极端工况下几何精确梁理论和CR理论的适用性研究

Research on the Applicability of Geometrically Exact Beam Theory and CR Theory Under Extreme Conditions

几何精确梁理论和共旋(Co-Rotational,CR)梁理论是模拟柔性梁动力响应的两种常用方法,然而,针对极端工况下这两种方法的仿真对比分析尚不充分.本文首先基于几何精确梁理论和共旋坐标理论推导了空间大转动梁的刚度矩阵、质量矩阵以及非线性求解的迭代方法,并从理论上讨论了两种方法的特点.最后编写了两种方法的有限元仿真程序,并通过具体算例对比了它们的收敛性和计算效率.仿真结果表明:在足够的单元数量下,两种方法都能提供准确的数值结果;当单元数量较少时,共旋公式的数值结果比几何精确梁理论更精确,并且在相对少量的单元情况下仍能提供足够的精度,这是由于两种方法在迭代过程中对变量更新的不同处理方式所致;在相同单元数量下,几何精确梁理论展现出更高的计算效率和更好的收敛性.本文的研究结果对于柔性梁的数值仿真具有重要的参考价值.

Geometrically exact beam theory and co-rotational(CR)beam theory are two commonly used methods for simulating the dynamic response of flexible beams.However,the comparative analyses of these two methods under extreme conditions are not yet sufficient.In this paper,we first derive the stiffness matrix,mass matrix,and iterative methods for the nonlinear solutions of spatial beams undergoing large rotations based on the geometrically exact beam theory and the CR coordinate theory.The characteristics of the two methods are discussed theoretically.Finally,finite element simulation programs for both methods are developed,and their convergence and computational efficiency are compared through specific examples.The simulation results show that both methods can provide accurate numerical results with sufficient element numbers.When the number of elements is small,the CR formulation yields more accurate numerical results than the geometrically exact beam theory,and can still provide sufficient accuracy with a relatively small number of elements.This is due to the different treatment of variable updates during the iteration process in the two methods.With the same number of elements,the geometrically exact beam theory demonstrates higher computational efficiency and better convergence.The research findings of this paper provide important reference for the numerical simulation of flexible beams.