Fourier series solution for free vibration of double-walled nanotubes within the framework of nonlocal strain gradient theory

Abstract

In this study, the free vibration behavior of double-walled nanotubes is examined within the framework of the nonlocal strain gradient theory. The governing equations and moment expressions are derived, and a semi-analytical solution based on Fourier series expansion combined with Stokes transformation is employed for a system modeled with rotation-restricting boundary springs. Convergence analysis confirms that a 500-term approximation provides sufficient accuracy. The results indicate strong agreement with existing studies on both single- and double-walled nanotubes, validating the proposed formulation. A comprehensive parametric investigation is conducted, considering the effects of dimensionless frequencies, interlayer coupling, scale parameters, boundary conditions, and vibration modes. The findings indicate that the nonlocal scale parameter reduces system stiffness and frequencies, whereas the strain gradient parameter enhances stiffness and increases frequencies. These influences become more pronounced under stiffer boundary conditions and higher vibration modes. Additionally, increasing the interlayer coupling coefficient leads to a rise in frequencies, with the system asymptotically approaching the dynamic response of single-walled nanotubes.