Coupled effect of in-plane magnetic field and size effect on vibration properties of the completely free double-layered nanoplate system

The coupled effects of an in-plane magnetic field and the size effect on free vibration of a completely free magnetically affected orthotropic double-layered nanoplate system (DLNS) embedded in elastic media are examined by a Hamiltonian-based method combined with Eringen's nonlocal elasticity theory. The Lorentz magnetic force exerted on the DLNS is derived based on Maxwell's equations and Lorentz's formula. By introducing a full-state vector as the primary unknown, the classical governing equation is converted into its Hamiltonian form so that exact solutions of the DLNS are obtained by symplectic expansion and a superposition of boundaries. Analytical frequency equation and vibration mode functions are obtained simultaneously. Numerical results illustrate that, by changing the magnetic parameter, the natural frequencies of the DLNS show two opposite variation trends with increasing nonlocal parameter; and the vibration modes are re-arranged. These findings demonstrate that applying an in-plane magnetic field is an efficient way to control the natural frequencies and vibration modes of such magnetically affected DLNS.