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A cutting sequence optimization algorithm to reduce the workpiece deformation in thin-wall machining
Accepted Manuscript Title: A cutting sequence optimization algorithm to reduce the workpiece deformation in thin-wall machining Authors: Jun Wang, Soichi Ibaraki, Atsushi Matsubara PII: DOI: Reference:
S0141-6359(17)30379-3 http://dx.doi.org/doi:10.1016/j.precisioneng.2017.07.006 PRE 6616
To appear in:
Precision Engineering
Received date: Revised date: Accepted date:
23-3-2017 25-5-2017 22-6-2017
Please cite this article as: Wang Jun, Ibaraki Soichi, Matsubara Atsushi.A cutting sequence optimization algorithm to reduce the workpiece deformation in thin-wall machining.Precision Engineering http://dx.doi.org/10.1016/j.precisioneng.2017.07.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Type of contribution: Original paper
A cutting sequence optimization algorithm to reduce the workpiece deformation in thin-wall machining
Jun Wang a, Soichi Ibaraki b*, Atsushi Matsubara a
(a) Department of Micro Engineering, Kyoto University, Katsura Nishikyo-ku, Kyoto, 615-8540 Japan (b)
Department
of
Mechanical
Systems
Engineering,
Hiroshima
University,
Kagamiyama,
Higashi-Hiroshima, 739-8511 Japan
* Corresponding author; Soichi Ibaraki, Tel.: +81 0824247580; fax: +81 0824247580; E-mail address: [email protected] (S. Ibaraki)
Highlights
End milling of workpiece of lower stiffness is often subject to larger machining error.
Optimization of block removal reduces deformation in thin-wall milling.
No division of blocks in the feed direction avoids cutter marks and severe vibration.
Sub-division of blocks in the tool’s axial direction reduces deformation further.
Abstract A thin-wall part of lower stiffness can be subject to significant deformation during its cutting process. This study proposes a cutting process optimization algorithm to reduce the workpiece deformation. First,
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the volume to be removed is divided into a set of blocks. The proposed algorithm starts from the finished workpiece shape, with all the blocks removed. The objective of the proposed algorithm is to find a sequence of adding the blocks, such that the workpiece deformation is always smaller than the given threshold value when the cutting forces is imposed at each step. The workpiece deformation at each step is simulated by using the FEM (finite element method) simulation. By inverting the sequence of adding the blocks, the optimized sequence to remove the blocks can be obtained. Additionally, the block size can be modified to reduce the axial depth of cut to further reduce the workpiece deformation, or to increase the radial depth of cut to enhance the efficiency. Experiments are conducted to confirm the effectiveness of the algorithm to reduce the maximum workpiece deformation during the entire cutting process.
Keywords Thin wall, cutting process, tool path, displacement, CAM (computer-aided manufacturing)
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Introduction Machining a workpiece of lower rigidity is often difficult because its deformation or vibration may
occur due to the cutting force. A typical example is a thin-walled structure. When it is machined by side milling with a contour-parallel tool path, the workpiece’s stiffness is gradually reduced as the material removal proceeds. When the workpiece’s thickness reaches several millimeters, its deflection may increase significantly. To address such an issue in thin-wall machining, many strategies have been investigated, which can be categorized as follows. (1) Optimization of cutting conditions [1] or tool geometries [2] (2) Optimization of the cutting path [3–5] (3) Fixture properly designed [6–8] (4) Simultaneous milling by two spindles from both sides of a thin-wall workpiece [9]
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The approaches (3) and (4) are promising to improve the productivity. However, they require special devices or know-hows. The approaches (1) and (2) may be preferred for the feasibility. The prediction of the workpiece deformation using finite element analysis, as well as the modelling of cutting forces, have been advanced [10–12], which enhances the feasibility of the approach (1). In such an approach, if the cutting path is not properly designed, the cutting depth and the feed rate are often reduced, leading to the cutting efficiency significantly reduced. The experience-based “8:1 rule” has been used in the machining process of aircraft components [13]. According to this rule, to finish a thin wall of width 1 mm by side milling, the axial depth of cut must be 8 mm or smaller. Another empirical way is based on the cutting path design to make "sacrifice" ribs in the workpiece stock that will be finally removed. Smith et al. [14] proposed a tool path design making sacrifice ribs to support a thin-wall workpiece of an aircraft component. In this design, the tool path always leaves a supporting part in the side opposite to the surface to be machined. These cutting processes planning schemes are based on experience, where a solution is obtained in try-and-error manner. For the cutting process planning, automatic programming software or CAM software are widely available. The automatic programming software is equipped on a CNC controller of a machine tool, which generates a cutting path automatically when the machine operator selects features, path patterns, cutting tools, and conditions. Using the CAM software, a cutting path is also automatically generated when appropriate cutting conditions are input. The tool path design is based only on the workpiece’ final geometry. When the rigidity of the workpiece is lower, the displacement of the cutting point should be considered. However, such a tool path design, taking the change in the workpiece rigidity into consideration, is difficult. The authors' group proposed a tool path design algorithm that can minimize the workpiece displacement at the cutting point [4,5]. This algorithm employs the reversed design process. First, the entire stock to be removed is discretized as a set of blocks. The algorithm starts from the finished
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workpiece shape, with all the blocks removed. At each step, a block is added to the location where the workpiece displacement due to the cutting force is minimized. Finally, the sequence to add the blocks is reversed to obtain the removal process. Although our previous works [4, 5] showed a reasonable material removal process designed by the proposed scheme, it cannot assure the maximum workpiece displacement over the entire process is minimized. Furthermore, the productivity of the machining process with the calculated path can be significantly lower. To obtain more practical removal paths, this paper presents an improved algorithm to optimize the block removal sequence for the application to thin-wall milling.
