Nonlinear free vibrations of curved double walled carbon nanotubes using differential quadrature method

• Vibrations of curved DWCNTs with geometric & vdW force nonlinearities using DQM are studied.
• Different boundary conditions (BCs) are formulated and studied using DQM.
• Nonlinear mode shape of the CNT cannot be represented by a single linear eigenfunction.
• For symmetric BCs, only the odd modes of the linear system are present in the solution.
• For asymmetrical BCs, both odd and even modes of the linear system are present.

Nonlinear free vibration analysis of curved double-walled carbon nanotubes (DWNTs) embedded in an elastic medium is studied in this study. Nonlinearities considered are due to large deflection of carbon nanotubes (geometric nonlinearity) and nonlinear interlayer van der Waals forces between inner and outer tubes. The differential quadrature method (DQM) is utilized to discretize the partial differential equations of motion in spatial domain, which resulted in a nonlinear set of algebraic equations of motion. The effect of nonlinearities, different end conditions, initial curvature, and stiffness of the surrounding elastic medium, and vibrational modes on the nonlinear free vibration of DWCNTs is studied. Results show that it is possible to detect different vibration modes occurring at a single vibration frequency when CNTs vibrate in the out-of-phase vibration mode. Moreover, it is observed that boundary conditions have significant effect on the nonlinear natural frequencies of the DWCNT including multiple solutions.

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