The present research deals with the small and large amplitude free vibrations of thick functionally graded material (FGM) beams. Beams are resting on an elastic foundation which consists of the Winkler layer, shear layer and nonlinear springs. The effect of uniform temperature elevation is also included into the formulation. A third-order shear deformation beam theory is adopted as the basic assumption of the formulation which satisfies the traction-free boundary conditions on the top and bottom surfaces. Using two different formulations, i.e. the mid-surface-based formulation and the neutral surface-based formulation, the governing equations of motion are established. For the case of beams with immovable edges in in-plane direction and simply supported in flexure, the governing equations are given in a new presentation. These equations are solved using the two-step perturbation technique. Closed-form expressions are provided to obtain the small and large amplitude natural frequencies of the FGM beam resting on nonlinear elastic foundation and operating in thermal environment. Numerical results are given to discuss the effects of thermal environment, elastic foundation parameters, different formulations, and the geometrical parameters.

中文翻译：

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基于中性/中平面公式的热场非线性弹性基础上 FGM 梁的大振幅自由振动

目前的研究涉及厚功能梯度材料 (FGM) 梁的小振幅和大振幅自由振动。梁位于弹性基础上，该基础由 Winkler 层、剪切层和非线性弹簧组成。均匀升温的影响也包括在配方中。采用三阶剪切变形梁理论作为公式的基本假设，满足顶面和底面的无牵引边界条件。使用两种不同的公式，即基于中间面的公式和基于中性面的公式，建立了运动控制方程。对于在平面方向上具有不可移动边缘且仅受挠曲支撑的梁的情况，在新的演示文稿中给出了控制方程。这些方程使用两步微扰技术求解。提供闭式表达式以获得基于非线性弹性基础并在热环境中运行的 FGM 梁的小振幅和大振幅固有频率。给出数值结果以讨论热环境、弹性基础参数、不同公式和几何参数的影响。