Estimation andexperimentalvalidationofcuttingforcesinball endmilling

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Estimation andexperimentalvalidationofcuttingforcesinball endmilling Estimation andexperimentalvalidationofcuttingforcesinball endmilling Document Transcript

  • ARTICLE IN PRESS International Journal of Machine Tools & Manufacture 49 (2009) 1238–1244 Contents lists available at ScienceDirect International Journal of Machine Tools & Manufacture journal homepage: www.elsevier.com/locate/ijmactoolShort CommunicationEstimation and experimental validation of cutting forces in ball-end millingof sculptured surfacesYuwen Sun Ã, Fei Ren, Dongming Guo, Zhenyuan JiaKey Laboratory for Precision and Non-Traditional Machining Technology of the Ministry of Education, Dalian University of Technology, Dalian 116024, Chinaa r t i c l e in f o a b s t r a c tArticle history: Chip thickness calculation has a key important effect on the prediction accuracy of accompanied cuttingReceived 29 April 2009 forces in milling process. This paper presents a mechanistic method for estimating cutting force in ball-Received in revised form end milling of sculptured surfaces for any cases of toolpaths and varying feedrate by incorporation into a31 July 2009 new chip thickness model. Based on the given cutter location path and feedrate scheduling strategy, theAccepted 31 July 2009Available online 12 August 2009 trace modeling of the cutting edge used to determine the undeformed chip area is resulted from the relative part-tool motion in milling. Issues, such as the selection of the tooth tip and the computation ofKeywords: the preceding cutting path for the tooth tip, are also discussed in detail to ensure the accuracy of chipCutting forces thickness calculation. Under different chip thicknesses cutting coefficients are regressed with goodChip thickness agreements to calibrated values. Validation tests are carried out on a sculptured surface with curvedSculptured surface machining toolpaths under practical cutting conditions. Comparisons of simulated and experimental results showBall-end mill the effectiveness of the proposed method. & 2009 Elsevier Ltd. All rights reserved.1. Introduction instantaneous chip load and cutting forces. Lamikiz et al. [4] estimated the cutting force in inclined surface machining based Ball-end milling is widely used in machining parts with curved on a semi-mechanistic force model. The undeformed chip for thegeometries such as die & mould, propellers and turbine blades. slope cutting was calculated as the same as the horizontal case byRegardless of the emergence of many advanced CAM systems, means of a special reference system, composed by threemachining of complicated surfaces is still identified as a challenge. directions: feed direction, normal to machining surface and vectorThis partly contributes to high demand for tolerance, roughness or cross-product of both. The coefficients of the semi-mechanisticproductivity of machined parts and partly to the machinability of force model were obtained from horizontal slot cutting tests withdifficult-to-cut materials. For this end, cutting force modeling has different cutting conditions. Imani et al. [5] developed a simula-become an essential step to understand the behavior of cutting tion system for ball-end milling. A modified chip model wasprocess and further to ensure the stability of machining system represented based on the effect of vertical component of feed onand the optimization of process parameters. the chip thickness. And a commercial solid modeler was used to Some strategies have been addressed for the prediction of automatically extract the critical geometric information requiredcutting forces. Kim et al. [1] analyzed the relationship between for the physical simulation. Naserian et al. [6] introduced a staticundeformed chip geometry and the cutter feed inclination angle. rigid force model to estimate cutting forces of sculptured surface.Cutting forces acting on the engaged cutting edge elements were In the model, the approximated equation of chip thickness wascalculated using an empirical method. Then the resultant cutting derived from the same fundamental basis as in [5]. Most of theforce was calculated by numerical integration of cutting forces past researches are based on the premise that cutting force isacting on the engaged cutting edge elements. Fontaine et al. [2] viewed as a product of a coefficient and undeformed chipresearched the effect of tool–surface inclination on cutting forces thickness. Some methods have been proposed for calibratingin ball-end milling, and presented a milling force model based on milling force coefficients by different authors [7–9], and the chipa thermo-mechanical modeling of oblique cutting. Lazoglu [3] thickness is basically calculated with the classic approximationpresented a new mechanistic model, which has the ability to formula tn ¼ fz sin c sin k.calculate the workpiece/cutter intersection domain automatically, The increasing number of researches [10–20] on simulation offor the prediction of cutting forces in ball-end milling. Further- milling process highlights the importance of cutting force modelmore, an analytical approach was used to determine the for machining process plan and optimization. Undeformed chip thickness has become a critical factor of affecting the prediction accuracy of cutting forces. Li et al. [21] found the classical chip à Corresponding author. thickness model assumes that the tooth path is circular and thus E-mail addresses: xiands@dlut.edu.cn (Y. Sun), renfei_dl@yahoo.com.cn (F. Ren). lacks accuracy. They developed a new model for the undeformed0890-6955/$ - see front matter & 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijmachtools.2009.07.015
