Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 54, 3, pp. 691-703, Warsaw 2016 DOI: 10.15632/jtam-pl.54.3.691 CUTTING PARAMETERS AND VIBRATIONS ANALYSIS OF MAGNETIC BEARING SPINDLE IN MILLING PROCESS Amel Bouaziz, Maher Barkallah, Slim Bouaziz Laboratory of Mechanical Modeling and Production, National School of Engineers of Sfax, University of Sfax, Tunisia e-mail: slim.bouaziz1@gmail.com Jean Yves Choley Laboratory of Engineering in Mechanical Systems and Materials, SUPMECA, Saint-Ouen, France Mohamed Haddar Laboratory of Mechanical Modeling and Production, National School of Engineers of Sfax, University of Sfax, Tunisia Inmodernproduction,milling is considered thewidespreadcutting process in the formatting field. It remains important to study thismanufacturing process as it can be subject to some parasitic phenomena that can degrade surface roughness of themachined part, increase tool wear and reduce spindle life span. In fact, the best quality work piece is obtained with a suitable choice of parameters and cutting conditions. In another hand, the study of tool vibrations and the cutting force attitude is related to the study of bearings as they present an essential part in the spindle system. In this work, a modeling of a High Speed Milling (HSM) spindle supported by two pair of ActiveMagnetic Bearings (AMB) is presented.The spindle is modeled by Timoshenko beam finite elements where six degrees of freedom are taken into account.The rigid displacements are also introduced in themodeling.Gyroscopic and centrifugal terms are included in the general equation. The bearings reaction forces are modeled as linear functions of journal displacement and velocity in the bearing clearance. A cutting force model for peripheral milling is proposed to estimate the tool-tip dynamic responses as well as dynamic cutting forces which are also numerically investigated. The timehistoryof response, orbit, FFTdiagramat the tool-tip center and the bearingsdynamic coefficients are plotted to analyze dynamic behavior of the spindle. Keywords: milling process, cutting forces, chip thickness, dynamic coefficients, orbit 1. Introduction During the last years, High SpeedMilling (HSM) process have become one of themost popular in the industry of shaping and producingmechanical parts andmolds. It allows having complex forms thanks to the variety of cutting tools composed of several cutting edges and driven by rotational motion. There aremultiple parameters that influence the forces acting on the cutter. Their knowledge and prediction become important in order to favor the adaptation of cutting tools conditions. The last researches were concentrated on studying the milling spindle with different types of bearings, especiallyAMBas they presentmany advantages deducedbyKnospe (2007). He showed thatAMBwere characterized by their accuracy, high robustness to shock and high rotational speed, which was also proved by Kimman et al. (2010) and Gourc et al. (2011). Also, among the special feature of AMB, the possibility to vary the number of electromagnets is treated in literature. Indeed, Bouaziz et al. (2011, 2013) studied that variation effect on the dynamic behavior of a rigid rotor with a misalignment defect. They demonstrated that the vibratory level of the rotor decreased with an increase in the electromagnets number. Belhadj Messaoud et al. (2011) presented the effect of the air gap and rotor speed on electromagnetic forces. From the results obtained in that work, they concluded that the electromagnetic forces 692 A. Bouaziz et al. intensity increased when the air gap decreased, but it remained unresponsive according to the rotor speed level. In addition, it is possible to use various components of AMB such as sensors and feedback currents to predict cutting forces. In fact, Auchet et al. (2004) expanded a new method to measure cutting forces by analyzing command voltages of AMB. This cutting force has an influence ondimensional accuracy due to the tool andworkpiece deflection in peripheralmilling. For this, a theoretical dynamic cutting forcemodel was presented by Liu et al. (2002) including the size effect of not