Nonlinear vibration of nonlocal strain gradient nanotubes under longitudinal magnetic field

The nonlinear free vibration of embedded nanotubes under longitudinal magnetic field is studied in this paper. The governing equation for the nanotube is formulated by employing Euler – Bernoulli beam model and the nonlocal strain gradient theory. The analytical expression of the nonlinear frequency of the nanotube is obtained by using Galerkin method and the equivalent linearization method with the weighted averaging value. The accuracy of the obtained solution has been verified by comparison with the published solutions and the exact solution. The influences of the nonlocal parameter, material length scale parameter, aspect ratio, diameter ratio, Winkler parameter and longitudinal magnetic field on the nonlinear vibration responses of the nanotubes with pinned-pinned and clamped-clamped boundary conditions are investigated and discussed.