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Algorithm to remove blocks to minimize the workpiece deformation
2.1 Previously proposed algorithm Koike et al. [4,5] proposed an algorithm that generates the cutting sequence by adding materials in a reversed manner from a final workpiece shape to initial unprocessed workpiece shape. This subsection briefly reviews this algorithm by showing an example. Consider the workpiece shown in Fig. 1(a) machined by removing the blocks B1 to BNb (Nb = 12 in this example). Each block may represent either a volume removed by a single cut (see Fig. 2(a)) or that cut by a repetitive contour-parallel path (see Fig. 2(b)). Suppose that the displacement of the workpiece at the top edge of the i-th block is represented by a function d(Bi, f, ), when the force, f, is given to the block Bi. represents the geometry of the workpiece with remaining blocks. For example, when the blocks B 10, B11, B12 are not removed yet, can be written as ={ B10, B11, B12 } (see Fig. 1(b)). Throughout this paper, d(Bi, f, ) is calculated by the FEM (finite element method). This paper only discusses the displacement normal to the thin-wall surface. The optimization of block removal sequence proposed by Koike et al. [4,5] can be summarized as follows: (1) The volume to be removed is divided into blocks, Bi (i = 1, …, Nb). The blocks are given in priori.
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(2) The algorithm starts from the finished workpiece geometry with all the blocks removed, i.e. k = {null}, where k=0. (3) Calculate the displacement, d(Bi, f, k), for all the “candidate” blocks, B i. For example, in Fig. 1(b), when k is null, blocks 4, 5, 6, 10, 11, 12 can be candidates. (4) Choose ik that minimizes d(Bik , f, k). Update k to k+1 by adding the block Bi to k. (5) kk+1 and repeat (3)-(4) until all the blocks are added.
The basic idea behind this algorithm can be described as follows: this algorithm aims to reduce the largest deformation by modifying the sequence to remove the blocks. This objective can be written as:
min max 𝑑(B𝑖𝑘 , 𝑓, 𝑘 ) {ik }
𝑘
(1)
By choosing the block to minimize the displacement at every cutting step in Step (4), a block is potentially added to more stiff positions, which in turn will support the blocks at subsequent steps. As a result, the less stiff blocks will have higher stiffness.
2.2 Issues with the previous algorithm The issues with previously proposed algorithm are as follows: (a) The objective in Eq. (1) is not explicitly taken into consideration in this algorithm, and thus the removal sequence is not guaranteed to achieve the objective in Eq. (1). It is just an ad-hoc way to obtain the removal sequence to achieve the objective in Eq. (1). (b) In this algorithm, all the blocks have the same size. Clearly, the deformation of the workpiece can be reduced by reducing the axial depth of cut or the radial depth of cut. It can be done by changing the size of some blocks. (c) This algorithm usually generates cutting paths that are not parallel to the finished workpiece surface
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(see an example to be presented in Section 3). It results in more air-cut, which could significantly reduce the cutting efficiency. Furthermore, when the tool’s feed direction is changed in the finishing path, it often leaves an abrupt cutter mark on the finished workpiece surface.
2.3 Proposed algorithm (a) Main algorithm for optimization of block removal sequence To address the issues discussed in Section 2.2, the algorithm is improved. The objective of the proposed algorithm is to find the sequence of removing the blocks B1 to BNb such that the workpiece displacement for all removal step, d(Bik, f, k), is always less than the given threshold value, dc, for any step k. Namely, the objective can be written as:
Find {𝑖𝑘 }𝑘=1,⋯,𝑁𝑏 such that 𝑑(B𝑖𝑘 , 𝑓, 𝑘 ) < 𝑑𝑐 for any 𝑘
(2)
The proposed algorithm is as follows: (a-1)
The volume to be removed is divided into blocks, Bi (i=1, …, Nb). The blocks are given in priori.
(a-2)
The algorithm starts from the finished workpiece geometry with all the blocks removed, i.e. k = {null} where k = 0.
(a-3)
Calculate the displacement, d ( Bi, f, k ), for all the “candidate” blocks, Bi.
(a-4)
Find all ik that meets d ( Bik , f, k ) < dc. If there exist multiple ik, then add them all. Their sequence can be arbitrarily chosen. Update k to k+1 by adding all the blocks Bik..
(a-5)
If there exists no ik, it means that there is no such block sequence that achieves the objective in Eq. (2). The calculation finishes without any solution. If it is acceptable to change the axial depth of cut to reduce the workpiece deformation, go to the algorithm (b).
(a-6)
kk+1 and repeat (a-3)-(a-5) until all the blocks are added.
(a-7)
After this algorithm is finished, the blocks can be modified to increase the radial depth of cut for
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enhancing the machining efficiently. Go to the algorithm (c).
Unlike the past algorithm in Section 2.1, this improved algorithm guarantees that the workpiece displacement at each removal step, d ( Bi, f, k ), is always smaller than the threshold, dc. To obtain the minimum dc that can be achieved, the algorithm (a) should be iterated with reducing dc until there exists no solution. This iterative calculation gives a solution to the problem in Eq. (2). The proposed algorithm addresses the issue (a) in Section 2.2.
