Stability of a two degrees of freedom model of the turning process is considered. An accurate modeling of the surface regeneration shows that the regenerative delay, determined by the combination of the workpiece rotation and the tool vibrations, is in fact statedependent. For that reason, the mathematical model considered in this paper is a delay-differential equation with state-dependent time delay. In order to study linearized stability of stationary cutting processes, an associated linear system, corresponding to the statedependent delay equation, is derived. Stability analysis of this linear system is performed analytically.A comparison between the state-dependent delay model and the previously used constant or timeperiodic delay models shows that the incorporation of the state-dependent delay into the model slightly affects the linear stability properties of the system in certain parameter domains.

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Traditional models of regenerative machine tool chatter use constant time delays assuming that the period between two subsequent cuts is a constant determined definitely by the spindle speed. These models result in delay-differential equations with constant time delay. If the vibrations of the tool relative to the workpiece are also included in the surface regeneration model, then the resulted time delay is not constant, but it depends on the actual and a delayed position of the tool. In this case, the governing equation is a delay-differential equation with state dependent time delay. Equations with state dependent delays can not be linearized in the traditional sense, but there exists linear equations that can be associated to them. This way, the local behavior of the system with state dependent delays can be investigated. In this study, a two degree of freedom model is presented for milling process. A thorough modeling of the regeneration effect results in the governing delay-differential equation with state dependent time delay. It is shown through the linearization of the nonlinear equation that an additional term arises in the linearized equation of motion due to the state-dependency of the time delay.

We consider a class of linear delay equations with perturbed time lags and present conditions which guarantee that the asymptotic stability of the trivial solution of the equation at hand is preserved under these perturbations. As an example we show how our results can be used to obtain an estimate on the maximum allowable sampling interval in the stabilization of a hybrid system with feedback delays. We also present applications of our perturbation theorem to obtain stability conditions for delay equations with multiple delays. ᮊ 1998 Academic Press

International audienc

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