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Vibration suppression and higher branch responses of beam with parallel nonlinear energy sinks

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Abstract

The effects of nonlinear energy sink (NES) on vibration suppression of a simply supported beam are investigated in this work. The slow flow equations of the system are derived by using complexification–averaging method, and the validity of the derivation is verified. By comparing the vibration absorption of single and parallel NESs of equal mass, it is found that the latter exhibits superior vibration absorption performance. In addition, the parallel NES can eliminate higher branch responses of the system under the harmonic load. Furthermore, it is found that parallel NES can eliminate the higher branches of the system more effectively by tuning nonlinear stiffness and damping. Moreover, the thermal effect on natural frequencies of the simply supported beam is considered, and the influences of the parallel NES’s parameters on the energy dissipation rate under shock load are investigated. The nonlinear responses of the simply supported beam with parallel NES under harmonic load and with the increase of temperature are described.

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Acknowledgements

This research was supported by the National Natural Science Foundation of China (Nos. 11402170 and 11402165) and Tianjin Natural Science Foundation of China (Nos. 17JCYBJC18800 and 17JCZDJC38500).

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Authors and Affiliations

  1. Tianjin Key Laboratory of the Design and Intelligent Control of the Advanced Mechatronical System, Tianjin University of Technology, Tianjin, 300384, People’s Republic of China

    J. E. Chen, W. He & J. Liu

  2. National Demonstration Center for Experimental Mechanical and Electrical Engineering Education, Tianjin University of Technology, Tianjin, 300384, People’s Republic of China

    J. E. Chen, W. He & J. Liu

  3. College of Mechanical Engineering, Beijing University of Technology, Beijing, 100124, People’s Republic of China

    W. Zhang & M. H. Yao

  4. School of Science, Tianjin Chengjian University, Tianjin, 300384, People’s Republic of China

    M. Sun

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  1. J. E. Chen
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  2. W. He
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Correspondence to J. E. Chen.

