On the dynamics of a thin elastica

We revisit the two degrees of freedom model of the thin elastica presented by Cusumano and Moon (1995) [3]. We observe that for the corresponding experimental system (Cusumano and Moon, 1995 [3]), the ratio of the two natural frequencies of the system was ≈44≈44 which can be considered to be of O(1/ε)O(1/ε), where ε⪡1ε⪡1. The presence of such a vast difference between the frequencies motivates the study of the system using the method of direct partition of motion (DPM), in conjunction with a rescaling of fast time in a manner that is inspired by the WKB method, similar to what was done in Sheheitli and Rand (to appear) [8]. Using this procedure, we obtain an approximate expression for the solutions corresponding to non-local modes of the type observed in the experiments (Cusumano and Moon, 1995 [2]). In addition, we show that these non-local modes will exist for energy values larger than a critical energy value that is expressed in terms of the parameters. The formal approximate solution is validated by comparison with numerical integration.

► A model of a thin elastica is studied using the method of direct partition of motion.
► A rescaling of fast time, as in the WKB method, is utilized to analyze the fast dynamics.
► A critical energy value is found at which a bifurcation of periodic orbits occur.
► An explicit expression for the bifurcating non-local modes is derived.