deformed chip thickness, the impact of the effective rake angle and the chip flow angle. Numerical simulation and prediction of cutting forces in five-axis milling processes with cutter run-out and based on tool motion analysis was presented by Sun and Guo (2011). The predicted cutting forces illustrate good accordance with experimental results that the tool orientation angle and cutting depth vary continuously for both the specified cutting conditions and milling cases. Relative to classical methods, that proposed method permits one to predict cutting forces with a higher accuracy, and it is able to be directly used in five-axis milling. Lai (2000) studied the influenceof dynamic radii, cutting feed rate, and radial andaxial depthsof cut onmilling forces. He concluded that the chip thickness presented themost significant influence. In the samecontext,Klocke et al. (2009) investigated the influenceof cuttingparameters inmicro milling on the surface quality and tool life such as cutting speed and feed per tooth.As a results, they showed that the feed per tooth and feed rate extremely affected the surface quality inmicro milling. In fact, to increase the surface quality, it is necessary to decrease the feed rate value. From an analytical prediction of the cutting force, Fontaine et al. (2007) presented amethod to optimize themilling tools geometry.The influenceof geometrical parameters likehelix angle, rake angle and tool tip envelope radiuswas studied.The author demonstrates the ability of that type of model to provide optimization criteria for the design and selection of cutting tools. Budak (2006a,b) presented the milling force, workpiece and tool deflection, form error and stability models. From the used method, he checked the process constraints and selected the optimal cutting conditions. Concerning the machining process stability, Faassen et al. (2003) developed a dynamic model for milling process in which the stability lobes were generated. This model structurally predicts the stability limit slightly too conservative. Another new to the dynamical modelingofAMBto identifymachining stabilitywaspresentedbyGourc et al. (2011). Fromthat model, the authors concluded that the machining process stability was sensitive to the position of nodes of mode of the flexible rotor. Also, they confirmed that it was important to take in consideration strong forced vibrations as they could cause loss off safety. The stability of a high speed spindle system in the presence of the gyroscopic effect was investigated by Movahhedy and Mosaddegh (2006). In that study, it has found that the stability lobe predictions based on stationary FRFs were not conservative when the gyroscopic effects were respectable and that the gyroscopic effects became significant only at very high speeds compared with conventional speeds. Also, Gagnol et al. (2007) developed and experimentally validated an integrated spindle finite-element model in order to characterize the dynamic behavior of amotorizedmachine tool spindle. They demonstrated the dependence of dynamic stiffness on spindle speed. Using this model, a new stability lobes diagram was proposed. A research performed by Zatarain et al. (2006) showed thatmill helix angle could play an important role in instability due to repetitive impact driven chatter. Inorder to predict the occurrence of chatter vibrations, LacerdaandLima (2004) applied ananalyticalmethod inwhich the time-varyingdirectional dynamicmilling forces coefficientswere expanded inFourier series and integrated alongwidth of the cut boundby entry and exit angles. Wan et al. (2010) proposed a unified method for predicting stability lobes of the milling process with multiple delays. It was found that feed per tooth had great influence on the stability lobes when cutter run out occurred. In this paper, a HSM spindle with AMB is modeled by the finite element method based on the Timoshenko beam theory and used by Nelson andMcVough (1976) and Nelson (1980). Six Cutting parameters and vibrations analysis of magnetic bearing spindle... 