Delayed feedback control and parameter continuation of multistability in a nonsmooth hydraulic rock drill model.

Abstract

In response to the complex multistable behavior observed in hydraulic rock drills during the drilling process, this study first establishes a four-degree-of-freedom physical model based on dry friction rock mechanics theory. The motion trajectory is classified into three states: non-viscous, impact viscous, and buffer viscous. Using the impact frequency ω as the bifurcation parameter, multistable attractors p0q1 and p1q2 are identified in the system when ω = 9. To control the multistability, a delayed feedback control method is applied, in which the infinite-dimensional delay differential equations are approximated by finite-dimensional ordinary differential equations. The reliability of this approximation is validated through a distance function. When the control gain K = 9 and the delay time τd = 0.35, both attractors p0q1 and p1q2 are successfully converted into a single p0q1 attractor. Next, the pseudo-arclength continuation method and Floquet theory are employed to investigate parameter continuation and parameter domains. The period-doubling bifurcation points PD1 and PD2 divide the parameter space of K and τd into three distinct regions. Crossing these regions induces a supercritical period-doubling bifurcation. For constant K, a smaller τd leads to an increased number of collisions and periodic motions in the system. Simulation results demonstrate that by tuning the delay parameters, the multistability during the drilling process can be effectively controlled, thereby enhancing drilling efficiency and stability. Finally, rock drilling experiments confirm the validity of the model and the presence of multistability. When drilling into rocks with high hardness and brittleness, multistable motions are more likely to occur.