  • ARTICLE IN PRESS Y. Sun et al. / International Journal of Machine Tools & Manufacture 49 (2009) 1238–1244 1239 Nomenclature j index of discrete element of cutting edge i0 nominal helix angle (oT, x, y, z) Tool Reference System jc flute spacing angle (oG,X,Y,Z) Globe Reference System Nf number of cutting flute i index of cutting edge F feed speed R0 tool radius N rotational speed of spindle y cutter rotational angle fz feed per tooth b lag angle k positioning angle between a point on the flute and dA undeformed chip area the z-axis in vertical plane Kt, Kr, Ka cutting force coefficients (XQ, YQ, ZQ) position of intersection point Q in globe reference c rotational angle of a point on the cutting edge system dFt, dFr, dFa differential cutting forces in tangential, radial, and (XP, YP, ZP) position of current cutting point P in globe reference axial directions system systemchip thickness in horizontal milling. A transcendental equation cij ðy; zÞ ¼ y þ ði À 1Þjc þ bj ðzÞ ð2aÞwas then derived to calculate the underformed chip thickness.Kumanchik and Schmitz [22] also gave an analytic expression for jc ¼ 2p=Nf ð2bÞchip thickness while considering factors such as the cycloidalmotion of teeth, and uneven teeth spacing. In the model, line feed, bj ðzÞ ¼ zj tanði0 Þ=R0 ð2cÞtool rotational speed, and radius associated with milling werecombined into a single, non-dimensional parameter. Sai et al. [23] where zj denotes the z coordinate component of the jth segment ofnoted very little work has been done in research of modeling of the cutting edge in Tool Reference System, tn is the undeformedchip thickness in circular interpolation. They described a method chip thickness, and db is the height of axial disc segment.for calculating the instantaneous undeformed chip thickness inface milling case of circular interpolation and it was compared 2.1. Undeformed chip thickness calculationwith the case of linear interpolation. Serval improvements have been proposed in linear or arc At the moment during the tool-part engagement, as shown ininterpolation for some specificed ball-end milling cases. However, Fig. 2, the undeformed chip thickness can be calculated with thethere are few literaterures on modeling of chip thickness following form by finding the intersection point Q between thefor curved geometrics, varying feed and toolpath in parameteric path left by the previous cutting edge and line segment CPinterploation. Despite the influence of factors in real milling, perpendicular to the cutter axis C–C and through the currentfrom theoretical analysis the existing chip thickness models cutting point Pinevitablely brings errors to the solution in ideal status in millingfreeform surface along curved path with adaptive feed. It is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiessential to further improve the prediction accuracy of cutting tn ¼ JQPJ ¼ ðXP À XQ Þ2 þ ðYP À YQ Þ2 ð3Þforces and algorithms to concrete implementions for sculpturedsurface machining. Hence, we present an approach to estimating In this manner, two fundamental issues have to be dealt with.cutting forces based on a new undeformed chip thickness model One is the preceding trace modeling of cutting edge. The other isderived from the relative tool-part motion analysis in milling. the intersection computation of the trace with a line segment defined by the point and related cutter axis vector. Following the2. Proposed method The prediction of cutting forces consists of three steps in thedeveloped force model. First, a new chip thickness model isproposed by analyzing the relative tool-part motion so that it isable to handle the combined effects of toolpath pattern, feedrateschedule and tool geometry. In the second step a specialprocedure to Z-map model is applied for efficient extraction ofthe engaged cutting edge. At last, differential cutting forces ofeach engaged segment are obtained and integrated to determinethe resultant cutting force based on the calibrated cutting forcecoefficients. As shown in Fig. 1, according to the premise that the cuttingforce is proportional to undeformed chip area, the tangential (dFt),radial (dFr), axial (dFa) components of differential cutting force aremodeled as follows:dFm ðyÞ ¼ Km dAðcij ; y; kÞ ð1Þ 2where m ¼ t, r, a, Km (N/mm ) denote the calibrated milling forcecoefficients, dA can be calculated asdAðcij ; y; kÞ ¼ tn ðcij ; y; kÞ db ð2Þ Fig. 1. Geometrical model of ball-end milling process.
