ANALYSIS OF THE INFLUENCE OF RADIAL DEPTH OF CUT ON THE STABILITY OF THE PARTS: CASE OF PERIPHERAL MILLING

International Journal of Mechanical Engineering and Technology (IJMET) Volume 8, Issue 5, May 2017, pp. 730 743, Article ID: IJMET_08_05_079 Available online at http://www.ia aeme.com/ijmet/issues.asp?jtype=ijmet&vtyp pe=8&itype=5 ISSN Print: 0976-6340 and ISSN Online: 0976-6359 IAEME Publication Scopus Indexed ANALYSIS OF THE INFLUENCE OF THE RADIAL DEPTH OF CUT ON THE STABILITY OF THE PARTS: CASE OF PERIPHERAL MILLING Mohamed SALIH, Mohamed OUBREK, Abdelouahhab SALIH, and Mourad TAHA JANAN MOHAMMED V UNIVERSITY IN RABAT, MOROCCO ENSET, Moroccan laboratory of the innovation and the Industrial Performance LaMIPI, Research team under Instrumentation, Measurement and Tests IME, Rabat, Morocco. ABSTRACT Manufacturers aim to produce mechanical parts at lower costs and high quality. Yet, the productivity and geometric precision of these manufactured parts could be reduced due to vibrations. The main objective of this research paper is to study the influence of the dynamic cutting effects on the stability of the parts during peripheral milling. Faced with thesee vibratory problems in machining, two main approaches have been developed. The first consists in predicting vibrations through increasingly sophisticated modeling. The second is based on experimental work, allowing to understand the behavior of the tool during machining thanks to the instrumentation of the machines. The scope of our work is listed in the first approach. A general model was developed to predict the factors affecting the stability of parts during peripheral milling operation. We have considered the variation of the radial depth of cut to study its effects on the stability of parts. The equations deduced from our developed model allow us to predict the influence of the depth of the cut on the stability of the manufactured parts. Our case study has proved that the deeper the cut goes the more the produced parts vibrate. The developed model can be used to improve the peripheral milling processs in order to reduce the cost of production and to improve the dimensional quality of the surface of the parts obtained. Keywords: Machining dynamics; equation of motion; Lagrange; cutting forces in peripheral milling. Cite this Article: Mohamed SALIH, Mohamed OUBREK, Abdelouahhab SALIH, and Mourad TAHA JANAN Analysis of the Influence of the Radial Depth of Cut on the Stability of The Parts: Case of Peripheral Milling. International Journal of Mechanical Engineering and Technology, 8(5), 2017, pp. 730 743. http://www.iaeme.com/ijmet/issues.asp?jtype=ijmet&vtype=8&i IType=5 http://www.iaeme.com/ijmet/index.asp 730 editor@iaeme.com

Mohamed SALIH, Mohamed OUBREK, Abdelouahhab SALIH, and Mourad TAHA JANAN 1. INTRODUCTION Given the importance of the milling process in mechanical manufacturing, it is necessary to alleviate the machining process in order to reduce its cost and to enhance the dimensional precision and the surface quality of the parts obtained. This improvement requires the use of means, tools, as well as test benches which is very expensive and time consuming. To avoid all of this, it is very useful to predict the vibratory behavior of the Piece-Tool-Machine (PTM) system so as to determine the factors that influence its stability. In this context, several works evoked the prediction of vibrations in milling, but few proposed models that are adapted for our specific case which is the prediction of the vibrations of the part during a peripheral milling. During machining, the PTM system forms a closed loop system where all elements interact with each other. The internal stresses are mechanical stresses (inertial forces caused by the displacements of the table, presence of unbalance at the spindle, etc.) and thermal stresses (heat generated by cutting, heating of moving parts, etc.). The system is also submitted to external stresses due to the environment of the machine (changes in ambient temperature, vibration). All these stresses have an influence on the machining process. The present work constitute a third part of the work cited in[1] and [2], in which we studied the dynamic response of the part during a peripheral milling for different values of feed per tooth (ft) and helix angle. In the case of up-milling, we observed that the increase in the feed per tooth and the decrease in the helix angle lead to an increase in the values of the displacements and the speeds of the supports' displacements. The considered model in [1] and [2] consists in studying the impacts of the dynamic cutting effects on the peripheral milling of the parts with the following assumptions: Flexible part and rigid tool - worktable. The displacement of the part in due production is a displacement of the contact surfaces between the part and the worktable. It occurs in an arbitrary manner along the axes ( ),( ) and ( ). The tool-feed direction is in the positive direction of the ( ). The axis ( ) is in parallel with the tool axis. The negative direction of ( ) is perpendicular on the tool-feed direction which respects the rule of the right hand. The milling cutter is discretized in(n tra ): number of slices along its axis. The objective of our work is to study the nonlinear dynamic behavior of the part in due course of production during a peripheral machining by means of a theoretical study method that is implemented numerically. The first part of this work will be reserved to presenting the model used in [1] and [2]; a prismatic part placed on a standard assembly "3_2_1", whose supports are modeled by a damping spring. In this section, we will present the formulation of the equations of motion governing the behavior of the part when machining in peripheral milling. In the aftermath, we will determine the cutting effort in peripheral milling starting from the theoretical model of [5] and [10] based on the discretization of the tool in several slices of equal thickness along the z axis. This will allow us to predict and simulate cutting forces in peripheral milling. Then, we will present the influence of the radial depth of cut on the cutting forces. Finally, we will evaluate the influence of radial depth of cut on the vibrations of the part in due production. http://www.iaeme.com/ijmet/index.asp 731 editor@iaeme.com

