The issue of energy harvesting has received an active interest during the past decade. To enrich the bandwidth of energy harvester, researchers have developed many significant methods, such as tuned linear resonance frequency,^[1] multimodal,^[2] cubic nonlinearity,^[3–10] bistable,^[11–15] tristable,^[16] multistable,^[17–19] stochastic resonance,^[20,21] and internal resonance.^[22,33] However, all the internal resonance energy harvesting systems assume that the second modal is approximately twice or thrice its first mode, but no investigation focused on the one-to-one internal resonance. To enhance the family of energy harvesting, this work develops a one-to-one internal resonance energy harvesting to realize wideband vibration.

Electromechanical systems are involved in many engineering devices, such as vibrating sieves,^[34] industrial mixers,^[35] industrial shakers,^[36] and hydro-turbine systems.^[37] Much research has been carried out to understand the dynamical behaviors of such systems which lead to various interesting phenomena such as frequency entrainment,^[38] harmonic oscillations,^[39] chaotic behavior,^[40] feedback control^[41] and bursting oscillations.^[42] One-to-one internal resonance has been the subject of great deal of recent research. Physical systems with a one-to-one frequency relationship have been extensively reported. Some of these systems, such as the buckled beams,^[43] composite laminated rectangular plates^[44–51] and nanoresonators.^[52] Nevertheless, one-to-one internal resonance harvesting is not found.

In this work, we establish an innovative energy harvesting via the one-to-one internal resonance. An electromechanical transducer energy harvester with such a resonance is designed. Then the equations of the electromechanical coupling system is established. The motion response of the system is computed and the study demonstrates that such an internal resonance owns an advanced frequency bandwidth.

The model of an electromechanical transducer energy harvester, as shown in Fig. 1, consists of a Duffing electrical oscillator magnetically coupled to an attached linear mechanical oscillator via an electromechanical coupling mechanism. By introducing Kirchhoff’s voltage law for the circuit part and Newton’s second law for the mechanical part, the motion equations of the electromechanical coupling system can be written as

$$ \begin{eqnarray}\begin{array}{lll} & & {L}_{{\rm{ind}}1}\ddot{q}+{R}_{1}\dot{q}+\displaystyle \frac{q}{{C}_{0}}+{a}_{3}{q}^{3}+l{B}_{1}\dot{x}={V}_{0}\cos (\varOmega \tau ),\end{array}\end{eqnarray}$$ (1)

$$ \begin{eqnarray}\begin{array}{lll} & & m\ddot{x}+c\dot{x}+kx-l{B}_{1}\dot{q}+{B}_{2}I{L}_{{\rm{coil}}}=0,\end{array}\end{eqnarray}$$ (2)

$$ \begin{eqnarray}\begin{array}{lll} & & {L}_{{\rm{ind}}2}\dot{I}+{R}_{2}I-{B}_{2}{L}_{{\rm{coil}}}\dot{x}=0,\end{array}\end{eqnarray}$$ (3)

Note that the driving force described in Fig. 1 is a harmoniously voltage excitation, and the nonlinear capacitor is realized via the varactor diode.^[39] The nonlinear term is proposed by introducing a capacitor represented by Fig. 2, an amplifier is utilized to increase the amplitude of the electric signal. The coupled term lB1x˙ in the circuit equation (1) is the Lorenz electromechanical force, the coupled term lB1q˙ in the mechanical equation (2) is the Laplace force, and the B₂IL_coil term is the electro-magnetic force which is produced by the coil through the magnetic field. The device is feasible and available in electromechanical engineering.^[38–42] Moreover, the model shown in Fig. 1 has also been verified by experiment in Refs. [40,41].

