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Chaotic dynamics and forced harmonic vibration analysis of magneto-electro-viscoelastic multiscale composite nanobeam

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Abstract

In this article, the damping forced harmonic vibration characteristics of magneto-electro-viscoelastic (MEV) nanobeam embedded in viscoelastic foundation is evaluated based on nonlocal strain gradient elasticity theory. The viscoelastic foundation consists of Winkler–Pasternak layer. The governing equations of nonlocal strain gradient viscoelastic nanobeam in the framework of refined shear deformable beam theory are obtained using Hamilton’s principle and solved implementing an analytical solution. In addition, a parametric study is presented to examine the effect of the nonlocal strain gradient parameter, magneto-electro-mechanical loadings, and aspect ratio on the vibration characteristics of nanobeam. From the numerical evaluation, it is revealed that the effect of electric and magnetic loading on the natural frequency has a predominant influence.

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Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Faculty of Engineering, Imam Khomeini International University, Qazvin, Iran

    Farzad Ebrahimi & Mahsa karimiasl

  2. Department of Aerospace Engineering, Indian Institute of Science, Bangalore, 560012, India

    Vinyas Mahesh

  3. Composite Research: Experimentation, Analysis and Modeling (CREAM) Lab, Department of Mechanical Engineering, Nitte Meenakshi Institute of Technology, Bangalore, 560064, India

    Vinyas Mahesh

Authors
  1. Farzad Ebrahimi
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  2. Mahsa karimiasl
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  3. Vinyas Mahesh
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Correspondence to Farzad Ebrahimi.

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Appendix

Appendix

\(M_{11} = I_{0} \left( {\mu \bar{X}_{13} - \bar{X}_{11} } \right)\)

\(M_{12} = I_{1} \left( {\bar{X}_{11} - \mu \bar{X}_{13} } \right)\)

\(M_{13} = I_{2} \left( {\bar{X}_{11} - \mu \bar{X}_{13} } \right)\)

\(M_{21} = I_{1} \left( {\mu \bar{X}_{40} - \bar{X}_{20} } \right)\)

\(M_{22} = I_{0} \left( {\mu \bar{X}_{20} - \bar{X}_{00} } \right) + I_{4} \left( {\bar{X}_{20} - \mu \bar{X}_{40} } \right)\)

\(M_{23} = I_{0} \left( {\mu \bar{X}_{20} - \bar{X}_{00} } \right) + I_{3} \left( {\bar{X}_{20} - \mu \bar{X}_{40} } \right)\)

\(M_{31} = I_{2} \left( {\mu \bar{X}_{40} - \bar{X}_{20} } \right)\)

\(M_{32} = I_{0} \left( {\mu \bar{X}_{20} - \bar{X}_{00} } \right) + I_{3} \left( {\bar{X}_{20} - \mu \bar{X}_{40} } \right)\)

\(M_{33} = I_{0} \left( {\mu \bar{X}_{20} - \bar{X}_{00} } \right) + I_{5} \left( {\bar{X}_{20} - \mu \bar{X}_{40} } \right)\)

 

\(C_{11} = i{\text{g}}_{0} J_{11} \left( {\lambda \bar{X}_{15} - \bar{X}_{13} } \right)\)

\(C_{12} = i{\text{g}}_{0} J_{11}^{z} \left( {\bar{X}_{13} - \lambda \bar{X}_{15} } \right)\)

\(C_{13} = i{\text{g}}_{0} J_{11}^{f} \left( {\bar{X}_{13} - \lambda \bar{X}_{15} } \right)\)

\(C_{21} = i{\text{g}}_{0} J_{11}^{z} \left( {\lambda \bar{X}_{60} - \bar{X}_{40} } \right)\)

\(C_{22} = i\left( {\bar{X}_{00} - \mu \bar{X}_{20} } \right)c_{d} + i{\text{g}}_{0} J_{11}^{zz} \left( {\bar{X}_{40} - \lambda \bar{X}_{60} } \right)\)

\(C_{23} = i\left( {\bar{X}_{00} - \mu \bar{X}_{20} } \right)c_{d} + i{\text{g}}_{0} J_{11}^{zf} \left( {\bar{X}_{40} - \lambda \bar{X}_{60} } \right)\)

\(C_{31} = i{\text{g}}_{0} J_{11}^{f} \left( {\lambda \bar{X}_{60} - \bar{X}_{40} } \right)\)

\(C_{32} = i\left( {\bar{X}_{00} - \mu \bar{X}_{20} } \right)c_{d} + i{\text{g}}_{0} J_{11}^{zf} \left( {\bar{X}_{40} - \lambda \bar{X}_{60} } \right)\)

\(C_{33} = i\left( {\bar{X}_{00} - \mu \bar{X}_{20} } \right)c_{d} + i{\text{g}}_{0} J_{11}^{ff} \left( {\bar{X}_{40} - \lambda \bar{X}_{60} } \right) + i{\text{g}}_{0} K_{55} \left( {\lambda \bar{X}_{40} - \bar{X}_{20} } \right)\)

\(k_{11} = J_{11} \left( {\lambda \bar{X}_{15} - \bar{X}_{13} } \right)\)

\(k_{12} = J_{11}^{z} \left( {\bar{X}_{13} - \lambda \bar{X}_{15} } \right)\)

\(k_{13} = J_{11}^{f} \left( {\bar{X}_{13} - \lambda \bar{X}_{15} } \right)\)

