Free vibration behavior of rotating bidirectional functionally graded nano-beams is studied based on Eringen’s nonlocal theory. The beam material consists of ceramic and metal constituents, and the material is graded across the length and thickness directions. The mathematical formulation is framed on Euler–Bernoulli beam theory, and the system of governing equations is derived in variational form using Hamilton’s principle. The governing equations are discretized and transformed to an eigen value problem using Ritz method. The model is formulated to study the flapping and lead-lag motions due to free vibration. The model is verified with the available numerical results. The numerical results are presented in non-dimensional frequency-speed plane to study the influence of normalized nonlocal parameter, length gradient parameter, thickness gradient parameter, root radius parameter, and section aspect ratio. Some normalized mode shapes are presented to illustrate the mode switching phenomenon. The mathematical model of a nonlocal rotating bidirectional functionally graded material nano-beam is presented for the first time through this work and the reported results are new of its kind.

中文翻译：

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基于Eringen非局域理论的旋转双向功能梯度纳米梁自由振动行为研究

基于Eringen非局域理论研究了旋转双向功能梯度纳米梁的自由振动行为。梁材料由陶瓷和金属成分组成，材料在长度和厚度方向上进行分级。数学公式以欧拉-伯努利梁理论为框架，控制方程组是使用哈密顿原理以变分形式导出的。使用 Ritz 方法将控制方程离散化并转换为特征值问题。该模型用于研究由于自由振动引起的扑动和超前滞后运动。该模型通过可用的数值结果进行验证。将数值结果呈现在无量纲频率-速度平面上，研究归一化非局部参数、长度梯度参数、厚度梯度参数、根半径参数和截面纵横比。提出了一些归一化的模式形状来说明模式切换现象。通过这项工作首次提出了非局部旋转双向功能梯度材料纳米束的数学模型，并且报告的结果是同类新的。