The charge–voltage characteristics represented in Fig. 2 can be written as^[42]

$$ \begin{eqnarray}{V}_{C}=\displaystyle \frac{q}{{C}_{1}}+n{V}_{{\rm{C}}0}{\sinh }^{-1}\left(-\displaystyle \frac{q}{2{R}_{C}{i}_{0}{C}_{2}}\right),\end{eqnarray}$$ (4)

Consider a small charge in the circuit, the nonlinear item in Eq. (4) can be expanded as^[42]

$$ \begin{eqnarray}{V}_{C}=\displaystyle \frac{q}{{C}_{0}}+{a}_{3}{q}^{3},\end{eqnarray}$$ (5)

The coupled differential Eqs. (1)–(3) can be simplified to the following dimensionless equations:

$$ \begin{eqnarray}\begin{array}{lll} & & {\ddot{y}}_{1}+{\varsigma }_{1}{\dot{y}}_{1}+{y}_{1}+\alpha {y}_{1}^{3}+{\lambda }_{1}{\dot{y}}_{2}=f\cos (\omega t),\end{array}\end{eqnarray}$$ (6)

$$ \begin{eqnarray}\begin{array}{lll} & & {\ddot{y}}_{2}+{\varsigma }_{2}{\dot{y}}_{2}+\eta {y}_{2}-{\lambda }_{2}{\dot{y}}_{1}+{\lambda }_{3}{y}_{3}=0,\end{array}\end{eqnarray}$$ (7)

$$ \begin{eqnarray}\begin{array}{lll} & & {\dot{y}}_{3}+\chi {y}_{3}-{\dot{y}}_{2}=0,\end{array}\end{eqnarray}$$ (8)

 1

We introduce the small transformations of the damping, the external excitation, the non-linearity and the coupling coefficients as follows:

$$ \begin{eqnarray}{\zeta }_{i}\leftrightarrow \varepsilon {\zeta }_{i},\ \ \ f\leftrightarrow \varepsilon f,\ \ \ \alpha \leftrightarrow \varepsilon \alpha,\ \ \ {\lambda }_{i}\leftrightarrow \varepsilon {\lambda }_{i}.\end{eqnarray}$$ (9)

For the case of one-to-one internal resonance accompanied by primary resonance, we put

$$ \begin{eqnarray}\omega =1+\varepsilon {\sigma }_{0},\begin{array}{cc} & \eta =1+\varepsilon {\sigma }_{1},\end{array}\end{eqnarray}$$ (10)

Employing the method of multiple scales to Eqs. (6)–(8), we can approximate the system solution as

$$ \begin{eqnarray}\begin{array}{lll} & & {y}_{1}(t;\varepsilon )={y}_{11}({T}_{0},{T}_{1})+\varepsilon {y}_{12}({T}_{0},{T}_{1})+O({\varepsilon }^{2}),\\ & & {y}_{2}(t;\varepsilon )={y}_{21}({T}_{0},{T}_{1})+\varepsilon {y}_{22}({T}_{0},{T}_{1})+O({\varepsilon }^{2}),\\ & & {y}_{3}(t;\varepsilon )={y}_{31}({T}_{0},{T}_{1})+\varepsilon {y}_{32}({T}_{0},{T}_{1})+O({\varepsilon }^{2}).\end{array}\end{eqnarray}$$ (11)

$$ \begin{eqnarray}\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}={D}_{0}+\varepsilon {D}_{1}+\cdots,\begin{array}{cc} & \end{array}\displaystyle \frac{{{\rm{d}}}^{2}}{{\rm{d}}{t}^{2}}={D}_{0}^{2}+2\varepsilon {D}_{0}{D}_{1}+\cdots,\end{eqnarray}$$ (12)

Setting Eqs. (9)–(12) into Eqs. (6)–(8) and equating terms of ε⁰ and ε¹ under the generating equations yield

$$ \begin{eqnarray}\begin{array}{lll} & & {D}_{0}^{2}{y}_{11}+{y}_{11}=0,\\ & & {D}_{0}^{2}{y}_{21}+{\eta }^{2}{y}_{21}=0,\\ & & {D}_{0}{y}_{31}+\chi {y}_{31}={D}_{0}{y}_{21},\end{array}\end{eqnarray}$$ (13)

$$ \begin{eqnarray}\begin{array}{lll} & & {D}_{0}^{2}{y}_{12}+{y}_{12}=-2{D}_{0}{D}_{1}{y}_{11}-{\varsigma }_{1}{D}_{0}{y}_{11}-\alpha {y}_{11}^{3}\\ & & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -{\lambda }_{1}{D}_{0}{y}_{21}+f\cos \omega t,\\ & & {D}_{0}^{2}{y}_{22}+{\eta }^{2}{y}_{22}=-2{D}_{0}{D}_{1}{y}_{21}-{\varsigma }_{2}{D}_{0}{y}_{21}+{\lambda }_{2}{D}_{0}{y}_{11}-{\lambda }_{3}{y}_{31},\\ & & {D}_{0}{y}_{32}+\chi {y}_{32}={D}_{0}{y}_{22}+{D}_{1}{y}_{21}-{D}_{1}{y}_{31}.\end{array}\end{eqnarray}$$ (14)

