On the Nonlinear Deformation Geometry of Euler-Bernoulli Beams.
NATIONAL AERONUATICS AND SPACE ADMINISTRATION MOFFETT FIELD CA AMES RESEARCH CENTER
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Nonlinear expressions are developed to relate the orientation of the deformed-beam cross section, torsion, local components of bending curvature, angular velocity, and virtual rotation to deformation variables. These expressions are developed in an exact manner in terms of a quasi-coordinate in the space domain for the torsion variable. The entire formulation is independent of the sequence of the three rotations used to describe the orientation of the deformed-beam cross section. For more common cases in the literature in which one of the three rotation angles is used as the torsion variable, the resulting equations depend on the choice of the three angles. Differences in the equations, however, are demonstrated to be in form only. The present deformed-beam kinematic quantities are proven to be equivalent to those derived from various rotation sequences by identifying appropriate changes of variable based on fundamental uniqueness properties of the deformed beam geometry. This development helps to clarify the issues raised in the literature concerning the choice of the angles. The torsion variable used herein is also shown to be mathematically analogous to an axial deflection variable that has been commonly used in the literature. Both variables are quasi-coordinates in the space domain and have been used in derivations based on Hamiltons principle, despite lack of rigorous justification. Rigorous applicability of Hamiltons principle to systems described by a class of quasi-coordinates that includes these variables is formally established. Author