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Effect of rotary speed modulation on the stability of rotary drilling
Last modified: 2017-12-29
Abstract
In the current work, we investigate the dynamics of a lumped parameter model corresponding to axial and torsional modes of
rotary drilling with periodic rotary speed modulation. It had been observed that for the case of turning, spindle-speed modulation
increases the regime of operating parameters corresponding to stable cutting but rotary speed modulation has not been explored
for deep-hole drilling applications yet. One possible reason for this could be the complexity in modeling the cutting process as a
state-dependent delayed differential equation for rotary drilling wherein the equation governing the delay involves the rotational
speed. To get over this complexity, we have used a functional description of the cut profile to model the regenerative effect
instead of the traditional state-dependent delay model, which allows easier and straightforward incorporation of rotary speed
modulation. It has to be noted that due to the rotary speed modulation, there are number of possible solutions for steady-drilling.
In the current work, we are considering the case wherein the steady depth of cut remains the same even with the periodically varying
steady twist of the drill-string. The evolution of the cut surface is governed by a nonlinear partial differential equation with time
periodic coefficients coupled with ordinary differential equations for axial and torsional motions of the drill-string. The governing
equations are non-dimensionalized (to reduce the number of effective parameters) after shifting the origin to the steady state.
We first linearize the coupled PDE-ODE model and then obtain a finite-dimensional approximation to study the stability of this
steady state. Floquet theory has been employed for this purpose and stability charts have been obtained for different modulation
amplitude and frequency. It has been observed that for a given value of modulation amplitude, the stable regime increases as the
modulation frequency approaches the nondimensional natural frequency corresponding to the torsional mode.
rotary drilling with periodic rotary speed modulation. It had been observed that for the case of turning, spindle-speed modulation
increases the regime of operating parameters corresponding to stable cutting but rotary speed modulation has not been explored
for deep-hole drilling applications yet. One possible reason for this could be the complexity in modeling the cutting process as a
state-dependent delayed differential equation for rotary drilling wherein the equation governing the delay involves the rotational
speed. To get over this complexity, we have used a functional description of the cut profile to model the regenerative effect
instead of the traditional state-dependent delay model, which allows easier and straightforward incorporation of rotary speed
modulation. It has to be noted that due to the rotary speed modulation, there are number of possible solutions for steady-drilling.
In the current work, we are considering the case wherein the steady depth of cut remains the same even with the periodically varying
steady twist of the drill-string. The evolution of the cut surface is governed by a nonlinear partial differential equation with time
periodic coefficients coupled with ordinary differential equations for axial and torsional motions of the drill-string. The governing
equations are non-dimensionalized (to reduce the number of effective parameters) after shifting the origin to the steady state.
We first linearize the coupled PDE-ODE model and then obtain a finite-dimensional approximation to study the stability of this
steady state. Floquet theory has been employed for this purpose and stability charts have been obtained for different modulation
amplitude and frequency. It has been observed that for a given value of modulation amplitude, the stable regime increases as the
modulation frequency approaches the nondimensional natural frequency corresponding to the torsional mode.