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Vibration energy harvesting with a nonlinear structure

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Abstract

In the paper, beneficial nonlinearities incurred by an X-shape structure are explored for advantageous vibration energy harvesting performance. To this aim, a nonlinear structure beneficial for vibration energy harvesting is proposed, which is composed by X-shape supporting structures and a rigid body. By designing structure nonlinearities, which are determined by several key structure parameters, the power output peak of the harvesting system can be much improved and the effective frequency bandwidth for energy harvesting can be obviously increased, especially at the low frequency range. A coupling effect can be created among nonlinear stiffness and damping characteristics by constructing a 2-DOF vibration system, which has great influence on the energy harvesting performance. The proposed nonlinear energy harvesting systems can obviously outperform the corresponding linear systems in the whole frequency range and also demonstrate advantages compared with some other existing nonlinear energy harvesting systems in the literature. The results in this study provide a novel and practical method for the design of effective and efficient energy harvesting systems (especially in the low frequency range).

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Acknowledgments

The authors gratefully acknowledge the support from a GRF project of Hong Kong RGC (Ref No.15206514), NSFC projects (No. 61374041 and 11402067) of China, Internal Competitive Research Grants of Hong Kong Polytechnic University, and a grant from the Innovation and Technology Commission of the HKSAR Government to the Hong Kong Branch of National Rail Transit Electrification and Automation Engineering Technology Research Center.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Hong Kong Polytechnic University, Hung Hom, HK, People’s Republic of China

    Chunchuan Liu & Xingjian Jing

  2. Hong Kong Branch of National Rail Transit Electrification and Automation Engineering Technology Research Center, Hong Kong, People’s Republic of China

    Chunchuan Liu

  3. School of Mechatronics Engineering, Harbin Institute of Technology, Harbin, People’s Republic of China

    Chunchuan Liu

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  1. Chunchuan Liu
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  2. Xingjian Jing
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Correspondence to Xingjian Jing.