(b) Sub-division of blocks to reduce the axial depth of cut When there exists no block satisfying d ( Bik , f, k ) < dc in Step (a-5) in the algorithm (a), candidate blocks can be divided to reduce the axial depth cut: (b-1) All the candidate blocks, Bi, are divided into two sub-blocks in the axial direction. Denote them as B i1 (lower) and Bi2 (upper) (see Fig. 3). (b-2) Find all i1k from lower sub-blocks that meets d(Bi1k , f, k) < dc. If there exist multiple i1k, then add them all. Their sequence can be arbitrarily chosen. Update k to k+1 by adding all the blocks Bi1k.. (b-3) If lower sub-block B i1 is added at the k-th step, then upper sub-block B i2 will be in the candidate blocks in the (k+1)-th step. (b-4) Go back to the algorithm (a).
By sub-dividing each block, the axial depth of cut is reduced, which naturally reduces the workpiece deformation. This algorithm is added to address the issue (b) discussed in Section 2.2.
(c) Post-optimization modification to increase the radial depth of cut
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In contrast to the case in (b), when the displacement at block Bik, d ( Bik , f, k ), is significantly smaller than the threshold dc, two blocks adjacent to each other in the direction normal to the feed direction can be combined, as illustrated in Fig. 4. It doubles the radial depth cut, and then enhances the machining efficiency. Here it is assumed that each block is removed by a single cut (see Fig. 2(a)). This can be done after the main algorithm (a) is finished as a post-optimization modification of the designed removal sequence. This modification is also to address the issue (b) discussed in Section 2.2.
(d) Design of blocks to be removed To apply the algorithm (a), the removed volume must be divided into the blocks in priori. It must be designed properly, taking the machining process into consideration. As discussed in Section 2.2 (c), when the tool’s feed direction is changed in the finishing path, it often leaves an abrupt cutter mark on the finished workpiece surface (see the case study to be presented in Section 3). When it is not acceptable, the blocks should not be divided in the feeding direction, as illustrated in Fig. 5. Then, the tool path will be parallel to the finished workpiece surface. The same algorithm (a) can be applied to determine the removal sequence. This addresses the issue (c) in Section 2.2.
Remark 1: application to workpieces of free-form surface Throughout this paper, as illustrated in Fig. 1(a), the finished workpiece is a thin vertical wall and the removed stock is divided into rectangular blocks. This simplification is to illustrate the algorithm, and this paper’s basic concept can be applied to workpieces of arbitrary free-form surface. Figure 6 illustrates such an example. The geometry of each block is designed by shifting the finished workpiece’s surface. Each block can be removed by contour-parallel or direction parallel tool paths.
Remark 2: calculation of cutting force and workpiece deformation
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In end milling processes, the instantaneous cutting force varies with tool rotation. Furthermore, within each block, since the workpiece’s stiffness varies as the tool is fed, the cutting force may vary. In the algorithm (a), the cutting force when removing the block Bi is represented by a single value, f. This is to simplify the calculation of d ( Bi, f, k ), and this value must be chosen carefully. Figure 7 (a) illustrates the side milling with a helix mill. Figure 7 (b) shows an example of the measured cutting force profile in the direction normal to the feed direction. As has been discussed in well-established cutting mechanism models, e.g. [15], a cutter tooth immerses into workpiece at position P1, which corresponds to the force value F1, and it exits at position P2 which corresponds to the force F2 . To predict the deformation at the point shown in Fig. 1 (b), f = F2 is used in the calculation of d (Bi, f, k ). In the following case study, the value of f was determined by a cutting test. Within each block, the workpiece’s deformation can vary due to the change in the remained stock’s stiffness. This can be particularly significant when blocks are not divided in the feed direction as shown in Fig. 5. In such a case, the suggested procedure to calculate the deformation, d(Bi, f, k ), is as follows: sub-divide the block Bi in the feed direction as shown in Fig. 8 (denote sub-blocks by Bij). Then, calculate the deformation d ( Bij, f, k ) at each sub-block. Take the maximum deformation as the representative deformation at the block Bi, i.e. 𝑑 ( B𝑖 , 𝑓, 𝑘 ) = max(𝑑( B𝑖𝑗 , 𝑓, 𝑘 )). 𝑗
For the simplification, this paper assumes that the workpiece deformation during the cutting, d(Bi, f,
k ), is copied as the thickness error of the finished workpiece, as shown in Fig. 9. In other words, when the finished workpiece’s thickness at the top edge of the block Bi is denoted by t(Bi), it can be represented by
𝑡(Bi ) = 𝑡 ∗ (Bi ) + 𝑑( B𝑖 , 𝑓, 𝑘 )
(3)
where t*(Bi) represents the nominal thickness. For a vertical thin wall, it is a constant value for any blocks.
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3
Case study
3.1 Test objective and common conditions The objective of the cutting experiments presented in this section is to study the effectiveness of the proposed tool path design schemes in reducing the geometric error of the machined thin wall workpiece. The geometry of the initial workpiece (part blank) and the finished thin-wall workpiece are shown in Fig. 10. A square end mill is used. The tool's axial direction is in the Z direction. The cutting conditions are shown in Table 1. The radial depth of cut, Rdn, and the axial depth of cut, Adn, shown in Table 1 are “nominal” values adopted in Strategies 1 and 2. They may be modified in Strategy 3 (further details will be given in Section 3.2). The cutting force was measured in advance by using a dynamometer in a cutting test under the cutting conditions shown in Table 1. The workpiece was sufficiently thick such that the influence of its deformation was negligible. The cutting force corresponding to the tool engagement’s top edge point (see Fig. 7) was measured as 𝐹 = (𝐹𝑥 , 𝐹𝑦 , 𝐹𝑧 ) = (−9, 20, 3)N. This value is used throughout the tool path design to be presented in Section 3.2. As the machining process proceeds, the remained workpiece becomes gradually thinner, which likely changes the cutting force. This change is, however, considered minor and ignored in the tool path design. The authors’ previous work [16] presented the modelling of the influence of the workpiece’s thickness on the cutting force.
