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 which can be considered to be of Oð1=eÞ, where e51. 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. & 2012 Elsevier Ltd. All rights reserved.