Finite element formulation and analysis of a functionally graded Timoshenko beam subjected to an accelerating mass including inertial effects of the mass

Abstract This study describes a new finite element method that can be used to analyse transverse and axial vibrations of a Functionally Graded Material (FGM) beam under an accelerating / decelerating mass. The differential equations of the FGM beam are obtained using First-order Shear Deformation Theory (FSDT). In these equations, the interaction terms of mass inertia are derived from the second-order exact differentiation of displacement functions with respect to mass contact point. The FGM beam is made of two different materials (Steel and Alumina Al2O3), which vary in thickness with a power law. Including the effects of neutral axis shift and mass inertia, the proposed method can be used when the dynamic behaviour of the FGM Timoshenko beams is to be analysed in transverse and axial directions, depending on the interaction with the acceleration of the moving loads. After validating this work with literature studies, new investigations and findings are presented for both moving load and mass assumptions. In addition, the obtained results of Timoshenko Beam (TBT) and Euler Bernoulli beam theory (EBT) are compared for FGM beams with various speeds and accelerations of moving mass.摘要 本研究提出一种新型有限元方法，用于分析加速/减速运动质量作用下功能梯度材料（Functionally Graded Material, FGM）梁的横向与轴向振动。本研究基于一阶剪切变形理论（First-order Shear Deformation Theory, FSDT）推导得到功能梯度材料梁的控制微分方程组；其中，质量惯性交互项通过位移函数对质量接触点的二阶精确微分推导得出。所研究的功能梯度材料梁由钢与氧化铝Al₂O₃两种材料构成，其材料组分沿厚度方向遵循幂律分布。该方法考虑中性轴偏移与质量惯性效应，可用于分析运动载荷加速度交互作用下功能梯度材料铁木辛柯梁（Timoshenko Beam, TBT）的横向与轴向动态特性。本研究通过与已有文献结果对比验证了方法的正确性，并针对移动载荷与移动质量两种假设场景开展了全新的研究并给出相关结论。此外，针对不同运动质量速度与加速度工况，对比了基于铁木辛柯梁（Timoshenko Beam, TBT）与欧拉-伯努利梁理论（Euler Bernoulli beam theory, EBT）得到的功能梯度材料梁计算结果。