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基于无网格法的中心刚体-FGM微梁动力学建模和仿真

DYNAMIC MODELING AND SIMULATION OF A HUB-FGM MICRO-BEAM BASED ON MESHLESS METHOD

  • 摘要: 采用无网格点插值法(PIM)和无网格径向基点插值法(RPIM)描述柔性体微梁的变形场, 对旋转功能梯度材料(FGM)微梁的动力学特性进行研究. 基于修正偶应力理论和刚柔耦合动力学建模理论, 在浮动坐标系中描述变形位移, 计及由横向弯曲变形所引起的纵向缩短量, 即非线性耦合变形项, 在系统动能中保留与之有关的所有高阶量, 在系统势能中加入了考虑尺度效应的偶应力张量和曲率张量, 利用第二类Lagrange方程, 建立了旋转FGM微梁的高次刚柔耦合非线性动力学模型. 通过仿真算例, 比较了一次近似模型和高次模型的适用范围, 同时验证了本文所建模型的正确性. 进一步研究了材料梯度变化、材料特征长度参数变化以及转动规律等因素对旋转微梁瞬态动力学响应和稳态自由振动的影响规律. 当梁的厚度减小到与材料的特征长度参数接近时, 尺度效应对FGM微梁系统的瞬态动力学响应和稳态自由振动的影响不可忽略. 将无网格法的仿真结果与传统有限元法和假设模态法进行比较分析, 表明了无网格法作为一种柔性体离散方法在刚柔耦合多体系统动力学的研究中具有可推广性, 丰富了柔性多体系统动力学领域的离散方法.

     

    Abstract: This paper presents a comprehensive investigation into the dynamic characteristics of rotating functionally graded material (FGM) micro-beams, employing the meshless point interpolation method (PIM) and radial point interpolation method (RPIM) to describe the deformation field of flexible micro-beams. Based on the modified couple stress theory and the rigid-flexible coupled dynamic theory, the deformation displacement is formulated in a floating coordinate system. To ensure high accuracy in the dynamic model, the longitudinal shortening caused by transverse bending deformation which is called nonlinear coupling terms are taken into account, and all high-order terms that have been overlooked in existing research are retained in the formulation of the system’s kinetic energy. Concurrently, the potential energy expression is enhanced by including the couple stress tensor and curvature tensor, which are fundamental to accounting for size effects predicted by the modified couple stress theory. Utilizing the Lagrange’s equation of the second kind, the high-order rigid-flexible coupled (HOC) dynamic model of a rotating FGM micro-beam is systematically derived. Through a series of numerical simulations, the applicability and limitations of the first-order approximation coupled (FOAC) dynamic model are compared against those of the proposed HOC dynamic model, thereby validating the accuracy and necessity of the established comprehensive model. The factors affecting the dynamic characteristics and natural frequencies of FGM micro-beam such as the functionally gradient index, size-dependency and rotation law are discussed. When the thickness of the beam is reduced to approach the characteristic length parameters of the material, size effect on the dynamic characteristics and natural frequencies of FGM micro-beam cannot be ignored. The simulation results of the meshless method are compared with the traditional finite element method (FEM) and the assumed mode method (AMM). It is shown that the meshless method as a discrete method can be extended in the study of rigid-flexible coupled multi-body system dynamics, which enriches the discrete methods in the field of flexible multi-body system dynamics.

     

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