  • ARTICLE IN PRESS1240 Y. Sun et al. / International Journal of Machine Tools & Manufacture 49 (2009) 1238–1244 Current cutter axis Current trace Preceding trace C Q t n P Z Cutter geometry at the Y X cutting moment OG Z = ZP section plane Cutting Direction Fig. 2. Schematic diagram of undeformed chip thickness calculation.kinematics of the cutter with respect to the workpiece, the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðdXCL =duÞ2 þ ðdYCL =duÞ2 þ ðdZCL =duÞ2trajectory surface of cutting edge can be derived which naturally du ¼ dt ð8cÞcontains the combined effects of feedrate, toolpath patterns FðuÞand cutter geometry. The preceding trace at the z-plane isjust the intersection curve of the trajectory surface of the 2.1.2. Preceding trace for a point on the cutting edgepreceding cutting edge and the plane. Then, the intersection For a point on the current engaged cutting edge, there are twopoint is determined using the line segment/curve intersection necessary and sufficient conditions to determine those points onalgorithm. the preceding trace: (I) satisfying Eq. (4) and (II) having the same Z coordinate component as the specified point on the current2.1.1. Trajectory of cutting edge in milling engaged cutting edge in Globe Reference System. Based on Let rCL(u) ¼ {XCL(u),YCL(u),ZCL(u)} be a given cutter location Eqs. (4), (8a)–(8c) and Z coordinate component of the engagedpath along which the cutter performs the rotational and point, a mathematical equation can be derived to describe thetranslational motion. F(u) is the feed velocity of the cutter with preceding trace for the point P whose coordinate components arerespect to path parameter u. According to the relative tool-part labeled as XP, YP and ZP in Global Reference System.motion analysis, the kinematic equation of an arbitrary point on rPT ðt Ã Þ ¼ rCL ðuðt à ÞÞ þ Bðyðt à ÞÞCðzðt à ÞÞ P ð9Þthe cutting edge defined in Global Reference System is establishedas follows: where rPT ðt Ã Þ represents the preceding trace for P with regard to P CEr ðtÞ ¼ rCL ðuðtÞÞ þ BðyðtÞÞCðzÞ ð4Þ parameter t*; z(t*) denotes the z coordinate component of the point on the preceding trace in Tool Reference System.where rCE(t) represents the trajectory of the point at the zðtÃ Þ ¼ ZP À ZCL ðuðt à ÞÞ þ R0 ð10Þtime moment t; B(y) is the rotation matrix of the cutter withy ¼ 2pNt. In Tool Reference System, the position of the point is where ZCL(u(t*)) represents the Z coordinate component of thegiven by cutter location point at t* in Global Reference System; t*A[t0,t], t 2  3 denotes the moment when the cutting point P is intersected with z tan i0 6 RðzÞ sin ði À 1Þjc þ 7 the workpiece. R0 6 6  7 7 iÃCðzÞ ¼ 6 z tan i0 7 ð5Þ t0 ¼ t À ð11Þ 6 RðzÞ cos ði À 1Þjc þ 7 N 4 R0 5 z where i* represents the specified number of backward revolutions of the cutter and satisfies i*A{1, 2, y}.where z is the z coordinate component of the point in ToolReference System, R(z) is the radius of the section circle of thecutter at the z-plane perpendicular to the tool axis vector, as 2.1.3. Intersection point calculationprovided in the following form: As shown in Fig. 2, to calculate the undeformed chip thickness, the intersection point Q between the preceding trace and the line8> RðzÞ ¼ R0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi zZR0 segment CP must be known first. Let rCE be a vector of the point P P<  2 zoR0 ð6Þ intersected with the workpiece at t. rP is the corresponding cutter CL> RðzÞ ¼ R0 1 À z À 1: location point at the moment. Then the following expression can R0 be derived to construct the line segment To guarantee the synchronization of two motion processes, it is sðtÞ ¼ tfrP À ta ½ðrP À rCE Þ Á ta Šg þ ð1 À tÞrCE CL CL P P ð12Þnecessary to derive the relationship between parameter u andtime t. The mathematical formulation is given as where s(t) represents the line segment with regard to t,tA[0,1]; ta 0 denotes the cutter axis vector and satisfies ta ¼ [0,0,1]T in 3-axisdrCL ðuÞ drCL du r CL ðuÞ ¼ Á ¼ FðuÞ 0 ð7Þ milling. Thus, the intersection point is derived as follows: dt du dt jr CL ðuÞj rPT ðt Ã Þ À sðtÞ ¼ 0 P ð13Þ 0 du r0 CL ðuÞr CL ðuÞ ¼ FðuÞ 0 ð8aÞ dt jr CL ðuÞj Due to the implicit relationship between t* and t, numerical approach is needed to solve the equation. In this case, a geometric 0jr CL ðuÞjdu ¼ FðuÞdt ð8bÞ transformation is performed on the preceding trace and the line View slide