Appendices

Appendix 1

$$\begin{aligned}&f_{11} ({\tilde{y}},{\tilde{\psi }})=\frac{l_1 \cos \theta _1 \left( {l_1 \sin \theta _1 +\frac{{\tilde{y}}-{\tilde{\psi }}d_1 }{2n_1 +1}} \right) }{(2n_1 +1)\sqrt{l_1^2 -\left( {l_1 \sin \theta _1 +\frac{{\tilde{y}}-{\tilde{\psi }}d_1 }{2n_1 +1}} \right) ^{2}}}\nonumber \\&\quad -\frac{\left( {l_1 \sin \theta _1 +\frac{{\tilde{y}}-{\tilde{\psi }}d_1 }{2n_1 +1}} \right) }{(2n_1 +1)}, \end{aligned}$$
(43)
$$\begin{aligned}&f_{12} ({\tilde{y}},{\tilde{\psi }})=\frac{l_2 \cos \theta _2 \left( {l_2 \sin \theta _2 +\frac{{\tilde{y}}+{\tilde{\psi }}d_1 }{2n_2 +1}} \right) }{(2n_2 +1)\sqrt{l_2^2 -\left( {l_2 \sin \theta _2 +\frac{{\tilde{y}}+{\tilde{\psi }}d_1 }{2n_2 +1}} \right) ^{2}}}\nonumber \\&-\frac{\left( {l_2 \sin \theta _2 +\frac{{\tilde{y}}+{\tilde{\psi }}d_1 }{2n_2 +1}} \right) }{2n_2 +1}, \end{aligned}$$
(44)
$$\begin{aligned}&f_{21} ({\tilde{y}},{\tilde{\psi }})=-\frac{dl_1 \cos \theta _1 \left( {l_1 \sin \theta _1 +\frac{{\tilde{y}}-{\tilde{\psi }}d_1 }{2n_1 +1}} \right) }{(2n_1 +1)\sqrt{l_1^2 -\left( {l_1 \sin \theta _1 +\frac{{\tilde{y}}-{\tilde{\psi }}d_1 }{2n_1 +1}} \right) ^{2}}}\nonumber \\&+\,\frac{d\left( {l_1 \sin \theta _1 +\frac{{\tilde{y}}-{\tilde{\psi }}d_1 }{2n_1 +1}} \right) }{2n_1 +1}, \end{aligned}$$
(45)
$$\begin{aligned}&f_{22} ({\tilde{y}},{\tilde{\psi }})=\frac{dl_2 \cos \theta _2 \left( {l_2 \sin \theta _2 +\frac{{\tilde{y}}+{\tilde{\psi }}d_1 }{2n_2 +1}} \right) }{(2n_2 +1)\sqrt{l_2^2 -\left( {l_2 \sin \theta _2 +\frac{{\tilde{y}}+{\tilde{\psi }}d_1 }{2n_2 +1}} \right) ^{2}}}\nonumber \\&-\frac{d\left( {l_2 \sin \theta _2 +\frac{{\tilde{y}}+{\tilde{\psi }}d_1 }{2n_2 +1}} \right) }{2n_2 +1}, \end{aligned}$$
(46)
$$\begin{aligned}&g_{11} ({\tilde{y}},{\tilde{\psi }})=\frac{\left( {l_1 \sin \theta _1 +\frac{{\tilde{y}}-{\tilde{\psi }}d_1 }{2n_1 +1}} \right) ^{2}}{(2n_1 +1)^{2}\left[ {l_1^2 -\left( {l_1 \sin \theta _1 +\frac{{\tilde{y}}-{\tilde{\psi }}d_1 }{2n_1 +1}} \right) ^{2}} \right] }, \end{aligned}$$
(47)
$$\begin{aligned}&g_{12} ({\tilde{y}},{\tilde{\psi }})=\frac{\left( {l_2 \sin \theta _2 +\frac{{\tilde{y}}+{\tilde{\psi }}d_1 }{2n_2 +1}} \right) ^{2}}{(2n_2 +1)^{2}\left[ {l_2^2 -\left( {l_2 \sin \theta _2 +\frac{{\tilde{y}}+{\tilde{\psi }}d_1 }{2n_2 +1}} \right) ^{2}} \right] }, \end{aligned}$$
(48)
$$\begin{aligned}&g_{21} ({\tilde{y}},{\tilde{\psi }})=\frac{d_1^2 \left( {l_1 \sin \theta _1 +\frac{{\tilde{y}}-{\tilde{\psi }}d_1 }{2n_1 +1}} \right) ^{2}}{(2n_1 +1)^{2}\left[ {l_1^2 -\left( {l_1 \sin \theta _1 +\frac{{\tilde{y}}-{\tilde{\psi }}d_1 }{2n_1 +1}} \right) ^{2}} \right] },\nonumber \\ \end{aligned}$$
(49)
$$\begin{aligned}&g_{22} ({\tilde{y}},{\tilde{\psi }})=\frac{d_1^2 \left( {l_2 \sin \theta _2 +\frac{{\tilde{y}}+{\tilde{\psi }}d_1 }{2n_2 +1}} \right) ^{2}}{(2n_2 +1)^{2}\left[ {l_2^2 -\left( {l_2 \sin \theta _2 +\frac{{\tilde{y}}+{\tilde{\psi }}d_1 }{2n_2 +1}} \right) ^{2}} \right] }, \end{aligned}$$
(50)
$$\begin{aligned}&g_{01} ({\tilde{y}},{\tilde{\psi }})=-\frac{d_1 \left( {l_1 \sin \theta _1 +\frac{{\tilde{y}}-{\tilde{\psi }}d_1 }{2n_1 +1}} \right) ^{2}}{(2n_1 +1)^{2}\left[ {l_1^2 -\left( {l_1 \sin \theta _1 +\frac{{\tilde{y}}-{\tilde{\psi }}d_1 }{2n_1 +1}} \right) ^{2}} \right] }, \end{aligned}$$
(51)
$$\begin{aligned}&g_{02} ({\tilde{y}},{\tilde{\psi }})=\frac{d_1 \left( {l_2 \sin \theta _2 +\frac{{\tilde{y}}+{\tilde{\psi }}d_1 }{2n_2 +1}} \right) ^{2}}{(2n_2 +1)^{2}\left[ {l_2^2 -\left( {l_2 \sin \theta _2 +\frac{{\tilde{y}}+{\tilde{\psi }}d_1 }{2n_2 +1}} \right) ^{2}} \right] },\nonumber \\ \end{aligned}$$
(52)

