Torsional vibrations of functionally graded restrained nanotubes

Abstract

The torsional free vibration response of a functionally graded (FG) restrained nanotube is studied through an exact analytical solution method. Fourier sine series is utilized with the Stokes' transformation for the FG nanotube with arbitrary boundary conditions. To provide generality to the solution of the problem, regardless of whether it is rigid or non-rigid two rotational (torsional direction) springs in torsional direction are attached to a FG nanorod at two ends. The nonlocal elasticity theory is employed to derive the higher order differential equations with non-local boundary conditions of the functionally graded nanotube. The present model can account for the torsional rotation mechanism at the ends and the nonlocal effects of the atomic range force by introducing torsional spring coefficients and nonlocal parameter. A coefficient matrix including these parameters is obtained for torsional vibration. The characteristic equation of this matrix, which gives the free torsional frequencies, is computed and solved for the dynamical response of the functionally graded nanotubes. In addition, eigen value solutions expressed exact analytically are tabulated to figure out the effects of FG index, length scale parameter, torsional restraints on the free vibration characteristics of the FG nanorod. Analytical results include validation and comparison with previously published papers in the literature are presented for the free vibration response that fully demonstrates the accuracy of the present study.