Efficient algorithms for calculations of static form errors in peripheral milling

Journal of Materials Processing Technology 171 (2006) 156 165 Efficient algorithms for calculations of static form errors in peripheral milling M. Wan, W.H. Zhang Sino-French Laboratory of Concurrent Engineering, School of Mechatronic Engineering, Northwestern Polytechnical University, P.O. Box 552, 710072 Xi an, Shaanxi, China Received 30 March 2004; accepted 15 July 2005 Abstract This paper aims at developing numerical algorithms to predict the form errors in peripheral milling of thin-walled workpieces using finite element method. Some crucial algorithms associated with the judgment of contacts between the cutter and the workpiece, iterative corrections of radial and axial cutting depths as well as the workpiece s rigidity without remeshing are developed and presented in detail. By means of numerical tests, it is shown that the proposed approach is efficient and flexible. 3D irregular volume elements such as tetrahedral elements, prismatic elements, hexahedral elements or a combination of them can be freely used to predict form errors of machining workpieces. 2005 Elsevier B.V. All rights reserved. Keywords: Peripheral milling; Form errors; Finite element method 1. Introduction Peripheral milling is a common machining means used extensively in the aerospace manufacturing industry. With the advent of monolithic thin-walled parts, a lot of materials need to be removed from the initial configuration of the workpiece. To achieve acceptable dimension accuracy, it is recognized that deflections of both the cutter and workpiece have a direct impact upon the right profile of the workpiece and will be no longer negligible. Traditionally, the usual approach of remedying the machining precision was to validate the NC program by an expensive trial-error cutting. An alternative approach is to simulate numerically the milling process a priori. It is desired that a quasi-net-shaping will be obtained practically with optimal cutting parameters in perhaps one pass without grinding and polishing. To this end, the prediction of surface form errors due to deflections of the cutter and the workpiece is of great value and has been attracting many researchers [1 7]. Smith and Tlusty [1] classified a variety of available cutting force models concerning the milling process. Kline et Corresponding author. Tel.: +86 29 88495774; fax: +86 29 88495774. E-mail address: zhangwh@nwpu.edu.cn (W.H. Zhang). 0924-0136/$ see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2005.07.001 al. [2] have considered the cutter as a cantilevered beam with concentrated cutting forces to compute its deflection and applied the finite element method to predict the cutting deflections of a rectangular plate. It follows that normal cutting forces dominate form errors of the workpiece. Budak and Altintas [3], Shirase and Altintas [4] also used the beam theory to analyze the form errors caused by the static deformation of the slender helical end mill. The modeling was firstly made by dividing the cutter into a set of elements with equal length, and the errors were then calculated by summing the normal deflections caused by all element-cutting forces in the machined positions. In addition, Shirase and Altintas [4] showed that variable pitch cutters have considerable effects of reducing dimensional form errors in milling process. In all above formulations, however, influences of both tool and workpiece deflections upon the chip thickness have not been taken into account. Improvements were made by Sutherland and DeVor [5] who considered the effects of static deflections of the cutter in estimating the instantaneous uncut chip thickness. Thereafter, Budak and Altintas [6], Tsai and Liao [7] used iteration schemes to predict the cutting forces and form error distributions on the thin-walled rectangle. Due to the material removal and corrections of nominal cutting parameters such as radial cutting depth, the rigidity diminution of the 转载

