We study the geometrically nonlinear vibrations of rectangular micro/nanoplates with an account of small-scale effects in a framework of the nonlocal elasticity theory. Hamilton’s principle yields the governing system of nonlinear partial differential equations (PDEs) based on the Kirchhoff-Love hypotheses and geometric nonlinearity introduced through the von Kármán theory.

Next, we employed a strategy to get coupled nonlinear and non-autonomous system of ordinary differential equations (ODEs) due to the concept of both reduced order modelling (ROM) truncated to the double-mode approximation of the plate deflection and the Bubnov-Galerkin procedure. The latter one allowed for the transformation of the initial problem to that with separated time and position functions and the used approach was validated.

Then we have studied the reduced model of second-order nonlinear ODEs with coupled geometric nonlinearities through the multiple scales method (MSM) in time domain. Both resonant and non-resonant vibrations have been considered with emphasis put on the small-scale effects and the approximation accuracy. In spite of numerous novel results regarding parametric analysis carried out by combined analytical-numerical approach, we have detected ambiguous and unambiguous back-bone curves which allow one to predict and possibly avoid the pull-in phenomena related to the primary problem governed by the nonlinear vibrations of micro/nanoplates.

我们研究了矩形微/纳米板的几何非线性振动，并在非局部弹性理论的框架中考虑了小尺度效应。哈密​​顿原理产生了非线性偏微分方程 (PDE) 的控制系统，该系统基于通过 von Kármán 理论引入的 Kirchhoff-Love 假设和几何非线性。

接下来，由于将降阶建模 (ROM) 截断为板偏转的双模近似和 Bubnov-Galerkin 的概念，我们采用了一种策略来获得耦合非线性和非自治常微分方程 (ODE) 系统程序。后一个允许将初始问题转换为具有分离时间和位置函数的问题，并且所使用的方法得到了验证。

然后我们在时域中通过多尺度方法（MSM）研究了具有耦合几何非线性的二阶非线性常微分方程的简化模型。已经考虑了共振和非共振振动，重点放在小尺度效应和近似精度上。尽管通过结合分析-数值方法进行的参数分析有许多新结果，但我们已经检测到模糊和明确的主干曲线，这些曲线允许人们预测并可能避免与非线性控制的主要问题相关的拉入现象微/纳米板的振动。