693 degrees of freedom are considered. Rigid displacements are also taken in account (Hentati et al., 2013). AMBare presented as spring and damper elements. Peripheralmilling process ismodeled and cutting forces are formulated.Thedynamic responseat the spindle tool-tip, cutting forces in x-, y- and z- directions and the influence of some cutting parameters are predicted.The attitude of the used AMB is then investigated. 2. Modeling of AMB spindle machining 2.1. Mechanical model of the spindle The studied spindle model is presented in Fig. 1. The modeling is based on using a new approach developed byHentati et al. (2013). Thismethod is based on coupling both elastic and rigid spindle deformations. So, the shaft is discretized into 23 Timoshenko beam elements with different circular sections, where six elastic degrees of freedom are taken in account. Six degrees of rigid motion are also considered. Thus, the gyroscopic effect and centrifugal force will be taken into account in this study. The unbalance is not treated. The tool holder and cutter are included as parts of the spindle system in the specifically developed finite element model. Fig. 1. Modeling of the AMB spindle machining In this AMB spindle model we make use of a classical bearing configuration. Two radial AMBs and an axial bearing suspend the rotor in the central position. TheAMB consists of four electromagnets symmetrically placed relative to the rotor. Electromagnetic forces produced by every pair of electromagnets in the x- and y-directions are presented in the following equation (Bouaziz et al., 2011) fj(Ij,uj)=−a [ (I0− Ij e0−uj )2 − (I0+ Ij e0+uj )2 ] j=x,y (2.1) where uj represents small deformations in the j-th direction and e0 is the nominal air gap between the shaft and the stator, a is the global magnetic permeability, and it is expressed as follows a= µ0Sn 2I20 4 cosθ µ0, n, S and θ are respectively permeability of vacuum, windings number, cross sectional area and half angle between the poles of electromagnets. A proportional derivative controller (PD) is used to determine the control current expressed as Ij = kpuj +kdu̇j j=x,y (2.2) 694 A. Bouaziz et al. where u̇j is the velocity corresponding to the small deformation uj, kp presents the proportional gain. It is assumed to be in periodic form for better stability and controllability of motion of the spindle AMB system (Bouaziz et al., 2011, 2013; Amer and Hegazy, 2007) kp = k0+k1cosωt+k2cos2ωt (2.3) and kd denotes the derivative gain. In this model and as presented in Fig. 1, the electromagnetic field is modeled by stiffness and damping coefficients. The nonlinear electromagnetic forces at each bearing can be written in matrix form as follows (Bouaziz et al., 2011) [Kij] { ux uy } +[Cij] { u̇x u̇y } = { fx fy } (2.4) where [Kij] is the bearing stiffness matrix, [Cij] represents the bearing dampingmatrix [Kij] = [ Kxx 0 0 Kyy ] [Cij] = [ Cxx 0 0 Cyy ] (2.5) The axial bearing consists of two magnetic actuators on each side of the thrust element. The magnetic actuators in this setup are reluctance type actuators having a circular u-shaped core with a tangentially wound coil. The resulting force, fz of the axial bearing is linearized as fab =KizIz +Kzuz (2.6) where Kiz, Iz and Kz are the force current dependencies, the control current of the actuator and the negative stiffness of the axial bearing respectively, uz is the axial displacement of the shaft. 2.2. Model formulation of the cutting forces Figure 2 presents a cross-sectional view of the cutting forcemodel in peripheralmilling. The cutting force acting on the tool and the workpiece vary depending on chip thickness, shock of engagement, specific cutting pressure and generated vibrations. They appear onlywhen the tool is in contact with the part (cutting region). The cutting force is characterized by a tangential component Ft orthogonal to the specific segment of the cutting edge. This force is assumed to be proportional to the chip thickness and axial depth of the cut. The radial component Fr is proportional to Ft and orthogonal to both the cutting edge segment and the z-axis. The third component is the axial forcFa. This one is typically much smaller than eitherFt orFr and does not contribute greatly to the bendingmoment produced on the cutter. Fig. 2. Cross-sectional view of a peripheral milling process showing different forces Cutting parameters and vibrations analysis of magnetic bearing spindle... 