Longitudinal vibration analysis of carbon nanotubes with elastically restrained ends using doublet mechanics

In the vast majority of studies, the supporting conditions of nano-sized rods have been all restricted to clamped, free, or simply supported cases in nature. Consequently, the boundary conditions should be approximated and described by axial spring restraints at the end. In this study, free axial vibration analysis of axially restrained carbon nanotubes (CNTs) is studied within the framework of doublet mechanics theory. Fourier sine series are utilized for describing the axial deflection of the carbon nanotube. An eigenvalue approximation is constructed for vibrational modes with the aid of Stokes' transformation to deformable axial springs. This unclassified approximation bridges the gap between the deformable and rigid boundary conditions. The comparison studies are carried out to verify the efficiency and accuracy of the proposed analytical model by assigning proper values to elastic spring coefficients. The results indicate that the axial springs and small scale parameter of carbon nanotube have considerable effects on the axial vibration behavior of nanotubes NTs. Similarly the dependencies of the vibration frequencies on material scale parameter and axial restraints are significant. Similar higher order effects are predicted for other nano or micro structures, all of that confirmed the smaller is stiffer phenomenon.