Ultrasonic Additive Manufacturing is a solid state manufacturing process that combines ultrasonic welding of layers of thin metal foil with contour milling. Bonding between two foils is accomplished by holding the foils together under pressure and applying high-frequency excitations normal to the pressure direction. The accepted explanation for bonding is that stresses due to both compression and friction stemming from the interfacial motion between the foils result in plasticity and ultimately produce a metallurgical bond. The process however, has been shown to have a critical shortcoming in its operation; namely, the presence of a range of build heights within which bonding cannot be initiated. To better understand the reasons for this anomaly, this paper simplifies the process into a lumped parameter dry friction oscillator and shows that complex stick-slip motions of the build feature near or above its resonance frequency may explain bond degradation. Specifically, it is shown through bifurcation maps obtained for different process parameters that, at the critical build heights, the feature exhibits pure stick motions due to primary resonant interactions between the external excitation and the feature. Furthermore, complex aperiodic responses are observed at build heights above resonance (short features). In such scenarios, bonding cannot be initiated because no or non-uniform interfacial motions occur between the tape and the feature. It is also observed that, once the height of the build feature increases beyond the critical value corresponding to resonance, periodic uniform responses essential for bonding, are recovered. These results corroborates previous experimental findings which demonstrate that bonding can be hard to initiate near or slightly above resonance (at or slightly below a critical height) but can be reinitiated below resonance (above the critical height).