State-dependent distributed-delay model of orthogonal cutting

In this paper we present a model of turning operations with state-dependent distributed time delay. We apply the theory of regenerative machine tool chat- ter and describe the dynamics of the tool-workpiece sys- tem during cutting by delay-diferential equations. We model the cutting-force as the resultant of a force sys- tem distributed along the rake face of the tool, which results in a short distributed delay in the governing equation superimposed on the large regenerative de- lay. According to the literature on stress distribution along the rake face, the length of the chip-tool inter- face, where the distributed cutting-force system is act- ing, is function of the chip thickness, which depends on the vibrations of the tool-workpiece system due to the regenerative efect. Therefore, the additional short de- lay is state-dependent. It is shown that involving state- dependent delay in the model does not afect linear sta- bility properties, but does afect the nonlinear dynamics of the cutting process. Namely, the sense of the Hopf bi- furcation along the stability boundaries may turn from sub- to supercritical at certain spindle speed regions

Nonlinear Drillstring Dynamics Analysis

This paper studies the dynamical response of a rotary drilling system with a drag bit, using a lumped parameter model that takes into consideration the axial and torsional vibration modes of the bit. These vibrations are coupled through a bit-rock interaction law. At the bit-rock interface, the cutting process introduces a state-dependent delay, while the frictional process is responsible for discontinuous right-hand sides in the equations governing the motion of the bit. This complex system is characterized by a fast axial dynamics compared to the slow torsional dynamics. A dimensionless formulation exhibits a large parameter in the axial equation, enabling a two-time-scales analysis that uses a combination of averaging methods and a singular perturbation approach. An approximate model of the decoupled axial dynamics permits us to derive a pseudoanalytical 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 work. 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

Axial Stick-slip Limit Cycling in Drill-string Dynamics with Delay

In a novel approach to model stick-slip vibrations occurring when drilling with drag bits, the axial and torsional dynamics are coupled through the boundary conditions via a state-dependent delay. Moreover, friction is modelled by a rate-independent discontinuous term. A regime characterized by a low amplitude of the torsional vibrations and a high drilling efficiency is numerically observed for some sets of parameters. In this regime, the axial fast vibrations have a stabilizing effect on the torsional equilibrium. To understand this stabilizing mechanism, we are studying the decoupled axial equation obtained by freezing the delay. This approximation reflects the small variations of the delay when the bit experiences small torsional vibrations. Axial periodic solutions may be analysed independently. Particularities of this equation lie in the presence of a delayed term and a non-smooth non linearity. In this paper, we apply different well-known methods to study the periodic orbits of the axial dynamics. The results and limitations of semi-analytical (Describing Functions Method) and numerical procedures (Finite Difference Method, ShootingMethod) are exposed here. We use these numerical techniques to investigate some particular properties of the system, such as the dependency of period time with the delay

Laparoscopic transhiatal esophago-gastrectomy after corrosive injury

SCOPUS: ar.jinfo:eu-repo/semantics/publishe