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Nonlinear vibration energy harvesting with adjustable stiffness, damping and inertia

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Abstract

A novel nonlinear structure with adjustable stiffness, damping and inertia is proposed and studied for vibration energy harvesting. The system consists of an adjustable-inertia system and X-shaped supporting structures. The novelty of the adjustable-inertia design is to enhance the mode coupling property between two orthogonal motion directions, i.e., the translational and rotational directions, which is very helpful for the improvement of the vibration energy harvesting performance. Weakly nonlinear stiffness and damping characteristics can be introduced by the X-shaped supporting structures. Combining the mode coupling effect above and the nonlinear stiffness and damping characteristics of the X-shaped structures, the vibration energy harvesting performance can be significantly enhanced, in both the low frequency range and broadband spectrum. The proposed 2-DOF nonlinear vibration energy harvesting structure can outperform the corresponding 2-DOF linear system and the existing nonlinear harvesting systems. The results in this study provide a novel and effective method for passive structure design of vibration energy harvesting systems to improve efficiency in the low frequency range.

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Acknowledgements

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, 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, and internal Research Grants of Hong Kong Polytechnic University.

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Authors and Affiliations

  1. College of Aerospace and Civil Engineering, Harbin Engineering University, Harbin, People’s Republic of China

    Chunchuan Liu

  2. Department of Mechanical Engineering, Hong Kong Polytechnic University, Kowloon, Hong Kong

    Chunchuan Liu & Xingjian Jing

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

    Chunchuan Liu & Xingjian Jing

  4. Hong Kong Polytechnic University Shenzhen Research Institute, Shenzhen, People’s Republic of China

    Chunchuan Liu & Xingjian Jing

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

Appendix

Appendix

$$\begin{aligned} \Gamma _{11}= & {} \frac{4\tan ^{2}\theta _1 }{(2n_1 +1)^{2}}+\frac{4\tan ^{2}\theta _2 }{(2n_2 +1)^{2}}, \end{aligned}$$
(26)
$$\begin{aligned} \Gamma _{12}= & {} \frac{4d_2 \tan ^{2}\theta _2 }{(2n_2 +1)^{2}}-\frac{4d_1 \tan ^{2}\theta _1 }{(2n_1 +1)^{2}}, \end{aligned}$$
(27)
$$\begin{aligned} \Gamma _{20}= & {} \frac{6\tan \theta _1 }{(2n_1 +1)^{3}l_1 \cos ^{3}\theta _1 }+\frac{6\tan \theta _2 }{(2n_2 +1)^{2}l_2 \cos ^{3}\theta _2 },\nonumber \\ \end{aligned}$$
(28)
$$\begin{aligned} \Gamma _{21}= & {} \frac{6d_1^2 \tan \theta _1 }{(2n_1 +1)^{3}l_1 \cos ^{3}\theta _1 }+\frac{6d_2^2 \tan \theta _2 }{(2n_2 +1)^{2}l_2 \cos ^{3}\theta _2 },\nonumber \\ \end{aligned}$$
(29)
$$\begin{aligned} \Gamma _{22}= & {} \frac{6d_2 \tan \theta _2 }{(2n_2 +1)^{3}l_2 \cos ^{3}\theta _2 }-\frac{6d_1 \tan \theta _1 }{(2n_1 +1)^{3}l_1 \cos ^{3}\theta _1 },\nonumber \\ \end{aligned}$$
(30)
$$\begin{aligned} \Gamma _{30}= & {} \frac{2\left( {3-2\cos 2\theta _1 } \right) }{(2n_1 +1)^{4}l_1^2 \cos ^{6}\theta _1 }+\frac{2\left( {3-2\cos 2\theta _2 } \right) }{(2n_2 +1)^{4}l_2^2 \cos \theta _2^6 },\nonumber \\ \end{aligned}$$
(31)
$$\begin{aligned} \Gamma _{31}= & {} \frac{2d_1^2 \left( {3-2\cos 2\theta _1 } \right) }{(2n_1 +1)^{4}l_1^2 \cos ^{6}\theta _1 }+\frac{2d_2^2 \left( {3-2\cos 2\theta _2 } \right) }{(2n_2 +1)^{4}l_2^2 \cos \theta _2^6 },\nonumber \\ \end{aligned}$$
(32)
$$\begin{aligned} \Gamma _{32}= & {} \frac{2d_2 \left( {3-2\cos 2\theta _2 } \right) }{(2n_1 +1)^{4}l_2^2 \cos ^{6}\theta _2 }-\frac{2d_1 \left( {3-2\cos 2\theta _1 } \right) }{(2n_1 +1)^{4}l_1^2 \cos ^{6}\theta _1 },\nonumber \\ \end{aligned}$$
(33)
$$\begin{aligned} \Gamma _{33}= & {} \frac{2d_2^3 \left( {3-2\cos 2\theta _2 } \right) }{(2n_1 +1)^{4}l_2^2 \cos ^{6}\theta _2 }-\frac{2d_1^3 \left( {3-2\cos 2\theta _1 } \right) }{(2n_1 +1)^{4}l_1^2 \cos ^{6}\theta _1 },\nonumber \\ \end{aligned}$$
(34)