$$ \begin{eqnarray}\begin{array}{lll} & & {y}_{11}={A}_{1}({T}_{1})\exp ({\rm{i}}{T}_{0})+{\rm{c}}.{\rm{c}}.,\\ & & {y}_{21}={A}_{2}({T}_{1})\exp ({\rm{i}}\eta {T}_{0})+{\rm{c}}.{\rm{c}}.,\\ & & {y}_{31}=\displaystyle \frac{{\rm{i}}\eta {A}_{2}({T}_{1})}{\chi +{\rm{i}}\eta }\exp ({\rm{i}}\eta {T}_{0})+B({T}_{1})\exp (-\chi {T}_{0})+{\rm{c}}.{\rm{c}}.\ \ \ \ \end{array}\end{eqnarray}$$ (15)

$$ \begin{eqnarray}\begin{array}{lll}{D}_{1}{A}_{1} & = & {\varGamma }_{11}{A}_{1}+{\varGamma }_{12}{A}_{1}^{2}{\bar{A}}_{1}+{\varGamma }_{13}{A}_{2}\exp [{\rm{i}}(\eta -1){T}_{0}]\\ & & +f{\varGamma }_{14}\exp [i(\omega -1){T}_{0}],\\ {D}_{1}{A}_{2} & = & {\varGamma }_{21}{A}_{2}+{\varGamma }_{22}{A}_{1}\exp [{\rm{i}}(1-\eta ){T}_{0}],\end{array}\end{eqnarray}$$ (16)

 1

$$ \begin{eqnarray}{A}_{n}({T}_{1})=\displaystyle \frac{1}{2}{a}_{n}({T}_{1})\exp [{\rm{i}}{\theta }_{n}({T}_{1})].\end{eqnarray}$$ (17)

$$ \begin{eqnarray}\begin{array}{lll}{D}_{1}{a}_{1} & = & {\rm{Re}}({\varGamma }_{11}){a}_{1}+\displaystyle \frac{1}{4}{a}_{1}^{3}{\rm{Re}}({\varGamma }_{12})\\ & & +{a}_{2}[{\rm{Re}}({\varGamma }_{13})\cos {\gamma }_{1}-{\rm{Im}}({\varGamma }_{13})\sin {\gamma }_{1}]\\ & & +2f[{\rm{Re}}({\varGamma }_{14})\cos {\gamma }_{2}-{\rm{Im}}({\varGamma }_{14})\sin {\gamma }_{2}],\end{array}\end{eqnarray}$$ (18)

$$ \begin{eqnarray}\begin{array}{lll}{D}_{1}{a}_{2} & = & {\rm{Re}}({\varGamma }_{21}){a}_{2}+{a}_{1}[{\rm{Re}}({\varGamma }_{22})\cos {\gamma }_{1}+{\rm{Im}}({\varGamma }_{22})\sin {\gamma }_{1}],\end{array}\end{eqnarray}$$ (19)

$$ \begin{eqnarray}\begin{array}{lll}{D}_{1}{\gamma }_{1} & = & {\sigma }_{1}+{\rm{Im}}({\varGamma }_{21})-\displaystyle \frac{{a}_{1}}{{a}_{2}}[{\rm{Re}}({\varGamma }_{22})\sin {\gamma }_{1}\\ & & -{\rm{Im}}({\varGamma }_{22})\cos {\gamma }_{1}]-{\rm{Im}}({\varGamma }_{11})\\ & & -\displaystyle \frac{1}{4}{a}_{1}^{2}{\rm{Im}}({\varGamma }_{12})-\displaystyle \frac{{a}_{2}}{{a}_{1}}[{\rm{Re}}({\varGamma }_{13})\sin {\gamma }_{1}+{\rm{Im}}({\varGamma }_{13})\cos {\gamma }_{1}]\\ & & -\displaystyle \frac{2f}{{a}_{1}}[{\rm{Re}}({\varGamma }_{14})\sin {\gamma }_{2}+{\rm{Im}}({\varGamma }_{14})\cos {\gamma }_{2}],\end{array}\end{eqnarray}$$ (20)

