Kinematic Stability of Variable Cross-section Beam with Large Deformation by using the Absolute Nodal Coordinate Formulation

Abstract

The motion accuracy and dynamic performance of the flexible beams used in space structures are heavily affected by their large deformations during the motion, which leads to an unpredictability of the trajectory of the ends. Structural instability may occur in some extreme cases. The shape of the cross-sections and the material properties are important factors that influence the deformation of the flexible beams. In this paper, the boundary features of the variable cross-section beams are taken into consideration and the stiffness matrix is derived based on the nonlinear continuum mechanics approach. The dynamic model of the beam is established based on the absolute nodal coordinate formulation. The state-space equation of the moving beam is developed based on the Lyapunov theory. A criterion method of the kinematic stability of the flexible beams is proposed, by which the effect of material properties and variable cross-sections is investigated. The results indicate that with a small Young's modulus, the variable cross-section beam shows a better stability than the constant cross-section beam. As the Young's modulus increases, the stability time of the constant cross-section increases much more than the variable cross-section beam. When the Young's modulus reaches a certain value, the kinematic stability of the constant cross-section beam does not change any more.