Spatial chaos of 3-D elastica with the Kirchhoff gyrostat analogy using Melnikov integrals

Abstract

The Kirchhoff kinetic analogy, from which the similarity between the governing equations for the static spatial equilibrium of a 3-D elastica and those for the temporal dynamics of a rigid body is constituted, is revisited. The Melnikov integrals for detecting chaos cannot be easily formed for an elastica. We shall modify the previous procedure of using the Melnikov integrals for detecting temporal chaos for a gyrostat to solve the spatial chaos problem of an elastica. One way to find the disturbed Hamiltonian equations of the equilibrium equations of the elastica are by means of Deprit's variables. Using the Melnikov integrals, the spatially chaotic deformation patterns can be resulted from the homoclinic transversal intersections of the stable and unstable manifolds at a saddle point in the Poincare map when the elastica is disturbed by constant stress-resultants. The Melnikov integrals are integrated for detecting homoclinic intersections. The equations governing the evolution of the stress-couples and the stress-resultants are numerically integrated using the fourth Runge–Kutta algorithms to crosscheck the analytical results. The phase portraits of the Poincare sections are created on a number of hyper-planes to demonstrate the chaotic patterns of the stationary spatial deformations along the centreline of the elastica under weightless conditions. The bounded, non-periodic solutions to the equilibrium of the elastica clearly show the existence of spatially chaotic deformation patterns. The findings in this paper are useful in many structures in which the elastica are important. Examples are the mathematical modelling of the DNA super-coiling, submarine cables and the design of the tethered satellites.