<p>A commonly used hydraulic damper is simplified and equivalently modeled as a single-degree-of-freedom vibro-impact system coupled with multiple non-smooth factors. Typical periodic motion patterns within a complete excitation cycle are identified, and the global distribution of periodic motions is obtained through global dynamic analysis. Particular attention is given to the occurrence conditions of grazing motions and their influence on periodic motion transitions within the sticking region. By integrating cell mapping, Lyapunov exponent spectrum, and parameter continuation algorithms, the transition processes of basic periodic motions and multi-periodic coexistence phenomena are systematically studied. The results show that within certain parameter ranges, the system exhibits sticking motion characteristics, with periodic motions displaying regular distribution patterns. Transitions between periodic motions are mainly induced by grazing bifurcations. The period-adding transition process presents two types of self-similar sequences with different boundary structures. Due to differences in grazing–saddle-node bifurcation positions, two distinct forms of multi-periodic coexistence regions are observed. These findings provide theoretical support for the design and parameter optimization of hydraulic dampers.</p>