Appendix

Appendix

$$\begin{aligned} J_{11}= & {} -S_1 , \quad J_{12} =\Omega -(S_2 +S_3 )\frac{1}{\Omega }, \\ J_{13}= & {} 0, \quad J_{14} =0, \\ J_{15}= & {} -\left( \sqrt{2}S_5 +\frac{3\sqrt{2}S_4 a_{30} b_{30} }{4\Omega ^{3}}\right) \sin \left( \frac{\pi d_1 }{l}\right) , \\ J_{16}= & {} -\frac{3\sqrt{2}S_4 a_{30} ^{2}(a_{30} ^{2}-b_{30} ^{2})}{4\Omega ^{3}}\sin \left( \frac{\pi d_1}{l}\right) , \\ J_{17}= & {} -\left( \sqrt{2}S_7 +\frac{3\sqrt{2}S_6 a_{40} b_{40} }{2\Omega ^{3}}\right) \sin \left( \frac{\pi d_2 }{l}\right) , \\ J_{18}= & {} -\left( \frac{3\sqrt{2}S_6 a_{40} ^{2}+9\sqrt{2}S_6 b_{40} ^{2}}{4\Omega ^{3}}\right) \sin \left( \frac{\pi d_2 }{l}\right) , \\ J_{21}= & {} (S_9 +S_{10} )\frac{1}{\Omega }-\Omega , \\ J_{22}= & {} -S_1 , \quad J_{23} =0, \quad J_{24} =0,\\ J_{25}= & {} \left( \frac{9\sqrt{2}S_4 a_{30} ^{2}+3\sqrt{2}S_4 b_{30} ^{2}}{4\Omega ^{3}}\right) \sin \left( \frac{\pi d_1 }{l}\right) , \\ J_{26}= & {} \left( \frac{3\sqrt{2}S_4 a_{30} b_{30} }{2\Omega ^{3}}-\sqrt{2}S_5\right) \sin \left( \frac{\pi d_1 }{l}\right) ,\\ J_{27}= & {} \frac{\left( 3\sqrt{2}S_6 b_{40} ^{2}+9\sqrt{2}S_6 a_{40} ^{2}\right) }{4\Omega ^{3}}\sin \left( \frac{\pi d_2 }{l}\right) ,\\ J_{28}= & {} \left( \frac{3\sqrt{2}S_6 a_{40} b_{40} }{2\Omega ^{3}}-\sqrt{2}S_7 \right) \sin \left( \frac{\pi d_2 }{l}\right) ,\\ J_{31}= & {} 0, \quad J_{32} =0,\\\end{aligned}$$
$$\begin{aligned} J_{33}= & {} -\left( \frac{\hbox {3}\sqrt{2}S_6 a_{40} b_{40} }{2\Omega ^{3}}+\sqrt{2}S_7 \right) \sin \left( \frac{2\pi d_2 }{l}\right) ,\\ J_{34}= & {} -\frac{(4\sqrt{2}S_6 a_{40} ^{2}+9\sqrt{2}S_6 b_{40} ^{2})}{4\Omega ^{3}}\sin \left( \frac{2\pi d_2 }{l}\right) ,\\ J_{35}= & {} -\mu , \quad J_{36} =\Omega -(S_9 +S_{10} )\frac{1}{\Omega }, \\ J_{37}= & {} -\left( \frac{3\sqrt{2}S_4 a_{30} b_{30} }{2\Omega ^{3}}+\sqrt{2}S_5 \right) \sin \left( \frac{2\pi d_1 }{l}\right) , \\ J_{38}= & {} -\frac{\hbox {(3}\sqrt{2}S_4 a_{30} ^{2}+\hbox {9}\sqrt{2}S_4 b_{30} ^{2})}{4\Omega ^{3}}\sin \left( \frac{2\pi d_1 }{l}\right) , \\ J_{41}= & {} 0, \quad J_{42} =0,\\ J_{43}= & {} (S_9 +S_{10} )\frac{1}{\Omega }-\Omega , \quad J_{44} =-S_1, \\ J_{45}= & {} \left( \frac{\hbox {3}\sqrt{2}S_4 b_{30} ^{2}\hbox {+9}\sqrt{2}S_4 a_{30} ^{2}}{4\Omega ^{3}}\right) \sin \left( \frac{2\pi d_1 }{l}\right) , \\ J_{46}= & {} \left( \frac{\hbox {3}\sqrt{2}S_4 a_{30} b_{30} }{2\Omega ^{3}}-\sqrt{2}S_5 \right) \sin \left( \frac{2\pi d_1 }{l}\right) , \\ J_{47}= & {} \left( \frac{\hbox {3}\sqrt{2}S_6 b_{40} ^{2}+\hbox {9}\sqrt{2}S_6 a_{40} ^{2}}{4\Omega ^{3}}\right) \sin \left( \frac{2\pi d_2 }{l}\right) , \\ J_{48}= & {} \left( \frac{\hbox {3}\sqrt{2}S_6 a_{40} b_{40} }{2\Omega ^{3}}-\sqrt{2}S_7 \right) \sin \left( \frac{2\pi d_2 }{l}\right) , \\ J_{51}= & {} -\sqrt{2}S_1 \sin \left( \frac{\pi d_1 }{l}\right) ,\\ \end{aligned}$$
