Two Differential Equations for Investigating the Vibration of Conductive Nanoplates in a Constant In-Plane Magnetic Field Based on the Energy Conservation Principle and the Local Equilibrium Equations

Abstract

Abstract In this paper, we investigate the movement of nanoplates using two approaches: extracting its differential equation, and extracting an integral equation based on the energy conservation principle. To extract the differential equation describing the free vibration of nanoplates in constant in-plane magnetic fields, we first use the theories developed by Kirchhoff and Mendelian to investigate the deformation of the nanoplates. Then, we use Lorentz force to calculate the electromagnetic force, and we use the Eringen’s non-local theory to consider the non-local effects. The extracted equation has an exact solution for calculating the natural frequency of rectangle nanoplates with simply support boundary conditions. To extract the differential equation based on the energy conservation principle, we calculate the stresses based on local equilibrium equations. These stresses are then used to discover the relationship between inner moments and mid-plane deformation. After that, based on the energy conservation principle, an equation describing the vibration is obtained. Finally, based on the extracted equation, the curvatures are calculated so that the Eringen’s non-local theory is satisfied. These curvatures are used to calculate the elastic potential energy and rate of work done for the applied magnetic field. For a rectangular plate with simply support, the results indicate that the two equations are consistent with each other in predicting the frequency. However, as the power of the applied field increases, the existence of magnetic viscosity is predicted, and the difference between the results of these equations will become significant.