3.2 Tool path design strategies The following three tool path design strategies are compared: (1) Strategy 1: Conventional tool path: The conventional contour-parallel path shown in Fig.11 is used. The radial depth of cut is 0.2 mm (see Table 1), which means total 20 paths in the X-direction. The axial depth of cut is 15 mm (see Table 1),
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which means total 3 layers in the Z-direction. The feed direction is parallel to the finished workpiece’s surface (from Y- to Y+). The entire stock is removed by total 60 paths.
(2) Strategy 2: Tool path designed based on the block removal sequence optimized by the algorithm (a): The block removal sequence is optimized by using the algorithm (a) in Section 2.3. The volume to be removed is divided into 5 blocks in the feed (Y) direction, 3 blocks in the axial (Z) direction, and 4 blocks in the direction normal to the feed direction (X) (refer to Fig. 13). In other words, the size of each block is 12015 mm (XYZ). The radial depth of cut is 0.2 mm as shown in Table 1. In other words, each block is removed by 5 repetitive contour-parallel paths (see Fig. 2(b)). The algorithm (a) was iteratively performed, reducing the threshold value, dc, from 0.1 mm to 0.06 mm. To calculate the workpiece deformation d ( Bi, f, k ) in step (a-3) in the algorithm (a), the FEM model was built in COMSOL and the calculation was run with MATLAB. Figure 12 shows the block removal sequence calculated by applying the algorithm (a) for the case of dc = 0.08 mm. The circled numbers represent each block, and the blocks are removed from 1 to 60. Figure 13 illustrates the block removal sequence. There was no solution for dc = 0.06 mm. Remark: To enhance the efficiency, when there are two adjacent blocks to be removed consequentially, they are removed altogether as shown in Fig. 14, instead of removing them one by one.
(3) Strategy 3: Tool path designed with no block division in the feed direction ((d) in Section 2.3) and with the sub-division of blocks (b) and the post-optimization modification (c) For the reason discussed in (d) in Section 2.3, the volume to be cut is not divided in the feed direction as shown in Fig. 15(a). The algorithm (a) is applied to 12 blocks. In the application of the algorithm (a), the maximum displacement within each block is calculated by sub-dividing it into five blocks and
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calculating the deformation at each sub-block, as discussed in Remark 2 of Section 2.3(d). In the application of the algorithm (a), the target maximum deformation was set at dc = 0.04 mm. This is lower than in Strategy (2). To achieve it, the sub-division of blocks to reduce the axial depth of cut, presented in the algorithm (b) in Section 2.3, and the post-optimization modification to increase the radial depth of cut, presented in (c) in Section 2.3, were applied. Figure 15(b) shows the designed block removal sequence. The following combinations of cutting depths are included in Fig. 15(c): 1) Ad = 3.75mm, Rd = 0.2 mm, 2) Ad = 7.5 mm, Rd = 0.2 mm, and 3) Ad = 15 mm, Rd = 0.4 mm; 4) Ad = 15 mm, Rd = 0.2 mm.
3.3 Comparison in simulated workpiece displacement profiles Figure 16 shows the simulated displacement histories for Strategies 1 to 3. The horizontal axis for Strategy 1 represents the number of removed layers. The horizontal axis for Strategies 2 and 3 represent the index number of blocks shown in Figs. 12 and 15. The vertical axis shows the simulated workpiece deformation, d(Bik , f, k) (calculated in Step (a-3) in the algorithm (a)). Figure 17 shows the predicted thickness profiles of the finished workpiece, calculated by Eq. (3), at lines L 1 to L3 (see Fig. 19 for the location of L1 to L3). They are calculated by using the same FEM model as the one used in the algorithm (a). For Strategy 2 in Fig. 16, it can be observed that the simulated workpiece deformation, d(Bik , f, k), is smaller than the given threshold, dc = 0.08 mm for all the blocks. In Strategy 3, the simulated deformation is smaller than dc = 0.04 mm for all the blocks.
3.4 Experimental setup The part blank shown in Fig. 10(a) was machined with Strategies 1 to 3. The experimental setup is shown in Fig. 18. A five axis machining center, NMV3000 DCG by DMG MORI Co., Ltd., and a straight end mill, CA-MFE14 by OSG Corp., were used. The specifications of the tool and cutting conditions are
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shown in Table 1. Measurement of the workpiece displacement during the cutting operation: During the cutting process, the workpiece displacement was continuously measured by a triangulation-based laser displacement sensor (LK-G10 by Keyence Corp.) at three positions P1, P2, and P3 as shown in Fig. 19. Measurement of the finished workpiece’s thickness profile: The thickness profiles of the finished workpieces were also measured on-the-machine. An interferometry-based laser displacement sensor (SI-F10 by Keyence Corp.) was used. The sensor mounted on the spindle scans the surface of a gauge block and the finished workpiece surface from both sides, as shown in Fig. 20. The thickness of the block gauge was pre-calibrated. By the measured profiles on the finished workpiece, D1 and D2 in Fig. 20, and those on the gauge block, d1 and d2, the thickness of the finished workpiece, T, can be measured. X1, X2, x1, x2 are spindle positions in X in the machine tool coordinates. Three lines on the finished workpiece, L1 to L3 in Fig. 19, were scanned by feeding the sensor to the Y-direction.