$$\begin{aligned} J_{52}= & {} -\frac{\sqrt{2}(S_2 +S_3 )}{\Omega }\sin \left( \frac{\pi d_1}{l}\right) , \\ J_{53}= & {} -\sqrt{2}S_1 \sin \left( \frac{2\pi d_1 }{l}\right) , \\ J_{54}= & {} -(S_9 +S_{10} )\frac{\sqrt{2}}{\Omega }\sin \left( \frac{2\pi d_1 }{l}\right) ,\\ J_{55}= & {} -\left( \frac{3S_4 a_{30} b_{30} }{\Omega ^{3}}+2\right) \sin ^{2}\left( \frac{\pi d_1 }{l}\right) -\frac{S_5 }{\varepsilon _1 }\\&-\left( 2S_5+\frac{3S_4 a_{30} b_{30} }{\Omega ^{3}}\right) \sin ^{2}\left( \frac{2\pi d_1 }{l}\right) -\frac{3S_4 a_{30} b_{30} }{2\varepsilon _1 \Omega ^{3}},\\ J_{56}= & {} \Omega -\frac{\left( 3S_4 a_{30} ^{2}+9S_4 b_{30} ^{2}\right) }{4\varepsilon _1 \Omega ^{3}}\\&-\left( \frac{9S_4 b_{30} ^{2}+3S_4 a_{30} ^{2}}{2\Omega ^{3}}\right) \sin ^{2}\left( \frac{2\pi d_1}{l}\right) \\&-\left( \frac{9S_4 b_{30} ^{2}+3S_4 a_{30} ^{2}}{2\Omega ^{3}}\right) \sin ^{2}\left( \frac{\pi d_1 }{l}\right) ,\\ J_{57}= & {} -2S_7 \sin \left( \frac{\pi d_1 }{l}\right) \sin \left( \frac{\pi d_2}{l}\right) \\&-2S_7 \sin \left( \frac{2\pi d_1 }{l}\right) \sin \left( \frac{2\pi d_2}{l}\right) \\&-\frac{3S_6 a_{40} b_{40} }{\Omega ^{3}}\sin \left( \frac{\pi d_1 }{l}\right) \sin \left( \frac{\pi d_2 }{l}\right) \\&-\frac{3S_6 a_{40} b_{40} }{\Omega ^{3}}\sin \left( \frac{2\pi d_1 }{l}\right) \sin \left( \frac{2\pi d_2 }{l}\right) ,\\ J_{58}= & {} -\frac{(9S_6 b_{40} ^{2}+3S_6 a_{40} ^{2})}{2\Omega ^{3}}\sin \left( \frac{\pi d_1 }{l}\right) \sin \left( \frac{\pi d_2 }{l}\right) \\&-\frac{\left( 9S_6b_{40} ^{2}+3S_6 a_{40} ^{2}\right) }{2\Omega ^{3}}\sin \left( \frac{2\pi d_1 }{l}\right) \sin \left( \frac{2\pi d_2 }{l}\right) ,\\ \end{aligned}$$
$$\begin{aligned} J_{61}= & {} \frac{\sqrt{2}(S_2 +S_3 )}{\Omega }\sin \left( \frac{\pi d_1 }{l}\right) ,\\ J_{62}= & {} -\sqrt{2}S_1 \sin (\frac{\pi d_1 }{l}),\\ J_{63}= & {} -\frac{\sqrt{2}(S_9 +S_{10} )}{\Omega }\sin \left( \frac{2\pi d_1 }{l}\right) ,\\ J_{64}= & {} -\sqrt{2}S_6 \sin \left( \frac{2\pi d_1 }{l}\right) ,\\ J_{65}= & {} -\Omega +\frac{9S_4 a_{30} ^{2}+3S_4 b_{30} ^{2}}{4\varepsilon _1 \Omega ^{3}} \\&+\frac{(9S_4 a_{30} ^{2}+3S_4 b_{30} ^{2})}{2\Omega ^{3}}\sin ^{2}(\frac{\pi