$$ \begin{eqnarray}\begin{array}{lll}{D}_{1}{\gamma }_{2} & = & {\sigma }_{0}-{\rm{Im}}({\varGamma }_{11})-\displaystyle \frac{1}{4}{a}_{1}^{2}{\rm{Im}}({\varGamma }_{12})\\ & & -\displaystyle \frac{{a}_{2}}{{a}_{1}}[{\rm{Re}}({\varGamma }_{13})\sin {\gamma }_{1}+{\rm{Im}}({\varGamma }_{13})\cos {\gamma }_{1}]\\ & & -\displaystyle \frac{2f}{{a}_{1}}[{\rm{Re}}({\varGamma }_{14})\sin {\gamma }_{2}+{\rm{Im}}({\varGamma }_{14})\cos {\gamma }_{2}],\end{array}\end{eqnarray}$$ (21)

 1

To proceed further we discuss the steady-state response, and find

$$ \begin{eqnarray}\begin{array}{lll}0 & = & {\rm{Re}}({\varGamma }_{11}){a}_{1}+\displaystyle \frac{1}{4}{a}_{1}^{3}{\rm{Re}}({\varGamma }_{12})\\ & & +{a}_{2}[{\rm{Re}}({\varGamma }_{13})\cos {\gamma }_{1}-{\rm{Im}}({\varGamma }_{13})\sin {\gamma }_{1}]\\ & & +2f[{\rm{Re}}({\varGamma }_{14})\cos {\gamma }_{2}-{\rm{Im}}({\varGamma }_{14})\sin {\gamma }_{2}],\end{array}\end{eqnarray}$$ (22)

$$ \begin{eqnarray}\begin{array}{lll}0 & = & {\rm{Re}}({\varGamma }_{21}){a}_{2}+{a}_{1}[{\rm{Re}}({\varGamma }_{22})\cos {\gamma }_{1}+{\rm{Im}}({\varGamma }_{22})\sin {\gamma }_{1}],\end{array}\end{eqnarray}$$ (23)

$$ \begin{eqnarray}\begin{array}{lll}0 & = & {\sigma }_{1}+{\rm{Im}}({\varGamma }_{21})-\displaystyle \frac{{a}_{1}}{{a}_{2}}[{\rm{Re}}({\varGamma }_{22})\sin {\gamma }_{1}\\ & & -{\rm{Im}}({\varGamma }_{22})\cos {\gamma }_{1}]-{\rm{Im}}({\varGamma }_{11})\\ & & -\displaystyle \frac{1}{4}{a}_{1}^{2}{\rm{Im}}({\varGamma }_{12})-\displaystyle \frac{{a}_{2}}{{a}_{1}}[{\rm{Re}}({\varGamma }_{13})\sin {\gamma }_{1}+{\rm{Im}}({\varGamma }_{13})\cos {\gamma }_{1}]\\ & & -\displaystyle \frac{2f}{{a}_{1}}[{\rm{Re}}({\varGamma }_{14})\sin {\gamma }_{2}+{\rm{Im}}({\varGamma }_{14})\cos {\gamma }_{2}],\end{array}\end{eqnarray}$$ (24)

$$ \begin{eqnarray}\begin{array}{lll}0 & = & {\sigma }_{0}-{\rm{Im}}({\varGamma }_{11})-\displaystyle \frac{1}{4}{a}_{1}^{2}{\rm{Im}}({\varGamma }_{12})\\ & & -\displaystyle \frac{{a}_{2}}{{a}_{1}}[{\rm{Re}}({\varGamma }_{13})\sin {\gamma }_{1}+{\rm{Im}}({\varGamma }_{13})\cos {\gamma }_{1}]\\ & & -\displaystyle \frac{2f}{{a}_{1}}[{\rm{Re}}({\varGamma }_{14})\sin {\gamma }_{2}+{\rm{Im}}({\varGamma }_{14})\cos {\gamma }_{2}].\end{array}\end{eqnarray}$$ (25)

