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Chinese Physics B
Vol. 29, Issue 10, (2020)

Broadband energy harvesting based on one-to-one internal resonance

Wen-An Jiang¹, Xin-Dong Ma¹, Xiu-Jing Han^1,†, Li-Qun Chen^2,3, and Qin-Sheng Bi¹

Author Affiliations

¹Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 2203, China

²School of Science, Harbin Institute of Technology, Shenzhen 518055, China

³Department of Mechanics, Shanghai University, Shanghai 200072, China

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DOI: 10.1088/1674-1056/aba5fd Cite this Article

Wen-An Jiang, Xin-Dong Ma, Xiu-Jing Han, Li-Qun Chen, Qin-Sheng Bi. Broadband energy harvesting based on one-to-one internal resonance[J]. Chinese Physics B, 2020, 29(10): Copy Citation Text

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Abstract

We design an electromechanical transducer harvesting system with one-to-one internal resonance that can emerge a broader spectrum vibrations. The novel harvester is composed of a Duffing electrical circuit coupled to a mobile rod, and the coupling between both components is realized via the electromagnetic force. Approximate analytical solutions of the electromechanical system are carried out by introducing the multiple scales analysis, also the nonlinear modulation equation for one-to-one internal resonance is obtained. The character of broadband harvesting performance are analyzed, the two peaks and one jump phenomenon bending to the right for variation of control parameters are observed. It is shown that an advanced bandwidth over a corresponding linear model that does not possess a modal energy interchange.

Keywords

broadband energy harvesting internal resonance nonlinear modal interactions

\begin{array}{r} \begin{array}{lll} L_{i n d 1} \ddot{q} + R_{1} \dot{q} + \frac{q}{C_{0}} + a_{3} q^{3} + l B_{1} \dot{x} = V_{0} \cos (Ω τ), \end{array} \end{array}

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\begin{array}{r} \begin{array}{lll} m \ddot{x} + c \dot{x} + k x - l B_{1} \dot{q} + B_{2} I L_{c o i l} = 0, \end{array} \end{array}

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\begin{array}{r} \begin{array}{lll} L_{i n d 2} \dot{I} + R_{2} I - B_{2} L_{c o i l} \dot{x} = 0, \end{array} \end{array}

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\begin{array}{r} V_{C} = \frac{q}{C_{1}} + n V_{C 0} \sinh^{- 1} (- \frac{q}{2 R_{C} i_{0} C_{2}}), \end{array}

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\begin{array}{r} V_{C} = \frac{q}{C_{0}} + a_{3} q^{3}, \end{array}

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\begin{array}{r} \begin{array}{lll} {\ddot{y}}_{1} + ς_{1} {\dot{y}}_{1} + y_{1} + α y_{1}^{3} + λ_{1} {\dot{y}}_{2} = f \cos (ω t), \end{array} \end{array}

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\begin{array}{r} \begin{array}{lll} {\ddot{y}}_{2} + ς_{2} {\dot{y}}_{2} + η y_{2} - λ_{2} {\dot{y}}_{1} + λ_{3} y_{3} = 0, \end{array} \end{array}

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\begin{array}{r} \begin{array}{lll} {\dot{y}}_{3} + χ y_{3} - {\dot{y}}_{2} = 0, \end{array} \end{array}

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\begin{array}{r} ζ_{i} \leftrightarrow ε ζ_{i}, f \leftrightarrow ε f, α \leftrightarrow ε α, λ_{i} \leftrightarrow ε λ_{i} . \end{array}

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\begin{array}{r} ω = 1 + ε σ_{0}, \begin{array}{cc} η = 1 + ε σ_{1}, \end{array} \end{array}

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\begin{array}{r} \begin{array}{lll} y_{1} (t; ε) = y_{11} (T_{0}, T_{1}) + ε y_{12} (T_{0}, T_{1}) + O (ε^{2}), \\ y_{2} (t; ε) = y_{21} (T_{0}, T_{1}) + ε y_{22} (T_{0}, T_{1}) + O (ε^{2}), \\ y_{3} (t; ε) = y_{31} (T_{0}, T_{1}) + ε y_{32} (T_{0}, T_{1}) + O (ε^{2}) . \end{array} \end{array}

