Mechanics and Mechanical Engineering Vol. 15, No. 4 (2011) 131–145 c ⃝ Technical University of Lodz Regenerative Model of Cutting Process with Nonlinear Duﬃng Oscillator Rafal Rusinek Andrzej Weremczuk Jerzy Warmi´ nski Lublin University of Technology Department of Applied Mechanics Nadbystrzycka 36, 20–618 Lublin, Poland r.rusinek@pollub.pl a.weremczuk@pollub.pl j.warminski@pollub.pl Received (16 December 2011) Revised (16 January 2012) Accepted (4 February 2012) The paper presents analytical and numerical study of externally forced Duﬃng oscillator with delayed displacement feedback. By using analytical method, stability lobes for a linear oscillator with time delay is determined and the fundamental resonance of the Duﬃng oscillator with time delay is calculated by means of the multiple scale method. Next, an inﬂuence of delayed displacement feedback on the classical Duﬃng oscillator is examined and an eﬀect of cubic nonlinearity on the stability of the regenerative model of cutting process is analysed. Keywords : Nonlinear vibrations, Duﬃng oscillator, time delay, regenerative chatter 1. Introduction Recent years have provided numerous studies on time delay systems and their appli- cations in various ﬁelds. The delayed state feedback is often used to active control of mechanical vibrations, specially of nonlinear systems [1]. As shown in [2] time delay eﬀect can stabilize the unstable periodic motions in linear oscillators and also can control local bifurcations of nonlinear systems through improving the sta- bility of periodic motion. Therefore, the time delay systems are more and more popular not only to control processes but to model various physical phenomena as well. The delayed state feedback is commonly being made use to model ma- chining processes with, so called, regenerative eﬀect that is very harmful because can produced self–excited chatter vibrations [3–10]. Vibrations become one of the most common limitations for productivity and quality of ﬁnal surface in milling,