  • ARTICLE IN PRESS Y. Sun et al. / International Journal of Machine Tools & Manufacture 49 (2009) 1238–1244 1241segment to convert the issue of curve/line segment intersection Meshed work-pieceinto that of determining the root of an equation. Let b be the CL-pathvector angle between the direction vector of line segment CP and Cthe X component of Global Reference System, then " ! # wP À rCE Pb ¼ cosÀ1 Á ðvX Þ 0rbrp ð14Þ jwP À rCE j Pwith vX ¼ [0,0,1]T. Since both preceding trace rPT ðtÃÞ and line P B ¯segment s(t) are located on the plane perpendicular to the Z-axis Dof the Global Reference System in 3-axis milling, the intersectionpoint can be determined by solving the equation Yr PT ¼ 0 with ^Psubject to Xr PT rjwP À rCE j where Xr PT and Yr PT are coordinate ^P P ^P ^P Discrete cuttingcomponents of a point on transformed preceding trace in Global pointsReference System A^ PTr P ¼ RotðZ; bÞ Á ðrPT À rP Þ P CL ð15Þ Points engaged in cutting Projection of meshedwith 2 3 C work-piece cos ðbÞ sin ðbÞ 0 6 7RotðZ; bÞ ¼ 4 Àsin ðbÞ cos ðbÞ 05 B 0 0 1where b ¼ Àb if the point P is in quadrant I; b ¼ bÀp in Dquadrant II; b ¼ pÀb in quadrant III and b ¼ b in quadrant IV. ZThe iteration scheme can be used to solve Eq. (13). Thetransformed point P and the negative X direction are selected X A Grids connectingas starting iteration point and search direction, respectively. In Y with incut edge oGspecial cases, more than one intersection points may exist.The Bezier clipping technology [24] can be used to find all the Fig. 3. Illustration of the Z-map model.intersection points (Q1, Q2, y) and a set of possible undeformedchip thicknesses are obtained (tn1, tn2, y). The desired chipthickness is determined by l where the matrix A is defined astn ¼ mintn ðl ¼ 1; 2; . . .Þ ð16Þ 2 3 Àcos ðcÞ Àsin ðcÞsin ðkÞ sin ðcÞcos ðkÞ2.2. Engaged cutting edge 6 ÀcosðcÞsinðkÞ cosðcÞcosðkÞ 7 A ¼ 4 sinðcÞ 5 0 cosðkÞ sinðkÞ Z-Map model is used to determine whether a differentialcutter element is intersected with the workpiece at the momentof machining. The workpiece is meshed into small grids whose To obtain the resultant force, it is necessary to perform aprojection into the XoGY plane is square. In general cases, numerical integration along the cutting edge engaged in cuttingthe engaged cutter element can be achieved according to the process. The engagement conditions of differential cuttingdifference between the cutter element and the projection of the elements are used for the estimation of boundaries of theinstantaneous workpiece height into the cutter element. integration. By summing up the differential cutting forces for allWith consideration of adaptive feedrate schedule, varied work- in-cutting differential cutting elements, the total cutting forcespiece geometry feature and curved toolpath pattern, more are finally determined.than one cutter element may cut a grid for each tool movement.In this case, re-computing each projection of instantaneous 2.4. Cutting coefficient calibrationworkpiece heights into cutter elements is time consuming.To counteract the situation, in computing the mesh grids canbe viewed as a set of planes. For each mesh grid, the engaged Precondition to acquiring the cutting force is that the cuttingcutter elements are those not only they are below the related coefficients should be known. For a specified cutter, part materialplane, but also their projections are located into the correspond- and cutting conditions, the cutting coefficients can be calibrateding squares as shown in Fig. 3. Repeating the process, the lengths with experimental data. In this work, one planar milling test wasof cutting edges and the workpiece surface topography can be conducted in milling aluminum 2024-T6 with a vertical CNCobtained. milling machine. Cutting parameters are