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Nonlinear modal analysis of multi-walled nanotube oscillations using nonlocal anisotropic elastic shell model

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Abstract

System of nonlinear partial differential equations, which describes the multi-walled carbon nanotube nonlinear oscillations, is derived. The Sanders–Koiter nonlinear shell theory and the nonlocal anisotropic Hooke’s law are used. Three kinds of nonlinearities are accounted. First of all, the van der Waals forces are considered as nonlinear functions of the radial displacements. Secondly, the nanotube walls displacements have moderate values, which are described by the geometrically nonlinear shell theory. Thirdly, as the stress resultants are the nonlinear functions of the displacements, the additional nonlinear terms are accounted in the equations of motions. These terms are derived from the natural boundary conditions, which are used in the weighted residual method. The finite degrees of freedom nonlinear dynamical system is derived to describe the oscillations of nanostructure. The multi-mode invariant manifolds are used to describe the free nonlinear oscillations, as the dynamical systems have the internal resonances 1:1. The motions on the invariant manifolds are described by two degrees of freedom nonlinear dynamical systems, which are studied by the multiple scales method. The backbone curves of the nonlinear modes are analyzed. As follows from the results of the numerical simulations, the eigenmode of low eigenfrequency has commensurable longitudinal, transversal and circumference displacements. In this case, the nonlinear parts of the van der Waals forces harden essentially the backbone curve of the oscillations.

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The data generated during the current study are available from the corresponding author on reasonable request.

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Authors and Affiliations

  1. Podgorny Institute for Mechanical Engineering, National Academy of Science of Ukraine, Kharkiv, Ukraine

    K. Avramov

  2. Department of Aircraft Strength, National Aerospace University N.Ye. Zhukovsky “KhAI”, Kharkiv, Ukraine

    K. Avramov

  3. Department of Technical Systems, Kharkiv National University of Radio Electronics, Kharkiv, Ukraine

    K. Avramov & I. Grebennik

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Appendix 1: Functions and operators of system (22–24)

Appendix 1: Functions and operators of system (22–24)

The functions and the operators of Eqs. (22–24) are the following:

