基于SO(3)群几何精确Kirchhoff梁理论的细长梁动力学

Dynamics of slender beam based on geometrically exact Kirchhoff beam theory formulated on SO(3)group

本研究旨在开发一种高效的细长梁单元,精确描述细长柔性结构的大变形和大旋转,在现有细长梁单元研究的基础上,提出一种基于SO(3)群活动标架的二阶几何精确Kirchhoff梁单元(GEKBF),用于细长梁的动力学研究.首先,在此标架中推导出运动学描述和几何、本构和动态平衡方程及其线性化.然后,引入相对扭转角的二阶插值函数,以提高单元的收敛性,通过选择合适的标架来开发旋转矢量参数化.它可以显著减少由刚体旋转引起的几何非线性,并提高描述当前标架下单元旋转公式的计算效率.此外,考虑了参考标架的影响,并使用空间平行移动而非时间平行移动,简化了弹性力和雅可比矩阵的推导.使用李群广义α-方法求解李群上的半离散运动方程.最后,通过数值实例验证了GEKBF的准确性,包括静力学和动力学.

A geometrically exact Kirchhoff beam formulation(GEKBF)established on the Lie group SO(3)for simulating the dynamics of a slender beam is proposed.The kinematic description,dynamic equilibrium equations and their linearization are derived in this framework.Then,a second-order interpolation function for the torsion angle is introduced to improve the convergence of the element.Next,the rotation vector parameterization is developed to reduce the geometric nonlinearity caused by the rotation of the rigid body.In addition,the influence of the reference frame is considered,and the derivation of the elastic force and Jacobian matrix is simplified using the spatial-parallel transport.The semi-discrete equations of motion are in the form of a second-order ordinary differential equation on Lie group,which are solved using the Lie group generalizedα-method.Finally,the accuracy of the proposed formulation is verified using several numerical examples.