Abstract A mathematical model for investigating the asymmetric as well as the axisymmetric free vibration behavior of a rotating annular micro-disk is presented for the first time. The disk is assumed to be functionally-graded (FG) along the radial and thickness directions, and is considered to be operating in high-temperature environment. An energy based approach involving minimum potential energy principle and Hamilton's principle is used to derive the governing equations of motion considering Kirchhoff plate theory. The size-effect is addressed employing modified couple stress theory. A novel tangent stiffness based formulation is employed to derive the governing equations of vibratory motion in the neighborhood of the centrifugally and thermally deformed disk configuration. The governing equations are solved following Ritz method. The model captures both the axisymmetric and asymmetric flexural vibration modes, as well as the torsional mode. The model is successfully validated with the available results for some reduced problems. Numerical results are presented in tabular and graphical form for various material and geometric parameters, and some illustrative mode shape plots are presented showing the mode-switching phenomenon. The work presents a generalized model which can be reduced to theoretically model a wide variety of practical problems.

中文翻译：

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热环境中弯曲和扭转模式下旋转双向功能梯度微盘的自由振动行为

摘要 首次提出了研究旋转环形微盘的非对称和轴对称自由振动行为的数学模型。假设磁盘沿径向和厚度方向功能分级（FG），并被认为是在高温环境中运行。基于能量的方法涉及最小势能原理和哈密顿原理，用于推导出考虑基尔霍夫板理论的运动控制方程。使用修正的偶应力理论来解决尺寸效应。采用一种新的基于切线刚度的公式来推导离心和热变形圆盘结构附近的振动运动控制方程。控制方程按照 Ritz 方法求解。该模型捕获轴对称和非对称弯曲振动模式以及扭转模式。该模型已通过一些减少问题的可用结果成功验证。各种材料和几何参数的数值结果以表格和图形形式呈现，并呈现了一些说明性的模式形状图，显示了模式切换现象。这项工作提出了一个广义模型，该模型可以简化为理论上模拟各种实际问题。并提供了一些说明性的模式形状图，显示了模式切换现象。这项工作提出了一个广义模型，该模型可以简化为理论上模拟各种实际问题。并提供了一些说明性的模式形状图，显示了模式切换现象。这项工作提出了一个广义模型，该模型可以简化为理论上模拟各种实际问题。