A geometrically-exact momentum-based non-linear theory applicable to beams in non-inertial frames
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© 2019 A geometrically-exact non-linear beam model is developed based on conservation of momentum for application to arbitrarily-shaped beams having large deformations and large overall motions. Coordinate transformations are used to derive the non-linear inertial forces and moments and the non-linear relationships between displacements and strains, enabling rigorous consideration of kinematic and geometric nonlinearities. General non-linear equations of motion are first derived in a differential form, and then are simplified using practical engineering assumptions, including developments of a linearized model for small angles and small strains and a discretized model using finite volumes. A separation of displacements technique is proposed, which enables non-linear beam dynamics to be rigorously reduced to a series of piecewise-linear models. The proposed model is unique amongst existing geometrically-exact beam formulations in that it is formulated using dynamic quantities relative to intermediate non-inertial coordinates, allowing general Lagrangian formulations to be established in floating frames with inertial coupling effects, which is compatible with multibody models. The practical value of these theoretical developments is demonstrated through numerical implementation and convergence analyses. The piecewise-linear dynamic solver in which displacements are computed from an evolving configuration is shown to capture non-linear dynamic behaviors using linear solutions without iteration at each time-step.
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