3.5 Measured and thickness profiles of the finished workpieces Figure 21 shows the thickness profiles of the finished workpieces measured as shown in Section 3.4. In Fig. 21(a), three profiles measured at the lines L1 to L3 are plotted in the 3D view, and they are compared between Strategies 1 to 3. In Fig. 21(b-1) to (b-3), they are presented in the XY plane for clearer comparison. In Fig. 21(b-1) to (b-3), the workpiece’s displacements (the values are added with nominal thickness 1 mm) when the tool pass the measured points P1 to P3 (in the opposite side) in the finishing cut are also shown by the marks ●▲■. The displacement in Strategy 2 is missing in Fig. 21(b-3) because severe chatters happened there. An example displacement profile measured at P 1 in the finishing process in Strategy 1 is shown in Fig. 22. The tool passes the same Y position as the sensor at 14.4 sec after the finishing process starts. The peak displacement within 0.01 sec around this time is taken and shown in the
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mark '●' in Fig. 21(b) (Strategy 1).
3.6 Discussion on experimental results (1) On the comparison in thickness profiles in Strategies 1 to 3: Figure 21 clearly shows that Strategy 3 minimized the error in the finished workpiece’s thickness profile. From the nominal thickness 1 mm, Strategy 1 resulted in the geometric error at maximum 0.2 mm (see Fig. 21 (b-1)). In Strategy 3, the maximum error was reduced to 0.036 mm. Strategy 2 also reduced the workpiece’s average geometric error significantly compared to Strategy 1. However, the entry and the exit of the tool to the finishing path caused significant “bump” in the surface profile, as can be observed e.g. at Y=20 and 80 mm in Fig. 21(b-1). This is predicted in the discussion in Section 2.3 (d). Furthermore, the workpiece finished by Strategy 2 was partially subject to the chattering (see also (5) in this subsection). In Strategy 3, the thickness profiles are partially below the nominal thickness 1 mm, which means the workpiece is over-cut. This is potentially caused by the workpiece’s vibration.
(2) On the difference between the measured surface profile and the workpiece’s displacement during the cutting: Figure 21 (b-1) to (b-3) also show the workpiece’s displacement measured during the cutting. It can be observed that Strategies 2 and 3 reduced this displacement compared to Strategy 1. However, they are significantly different from the surface profiles of the finished workpieces. This difference can be potentially caused by 1) the tool deflection caused by the cutting force, 2) the measurement error of the laser displacement sensor, including the error caused by the limited measurement bandwidth, and 3) the contribution of the material removal process, including the “spring-back” of the material which typically occurs when the radial depth of cut is smaller.
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(3) On the difference between measured and simulated surface profiles of the finished workpieces: The FEM simulation shown in Fig. 17 predicts that Strategies 2 and 3 can reduce the maximum deformation by about 20 % and 63 %, respectively, compared to Strategy 1. This is similar in actual surface profiles shown in Fig. 21(b-1), where 21% and 82% reduction can be observed. However, the difference between simulated and measured surface profiles are quantitatively large. This difference can be potentially caused by 1) the difference in the cutting force in simulation and actual cutting. In simulation the force was taken as a constant value (see Section 3.1), while in the actual cutting the force varies as was discussed in Remark 2 of Section 2.3 (see also [16]), 2) The contribution of the material removal process, including the “spring-back” of the material, or the tool’s deflection due to the cutting force. In other words, Eq. (3), assuming that the workpiece instantaneous deformation is copied as the thickness error of the finished workpiece, may have significant error. The causes for the estimation error of the finished workpiece’s thickness (the difference between Figs. 17 and 21) are not fully investigated yet. Further investigation on such issues will be planned to build a more accurate simulation model for predicting the workpiece profile.
(4) On the efficiency: The efficiency of three cutting strategies is also compared as shown in Fig. 23. The cutting time is simulated by using a machining simulator with the NC programs used in the experiments. The cutting time for Strategy 2 is longer than Strategy 1 by about 11 min. This is predicted in the discussion in Section 2.3 (d). On the other hand, the cutting time for Strategy 3 is longer only by 2 min.
(5) On the occurrence of the chatter:
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Figure 24 shows the appearance of the finished surfaces. The workpiece by Strategy 2 shows clear chatter marks. It also shows abrupt cutter marks (“bump”) caused by the entry and the exit of the tool to each block. Interestingly, the chatter mark on the Strategy 2 workpiece can be observed on the third (lowermost) layer, where the workpiece stiffness is predicted higher than the first layer. At the 3 rd layer, the chatter seemed to start when the cutter passes this “bump” in higher (2nd or 1st) layers. This can be a critical issue for Strategy 2, as was discussed in Section 2.3 (d).
(6) On the measurement of workpiece displacement during the cutting process In Fig. 22, the workpiece’s displacement profile measured by the triangulation-based laser displacement sensor shows low-frequency periodical variation (its period is about 1.2 sec). This may be caused by the beat of the vibrations of slightly different frequencies, or the aliasing. The workpiece’s displacements shown in Fig. 21 (b-1) to (b-3) may contain the measurement error caused by this (up to 15%).
(7) On the calculation time of block removal sequence The calculation time to obtain the optimal block removal sequence for Strategy 2 was 20 minutes (1229 seconds) while for Strategy 3 it was 97 minutes (5860 seconds). Most of the calculation time was consumed for the calculation of the workpiece displacement, d ( Bi, f, k ) in Step (a-3) in Section 2.3 (a), by the FEM calculation. If this calculation can be simplified by e.g. using an analytical model, then the entire calculation time would be significantly reduced.