Appendix 2

$$\begin{aligned} \varGamma _{11}= & {} \frac{4\tan ^{2}\theta _1 }{(2n_1 +1)^{2}}+\frac{4\tan ^{2}\theta _2 }{(2n_2 +1)^{2}}, \end{aligned}$$
(53)
$$\begin{aligned} \varGamma _{12}= & {} \frac{4\tan ^{2}\theta _2 }{(2n_2 +1)^{2}}-\frac{4\tan ^{2}\theta _1 }{(2n_1 +1)^{2}},\end{aligned}$$
(54)
$$\begin{aligned} \varGamma _{21}= & {} \frac{6\tan \theta _1 }{(2n_1 +1)^{3}l_1^1 \cos ^{3}\theta _1 }\nonumber \\&+\frac{6\tan \theta _2 }{(2n_2 +1)^{3}l_2 \cos ^{3}\theta _2 },\end{aligned}$$
(55)
$$\begin{aligned} \varGamma _{22}= & {} \frac{6\tan \theta _2 }{(2n_2 +1)^{3}l_2 \cos ^{3}\theta _2 }\nonumber \\&-\frac{6\tan \theta _1 }{(2n_1 +1)^{3}l_1^1 \cos ^{3}\theta _1 },\end{aligned}$$
(56)
$$\begin{aligned} \varGamma _{31}= & {} \frac{2\left( {3-2\cos 2\theta _1 } \right) }{(2n_1 +1)^{4}l_1^2 \cos ^{6}\theta _1 }\nonumber \\&+\frac{2\left( {3-2\cos 2\theta _2 } \right) }{(2n_2 +1)^{4}l_2^2 \cos \theta _2^6 },\end{aligned}$$
(57)
$$\begin{aligned} \varGamma _{32}= & {} \frac{2\left( {3-2\cos 2\theta _2 } \right) }{(2n_1 +1)^{4}l_2^2 \cos ^{6}\theta _2 }\nonumber \\&-\frac{2\left( {3-2\cos 2\theta _1 } \right) }{(2n_1 +1)^{4}l_1^2 \cos ^{6}\theta _1 },\end{aligned}$$
(58)
$$\begin{aligned} \varLambda _{01}= & {} \frac{4\tan ^{2}\theta _1 }{(2n_1 +1)^{2}}+\frac{4\tan ^{2}\theta _2 }{(2n_2 +1)^{2}},\end{aligned}$$
(59)
$$\begin{aligned} \varLambda _{02}= & {} \frac{4\tan ^{2}\theta _2 }{(2n_2 +1)^{2}}-\frac{4\tan ^{2}\theta _1 }{(2n_1 +1)^{2}},\end{aligned}$$
(60)
$$\begin{aligned} \varLambda _{11}= & {} \frac{8\tan \theta _1 }{(2n_1 +1)^{3}l_1 \cos ^{3}\theta _1 }\nonumber \\&+\frac{8\tan \theta _2 }{(2n_2 +1)^{3}l_2 \cos ^{3}\theta _2 },\end{aligned}$$
(61)
$$\begin{aligned} \varLambda _{12}= & {} \frac{8\tan \theta _2 }{(2n_2 +1)^{3}l_2 \cos ^{3}\theta _2 }\nonumber \\&-\frac{8\tan \theta _1 }{(2n_1 +1)^{3}l_1 \cos ^{3}\theta _1 },\end{aligned}$$
(62)
$$\begin{aligned} \varLambda _{21}= & {} \frac{4(1+3\sin ^{2}\theta _1 )}{(2n_1 +1)^{4}l_1^2 \cos ^{6}\theta _1 }\nonumber \\&+\frac{4(1+3\sin ^{2}\theta _2 )}{(2n_1 +1)^{4}l_2^2 \cos ^{6}\theta _2 },\end{aligned}$$
(63)
$$\begin{aligned} \varLambda _{22}= & {} \frac{4(1+3\sin ^{2}\theta _2 )}{(2n_2 +1)^{4}l_2^2 \cos ^{6}\theta _2 }\nonumber \\&-\frac{4(1+3\sin ^{2}\theta _1 )}{(2n_1 +1)^{4}l_1^2 \cos ^{6}\theta _1 }. \end{aligned}$$
(64)