\(k_{14} = K_{31}^{e} \left( {\lambda \bar{X}_{13} - \bar{X}_{11} } \right)\)

\(k_{15} = K_{31}^{m} \left( {\lambda \bar{X}_{13} - \bar{X}_{11} } \right)\)

\(k_{21} = J_{11}^{z} \left( {\lambda \bar{X}_{60} - \bar{X}_{40} } \right)\)

\(\begin{aligned} k_{22} & \, & = \left( {N^{E} + N^{H} } \right)\left( {\bar{X}_{20} - \mu \bar{X}_{40} } \right) \\ & \quad + k_{w} \left( {\bar{X}_{00} - \mu \bar{X}_{20} } \right) \\ \quad + k_{p} \left( {\mu \bar{X}_{40} - \bar{X}_{20} } \right) \\ \quad + J_{11}^{zz} \left( {\bar{X}_{40} - \lambda \bar{X}_{60} } \right) \\ \end{aligned}\)

\(\begin{aligned} k_{23} & & = \left( {N^{E} + N^{H} } \right)\left( {\bar{X}_{20} - \mu \bar{X}_{40} } \right) \\ \quad + k_{w} \left( {\bar{X}_{00} - \mu \bar{X}_{20} } \right) \\ \quad + k_{p} \left( {\mu \bar{X}_{40} - \bar{X}_{20} } \right) \\ \quad + J_{11}^{zf} \left( {\bar{X}_{40} - \lambda \bar{X}_{60} } \right) \\ \end{aligned}\)

\(k_{24} = X_{31}^{e} \left( {\lambda \bar{X}_{40} - \bar{X}_{20} } \right)\)

\(k_{25} = X_{31}^{m} \left( {\lambda \bar{X}_{40} - \bar{X}_{20} } \right)\)

\(k_{31} = J_{11}^{f} \left( {\lambda \bar{X}_{60} - \bar{X}_{40} } \right)\)

\(\begin{aligned} k_{32} & = \left( {N^{E} + N^{H} } \right)\left( {\bar{X}_{20} - \mu \bar{X}_{40} } \right) + k_{w} \left( {\bar{X}_{00} - \mu \bar{X}_{20} } \right) \\ + k_{p} \left( {\mu \bar{X}_{40} - \bar{X}_{20} } \right) + J_{11}^{zf} \left( {\bar{X}_{40} - \lambda \bar{X}_{60} } \right) \\ \end{aligned}\)

\(\begin{array}{ll} k_{33} & = \left( {N^{E} + N^{H} } \right)\left( {\bar{X}_{20} - \mu \bar{X}_{40} } \right) \\ &+ k_{w} \left( {\bar{X}_{00} - \mu \bar{X}_{20} } \right) \\ &+ k_{p} \left( {\mu \bar{X}_{40} - \bar{X}_{20} } \right) \\ & + K_{55} \left( {\lambda \bar{X}_{40} - \bar{X}_{20} } \right) \\ & + J_{11}^{ff} \left( {\bar{X}_{40} - \lambda \bar{X}_{60} } \right) \\ \end{array}\)

\(k_{34} = \left( {Y_{31}^{e} - X_{15} } \right)\left( {\lambda \bar{X}_{40} - \bar{X}_{20} } \right)\)

\(k_{35} = \left( {Y_{31}^{m} - Y_{15} } \right)\left( {\lambda \bar{X}_{40} - \bar{X}_{20} } \right)\)

\(k_{41} = K_{31}^{e} \left( {\bar{X}_{20} - \lambda \bar{X}_{40} } \right)\)

\(k_{42} = X_{31}^{e} \left( {\lambda \bar{X}_{40} - \bar{X}_{20} } \right)\)

\(k_{43} = \left( {X_{15} - Y_{31}^{e} } \right)\left( {\bar{X}_{20} - \lambda \bar{X}_{40} } \right)\)

\(k_{44} = X_{11} \left( {\bar{X}_{20} - \lambda \bar{X}_{40} } \right) + X_{33} \left( {\lambda \bar{X}_{20} - \bar{X}_{00} } \right)\)

\(k_{45} = Y_{11} \left( {\bar{X}_{20} - \lambda \bar{X}_{40} } \right) + Y_{33} \left( {\lambda \bar{X}_{20} - \bar{X}_{00} } \right)\)

\(k_{51} = K_{31}^{m} \left( {\bar{X}_{20} - \lambda \bar{X}_{40} } \right)\)

\(k_{52} = X_{31}^{m} \left( {\lambda \bar{X}_{40} - \bar{X}_{20} } \right)\)

\(k_{53} = \left( {Y_{15} - Y_{31}^{m} } \right)\left( {\bar{X}_{20} - \lambda \bar{X}_{40} } \right)\)

\(k_{54} = Y_{11} \left( {\bar{X}_{20} - \lambda \bar{X}_{40} } \right) + Y_{33} \left( {\lambda \bar{X}_{20} - \bar{X}_{00} } \right)\)

\(k_{55} = K_{11} \left( {\bar{X}_{20} - \lambda \bar{X}_{40} } \right) + K_{33} \left( {\lambda \bar{X}_{20} - \bar{X}_{00} } \right)\)

F e33  = F m33  = 0

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Ebrahimi, F., karimiasl, M. & Mahesh, V. Chaotic dynamics and forced harmonic vibration analysis of magneto-electro-viscoelastic multiscale composite nanobeam. Engineering with Computers 37, 937–950 (2021). https://doi.org/10.1007/s00366-019-00865-3

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