Nonlinear frequency analysis of buckled nanobeams in the presence of longitudinal magnetic field

The nonlinear free vibration around the postbuckling configuration of nonlocal nanobeams with pinned—pinned boundary conditions are analytically investigated, considering the geometric nonlinearity arising from the mid-plane stretching and particularly the effect of a longitudinal magnetic field. When the applied axial force is small, the nanobeam is stable and its free vibration is around the straight equilibrium position. As the axial force becomes large and is beyond a certain critical value (critical axial force), however, either positive or negative non-trivial equilibrium configuration occurs and the free vibration is around the buckled configuration. The governing equation for the nanobeam before buckling exhibits a cubic nonlinearity while for a buckled nanobeam it contains both cubic and quadratic non-linearities. Based on the nonlinear governing equations, approximate analytical solutions for the postbuckling configuration in terms of the applied axial force are evaluated, showing that the magnetic field parameter has a great effect on the critical axial force, buckled displacement and linear frequencies of the nanobeam system. More interestingly, the nonlinear frequency is found to be generally higher than the linear frequency in the case of prebuckling regime while it could be lower than the linear one in the case of postbuckling regime. The nonlinear frequency can be significantly influenced by the magnetic field, the nonlinear coefficient associated with the mid-plane stretching, and the initial vibration amplitude.