Dynamics of beams undergoing large rotations accounting for arbitrary axial deformation

It is well-known that flexible beams become stiffer when subjected to high-speed rotations. This is due to the membrane-bending coupling resulting from the large displacements of the beam cross section. This effect, oftencalled geometric stiffening, has been largely discussed in the past two decades. Several methodologies have been proposed in the literature to account for the stiffening effect in the dynamics equations. However, considerable effort is generally expended to derive linear models using steady-state assumptions and membrane-bending decoupling. First a brief review of the open literature on this subject is presented. Then, a general nonlinear model is formulated using a nonlinear strain-displacement relation. This model is used to analyze deeply simplified models found in the literature. In particular, the assumption of steady-state values for the centrifugal load is analyzed, and its consequences are discussed. Thereafter, four finite element models are proposed, one based on nonlinear theory and the others on simplified linear theories. These models are then applied to the study of a flexible beam undergoing prescribed high-speed large rotations. The analyses show that one must account for the geometric stiffening effect to obtain realistic results. In addition, it is shown that models disregarding the axial displacement dynamics lead to erroneous results for the axial stress in the beam, which may be of primary importance in structural integrity analysis. Hence, in the general case, geometric stiffening must be accounted for in association with the inclusion of full axial-transverse displacements coupling dynamics in the model.