Equations (22)–(25) determine the modulation relationships between response and frequency. To calculate these solutions, we numerically compute the generating nonlinear algebraic equations by the Mathematica software. In the meantime, the output current response can be established as

$$ \begin{eqnarray}{y}_{3}=\displaystyle \frac{\eta }{\sqrt{{\chi }^{2}+{\eta }^{2}}}{a}_{2}.\end{eqnarray}$$ (26)

$$ \begin{eqnarray}\begin{array}{lll} & & {(\begin{array}{cccc}{D}_{1}\Delta {a}_{1} & {D}_{1}\Delta {a}_{2} & {D}_{1}\Delta {\gamma }_{1} & {D}_{1}\Delta {\gamma }_{2}\end{array})}^{{\rm{T}}}\\ & = & {\boldsymbol{J}}{(\begin{array}{cccc}\Delta {a}_{1} & \Delta {a}_{2} & \Delta {\gamma }_{1} & \Delta {\gamma }_{2}\end{array})}^{{\rm{T}}},\end{array}\end{eqnarray}$$ (27)

Since the most important attribution of energy harvesting is the collected bandwidth, which can enhance the power transduction efficiency of energy harvesters under broadband external ambient environmental excitations, thereby collecting power on a wide spectrum of external frequency is our most pressing problem. In light of this challenge, the amplitude-frequency curves of one-to-one internal resonance will be discussed in this section. To describe the influence of the design parameters, we plot three sets of response curves in Figs. 3–8 for six physical parameters. The geometric characteristics of the energy harvester are denoted as m = 0.05 kg, C₀ = 2200 μF, Q₀ = 0.1C, a₃ = 43182 V/C³, c = 0.1016 N ⋅ m/s, L_ind1 = 1.1 H, B₁ = 20 T, l = 0.2 m, R₁ = 1.118 Ω, R₂ = 20.3279 Ω, B₂ = 1 T, L_ind2 = 1 H, L_coil = 2 m and V₀ = 10 V. Consequently, one-to-one internal resonance occurs together with ω₁ = 20.3279 s⁻¹ and ω₂ = 20.3279 s⁻¹. Moreover, the internal detuning value σ₁ is zero. The motion response curves will be plotted in the following figures, where solid lines denote stable vibrations and dotted lines employ the unstable motions.

Having observed the favorable effect of excitation levels on the response of system, Fig. 3 shows the frequency response curves of one-to-one internal resonance for three different amplitudes of external voltage source (V₀ = 10 V, 7.5 V and 5 V), in which the height and the bandwidth are increased when the amplitudes of external voltage source is extended. There are obviously two mountain peaks in the response curves, also the motion curves have a multi-valued regions. Subsequently, there is one peak bending to the right indicating a hardening-type nonlinearity. This is because the modal interactions under the one-to-one internal resonance can result in an energy exchange between the first modal frequency and the second mode. In the meantime, the two mountain peaks and the jumping bring about broadband energy harvesting. In addition, the motion amplitude and the response bandwidth increase with the voltage source. In energy harvesting, the broadband feature of resonance frequency is desired.

Figure 4 depicts the frequency response curve of one-to-one internal resonance with three different levels of cubic nonlinear coefficients (a₃ = 43182 V/C³, 32387 V/C³ and 21591 V/C³). In Fig. 4, as nonlinearity is decreased, the response amplitude of left peak is increased slightly, while the response amplitude of right peak is increased obviously. In addition, the bandwidth is extended obviously. Consequently, in the one-to-one internal resonance energy harvesting, decrease in the cubic nonlinear terms becomes more significant. It is worth noticing that the result is different from the classical single degree of freedom Duffing energy harvester.^[3–10]

Figure 5 plots the frequency response curve of one-to-one internal resonance with three different resistors (R₁ = 1.118 Ω, 2.236 Ω and 3.354 Ω). The frequency response diagrams of displacement and current show that the influence of one-to-one internal resonance becomes more pronounced with reduced resistors. In addition to the increase in the mountain peak motion amplitude, the bandwidth of stable solutions also increases as the resistors is decreased. Therefore, light resistors are preferred for enhanced bandwidth of effective operation for the harvester with one-to-one internal resonance.

Figure 6 shows the frequency response curve of one-to-one internal resonance along with different damping values (c = 0.1016 N ⋅ m/s, 0.2032 N ⋅ m/s and 0.3048N ⋅ m/s). In Fig. 6, as the damping coefficient c increases, the peak height and the bandwidth are decreased.