Analysis of the Influence of the Radial Depth of Cut on the Stability of The Parts: Case of Peripheral Milling 2. DYNAMIC OF FITTING IN PERIPHERAL MILLING The complexity of the simulation of a milling operation depends mainly on the geometry of the tool and the effect of the surface generation being machined. Our approach consists in studying the behavior of the part during the peripheral machining and describing the vibration phenomenon of the part during machining. The model described in [1] and [2] is developed for processing the results of the simulation of the dynamic response of the cut which causes the displacement of the part. The complexity of the task is to proximate the real needs as effectively as possible. Indeed, the importance of geometric modeling is matched to the choice of the model, which will allow us to determine all the geometrical parameters necessary for the temporal integration. The developed model considered in this work consists in a peripheral milling of a prismatic part. The tool is a cylindrical milling cutter of diameter Do, length Lo and number of teeth NT, driven by a cutting movement Mc of which N is the rotation frequency around its axis (z). Figure 1 Modeling of peripheral milling in dynamic rate. The position of each support is marked by: = (1); = (2); = (3) = + (4); = + (5); = + (6) To the tool, we link the referencer :,,,, To the part, we link the reference R :,,, ; To the worktable, we link the referencer :,,, ; The center of gravity of the part is marked by: = + (7) http://www.iaeme.com/ijmet/index.asp 732 editor@iaeme.com

Mohamed SALIH, Mohamed OUBREK, Abdelouahhab SALIH, and Mourad TAHA JANAN The point of application of the clamping force S is marked by: = + (8) The aim of the model referred in [1] and [2] is to study the dynamic behavior of the part - worktable, which is modeled by a damping spring mass system applied to the mounting supports, the scalar equations of movement are: M. θ +.. +. +. = M. φ +. +. = M. ψ +. +. = (9) M. X +. ψ +. +. = M. Y +. θ +. +. = M. Z +. θ +. +. = The above equations represent the equations of motion of the part on an isostatic bridle; three rotations (Euler angles) and three translations of direction parallel to the axes x, and y z. M, M and M are respectively the mass of the part and the inertial operators: M = 1 12 M B + C (10) M = 1 12 M A + B (11),,,,,,,,,, et are the dynamic parameters of the part worktable =. (12) =. (13) =. (14) =. (15) =. (16) =. (17),,,,, represent the components of the cutting forces according to each unknown kinematic = ; (18) = ; (19) = + ; (20) =. ; 1 cos ; + (21) = 1 cos ; + ; + sin ; ; (22) http://www.iaeme.com/ijmet/index.asp 733 editor@iaeme.com

