Numerical Prediction of Static Weihong Zhang1 Form Errors...

theeling,forson the

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Min Wan

Weihong Zhang1

Kepeng Qiu

Tong Gao

Sino-French Laboratory of ConcurrentEngineering,

School of Electromechanics,Northwestern Polytechnical University,

P.O. Box 552,710072 Xi’an, Shaanxi, China

Yonghong YangXi’an Aircraft International Corporation,

710089 Xi’an, Shaanxi, China

Numerical Prediction of StaticForm Errors in Peripheral Millingof Thin-Walled Workpieces WithIrregular MeshesThe finite element formulation is studied in this paper to predict static form errors inperipheral milling of complex thin-walled workpieces. Key issues such as cutter modfinite element discretization of cutting forces, tool–workpiece coupling and variation othe workpiece’s rigidity in milling are investigated. To be able to predict static form erron the machined surface of complex form, considerable improvements are madeproper modeling of the material removal in milling and the iterative calculationstool-workpiece deflections. A general simulation approach is developed based oirregular finite element meshes. By using illustrative examples, rigid and flexible moare compared with existing ones to show the validity of the approach.@DOI: 10.1115/1.1828055#

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1 IntroductionAdvanced manufacturing technologies constitute a basis

productivity improvements in aeronautical and aerospace mafacturing industries. The peripheral milling of thin-walled strutural components such as entire girders, aero-engine bladesturbine disks is an important machining process in these indtries. Due to the weak rigidity of workpieces, deflections inducby cutting forces are inevitable to cause form errors that wdeteriorate extremely the accuracy and quality of the workpiecthe form errors violate seriously the dimensional tolerance,milling process will lead to waste products. Therefore, reliamachining technologies must be employed to obtain consispart shapes and the machining accuracy. This can be realizeficiently by numerical simulations combining the finite elememethod with cutting mechanics, and furthermore by optimizcutting parameters to improve the cutting process.

The prediction of cutting forces is the basis of the milling prcess simulation, the accuracy of which depends upon the avaicutting force models and the determination of involved coecients. Smith and Tlusty@1# gave an overview of related millingmodels. Koenigsberger and Sabberwal@2# and Altintas andSpence@3# proposed that the cutting coefficients are power futions of the instantaneous uncut chip thickness removed bycutting edges. For the sake of simplicity, cutting coefficientsassumed to depend upon the average chip thickness. In this mcombinational effects of the shearing on the rake face andrubbing or plowing at the cutting edge are represented by a sicoefficient. Later, Budak et al.@4# used two independent coefficients to represent the shearing and ploughing effects separSuch coefficients are experimentally calibrated with off-line tesAn on-line method was also proposed by Zheng and Wang@5#.Besides, a more complicated model is proposed by Yu¨cesan andAltintas @6# who accounted for the effects of the friction, pressuas well as the force flow angle.

As we know, deflections of the cutter and of the workpieaffect directly the uncut chip thickness and cutting forces.method of estimating the instantaneous uncut chip thicknessgiven by Sutherland and DeVor@7#, who considered the effect otool defections upon the local chip thickness, but did not take iaccount the influence of workpiece deflections. Another import

1Corresponding author.Contributed by the Manufacturing Engineering Division for publication in t

JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript receivedJuly 7, 2003; Revised March 1, 2004. Associate Editor: D.-W. Cho.

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issue is concerned with the formulation of the regenerative moIt means that the variation of the instantaneous chip thicknmanifests in such a way that the additional material left byprevious teeth will be removed by the next one. Within this scoBudak and Altintas@8# demonstrated theoretically that the chthickness predicted by the regenerative model converges tonominal value after several revolutions for a static peripheral ming process. Based on this result, only the radial cutting deneed be corrected, as used in@9,10#, to reflect the deflection feedback effect.

A literature review shows that in a milling process without chter, the static form error mainly caused by elastic deflections ofcutter and of the workpiece is often the dominant defect whmilling the thin-walled workpiece made up of titanium or alumnum alloys at low spindle speed@6,9#. Kline et al.@11# studied theprediction of surface form errors in the peripheral milling ofclamped-clamped-clamped-free rectangle plate. The cutter is melled as a continuous cantilevered beam and the plate iscretized by the FEM. In the meantime, cutting forces are assuto be concentrated forces in the calculation of the cutter and wpiece deflections. Budak and Altintas@12#, Shirase et al.@13# stud-ied the form errors uniquely caused by the deflection of the cuthat is modelled as an assemblage of discrete elements with elength. Thus, cutting forces are discretized onto the element noto calculate the deflection of the cutter. This approximationvalid when the workpiece has relatively a large rigidity. Zhaet al. @14# determined the surface errors by evaluating the defltions of both the cutter and the workpiece without consideringcoupling effect between cutter and workpiece. Budak and Altin@10#, Tsai and Liao@9# developed iteration schemes to retain tcoupling effect of deflections between the cutter and workpiecewell as the rigidity diminution of the workpiece due to the matrial removal. Meanwhile, the workpiece is meshed by one laye8-node and 12-node isoparametric volume elements alongthickness direction, respectively. Nevertheless, the genermesh of the workpiece has to coincide, element to element, wthat of the cutter. In addition, the stiffness reduction of the wopiece due to the material removal can be simulated by varynodal coordinates of such a one-layer element. This requirembecomes a main limitation in the modeling step, especially inmodeling process of complex workpieces.