695 The milling force variation against cutter rotation can be predicted by calculating Ft, Fr and Fa for different values of φj Ft =KtapH ( φj(t) ) Fr =KrFt Fa =KaFt (2.7) where H ( φj(t) ) is the instantaneous chip thickness, φj(t) is the rotation angle of tooth j, me- asured from the positive y-axis as shown in Fig. 2. ap, Kt, Kr and Ka are axial depth and the specific coefficients of the cut. The resulting chip thicknessH ( φj(t) ) is composedof a static part (stationary part)Hs ( φj(t) ) due to the rigid bodymotion of the cutter and a dynamic componentHd ( φj(t) ) which is caused by vibrations of the tool at the present and previous tooth periods. Hs ( φj(t) ) and Hd ( φj(t) ) assume the following form Hs ( φj(t) ) = fz sin ( φj(t) ) Hd ( φj(t) ) = [ux(t)−ux(t− τ)]sin ( φj(t) ) − [uy(t)−uy(t− τ)]cos ( φj(t) ) (2.8) where ux(t) and uy(t) represent deflections of the tool-tip at the present time, ux(t− τ) and uy(t−τ) are deflections of the tool-tip at the previous time, τ is the tooth passing period time, defined as τ =60/NZ andN,Z are respectively the spindle speed and the teeth number of the cutter. The rotation angle φj(t) is expressed as follows φj(t)=Ωt+ jΦp j=0,1, . . . ,Z−1 (2.9) whereΩ is the angular velocity in rad/s andΦp is the angle between two subsequent teeth (pitch angle), expressed asΦp =2π/Z. 2.3. Equation of motion As the modeling of the spindle is based on the coupling of rigid displacements and small elastic deformations, the total displacement vector is expressed as follows Q= [U1,V1,W1,θx1,θy1,θz1, . . . ,Ui,Vi,Wi,θxi,θyi,θzi,XA,YA,ZA,αx,αy,αz] T (2.10) i is the number of nodes, (U1,V1,W1,θx1,θy1,θz1, . . . ,Ui,Vi,Wi,θxi,θyi,θzi) presents nodal di- splacements, (XA,YA,ZA,αx,αy,αz) are the displacements of rigid motion. Applying the Lagrange formalism for kinetic and potential energies, the global equation of motion is MQ̈+(G+D+Cb(t))Q̇+(K+Kb(t))Q=Fc(x,y,z)(t,Q)+ fab(t,Q,Q̇) (2.11) whereM is the global mass matrix,G – the global gyroscopic matrix M= [ MF MRF MTRF MR ] G=2Ω [ GF GRF −GTRF GR ] F and R subscripts respectively represent the flexible or rigid part. D = αM+βK presents the damping matrix numerically constructed as a linear combination of the mass and stiffness matrix, whereα and β are the damping coefficients.Cb(t) is the variable matrix containing the damping coefficients of bearings Cb(t)=             0 · · 0 · · 0 · · · · · · 0 · Cxx 0 · · · · 0 0 Cyy · · · · · · · 0 Cxx 0 · · · · · Cyy 0 0 · · · 0 · 0             696 A. Bouaziz et al. K is the global stiffness matrix,Kc – the global centrifugal matrix K= [ KF 0 0 0 ] −Ω2 [ CF 0 0 0 ] ︸ ︷︷ ︸ Kc Kb(t) is the variable matrix containing the stiffness coefficients of bearings Kb(t)=             0 · · 0 · · 0 · · · · · · 0 · Kxx 0 · · · · 0 0 Kyy · · · · · · · 0 Kxx 0 · · · · · 0 Kyy 0 0 · · · 0 · 0             and Fc(x,y,z)(t,Q) denotes the cutting forces in the x-, y- and z-directions, respectively, fab(t,Q,Q̇) is the electromagnetic force vector exerted by the axial bearing. 3. Results and discussions In this Section, simulations are based on the spindle system with parameters listed in Tables 1 and 2. The general dynamic equation is solved by themethod of resolution byNewmark coupled with Newton Raphson. Table 1. Spindle parameters Parameter Symbol Value Unit Permeability of vacuum µ0 4π·10 −7 Wb/Am Air gap between stator and shaft e0 0.8 mm Effective cross-sectional area of one electromagnet S 200 mm2 Number of windings around the core n 300 – Half angle between poles of electromagnet θ 22.5 deg Bias current I0 3 A Rotor angular velocity N 20000 rpm Rotor length L 651.95 mm Stiffness coefficients Kxx,Kxy,Kyx,Kyy – N/m Damping coefficients Cxx,Cxy,Cyx,Cyy – N·s/m Derivative gain kd 42.4 As/m Proportional gain constant k0 4520 – k1 14869 – k2 14869 – Young’s modulus E 2.1 ·1011 Pa Density ρ 7.85 g/cm3 Poisson’s ratio ν 0.3 – Moment of inertia I 0.136 kg·m2 The time responses of the tool tip are plotted in Fig. 3. FromFig. 