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Conclusion The proposed algorithm optimizes the sequence to remove blocks in cutting operations, such that the
maximum deformation of the workpiece in the finishing process is minimized (Strategy 2 in Section 3). If
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the blocks are divided in the feed direction, the tool must change its direction on the finishing path to remove each block. This may cause abrupt cutter marks or severe vibration. To address this issue, Strategy 3 does not divide the blocks in the feed direction, which gives the tool path parallel to the finished workpiece surface. To reduce the workpiece deformation, sub-division of the blocks in the tool’s axial direction can be done to reduce the axial depth of cut. The post-optimization modification can increase the radial cutting depth to improve the cutting efficiency. The effectiveness of the proposed tool path design strategies in reducing the geometric error of the finished thin-wall workpiece was studied by cutting experiments.
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http://www.malinc.com/wp-content/uploads/2014/05/CutPro_Guide.pdf
[14]
Smith S, Wilhelm R, Dutterer B, Cherukuri H, Goel G. Sacrificial structure preforms for thin part machining. CIRP Ann - Manuf Technol 2012;61:379–82.
[15]
Budak E, Altintas Y. Modeling and avoidance of static form errors in peripheral milling of plates. Int J Mach Tools Manuf 1995;35:459–76.
[16]
Wang J, Ibaraki S, Matsubara A, Shida K, Yamada T. FEM-Based Simulation for Workpiece Deformation in Thin-Wall Milling. Int J Autom Technol 2015;9:122–8.
19
Displacement d (B5 , f, Γ )
B4
Force f
B5
B1
B6 B2 B3
B10
B7
B11
B8
B12
B9 (a)
(b)
Γi = { B10 , B11 , B12 }
Fig. 1. An example problem setup: (a) the volume to be removed is divided into a set of blocks, B1 to B12, (b) the definition of the simulated workpiece displacement, d(Bi, f, k).
20
Tool
(a)
(b)
Fig. 2. Possible removal of each block. The tool path to remove each block can be designed arbitrarily: (a) a block removed by a single cut; (b) a block removed by a repetitive contour-parallel path.
21
B42 B62
B41
B61
Fig. 3. Sub-division of blocks when there exists no candidate block that can be added with d ( Bik , f, k)
< dc
22
Fig. 4. Post-optimization modification to increase the radial depth of cut
23
Feeding
Fig. 5. Blocks are not divided in the feeding direction
24
Thin wall
Fig. 6. Example block division for a free-form surface workpiece
25
Tool P2 P1
Last cut-layer of block
F1
F2
60
Simulation Experimental
50 40
Cutting force (N)
Block
30 20 10 0 -10 -20
(a)
Rotation angle (104 deg)
(b)
Fig. 7. Cutting force value used in simulation: (a) geometry of milling; (b) force pattern
26
d( Bi5, f, ) Bi5
d( Bi1, f, )
Bi1
Bi
Fig. 8. Sub-division of a block to calculate the maximum deformation in the block Bi
27
End mill
Deformation d( Bi, f, Γk )
Nominal thickness t*(Bi)
Predicted Thickness t(Bi)
Workpiece
(a)
(b)
Fig.9. The relationship of the deformation in finishing cut and the thickness of the finished workpiece: (a) the deformation in the cutting; (b) the finished workpiece
28
5
1
unit : mm
100
45
65
z (a)
y x (b)
Fig. 10. Geometry of workpiece: (a) initial shape; (b) final shape
29
z
y x
(a)
(b)
(c)
(d)
Fig.11 Strategy 1: conventional contour-parallel path: (a) 2 layers are removed; (b) 20 layers are removed; (c) 22 layers are removed; (d) 60 layers are removed
30
Block layer in Z
Sequence when e080 22 17 21 15 1210 7 20 14 16 5 2 11 13 6 8 9 4 1 3 50 4240 32 49 30 2725 19 41 4831 38 39 29 26 2818 23 24 59 6054 55 58 46 4736 37 56 5751 52 53 45 43 4433 34 35 z y x
Block layer Y mm; Fig. 12. The calculated block removal sequence of Strategy 2, dc =in 0.08 Block layer in X
31
z
y x
(a)
(b)
(c)
(d)
Fig. 13. Strategy 2: the block removal sequence designed by the proposed algorithm. The sequence corresponds to Fig. 12: (a) 1st to 2nd blocks are removed; (b) 1st to 19th blocks are removed; (c) 1st to 42nd blocks are removed; (d) 1st to 60th blocks are removed.
