Nonlinear free vibration analysis of nanobeams under magnetic field based on nonlocal elasticity theory

[[abstract]]In the present study, we perform the nonlinear free vibration analysis of nanobeams under longitudinal magnetic field based on Eringen’s nonlocal elasticity and Euler-Bernoulli beam theory. To capture the small scale effect of the nanobeams, we adopt the nonlocal beam model with the nonlocal parameter. We use the Hamilton’s principle and von-K?rm?n’s nonlinear strain-displacement relationships to derive the governing equations of nanobeams subject to magnetic field. In order to solve the equations, we utilize the Galerkin method and He’s variational method to come up with an approximate analytical solution for the nonlinear frequency of the nanobeams under magnetic field. In the numerical results, the nonlinear frequency ratio is presented for various values of the dimensionless nonlocal parameter and the dimensionless amplitude. Finally, we investigate the effect of nonlocal parameter on the nonlinear frequency ratio; moreover, we study and discuss the effect of magnetic field on the nonlinear free vibration behavior of nanobeams