Free transverse vibration of an axially loaded non-prismatic single-walled carbon nanotube embedded in a two-parameter elastic medium

The free vibration analysis of a single-walled carbon nanotube (SWCNT) is a requisite analysis for the optimal design of nanolevel devices and systems. However, the natural frequencies of structures from free vibration analysis depend, among other things, on their geometry, boundary conditions, material property and the nature of the surrounding medium. Thus in this paper, a mathematical model is presented for the vibration analysis of a geometrically tapered SWCNT. The SWCNT is idealized as a nonlocal Rayleigh (NLR) beam that is axially loaded and embedded in a deformable surrounding medium. A polynomial power law that defines the range of tapers along the longitudinal axis of the continuum is proposed. The variable coefficient governing differential equation of the tapered SWCNT is solved with the Bubnov–Galerkin method. Numerical values, graphical plots and the Pareto charts of the influence of the parameters in the model on the natural frequency of the NLR beam are presented. The presence of the taper ratio alters the natural frequencies of the SWCNT for the values of the radius of gyration considered. The analysis further shows that the influence of the nonlocal parameter with the Pasternak shear modulus is more profound than with the Winkler modulus at the nanoscale level.