M. Wan, W.H. Zhang / Journal of Materials Processing Technology 171 (2006) 156 165 157 workpiece is retained by moving element nodes on the cutting surface. However, the inconvenient lies in that the finite element mesh of the workpiece has to be a structured one and is limited to have only one layer of 8-node or 12-node isoparametric volume elements along the thickness direction. In this work, a general approach is presented for calculations of surface form errors in peripheral milling of thinwalled workpiece. An equivalent cantilevered beam is used to model the helical end mill. The FE simulation method incorporating cutting mechanics is used to analyze the deformations of the workpiece. To suit the geometric complexity of the workpiece, the proposed approach offers a great flexibility of employing an irregular finite element mesh for the workpiece modeling and this can be done independently of the cutter. Furthermore, based on the recent work of authors [8], efforts are focused on the development of efficient numerical algorithms considering the feedback influence of the cutter and workpiece deflections. A determinant algorithm is presented to judge the contact between a cutter element and the workpiece. More importantly, a new iteration algorithm using the concept of correction factor is developed to determine accurately the radial and axial cutting depths. Besides, to update the rigidity of the workpiece induced by the material removal, the idea of softening material as used in structural topology optimization is adopted to keep the same finite element mesh during the whole milling process without remeshing. Furthermore, the workpiece can be meshed by more than one layer of finite elements along the workpiece thickness depending upon the computing accuracy. Finally, all these novelties are integrated with an available finite element analysis package and the validation of the simulation procedure is demonstrated with the help of numerical tests. The coordinate system of the workpiece can be defined arbitrarily and is independent of that of the cutter. 3D irregular finite element meshes and element types such as tetrahedral element, prismatic element, hexahedral element or a combination of them can be freely used to discretize the workpiece, e.g., the structure of the workpiece in Fig. 1. Due to the independent modeling of the cutter and workpiece, a coherence description is made to determine their relative position so that a geometric relationship between the cutting edge and machined surface can be easily identified for the cutting force discretization. One can refer to another relevant work of authors [8] for details about the modeling process. 2.2. Basic cutting force model As shown in Fig. 1, the tangential, radial and axial cutting forces acting on cutter element {i, j} are calculated in terms of nominal cutting parameters as follows [10 12] F i,j,t = k T A i,j F i,j,r = r 1 F i,j,t (1) F i,j,z = r 2 F i,j,t where k T, r 1, r 2 are the cutting force coefficients determined experimentally [12 14]. A i,j = b i,j f i,j, in which b i,j is the axial 2. Numerical modeling of peripheral milling process 2.1. Modeling of the cutter and the workpiece To be able to calculate the deflections of complex structures in practical milling process, it is necessary to have a flexible and reliable modeling scheme. The cutter and the workpiece will be modeled independently, i.e., coordinate systems, meshing methods and element types are selected separately according to their own structure characteristics. In our study, the cutter structure is shown in Fig. 1. X 0 Y 0 Z 0 is the local coordinate system attached to the cutter for cutter modeling. x 0 y 0 z 0 is another local coordinate system which moves with the cutter. Axes x 0 and z 0 are aligned with the normal direction of the machined surface and the cutter axis, respectively. Here, the helical end mill is modeled as an equivalent cantilevered beam (see Ref. [9]) with equal elements along the axial direction. Notation (i, j) designates the cutter node which is the intersection between one horizontal mesh line and the ith cutting edge, while notation {i, j} means the cutter element which is the cutting edge segment between cutter node (i, j) and cutter node (i, j + 1). Fig. 1. Modeling of the cutter and the workpiece.

