In this work the chaotic dynamics of flexible curvilinear Euler–Bernoulli micro‐beams embedded into a stationary temperature field is investigated. The temperature field is modelled based on a the Duhamel–Neumann theory and is free from the restrictions on the temperature field distribution along beam thickness. The von Kármán geometric strain–stress relations are employed. The governing nonlinear PDEs are yielded by the Hamilton principle with an account of the modified couple stress theory. The finite dimension problem is truncated to a finite system of nonlinear ODEs using the finite difference method (FDM) and then the Cauchy problem is solved with a help of the Runge–Kutta method. Action of the 2D thermal field is defined by solution to the heat transfer PDE which is also solved by FDM of the second order of accuracy. The so‐called charts of vibration regimes (amplitude‐frequency planes) are constructed. In particular, novel features of nonlinear (chaotic) dynamics versus the change of the magnitude of the size (length) dependent parameter are reported.

中文翻译：

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稳态热场中与尺寸有关的曲线Euler-Bernoulli光束谐振器（MEMS）的混沌动力学

在这项工作中，研究了嵌入到固定温度场中的柔性曲线Euler–Bernoulli微束的混沌动力学。温度场是基于Duhamel–Neumann理论建模的，并且不受沿光束厚度的温度场分布的限制。使用了vonKármán几何应变-应力关系。通过修正偶应力理论，由汉密尔顿原理得出控制非线性PDE。使用有限差分法（FDM）将有限维问题截断为非线性ODE的有限系统，然后在Runge–Kutta方法的帮助下解决Cauchy问题。二维热场的作用由传热PDE的解决方案定义，这也可以通过二级精度的FDM解决。构造了所谓的振动状态图（幅频平面）。特别是，非线性（混沌）动力学相对于尺寸（长度）相关参数的大小变化的新颖特征得到了报道。