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 …

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 …