Regarding the inflexibilities of existing simulation methods,general approach is developed with an enhancement of the ro

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FEBRUARY 2005, Vol. 127 Õ 13005 by ASME

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ness of the numerical procedure and iterative algorithms. Aavailable cutting force models can be integrated into the propoapproach for estimating the forces and the deflection errors. Inpaper, to suit the geometric complexity of the workpiece, the lais modeled with irregular finite element meshes that can be gerated independently of the cutter. Besides, based on the idethe artificial power law used in structural topology optimizatiothe rigidity variation of the workpiece due to the material remois updated without remeshing. The proposed approach is finintegrated with an available finite element analysis packageensure the validity of the method, numerical examples are sowith different cutting models and results are compared withsolutions given in available references.

2 Finite Element Modeling of the Cutter–WorkpieceSystem

To have a flexible and reliable modeling scheme, we proposmodel the cutter and workpiece separately, i.e., coordinatetems, meshing methods and element types are selected inddently. A coordination of the cutter and the workpiece will bfinally needed and this will be carried out by a coherent geomedescription.

2.1 FE Model of the Cutter. As shown in Fig. 1, a helicafluted end mill is divided into axial segments of equal lengX0Y0Z0 is the local coordinate system of the cutter and its oriis located in the center of the cross-section with a distance ofL tothe free end of the cutter.L is the axial length of the tooth.Z0 issuperposing the axis of the cutter and its direction is oriented fthe origin to the free end of the cutter.Y0 passes through theintersection between a prescribed cutting edge and the crsection where the origin is located.

To evaluate the cutter’s deflection, the helical end mill is moeled as a cantilevered elastic beam with a cylindrical crosection. The equivalent diameter isde5s•d0 according to themodel of Kops and Vo@15#. Note thats ('0.8) is the scale factordue to helical flutes andd0 is the cutter’s diameter. Besides, itassumed that the stiffness of the collet clamping the cutter isproximated by a linear spring between the spindle of the machand the end mill shank. Hence, cutter’s deflections at any pointhe cutter induced by the cutting force are analytically calculaby means of the formula given by Budak and Altintas@10#.

Fig. 1 Cutter’s finite element model

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For the convenience of study, intersections between horizomesh lines and cutting edges of the cutter are called cutter nand the segment between any two adjacent mesh lines is ccutter element. As shown in Fig. 1, cutter node is symbolized( i , j ) with i and j to be the tooth number and node number,spectively. The cutter element between cutter node (i , j ) and( i , j 11) is represented by the symbol$ i , j %. Here, cutting edgesare numbered in sequence that begins with 1 and followsopposite direction of the helical angle. Hence, the first one isintersecting with axisY0 . The cutter node of thei th tooth isnumbered in sequence that begins with 0 from the cutter’send. Therefore, the coordinate of cutter node (i , j ) in X0Y0Z0 isexpressed as

X0,i , j50.5•d0•cosg i , j

Y0,i , j50.5•d0•sing i , j

Z0,i , j5L2 j •L/N (1)

Fig. 2 Coherence description of the cutter and the workpiece.„a… Milling of plane structure; „b… milling of curved surfacestructure.

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whereN is the total number of cutter elements equal to the mamal node number.g i , j is the clockwise rotation angle of cuttenode (i , j ) from the positive direction ofX0 to the positive direc-tion of Y0 .

2.2 FE Model of the Workpiece. To have a flexible mod-eling process, the coordinate system attached to the workpiecebe defined independently of that of the cutter. 3D irregular finelements of different shapes such as tetrahedral elements, pelements, hexahedral elements or a combination of them maadopted in practical modeling. In this work, the workpiece is atomatically discretized by available mesh generators of the ufinite element system. In addition, fixture effects are propemodeled by corresponding boundary conditions.

