The Paper is devoted to the study of the oblique-impact vibrating system that is a double compound pendulum under the harmonic moment excitation with the end displacement limit. The specific relations between the pre-impact state and the post-impact state are presented in the case when the friction in the contact surface is not considered. Detailed numerical studies are presented for the transitional dynamics of the steady-state motions of the system with the variation of the excitation and the system parameters, with the illustrations given for rich nonlinear phenomena, such as the bifurcations of periodic vibro-impact motions and the chaotic vibro-impact motions. The dynamic complexity of the oblique-impact vibrating system is showed in these illustrations.