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Nonlinear vibration absorption of laminated composite beams in complex environment

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Abstract

Nonlinear vibration absorption of a laminated composite beam is investigated with the account of complex environment (moisture and temperature). A passive efficient nonlinear energy sink (NES) vibration absorber is used to control the transverse vibration. The generalized Hamilton principle is applied to derive a dynamic model of the laminated composite beam coupled with the NES. Numerical simulations reveal the effects of temperature, moisture, and laying angle on natural frequencies. It is numerically found that the NES can rapidly reduce the vibration amplitude. Then, approximate analytical solutions are sought via the harmonic balance method. The approximate analytical solutions are confirmed by the numerical solutions. Amplitude–frequency response curves show that the NES can reduce the amplitude to very low values for various temperatures, moisture levels, and laying angles. In a certain ranges of the NES parameters, different control effects are determined via an approximate analysis. It is demonstrated that the NES is a promising approach to control vibration of a laminated composite beam in complex environment.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Project Nos. 11772205, 11872159), the Scientific Research Fund of LiaoNing Provincial Education Department (No. L201703), and LiaoNing Revitalization Talents Program (XLYC1807172).

Author information

Authors and Affiliations

  1. College of Aerospace Engineering, Shenyang Aerospace University, Shenyang, 110136, China

    Ye-Wei Zhang, Shuai Hou, Jian Zang, Zhi-Yu Ni & Ying-Yuan Teng

  2. Science and Technology on Reliability and Environment Engineering Laboratory, Beijing Institute of Structure and Environment Engineering, Beijing, 100076, China

    Zhong Zhang

  3. Department of Mechanics, Harbin Institute of Technology, Shenzhen, 518055, China

    Li-Qun Chen

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  1. Ye-Wei Zhang
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Appendix A

Appendix A

The nonlinear ordinary differential equations with different N.

For \(N=1\)

$$\begin{aligned}&M_1 \ddot{q}_1 +C_1 \dot{q}_1 +K_1 q_1 +S_1 q_1 ^{3}-A_1 \sin (\omega \bar{{t}})\nonumber \\&\quad -\left( {k(\bar{{y}}-\phi _1 (d_1 )q_1 )^{3}+c(\dot{\bar{{y}}}-\phi _1 (d_1 )\dot{q}_1 )} \right) \phi _1 (d_1 )=0 \end{aligned}$$
(A1a)
$$\begin{aligned}&\varepsilon \ddot{\bar{{y}}}+k(\bar{{y}}-\phi _1 (d_n )q_1 )^{3}+c(\dot{\bar{{y}}}-\phi _1 (d_n )\dot{q}_1 )=0 \end{aligned}$$
(A1b)
$$\begin{aligned}&M_1 =\int _0^1 {\phi _1 (\bar{{x}})\phi _1 (\bar{{x}})} \mathrm{d}\bar{{x}}-\bar{{m}}\int _0^1 {\phi _1 ^{\prime \prime }(\bar{{x}})\phi _1 (\bar{{x}})} \mathrm{d}\bar{{x}} \nonumber \\&C_1 =\eta \int _0^1 {\phi _1 (\bar{{x}})\phi _1 (\bar{{x}})} \mathrm{d}\bar{{x}} \nonumber \\&K_1 =\bar{{M}}\int _0^1 {\phi _1 ^{\prime \prime }(\bar{{x}})\phi _1 (\bar{{x}})} \mathrm{d}\bar{{x}}+\int _0^1 {\phi _1 ^{(4)}(\bar{{x}})\phi _1 (\bar{{x}})d} \bar{{x}} \nonumber \\&S_1 =-\frac{3}{2}\bar{{P}}\int _0^1 {\left( {\phi _1 ^{\prime }(\bar{{x}})} \right) } ^{3}\phi _s (\bar{{x}})\mathrm{d}\bar{{x}} \nonumber \\&A_1 =\int _0^1 {f\delta (\bar{{x}}-d_0 )\phi _1 (d_0 )} \phi _1 (\bar{{x}})\mathrm{d}x \end{aligned}$$
(A1c)