Figure 7 plots the frequency response curve of one-to-one internal resonance with different magnetic strengths B₁ (B₁ = 5 T, 10 T and 20 T). The results indicate that the peak and the bandwidth of displacement and current frequency response curves increase monotonically with increasing magnetic strengths B₁.

Figure 8 depicts the frequency response curve of one-to-one internal resonance with different magnetic strengths B₂ (B₂ = 1 T, 2 T and 3 T). The results indicate that the peak height and the bandwidth of displacement frequency response curves increase with decreasing magnetic strengths B₂. In addition, as the magnetic strengths B₂ is decreased, the bandwidth of current frequency response curves is increased, while the peak of current frequency response is decreased slightly.

Having observed the broadband electromechanical response of the one-to-one internal resonance electromechanical transducer coupling a linear oscillator energy harvester described by Eqs. (1)–(3), comparisons can be made against the conventional linear configuration.

The electromechanical equations of the linear RLC circuit with a linear oscillator energy harvester are

$$ \begin{eqnarray}\begin{array}{lll} & & {L}_{{\rm{ind}}1}\ddot{q}+{R}_{1}\dot{q}+\displaystyle \frac{q}{{C}_{0}}+l{B}_{1}\dot{x}={V}_{0}\cos (\varOmega \tau ),\end{array}\end{eqnarray}$$ (28)

$$ \begin{eqnarray}\begin{array}{lll} & & m\ddot{x}+c\dot{x}+kx-l{B}_{1}\dot{q}+{B}_{2}I{L}_{{\rm{coil}}}=0,\end{array}\end{eqnarray}$$ (29)

$$ \begin{eqnarray}\begin{array}{lll} & & {L}_{{\rm{ind}}2}\dot{I}+{R}_{2}I-{B}_{2}{L}_{{\rm{coil}}}\dot{x}=0.\end{array}\end{eqnarray}$$ (30)

For the same system parameters, initial conditions, and the forcing amplitude of the one-to-one internal resonance harvester, one can calculate the frequency motion diagrams of the linear configuration via Eqs. (28)–(30). Figure 9 shows the motion response diagrams of one-to-one internal resonance and the linear configurations. In the meantime, we suppose that a minimum steady-state current of 0.1 A should be kept during the system operation, and the corresponding bandwidth of frequency response is defined under the minimum current. As can be seen from the steady-state motion responses appearing in this figure, the bandwidth of the one-to-one internal resonance harvester can be much larger than that of the linear configuration.

To confirm the accuracy of approximately analytical steady-state values, Eqs. (6)–(8) are numerically calculated by utilizing the fourth-order Runge–Kutta algorithm. Numerical values from Eqs. (6)–(8) are recorded in Fig. 10 as red solid dots. Investigating Fig. 10, the steady-state responses from the analysis and the simulation have the greater consistency. In the meantime, the simulation values show the jumping phenomena emerged in the primary resonance and one-to-one internal resonances of electromechanical transducer electromagnetic harvester.

We have confirmed that developing multimodal nonlinear energy harvester coupling with one-to-one internal resonance can be employed to extend the steady-state bandwidth of the harvester. As a simple proof, an electromechanical transducer energy harvester along with a linear oscillator is exploited to address one-to-one internal resonance. Specifically, the multiple-scale analysis is utilized to employ the motion response, and compared the performance of the presented internal resonance model and the correspond linear location. The accuracy of approximately analytical values is checked via numerical integration. The investigation acquires the following observations.

(1) An electromechanical transducer shaped energy harvester constructed by a nonlinear RLC circuit is utilized to realize one-to-one internal resonance.

(2) The motion responses are analytically observed to predict two peaks and one jump phenomenon, one peak bending to the right implicating a hardening nonlinearity.

(3) The amplitude and the bandwidth of motion response increase with the excitation amplitudes of voltage source and magnetic strengths B₁, while decrease with increasing cubic nonlinear coefficients a₃, resistor R₁, damping coefficient c, and magnetic strengths B₂.

(4) The analytical forecast is supported by the numerical results.

(5) The advancement of the presented internal resonance technique is contrasted with its linear counterparts.