In this paper, a new model of the progression phase of a drifting oscillator is proposed. This is to account more accurately for the penetration of an impactor through elasto-plastic solids under a combination of a static and a harmonic excitation. First, the dynamic response of the semi-infinite elasto-plastic medium subjected to repeated impacts by a rigid impactor with conical or spherical contacting surfaces is considered in order to formulate the relevant force-penetration relationship during the loading and unloading phases of the contact. These relationships are then used to develop a physical and mathematical model of a new drifting oscillator, where the time histories of the progression through the medium include both the loading and unloading phases. A nonlinear dynamic analysis of the system was performed and it confirms that the maximum progressive motion of the oscillator occurs when the system exhibits period one motion. The dynamic response for both contact geometries (conical or spherical) show a topological similarity for a range of the static loads.