A high-precision co-rotational formulation of 3D beam elements for dynamic analysis of flexible multibody systems
A high-precision co-rotational formulation of 3D beam elements for dynamic analysis of flexible multibody systems
复制标题
用于柔性多体系统动态分析的 3D 梁单元的高精度同转公式
DOI：
10.1016/j.cma.2019.112701
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发表时间：
2020
影响因子：
7.2
通讯作者：
Jinshuai Xu
中科院分区：
文献类型：
--
作者：
Gang Wang;Zhaohui Qi;Jinshuai Xu
计算惯性力是大位移大转动柔性多体系统动力学分析的核心和复杂问题。在许多共转方法中，惯性项和内力都采用三次插值法来表示，其中惯性项的推导可能比较复杂，数值积分时需要几个高斯点。本文提出了一种用于柔性制造系统动力分析的三维梁单元的高精度共转列式。为了避免空间有限旋转中的奇异性问题，采用了旋转矢量及其补码互换的方法。与传统的共转公式不同，本文的控制方程是基于虚功率原理建立的，不需要改变旋转矩阵。在所提出的共转公式框架下，可以直接使用小变形梁单元的刚度矩阵，包括Euler-Bernoulli梁单元和Timoshenko-Reissner梁单元。更本质地说，梁单元的惯性项是通过将梁单元离散成在左端节点和右端节点的三个集中质量以及在梁单元的中点的辅助节点来表示的，从而得到集中质量矩阵。这种等价性保证了梁单元在局部坐标系中承受小的弹性变形时，总质量是完全精确的，转动惯量是高精度的。在共转公式的局部坐标系中，可以保证小的弹性变形，从而使惯性力可以解析地表示出来，而不需要高斯积分。最后，通过四个数值算例验证了该公式相对于几何精确梁理论和前人的同旋梁公式的准确性。即使使用少量的单元，所提出的方法也可以从比较中给出高精度的数值结果
Evaluating inertia forces is a central and complicated task for dynamic analysis of flexible multibody systems (FMS) involving large displacements and rotations. In a number of co-rotational approaches, cubic interpolations are adopted to formulate both inertia and internal forces, where the inertia term may be complicated in derivation and several Gauss points are needed in the numerical integration. The paper presents a high-precision co-rotational formulation of 3D beam elements for dynamic analysis of FMS. The method of switching the rotation vector and its complement is adopted to avoid the singularity problem in spatial finite rotations. In contrast to the traditional co-rotational formulation, the governing equations in the paper are formulated based on the principle of virtual power, without requiring the variation of rotation matrix. In the framework of the proposed co-rotational formulation, the stiffness matrix of small-deformation beam element can be used directly, including the Euler–Bernoulli and Timoshenko–Reissner beam elements. More essentially, the inertia terms of the beam element are formulated by discretizing the beam element into three lumped masses at the left and right end nodes, and an auxiliary node at the middle point of the beam element, resulting in a lumped mass matrix. The equivalence ensures that the total mass is completely accurate and the moment of inertia is high-precision if the beam element undergoes small elastic deformation in the local coordinate system. In the local coordinate system of co-rotational formulations, small elastic deformation can be guaranteed so that the inertia forces are formulated analytically without needing the Gauss integration. Finally, four numerical examples are considered to evaluate the accuracy of the formulation against to the geometrically exact beam theory and the previous co-rotational beam formulations. The proposed method can give high-precision numerical results from the comparison, even if using a small number of elements