Non-linear dynamics of a shallow arch under periodic excitation —I.1:2 internal resonance

Abstract

The weakly non-linear resonance response of a two-degree-of-freedom shallow arch subjected to simple harmonic excitation is examined in detail for the case of 1:2 internal resonance. The method of averaging is used to yield a set of autonomous equations of the first-order approximations to the response of the system. The averaged equations are then examined to determine their bifurcation behavior. Our analysis indicates that by varying the detuning para- meters from the exact external and internal resonance conditions, the coupled mode response can undergo a Hopf bifurcation to limit cycle motion. It is also shown that the limit cycles quickly undergo period-doubling bifurcation, giving rise to chaos. In order to study the global bifurcation behavior, the Melnikov method is used to determine the analytical results for the critical parameter at which the dynamical system possesses a Smale horseshoe type of chaos.