基于绝对节点坐标法的大变形柔性梁几种动力学模型研究

对在平面内大范围转动的大变形柔性梁动力学进行了研究, 基于绝对节点坐标法建立了一种新的大变形柔性梁的非线性动力学模型. 该动力学模型中考虑了柔性梁的轴向拉伸变形和横向弯曲变形, 利用Green-Lagrangian应变张量计算柔性梁的轴向应变及应变能, 利用曲率的精确表达式计算柔性梁的横向弯曲变形能. 运用拉格朗日恒等式给出了柔性梁横向弯曲变形能新的表达式, 该变形能表达式更加简洁, 通过新的变形能表达式得到了新的弹性力模型, 由此得到的动力学方程可以精确地描述柔性梁的几何大变形问题. 通过与高次耦合模型以及ANSYS中BEAM188非线性梁单元模型的比较, 验证了本模型在计算大变形时的正确性以及高次耦合模型在处理大变形问题时的不足. 进一步研究发现, 新的广义弹性力模型可以适当地简化, 给出了两种简化模型, 根据不同模型的计算效率以及计算精度的比较确定了不同模型的适用范围.

Several dynamic models of a large deformation flexible beam based on the absolute nodal coordinate formulation

With the development of space technology, flexible appendages such as lightweight manipulators and satellite antennas, often appear in spacecrafts. Usually, the large overall motion of the flexible appendage will bring about large deformation problem. And there is a strong nonlinear coupling between the large overall motion and deformation of the flexible appendage, which brings about a large challenge to the precise control of the spacecraft. Dynamics of a rotating flexible planar beam with large deformation is investigated in this paper. A new nonlinear dynamic model of a flexible beam with large deformation is established based on an absolute node coordinate formulation (ANCF). The longitudinal and bending deformations of the flexible beam are both considered in the model. The longitudinal strain energy and bending strain energy of the beam can be calculated by using Green-Lagrangian strain tensor and the exact expression of the flexible beam curvature, respectively. A new concise expression of the bending deformation energy can be obtained by using the Lagrange identical equation. The new elastic force model is derived from the new expression of the deformation energy. The dynamic equations of the present model can precisely deal with the large deformation problem of flexible beams. Then, simulation results from three dynamic models, including the ANCF model, the high order coupling model (HOC model), and the BEAM188 model in ANSYS, are compared to prove the validity of the ANCF model proposed in this paper. And we can also find the deficiency of the HOC model from the simulation. Further study shows that the new generalized elastic force model can be simplified properly. Two simplified models are presented in this paper. The applicabilities of the simplified models are pointed out from the viewpoints of computational efficiency and accuracy. A dimensionless parameter denoted as π is introduced to describe the extent to which a flexible beam pendulum undergoing free falling motion is deformed. The deformation of the flexible beam increases as π increases. Considering the calculating efficiency of the dynamic model, when π is small, simplified model I is chosen preferentially; when π is big, simplified model Ⅱ is adopted preferentially.