$$\begin{aligned} \Pi _{10}= & {} \frac{4d_2 \tan ^{2}\theta _2 }{(2n_2 +1)^{2}}-\frac{4d_1 \tan ^{2}\theta _1 }{(2n_1 +1)^{2}}, \end{aligned}$$
(35)
$$\begin{aligned} \Pi _{11}= & {} \frac{4d_1^2 \tan ^{2}\theta _1 }{(2n_1 +1)^{2}}+\frac{4d_2^2 \tan ^{2}\theta _2 }{(2n_2 +1)^{2}}, \end{aligned}$$
(36)
$$\begin{aligned} \Pi _{20}= & {} \frac{6d_2 \tan \theta _2 }{(2n_2 +1)^{3}l_2 \cos ^{3}\theta _2 }-\frac{6d_1 \tan \theta _1 }{(2n_1 +1)^{3}l_1^1 \cos ^{3}\theta _1 },\nonumber \\ \end{aligned}$$
(37)
$$\begin{aligned} \Pi _{21}= & {} \frac{6\tan \theta _2 d_2^3 }{(2n_2 +1)^{3}l_2 \cos ^{3}\theta _2 }-\frac{6\tan \theta _1 d_1^3 }{(2n_1 +1)^{3}l_1^1 \cos ^{3}\theta _1 },\nonumber \\ \end{aligned}$$
(38)
$$\begin{aligned} \Pi _{22}= & {} \frac{6d_1^2 \tan \theta _1 }{(2n_1 +1)^{3}l_1^1 \cos ^{3}\theta _1 }+\frac{6d_2^2 \tan \theta _2 }{(2n_2 +1)^{2}l_2 \cos ^{3}\theta _2 },\nonumber \\ \end{aligned}$$
(39)
$$\begin{aligned} \Pi _{30}= & {} \frac{2\left( {3-2\cos 2\theta _2 } \right) d_2 }{(2n_1 +1)^{4}l_2^2 \cos ^{6}\theta _2 }-\frac{2\left( {3-2\cos 2\theta _1 } \right) d_1 }{(2n_1 +1)^{4}l_1^2 \cos ^{6}\theta _1 },\nonumber \\ \end{aligned}$$
(40)
$$\begin{aligned} \Pi _{31}= & {} \frac{2\left( {3-2\cos 2\theta _2 } \right) d_2^3 }{(2n_1 +1)^{4}l_2^2 \cos ^{6}\theta _2 }-\frac{2\left( {3-2\cos 2\theta _1 } \right) d_1^3 }{(2n_1 +1)^{4}l_1^2 \cos ^{6}\theta _1 },\nonumber \\ \end{aligned}$$
(41)
$$\begin{aligned} \Pi _{32}= & {} \frac{2\left( {3-2\cos 2\theta _1 } \right) d_1^2 }{(2n_1 +1)^{4}l_1^2 \cos ^{6}\theta _1 }+\frac{2\left( {3-2\cos 2\theta _2 } \right) d_2^2 }{(2n_2 +1)^{4}l_2^2 \cos \theta _2^6 },\nonumber \\ \end{aligned}$$
(42)
$$\begin{aligned} \Pi _{33}= & {} \frac{2\left( {3-2\cos 2\theta _1 } \right) d_1^4 }{(2n_1 +1)^{4}l_1^2 \cos ^{6}\theta _1 }+\frac{2\left( {3-2\cos 2\theta _2 } \right) d_2^4 }{(2n_2 +1)^{4}l_2^2 \cos \theta _2^6 },\nonumber \\ \end{aligned}$$
(43)
$$\begin{aligned} \Lambda _{01}= & {} \frac{4\tan ^{2}\theta _1 }{(2n_1 +1)^{2}}+\frac{4\tan ^{2}\theta _2 }{(2n_2 +1)^{2}}, \end{aligned}$$
(44)
$$\begin{aligned} \Lambda _{02}= & {} \frac{4d_2 \tan ^{2}\theta _2 }{(2n_2 +1)^{2}}-\frac{4d_1 \tan ^{2}\theta _1 }{(2n_1 +1)^{2}}, \end{aligned}$$
(45)
$$\begin{aligned} \Lambda _{11}= & {} \frac{8\tan \theta _1 }{(2n_1 +1)^{3}l_1 \cos ^{3}\theta _1 }+\frac{8\tan \theta _2 }{(2n_2 +1)^{3}l_2 \cos ^{3}\theta _2 },\nonumber \\ \end{aligned}$$
(46)
$$\begin{aligned} \Lambda _{12}= & {} \frac{8d_2 \tan \theta _2 }{(2n_2 +1)^{3}l_2 \cos ^{3}\theta _2 }-\frac{8d_1 \tan \theta _1 }{(2n_1 +1)^{3}l_1 \cos ^{3}\theta _1 },\nonumber \\ \end{aligned}$$
(47)
$$\begin{aligned} \Lambda _{20}= & {} \frac{4(1+3\sin ^{2}\theta _1 )}{(2n_1 +1)^{4}l_1^2 \cos ^{6}\theta _1 }+\frac{4(1+3\sin ^{2}\theta _2 )}{(2n_1 +1)^{4}l_2^2 \cos ^{6}\theta _2 },\nonumber \\ \end{aligned}$$
(48)
$$\begin{aligned} \Lambda _{21}= & {} \frac{4(1+3\sin ^{2}\theta _1 )d_1^2 }{(2n_1 +1)^{4}l_1^2 \cos ^{6}\theta _1 }+\frac{4(1+3\sin ^{2}\theta _2 )d_2^2 }{(2n_1 +1)^{4}l_2^2 \cos ^{6}\theta _2 }, \nonumber \\\end{aligned}$$
(49)
$$\begin{aligned} \Lambda _{22}= & {} \frac{4(1+3\sin ^{2}\theta _2 )d_2 }{(2n_2 +1)^{4}l_2^2 \cos ^{6}\theta _2 }-\frac{4(1+3\sin ^{2}\theta _1 )d_1 }{(2n_1 +1)^{4}l_1^2 \cos ^{6}\theta _1 }. \nonumber \\\end{aligned}$$
(50)
$$\begin{aligned} {\Lambda }'_{01}= & {} \frac{4d_1^2 \tan ^{2}\theta _1 }{(2n_1 +1)^{2}}+\frac{4d_2^2 \tan ^{2}\theta _2 }{(2n_2 +1)^{2}}, \end{aligned}$$
(51)
$$\begin{aligned} {\Lambda }'_{11}= & {} \frac{8d_1^2 \tan \theta _1 }{(2n_1 +1)^{3}l_1 \cos ^{3}\theta _1 }+\frac{8d_2^2 \tan \theta _2 }{(2n_2 +1)^{3}l_2 \cos ^{3}\theta _2 }, \nonumber \\\end{aligned}$$
(52)
$$\begin{aligned} {\Lambda }'_{12}= & {} \frac{8d_2^3 \tan \theta _2 }{(2n_2 +1)^{3}l_2 \cos ^{3}\theta _2 }-\frac{8d_1^3 \tan \theta _1 }{(2n_1 +1)^{3}l_1 \cos ^{3}\theta _1 },\nonumber \\ \end{aligned}$$
(53)
$$\begin{aligned} {\Lambda }'_{21}= & {} \frac{4(1+3\sin ^{2}\theta _1 )d_1^4 }{(2n_1 +1)^{4}l_1^2 \cos ^{6}\theta _1 }+\frac{4(1+3\sin ^{2}\theta _2 )d_2^4 }{(2n_1 +1)^{4}l_2^2 \cos ^{6}\theta _2 }, \nonumber \\\end{aligned}$$
(54)
$$\begin{aligned} {\Lambda }'_{22}= & {} \frac{4(1+3\sin ^{2}\theta _2 )d_2^3 }{(2n_2 +1)^{4}l_2^2 \cos ^{6}\theta _2 }-\frac{4(1+3\sin ^{2}\theta _1 )d_1^3 }{(2n_1 +1)^{4}l_1^2 \cos ^{6}\theta _1 },\nonumber \\ \end{aligned}$$
(55)