3a, some transient effects canbeobservedduring thefirst cyclewhile a steady state is achieved after that.Theirmagnitude is close. In fact, before the cutter is fully engaged, thearc of engagement increases graduallywhile Cutting parameters and vibrations analysis of magnetic bearing spindle... 697 Table 2.Cutting parameters Parameter Symbol Value Unit Feed per tooth fz 0.16 mm Axial depth of cut ap 5 mm Tangential cutting coefficient Kt 644 N/mm 2 Radial cutting coefficient Kr 0.38 N/mm 2 Axial cutting coefficient Ka 0.25 N/mm 2 Teeth number Z 2 – Fig. 3. The time response of the tool tip: (a) displacement of the tool tip, (b) orbit of the tool tip the cutter is entering progressively in the cutting zone. Consequently, this leads to a gradual increase of vibrations. The orbit of the tool tip has an elliptical shape explained by the introduction of flexible be- arings (stiffness coefficient) thatmake the systemasymmetric.Therefore,wenote thatvibrations in the x- and y-directions are different from the elliptic trajectory. The Fast Fourier Transformation (FFT) diagram for the x-response of the tool-tip with two teeth at a spindle speed of 20000rpm is shown in Fig. 4. It is found that two frequencies govern the behavior of the tool-tip response. The major peak corresponds to the frequency of 2Fr (666.66Hz) which occurs with the cutting force frequency. An obvious low frequency peak, corresponding with the rotation frequency, is also found. Fig. 4. Response of the tool-tip in the x-direction with 2 teeth 698 A. Bouaziz et al. The dynamic cutting force in the x-, y- and z-directions for a two teeth cutter are presented in Fig. 5. It can be seen that the cutting force is constant as the cutter is always in contact with the matter and where the evolution of radial engagement is continuous. All the cutting components have periodic and sinusoidal behavior with a period of time equal to the half of the rotation period (0.5Tr). Fig. 5. Cutting forces in the x-, y- and z-directions as function of time The following part of the study is devoted to presentation of the impact of some parameters involved in the cutting forces such as the teeth number, feed per tooth and speed rotation. Figure 6 presents variation of the cutting forces for different values of feed per tooth: 0.1, 0.2 and 0.3mm. It appears that the components in thex- and y-directions increase with an increase in the feed. It is worth noticing that when fz = 0.1mm, the maximum value of Fy is greater than the half value as fz =0.3mm. This evolution is logic and is explained by the fact that the feed is involved in the cut section. In fact, if the feed increases, the cutting section also increases and, therefore, the cutting force increases. This result was found by Liu et al. (2002) who also revealed that this variation was relative to the size effect of the chip thickness. In addition, we note that the Fy component rises with a greater rate. Fig. 6. Predicted cutting forces for different feed per tooth: (a) x-direction, (b) y-direction Figure 7 shows axial depth of the cut effects the cutting force value. When this parameter changes, the cutting force vary significantly. It increases when the cutting depth increases. This raise is explained by the increase of the width of chip. For these values of axial cutting depth, we find that the cutting forces Fy are always more important than the cutting force Fx. Figure 8a shows the instantaneous cutting forces for the tooth numberZ =3. It can be seen from variation of Fy that the cutting operation starts when the second tooth comes out, with the maximum value reaching 580N. Cutting parameters and vibrations analysis of magnetic bearing spindle... 