Analysis of the Influence of the Radial Depth of Cut on the Stability of The Parts: Case of Peripheral Milling = 1 cos ; + ; + sin ; ; (23) The obtained equations are the equations of motion of the part under construction described as a function of the components of the cutting forces. These forces will be determined by a discretization of the tool into several elementary sections along its main axis [5] and [10]. 3. FORMALIZATION OF THE CUTTING FORCES MODEL The theoretical model for determining the cutting forces presented in this work is based on the theory of predictive machining [10]. The tool is discretized in N tra elementary slices of constant thickness dz, perpendicular to its principal axis, and animated by a cutting motion Mc that is the rotation frequency around its ax (Z) [Figure 1]. The force applied on a cutting edge is obtained by the summation of the components of the forces occurring on each slice. A summation on all the edges involved in the material makes it possible to obtain the overall effort on the tool at a given instant. Figure 2: Modeling of cutting forces. The angular position of a tooth in contact with the part to be produced is determined as a function of the axial depth of cut A and radial depth A, the number of teeth NT, the diameter of the tool D and the angle of Helix β. The instantaneous thickness of the chip at a certain point on the cutting edge can be approximated as follows (Eq.24): h, = f sin, (24) Wheref is the feed per tooth and ; is the angular position with respect to the axis of the j th tooth at the elementary disc k. Its value changes along the axial direction as follows (Eq.25):, () = 2 ( 1) ( 1) (25) http://www.iaeme.com/ijmet/index.asp 734 editor@iaeme.com

Mohamed SALIH, Mohamed OUBREK, Abdelouahhab SALIH, and Mourad TAHA JANAN N is the rotational speed of the tool (rpm), t is the time (min), D is the diameter of the milling cutter. The cutting actions can be modeled as oblique cut, using the predictive machining theory developed by Oxley [10]. The cutting forces can be calculated from the data of the properties of the machined material, the geometry of the tool and the cutting parameters. The basis of this theory is the analysis of the stress distribution along the shear plane and the tool-chip interface in terms of the shear plane angle s and the properties of the machined material. Figure 3 Cutting forces model for oblique cutting [10]. According to Oxley [10], for an oblique cutting process, the actions of the cutting forces in the cutting, radial feed: (O1), (O2) and (O3) respectively, as illustrated in [Figure 3], are given by the equations 26 to 28. O = F (26) O = F cos(β ) + F sin(β ) (27) O = F sin(β ) F cos(β ) (28) If we consider β = 0; then F C, F T and F R, will be components of the cutting force in radial cutting feed, the components can therefore be determined respectively using the equations 29 to 31. F = R cos(λ α ) (29) F = R sin(λ α ) (30) F = (()() ( ) ( )) ( ) ( ) (31) () ( ) ( )() With λ representing the average friction angle at the tool-chip interface, α is the cutting angle, δ the angle of inclination of the tool and η is the angle of flow of the chip. These components of the oblique cutting force are calculated from the resultant R of the forces in the shear plane and at the tool-chip interface. This resultant R is determined by equation (Eq.32) as follows: R = ( ) ( ) The factor k representing the shear stress in the primary shear zone, h the undeformed cutting thickness, θ the angle between the resultant of the cutting forces R and the shear plane and w is the cutting width which is determined in the following equation (Eq.33): w = ( ) Since the cutting action of each segment is considered as an oblique cut with a cutting angle equal to the helix angle, the resulting force is given by the equation (Eq.34): (32) (33) http://www.iaeme.com/ijmet/index.asp 735 editor@iaeme.com

Analysis of the Influence of the Radial Depth of Cut on the Stability of The Parts: Case of Peripheral Milling R =,, ( ) ( ).( ) Following [13], the thickness of the chip varies according to time or position of the tool in the piece, and in static mode the thickness of the chip is presented in the following equation (Eq.35): h, = f sin, if <, < (35) h, = 0 if not With representing the angle of entry into the material and the exit angle. These angles are given for the opposing milling by equation (Eq.36), and for up milling by (Eq.37), where D represents the diameter of the mill and Ar as the radial depth of cut: = 0 = arccos((d 2A )/D) (36) = π arccos((d 2A )/D) (37) = π To predict the cutting forces using a helical milling cutter and to express the geometry of the milling cutter based on the works [5], the prediction of the cutting forces is determined from the equations (Eq.38) for the up-milling and equations (Eq.39) to down-milling: N tra N T F x t = (O j,k cos( j,k ) + O j,k sin( j,k )) k j N F y t = tra N T ( O j,k sin( j,k ) + O j,k cos( j,k )) (38) k j N F z t = tra N T O k j j,k (34) N tra N T F x t = (O j,k cos( j,k ) + O j,k sin( j,k )) k j N F y t = tra N T (O j,k sin( j,k ) O j,k cos( j,k )) (39) k j N F z t = tra N T O k j j,k 4. RESULTS OF NUMERICAL SIMULATION OF CUTTING FORCES In this part, the simulation results for the case of peripheral up-milling in the case where the milling cutter is in the middle of the part under construction will be presented. The cutting forces Fx, Fy and Fz are simulated for 4 radial depth of cut values and with the following parameters (Table1): Table 1 Simulation data N [trs/s] D[mm] NT [tooth] ft[mm/tooth] Aa [mm] Ar[mm] β h K AB [N/m2] 2 10 1 0,1 50 10; 12; 14 and 16 0 460 x 106 http://www.iaeme.com/ijmet/index.asp 736 editor@iaeme.com