where the coefficients \(\Gamma _{11}\), \(\Gamma _{12}\), \(\Gamma _{20}\), \(\Gamma _{21}\), \(\Gamma _{22}\), \(\Gamma _{30}\), \(\Gamma _{31}\), \(\Gamma _{32}\), \(\Gamma _{33}\), \(\Pi _{10}\), \(\Pi _{11}\), \(\Pi _{20}\), \(\Pi _{21}\), \(\Pi _{22}\), \(\Pi _{30}\), \(\Pi _{31}\), \(\Pi _{32}\), \(\Pi _{33}\), \(\Lambda _{01}\), \(\Lambda _{02}\), \(\Lambda _{11}\), \(\Lambda _{12}\), \(\Lambda _{20}\), \(\Lambda _{21}\), \(\Lambda _{22}\), \({\Lambda }'_{01}\), \({\Lambda }'_{11} \), \({\Lambda }'_{12}\), \({\Lambda }'_{21} \) and \({\Lambda }'_{22} \) listed in Appendix A are the constant coefficients composed by the structure parameters \(n_{1}\), \(n_{2}\), \(d_{1}\), \(d_{2}\), \(\theta _{1}\) and \(\theta _{2}\).

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Liu, C., Jing, X. Nonlinear vibration energy harvesting with adjustable stiffness, damping and inertia. Nonlinear Dyn 88, 79–95 (2017). https://doi.org/10.1007/s11071-016-3231-1

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