Appendix 3

$$\begin{aligned}&A_0 \varGamma _{11} \!+\!d_1 B_0 \varGamma _{12} \!+\!\left( A_0^2 \!+\!B_0^2 d_1^2 \!+\!\frac{A_1^2 \!+\!d_1^2 B_1^2 }{2}\right) \varGamma _{21}\nonumber \\&\quad +\,d_1 A_0 B_0 \varGamma _{22} +\left( A_0^3 +3d_1^2 A_0 B_0^2 \right. \nonumber \\&\left. \quad +\,\frac{3A_0 A_1^2 +3d_1^2 A_0 B_1^2 }{2}\right) \varGamma _{31} +\left( 3d_1 B_0 A_0^2 +d_1^3 B_0^3\right. \nonumber \\&\left. \quad +\,\frac{3d_1 A_1^2 B_0 +3d_1^3 B_0 B_1^2 }{2}\right) \varGamma _{32} +\left( d_1 A_1 B_1 \varGamma _{22} \right. \nonumber \\&\left. \quad +\,6d_1^2 A_1 B_0 B_1 \varGamma _{31} +6d_1 A_0 A_1 B_1 \varGamma _{32} \right) \nonumber \\&\quad \times \frac{\cos (\varphi _1 -\varphi _2 )}{2}=0, \end{aligned}$$
(65)
$$\begin{aligned}&A_0 \varGamma _{12} +d_1 B_0 \varGamma _{11} +d_1 A_0 B_0 \varGamma _{21}\nonumber \\&\left. \quad +\,\left( A_0^2 +d_1^2 B_0^2 +\frac{A_1^2 +d_1^2 B_1^2 }{2}\right) \right. \nonumber \\&\varGamma _{22} +\left( 3d_1 B_0 A_0^2 +d_1^3 B_0^3\right. \nonumber \\&\left. \quad +\,\frac{3d_1 A_1^2 B_0 +3d_1^3 B_0 B_1^2 }{2}\right) \varGamma _{31} +\left( A_0^3 +3d_1^2 A_0 B_0^2\right. \nonumber \\&\left. \quad +\,\frac{3A_0 A_1^2 +3d_1^2 A_0 B_1^2 }{2}\right) \varGamma _{32} \nonumber \\&\quad +\,\left( \frac{1}{2}d_1 A_1 B_1 \varGamma _{21} +3d_1^2 A_1 B_0 B_1 \varGamma _{32}\nonumber \right. \\&\left. \quad +\,3d_1 A_0 A_1 B_1 \varGamma _{31}\right) \cos (\varphi _1 -\varphi _2)=0, \end{aligned}$$
(66)
$$\begin{aligned}&A_1 \varGamma _{11} +2A_0 A_1 \varGamma _{21} +d_1 B_0 A_1 \varGamma _{22}\nonumber \\&\quad +\,\left( 3A_0^2 A_1 +3d_1^2 A_1 B_0^2 \right. \nonumber \\&\left. \quad +\,\frac{A_1^3 }{4}+\frac{3}{2}d_1^2 A_1 B_1^2 \right) \varGamma _{31} +6d_1 A_0 A_1 B_0 \varGamma _{32} \nonumber \\&\quad +\left[ d_1 B_1 \varGamma _{12} +2d_1^2 B_0 B_1 \varGamma _{21} +d_1 A_0 B_1 \varGamma _{22}\right. \nonumber \\&\left. \quad +\,6d_1^2 A_0 B_0 B_1 \varGamma _{31} +\left( 3d_1 A_0^2 B_1\right. \right. \nonumber \\&\left. \left. \quad +\,3d_1^3 B_0^2 B_1 +\frac{9}{4}d_1 A_1^2 B_1 \right. \right. \nonumber \\&\left. \left. \quad +\,\frac{1}{4}d_1^3 B_1^3 \right) \varGamma _{32} \right] \cos (\varphi _1 -\varphi _2 )\nonumber \\&\quad +\,\frac{3}{4}d_1^2 A_1 B_1^2 \varGamma _{31} \cos 2(\varphi _1 -\varphi _2 )\nonumber \\&\quad +\,2\xi \varOmega d_1 \left[ B_1 \varLambda _{02} +A_0 B_1 \varLambda _{12} \right. \nonumber \\&\left. \quad +\,d_1 B_0 B_1 \varLambda _{11} +2d_1 A_0 B_0 B_1 \varLambda _{21} +\left( A_0^2 B_1 \right. \right. \nonumber \\&\left. \left. \quad +\,d_1^2 B_0^2 B_1 +\frac{3}{4}A_1^2 B_1 +\frac{1}{4}d_1^2 B_1^3 \right) \varLambda _{22} \right] \sin (\varphi _1 -\varphi _2 ) \nonumber \\&\quad +\,\frac{1}{2}\xi \varOmega d_1^2 B_1^2 A_1 \varLambda _{21} \sin 2(\varphi _1 -\varphi _2 )\nonumber \\&\quad -\varOmega ^{2}A_1 -\varOmega ^{2}Z_0 \cos \varphi _1 =0, \end{aligned}$$
