Abstract A dynamic analysis of rotating functionally gradient (FG) beams is presented for capturing the steady bending deformation by using a novel floating frame reference (FFR) formulation. Usually, the cross section of bending beams should rotate round the point at the neutral axis while centrifugal inertial forces are supposed to act on centroid axis. Due to material inhomogeneity of FG beams, centroid and neutral axes may be in different positions, which leads to the eccentricity of centrifugal forces. Thus, centrifugal forces can be divided into three componets: transverse component, axial component and force moment acting on the points of the neutral axis, in which transverse component and force moment can make the beam produce the steady bending deformation. However, this speculation has not been presented and discussed in existing literatures. To this end, a novel FFR formulation of rotating FG beams is especially developed considering centroid and neutral axes. The FFR and its nodal coordinates are used to determine the displacement field, in which kinetic and elastic energies can be accurately formulated according to centroid and neutral axes, respectively. By using the Lagrange's equations of the second kind, the nonlinear dynamic equations are derived for transient dynamics problems of rotating FG beams. Simplifying the nonlinear dynamic equations obtains the equilibrium equations about inertial and elastic forces. The equilibrium equations can be solved to capture the steady bending deformation. Based on the steady bending state, the nonlinear dynamic equations are linearized to obtain eigen-frequency equations. Transient responses obtained from the nonlinear dynamic equations and frequencies obtained from the eigen-frequency equations are compared with available results in existing literatures. Finally, effects of material gradient index and angular speed on the steady bending deformation and vibration characteristics are investigated in detail.

中文翻译：

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用于捕获稳定弯曲变形的旋转 FG 梁的动力学建模和分析

摘要 对旋转功能梯度 (FG) 梁进行动态分析，通过使用新型浮动框架参考 (FFR) 公式捕获稳定弯曲变形。通常，弯曲梁的横截面应围绕中性轴的点旋转，而离心惯性力应作用在质心轴上。由于 FG 梁的材料不均匀性，质心和中性轴可能处于不同的位置，从而导致离心力的偏心。因此，离心力可分为三个分量：横向分量、轴向分量和作用在中性轴点上的力矩，其中横向分量和力矩可使梁产生稳定的弯曲变形。然而，这种推测在现有文献中没有提出和讨论。为此，考虑了质心和中性轴，特别开发了旋转 FG 光束的新型 FFR 公式。FFR 及其节点坐标用于确定位移场，其中动能和弹性能可以分别根据质心和中性轴精确公式化。利用第二类拉格朗日方程，推导了旋转FG梁瞬态动力学问题的非线性动力学方程。简化非线性动力学方程，得到关于惯性力和弹性力的平衡方程。可以求解平衡方程来捕捉稳定的弯曲变形。基于稳态弯曲状态，非线性动力学方程被线性化以获得特征频率方程。将从非线性动力学方程获得的瞬态响应和从特征频率方程获得的频率与现有文献中的可用结果进行比较。最后，详细研究了材料梯度指数和角速度对稳态弯曲变形和振动特性的影响。