32
block to be removed after NO. k block NO. k
Remove two blocks together
Fig. 14. Example of adjacent blocks that are removed together
33
z
Ad = 3.75 mm Rd = 0.2mm
y x 4
Ad = 15 mm Rd = 0.2mm
5
6
32
10
1
98
7
11
Ad = 7.5 mm Rd = 0.2mm
15
16 14
(a)
13
12
(b)
(c)
Ad = 15 mm Rd = 0.4mm
Fig. 15. Strategy 3: tool path designed with no block division in the feed direction: (a) initial block division; (b) finished block division with removal sequence index; (c) the axial depth of cut, Ad, and the radial depth of cut, Rd, for each block
34
Fig. 16. Simulated workpiece displacements at each block in Strategies 1 to 3
35
Strategy 1
Strategy 2
Strategy 3
1.1
20%
(a)
63%
1.05
Thickness mm
1 0
20
40
60
80
100
20
40
60
80
100
20
40
60
80
100
1.1 (b) 1.05 1 0 1.1 (c) 1.05 1 0
Y mm
Fig. 17. Simulated thickness profiles of the finished workpiece at lines L1 to L3: (a) at the line L1; (b) at the line L2; (c) at the line L3
36
Laser sensor LK-G10 Tool
Z Workpiece
Y X
Fig. 18. Experimental setup
37
10 P1 P2
1 L1
16 31
P3
L2 L3 z
unit: mm y
Fig. 19.Positions on the workpiece’s surface measured by the laser displacement sensor (P 1 to P3) and the scanned lines to measure the finished workpiece’s thickness profiles (L1 to L3)
38
X2
X1 T D2
lasor sensor SI-F10
D1
workpiece d2 d1
Z X
x2 gauge block
t
Spindle P
x1
Fig. 20. Setup for the thickness error measurement of finished workpieces
39
Strategy 1 Strategy 2 Strategy 3
1.2 0 1.15
L1 Thickness mm
Z mm
-10
-20
L2 -30
-40
1
1
0
10
Y mm
Strategy 1 Strategy 2 Strategy 3
30
40 50 60 Y coordinate mm
70
80
90
70
80
90
Strategy 1 Strategy 2 Strategy 3
1.15
1.15
1.1
Thickness mm
Thickness mm
20
(b-1)
Thickness mm
1.2
Displacement in cutting
100
50
1.1 1.2
(a)
82% 1.1
1.05
Strategy 1 Strategy 2 Strategy 3
L3
0.9
21%
1.1
Displacement in cutting
Displacement in cutting 1.05
1
1.05 0.95 1
0.9 10
(b-2)
20
30
40 50 60 Y coordinate mm
70
80
90
10
(b-3)
20
30
40 50 60 Y coordinate mm
Fig. 21. Measured thickness profiles of the finished workpieces. The marks, ●▲■, represents the workpiece displacement measured during the cutting in Strategies 1 to 3 (at the points P 1 to P3 in Fig. 19): (a) 3D view; (b-1) at the line L1; (b-2) at the line L2; (b-3) at the line L3
40
70 16 s
60 Peak value at 14.4s from cutting starts.
Displacement m
50 40
Cutting finishes
30
Cutting starts
20 10 0 -10
6
8
10
12
14 Time s
16
18
20
22
Fig. 22. The workpiece displacement measured at the point P 1 during the finishing process in Strategy 1 (an example).
41
34 min
25 min
23 min
Strategy 1
Strategy 2
Strategy 3
Fig. 23. The overall cutting time in three strategies
42
Boundaries in Y direction
Boundaries between layers in Z “bump” Z Y (a)
(b)
(c)
Fig. 24. Appearances of finished workpiece: (a) Strategy 1; (b) Strategy 2; (c) Strategy 3
43
Table 1
Cutting conditions
Radial depth of cut (nominal), Rdn
0.2 mm
Axial depth of cut (nominal), Adn
15 mm
Cutting direction
up-cut
Spindle speed
2275 min-1
Feed per tooth
0.055 mm
Material of workpiece
Aluminum alloy, JIS A5052
Cutting fluid
None
Tool diameter Φ
14 mm
Tool helix angle β
40°
Tooth number z
3
Tool flute length
21 mm
44
S0141-6359(17)30379-3 http://dx.doi.org/doi:10.1016/j.precisioneng.2017.07.006 PRE 6616
To appear in:
Precision Engineering
Received date: Revised date: Accepted date:
23-3-2017 25-5-2017 22-6-2017
Please cite this article as: Wang Jun, Ibaraki Soichi, Matsubara Atsushi.A cutting sequence optimization algorithm to reduce the workpiece deformation in thin-wall machining.Precision Engineering http://dx.doi.org/10.1016/j.precisioneng.2017.07.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Type of contribution: Original paper
A cutting sequence optimization algorithm to reduce the workpiece deformation in thin-wall machining
Jun Wang a, Soichi Ibaraki b*, Atsushi Matsubara a
(a) Department of Micro Engineering, Kyoto University, Katsura Nishikyo-ku, Kyoto, 615-8540 Japan (b)
Department
of
Mechanical
Systems
Engineering,
Hiroshima
University,
Kagamiyama,
Higashi-Hiroshima, 739-8511 Japan
* Corresponding author; Soichi Ibaraki, Tel.: +81 0824247580; fax: +81 0824247580; E-mail address: [email protected] (S. Ibaraki)
Highlights
End milling of workpiece of lower stiffness is often subject to larger machining error.
Optimization of block removal reduces deformation in thin-wall milling.
No division of blocks in the feed direction avoids cutter marks and severe vibration.
Sub-division of blocks in the tool’s axial direction reduces deformation further.
Abstract A thin-wall part of lower stiffness can be subject to significant deformation during its cutting process. This study proposes a cutting process optimization algorithm to reduce the workpiece deformation. First,
1
the volume to be removed is divided into a set of blocks. The proposed algorithm starts from the finished workpiece shape, with all the blocks removed. The objective of the proposed algorithm is to find a sequence of adding the blocks, such that the workpiece deformation is always smaller than the given threshold value when the cutting forces is imposed at each step. The workpiece deformation at each step is simulated by using the FEM (finite element method) simulation. By inverting the sequence of adding the blocks, the optimized sequence to remove the blocks can be obtained. Additionally, the block size can be modified to reduce the axial depth of cut to further reduce the workpiece deformation, or to increase the radial depth of cut to enhance the efficiency. Experiments are conducted to confirm the effectiveness of the algorithm to reduce the maximum workpiece deformation during the entire cutting process.