For \(N=2\)

$$\begin{aligned}&M_{11} \ddot{q}_1 +C_{11} \dot{q}_1 +K_{11} q_1 +S_{11} q_1 ^{3}\nonumber \\&\quad +\,R_{11} q_2 ^{2}q_1 -A_{11} \sin (\omega \bar{{t}})\nonumber \\&\quad -\, \left( k(\bar{{y}}-\phi _1 (d_n )q_1 -\phi _2 (d_n )q_2 )^{3}\right. \nonumber \\&\left. \quad +\,c(\dot{\bar{{y}}}-\phi _1 (d_n )\dot{q}_1 -\phi _2 (d_n )\dot{q}_2 ) \right) \phi _1 (d_n )=0 \end{aligned}$$
(A2a)
$$\begin{aligned}&M_{22} \ddot{q}_2 +C_{22} \dot{q}_2 +K_{22} q_2 +S_{22} q_2 ^{3}+R_{22} q_1 ^{2}q_2 \nonumber \\&\quad -\, \left( k(\bar{{y}}-\phi _1 (d_n )q_1 -\phi _2 (d_n )q_2 )^{3}\right. \nonumber \\&\left. \quad +\,c(\dot{\bar{{y}}}-\phi _1 (d_n )\dot{q}_1 -\phi _2 (d_n )\dot{q}_2 ) \right) \phi _2 (d_n )=0 \end{aligned}$$
(A2b)
$$\begin{aligned}&\varepsilon \ddot{\bar{{y}}}+k(\bar{{y}}-\phi _1 (d_n )q_1 -\phi _2 (d_n )q_2 )^{3}\nonumber \\&\quad +\,c(\dot{\bar{{y}}}-\phi _1 (d_n )\dot{q}_1 -\phi _2 (d_n )\dot{q}_2 )=0 \end{aligned}$$
(A2c)
$$\begin{aligned}&M_{11} =\int _0^1 {\phi _1 ^{2}(\bar{{x}})} \mathrm{d}\bar{{x}}-\bar{{m}}\int _0^1 {\phi _1 } (\bar{{x}})\phi _1 ^{\prime \prime }(\bar{{x}})\mathrm{d}\bar{{x}} \nonumber \\&M_{22} =\int _0^1 {\phi _2 ^{2}(\bar{{x}})} \mathrm{d}\bar{{x}}-\bar{{m}}\int _0^1 {\phi _2 (\bar{{x}})} \phi _2 ^{\prime \prime }(\bar{{x}})\mathrm{d}\bar{{x}} \nonumber \\&C_{11} =\eta \int _0^1 {\phi _1 (\bar{{x}})\phi _1 (\bar{{x}})} \mathrm{d}\bar{{x}} \nonumber \\&C_{22} =\eta \int _0^1 {\phi _2 (\bar{{x}})\phi _2 (\bar{{x}})} \mathrm{d}\bar{{x}} \nonumber \\&K_{11} =\bar{{M}}\int _0^1 {\phi _1 (\bar{{x}})\phi _1 ^{\prime \prime }(\bar{{x}})\mathrm{d}\bar{{x}}+\int _0^1 {\phi _1 (\bar{{x}})\phi _1 ^{(4)}(\bar{{x}})\mathrm{d}\bar{{x}}} } \nonumber \\&K_{22} =\bar{{M}}\int _0^1 {\phi _2 (\bar{{x}})\phi _2 ^{\prime \prime }(\bar{{x}})\mathrm{d}\bar{{x}}+\int _0^1 {\phi _2 (\bar{{x}})\phi _2 ^{(4)}(\bar{{x}})\mathrm{d}\bar{{x}}} } \nonumber \\&R_{11} =-\frac{3}{2}\bar{{P}}\int _0^1 {\phi _1 } (\bar{{x}})\left( \phi _1 ^{\prime \prime }(\bar{{x}})\phi _2 ^{\prime 2}(\bar{{x}})\right. \nonumber \\&\left. \quad \quad +\,2\phi _1 ^{\prime }(\bar{{x}})\phi _2 ^{\prime }(\bar{{x}})\phi _2 ^{\prime \prime }(\bar{{x}}) \right) \mathrm{d}\bar{{x}} \nonumber \\&R_{22} =-\frac{3}{2}\bar{{P}}\int _0^1 {\phi _2 } (\bar{{x}})\left( \phi _1 ^{\prime 2}(\bar{{x}})\phi _2 ^{\prime \prime }(\bar{{x}})\right. \nonumber \\&\left. \quad \quad +\,2\phi _1 ^{\prime }(\bar{{x}})\phi _1 ^{\prime \prime }(\bar{{x}})\phi _2 ^{\prime }(\bar{{x}}) \right) \mathrm{d}\bar{{x}}\nonumber \\&S_{11} =-\frac{3}{2}\bar{{P}}\int _0^1 {\phi _1 (\bar{{x}})\phi _1 ^{\prime 2}(\bar{{x}})\phi _1 ^{\prime \prime }(\bar{{x}})\mathrm{d}\bar{{x}}} \nonumber \\&S_{22} =-\frac{3}{2}\bar{{P}}\int _0^1 {\phi _2 (\bar{{x}})\phi _2 ^{\prime 2}(\bar{{x}})\phi _2 ^{\prime \prime }(\bar{{x}})\mathrm{d}\bar{{x}}} \nonumber \\&A_{11} =\int _0^1 {f\delta (\bar{{x}}-d_0 )\phi _1 (d_0 )} \phi _1 (\bar{{x}})\mathrm{d}x \end{aligned}$$
(A2d)

