Dynamics of a nanotube supported on the lateral and rotational restrained using doublet mechanics theory

In this study, the vibration behavior of a carbon nanotube on an elastic foundation and supported by deformable springs is investigated. In order to include small-size effects in the solution of the problem, the doublet mechanics theory, an atomistic model, is preferred. Since deformable springs are at the ends, the lateral deflection function is proposed in three different regions, using constant coefficients at the boundaries and a Fourier sine series in the region. Then the Stokes' transform is applied to force the boundary conditions into the desired form. After the mathematical arrangements, an eigenvalue problem including small-size effects, elastic foundation effect, and deformable springs is derived. The accuracy of the model is checked by comparing the results obtained with the results obtained in the literature. After receiving a good match with the literature, the effects of various parameters on the vibration frequencies are analyzed on a series of graphical figures and tables.