158 M. Wan, W.H. Zhang / Journal of Materials Processing Technology 171 (2006) 156 165 length of the cutting cutter element {i, j}. f i,j = f z sin θ i,j is the instantaneous uncut chip thickness with f z being the feed per tooth. The absolute value of sin θ i,j is used to prevent the negative value in up milling due to the definition of θ i,j. Here, θ i,j is the immersion angle determined clockwise from the exterior normal of the machined surface to the cutting position of cutter element {i, j}. Since each cutter element is relatively small, the related cutting position is approximately assumed to be at the middle point of cutter element {i, j} without considering the variation of the helix angle β in each element. In practical milling process, some cutter elements may partially contact the workpiece. As shown in Fig. 2, only part BC of cutter element {i, j} is engaged with the workpiece. Therefore, A i,j can be estimated by the following relation A i,j = u i,j b i,j f i,j (2) where u i,j is a correction factor of cutter element {i, j} defined as u i,j = b i,j (3) b i,j with b i,j being the axial length of cutter element {i, j} in contact. For example, one has u i,j = EC/ED for the correction factor of AB shown in Fig. 2. After cutting forces are obtained for all engaged cutter elements, they will be then discretized averagely to their adjacent cutter nodes. The obtained nodal forces are further applied equivalently to the workpiece by projecting them onto the nodes of the machined surface [8]. 2.3. Calculations of θ i,j and u i,j Obviously, values of θ i,j and u i,j must be known in advance before computing the cutting forces. To do this, it is necessary to judge whether the concerned cutter element is engaged with the workpiece. With a given initial configuration of the cutter in the milling process, e.g., down milling shown in Fig. 3(a), we can see that cutter nodes may be in contact with the workpiece only when rotating to the right side of axis x 0. Hence, an angular zone can be defined for each cutter node (i, j)by [D l i,j,du i,j ] (4) where Di,j l and Du i,j are the entering and leaving angles of cutter node (i, j) with respect to the right (down milling) or left (up milling) side of axis x 0, respectively. Note that both bounds will remain unchanged provided that the cutter rotates with entire periods. As shown in Fig. 3, the initial position of cutter node A (i, j) can be geometrically described by α i,j that is defined as an anticlockwise rotation angle from the positive direction of x 0 to the negative direction of y 0 whenever it is Fig. 2. Illustrations of correction factor of the cutting forces. Fig. 3. Definitions of contact zones. (a) Down milling. (b) Up milling.

M. Wan, W.H. Zhang / Journal of Materials Processing Technology 171 (2006) 156 165 159 concerned with down milling or up milling. α i,j reads α i,j = α 1,0 + 2π(i 1) N f + 2jb i,j tan β d 0 (5) where N f is the tooth number, d 0 is the cutter diameter, and α 1,0 is the angular value of node (1, 0) in the initial configuration of the cutter. Clearly, the determination of D l i,j and D u i,j depends upon α i,j and can be made as follows. 2.3.1. Down milling process If cutter node (i, j) is to the left side of axis x 0, e.g., at node AinFig. 3(a) with α i,j π, then D l i,j = α i,j, D u i,j = α i,j + π (6) If cutter node (i, j) is to the right side of axis x 0, e.g., at node A in Fig. 3(a) with α i,j > π, then two solutions exist Di,j l = α i,j, Di,j u = 2π (7.1) and D l i,j = 0, Du i,j = α i,j π (7.2) 2.3.2. Up milling process The angular zone in the up milling is to the left side of axis x 0, as shown in Fig. 3(b). We have then D l i,j = α i,j π, D u i,j = α i,j if α i,j π (8) or alternatively D l i,j = 0, Du i,j = α i,j if α i,j <π (9.1) and D l i,j = α i,j + π, D u i,j = 2π if α i,j <π (9.2) Now, it becomes easier to determine whether the cutter element {i, j} is engaged with the workpiece. Suppose that Rot denotes an instantaneous cutter rotation angle, which equals also the angle from the starting position of cutter node (i, j) to its ending position at that time and that the angle measured clockwise from x 0 to the ending position of cutter node (i, j) isψ i,j. Rot can be then limited to Rot [0, 2π] in consideration of the periodicity. Therefore, if Di,j l Rot Di,j u, it means that cutter node (i, j) may be engaged with the workpiece. Assume that φ i,j st and φex i,j represent the start and exit immersion angles of cutter node (i, j) measured in its section, respectively. Cutter node (i, j) is therefore engaged with the workpiece provided that φ i,j st ψ i,j φex i,j (10) Precisely, calculations of ψ i,j can be performed as follows: In down milling, when Eqs. (6) or (7.1) is retained, one has ψ i,j = Rot α i,j (11) Alternatively, if Eq. (7.2) is verified, one has ψ i,j = Rot α i,j + 2π (12) As shown in Fig. 3(a), φex i,j = π while φ i,j st the radial cutting depth R r (i, j) with φ i,j st = arccos ( 2 R r(i, j) d 0 1 ) depends on (13) In up milling, one has ( φ i,j st = π, φex i,j = 2π arccos 2 R ) r(i, j) 1 d 0 (14) If Eqs. (8) or (9.1) is retained, one has ψ i,j = Rot α i,j + 2π (15) Otherwise, if Eq. (9.2) is retained, one has ψ i,j = Rot α i,j (16) Therefore, it concludes that cutter element {i, j} contacts the workpiece as long as both cutter nodes (i, j) and (i, j +1) are engaged with the workpiece. Then, following relations hold tan β θ i,j = ψ i,j b i,j (17) d 0 and u i,j = 1 (18) If neither cutter node (i, j) nor (i, j + 1) is engaged with the workpiece, cutting forces will not be necessarily computed since u i,j =0. If only cutter node (i, j) contacts the workpiece instead of (i, j + 1), it implies that cutter element {i, j} is partially engaged with the workpiece so that b i,j can be obtained approximately by b i,j = 1 2 (ψ i,j φ i,j st ) d 0 (19) tan β If cutter node (i, j + 1) contacts the workpiece instead of (i, j), then b i,j = 1 2 (φi,j+1 ex ψ i,j ) d 0 (20) tan β Finally, note that u i,j and θ i,j will still be evaluated by means of Eqs. (3) and (17), respectively when the cutter element is partially engaged in last two cases. 3. Iteration algorithms for flexible model The basic cutting force model is a rigid one that uses nominal cutting parameters directly. However, in milling of thin-walled workpieces, the cutter/workpiece deflections and

160 M. Wan, W.H. Zhang / Journal of Materials Processing Technology 171 (2006) 156 165 the rigidity variation of the workpiece induced by material removal have considerable feedback influences upon cutting forces. Hence, flexible models are required to compute cutting forces more accurately. This needs to correct both nominal cutting parameters and the workpiece rigidity in an iterative way. 3.1. Corrections of radial cutting depth As a result of the deflections, both the instantaneous uncut chip thickness and radial cutting depth need to be corrected. However, Budak and Altintas [15] theoretically showed that for a static peripheral milling process free of chatter, the chip thickness variation induced by deflections converges to the nominal value in a few rotation periods. Hence, only the correction of radial cutting depth is desirable in flexible model. For any cutter element {i, j}, the actual radial cutting depth is iteratively corrected by R (k+1) r (i, j) = R r [δ (k) (i, j) + δ (k) (i, j)] (21) t in which R r is the nominal radial cutting depth, R (k+1) r (i, j) is the corrected radial cutting depth in the (k + 1)th iteration. δ (k) t (i, j) and δ (k) w (i, j) denote deflections of the cutter and the workpiece corresponding to cutter element {i, j} in the kth iteration, respectively. δ (k) t (i, j) is obtained by the cantilevered theory as used in Refs. [4,6]. To reduce the FEA computing time, δ (k) w (i, j) is evaluated based on the unit force method. That is, values of δ (k) w (i, j) are instantaneously updated by scaling in the iteration procedure. During the iterating procedure of Eq. (21), following phenomena may be observed numerically. The increase of radial cutting depth in the previous iteration will directly lead to an increase of cutting forces and deflections in the current iteration. Consequently, the radial cutting depth will decrease after one iteration step. In the subsequent iterative steps, such a decrease will lead to the increase of radial cutting depth. These phenomena can be schematically seen in Table 1. Therefore, the iteration scheme of Eq. (21) may be divergent and values of the radial cutting depth will be in oscillation. Concretely, two cases may happen in the event of divergence. Following is the investigation of down milling process: Case 1 Oscillations of a single correction factor u i,j related to cutter element {i, j}. Generally, this happens to the last engaged cutter element {i, j}. To solve this problem, the Table 1 Variations of cutting forces and radial cutting depth Iteration no., k w Variation tendency of cutting forces 1 2 3 4......... Variation tendency of R (k) r (i, j) following sub-iteration scheme is adopted to stabilize u i,j instead of continuing Eq. (21) directly ũ (k+1) i,j u (k+1) i,j s.t. 0 <ξ 1 ũ (k+1) i,j = ξ(u (k+1) i,j ũ (k) i,j ) + ũ(k) i,j = b (k+1) i,j, k = 0, 1,...,n b i,j u (k+1) i,j ε (22) where ũ (k+1) i,j is the corrected value of the correction factor in the (k + 1)th iteration. u (k+1) i,j and b (k+1) i,j are the actual values of the correction factor and the axial contacting length related to cutter element {i, j}, respectively. ξ and ε are the weighted parameter given a priori and the prescribed tolerance used to stop iterations, respectively. Case 2 Oscillations of multiple correction factors. In this case, for some cutter elements, e.g., the cutter elements {i, p}, {i, p +1},...,{i, j} of the ith cutting edge, may be engaged with the machined surface in the current iteration and separate in the next one. To solve this problem, the following procedure is proposed to ensure the convergence of the iteration scheme: (a) Assume that cutter elements from {i, p +1} to {i, j} are not engaged in cut. So, set R (k) r (i, t) = 0 for all (p +1 t j ). (b) Start the iterative process of Eq. (22) for the cutter element {i, p} with ũ (k+1) i,p until the convergence is reached. (c) If 0 ũ (k+1) i,p < 1, the convergence achieves for all cutter elements and stop the iteration. Otherwise, continue step (b) by setting p = p + 1 and R (k) r (i, p) = R (k) r (i, p 1). In the up milling process, Case 1 is still applicable. Instead, the oscillation will happen for cutter elements near the cutter tip in Case 2. Therefore, oscillating cutter elements, e.g., {i, p}, {i, p 1},...,{i, j} are needed to be identified sequentially as performed in down milling. The whole flowchart that describes this iteration scheme is shown in Fig. 4. From this solution routine, we can see that after the radial cutting depth is corrected, the axial cutting depth will be corrected correspondingly. 3.2. Correction of finite-element mesh due to material removal To consider the rigidity change of the workpiece due to the material removal, the idea of softening materials as used in structural topology optimization (see Ref. [16]) is implemented. The basic idea of this technique is to correct the element stiffness matrix in terms of its volume variation

M. Wan, W.H. Zhang / Journal of Materials Processing Technology 171 (2006) 156 165 161 Fig. 5. Correction of workpiece rigidity due to material removal. (a) Corrections of the element stiffness. (b) Partially cut off cases for tetrahedral element. Fig. 4. The routine to correct radial cutting depth. without remeshing so that K i = η i K i (23) where K i is the nominal stiffness matrix of element i. η i denotes the ratio of volume variation of element i after sweeping with η i = V i (10 6 = ε 1 η i 1) (24) V i in which V i and V i designate the remaining and nominal volume of element i before and after cutting, respectively. Here, a lower bound ε 1 is used to prevent the singularity of the element stiffness matrix when the material is completely removed in milling. The determination of η i is made in two basic steps: Identification of the element status of the workpiece. An element status depends on its relative position with respect to actual radial cutting depths. For an element, if distances calculated from all its nodes to the machined surface are less than the actual radial cutting depths at corresponding machined surface positions, it means that this element is cut off completely; if distances are more than the corresponding radial cutting depths, the element is not cut off at all; if only some distances are less than the corresponding radial cutting depths, the element is cut off partially. As shown in Fig. 5(a), distances of nodes P 0 and P 1 to the machined surface are l 0 and l 1, respectively. P 0 and P 1 have a common projection to P n so that the same radial cutting depth R 0 can be used. In this case, element i is cut off completely with η i = ε because distances of all attached nodes are less than corresponding radial depths. Element j is partially cut off because l 0 < R 0 and l 1 > R 0. Element k is not cut at all so that η k =1. Identification of the cutting boundary for partially cut off elements. Based on the obtained radial cutting depths from Eq. (21), the cutting boundary can be determined by finding

