Analysis of resonant/nonresonant vibrations of simply-supported Kirchhoff nanoplates under in–plane magnetic field based on a strongly coupled two-mode model

Abstract

The paper aims to study the nonlinear vibrations of simply supported square and rectangular nanoplates in the 2D magnetic field. The investigated model is developed in the framework of the von Kármán nonlinear theory, whereas nonlocal effects are taken into consideration due to the Eringen nonlocal theory of elasticity. PDEs governing system dynamics include the stress function. Both an in-plane magnetic field caused by the Lorentz force yielded by Maxwell's equations, as well as a transverse harmonical excitation are taken into account. The obtained results are based on the Bubnov–Galerkin approach and the two-mode deflection approximation. The latter reduces the problem of infinite dimensions to the system of coupled nonlinear ODEs, which is investigated by the multiple scale method (MSM). The employed MSM in the framework of Mathematica symbolic computations yielded the analytical approximate solutions, which were validated via the numerical Adams method. Both nonresonant and external/internal resonances exhibited by the studied nanoplates are thoroughly analysed. Numerous novel nonlinear phenomena are detected and analysed, with emphasis put on their physical interpretation.