$${\widetilde{Q}}_{j}^{(i)}\left(\cdot \right)=\frac{{\widetilde{Y}}_{j2}^{\left(i\right)}}{{\delta }_{i}}\left(\cdot \right);\quad i=1,\dots ,N;j=\mathrm{1,2};3;$$
$${\widetilde{Q}}_{4}^{(i)}\left({\widetilde{u}}_{i},{\widetilde{v}}_{i},{\widetilde{w}}_{i}\right)=\frac{1}{{\delta }_{i}}\left\{{\alpha }_{i}^{2}\frac{{\partial }^{2}}{{\partial \eta }^{2}}\sum_{j=1}^{3}{\widetilde{X}}_{1j}^{\left(i\right)}{\widetilde{k}}_{j}^{\left(i\right)}+\right.\frac{{\partial }^{2}}{{\partial \theta }^{2}}\sum_{j=1}^{3}{\widetilde{X}}_{2j}^{\left(i\right)}{\widetilde{k}}_{j}^{\left(i\right)}+\left.\frac{2{\alpha }_{i}{\partial }^{2}}{\partial \theta \partial \eta }\sum_{j=1}^{3}{\widetilde{X}}_{3j}^{\left(i\right)}{\widetilde{k}}_{j}^{\left(i\right)}\right\};$$
$${\widetilde{\mathfrak{L}}}_{j}^{\left(i\right)}\left(\cdot \right)={\alpha }_{1}\frac{\partial }{\partial \eta }\left[{\widetilde{Y}}_{1j}^{\left(i\right)}(\cdot )\right]+\frac{\partial }{{\delta }_{i}\partial \theta }\left[{\widetilde{Y}}_{j3}^{\left(i\right)}(\cdot )\right];\quad i=1,\dots ,N;j=1,\dots ,3;$$
$${\widetilde{\mathfrak{L}}}_{4}^{\left(i\right)}\left({\widetilde{u}}_{i},{\widetilde{v}}_{i},{\widetilde{w}}_{i}\right)=\frac{1}{2{\delta }_{i}}\frac{\partial }{\partial \theta }\sum_{j=1}^{3}{\widetilde{X}}_{3j}^{\left(i\right)}{\widetilde{k}}_{j}^{\left(i\right)};$$
$${\widetilde{P}}_{j}^{\left(i\right)}\left(\cdot \right)=\frac{\partial }{{\delta }_{i}\partial \theta }\left[{\widetilde{Y}}_{2j}^{\left(i\right)}(\cdot )\right]+{\alpha }_{1}\frac{\partial }{\partial \eta }\left[{\widetilde{Y}}_{j3}^{\left(i\right)}(\cdot )\right];i=1,\dots ,N;j=\mathrm{1,2},3;$$
$${\widetilde{P}}_{4}^{\left(i\right)}\left({\widetilde{u}}_{i},{\widetilde{v}}_{i},{\widetilde{w}}_{i}\right)=\frac{1}{{\delta }_{i}}\left\{\frac{\partial }{\partial \theta }\sum_{j=1}^{3}{\widetilde{X}}_{2j}^{\left(i\right)}{\widetilde{k}}_{j}^{\left(i\right)}+\frac{3{\alpha }_{i}}{2}\left.\frac{\partial }{\partial \eta }\sum_{j=1}^{3}{\widetilde{X}}_{3j}^{\left(i\right)}{\widetilde{k}}_{j}^{\left(i\right)}\right\}\right.;$$
$${\widetilde{F}}_{W}^{\left(i\right)}=-{\widetilde{Q}}_{1}^{\left(i\right)}\left({\widetilde{\varepsilon }}_{i,X,0}^{\left(NL\right)}\right)-{\widetilde{Q}}_{2}^{\left(i\right)}\left({\widetilde{\varepsilon }}_{i,\theta ,0}^{\left(NL\right)}\right)-{\widetilde{Q}}_{3}^{\left(i\right)}\left({\widetilde{\gamma }}_{i,X\theta ,0}^{\left(NL\right)}\right)+\widetilde{\Lambda }\left\{{\alpha }_{1}{\alpha }_{i}\frac{\partial }{\partial \eta }\right.\left({\widetilde{N}}_{XX}^{\left(i\right)}\frac{\partial {\widetilde{w}}_{i}}{\partial \eta }\right)+\frac{\partial }{{\delta }_{i}\partial \theta }\left[{\widetilde{N}}_{\theta \theta }^{\left(i\right)}\left(\frac{\partial {\widetilde{w}}_{i}}{\partial \theta }-{\widetilde{v}}_{i}\right)\right]+{\alpha }_{1}\frac{\partial }{\partial \eta }\left[{\widetilde{N}}_{X\theta }^{\left(i\right)}\left(\frac{\partial {\widetilde{w}}_{i}}{\partial \theta }-{\widetilde{v}}_{i}\right)\right]+{\alpha }_{1}\left.\frac{\partial }{\partial \theta }\left({\widetilde{N}}_{X\theta }^{\left(i\right)}\frac{\partial {\widetilde{w}}_{i}}{\partial \eta }\right)\right\};$$
$${\widetilde{F}}_{U}^{\left(i\right)}={\widetilde{\mathfrak{L}}}_{1}^{\left(i\right)}\left({\widetilde{\varepsilon }}_{i,X,0}^{\left(NL\right)}\right)+{\widetilde{\mathfrak{L}}}_{2}^{\left(i\right)}\left({\widetilde{\varepsilon }}_{i,\theta ,0}^{\left(NL\right)}\right)+{\widetilde{\mathfrak{L}}}_{3}^{\left(i\right)}\left({\widetilde{\gamma }}_{i,X\theta ,0}^{\left(NL\right)}\right)-\widetilde{\Lambda }\left.\left\{\frac{\partial }{\partial \theta }\left[\frac{{\widetilde{N}}_{XX}^{(i)}+{\widetilde{N}}_{\theta \theta }^{(i)}}{4{\delta }_{i}}\right.\right.\left.\left({\alpha }_{i}\frac{\partial {\widetilde{v}}_{i}}{\partial \eta }-\frac{\partial {\widetilde{u}}_{i}}{\partial \theta }\right)\right]\right\};$$
$${\widetilde{F}}_{V}^{\left(i\right)}={\widetilde{P}}_{1}^{\left(i\right)}\left({\widetilde{\varepsilon }}_{i,X,0}^{\left(NL\right)}\right)+{\widetilde{P}}_{2}^{\left(i\right)}\left({\widetilde{\varepsilon }}_{i,\theta ,0}^{\left(NL\right)}\right)+{\widetilde{P}}_{3}^{\left(i\right)}\left({\widetilde{\gamma }}_{i,X\theta ,0}^{\left(NL\right)}\right)+\widetilde{\Lambda }\left.\left\{{\alpha }_{1}\frac{\partial }{\partial \eta }\left[\left.\frac{{\widetilde{N}}_{XX}^{(i)}+{\widetilde{N}}_{\theta \theta }^{(i)}}{4}\left({\alpha }_{i}\frac{\partial {\widetilde{v}}_{i}}{\partial \eta }-\frac{\partial {\widetilde{u}}_{i}}{\partial \theta }\right)\right]+\frac{{\widetilde{N}}_{\theta \theta }^{(i)}}{{\delta }_{i}}\left(\frac{\partial {\widetilde{w}}_{i}}{\partial \theta }-{\widetilde{v}}_{i}\right)+{\alpha }_{1}{\widetilde{N}}_{X\theta }^{(i)}\frac{\partial {\widetilde{w}}_{i}}{\partial \eta }\right.\right.\right\}.$$

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Avramov, K., Grebennik, I. Nonlinear modal analysis of multi-walled nanotube oscillations using nonlocal anisotropic elastic shell model. Nonlinear Dyn 111, 21587–21610 (2023). https://doi.org/10.1007/s11071-023-08978-7

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