Mohamed SALIH, Mohamed OUBREK, Abdelouahhab SALIH, and Mourad TAHA JANAN 4.1. Results Figure 4 Cutting force F x for up-milling. Figure 5 Cutting force F y for up-milling. Figure 6 Cutting force F z for up-milling. Figures [4, 5 and 6] show the influence of the radial depth on the cutting forces applied to the part during machining. This simulation shows that the increase in the radial depth of cut implies the increase of the values of the cutting forces, thus increasing the impact of the cutter on the material of the part in due production. http://www.iaeme.com/ijmet/index.asp 737 editor@iaeme.com

Analysis of the Influence of the Radial Depth of Cut on the Stability of The Parts: Case of Peripheral Milling 5. DYNAMIC SIMULATION RESULTS FOR PERIPHERAL MILLING In order to optimize the machining process, the cutting parameters should be meticulously selected. To observe the influence of these parameters, the results obtained by varying the values of the radial depth of cut shall be compared. All simulations were carried out with a frequency N = 21 [trs / s], a number of teeth of the tool NT = 1 tooth, an axial depth Aa = 50 mm and a helix angle β = 0, The only variable parameter is the radial depth of cut, which is chosen from [10; 12; 14; 16]. These results are obtained for a material 42CD4, different from that of the results carried out in the works [1] and [2]. 5.1 Results: Influence of Radial Depth of Cut These simulation time curves represent the results of displacements and displacement speeds of the supports P 6, P 1 and P 4 respectively on the axes, Figure 7 Move supports P6. http://www.iaeme.com/ijmet/index.asp 738 editor@iaeme.com

Mohamed SALIH, Mohamed OUBREK, Abdelouahhab SALIH, and Mourad TAHA JANAN Figure 8 Travel speed supports P6. Figure 9 Move supports P1. http://www.iaeme.com/ijmet/index.asp 739 editor@iaeme.com

Analysis of the Influence of the Radial Depth of Cut on the Stability of The Parts: Case of Peripheral Milling Figure 10: Travel speed supports P1. Figure 11 Move supports P4. http://www.iaeme.com/ijmet/index.asp 740 editor@iaeme.com

Mohamed SALIH, Mohamed OUBREK, Abdelouahhab SALIH, and Mourad TAHA JANAN Figure 12 Travel speed supports P4. 6. CONCLUSIONS In order to improve the productivity and increase the geometrical precision of the manufactured parts, a model of prediction of the influence of the depth of the pass on the positioning of the parts in peripheral milling has been developed. In this work, we investigated the influence of the radial depth of cut on the stability of the parts. We first saw the model of the dynamics of fitting the parts on milling developed in [1] and [2] then we moved to the determination of the equations of motion that govern the PTM system. This model has imposed the prediction of the cutting forces [5] and [10] which is based on the discretization of the tool into several elementary slices of the milling cutter along its main axis. To express the geometry of the milling cutter, each slice is considered oblique cutting actions of a unique number of tools equal to the number of milling cutter teeth. In this model it is assumed that the cutting forces applied to the milling tool are the sum of the cutting forces applied to all the segments of teeth for each slice and by the summation of the cutting forces for all the slices; The milling cutter forces are obtained for each given position of the milling cutter. Then we simulated these efforts for different depth of cut values, these results showed that the increase of the depth of cut values increases the values of the cutting forces. Finally, we predicted the dynamic behavior of the part in due production upon a standard molding "3-2-1" during the milling operation, for different depth of cut values. These predictions are represented in the form of time curves of simulation, which demonstrate the results of the displacements and speeds of displacement of the supports. Thanks to these results, we deduce that the increase of the depth of cut values leads to the increase of displacements and the displacement speeds of the supports much more in the direction xwhich is opposite to the part feed speed. http://www.iaeme.com/ijmet/index.asp 741 editor@iaeme.com

