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.

中文翻译：

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基于能量守恒原理和局部平衡方程研究导电纳米板在恒定平面磁场中的振动的两个微分方程

摘要

在本文中，我们使用两种方法研究纳米片的运动：提取其微分方程和基于能量守恒原理提取积分方程。为了提取描述恒定面内磁场中纳米片自由振动的微分方程，我们首先使用基尔霍夫和孟德尔发展的理论来研究纳米片的变形。然后，我们使用洛伦兹力来计算电磁力，我们使用 Eringen 的非局域理论来考虑非局域效应。提取的方程有一个精确的解来计算矩形纳米板的固有频率，并具有简单的支撑边界条件。根据能量守恒原理提取微分方程，我们根据局部平衡方程计算应力。然后使用这些应力来发现内部力矩和中平面变形之间的关系。之后，根据能量守恒原理，得到描述振动的方程。最后，基于提取的方程，计算曲率，以满足 Eringen 的非局部理论。这些曲率用于计算外加磁场的弹性势能和做功率。对于简单支撑的矩形板，结果表明这两个方程在预测频率上是一致的。然而，随着外加场功率的增加，预测磁粘性的存在，这些方程的结果之间的差异将变得显着。