appropriately selected to ensure single cutting edge engagement during machining so that it can guarantee the synchronization between measured cutting2.3. Cutting force estimation forces and those predicted. At a given cutter rotational angles y*, for each engaged differential cutter element (j ¼ 1, 2, y, q), the The tangential, radial and axial cutting forces of each cutter proposed method is used to determine the instantaneousdisk element are calculated with Eq. (1). In Tool Reference System, undeformed chip thickness t*n(j). By means of instantaneous Ãcomponents of differential cutting forces are expressed as average chip thickness, t n is calculated and then used to calibrate the coefficients with the corresponding measured instantaneousðdFx; dFy; dFzÞT ¼ A Á ðdFt; dFr; dFaÞT ð17Þ cutting forces. According to Eqs. (1), (2) and (17), coefficients View slide
  • ARTICLE IN PRESS1242 Y. Sun et al. / International Journal of Machine Tools & Manufacture 49 (2009) 1238–1244 Ãrelated to t n can be obtained as follows: calculated by the existing method are larger than the one 0 1À1 calculated by the proposed method, which may be resulted from q à à à T 1 @X A the difference between the circular cutting trace used in the ½FX ðy ÞFY ðy ÞFZ ðy ފT à à ýKt ðy ÞKr ðy ÞKa ðy ފ ¼ A ð18Þ t n db j¼1 existing method and the real trace generated by the relative tool- part motion. In terms of varied feed directions, in case III the distributions of errors dramatically change with respect to case I, When the cutter rotates to kjc (k ¼ 1, 2, y, Nf), the angular which shows that feed direction plays a vital role in definingposition of cutting edge is the same as the one at y*. Thus, the the chip geometry. The combined effects of the feed directionaverage value of measured cutting forces can be used to reduce and the sectional circle radius of the specified point are illustratedrandom errors. Repeat the process for different cutter rotational in case II.angles, relationship between cutter coefficients and the unde-formed chip thickness is subsequently established.3. Model validation Predicted by proposed method Measured 1200 Predicted by existing method3.1. Comparison validation 800 Cutting force (N) To investigate the differences between the proposed under- Fzformed chip thickness model and the classical one, three 400numerical cases are simulated where the cutter moves along thegiven toolpaths with R0 ¼ 12 mm, N ¼ 600 rpm and fz ¼ 0.25 mm 0feed per tooth. During the process, undeformed chip thickness of Fyspecified points on the cutting edge is calculated using the twomethods. Position angles k of the points are selected as -400k1 ¼ k3 ¼ 57.61 and k2 ¼ 34.21 where the subscript denotes casesI, II and III, respectively. Errors of the existing method with respect -800 Fxto the proposed one are illustrated in Fig. 4. From the figure it canbe seen that the error curves show different shapes andmagnitude ranges in the whole range of the rotational angle 0 60 120 180 240 300 360cA[0,p] of the specified point. Case I simulates the simplest Cutting rotational angle (Deg.)cutting process and the error varies following a quasi-sinusoid Fig. 5. Comparisons of measured cutting forces and predicted cutting forces withcurve. In most region of the rotational angle c, the results two methods. Approximation error patterns Designed toolpath & Preceding trace on the plane perpendicular to cutter axis from specified point 0.04 Case II approximation error (mm) 0.02 Case II Case III Tooth trajectory 0 Designed toolpath -0.02 Case I Toolpath Z Equation of the curve: -0.04 Case II ap pr Eap = tn - tn Y -0.06 Case III X -0.08 oG Chip thickness in 0 30 60 90 120 150 180 one tool revolution tooth position angle (Deg.) Position angle of specified points for Case I Case III Tooth trajectory Tooth trajectory Case I, II and III z Specified points for Case I and III Toolpath Toolpath 57.6 34.2 Specified point for Case II Chip thickness in Chip thickness in OT x one tool revolution one tool revolution Fig. 4. Error analysis between the existing and the proposed undeformed chip thickness model.
