Chaotic non-planar vibrations of the thin elastica, Part II: Derivation and analysis of a low-dimensional model

Here, we develop a simple model for the bending-torsion vibrations of the thin elastica; the experimental observations were described in Part I. A geometrically exact rod theory is developed: Dimensional analysis demonstrates that a curvature constraint not used in previous analyses is appropriate in our case, and this is used to develop a coupled set of non-linear integro-differential equations for the problem. Using an additional simplifying assumption on the spatial derivative of the torsional field variable, a simplified set of partial differential equations is derived. It is shown that a two-mode projection of these model partial differential equations can be related to an intuitively appealing two-degree-of-freedom mechanical system. Numerical experiments on the two-mode model show that it captures much of the behavior observed in the physical experiments on the thin elastica. In particular, the model possesses a family of bending-torsion non-linear modes with a frequency-amplitude characteristic much like that found experimentally, and the driven problem loses planar stability in a fashion analogous to that observed with the elastica.