Understanding the nonlinear dynamics of beam structures: A principal geodesic analysis approach

We present a framework to identify the main kinematic features that arise when considering the nonlinear dynamics of beam structures that takes advantage of the mathematical structure provided by the configuration space. This relies on: i) a finite-element formulation for geometrically exact beams; ii) a multibody formalism to deal with boundary conditions and to render complex structures; iii) a robust integration scheme; and, iv) a principal geodesic analysis to directly identify the main kinematic features. Our framework contributes to improve the understanding of the very complex nonlinear dynamics, and at the same time, provides some hints regarding the further model order reduction, but in a fully nonlinear setting. The proposed ideas are tested and their capabilities are illustrated with four examples: a swinging rod under gravity, a free oscillating clamped-free straight beam with pre-stress, a triple pendulum under gravity and a complete wind turbine.