The present paper studies the effect of an axial elastic tool (known as a shock sub), mounted downhole in the drill-string, on the occurrence of axial and torsional self-excited vibrations. In particular, we evaluate the feasibility of stabilizing the axial dynamics, dominated by a bilateral (feedback) coupling between the bit-rock interaction and the drill-string wave-equations, through the insertion of a passive down-hole tool. We consider the problem of unwanted drill-string vibrations and explain how these vibrations relate to the so-called axial instability using a distributed parameter (infinite dimensional) model. The equations describing the feedback system causing this instability are derived and then extended to accommodate for the inclusion of the effect of the shock sub. Conditions for the design parameters of the shock sub needed to avoid axial instability are then derived and their practical feasibility are considered.
- Distributed parameter systems
- Drill-string vibrations
- Hyperbolic systems
- Infinite dimensional systems