699 Fig. 7. Predicted cutting forces for various cutting depth: (a) x-direction, (b) y-direction Fig. 8. Cutting force in the x-, y- and z-directions: (a)Z =3 teeth, (b)Z =4 teeth Although the shape of cutting forces is affected and changed compared to that with two teeth, the distribution is still continuous. This difference is explained by the summation of cutting forces of teeth in attack with the mutter. So, it is possible that the number of teeth variation will significantly influence the accuracy of the finished part. Figure 8b also presents the predicted values of the cutting force for the same conditions but for a four teeth tool. It is noticed that the shapeandvalues for the cutting force componentsFx andFy are similar to those of the cutter with three teeth. Also, these levels for the cutting forces, for a four flutes, seems to be typical for peripheral milling as this result is approximately close to the experimental ones found byBudak (2006a). The percentage ratio of the difference between the absolutemaximum predicted andmeasured value relative to the measured value represents an error of 9%. The variation of dynamic coefficients (Kxx,Kyy) is presented in Fig. 9. It is clear that the curves have a periodic form with amplitude reaching 1.8 ·107N/m. The coefficient Kxx is less important than Kyy, that is why we have obtained in Fig. 3 vibrations in the x-direction more severe than in the y-direction. The damping coefficients, presented in Fig. 10, have low amplitude varying from about −4.1535 ·104 to−4.15345 ·104N·s/m with instability in the beginning. Figure 11 shows the effect of the nominal air gap on the dynamic coefficients. This result reveals that the amplitude of both stiffness and damping coefficients decrease with an increase in the nominal air gap. This result was also proved by Bouaziz et al. (2011). The impact of the bias current I0 on the dynamic coefficients is presented in Fig. 12. For different values of I0: I0 =5A, I0 =6A and I0 =6.5A, it is found thatKxx andCxx rise when 700 A. Bouaziz et al. Fig. 9. Stiffness coefficients of the bottomAMB in the x- and y-directions Fig. 10. Damping coefficients of the bottomAMB in the x- and y-directions Fig. 11. Dynamic coefficients variation at the bottomAMB in the x-direction as function of e0 I0 increases. Indeed, equation (2.1) shows that the electromagnetic forces are proportional to the bias current. Therefore, the damping and stiffness coefficients should be increased to minimize the fluctuations. Figure 13 presents variation of the stiffness and damping coefficients for rotational speeds of N = 20000rpm and N = 40000rpm, respectively. We remark that the damping coefficients increase in a noticeable way relative to the stiffness coefficients. This is explained by the fact that the vibrations generated with the highest rotational speed should be absorbed and reduced due to damping. A too small change is noted for the stiffness coefficients at the beginning. Cutting parameters and vibrations analysis of magnetic bearing spindle... 701 Fig. 12. Dynamic coefficients variation at the bottomAMB in the x-direction as function of I0 Fig. 13. Dynamic coefficients of the bottomAMB in the x-direction as function of rotation speed 4. Conclusion This study presents dynamical analysis of a high speed AMB spindle in the peripheral milling process. The spindle rotor ismodeled by finite elements using theTimoshenko beam theory. The rigid motions are also considered. A mechanistic model of the peripheral milling is presented including the influence of instantaneous chip thickness. To solve the general equations ofmotion, the Newmark coupled with the Newton Raphson numerical method is used. The solution gives the spindledynamic responseand explains variation of the cutting force.Analyzing thiswork,we conclude that the cutting force is related to the cutting parameters introduced in the modeling such as the effect of thickness of the formed chip, feed per tooth, cutting depth and the number of teeth in the cutter. So, it is necessary to select a suitable cutter with the determined flute number in order toobtain an ideal cutting forcedistribution.This ideality appearsfirst,whenthe absolute value of the cutting force perpendicular to the feed direction during the cutting process is as small as possible; secondly, when the cutting force distribution is continuous. Concerning the modeling of the bearing, it is clear that the dynamic coefficients are influenced by the air gap as they increase when this parameter decreases. 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