Non-linear dynamics of a shallow arch under periodic excitation —II. 1:1 internal resonance

Abstract

In this paper the work presented in Tien et al. [ Int. J. Non-Linear Mech. 29, 349–366 (1994)] is extended to study the dynamics of a shallow arch subjected to harmonic excitation in the presence of both external and 1:1 internal resonance. The method of averaging is used to yield a set of autonomous equations of the second-order approximations to the response of the system. The averaged equations are numerically examined to study the bifurcation behavior of the shallow arch system. In order to study the system with resonant fixed points, a new global perturbation technique developed by Kovacic and Wiggins [ Physica D 57, 185–225 (1992)] is used. This technique provides analytical results for the critical parameter values at which the dynamical system, through the Silnikov's type of homoclinic orbits, possesses a Smale horseshoe type of chaos.