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\begin{array}{r} \frac{d}{d t} = D_{0} + ε D_{1} + \dots, \begin{array}{cc}  \end{array} \frac{d^{2}}{d t^{2}} = D_{0}^{2} + 2 ε D_{0} D_{1} + \dots, \end{array}

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\begin{array}{r} \begin{array}{lll} D_{0}^{2} y_{11} + y_{11} = 0, \\ D_{0}^{2} y_{21} + η^{2} y_{21} = 0, \\ D_{0} y_{31} + χ y_{31} = D_{0} y_{21}, \end{array} \end{array}

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\begin{array}{r} \begin{array}{lll} D_{0}^{2} y_{12} + y_{12} = - 2 D_{0} D_{1} y_{11} - ς_{1} D_{0} y_{11} - α y_{11}^{3} \\ - λ_{1} D_{0} y_{21} + f \cos ω t, \\ D_{0}^{2} y_{22} + η^{2} y_{22} = - 2 D_{0} D_{1} y_{21} - ς_{2} D_{0} y_{21} + λ_{2} D_{0} y_{11} - λ_{3} y_{31}, \\ D_{0} y_{32} + χ y_{32} = D_{0} y_{22} + D_{1} y_{21} - D_{1} y_{31} . \end{array} \end{array}

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\begin{array}{r} \begin{array}{lll} y_{11} = A_{1} (T_{1}) \exp (i T_{0}) + c . c ., \\ y_{21} = A_{2} (T_{1}) \exp (i η T_{0}) + c . c ., \\ y_{31} = \frac{i η A_{2} (T_{1})}{χ + i η} \exp (i η T_{0}) + B (T_{1}) \exp (- χ T_{0}) + c . c . \end{array} \end{array}

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\begin{array}{r} \begin{array}{lll} D_{1} A_{1} & = & Γ_{11} A_{1} + Γ_{12} A_{1}^{2} {\bar{A}}_{1} + Γ_{13} A_{2} \exp [i (η - 1) T_{0}] \\ + f Γ_{14} \exp [i (ω - 1) T_{0}], \\ D_{1} A_{2} & = & Γ_{21} A_{2} + Γ_{22} A_{1} \exp [i (1 - η) T_{0}], \end{array} \end{array}

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\begin{array}{r} A_{n} (T_{1}) = \frac{1}{2} a_{n} (T_{1}) \exp [i θ_{n} (T_{1})] . \end{array}

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\begin{array}{r} \begin{array}{lll} D_{1} a_{1} & = & R e (Γ_{11}) a_{1} + \frac{1}{4} a_{1}^{3} R e (Γ_{12}) \\ + a_{2} [R e (Γ_{13}) \cos γ_{1} - I m (Γ_{13}) \sin γ_{1}] \\ + 2 f [R e (Γ_{14}) \cos γ_{2} - I m (Γ_{14}) \sin γ_{2}], \end{array} \end{array}

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\begin{array}{r} \begin{array}{lll} D_{1} a_{2} & = & R e (Γ_{21}) a_{2} + a_{1} [R e (Γ_{22}) \cos γ_{1} + I m (Γ_{22}) \sin γ_{1}], \end{array} \end{array}

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\begin{array}{r} \begin{array}{lll} D_{1} γ_{1} & = & σ_{1} + I m (Γ_{21}) - \frac{a_{1}}{a_{2}} [R e (Γ_{22}) \sin γ_{1} \\ - I m (Γ_{22}) \cos γ_{1}] - I m (Γ_{11}) \\ - \frac{1}{4} a_{1}^{2} I m (Γ_{12}) - \frac{a_{2}}{a_{1}} [R e (Γ_{13}) \sin γ_{1} + I m (Γ_{13}) \cos γ_{1}] \\ - \frac{2 f}{a_{1}} [R e (Γ_{14}) \sin γ_{2} + I m (Γ_{14}) \cos γ_{2}], \end{array} \end{array}

