beta_{opt}$ exists for fixed $\tau$ where the RMS voltage reaches the peak value. As $\tau$ increases, the $\beta_{opt}$ decreases and the corresponding peak value of the RMS voltage increases (see Fig. 15d). Thus, for fixed $\tau$, there exists an optimal $\beta_{opt}$ maximizing both the power conversion efficiency and the RMS voltage. One can choose the optimal delayed feedback combination ($\beta$,$\tau$) to enhance the energy harvesting performance effectively.

Fig. 15

The power conversion efficiency $\rho\%$ and the RMS voltage $V_{\text{rms}}$ as functions of the feedback gain $\beta$ and time delay $\tau$ for fixed $\sigma = 1$ and $\gamma = 0.5$

4 Conclusions

This paper presents the dynamical analysis of a Duffing-type VEH with time delay circuit driven by a narrow-band random excitation. The method of multiple scales is applied to obtain the analytical expressions of the steady-state responses near the principal resonance. The stable conditions of the first-order and the second-order steady-state moments are derived. Based on theoretical and numerical results, the effects of the system parameters, noise intensity, and time-delayed feedback on the harvesting performance are discussed. It is observed that the curves of the first-order and the second-order steady-state moments show right-bended resonance curves with the increase of frequency detuning. That is, the phenomenon of stochastic jump exists, and the operational bandwidth of nonlinear VEH is extended. Meanwhile, the feedback gain plays an important role in the minimization of design and enhancement of energy harvesting. For example, a negative feedback gain is more beneficial to miniaturize the device and enhance the mean output power than a positive one. Moreover, it is found that an optimal combination of time delay and feedback gain can maximize both the power conversion efficiency and the RMS voltage. As a result, a proper design of TDFC is proposed to improve the energy harvesting performance and achieve desirable optimization of the system parameter.