Superharmonic resonance analysis of nonlocal nano beam subjected to axial thermal and magnetic forces and resting on a nonlinear elastic foundation

In this article, the second and third order superharmonic resonances of a nonlocal Euler–Bernoulli beam are investigated. Eringen’s nonlocal elasticity theory that takes into account the effect of the scale parameter is utilized to derive the governing partial differential equation of motion. It is assumed that the nonlocal beam is resting on an elastic foundation with distributed quadratic and cubic nonlinearities, and is subjected to axial thermal and magnetic forces. A simply supported beam at the nano scale is considered in the analysis. The Glaerkin approach is applied to reduce the nonlinear partial differential equation into an ordinary differential equation. The method of multiple scales is employed to obtain analytical solutions for the superharmonic resonance response curves. The results reveal that the scale parameter, thermal and magnetic axial loads, and the values of the distributed quadratic and cubic nonlinearities of the foundation have a significant effect on the steady state amplitudes of the nonlocal beam. The results are presented over a selected range of physical parameters such as the scale effect parameter, foundation parameters, thermal and magnetic loads, and the excitation level.