ISSN: 2277-9655 [Mechkouri* et al., 6(10): October, 2017] Impact Factor: 4.116 IC™ Value: 3.00 CODEN: IJESS7 http: // www.ijesrt.com © International Journal of Engineering Sciences & Research Technology [329] IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY PREDICTION OF MULTI-DIMENSIONAL MILLING BEHAVIOR M. Hayani Mechkouri *1,2 , A. Chaaba *1 , G. Moraru *2 , P. Veron *2 *1 Ecole Nationale Supérieure d'Arts et Métiers, Marjane II, Beni M'Hamed B.P. 4024, Meknès, Morocco *2 LSIS-UMR CNRS 6168, CER ENSAM, 2 cours des Arts et Métiers, 13617 Aix-en-Provence cedex, France DOI: 10.5281/zenodo.1012535 ABSTRACT Milling process is a multi-dimensional cutting process accompanied by chatter vibrations because of material removal discontinuity. Also, chatter phenomenon has a great impact on final milled products quality. In this paper, an adequate criterion derived from mathematical literature is used to predict stability of multi-dimensional chatter milling. For that, firstly, the chatter milling system is represented by a system of three Degrees Of Freedom (DOF) and its dynamics is modeled by a system of Retarded Differential Equations (RDEs). Afterward, stability in Lyapunov sense is computed on the basis of the quasi-polynomial characteristic function in frequency domain. Finally, this method is used to predict stability of chatter milling process under different machining conditions and the output related to each case is verified by resolving the system of RDEs and visualizing the displacements in time domain using Matlab software. KEYWORDS: multi-dimensional, milling process, behavior, Chatter Vibrations, RDEs, Matlab software. I. INTRODUCTION Milling is a manufacturing process where material is removed by chip which is affected by vibrations. Chatter is one of the three vibration types [1], [2] and consists on remanufacturing of the manufactured surface. Firstly, the importance of chatter in vibrations behavior has been demonstrated experimentally by Nicolson [3] then Taylor warned industrials about its lower effect on machined piece quality in 1907 [4]. Effectively, based on mechanistic method, Tlusty proved the direct impact of chatter on chip thickness [5]. Afterwards, he developed a basic non- linear system of cutting force in machining chatter [6] which underlies the strong relationship between cutting force and chip thickness [7]. Whereas many works have been developed to avoid chatter [8] or to suppress it [9]- [10] by varying tool spindle speed, for example, others have been conducted to predict it in time [11] and frequency [12] domains. Since computing stability via simulation milling process in time domain is time consuming, researchers are putting greater focus on predicting stability in frequency domain. The most popular criterion used to achieve this target is the tau-decomposition criterion which stability lobes diagram is based on [13], [14]. Stability lobes diagram is still used to predict chatter in manufacturing by drilling [15], turning [16] and milling [17], [18] systems thanks to its ability to give general overview idea about process stability in constant machining conditions by considering only one degree of freedom at a time. Nevertheless, general milling is a multi-dimensional cutting process. Hence the importance of substituting the Tau-decomposition by a suitable criterion. Therefore, this works aims to analyze stability in the Lyapunov sense [19] of nonlinear differential equations describing chatter dynamical system in multi-dimensional milling process. The second section aims to demonstrate the crossing from milling kinematics to cutting forces. So, milling tool is discretized into elemental cutting edges where differential cutting forces are modeled, computed and summed. The third consists on representing a multi-dimensional chatter milling process by a system of three retarded differential equations (RDEs). After that, we prove that the characteristic function associated is a quasi-polynomial whose degree is superior to one. Then, looking back on theorems defining stability of RDEs on the basis of the characteristic quasi- polynomial root finder, some criteria are defined to determine the stability type (stable, asymptotically stable, unstable) of milling process. Hence, milling system behavior is determined in frequency domain. Afterwards, a case study of chatter milling process is borrowed from bibliography to apply the developed method of behavior prediction and Matlab software is used in computing phase. Finally, the outputs of this method are compared to milling simulation outcomes computed in time domain.