Longitudinally varying magnetic field influenced transverse vibration of embedded double-walled carbon nanotubes

•Free transverse vibrations of DWCNTs subjected to LVMFs are addressed.•Nonlocal Rayleigh, Timoshenko, and Reddy beam models are employed.•RKPM is adopted for solving the governing equations associated with each model.•Convergence checks and comparison studies are performed and good results are gained.•Roles of geometrical data and LVMF׳s strength on frequencies are studied.

Using nonlocal Rayleigh, Timoshenko, and higher-order beam theories, the free dynamic deflection of elastically supported double-walled carbon nanotubes (DWCNTs) subjected to a longitudinally varying magnetic field (LVMF) is examined. By employing reproducing kernel particle method (RKPM), the equations of motion of the magnetically affected DWCNT (MADWCNT) for each model are reduced to a set of algebraic equations. For four common boundary conditions, namely fully simple, fully clamped, simple-clamped, cantilevered supports, the dominant frequencies of the nanostructure are calculated. In particular cases, the predicted results by the RKPM are compared with those of the exact solution. Additionally, the convergence checks of the proposed numerical models are performed. The effects of the innermost radius, slenderness ratio, small-scale parameter, maximum strength of the LVMF, transverse and rotational stiffness of the surrounding medium on the fundamental frequency of the MADWCNT are addressed. The capabilities of the proposed models in predicting the characteristics of free vibration are also discussed. Further, the limitations of the local analysis as well as the classical beam theory in capturing the lateral vibrations of the MADWCNTs are revealed.