162 M. Wan, W.H. Zhang / Journal of Materials Processing Technology 171 (2006) 156 165 its intersection with all edges of elements being partially cut off. For example, when it is concerned with edge P 0 P 1 of element j, we can write R 0 = l 0 + (l 1 l 0 )t 1 with 0 <t 1 < 1 (25) If such a parameter t 1 exists, the intersection point is then P = P 0 + (P 1 P 0 )t 1 (26) As illustrated in Fig. 5(b), a summary of all possible cutting cases is presented for tetrahedral elements. Each element is partitioned into (I) and (II), and either of two parts can be retained as the remaining one. To simplify the calculation of volume variations, it is necessary to remark that element edges will be approximated by straight-line segments when they are curved ones. After the rigidity of the workpiece is modified, the same numerical procedure that stabilizing oscillations as described above will be used to evaluate the actual cutting forces. 4. Calculations of static form errors Deflections of the cutter and the workpiece can be combined to predict the overall static form errors with the adopted cutting force models. By definition, static form errors are the normal deviations of the finished surface from the desired surface. At any contact point G between the cutter tooth and the finished surface, the static form error e G equals then e G = δ t,g + δ w,g (27) in which δ t,g and δ w,g are the normal projections of the cutter deflection and of the workpiece deflection corresponding to G, respectively. To estimate values of δ t,g and δ w,g,an interpolation of displacements associated with a certain number of nodes around G is approximately used. As a result, form errors along one surface generation line can be obtained by repeating this computing process over one tooth cutting period with a given rotation step length. In summary, the milling trajectory is firstly split into a sequence of discrete cutting locations. Form errors along each surface generation line will be calculated at the current cutting location before the cutter shifts to the next one. At the same time, rigidities of elements swept between these two adjacent locations are corrected as illustrated in the previous section. This routine continues until analyses of all locations are finished. 5. Numerical experiments 5.1. Peripheral milling of a rectangular plate A rectangle plate as shown in Fig. 1 is adopted to validate the numerical simulation method. The first case corresponds to Test 1 studied in Ref. [7]. In this event, the workpiece material is aluminum alloy 7075-T6. The length of work- Fig. 6. Form errors in the middle position in down milling. (a) Predicted values by different models. (b) Predicted and experimental results from Ref. [7]. piece in the feed direction is 47.96 mm. The helical fluted end mill is single-fluted high-speed steel of Nachi Co-HSS with a diameter of 20 mm, helix angle of 30 and tool gauge length of 54.41 mm. The cutter is discretized into 55 axial elements. Young s moduli of the cutter and workpiece are 207 and 70 GPa, respectively. The nominal axial cutting depth is R z = 38.1 mm. The nominal radial cutting depth is R r =1mm for an initial plate thickness of 5.5 mm and α 1,0 = π. Detailed data can be found in Ref. [7]. These conditions are used both for down milling and for up milling. To have a clear insight of the difference between the rigid model and flexible model, a comparison of form errors is made in the middle feed position of the plate and results are plotted in Fig. 6(a). It can be seen that both results are almost the same in this case when compared with the results provided in Ref. [7] (see Fig. 6(b)). The reason is that the workpiece is relatively rigid so that the rigid cutting force model can predict the form error with sufficient accuracy. For up milling process, in Fig. 7, error curves evaluated by means of rigid model and flexible model in the middle feed position show that a good coherence also holds in up milling. Finally, the investigation is focused on the iteration history of correction factor u i,j. Here, two different intermediate positions of the cutter are selected for down milling and

M. Wan, W.H. Zhang / Journal of Materials Processing Technology 171 (2006) 156 165 163 Fig. 7. Form errors in the middle position in up milling. up milling, respectively. The iteration curves in Fig. 8 correspond to the position with u 1,35 and with u 1,2 when the cutter has a rotation of 86130.7443 and 86075.2895, respectively. We can see that the iteration scheme is very efficient. The correction factor is stabilized after a few iterations. The second test corresponds to Case 2 studied in Ref. [6]. It is concerned with the numerical simulation of the down milling process. The plate is now titanium alloy (Ti6Al4V) with an initial thickness of 2.45 mm smaller than the previous one. Now, the length of the workpiece in the feed direction becomes 63.5 mm. The helical fluted end mill is a singlefluted carbide end mill with a diameter of 19.05 mm, helix angle of 30 and tool gauge length