Nonlinear dynamics of geometrically exact beams modelled under the special euclidean lie group formulation

The dynamic analysis of flexible slender beams, which exhibit significant geometric nonlinearities due to large displacements and rotations, is critical for various industrial applications, including the aerospace, automotive, and renewable energy sectors. The use of mathematically rigorous and geometrically exact formulations, such as those based on Lie groups, is essential in modelling geometric nonlinearities accurately. However, current state-of-the-art dynamic analysis tools cannot be used on the nonlinear configuration space of Lie groups and require adaptation. This thesis addresses this challenge by developing both a single shooting and a multiple shooting method, together with a pseudo-arclength continuation method, which is compatible with geometrically exact beam formulations based on the Special Euclidean Lie group SE(3). Due to the correct kinematic representation of displacements and rotations in the Lie group framework of rigid body motion, and the local frame representation of the beam equations, the SE(3) model is geometrically exact and inherently shear locking free. The adapted shooting and continuation methods developed in this thesis enable the SE(3) model to be used in industrial applications. These methods are used herein to compute nonlinear normal modes and forced responses of geometrically nonlinear structures, in addition to performing stability analysis and bifurcation detection. The results affirm the suitability of the SE(3) formulation in modelling large amplitude motions, and confirm the effectiveness of the shooting and continuation methods in capturing nonlinear dynamic phenomena such as bifurcations and mode interactions with high accuracy, paving the way for its application in industrial design and analysis. This thesis therefore contributes to filling a critical gap in current industrial modelling capabilities, providing a rigorous and efficient approach for analysing nonlinear dynamics of flexible structures.