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Vibration energy harvesting under concurrent base and flow excitations with internal resonance

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Abstract

In this study, internal resonance is investigated to further explore the potential of energy harvesting under concurrent base and flow excitations. The effects of system parameters on the performance of energy harvester with three-to-one internal resonance are analyzed analytically. At first, a lumped-parameter model for the energy harvester, which consists of a two-degree-of-freedom airfoil with the piezoelectric coupling introduced to the plunging motion, is established by using a nonlinear quasi-steady aerodynamic model. Subsequently, the method of multiple scales is implemented to derive the approximate analytic solution of the energy harvesting system under three-to-one internal resonance. Then, the bifurcation characteristics of the energy harvester with respect to various system parameters are analyzed. Finally, the numerical solutions are presented to validate the accuracy of the approximate analytic solutions. The study shows that the harvested voltage and power of the energy harvester can be significantly improved in the presence of internal resonance. In addition, the analytic solutions of internal resonance and the bifurcation analysis can provide an essential reference for design of such a kind of energy harvester.

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Acknowledgements

This work was supported by the Starting Research Foundation of Nanjing Tech University under grant 3827400225.

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Authors and Affiliations

  1. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, China

    Haojie Liu

  2. School of Physical and Mathematical Sciences, Nanjing Tech University, Nanjing, 210016, People’s Republic of China

    Xiumin Gao

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  1. Haojie Liu
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  2. Xiumin Gao
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Correspondence to Xiumin Gao.