(67)
$$\begin{aligned}&\left[ d_1 B_1 \varGamma _{12} +2d_1^2 B_0 B_1 \varGamma _{21} +d_1 A_0 B_1 \varGamma _{22}\right. \nonumber \\&\left. \quad +\,6d_1^2 A_0 B_0 B_1 \varGamma _{31} + \left( 3d_1 A_0^2 B_1 \varGamma _{32}\right. \right. \nonumber \\&\left. \left. \quad +\,3d_1^3 B_0^2 B_1 +\frac{3}{4}d_1 A_1^2 B_1 \right. \right. \nonumber \\&\left. \left. \quad +\,\frac{1}{4}d_1^3 B_1^3 \right) \varGamma _{32} \right] \sin (\varphi _1 -\varphi _2 )\nonumber \\&\quad +\,\frac{3}{4}d_1^2 A_1 B_1^2 \varGamma _{31} \sin 2(\varphi _1 -\varphi _2 )\nonumber \\&\quad -\,2\xi \varOmega \left[ A_1 \varLambda _{01} +A_0 A_1 \varLambda _{11}\right. \nonumber \\&\left. \quad +\,d_1 B_0 A_1 \varLambda _{12} +\left( A_0^2 A_1 +d_1^2 B_0^2 A_1 +\frac{A_1^3 }{4}\right. \right. \nonumber \\&\left. \left. \quad +\,\frac{d_1^2 }{2}A_1 B_1^2 \right) \varLambda _{21} +2d_1 A_0 B_0 A_1 \right. \varLambda _{22} \nonumber \\&\left. \quad +\,\frac{d_1^2 }{4}A_1 B_1^2 \varLambda _{21} \cos 2(\varphi _1-\varphi _2 )\right] \nonumber \\&\quad -2\xi d_1 \varOmega \left[ B_1 \varLambda _{02} +A_0 B_1 \varLambda _{12}\right. \nonumber \\&\left. \quad +\,d_1 B_0 B_1 \varLambda _{11} +2d_1 A_0 B_0 B_1 \varLambda _{21} \right. \nonumber \\&\left. \quad +\left( A_0^2 B_1 +d_1^2 B_0^2 B_1 +\frac{d_1^2 }{4}B_1^3 \right) \varLambda _{22} \right] \cos (\varphi _1 -\varphi _2 )\nonumber \\&\quad -\varOmega ^{2}Z_0 \sin \varphi _1 =0 , \end{aligned}$$
(68)
$$\begin{aligned}&\beta d_1 \left\{ d_1 B_1 \varGamma _{11} +d_1 A_0 B_1 \varGamma _{21} +2d_1^2 B_0 B_1 \varGamma _{22}\right. \nonumber \\&\left. \quad +\left( 3d_1 A_0^2 B_1 +3d_1^3 B_0^2 B_1 +\frac{3}{2}d_1 \varGamma _{31} A_1^2 B_1 +\frac{1}{4}d_1^3 B_1^3 \right) \varGamma _{31} \right. \nonumber \\&\left. \quad +\,6d_1^2 A_0 B_0 B_1 \varGamma _{32} +\left[ A_1 \varGamma _{12} +d_1 B_0 A_1 \varGamma _{21}\right. \right. \nonumber \\&\left. \left. \quad +\,2A_0 A_1 \varGamma _{22} +6d_1 A_0 A_1 B_0 \varGamma _{31} +\left( 3A_0^2 A_1 +3d_1^2 A_1 B_0^2 \right. \right. \right. \nonumber \\&\left. \left. \left. \quad +\,\frac{A_1^3 }{4}+\frac{9}{4}d_1^2 A_1 B_1^2 \right) \varGamma _{32} \right] \cos (\varphi _1 -\varphi _2 )\right. \nonumber \\&\left. \quad +\,\frac{3}{4}d_1 A_1^2 B_1 \varGamma _{31} \cos 2(\varphi _1 -\varphi _2 )\right. \nonumber \\&\left. \quad -\,2\xi \varOmega \left[ A_1 \varLambda _{02} +d_1 B_0 A_1 \varLambda _{11} \right. \right. \nonumber \\&\left. \left. \quad +\,A_0 A_1 \varLambda _{12} +2d_1 A_0 B_0 A_1 \varLambda _{21} +\left( A_0^2 A_1 +d_1^2 B_0^2 A_1\right. \right. \right. \nonumber \\&\left. \left. \left. \quad +\,\frac{1}{4}A_1^3 +\frac{1}{2}d_1^2 B_1^2 A_1 \right) \varLambda _{22} \right] \sin (\varphi _1 -\varphi _2 ) \right. \nonumber \\&\left. \quad -\xi \varOmega d_1 A_1^2 B_1 \varLambda _{21} \sin 2(\varphi _1 -\varphi _2 )\right\} \nonumber \\&\quad -\varOmega ^{2}B_1 -\varOmega ^{2}\Psi _0 \cos \varphi _2=0, \end{aligned}$$
(69)
$$\begin{aligned}&\beta d_1 \left\{ \left[ A_1 \varGamma _{12} +2A_0 A_1 \varGamma _{22} +d_1 B_0 A_1 \varGamma _{21}\right. \right. \nonumber \\&\left. \left. \quad +\,6d_1 A_0 A_1 B_0 \varGamma _{31} -\frac{3}{4}d_1 A_1^2 B_1 \varGamma _{31} \sin 2(\varphi _1 -\varphi _2 ) \right. \right. \nonumber \\&\left. \left. \quad +\left( 3A_0^2 A_1 +3d_1^2 A_1 B_0^2 +\frac{A_1^3 }{4}\right) \varGamma _{32} \right] \sin (\varphi _1 -\varphi _2 )\right. \nonumber \\&\left. \quad +\,2\xi d_1 \varOmega \left( B_1 \varLambda _{01} +A_0 B_1 \varLambda _{11} +d_1 B_0 B_1 \varLambda _{12} \right. \right. \nonumber \\&\left. \left. \quad +\,A_0^2 B_1 \varLambda _{21} +d_1^2 B_0^2 B_1 \varLambda _{21} +2d_1 A_0 B_0 B_1 \varLambda _{22}\right. \right. \nonumber \\&\left. \left. \quad +\,\frac{1}{2}A_1^2 B_1 \varLambda _{21} +\frac{1}{4}d_1^2 B_1^3 \varLambda _{21} \right) +2\xi \varOmega \left[ \left( \varLambda _{02} A_1 \right. \right. \right. \nonumber \\&\left. \left. \left. \quad +\,A_0 A_1 \varLambda _{12} +d_1 B_0 A_1 \varLambda _{11} +A_0^2 A_1 \varLambda _{22} +d_1^2 B_0^2 A_1 \varLambda _{22}\right. \right. \right. \nonumber \\&\left. \left. \left. \quad +\,2d_1 A_0 B_0 A_1 \varLambda _{21} +\frac{A_1^3 }{4}\varLambda _{22} \right. \right. \right. \nonumber \\&\left. \left. \left. \quad +\,\frac{3}{4}d_1^2 B_1^2 A_1 \varLambda _{22} \right) \cos (\varphi _1 -\varphi _2 )\right. \right. \nonumber \\&\left. \left. \quad +\,\frac{1}{2}d_1 A_1^2 B_1 \varLambda _{21} \cos 2(\varphi _1 -\varphi _2 )\right] \right\} +\varOmega ^{2}\Psi _0 \sin \varphi _2 =0,\nonumber \\ \end{aligned}$$
(70)

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Liu, C., Jing, X. Vibration energy harvesting with a nonlinear structure. Nonlinear Dyn 84, 2079–2098 (2016). https://doi.org/10.1007/s11071-016-2630-7

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