2.3 Coherence Description of the Cutter–Workpiece Sys-tem. Due to the independent modeling of the cutter andworkpiece, a description of the geometric coherence must beto determine their relative positions. In this manner, a geome

Fig. 3 Illustration of the cutting force evaluation

Fig. 4 Illustration of the correction factor of the cutting force

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relationship between the cutting edge and machined surfacebe easily identified for the cutting force discretization.

In Fig. 2, supposeXYZ is the global coordinate system of thworkpiece.x0y0z0 is another local coordinate system of the cutwith x0 and z0 being aligned with the normal direction of thmachined surface and the cutter axis, respectively. Besix0y0z0 moves with the cutter and its origin is located in the cenof the cross-section with a distance of axial cutting depth tofree end of the cutter.xyz is the local coordinate system attacheto the workpiece. As shown in Fig. 2~a!, the origin of xyz islocated in thex0o0y0 plane for the down milling process of planparts. Axisx is aligned with the normal direction of the machinesurface, and axisz is parallel to axisz0 . In the down millingprocess of the cylindrical part shown in Fig. 2~b!, the origin ofxyz is supposed to be on the central line of the cylindrical part ait is also located in thex0o0y0 plane. The direction of axisx isfrom the origin to the first feed position in thex0o0y0 planewhereas axisz is parallel to axisz0 .

In order to determine relative positions of cutter nodes aworkpiece nodes, their coordinates are all transformed intoxyz coordinate system. The coordinate transformation of cunodes is realized by

xY5XY 0•T1 (2)

and the coordinate transformation of workpiece nodes is realby

xY5XY •T2 (3)

In Eq. ~2!, the first three terms ofXY 05(X0,i , j ,Y0,i , j ,Z0,i , j ,1) rep-resent the coordinates of cutter node (i , j ) in X0Y0Z0 while thefirst three terms ofxY5(x0,i , j ,y0,i , j ,z0,i , j ,1) represent the coordinates of cutter node (i , j ) after transformation. In Eq.~3!, the firstthree terms ofXY 5(Xk ,Yk ,Zk,1) represent the coordinates of thkth workpiece node inXYZ while the first three terms ofxY5(xk ,yk ,zk,1) represent the coordinates of thekth workpiecenode after transformation. The determination of transformatmatricesT1 and T2 can be made properly. For example, in ainstantaneous state of the down milling process of plane pasuppose thatRr is the nominal radial cutting depth,Rz is thenominal axial cutting depth,a is the clockwise angle formed fromx0 to X0 and l is the distance that the cutter feeds,T1 is thenexpressed as

T15F cosa sina 0 0

2sina cosa 0 0

0 0 1 0

0.5d02Rr l Rz2L 1

G (4)

Note thatX0Y0Z0 moves and rotates with the cutter.

Fig. 5 Cutting force discretization

FEBRUARY 2005, Vol. 127 Õ 15

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Now, suppose that (l x ,l y ,l z) represents the directional cosineof axesx, y, z in XYZ with l x5(ax ,bx ,cx)

T, l z5(az ,bz ,cz)T

and l y5 l z3 l x , the rotation matrix is thenT05( l x ,l y ,l z)T with

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where (a0 ,b0 ,c0) is the coordinate of thexyz’s origin in XYZ.

3 Cutting Force Modeling and Calculations of StaticForm Errors

During the cutting process, the cutting forces applied on acutter element$ i , j % are composed of three parts: axial cuttinforceFi , j ,Z , tangential cutting forceFi , j ,T and radial cutting forceFi , j ,R as shown in Fig. 3~a!. The cutting forces are assumed touniformly distributed along the cutter element. The instantanecutting forces associated with cutter element$ i , j % are expressedas @10,16,17#

Fi , j ,R5r 1•k•Ai , j

Fi , j ,T5k•Ai , j

Fi , j ,Z5r 2•k•Ai , j (6)

where k, r 1 , r 2 are cutting coefficients determined by expements. They depend upon the workpiece material, cutter geetry, cutter material and cutting conditions.Ai , j is the rake face-chip contact area of cutter element$ i , j %. In fact, any availablecutting force models for peripheral milling can be applied. If tinstantaneous chip thickness and the axial length for the cutcutter element$ i , j % are represented byf i , j and bi , j with bi , j5L/N, Ai , j can be then evaluated by

Ai , j5bi , j f i , j (7)

As shown in Fig. 3~b!, the following relation exists for the instantaneous uncut chip thickness

f i , j5 f z•sinu i , j (8)

where f z is the feed per tooth, andu i , j is the clockwise angledetermined fromx0 to the cutting position of cutter element$ i , j %.Due to the helical angleb, the cutting position is assumed to bapproximately at the middle point of cutter element$ i , j %. If thecutting is concerned with a down milling process, we can writ

u i , j5u1,022p~ i 21!/Nf22 jbi , j tanb/d0 (9)

whereNf is the tooth number andu1,0 is the initial value depend-ing on the starting position of the cutter.

In the milling process, the axial contact length of the cuttitooth is generally not an integral multiple ofL/N. As illustrated inFig. 4, partAC of cutter element$ i , j % is not in contact with theworkpiece but partBC is in contact with the workpiece. Therefore, the rake face-chip contact area is calculated based onfollowing relation

Ai , j5ui , jbi , j f i , j (10)

whereui , j is a correction factor of cutter element$ i , j % defined as

ui , j5bi , j8 /bi , j (11)

where bi , j8 is the axial length of cutter element$ i , j % being incontact. For example, the correction factor ofAB in Fig. 4 equalsHC/HF.

Now, by projecting the cutting forces associated with cutelement$ i , j % in Eq. ~6! into systemx0y0z0 , we have

Fi , j ,x5Fi , j ,T sinu i , j2Fi , j ,R cosu i , j


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Fi , j ,y52Fi , j ,T cosu i , j2Fi , j ,R sinu i , j

Fi , j ,z5Fi , j ,Z (12)

The resultant applied to the whole end-milling cutter can thbe obtained by summing the forces of all cutter elements.

The discretized cutting forcef i , j acting at cutter node (i , j ) isapproximately obtained as follows

f i , j5Fi , j /2 j 50

f i , j5Fi , j /21Fi , j 21/2 j 51,N21

f i , j5Fi , j 21/2 j 5N (13)

where Fi , j5(Fi , j ,x ,Fi , j ,y ,Fi , j ,z) is the cutting force associatewith cutter element$ i , j %. Hereinto, the first and third expressioncorrespond to two extreme nodes of the element.

Because the FE computing of the workpiece will be carriedin theXYZcoordinate system, the above discretized cutting formust also be converted intoXYZ so that

f i , j8 5 f i , j•T3 (14)

whereT3 is the transformation matrix fromx0y0z0 to XYZ. In thecase of milling plane plates as shown in Fig. 2~a!, T3 is equal toT0 .

To transfer cutting forces from the cutter to the workpiece,correspondence between each immersion cutter node and thechined surface node of the workpiece has to be identified instaneously. This can be done either by findingm surface nodes of

Fig. 6 Corrections of element stiffness

Table 1 Cutting parameters

Cutting parameters Test 1 Test 2 Test 3

k (N/mm2) 1938 207•h20.67 1938r 1 0.693 1.39•h20.043 0.693r 2 0 0 0

Axial cutting depth~mm!

38.1 34 30

Radial cutting depth~mm!

1 0.65 1

Feed~mm/tooth! 0.05 0.008 0.05Helix angle of cutter

~deg!30 30 30

Cutter diameter~mm!

20 19.05 20

Gauge length of cutter~mm!

54.41 55.6 54.41

Young’s moduli of cutter~GPa!

207 620 207

Workpiece material Al7075-T6 Ti6Al4V Al7075-T6Uncut plate thickness

~mm!5.5 2.45 5

Young’s moduli of workpiece~GPa!

70 110 70


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the workpiece that are near the immersion cutter node or by idtifying on the workpiece a properm-node grid, to which the im-mersion cutter node is projected. In this way, the cutting foracting on the immersion cutter node are discretized ontom nodesof the machined surface. For the purpose of convenience,approximation schemes are used here to interpolate cutting foinstead of using finite element shape functions.

• Distance-based discretization scheme. As shown in Fig. 5immersion cutter node (j ,p) is surrounded bym surface nodeswith m54. According to the distance of the immersion cuttnode to them surface nodes of the workpiece, the cutting forcediscretized approximately by

Fi5S 1

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1

dk, j ,pD f j ,p8 (15)

in which di , j ,p denotes the distance from the immersion cutnode (j ,p) to nodei of the workpiece andFi denotes the cuttingforce on thei th surface node after discretization.

• Averaging force discretization scheme. If the FE model ofworkpiece is established with a refined FE mesh, the cutting focan be discretized averagely with a sufficient accuracy by

Fi5f j,p8 /m (16)

Fig. 7 Modeling of the workpiece. „a… Mesh 1; „b… Mesh 2.


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However, it is known that deflections of the cutter and of tworkpiece as well as the rigidity change of the workpiece stroninfluence the cutting forces in the practical milling of thin-walleworkpieces. To ensure the computing accuracy, three diffemodels are studied and perfected in this work to suit the compity of the problem.