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Appendices

Appendix A

The details of \(a_{ij} \) in Eqs. (17) and (18) read

$$\begin{aligned} a_{11}= & {} 3\mathbf{p}_{12}^3 +6U_\infty \omega _1^2 b_0 \mathbf{p}_{11} \mathbf{p}_{12}^2 +3b_0^2 \omega _1^2 \mathbf{p}_{12}^3 \\&+\,3U_\infty ^2 \omega _1^2 \mathbf{p}_{11}^2 \mathbf{p}_{12} +\mathrm{i}(3\mathbf{p}_{12}^2 U_\infty \omega _1 \mathbf{p}_{11} +3b_0^3 \omega _1^3 \mathbf{p}_{12}^3 \\&+\,3U_\infty ^3 \omega _1^3 \mathbf{p}_{11}^3 +9U_\infty b_0^2 \omega _1^3 \mathbf{p}_{11} \mathbf{p}_{12}^2 \\&+\,9U_\infty ^2 \omega _1^3 b_0 \mathbf{p}_{11}^2 \mathbf{p}_{12} +3b_0 \omega _1 \mathbf{p}_{12}^3 ) \\= & {} \;a_{110} +\mathrm{i}a_{111} \\ a_{12}= & {} 6\mathbf{p}_{12} \mathbf{p}_{22}^2 +12U_\infty b_0 \omega _2^2 \mathbf{p}_{12} \mathbf{p}_{21} \mathbf{p}_{22} \\&+\,6U_\infty ^2 \omega _2^2 \mathbf{p}_{12} \mathbf{p}_{21}^2 +6b_0^2 \omega _2^2 \mathbf{p}_{12} \mathbf{p}_{22}^2 \\&+\,\mathrm{i}(6U_\infty \omega _1 \mathbf{p}_{11} \mathbf{p}_{22}^2+12U_\infty ^2 \omega _1 \omega _2^2 b_0 \mathbf{p}_{11} \mathbf{p}_{21} \mathbf{p}_{22} \\&+\,12U_\infty b_0^2 \omega _1 \omega _2^2 \mathbf{p}_{12} \mathbf{p}_{21} \mathbf{p}_{22} \\&+\,6U_\infty b_0^2 \omega _1 \omega _2^2 \mathbf{p}_{11} \mathbf{p}_{22}^2 \\&+\,6U_\infty ^3 \omega _1 \omega _2^2 \mathbf{p}_{11} \mathbf{p}_{21}^2 +6b_0 \omega _1 \mathbf{p}_{12} \mathbf{p}_{22}^2\\&+6b_0^3 \omega _1 \omega _2^2 \mathbf{p}_{12} \mathbf{p}_{22}^2 \\&+\,6U_\infty ^2 b_0 \omega _1 \omega _2^2 \mathbf{p}_{12} \mathbf{p}_{21}^2 ) \\= & {} a_{120} +\mathrm{i}a_{121} \\ a_{13}= & {} 6b_0^2 \omega _1 \omega _2 \mathbf{p}_{12}^2 \mathbf{p}_{22} +6U_\infty ^2 \omega _1 \omega _2 \mathbf{p}_{11} \mathbf{p}_{12} \mathbf{p}_{21} \\&-\,3U_\infty ^2 \omega _1^2 \mathbf{p}_{11}^2 \mathbf{p}_{22} -6U_\infty \omega _1^2 b_0 \mathbf{p}_{11} \mathbf{p}_{12} \mathbf{p}_{22} \\&-\,3b_0^2 \omega _1^2 \mathbf{p}_{12}^2 \mathbf{p}_{22} +6U_\infty b_0 \omega _1 \omega _2 \mathbf{p}_{11} \mathbf{p}_{12} \mathbf{p}_{22}\\&+\,6U_\infty b_0 \omega _1 \omega _2 \mathbf{p}_{12}^2 \mathbf{p}_{21} +3\mathbf{p}_{12}^2 \mathbf{p}_{22} \\&+\,\mathrm{i}(-6U_\infty \omega _1 \mathbf{p}_{11} \mathbf{p}_{12} \mathbf{p}_{22} +3b_0 \omega _2 \mathbf{p}_{12}^2 \mathbf{p}_{22} \\&-\,6U_\infty ^2 b_0 \omega _1^2 \omega _2 \mathbf{p}_{11} \mathbf{p}_{12} \mathbf{p}_{21} +3U_\infty \omega _2 \mathbf{p}_{12}^2 \mathbf{p}_{21} \\&-\,6U_\infty \omega _1^2 \omega _2 b_0^2 \mathbf{p}_{11} \mathbf{p}_{12} \mathbf{p}_{22} -6b_0 \omega _1 \mathbf{p}_{12}^2 \mathbf{p}_{22} \\&-\,3U_\infty ^2 b_0 \omega _1^2 \omega _2 \mathbf{p}_{11}^2 \mathbf{p}_{22} -3b_0^3 \omega _1^2 \omega _2 \mathbf{p}_{12}^2 \mathbf{p}_{22} \\&-\,3U_\infty ^3 \omega _1^2 \omega _2 \mathbf{p}_{21} \mathbf{p}_{11}^2 -3U_\infty b_0^2 \omega _1^2 \omega _2 \mathbf{p}_{12}^2 \mathbf{p}_{21} )\\= & {} a_{130} +\mathrm{i}a_{131} \\ a_{21}= & {} 6b_0^2 \omega _1^2 \mathbf{p}_{12}^2 \mathbf{p}_{22} +12U_\infty b_0 \omega _1^2 \mathbf{p}_{11} \mathbf{p}_{12} \mathbf{p}_{22} \\&+\,6U_\infty ^2 \omega _1^2 \mathbf{p}_{11}^2 \mathbf{p}_{22} +6\mathbf{p}_{12}^2 \mathbf{p}_{22} +\mathrm{i}(6b_0^3 \omega _1^2 \omega _2 \mathbf{p}_{12}^2 \mathbf{p}_{22} \\&+\,12U_\infty \omega _1^2 \omega _2 b_0^2 \mathbf{p}_{11} \mathbf{p}_{12} \mathbf{p}_{22} +6b_0 \omega _2 \mathbf{p}_{12}^2 \mathbf{p}_{22} \\&+\,6U_\infty b_0^2 \omega _1^2 \omega _2 \mathbf{p}_{12}^2 \mathbf{p}_{21} \;+6U_\infty \omega _2 \mathbf{p}_{12}^2 \mathbf{p}_{21} \\&+\,6U_\infty ^3 \omega _1^2 \omega _2 \mathbf{p}_{11}^2 \mathbf{p}_{21} +12U_\infty ^2 b_0 \omega _1^2 \omega _2 \mathbf{p}_{11} \mathbf{p}_{12} \mathbf{p}_{21} \\&+\,6U_\infty ^2 b_0 \omega _1^2 \omega _2 \mathbf{p}_{11}^2 \mathbf{p}_{22} ) \\&=a_{210} +\mathrm{i}a_{211} \\ a_{22}= & {} 3b_0^2 \omega _2^2 \mathbf{p}_{22}^3 +3\mathbf{p}_{22}^3 +3U_\infty ^2 \omega _2^2 \mathbf{p}_{22} \mathbf{p}_{21}^2 \\&+\,6U_\infty b_0 \omega _2^2 \mathbf{p}_{21} \mathbf{p}_{22}^2 +\mathrm{i}(3U_\infty ^3 \omega _2^3 \mathbf{p}_{21}^3 +3b_0 \omega _2 \mathbf{p}_{22}^3 \\&+\,9U_\infty b_0^2 \omega _2^3 \mathbf{p}_{21} \mathbf{p}_{22}^2 +3U_\infty \omega _2 \mathbf{p}_{21} \mathbf{p}_{22}^2 \\&+\,3b_0^3 \omega _2^3 \mathbf{p}_{22}^3 +9U_\infty ^2 b_0 \omega _2^3 \mathbf{p}_{22} \mathbf{p}_{21}^2 ) \\&=a_{220} +\mathrm{i}a_{221} \\ a_{31}= & {} 3b_0^2 \omega _1^2 \mathbf{p}_{12}^3 +6U_\infty b_0 \omega _1^2 \mathbf{p}_{12}^2 \mathbf{p}_{11} +3U_\infty ^2 \omega _1^2 \mathbf{p}_{12} \mathbf{p}_{11}^2 \\&-\,\mathbf{p}_{12}^3 +\mathrm{i}(U_\infty ^3 \omega _1^3 \mathbf{p}_{11}^3 -3b_0 \omega _1 \mathbf{p}_{12}^3 \\&+\,3U_\infty ^2 b_0 \omega _1^3 \mathbf{p}_{11}^2 \mathbf{p}_{12} -3U_\infty \omega _1 \mathbf{p}_{12}^2 \mathbf{p}_{11} +b_0^3 \omega _1^3 \mathbf{p}_{12}^3 \\&+\,3U_\infty \omega _1^3 b_0^2 \mathbf{p}_{12}^2 \mathbf{p}_{11} ) \\= & {} a_{310} +\mathrm{i}a_{311} \\ \end{aligned}$$

