State-dependent distributed-delay model of orthogonal cutting

Abstract

In this paper we present a model of turning operations with state-dependent distributed time delay. We apply the theory of regenerative machine tool chatter and describe the dynamics of the tool-workpiece system during cutting by delay-differential equations. We model the cutting-force as the resultant of a force system distributed along the rake face of the tool, which results in a short distributed delay in the governing equation superimposed on the large regenerative delay. According to the literature on stress distribution along the rake face, the length of the chip-tool interface, where the distributed cutting-force system is acting, is function of the chip thickness, which depends on the vibrations of the tool-workpiece system due to the regenerative effect. Therefore, the additional short delay is state-dependent. It is shown that involving statedependent delay in the model does not affect linear stability properties, but does affect the nonlinear dynamics of the cutting process. Namely, the sense of the Hopf bifurcation along the stability boundaries may turn from sub-to supercritical at certain spindle speed regions.

Bifurcation and chaotic vibration of frictional chatter in turning process

“…[2][3][4] Regenerative chatter occurs because of the unstable cutting force that caused by the variation in the effective chip thickness or the rate of penetration of the tool. [5][6][7][8] Chatter is considered to be an important factor that can affect cutting stability and machining productivity by Taylor. 9 Then, a fundamental theory of cutting chatter was first proposed by Tobias 10 and Koenigsberger and Tlusty 11 who provide the initial basic framework within which the cutting chatter problem is usually formulated.…”

On the segmentation-driven vibration in milling titanium alloy

“…4 According to the stress distribution along the rake face, a state-dependent distributed-delay model of orthogonal cutting was proposed to perform the bifurcation analysis. 5 A method was proposed to identify instantaneous dynamic characteristics during thin-walled part milling and analyze peripheral milling stability. 6 All these researches show that chatter is regarded as a feedback mechanism for growth of self-excited vibrations due to variations in chip thickness and cutting force and subsequent tool vibration.…”

Suppression of Hopf bifurcation in metal cutting by extrusion machining

“…Elias and Namboothiri [24] put forward a cross-recurrence plot methodology to recognize the transition from regular cutting to the chatter cutting. Molnár et al [25] presented a statedependent distributed-delay model of orthogonal cutting to describe the dynamics of the tool-workpiece system. Otto and Radons [5] analyzed the influence of tangential and torsional vibrations on the stability lobes in metal cutting.…”