3.1 Rigid Model „RM …. In this simplified model, it is as-sumed that deflections of the cutter and of the workpiece doaffect the magnitude of cutting forces and that the influence ofmaterial removal upon the rigidity of the workpiece is ignoreCutting forces will be calculated directly based on nominal cuttparameters as above.

3.2 Flexible Model With Invariable Rigidity of Workpiece„IRW …. This is an improved model as compared to the rigid oDeflections of the cutter and workpiece are taken into accoun

Fig. 8 Form errors of Test 1 in the middle position. „a… Pre-dicted values of two meshes by the VRW; „b… deviation betweenthe results produced from two meshes; „c… predicted and ex-perimental results from Ref. †9‡.


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the calculation of cutting forces while the rigidity change of tworkpiece induced by the material removal is not considered.der this assumption, nominal cutting parameters of cutter elem$ i , j % such as the instantaneous chip thicknessf i , j and radial cut-ting depthRr are less than the nominal values and need tocorrected.

Based on the results of previous work@8#, only the correction ofthe radial cutting depth is considered in this work because itrecognized that the chip thickness predicted by the regeneramodel converges to the nominal value after several revolutionsa static peripheral milling process.

In any cutting position, the actual radial cutting depth of cutelement$ i , j % has to be corrected by

Rr(k11)~ i , j !5Rr2@d t

(k)~ i , j !1dw(k)~ i , j !# (17)

whereRr(k11)( i , j ) is the corrected radial cutting depth in the (k

11)th iteration.d t(k)( i , j ) anddw

(k)( i , j ) denote deflections of cutter element$ i , j % and of the workpiece at the cutting position

Fig. 9 Form errors of Test 1 in the starting position. „a… Pre-dicted values by different models; „b… deviations obtained bysubtracting the VRW from the RM; „c… predicted and experimen-tal results from Ref. †9‡.


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the kth iteration, respectively. To improve the computing efciency of the iteration scheme, the unit load method will be usto compute deflections of the workpiece by FEM. That is, thecomputing is carried out just one time for each nodal load of uvalue and real node displacements of the workpiece will be tobtained by superposition and scaling depending upon thecutting forces in the iteration procedure. The validity of thimplementation is due to the assumption of linear elasticity forworkpiece deformation. As to the deflection of the cutter, itcalculated by the method given by Budak and Altintas@10#.

To solve Eq.~17!, the following phenomena may be observeAs the cutting force increases, deflections of the cutter andworkpiece will increase, too. Consequently, values of radial cting depth will decrease. Such a decrease will in turn lead todecrease of cutting force as well as deflections of the cutterthe workpiece. Therefore, the iteration scheme of Eq.~17! may bedivergent and values of the radial cutting depth will be in osciltion. Numerically, two cases may happen:

• Oscillations of correction factorui , j of cutter element$ i , j %.To avoid this problem, a new sub-iteration scheme is proposestabilizeui , j . First, an initial value ofui , j

(s) at steps is attributed toui , j with 0<ui , j

(s)<1. Therefore, the value of radial cutting depthevaluated based onui , j

(s) . From the new value of the radial cuttindepth, an updated value ofui , j

(s11) is subsequently derived. If the

absolute variation betweenui , j(s) and ui , j

(s11) satisfies a prescribedtolerance, the convergence of the iteration is achieved. Otherwan intermediate value betweenui , j

(s) and ui , j(s11) will be used to

repeat the above procedure until the convergence is reached• For certain cutting edges, e.g., thei th cutting edge, some

cutter elements may be engaged with the machined surface a

Fig. 10 Overall form errors. „a… Results predicted by the VRW;„b… experimental results from Ref. †9‡.


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current iteration and separate at the next one. To avoid this inbility, the proposed measure is to identify first the critical cutelement$ i ,p% which partitions possibly engaged cutter elemeinto two subgroups: one group$ i , j % with ( j <p) which holdsconstantly the contact with the machined surface during the ittion whereas the other with (j .p) is in fluctuation. In this way,iterations of Eq.~17! are carried out by assuming that the secoelement group is not engaged with the workpiece. After the cvergence is obtained,p is added by 1 and$ i ,p11% is now as-sumed to be engaged with the workpiece. Equation~17! will berepeated for next iterations until the convergence is obtainConversely, if the convergence is not achieved, it implies tcutter element$ i ,p11% is partially engaged so that a correctiofactor of ui ,p11,1 has to be used as illustrated in case 1.

