Abstract
In this paper modeling of a geometric nonlinear beam and the corresponding matrix representation of the model of 3rd order is considered. Especially in the case of lightweight robots for space applications undergoing large reference motion, the nonlinear dynamic behaviour of the beam should be modeled as exact as possible. Here coupling effects betweeen the elastic variables due to the geometric nonlinear beam kinematic are important. ‘Stiffning effects’ and parametrically excited effects between longitudinal and bending vibrations e.g., should also be considered.
Starting with the nonlinear beam kinematics, all equations up to terms of second order are considered. Using the principal of virtual work, nonlinear equations are given, which are discretized by Hermite polynomials in the next step. Considering also terms of second order in the elastic variables, a special technique handling this equations in the usual structural dynamics matrix representation is developed to preserve the couplings of higher order in contrast to the usual linearization. Additionally, the length-variability is considered. So the clearness and effectiveness of matrix methods is combined with beam theory of 3rd. order, this means quadratic terms for the elastic variables axe considered building up state-dependend matrices. The importance of the effects of those couplings axe shown by two simulation examples of a planar and a spatial lightweight, very flexible telescopic robot arm for space operations. It can be shown clearly, that beside the known stiffning effects due to nominal axial loads, effects of higher order may weaken the structure up to failure of the structure.