of 55.6 mm. The cutter is discretized into 56 axial elements. Young s moduli of the cutter and workpiece are 620 and 110 GPa, respectively. The nominal axial cutting depth is R z =34mmandα 1,0 = π. The nominal radial cutting depth is R r = 0.65 mm. Detailed data can be found in Ref. [6]. In this case, due to the decrease of the plate thickness, the plate becomes very flexible so that the two cutting force Fig. 9. Form errors in the middle position. (a) Predicted values by different models. (b) Predicted and experimental results from Ref. [6]. Fig. 10. Iteration history of correction factor u 1,29. models are not consistent any more. As shown in Fig. 9(a), considerable differences exist for form error predictions in the position where the cutter feeds 35 mm. Compared with the experiment results given in Fig. 9(b), the flexible model is shown to be the most reliable one because the material removal is not negligible. Likewise, iteration histories of correction factor u 1,29 are shown in Fig. 10 when the cutter has a rotation of 87.0062. 5.2. Down milling of a curved surface Fig. 8. Iteration history of correction factors. As shown in Fig. 11, the workpiece has a contour composed of circular arcs and straight-line segments. The bottom

164 M. Wan, W.H. Zhang / Journal of Materials Processing Technology 171 (2006) 156 165 Fig. 11. Down milling of a curved surface. Fig. 13. Iteration history of correction factor u 1,22. of the workpiece is clamped to simplify the modeling of boundary conditions. Due to the curved surface, the feed trajectory has to be designed as a combination of straight line and circular arcs. Here, the outer radius of the circular arc is 50 mm. The length of tool path on the plane part is 100 mm in the feed direction. The initial thickness of the workpiece is 5 mm and α 1,0 = π. Nominal axial and radial cutting depths are 30 and 1 mm, respectively. Other related parameters are chosen to be the same as in the first test. For down milling process, the distribution of form errors is plotted in Fig. 12 by using the flexible model. Due to the large rigidity at the connection zones between the curved surface and plane, form errors are relatively small. Besides, we can see that although the workpiece has a symmetrical geometry along the machining feed direction, the distribution of form errors is not strictly symmetric. Furthermore, the maximum form error produces at the start location instead of the final exit location because the effect of cutting forces is path dependent and not equivalent along the feed direction. Fig. 13 is the iteration history of correction factor u 1,22 for a cutter rotation of 46.7008. 6. Conclusions A general approach is developed to predict static form errors in peripheral milling of thin-walled structures using finite element method. Considering the geometric complexity of the workpiece in practical milling process, the developed method has the advantage of marching irregular finite element meshes for workpiece discretization and the flexibility of modeling the cutter and the workpiece, independently. Numerically, to ensure the reliability and the computing accuracy of the developed method, key algorithms are devised for cutting force modeling, identification of engaged cutter elements with the workpiece, iterative corrections of both radial and axial cutting depths with correction factors as well as the updating of the workpiece s rigidity without remeshing. By means of numerical examples, results show that cutting forces and form errors evaluated by the developed method march well the available experimental data. The integration of all developed key algorithms with the commercial finite element software SAMCEF provides a basic simulation tool for the peripheral milling of thin-walled workpieces. Acknowledgements This work is supported by the Doctorate Creation Foundation of Northwestern Polytechnical University (Grant No. CX200411), the Natural Science Foundation of Shaanxi Province (Grant No. 2004E 2 17) and the National Natural Science Foundation of China (Grant No. 50435020). References Fig. 12. Overall form errors predicted by the flexible model. [1] S. Smith, J. Tlusty, An overview of modeling and simulation of the milling process, ASME J. Eng. Ind. 113 (1991) 169 175. [2] W.A. Kline, R.E. DeVor, I.A. Shareef, The prediction of surface accuracy in end milling, ASME J. Eng. Ind. 104 (1982) 272 278. [3] E. Budak, Y. Altintas, Peripheral milling conditions for improved dimensional accuracy, Int. J. Mach. Tools Manuf. 34 (1994) 907 918.

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