Free vibration analysis of a single-walled carbon nanotube embedded in an elastic matrix under rotational restraints

Abstract

Free vibration analysis of a restrained carbon nanotube in an elastic matrix subjected to rotationally restrained boundary conditions is investigated based on Eringen's non-local elasticity. The analytical solution for free vibration frequencies and corresponding mode shapes of single-walled carbon nanotubes are established. Using Fourier series and Stokes' transformation, a useful coefficient matrix is derived. The present analytical formulation permits to have more efficient coefficient matrix for calculating the vibration frequencies of carbon nanotubes with different boundary conditions (rigid or restrained). The eigen values of this matrix give the vibration frequencies. Comparisons between the free vibration frequency results of the present solutions and previous works in the literature are performed. The calculated results show an excellent agreement with other solutions available in the literature.