耦合变形对大范围运动柔性梁动力学建模的影响

Dynamics modeling of the elastic beam undergoing large overall motion considering coupling effect in deformation

柔性梁在作大范围空间运动时,产生弯曲和扭转变形,这些变形的相互耦合形成了梁在纵向以及横向位移的二次耦合变量。本文考虑了变形产生的几何非线性效应对运动柔性梁的影响,在其三个方向的变形中均考虑了二次耦合变量,利用弹性旋转矩阵建立了准确的几何非线性变形方程,通过Lagrange方程导出系统的动力学方程。仿真结果表明,在大范围运动情况下,仅在纵向变形中计及了变形二次耦合量的一次动力学模型,与考虑了完全几何非线性变形的模型具有一定的差异。

A flexible beam undergoing large overall motions produces curved and twisted deformations. These deformations contribute to the second-order coupling terms of deformation in longitudinal and lateral displacement. Considering the second-order coupling terms of longitudinal deformation, an one-order coupling dynamic model can be obtained. In this paper, based on a geometrically exact nonlinear theory for curved and twisted flexible beam, the longitudinal extension, lateral and transversal deflections are taken into account to derive three-dimensional deformations. An elastic rotation matrix is employed to get the fully nonlinear model and then, the dynamic model and formulations can been implemented by modal function and Lagrange＇s equation. The new model includes more coupling terms. Then a rotating cantilever beam in three-dimensional space is investigated through numerical simulation. The simulation results of new model are different to one-order coupling dynamic model, when the beam is undergoing overall motions. The numerical comparison illustrates that the coupling terms can not be ignored, especially, when the spectrum of motion is increased.