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\begin{array}{r} \begin{array}{lll} D_{1} γ_{2} & = & σ_{0} - I m (Γ_{11}) - \frac{1}{4} a_{1}^{2} I m (Γ_{12}) \\ - \frac{a_{2}}{a_{1}} [R e (Γ_{13}) \sin γ_{1} + I m (Γ_{13}) \cos γ_{1}] \\ - \frac{2 f}{a_{1}} [R e (Γ_{14}) \sin γ_{2} + I m (Γ_{14}) \cos γ_{2}], \end{array} \end{array}

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\begin{array}{r} \begin{array}{lll} 0 & = & R e (Γ_{11}) a_{1} + \frac{1}{4} a_{1}^{3} R e (Γ_{12}) \\ + a_{2} [R e (Γ_{13}) \cos γ_{1} - I m (Γ_{13}) \sin γ_{1}] \\ + 2 f [R e (Γ_{14}) \cos γ_{2} - I m (Γ_{14}) \sin γ_{2}], \end{array} \end{array}

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\begin{array}{r} \begin{array}{lll} 0 & = & R e (Γ_{21}) a_{2} + a_{1} [R e (Γ_{22}) \cos γ_{1} + I m (Γ_{22}) \sin γ_{1}], \end{array} \end{array}

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\begin{array}{r} \begin{array}{lll} 0 & = & σ_{1} + I m (Γ_{21}) - \frac{a_{1}}{a_{2}} [R e (Γ_{22}) \sin γ_{1} \\ - I m (Γ_{22}) \cos γ_{1}] - I m (Γ_{11}) \\ - \frac{1}{4} a_{1}^{2} I m (Γ_{12}) - \frac{a_{2}}{a_{1}} [R e (Γ_{13}) \sin γ_{1} + I m (Γ_{13}) \cos γ_{1}] \\ - \frac{2 f}{a_{1}} [R e (Γ_{14}) \sin γ_{2} + I m (Γ_{14}) \cos γ_{2}], \end{array} \end{array}

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\begin{array}{r} \begin{array}{lll} 0 & = & σ_{0} - I m (Γ_{11}) - \frac{1}{4} a_{1}^{2} I m (Γ_{12}) \\ - \frac{a_{2}}{a_{1}} [R e (Γ_{13}) \sin γ_{1} + I m (Γ_{13}) \cos γ_{1}] \\ - \frac{2 f}{a_{1}} [R e (Γ_{14}) \sin γ_{2} + I m (Γ_{14}) \cos γ_{2}] . \end{array} \end{array}

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\begin{array}{r} y_{3} = \frac{η}{\sqrt{χ^{2} + η^{2}}} a_{2} . \end{array}

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\begin{array}{r} \begin{array}{lll} {(\begin{array}{cccc} D_{1} Δ a_{1} & D_{1} Δ a_{2} & D_{1} Δ γ_{1} & D_{1} Δ γ_{2} \end{array})}^{T} \\ = & \boldsymbol J {(\begin{array}{cccc} Δ a_{1} & Δ a_{2} & Δ γ_{1} & Δ γ_{2} \end{array})}^{T}, \end{array} \end{array}

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\begin{array}{r} \begin{array}{lll} L_{i n d 1} \ddot{q} + R_{1} \dot{q} + \frac{q}{C_{0}} + l B_{1} \dot{x} = V_{0} \cos (Ω τ), \end{array} \end{array}

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\begin{array}{r} \begin{array}{lll} m \ddot{x} + c \dot{x} + k x - l B_{1} \dot{q} + B_{2} I L_{c o i l} = 0, \end{array} \end{array}

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\begin{array}{r} \begin{array}{lll} L_{i n d 2} \dot{I} + R_{2} I - B_{2} L_{c o i l} \dot{x} = 0. \end{array} \end{array}

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Wen-An Jiang, Xin-Dong Ma, Xiu-Jing Han, Li-Qun Chen, Qin-Sheng Bi. Broadband energy harvesting based on one-to-one internal resonance[J]. Chinese Physics B, 2020, 29(10):

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