For \(N=3\)

$$\begin{aligned}&M_{11} \ddot{q}_1 +C_{11} \dot{q}_1 +K_{11} q_1 +S_{11} (q_1 ^{3}+3q_1 ^{2}q_2 \nonumber \\&\quad +\,8q_1 q_2 ^{2}+18q_1 q_3 ^{2}+12q_2 ^{2}q_3 )-A_{11} \sin (\omega \bar{{t}}) \nonumber \\&\quad -\,\left( k(\bar{{y}}-\phi _1 (d_n )q_1 -\phi _2 (d_n )q_2 -\phi _3 (d_n )q_3 )^{3}\right. \nonumber \\&\left. \quad +\,c(\dot{\bar{{y}}}-\phi _1 (d_n )\dot{q}_1 -\phi _2 (d_n )\dot{q}_2 -\phi _3 (d_n )\dot{q}_3 ) \right) \phi _1 (d_n )=0 \nonumber \\ \end{aligned}$$
(A3a)
$$\begin{aligned}&M_{22} \ddot{q}_2 +C_{22} \dot{q}_2 +K_{22} q_2 +S_{22} (2q_2 ^{3}\nonumber \\&\quad +\,9q_2 q_3 ^{2}+q_2 q_1 ^{2}+3q_1 q_2 q_3 )\nonumber \\&\quad - \,\left( k(\bar{{y}}-\phi _1 (d_n )q_1 -\phi _2 (d_n )q_2 -\phi _3 (d_n )q_3 )^{3}\right. \nonumber \\&\left. \quad +\,c(\dot{\bar{{y}}}-\phi _1 (d_n )\dot{q}_1 -\phi _2 (d_n )\dot{q}_2 -\phi _3 (d_n )\dot{q}_3 ) \right) \phi _2 (d_n )=0 \nonumber \\ \end{aligned}$$
(A3b)
$$\begin{aligned}&M_{33} \ddot{q}_3 +C_{33} \dot{q}_3 +K_{33} q_3 +S_{33} (81q_3 ^{3}+72q_3 q_2 ^{2}\nonumber \\&\quad +\,12q_1 q_2 ^{2}+18q_1 ^{2}q_3 +q_1 ^{3})-A_{33} \sin (\omega \bar{{t}}) \nonumber \\&\quad -\,\left( k(\bar{{y}}-\phi _1 (d_n )q_1 -\phi _2 (d_n )q_2 -\phi _3 (d_n )q_3 )^{3}\right. \nonumber \\&\left. \quad +\,c(\dot{\bar{{y}}}-\phi _1 (d_n )\dot{q}_1 -\phi _2 (d_n )\dot{q}_2 -\phi _3 (d_n )\dot{q}_3 ) \right) \phi _3 (d_n )=0 \nonumber \\ \end{aligned}$$
(A3c)
$$\begin{aligned}&\varepsilon \ddot{\bar{{y}}}+k(\bar{{y}}-\phi _1 (d_n )q_1 -\phi _2 (d_n )q_2 -\phi _3 (d_n )q_3 )^{3}\nonumber \\&\quad +\,c(\dot{\bar{{y}}}-\phi _1 (d_n )\dot{q}_1 -\phi _2 (d_n )\dot{q}_2 -\phi _3 (d_n )\dot{q}_3 )=0 \end{aligned}$$
(A3d)
