## Bifurcation and chaos in drillstring dynamics

G.H.M. van der Heijden

A model, consisting of a system of ordinary differential equations, is
discussed, describing the lateral vibrations of a stabilised
drillstring as is used in drilling oil wells. It turns out that the
nonlinearity introduced by the bearing clearance (the play between
stabiliser and borehole wall) gives rise to complicated dynamics. For
clearances less than (approximately) the mass eccentricity of the
drillstring synchronous forward whirl dominates asymptotic motion. For
clearances larger than the mass eccentricity, and relatively high rotary
speeds, whirl is no longer possible and one observes (in co-rotating
co-ordinates) several types of periodic solutions, each of them undergoing
a series of period-doubling bifurcations ending up in chaotic motion
described by a strange attractor with unusual dimensional properties.
Coulomb friction between stabiliser and wall causes the forward whirl
to become unstable at certain driving frequencies, resulting in
nonsynchronous self-excited oscillations of large amplitude. There are
several possibilities, then, for a transition from forward to backward
drillstring motion which may induce strongly fluctuating bending
moments.

*Chaos, Solitons & Fractals* **3**, 219-247 (1993)