Axial vibration of double-walled carbon nanotubes using double-nanorod model with van der Waals force under Pasternak medium and magnetic effects

The present study investigates the axial vibration of double-walled nanotubes. Using the nanorod continuum model with the van der Waals effect, the vibrational frequencies are studied. Aydogdu (Journal of Vibration and Control, Vol. 21, Issue 16, (2015), 3132-3154) proposed a reliable model for the study of axial vibration in a double-walled nanotube. This model provided a detailed investigation of axial vibration using van der Waals effects. But sometimes, the wrong equation might lead to erroneous scientific results. The incorrect term for axial vibration in the double-walled nanotube model is taken care of in the present study for the correct scientific inferences. Effectively, the axial vibrational frequencies appear without decoupling the continuum model as for primary and secondary nanotubes. The semi-analytical method estimates the axial vibrational frequencies of the double-walled nanotube as a coupled model. Two different boundary conditions like clamped-clamped and clamped-free support, are considered in this computation. The Pasternak medium support and magnetic effects influence the vibrational frequencies of the first and second nanotube for the first time. The Pasternak constant and magnetic parameters don't vary with the length of the nanotube for axial vibration. It means that still more understanding requires in modeling the Pasternak medium and magnetic force for the double-nanotube to model axial vibration.