$$\begin{aligned}&M_{11} =\int _0^1 {\phi _1 ^{2}(\bar{{x}})} \mathrm{d}\bar{{x}}-\bar{{m}}\int _0^1 {\phi _1 } (\bar{{x}})\phi _1 ^{\prime \prime }(\bar{{x}})\mathrm{d}\bar{{x}} \nonumber \\&M_{22} =\int _0^1 {\phi _2 ^{2}(\bar{{x}})} \mathrm{d}\bar{{x}}-\bar{{m}}\int _0^1 {\phi _2 (\bar{{x}})} \phi _2 ^{\prime \prime }(\bar{{x}})\mathrm{d}\bar{{x}} \nonumber \\&M_{33} =\int _0^1 {\phi _3 ^{2}(\bar{{x}})} \mathrm{d}\bar{{x}}-\bar{{m}}\int _0^1 {\phi _3 (\bar{{x}})} \phi _3 ^{\prime \prime }(\bar{{x}})\mathrm{d}\bar{{x}} \nonumber \\&C_{11} =\eta \int _0^1 {\phi _1 (\bar{{x}})\phi _1 (\bar{{x}})} \mathrm{d}\bar{{x}} \nonumber \\&C_{22} =\eta \int _0^1 {\phi _2 (\bar{{x}})\phi _2 (\bar{{x}})} \mathrm{d}\bar{{x}} \nonumber \\&C_{33} =\eta \int _0^1 {\phi _3 (\bar{{x}})\phi _3 (\bar{{x}})} \mathrm{d}\bar{{x}} \nonumber \\&K_{11} =\bar{{M}}\int _0^1 {\phi _1 (\bar{{x}})\phi _1 ^{\prime \prime }(\bar{{x}})\mathrm{d}\bar{{x}}+\int _0^1 {\phi _1 (\bar{{x}})\phi _1 ^{(4)}(\bar{{x}})\mathrm{d}\bar{{x}}} } \nonumber \\&K_{22} =\bar{{M}}\int _0^1 {\phi _2 (\bar{{x}})\phi _2 ^{\prime \prime }(\bar{{x}})\mathrm{d}\bar{{x}}+\int _0^1 {\phi _2 (\bar{{x}})\phi _2 ^{(4)}(\bar{{x}})\mathrm{d}\bar{{x}}} } \nonumber \\&K_{33} =\bar{{M}}\int _0^1 {\phi _3 (\bar{{x}})\phi _3 ^{\prime \prime }(\bar{{x}})\mathrm{d}\bar{{x}}+\int _0^1 {\phi _3 (\bar{{x}})\phi _3 ^{(4)}(\bar{{x}})\mathrm{d}\bar{{x}}} } \nonumber \\&A_{11} =\int _0^1 {f\delta (\bar{{x}}-d_0 )\phi _1 (d_0 )} \phi _1 (\bar{{x}})\mathrm{d}x \nonumber \\&A_{33} =\int _0^1 {f\delta (\bar{{x}}-d_0 )\phi _3 (d_0 )} \phi _3 (\bar{{x}})\mathrm{d}x \nonumber \\&S_{11} =\frac{9L\pi ^{4}}{h^{2}},\quad S_{22} =\frac{72L\pi ^{4}}{h^{2}},\quad S_{33} =\frac{9L\pi ^{4}}{h^{2}} \end{aligned}$$
(A3e)