3.3 Flexible Model With Variable Rigidity of Workpiece„VRW …. In this model, we take into account not only deflectioof the cutter and of the workpiece in the calculation of cuttiforces, but also the rigidity change of the workpiece due to

Fig. 11 Form errors of Test 2 in the middle position. „a… Pre-dicted values by different models; „b… deviations between dif-ferent models; „c… predicted and experimental results from Ref.†10‡.


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material removal. The latter is realized by implementing the ustechnique used in structural topology optimization~see Ref.@18#!.The basic idea is to correct the element stiffness in terms ovolume variation without remeshing so that

Ki5h i K i (18)

whereK i is the nominal stiffness matrix of elementi of the work-piece.h i denotes the ratio of volume variation of elementi aftersweeping with

h i5DVi /Vi ~10265«<h i<1! (19)

in which DVi andVi designate the remaining and nominal volumof elementi before and after cutting, respectively. Here, a lowbound of h i is used to prevent the singularity of the elemestiffness matrix when the material is completely removedmilling.

As shown in Fig. 6, elementi is cut off completely so thath i5«. Elementj is cut off partially so thath j has to be evaluated bydividing the nominal volume by the remaining volume shownhatched lines while elementk is uncut so thathk51. In this

Fig. 12 Cutting forces of Test 2. „a… Simulated values by dif-ferent models; „b… deviations between the RM and the VRW; „c…predicted and experimental results from Ref. †10‡.


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model formulation, it is necessary to note that the same issabout the iteration oscillation also exist as in the precedent.the same stabilization procedure can be used.

3.4 Numerical Calculations of Static Form Errors. A ma-chined position on the workpiece, i.e., a surface generation pis the intersection between the tooth’s transient trajectory andmachined surface. At such a machined position noted asQ, thestatic form erroreq will be defined as a normal projection odeflections of the cutter and of the workpiece such that

eq5d t,q1dw,q (20)

whered t,q anddw,q denote normal projections of displacementsthe cutter node and of the workpiece atQ. Geometrically, becauseQ does not superpose exactly the node of the cutter and wpiece,d t,q anddw,q will be evaluated approximately by averaginnodal displacements of a certain number of nodes aroundQ.

4 Numerical ApplicationsThree numerical tests have been carried out to demonstrat

validity of the proposed approach. The first two tests are seleto simulate two different milling processes of the rectangular pshown in Fig. 6. Test 1 corresponds to Test 1 studied in@9#,whereas Test 2 corresponds to Case 2 studied in@10#. To see thecapabilities of the proposed method, Test 3 is focused on a cuworkpiece that has to be meshed with irregular finite elementsto the presence of a hole. In this test, cutting coefficientschosen to be the same as those in Test 1. For all tests, the cuprocess is simulated by a single-fluted helical end mill. Soparameters such as cutting coefficients, cutting condition pareters, cutter geometry parameters, and so on, are listed in Tabwhere cutting coefficients in Test 1 and Test 3 are constants, wthe coefficients (k, r 1) are expressed as nonlinear power functioof average chip thicknessh in Test 2. Workpiece materials considered here are Ti6Al4V and Al7075-T6, which are extensivused for weak rigidity-structures, e.g., jet engine compresblades in aeronautical and aerospace industries. In the miprocess of such materials, low speeds are usually adopted to mmize chatter vibrations, for example, the spindle speed of aro500 rpm is used. In this case, elastic deflections of the workpand of the cutter constitute the main factors of the form errors.the form errors studied in this paper are based on the elasticterial model.

1. Test 1: the length of the workpiece in the feed direction47.96 mm: On one hand, to understand the mesh effects usimulation results, two different finite element meshes are studwith 3D hexahedral and prismatic elements as shown in Figs.~a!and 7~b!. Based on the VRW, distributions of form errors alonthe axial depth of cut are evaluated at the middle feed posiwith y547.96 mm/2. Results plotted in Fig. 8~a! show that bothmeshes lead to almost the same results. The maximum differbetween them is less than 3mm ~see Fig. 8~b!!. Note that axialdepth describes the axial position defined by the distance athe direction from the top of the workpiece to the bottom. Besidto validate the computation, these results are compared with tgiven in @9# ~see Fig. 8~c!!. A very good coherence exists.

On the other hand, to have a comparison of cutting force mels, form errors evaluated at the starting feed location ofy50 areplotted in Fig. 9~a! and compared with those given in@9# ~see Fig.9~c!!. By comparing these two figures, we can see that the mmum deviation between results predicted by the VRW and expmental ones are less than 10mm. Furthermore, it is interesting tonote that form errors predicted by the RM are slightly larger ththose based on the flexible model. Figure 9~b! represents the dif-ference between two models, which is less than 10mm. So, if theplate thickness is relatively large, both models are reliable w


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respect to the experimental results in this case. An overall pretation of the form error along the feed direction is illustratedFig. 10~a! and compared with the experimental results obtained@9# ~see Fig. 10~b!!.