Appendix B

The details of \(\alpha _i \) and \(\gamma _i \) in Eqs. (22–25) read

$$\begin{aligned} \alpha _2= & {} \mathbf{z}_1^\mathrm{T} \left( \mathbf{C}\omega _1 \mathbf{p}_1 +\mathbf{P}\frac{\chi _p \omega _1 R_0 }{\omega _1^2 +R_0^2 }{} \mathbf{p}_{11} \right) \\= & {} \frac{1}{N_{10} }\left( \Lambda _1 ^{\prime }\left( \omega _1 \Lambda _1 \left( {c_h -\frac{L_s }{U_\infty }} \right) -\omega _1 L_s b_0 \right. \right. \\&\quad \left. +\theta \Lambda _1 \frac{\chi _p \omega _1 R_0 }{\omega _1^2 +R_0^2 } \right) -\frac{T_s }{U_\infty }\omega _1 \Lambda _1 +\omega _1 c_\alpha \\&\quad \left. -\omega _1 T_s b_0 \right) \\ \alpha _3= & {} -\mathbf{z}_1^\mathrm{T} \mathbf{P}\frac{\chi _p \omega _1^2 }{\omega _1^2 +R_0^2 }{} \mathbf{p}_{11} =-\frac{1}{N_{10} }\Lambda _1 ^{\prime }\theta \Lambda _1 \\&\quad \frac{\omega _1^2 \chi _p }{\omega _1^2 +R_0^2 }\\ \alpha _4= & {} -\mathbf{z}_1^\mathrm{T} \mathbf{Q}a_{111} \\= & {} -\frac{1}{N_{10} }\left( {-\Lambda _1 ^{\prime }c_s L_s -c_s T_s } \right) (3\mathbf{p}_{12}^2 U_\infty \omega _1 \mathbf{p}_{11} \\&+\,3b_0^3 \omega _1^3 \mathbf{p}_{12}^3 +3U_\infty ^3 \omega _1^3 \mathbf{p}_{11}^3 +9U_\infty b_0^2 \omega _1^3 \mathbf{p}_{11} \mathbf{p}_{12}^2 \\&+\,9U_\infty ^2 \omega _1^3 b_0 \mathbf{p}_{11}^2 \mathbf{p}_{12} +3b_0 \omega _1 \mathbf{p}_{12}^3 ) \\ \alpha _5= & {} \mathbf{z}_1^\mathrm{T} \mathbf{Q}a_{110} \\= & {} \frac{1}{N_{10} }\left( {-\Lambda _1 ^{\prime }c_s L_s -c_s T_s } \right) \\&\quad \left( 3\mathbf{p}_{12}^3 +6U_\infty \omega _1^2 b_0 \mathbf{p}_{11} \mathbf{p}_{12}^2 +3b_0^2 \omega _1^2 \mathbf{p}_{12}^3\right. \\&\quad \left. +3U_\infty ^2 \omega _1^2 \mathbf{p}_{11}^2 \mathbf{p}_{12} \right) \\ \alpha _6= & {} -\mathbf{z}_1^\mathrm{T} \mathbf{Q}a_{121} \\= & {} -\frac{1}{N_{10} }\left( {-\Lambda _1 ^{\prime }c_s L_s -c_s T_s } \right) (12U_\infty ^2 \omega _1 \omega _2^2 b_0 \\&\quad \mathbf{p}_{11} \mathbf{p}_{21} \mathbf{p}_{22} +6b_0^3 \omega _1 \omega _2^2 \mathbf{p}_{12} \mathbf{p}_{22}^2 \\&+\,12U_\infty b_0^2 \omega _1 \omega _2^2 \mathbf{p}_{12} \mathbf{p}_{21} \mathbf{p}_{22} +6U_\infty ^3 \omega _1 \omega _2^2 \mathbf{p}_{11} \mathbf{p}_{21}^2\\&+6U_\infty b_0^2 \omega _1 \omega _2^2 \mathbf{p}_{11} \mathbf{p}_{22}^2 \\&+\,6U_\infty \omega _1 \mathbf{p}_{11} \mathbf{p}_{22}^2 +6U_\infty ^2 b_0 \omega _1 \omega _2^2 \mathbf{p}_{12} \mathbf{p}_{21}^2 \\&+\,6b_0 \omega _1 \mathbf{p}_{12} \mathbf{p}_{22}^2 ) \\ \alpha _7= & {} \mathbf{z}_1^\mathrm{T} \mathbf{Q}a_{120} \\= & {} \frac{1}{N_{10} }\left( {-\Lambda _1 ^{\prime }c_s L_s -c_s T_s } \right) \\&\left( 6\mathbf{p}_{12} \mathbf{p}_{22}^2 +12U_\infty b_0 \omega _2^2 \mathbf{p}_{12} \mathbf{p}_{21} \mathbf{p}_{22}\right. \\&\left. +6U_\infty ^2 \omega _2^2 \mathbf{p}_{12} \mathbf{p}_{21}^2 +6b_0^2 \omega _2^2 \mathbf{p}_{12} \mathbf{p}_{22}^2 \right) \\ \alpha _8= & {} -\mathbf{z}_1^\mathrm{T} \mathbf{Q}a_{131} \\= & {} -\frac{1}{N_{10} }\left( {-\Lambda _1 ^{\prime }c_s L_s -c_s T_s } \right) (-6U_\infty \omega _1 \mathbf{p}_{11} \mathbf{p}_{12} \mathbf{p}_{22} \\&+\,3b_0 \omega _2 \mathbf{p}_{12}^2 \mathbf{p}_{22} -6U_\infty ^2 b_0 \omega _1^2 \omega _2 \mathbf{p}_{11} \mathbf{p}_{12} \mathbf{p}_{21} \\&-\,6U_\infty \omega _1^2 \omega _2 b_0^2 \mathbf{p}_{11} \mathbf{p}_{12} \mathbf{p}_{22} +3U_\infty \omega _2 \mathbf{p}_{12}^2 \mathbf{p}_{21}\\&-\,3U_\infty ^3 \omega _1^2 \omega _2 \mathbf{p}_{21} \mathbf{p}_{11}^2 -6b_0 \omega _1 \mathbf{p}_{12}^2 \mathbf{p}_{22} \\&-\,3U_\infty b_0^2 \omega _1^2 \omega _2 \mathbf{p}_{12}^2 \mathbf{p}_{21} -3U_\infty ^2 b_0 \omega _1^2 \omega _2 \mathbf{p}_{11}^2 \mathbf{p}_{22}\\&-\,3b_0^3 \omega _1^2 \omega _2 \mathbf{p}_{12}^2 \mathbf{p}_{22}) \\ \alpha _9= & {} \mathbf{z}_1^\mathrm{T} \mathbf{Q}a_{130} \\= & {} \frac{1}{N_{10} }\left( {-\Lambda _1 ^{\prime }c_s L_s -c_s T_s } \right) (6b_0^2 \omega _1 \omega _2 \mathbf{p}_{12}^2 \mathbf{p}_{22}\\&-\,3b_0^2 \omega _1^2 \mathbf{p}_{12}^2 \mathbf{p}_{22} +6U_\infty ^2 \omega _1 \omega _2 \mathbf{p}_{11} \mathbf{p}_{12} \mathbf{p}_{21} +3\mathbf{p}_{12}^2 \mathbf{p}_{22} \\&-\,6U_\infty \omega _1^2 b_0 \mathbf{p}_{11} \mathbf{p}_{12} \mathbf{p}_{22} +6U_\infty b_0 \omega _1 \omega _2 \mathbf{p}_{11} \mathbf{p}_{12} \mathbf{p}_{22} \\&+\,6U_\infty b_0 \omega _1 \omega _2 \mathbf{p}_{12}^2 \mathbf{p}_{21} -3U_\infty ^2 \omega _1^2 \mathbf{p}_{11}^2 \mathbf{p}_{22} ) \\ \alpha _{10}= & {} \mathbf{z}_1^\mathrm{T} \mathbf{P}\frac{\chi _p \Omega ^{2}}{\Omega ^{2}+R_0^2 }z_0 =\frac{1}{N_{10} }\Lambda _1 ^{\prime }\theta \frac{\chi _p \Omega ^{2}}{\Omega ^{2}+R_0^2 }z_0\\ \alpha _{11}= & {} \mathbf{z}_1^\mathrm{T} \left( \mathbf{C}_1 \Omega z_0 +\mathbf{P}\frac{\chi _p \Omega R_0 }{\Omega ^{2}+R_0^2 }z_0 \right) \\= & {} \frac{1}{N_{10} }\Lambda _1 ^{\prime }\left( c_h \Omega +\theta \frac{\chi _p \Omega R_0 }{\Omega ^{2}+R_0^2 }\right) z_0\\ \gamma _2= & {} \mathbf{z}_2^\mathrm{T} (\mathbf{C}\omega _2 \mathbf{p}_2 +\mathbf{P}\frac{\chi _p \omega _2 R_0 }{\omega _2^2 +R_0^2 }{} \mathbf{p}_{21} )\\= & {} \frac{1}{N_{20} }\left( \Lambda _2 ^{\prime }\left( \omega _2 \Lambda _2 \left( {c_h -\frac{L_s }{U_\infty }} \right) -\omega _2 L_s b_0 \right. \right. \\&\quad \left. +\theta \Lambda _2 \frac{\chi _p \omega _2 R_0 }{\omega _2^2 +R_0^2 } \right) -\frac{T_s }{U_\infty }\omega _2 \Lambda _2 +\omega _2 c_\alpha \\&\quad \left. -\omega _2 T_s b_0 \right) \\ \gamma _3= & {} -\mathbf{z}_2^\mathrm{T} \mathbf{P}\frac{\chi _p \omega _2^2 }{\omega _2^2 +R_0^2 }{} \mathbf{p}_{21} =-\frac{1}{N_{20} }\Lambda _2 ^{\prime }\theta \Lambda _2 \\&\quad \frac{\omega _2^2 \chi _p }{\omega _2^2 +R_0^2 }\\ \gamma _4= & {} -\mathbf{z}_2^\mathrm{T} \mathbf{Q}a_{221} \\= & {} -\frac{1}{N_{20} }\left( {-\Lambda _2 ^{\prime }c_s L_s -c_s T_s } \right) (3U_\infty ^3 \omega _2^3 \mathbf{p}_{21}^3 \\&+\,3b_0 \omega _2 \mathbf{p}_{22}^3 +9U_\infty b_0^2 \omega _2^3 \mathbf{p}_{21} \mathbf{p}_{22}^2 +3U_\infty \omega _2 \mathbf{p}_{21} \mathbf{p}_{22}^2 \\&+\,9U_\infty ^2 b_0 \omega _2^3 \mathbf{p}_{22} \mathbf{p}_{21}^2 +3b_0^3 \omega _2^3 \mathbf{p}_{22}^3 ) \\ \gamma _5= & {} \mathbf{z}_2^\mathrm{T} \mathbf{Q}a_{220} \\= & {} \frac{1}{N_{20} }\left( {-\Lambda _2 ^{\prime }c_s L_s -c_s T_s } \right) \\&\quad ( 3b_0^2 \omega _2^2 \mathbf{p}_{22}^3 +3\mathbf{p}_{22}^3 +3U_\infty ^2 \omega _2^2 \mathbf{p}_{22} \mathbf{p}_{21}^2 \\&+\,6U_\infty b_0 \omega _2^2 \mathbf{p}_{21} \mathbf{p}_{22}^2 ) \\ \gamma _6= & {} -\mathbf{z}_2^\mathrm{T} \mathbf{Q}a_{211} \\= & {} -\frac{1}{N_{20} }\left( {-\Lambda _2 ^{\prime }c_s L_s -c_s T_s } \right) (12U_\infty \omega _1^2 \omega _2 b_0^2 \\&\mathbf{p}_{11} \mathbf{p}_{12} \mathbf{p}_{22} +6b_0^3 \omega _1^2 \omega _2 \mathbf{p}_{12}^2 \mathbf{p}_{22} +6b_0 \omega _2 \mathbf{p}_{12}^2 \mathbf{p}_{22} \\&+\,6U_\infty ^3 \omega _1^2 \omega _2 \mathbf{p}_{11}^2 \mathbf{p}_{21} +12U_\infty ^2 b_0 \omega _1^2 \omega _2 \mathbf{p}_{11} \mathbf{p}_{12} \mathbf{p}_{21}\\&+6U_\infty ^2 b_0 \omega _1^2 \omega _2 \mathbf{p}_{11}^2 \mathbf{p}_{22} +6U_\infty \omega _2 \mathbf{p}_{12}^2 \mathbf{p}_{21} \\&+\,6U_\infty b_0^2 \omega _1^2 \omega _2 \mathbf{p}_{12}^2 \mathbf{p}_{21} ) \\ \gamma _7= & {} \mathbf{z}_2^\mathrm{T} \mathbf{Q}a_{210} \\= & {} \frac{1}{N_{20} }\left( {-\Lambda _2 ^{\prime }c_s L_s -c_s T_s } \right) \left( 6b_0^2 \omega _1^2 \mathbf{p}_{12}^2 \mathbf{p}_{22}\right. \\&+\,12U_\infty b_0 \omega _1^2 \mathbf{p}_{11} \mathbf{p}_{12} \mathbf{p}_{22} +6U_\infty ^2 \omega _1^2 \mathbf{p}_{11}^2 \mathbf{p}_{22} \\&\left. +6\mathbf{p}_{12}^2 \mathbf{p}_{22} \right) \\ \gamma _8= & {} -\mathbf{z}_2^\mathrm{T} \mathbf{Q}a_{311} \\= & {} -\frac{1}{N_{20} }\left( {-\Lambda _2 ^{\prime }c_s L_s -c_s T_s } \right) (U_\infty ^3 \omega _1^3 \mathbf{p}_{11}^3 \\&+\,3U_\infty ^2 b_0 \omega _1^3 \mathbf{p}_{11}^2 \mathbf{p}_{12} -3b_0 \omega _1 \mathbf{p}_{12}^3 +b_0^3 \omega _1^3 \mathbf{p}_{12}^3 \\&-\,3U_\infty \omega _1 \mathbf{p}_{12}^2 \mathbf{p}_{11} +3U_\infty \omega _1^3 b_0^2 \mathbf{p}_{12}^2 \mathbf{p}_{11} ) \\ \gamma _9= & {} \mathbf{z}_2^\mathrm{T} \mathbf{Q}a_{310} \\= & {} \frac{1}{N_{20} }\left( {-\Lambda _2 ^{\prime }c_s L_s -c_s T_s } \right) \\&\quad (3b_0^2 \omega _1^2 \mathbf{p}_{12}^3 +6U_\infty b_0 \omega _1^2 \mathbf{p}_{12}^2 \mathbf{p}_{11} \\&+\,3U_\infty ^2 \omega _1^2 \mathbf{p}_{12} \mathbf{p}_{11}^2 -\mathbf{p}_{12}^3 ) \\ \end{aligned}$$

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Liu, H., Gao, X. Vibration energy harvesting under concurrent base and flow excitations with internal resonance. Nonlinear Dyn 96, 1067–1081 (2019). https://doi.org/10.1007/s11071-019-04839-4

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