For \(N =4\)

$$\begin{aligned}&M_{11} \ddot{q}_1 +C_{11} \dot{q}_1 +K_{11} q_1 +S_{11} (q_1 ^{3}+3q_1 ^{2}q_3 \nonumber \\&\quad +\,8q_1 q_2 ^{2}+16q_1 q_2 q_4 +18q_1 q_3 ^{2}+32q_1 q_4 ^{2}+12q_2 ^{2}q_3 \nonumber \\&\quad +\, 48q_2 q_3 q_4 )-A_{11} \sin (\omega \bar{{t}})-\left( k(\bar{{y}}-\,\sum _{r=1}^4 {\phi _r (d_n )q_r } )^{3}\right. \nonumber \\&\left. \quad +\,c(\dot{\bar{{y}}}-\sum _{r=1}^4 {\phi _r (d_n )\dot{q}_r } ) \right) \phi _1 (d_n )=0 \end{aligned}$$
(A4a)
$$\begin{aligned}&M_{22} \ddot{q}_2 +C_{22} \dot{q}_2 +K_{22} q_2 +S_{22} (q_1 ^{2}q_2 +q_1 ^{2}q_4 +3q_1 q_2 q_3\nonumber \\&\quad +\,6q_1 q_3 q_4 +2q_2 ^{3}+9q_2 q_3 ^{2}+16q_2 q_4 ^{2}\nonumber \\&\quad + \,9q_3 ^{2}q_4 )-\left( k(\bar{{y}}-\sum _{r=1}^4 {\phi _r (d_n )q_r } )^{3}\right. \nonumber \\&\left. \quad +\,c(\dot{\bar{{y}}}-\sum _{r=1}^4 {\phi _r (d_n )\dot{q}_r } ) \right) \phi _2 (d_n )=0 \end{aligned}$$
(A4b)
$$\begin{aligned}&M_{33} \ddot{q}_3 +C_{33} \dot{q}_3 +K_{33} q_3 +S_{33} (q_1 ^{3}+18q_1 ^{2}q_3 +12q_1 q_2 ^{2}\nonumber \\&\quad +\,48q_1 q_2 q_4 +72q_2 ^{2}q_3 +144q_2 q_3 q_4 +81q_3 ^{3}\nonumber \\&\quad +\,288q_3 q_4 ^{2})-A_{33} \sin (\omega \bar{{t}})-\left( k(\bar{{y}}-\sum _{r=1}^4 {\phi _r (d_n )q_r } )^{3}\right. \nonumber \\&\left. \quad +\,c(\dot{\bar{{y}}}-\sum _{r=1}^4 {\phi _r (d_n )\dot{q}_r } ) \right) \phi _3 (d_n )=0 \end{aligned}$$
(A4c)
$$\begin{aligned}&M_{44} \ddot{q}_4 +C_{44} \dot{q}_4 +K_{44} q_4 +S_{44} (q_1 ^{2}q_2 +4q_1 ^{2}q_4 \nonumber \\&\quad +\,6q_1 q_2 q_3 +16q_2 ^{2}q_4 +9q_2 q_3 ^{2}+36q_3 ^{2}q_4 +32q_4 ^{3}) \nonumber \\&\quad -\,\left( k(\bar{{y}}-\sum _{r=1}^4 {\phi _r (d_n )q_r } )^{3}\right. \nonumber \\&\quad \left. +\,c(\dot{\bar{{y}}}-\sum _{r=1}^4 {\phi _r (d_n )\dot{q}_r } ) \right) \phi _4 (d_n )=0 \end{aligned}$$
(A4d)
$$\begin{aligned}&\varepsilon \ddot{\bar{{y}}}+k(\bar{{y}}-\sum _{r=1}^4 {\phi _r (d_n )q_r } )^{3}+c(\dot{\bar{{y}}}-\sum _{r=1}^4 {\phi _r (d_n )\dot{q}_r } )=0 \end{aligned}$$
(A4e)
