The aim of this paper is to show how the concept of nonlinear normal modes (NNMs) can be used to characterize the nonlinear dynamical behaviors of double-walled carbon nanotubes (DWCNTs). DWCNTs are modeled as double simply supported elastic beams with van der Waals (vdW) forces between the inner and outer walls. The multiscale method for deriving the approximate solutions of NNMs is applicable to the nonlinear systems. According to the procedure, the typical features - coaxial and noncoaxial vibrations of the system are exhibited in the literature. Moreover, the case of 1:3 internal resonance is discussed in detail, which can give rise to much more complex phenomenon for DWCNTs systems. Meanwhile, the amplitude-time curves of the nonlinear vibration with different initial conditions are presented, and the amplitude-frequency characteristic curves of the nonlinear vibration are also obtained. © 2010.
Yan, Y., Wang, W., & Zhang, L. (2011). Applied multiscale method to analysis of nonlinear vibration for double-walled carbon nanotubes. Applied Mathematical Modelling, 35(5), 2279–2289. https://doi.org/10.1016/j.apm.2010.11.035