Longitudinal vibration of nanorods embedded in an elastic medium with elastic restraints at both ends is studied based on the non-local elasticity theory. Using Fourier sine series and Stokes' transformation, a coefficient matrix is obtained. It is very useful for calculating the vibrational frequencies of a nanorod with any type of boundary condition (rigid or restrained). Finally, carrying out some numerical computations, the effects of the elastic medium, non-local parameters and elastic restraints at both ends on the values of vibrational frequencies have been determined. The numerical results are validated through comparison of calculated values with those in the literature.

Free torsional vibration of cracked carbon nanotubes with elastic torsional boundary conditions is studied. Eringen’s nonlocal elasticity theory is used in the analysis. Two similar rotation functions are represented by two Fourier sine series. A coefficient matrix including torsional springs and crack parameter is derived by using Stokes’ transformation and nonlocal boundary conditions. This useful coefficient matrix can be used to obtain the torsional vibration frequencies of cracked nanotubes with restrained boundary conditions. Free torsional vibration frequencies are calculated by using Fourier sine series and compared with the finite element method and analytical solutions available in the literature. The effects of various parameters such as crack parameter, geometry of nanotubes, and deformable boundary conditions are discussed in detail.

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