Characterization of free vibration of elastically supported double-walled carbon nanotubes subjected to a longitudinally varying magnetic field

Free transverse vibrations of elastically supported double-walled carbon nanotubes (DWCNTs) subjected to axially varying magnetic fields are examined. Using nonlocal Rayleigh beam theory, the explicit expressions of the governing equations are obtained and then numerically solved via an efficient numerical scheme. For magnetically affected DWCNTs with simply supported, fully clamped, simple-clamped, and clamped-free ends, the flexural frequencies as well as the corresponding vibration modes are evaluated for different varying magnetic fields. The influences of the small-scale parameter and the magnetic field strength on the dominant flexural frequencies of the DWCNTs are explained and discussed. The results indicate that the vibration characteristics of DWCNTs can be significantly affected by the axially varying magnetic field. The role of variation of the axial magnetic field on the vibrational mode patterns of both the innermost and outermost tubes is also revealed. For a special applied magnetic field, the alteration from coaxial to noncoaxial vibration pattern is also reported. The obtained results display that the flexural frequencies magnify with the magnetic field strength. Generally, the variation of the magnetic field strength has more influence on the variation of the frequencies of DWCNTs with higher small-scale parameters. This matter is mainly attributed to the incorporation of the size effect into the nonlocal Lorentz forces.