Analysis of the Influence of the Radial Depth of Cut on the Stability of The Parts: Case of Peripheral Milling List of symbols: : Axial depth of cut. : Radial depth of cut D : Diameter of the milling cutter : Helix angle,,,,,,,,,, and Dynamic parameters of part worktable θ,φ et ψ : Euler angle, and : Cutting force along axes (x), (y) and (z) respectively ;, ; and ; : Components of the cutting force of the tooth segment of the jth tooth and the Kth slice linked to the fixed mark of the milling cutter : Main cutting force R: Resultant of cutting forces, and : Force in the direction of cutting, radial advance of the oblique cut,,,,, : Components of the cutting forces according to each kinematic unknown : Advance per tooth : Cutting movement : Cutting speed N: Rotation speed of the tool w: Cutting width : Thickness of a slice of the milling cutter. : Shear stress in the primary shear zone NT: Number of teeth h :Undeformed cutting thickness h, : Instant chip thickness : Entry angle in the material : Output angle of the material, : Angular position of the segment of the jth tooth and of the kth slice of the milling cutter α : Cutting angle : Angle of tilt of tool η : Chip flow angle : Angle between the resultant of the cutting forces R and the shear plane http://www.iaeme.com/ijmet/index.asp 742 editor@iaeme.com

Mohamed SALIH, Mohamed OUBREK, Abdelouahhab SALIH, and Mourad TAHA JANAN REFERENCES [1] Mohamed SALIH and Abdelouahhab SALIH, Taking Into Account of The Dynamic Effects of The Cut on The Laying of The Parts In Peripheral Milling. International Journal of Mechanical Engineering and Technology, 7(4), 2016, pp. 114 124. [2] Mohamed SALIH, Mohamed OUBREK and Abdelouahhab SALIH, Influence of Helix Angle on the Stability of Parts During Milling Peripheral. International Journal of Mechanical Engineering and Technology, 7(5), 2016, pp. 65 74. [3] A. Molinari, X. Soldani, and M. H. Miguélez, Adiabatic shear banding and scaling laws in chip formation with application to cutting of Ti 6Al 4V, Journal of the Mechanics and Physics of Solids, vol. 61, no. 11, pp. 2331 2359, Nov. 2013. [4] G. Campatelli, A. Scippa, Prediction of milling Cutting Force Coefficients for Aluminum 6082 T4. 5 th CIRP Conference on High Performance Cutting 2012; 563 568 [5] H.Z. Li, W.B. Zhang, X.P. Li. Modelling of cutting forces in helical end milling using a predictive machining theory. Department of Mechanical and Production Engineering, The National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapor, 2000. [6] J. Pei, Research on Physical Simulation in Cutting Process and Technical Parameters Optimization, Northeastern University, Shenyang, China, 2011, (Chinese). [7] L. Zhu, C. Zhu, J. Pei, X Li, and W. Wang. Prediction of Three-Dimensional Milling Forces Based on Finite Element, Hindawi Publishing Corporation Advances in Materials Science and Engineering Volume 2014, Article ID 918906. [8] Liu, X, et K Chen. Modelling the machining dynamics of peripheral milling. International Journal of Machine Tools & Manufacture; pp 1301-1320; vol 45, 2005 [9] Li XP, Nee AYC, Wong YS, Zheng HQ. Theoretical modelling and simulation of milling forces. Journal of materials Processing Technology 1999;89 90:266 72. [10] Oxley PLB. «Mechanics of Machining. Chichester: Ellie Horwood Limited, 1989. [11] Z. Fu, W. Yang, X. Wang, and J. Leopold, Analytical Modelling of Milling Forces for Helical End Milling Based on a Predictive Machining Theory, Procedia CIRP, vol. 31, pp. 258 263, 2015. [12] Tlusty, J, et F Ismail. Basic non linearity in machining chatter. Annals of the CIRP 24 (1), pp 463-466, 1981 [13] Zheng HQ, Li XP, Wong YS, Nee AYC. Theoretical modelling and simulation of cutting forces in face milling with cutter runout. International Journal of Machine Tools Manufacture 1999;39:2003 18. [14] Zhao, M.X. et B Balachandran. Dynamics and stability of milling process. International Journal of Solids and Structures, pp 2233-2248, vol 38, 2001. http://www.iaeme.com/ijmet/index.asp 743 editor@iaeme.com