In this study, the transverse vibration of a traveling beam made of functionally graded material was analyzed. The material gradation was assumed to vary continuously along the thickness direction of the beam in the form of power law exponent. The effect of the longitudinally varying tension due to axial acceleration was highlighted, and the dependence of the tension on the finite support rigidity was also considered. A complex governing equation of the functionally graded beam was derived by the Hamilton principle, in which the geometric nonlinearity, material properties and axial load were incorporated. The direct multiscale method was applied to the analysis process of an axially moving functionally graded beam with time-dependent velocity, and the natural frequency and solvability conditions were obtained. Based on the conditions, the stability boundaries of subharmonic resonance and combination resonance were obtained. It was found that the dynamic behavior of axial moving beams could be tuned by using the distribution law of the functional gradient parameters.

中文翻译：

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具有时变速度的轴向运动功能梯度梁的稳定性的直接多尺度分析

在这项研究中，分析了功能梯度材料制成的行进梁的横向振动。假定材料的灰度级以幂律指数的形式沿着光束的厚度方向连续变化。强调了由于轴向加速度引起的纵向变化张力的影响，并且还考虑了张力对有限支撑刚度的依赖性。利用汉密尔顿原理推导了功能梯度梁的复杂控制方程，其中考虑了几何非线性，材料特性和轴向载荷。将直接多尺度方法应用于具有速度随时间变化的轴向移动功能梯度梁的分析过程中，获得了固有频率和可解性条件。根据条件，得到了次谐波共振和组合共振的稳定性边界。研究发现，利用功能梯度参数的分布规律可以对轴向运动梁的动力特性进行调整。