2. Test 2: the initial thickness of the workpiece is now reduceto 2.45 mm and the length of the workpiece in the feed directis increased to 63.5 mm. A mesh of 3231734 volume elementsis used for the machined plate. In this test, the plate becomes mflexible so that three cutting force models are not consistent amore. For example, as shown in Fig. 11~a! and Fig. 11~b!, con-siderable differences exist for form error predictions aty535 mm along the feed direction. By comparing the results pduced by the VRW with experiment results given in Fig. 11~c!, themaximum deviation is less than 50mm. This indicates that theVRW is practically acceptable and more accurate than other mels. Furthermore, to have an idea about the simulation resultcutting forces produced by the VRW, the radial cutting depthevaluated iteratively and total cutting forces are then derivedshown in Fig. 12~a!. These results are very close to the measurdata shown in Fig. 12~c!. Figure 12~b! gives rise to the differenceof cutting forces between the RM and the VRW. Note that duethe different definition of coordinate systems, cutting force coponentsFx and Fy illustrated in Fig. 12~a! have to be permutedwhen compared with reference results in Fig. 12~c!.

3. Test 3: as shown in Fig. 13~a!, the workpiece has a contoucomposed of circular arcs and straight line segments. The boof the workpiece is clamped to simplify the modeling of bounda

Fig. 13 Test 3. „a… Finite element discretization of curvedworkpiece; „b… overall form errors by the VRW.


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conditions. Due to the presence of the hole, an irregular 3Dmesh has to be used and the feed trajectory has to be designa combination of straight lines and circular arcs. Here, the oradius of the circular arc is 10 mm. The length of the tool paththe plane part in the feed direction is 40 mm and the diametethe hole in the center of the plate is 10 mm.

By using VRW, the distribution of form errors is obtainedFig. 13~b!. It is observed that generated form errors vary smootalong the feed direction. Note that the concave part of Fig. 13~b!indicates the hole location where no material exists. The cutforce for one cutter revolution are numerically evaluatedmeans of VRW and RM in the middle feed position, respectiveResults are illustrated separately in Fig. 14~a! and Fig. 14~b!. Thesaddle of Figs. 14~a! and 14~b! indicates the state when the cuttinedge arrives in the hole position. Figure 14~b! illustrates quantita-tively that the deviation between both models is small. Thisbecause the workpiece is stiff enough with the current thicknFigures 15~a! and 15~c! show the form errors in two differencutting positions at 7.14 mm and 35.7 mm along the feed dirtion, respectively. Figures 15~b! and 15~d! indicate that differentmodels agree well as deviations are very small.

Finally, it is necessary to note that form error results canfurther used for error compensation in the industrial machinprocess. Such an error compensation is important to improvesurface quality. The procedure as explained in Fig. 16 canperformed as follows:

a. Evaluate the form errors with the proposed approach.b. Check the form errors with respect to the tolerance.c. Modify the nominal radial cutting depth iteratively if th

tolerance is violated.d. Create the compensated cutter path forNC codes by interpo-

lation.

Fig. 14 Cutting forces of Test 3. „a… Simulated values by dif-ferent models; „b… deviations obtained by subtracting the VRWfrom the RM


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5 ConclusionsA flexible numerical approach is developed to predict sta

form errors in the peripheral milling of thin-walled workpieceTo enhance the flexibility and the robustness of the simulatapproach, the utilization of irregular FE meshes is originally pposed and implemented considering the geometric complexitworkpieces in practical peripheral milling. That is, this approacan work with both regular and irregular meshes regardless ofsingle layer of FE elements or multi-layer of FE elements. Exing cutting force models are greatly improved by introducingvariable rigidity model without remeshing. The material remov

Fig. 15 Form errors of Test 3 in two different positions. „a…Predicted values in cutting positions at 7.14 mm; „b… deviationsbetween different models in „a…; „c… predicted values in cuttingpositions at 35.7 mm; „d… deviations between different modelsin „c…

Fig. 16 Flowchart of error compensation process


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is appropriately taken into account by implementing the artificpower law used in topology optimization. In addition to this, tconcept of the correction factor is proposed to describe the cpling effects between the cutter and workpiece, and an iterascheme is established to ensure the convergence when calcudeflections of the cutter and of the workpiece. To reduce the cputing time related to FEA, the iteration scheme is carriedbased on the unit load method. It is worthwhile to note there isadditional computing cost if the irregular FE mesh has the sanumber of d.o.f’s as the regular one.

