Axisymmetric vibrations of multi-walled carbon nanotubes (MWCNTs) with finite length are investigated in this paper. A multi-walled carbon nanotube is modeled with multiple elastic isotropic shells. Based on a continuum model and considering van der Waals forces between tubes, a two-dimensional finite element model is developed to obtain axisymmetric natural frequencies and mode shapes of MWCNTs. First, the axisymmetric vibrational characteristics of single-walled carbon nanotubes are investigated, and then, they are compared with those of MWCNTs. The effects of van der Waals forces on radial vibration of MWCNTs are also explained. Moreover, influences of end conditions on radial and axial natural frequencies of carbon nanotubes (CNTs) with wide range of length-to-thickness ratio are studied by considering the free-free (F-F) and simply supported (S-S) end conditions. Besides, it is focused on dependence of axisymmetric mode frequencies on dimensional parameters such as length, diameter, as well as number of layers of MWCNTs. Through this, explicit expressions are found for calculating the radial breathing mode and longitudinal mode frequencies of MWCNTs.

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