| Summary: | The research reported in this thesis builds on a novel model developed at
the University of Minnesota to analyze the self-excited vibrations that
occur when drilling with polycrystalline diamond cutter bits. The lumped
parameter model of the drilling system takes into consideration the axial
and the torsional vibrations of the bit. These vibrations are coupled
through a bit-rock interaction law. At the bit-rock interface, the cutting
process combined with the quasihelical motion of the bit leads to a
regenerative effect that introduces a coupling between the axial and
torsional modes of vibrations and a state-dependent delay in the governing
equations, while the frictional contact process is associated with
discontinuities in the boundary conditions when the bit sticks in its axial
and angular motion. The response of this complex system is characterized by
a fast axial dynamics superposed to the slow torsional dynamics.
A two time scales analysis that uses a combination of averaging methods and
a singular perturbation approach is proposed to study the dynamical response
of the system. An approximate model of the decoupled axial dynamics permits
to derive a pseudo analytical expression of the solution of the axial
equation. Its averaged behavior influences the slow torsional dynamics by
generating an apparent velocity weakening friction law that has been
proposed empirically in earlier works. The analytical expression of the
solution of the axial dynamics is used to derive an approximate analytical
expression of the velocity weakening friction law related to the physical
parameters of the system. This expression can be used to provide
recommendations on the operating parameters and the drillstring or the bit
design in order to reduce the amplitude of the torsional vibrations.
Moreover, it is an appropriate candidate model to replace empirical friction
laws encountered in torsional models used for control.
In this thesis, we also analyze the axial and torsional vibrations by basing
the model on a continuum representation of the drillstring rather than on
the low dimensional lumped parameter model. The dynamic response of the
drilling structure is computed using the finite element method. While the
general tendencies of the system response predicted by the discrete model
are confirmed by this computational model (for example that the occurrence
of stick-slip vibrations as well as the risk of bit bouncing are enhanced
with an increase of the weight-on-bit or a decrease of the rotational
speed), new features in the self-excited response of the drillstring are
detected. In particular, stick-slip vibrations are predicted to occur at
natural frequencies of the drillstring different from the fundamental one
(as sometimes observed in field operations), depending on the operating
parameters.
Finally, we describe the experimental strategy chosen for the validation of
the model and discuss results of tests conducted with DIVA, an analog
experimental set-up of the lumped
parameter model. Some results of the experiments conducted in an artificial
rock seem to validate the model studied here although the same experiments
obtained with natural rocks
were unsuccessful. Different problems with the design of the experimental
setup were identified. By using the outcome of the analysis of the uncoupled
dynamics, we could provide critical recommendation to elaborate and to
design a simpler and stiffer analog experiment (TAZ) used to study the self
excitation of the axial dynamics that ultimately lead to the excitation of
the torsional dynamics.
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