By using illustrative examples, it is shown that form errors acutting forces can be well predicted by the proposed fleximodel with the variable rigidity even though the plate thicknesvery thin. Simulation results are very close to experimental on

AcknowledgmentThis work is supported by the Doctorate Creation Foundat

of Northwestern Polytechnical University~Grant No. CX200411!,the National Natural Science Foundation of China~Grant No.50435020! and Graduate Starting Seed Foundation of Northweern Polytechnical University~Grant No. Z20030056!.

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the Milling Process,’’ ASME J. Eng. Ind.,113, pp. 169–175.@2# Koenigsberger, F., and Sabberwal, A. J. P., 1961, ‘‘An Investigation into

Cutting Force Pulsations During Milling Operations,’’ Int. J. Mach. Tool DeRes.,1, pp. 15–33.

@3# Altintas, Y., and Spence, A., 1991, ‘‘End Milling Force Algorithms for CADSystems,’’ CIRP Ann.,40, pp. 31–34.

@4# Budak, E., Altintas, Y., and Armarego, E. J. A., 1996, ‘‘Prediction of MillinForce Coefficients from Orthogonal Cutting Data,’’ASME J. Manuf. Sci. En118, pp. 216–224.


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@5# Zheng, C. M., and Wang, J.-J.J., 2003, ‘‘Estimation of In-process CuttConstants in Ball-End Milling,’’ Proc. Inst. Mech. Eng.,217, pp. 45–56.

@6# Yucesan, G., and Altintas, Y., 1994, ‘‘Improved Modeling of Cutting ForCoefficients in Peripheral Milling,’’ Int. J. Mach. Tools Manuf.,34, pp. 473–487.

@7# Sutherland, J. W., and DeVor, R. E., 1986, ‘‘An Improved Method for CuttiForce and Surface Error Prediction in Flexible End Milling Systems,’’ ASMJ. Eng. Ind.,108, pp. 269–279.

@8# Budak, E., and Altintas, Y., 1992, ‘‘Flexible Milling Force Model for ImproveSurface Error Predictions,’’Proceedings of Engineering System Design aAnalysis, Istanbul, Turkey, ASME, 47, pp. 89–94.

@9# Tsai, J. S., and Liao, C. L., 1999, ‘‘Finite-element Modeling of Static SurfaErrors in the Peripheral Milling of Thin-Walled Workpieces,’’ J. Mater. Process. Technol.,94, pp. 235–246.

@10# Budak, E., and Altintas, Y., 1995, ‘‘Modeling and Avoidance of Static ForErrors in Peripheral Milling of Plates,’’ Int. J. Mach. Tools Manuf.,35, pp.459–476.

@11# Kline, W. A., DeVor, R. E., and Shareef, I. A., 1982, ‘‘The Prediction oSurface Accuracy in End Milling,’’ ASME J. Eng. Ind.,104, pp. 272–278.

@12# Budak, E., and Altintas, Y., 1994, ‘‘Peripheral Milling Conditions for Improved Dimensional Accuracy,’’ Int. J. Mach. Tools Manuf.,34, pp. 907–918.

@13# Shirase, K., and Altintas, Y., 1996, ‘‘Cutting Force and Dimensional SurfaError Generation in Peripheral Milling with Variable Pitch Helical End Mills,Int. J. Mach. Tools Manuf.,36, pp. 567–584.

@14# Zhang, Z. H., Zheng, L., Li, Z. Z., and Zhang, B. P., 2001, ‘‘Analytical Modfor End Milling Surface Geometrical Error With Considering Cutting ForcTorque,’’ Chin. J. Mech. Eng.,37, pp. 6–10.

@15# Kops, L., and Vo, D. T., 1990, ‘‘Determination of the Equivalent Diameteran End Mill Based on Its Compliance,’’ CIRP Ann.,39, pp. 93–96.

@16# Ber, A., Rotberg, J., and Zombach, S., 1988, ‘‘A Method for Cutting ForEvaluation of End Mills,’’ CIRP Ann.,37, pp. 37–40.

@17# Zhang, W. H., Wan, M., and Qiu, K. P., 2002, ‘‘Numerical SimulationPeripheral Milling Process and Prediction of Form Errors,’’6th InternationalConference on Progress of Machining Technology, Xi’an, China, pp. 893–898.

@18# Rozvany, G. I. N., 2001, ‘‘Aims, Scope, Methods, History and Unified Termnology of Computer-Aided Topology Optimization in Structural MechanicsStruct. Multidisc. Optim.,21, pp. 90–108.


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