Nonlinear dynamics and continuation analysis of a four degree-of-freedom drifter-rock model

A rock contact model is introduced, and the rock drilling process of the hydraulic drifter is established as a four degree-of-freedom (DOF) mechanical model. The mechanical model is simplified using a nondimensionalization method, resulting in a compact form. The periodic trajectories of the mechanical model are segmented to establish a mathematical model. Non-stick period-1 trajectories are obtained. The angular frequency and vertical offset are used as control parameters for bifurcation and basins of attraction. One-parameter continuation and two-parameter domain are conducted. Results indicate that: When $0<\omega \le 2.34$, the model exhibits stick behavior. For $2.35\le \omega \le 20$, the model transitions to the non-stick mode. The fingered chaotic attractor emerges from stable periodic trajectories via period-doubling bifurcations. Period-doubling, saddle–node, and torus bifurcations are identified. To ensure operation on a period-1 trajectory, the angular frequency should be chosen within the range of $2.35<\omega <6.611$, and the vertical offset should be within $0.0467<b<0.2125$. The model exhibits multi-stability for $0.046<b<0.059$, where the $p_2{q}_3$ trajectory demonstrates stronger global stability compared to $p_2{q}_1$ and quasi-period trajectories. The hysteresis caused by the torus and saddle–node bifurcations is found in the continuation analysis. According to the two-parameter domain analysis, in the range $0<\omega \le 20$ and $0\le b\le 0.215$, the top three trajectory types in the proportion of the model are $p_0{q}_1$, $p_1{q}_1$, and $p_1{q}_2$. These results provide critical guidance for optimizing drilling parameters to avoid undesirable bifurcations and maintain stable periodic operation.