$$\begin{aligned}&M_{11} =\int _0^1 {\phi _1 ^{2}(\bar{{x}})} \mathrm{d}\bar{{x}}-\bar{{m}}\int _0^1 {\phi _1 } (\bar{{x}})\phi _1 ^{\prime \prime }(\bar{{x}})\mathrm{d}\bar{{x}} \nonumber \\&M_{22} =\int _0^1 {\phi _2 ^{2}(\bar{{x}})} \mathrm{d}\bar{{x}}-\bar{{m}}\int _0^1 {\phi _2 (\bar{{x}})} \phi _2 ^{\prime \prime }(\bar{{x}})\mathrm{d}\bar{{x}} \nonumber \\&M_{33} =\int _0^1 {\phi _3 ^{2}(\bar{{x}})} \mathrm{d}\bar{{x}}-\bar{{m}}\int _0^1 {\phi _3 (\bar{{x}})} \phi _3 ^{\prime \prime }(\bar{{x}})\mathrm{d}\bar{{x}} \nonumber \\&M_{44} =\int _0^1 {\phi _4 ^{2}(\bar{{x}})} \mathrm{d}\bar{{x}}-\bar{{m}}\int _0^1 {\phi _4 (\bar{{x}})} \phi _4 ^{\prime \prime }(\bar{{x}})\mathrm{d}\bar{{x}} \nonumber \\&C_{11} =\eta \int _0^1 {\phi _1 (\bar{{x}})\phi _1 (\bar{{x}})} \mathrm{d}\bar{{x}} \nonumber \\&C_{22} =\eta \int _0^1 {\phi _2 (\bar{{x}})\phi _2 (\bar{{x}})} \mathrm{d}\bar{{x}} \nonumber \\&C_{33} =\eta \int _0^1 {\phi _3 (\bar{{x}})\phi _3 (\bar{{x}})} \mathrm{d}\bar{{x}} \nonumber \\&C_{44} =\eta \int _0^1 {\phi _4 (\bar{{x}})\phi _4 (\bar{{x}})} \mathrm{d}\bar{{x}} \nonumber \\&K_{11} =\bar{{M}}\int _0^1 {\phi _1 (\bar{{x}})\phi _1 ^{\prime \prime }(\bar{{x}})\mathrm{d}\bar{{x}}+\int _0^1 {\phi _1 (\bar{{x}})\phi _1 ^{(4)}(\bar{{x}})\mathrm{d}\bar{{x}}} } \nonumber \\&K_{22} =\bar{{M}}\int _0^1 {\phi _2 (\bar{{x}})\phi _2 ^{\prime \prime }(\bar{{x}})\mathrm{d}\bar{{x}}+\int _0^1 {\phi _2 (\bar{{x}})\phi _2 ^{(4)}(\bar{{x}})\mathrm{d}\bar{{x}}} } \nonumber \\&K_{33} =\bar{{M}}\int _0^1 {\phi _3 (\bar{{x}})\phi _3 ^{\prime \prime }(\bar{{x}})\mathrm{d}\bar{{x}}+\int _0^1 {\phi _3 (\bar{{x}})\phi _3 ^{(4)}(\bar{{x}})\mathrm{d}\bar{{x}}} } \nonumber \\&K_{44} =\bar{{M}}\int _0^1 {\phi _4 (\bar{{x}})\phi _4 ^{\prime \prime }(\bar{{x}})\mathrm{d}\bar{{x}}+\int _0^1 {\phi _4 (\bar{{x}})\phi _4 ^{(4)}(\bar{{x}})\mathrm{d}\bar{{x}}} }\nonumber \nonumber \\&A_{11} =\int _0^1 {f\delta (\bar{{x}}-d_0 )\phi _1 (d_0 )} \phi _1 (\bar{{x}})\mathrm{d}x \nonumber \\&A_{33} =\int _0^1 {f\delta (\bar{{x}}-d_0 )\phi _3 (d_0 )} \phi _3 (\bar{{x}})\mathrm{d}x \nonumber \\&S_{11} =\frac{9L\pi ^{4}}{h^{2}},\quad S_{22} =\frac{72L\pi ^{4}}{h^{2}},\nonumber \\&S_{33}=\frac{9L\pi ^{4}}{h^{2}},\quad S_{44} =\frac{72L\pi ^{4}}{h^{2}} \end{aligned}$$
(A4f)

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Zhang, YW., Hou, S., Zhang, Z. et al. Nonlinear vibration absorption of laminated composite beams in complex environment. Nonlinear Dyn 99, 2605–2622 (2020). https://doi.org/10.1007/s11071-019-05442-3

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