d_1 }{l}) \\&+\frac{(9S_4a_{30} ^{2}+3S_4 b_{30} ^{2})}{2\Omega ^{3}}\sin ^{2}(\frac{2\pi d_1}{l}), \\ J_{66}= & {} -2S_5 \sin ^{2}\left( \frac{\pi d_1 }{l}\right) -\frac{S_5 }{\varepsilon _1 }-2S_5 \sin ^{2}\left( \frac{2\pi d_1 }{l}\right) \\&+\frac{3S_4 a_{30} b_{30} }{2\varepsilon _1 \Omega ^{3}}+\frac{3S_4 a_{30} b_{30} }{\Omega ^{3}}\sin ^{2}\left( \frac{\pi d_1 }{l}\right) \\&+\frac{3S_4 a_{30} b_{30} }{\Omega ^{3}}\sin ^{2}\left( \frac{2\pi d_1 }{l}\right) , \\ J_{67}= & {} \frac{\left( 3S_6 b_{40} ^{2}+9S_6 b_{40} ^{2}\right) }{2\Omega ^{3}}\sin (\frac{\pi d_1 }{l})\sin (\frac{\pi d_2 }{l}) \\&+\frac{(3S_6 b_{40} ^{2}+9S_6 b_{40} ^{2})}{2\Omega ^{3}}\sin (\frac{2\pi d_1 }{l})\sin (\frac{2\pi d_2 }{l}), \\ J_{68}= & {} -2S_7 \sin \left( \frac{\pi d_1 }{l}\right) \sin \left( \frac{\pi d_2}{l}\right) \\&-2S_7 \sin \left( \frac{2\pi d_1 }{l}\right) \sin \left( \frac{2\pi d_2}{l}\right) \\&+\frac{3S_6 a_{40} b_{40} }{\Omega ^{3}}\sin \left( \frac{\pi d_1 }{l}\right) \sin \left( \frac{\pi d_2 }{l}\right) \\&+\frac{3S_6 a_{40} b_{40} }{\Omega ^{3}}\sin \left( \frac{2\pi d_1 }{l}\right) \sin \left( \frac{2\pi d_2 }{l}\right) , \\ J_{71}= & {} -\sqrt{2}S_1 \sin \left( \frac{\pi d_2 }{l}\right) ,\\ \end{aligned}$$
$$\begin{aligned} J_{72}= & {} -\frac{\sqrt{2}(S_2 +S_3 )}{\Omega }\sin \left( \frac{\pi d_2 }{l}\right) ,\\ J_{73}= & {} -\sqrt{2}S_1 \sin \left( \frac{2\pi d_2 }{l}\right) ,\\ J_{74}= & {} -\frac{\sqrt{2}(S_9 +S_{10} )}{\Omega }\sin \left( \frac{2\pi d_1 }{l}\right) ,\\ J_{75}= & {} -2S_5 \sin (\frac{\pi d_1 }{l})\sin \left( \frac{\pi d_2}{l}\right) \\&-2S_5 \sin (\frac{2\pi d_1 }{l})\sin \left( \frac{2\pi d_2}{l}\right) \\&-\frac{3S_4 a_{30} b_{30} }{\Omega ^{3}}\sin \left( \frac{\pi d_1 }{l}\right) \sin \left( \frac{\pi d_2 }{l}\right) \\&-\frac{3S_4 a_{30} b_{30} }{\Omega ^{3}}\sin \left( \frac{2\pi d_1 }{l}\right) \sin \left( \frac{2\pi d_2 }{l}\right) , \\ J_{76}= & {} -\frac{(3S_4 b_{30} ^{2}+9S_4 b_{30} ^{2}}{2\Omega ^{3}}\sin \left( \frac{\pi d_1 }{l}\right) \sin \left( \frac{\pi d_2 }{l}\right) \\&-\frac{\left( 3S_4b_{30} ^{2}+9S_4 b_{30} ^{2}\right) }{2\Omega ^{3}}\sin \left( \frac{2\pi d_1 }{l}\right) \sin \left( \frac{2\pi d_2 }{l}\right) , \\ J_{77}= & {} -2S_7 \sin ^{2}\left( \frac{\pi d_2 }{l}\right) -\frac{S_7 }{\varepsilon _2 } \\&-2S_7 \sin ^{2}\left( \frac{2\pi d_2 }{l}\right) -\frac{3S_6 a_{40} b_{40} }{2\varepsilon _2 \Omega ^{3}} \\&-\frac{3S_6 a_{40} b_{40} }{\Omega ^{3}}\sin ^{2}\left( \frac{\pi d_2 }{l}\right) \\&-\frac{3S_6 a_{40} b_{40} }{\Omega ^{3}}\sin ^{2}\left( \frac{2\pi d_2 }{l}\right) ,\\ J_{78}= & {} \Omega +\frac{9S_6 a_{40} ^{2}+3S_6 b_{40} ^{2}}{4\varepsilon _2 \Omega ^{3}}\\&-\frac{\left( 9S_6 a_{40} ^{2}+3S_6 b_{40} ^{2}\right) }{2\Omega ^{3}}\sin ^{2}\left( \frac{\pi d_2 }{l}\right) \\&-\frac{\left( 9S_6a_{40} ^{2}+3S_6 b_{40} ^{2}\right) }{2\Omega ^{3}}\sin ^{2}\left( \frac{2\pi d_2}{l}\right) ,\\ \end{aligned}$$
$$\begin{aligned} J_{81}= & {} \frac{\sqrt{2}(S_9 +S_{10} )}{\Omega }\sin \left( \frac{\pi d_2 }{l}\right) ,\\ J_{82}= & {} -\sqrt{2}S_1 \sin \left( \frac{\pi d_2 }{l}\right) ,\\ J_{83}= & {} -\frac{\sqrt{2}(S_2 +S_3 )}{\Omega }\sin \left( \frac{2\pi d_2 }{l}\right) ,\\ J_{84}= & {} -\sqrt{2}S_4 \sin \left( \frac{2\pi d_2 }{l}\right) ,\\ J_{85}= & {} -\Omega +\frac{9S_6 a_{40} ^{2}+3S_6 b_{40} ^{2}}{4\varepsilon _2 \Omega ^{3}}\\&+\frac{\left( 9S_6 a_{40} ^{2}+3S_6 b_{40} ^{2}\right) }{2\Omega ^{3}}\sin ^{2}\left( \frac{\pi d_2 }{l}\right) \\&+\frac{\left( 9S_6a_{40} ^{2}+3S_6 b_{40} ^{2}\right) }{2\Omega ^{3}}\sin ^{2}\left( \frac{2\pi d_2}{l}\right) , \\ J_{86}= & {} -2S_7 \sin ^{2}(\frac{\pi d_2 }{l})-\frac{S_7 }{\varepsilon _2 }-2S_7 \sin ^{2}\left( \frac{2\pi d_2 }{l}\right) \\&+\frac{3S_6 a_{40} b_{40} }{2\varepsilon _2 \Omega ^{3}}+\frac{3S_6 a_{40} b_{40} }{\Omega ^{3}}\sin ^{2}\left( \frac{\pi d_2 }{l}\right) \\&+\frac{3S_6 a_{40} b_{40} }{\Omega ^{3}}\sin ^{2}\left( \frac{2\pi d_2 }{l}\right) , \\ J_{87}= & {} \frac{\left( 3S_4 b_{30} ^{2}+9S_4 b_{30} ^{2}\right) }{2\Omega ^{3}}\sin \left( \frac{\pi d_1 }{l}\right) \sin \left( \frac{\pi d_2 }{l}\right) \\&+\frac{(3S_4 b_{30} ^{2}+9S_4 b_{30} ^{2})}{2\Omega ^{3}}\sin \left( \frac{2\pi d_1 }{l}\right) \sin \left( \frac{2\pi d_2 }{l}\right) ,\\ J_{88}= & {} -2S_5 \sin \left( \frac{\pi d_1 }{l}\right) \sin \left( \frac{\pi d_2}{l}\right) \\&-2S_5 \sin \left( \frac{2\pi d_1 }{l}\right) \sin \left( \frac{2\pi d_2}{l}\right) \\&+\frac{3S_4 a_{30} b_{30} }{\Omega ^{3}}\sin (\frac{\pi d_1 }{l})\sin \left( \frac{\pi d_2 }{l}\right) \\&+\frac{3S_4 a_{30} b_{30} }{\Omega ^{3}}\sin \left( \frac{2\pi d_1 }{l}\right) \sin \left( \frac{2\pi d_2 }{l}\right) . \end{aligned}$$

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Chen, J.E., He, W., Zhang, W. et al. Vibration suppression and higher branch responses of beam with parallel nonlinear energy sinks. Nonlinear Dyn 91, 885–904 (2018). https://doi.org/10.1007/s11071-017-3917-z

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