  • ARTICLE IN PRESS Y. Sun et al. / International Journal of Machine Tools & Manufacture 49 (2009) 1238–1244 1243 Machined geometry Detail view of measured and predicted Cutting force (N) cutting forces Fx Fy Fz Measured Force Predicted Force Cutting force (N) Fig. 6. Measured cutting force, shape and predicted cutting forces, shapes in validation test. Compared with the existing model, more accurate chip geometry cutting forces. From the figures of measured cutting forces,is derived by the proposed method. To validate this, a standard it can be seen that there exist fluctuations of cutting forceshorizontal slot cutting experiment is conducted. Measured cutting within a relatively low magnitude range, which are mainlyforces and predicted cutting forces are shown in Fig. 5. It can be seen ascribed to the tool run-out, the dynamic characteristics of thethat cutting forces predicted by the proposed method agrees well cutting system, possible influences from the adjacent workingwith the measured results, and the maximum relative error of peak machines in real workshop environment, and the uncertaincutting forces is less 5%. However, using the existing model, the factors of the piezoelectric dynamometer. Regardless of theserelative errors are about 12% and even more. It also shows the error sources, results obtained from the experiments prove thenecessity of using the proposed method in sculptured surface validity of the proposed model in different machining cases.machining for varying feedrate and curved toolpaths. The magnitude and shape of predicted cutting forces have good agreements with those of measured forces in validation test, and the relative errors of peak cutting forces are controlled3.2. Curved surface milling validation within 10% in most milling regions. Unlike iso-parametric path, this kind of wave-like path has obvious change of feed Validation tests are conducted to testify the proposed method direction that is helpful in investigating the influence ofunder real machining conditions. During cutting tests, down- feed direction on the prediction accuracy of cutting forces.milling process are carried out without coolant. Workpiece From tested results we can see that in some cutting momentsmaterial is aluminum alloy 2024-T6. The spindle speed is set as the chip thickness calculated by the proposed method is different1000 rpm and the depth of cut is 1 mm. The cutter is made to from each other, even under the same cutter rotational anglemachine along sinusoidal type cutter paths on a curved surface and the engagement region of cutting edge and workpiece.expressed as That is to say, except for the local geometry of part and feedrate,rðu; vÞ ¼ ð125u; 20 cosðpvÞ; 10 sinðpvÞÞ where ðu; v 2 ½0; 1ŠÞ ð19Þ cutting direction also affects the shape and magnitude of cutting force. The proposed method has the ability to calculate the Fig. 6 shows the measured and predicted force signals for difference of the chip thickness under varying machiningsculptured surface machining as well as the detailed views of conditions.
  • ARTICLE IN PRESS1244 Y. Sun et al. / International Journal of Machine Tools & Manufacture 49 (2009) 1238–12444. Conclusions [4] A. Lamikiz, L.N. Lo’pez, J.A. de Lacalle, M.A. Salgado, Cutting force estimation in sculptured surface milling, International Journal of Machine Tools and Manufacture 44 (2004) 1511–1526. This paper represents a mechanistic approach to estimate [5] B.M. Imani, M.H. Sadeghi, M.A. Elbestawi, An improved process simulation forcutting forces in ball-end milling of sculptured surfaces. On the ball-end milling of sculptured surfaces, International Journal of Machine Toolsbasis of driving the kinematic trace expression of cutting edge in and Manufacture 38 (1998) 1089–1107. [6] R.S. Naserian, M.H. Sadeghi, H. 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