Keywords Thin wall, cutting process, tool path, displacement, CAM (computer-aided manufacturing)
1
Introduction Machining a workpiece of lower rigidity is often difficult because its deformation or vibration may
occur due to the cutting force. A typical example is a thin-walled structure. When it is machined by side milling with a contour-parallel tool path, the workpiece’s stiffness is gradually reduced as the material removal proceeds. When the workpiece’s thickness reaches several millimeters, its deflection may increase significantly. To address such an issue in thin-wall machining, many strategies have been investigated, which can be categorized as follows. (1) Optimization of cutting conditions [1] or tool geometries [2] (2) Optimization of the cutting path [3–5] (3) Fixture properly designed [6–8] (4) Simultaneous milling by two spindles from both sides of a thin-wall workpiece [9]
2
The approaches (3) and (4) are promising to improve the productivity. However, they require special devices or know-hows. The approaches (1) and (2) may be preferred for the feasibility. The prediction of the workpiece deformation using finite element analysis, as well as the modelling of cutting forces, have been advanced [10–12], which enhances the feasibility of the approach (1). In such an approach, if the cutting path is not properly designed, the cutting depth and the feed rate are often reduced, leading to the cutting efficiency significantly reduced. The experience-based “8:1 rule” has been used in the machining process of aircraft components [13]. According to this rule, to finish a thin wall of width 1 mm by side milling, the axial depth of cut must be 8 mm or smaller. Another empirical way is based on the cutting path design to make "sacrifice" ribs in the workpiece stock that will be finally removed. Smith et al. [14] proposed a tool path design making sacrifice ribs to support a thin-wall workpiece of an aircraft component. In this design, the tool path always leaves a supporting part in the side opposite to the surface to be machined. These cutting processes planning schemes are based on experience, where a solution is obtained in try-and-error manner. For the cutting process planning, automatic programming software or CAM software are widely available. The automatic programming software is equipped on a CNC controller of a machine tool, which generates a cutting path automatically when the machine operator selects features, path patterns, cutting tools, and conditions. Using the CAM software, a cutting path is also automatically generated when appropriate cutting conditions are input. The tool path design is based only on the workpiece’ final geometry. When the rigidity of the workpiece is lower, the displacement of the cutting point should be considered. However, such a tool path design, taking the change in the workpiece rigidity into consideration, is difficult. The authors' group proposed a tool path design algorithm that can minimize the workpiece displacement at the cutting point [4,5]. This algorithm employs the reversed design process. First, the entire stock to be removed is discretized as a set of blocks. The algorithm starts from the finished
3
workpiece shape, with all the blocks removed. At each step, a block is added to the location where the workpiece displacement due to the cutting force is minimized. Finally, the sequence to add the blocks is reversed to obtain the removal process. Although our previous works [4, 5] showed a reasonable material removal process designed by the proposed scheme, it cannot assure the maximum workpiece displacement over the entire process is minimized. Furthermore, the productivity of the machining process with the calculated path can be significantly lower. To obtain more practical removal paths, this paper presents an improved algorithm to optimize the block removal sequence for the application to thin-wall milling.
2
Algorithm to remove blocks to minimize the workpiece deformation
2.1 Previously proposed algorithm Koike et al. [4,5] proposed an algorithm that generates the cutting sequence by adding materials in a reversed manner from a final workpiece shape to initial unprocessed workpiece shape. This subsection briefly reviews this algorithm by showing an example. Consider the workpiece shown in Fig. 1(a) machined by removing the blocks B1 to BNb (Nb = 12 in this example). Each block may represent either a volume removed by a single cut (see Fig. 2(a)) or that cut by a repetitive contour-parallel path (see Fig. 2(b)). Suppose that the displacement of the workpiece at the top edge of the i-th block is represented by a function d(Bi, f, ), when the force, f, is given to the block Bi. represents the geometry of the workpiece with remaining blocks. For example, when the blocks B 10, B11, B12 are not removed yet, can be written as ={ B10, B11, B12 } (see Fig. 1(b)). Throughout this paper, d(Bi, f, ) is calculated by the FEM (finite element method). This paper only discusses the displacement normal to the thin-wall surface. The optimization of block removal sequence proposed by Koike et al. [4,5] can be summarized as follows: (1) The volume to be removed is divided into blocks, Bi (i = 1, …, Nb). The blocks are given in priori.
4
(2) The algorithm starts from the finished workpiece geometry with all the blocks removed, i.e. k = {null}, where k=0. (3) Calculate the displacement, d(Bi, f, k), for all the “candidate” blocks, B i. For example, in Fig. 1(b), when k is null, blocks 4, 5, 6, 10, 11, 12 can be candidates. (4) Choose ik that minimizes d(Bik , f, k). Update k to k+1 by adding the block Bi to k. (5) kk+1 and repeat (3)-(4) until all the blocks are added.
The basic idea behind this algorithm can be described as follows: this algorithm aims to reduce the largest deformation by modifying the sequence to remove the blocks. This objective can be written as:
min max 𝑑(B𝑖𝑘 , 𝑓, 𝑘 ) {ik }
𝑘
(1)
By choosing the block to minimize the displacement at every cutting step in Step (4), a block is potentially added to more stiff positions, which in turn will support the blocks at subsequent steps. As a result, the less stiff blocks will have higher stiffness.
2.2 Issues with the previous algorithm The issues with previously proposed algorithm are as follows: (a) The objective in Eq. (1) is not explicitly taken into consideration in this algorithm, and thus the removal sequence is not guaranteed to achieve the objective in Eq. (1). It is just an ad-hoc way to obtain the removal sequence to achieve the objective in Eq. (1). (b) In this algorithm, all the blocks have the same size. Clearly, the deformation of the workpiece can be reduced by reducing the axial depth of cut or the radial depth of cut. It can be done by changing the size of some blocks. (c) This algorithm usually generates cutting paths that are not parallel to the finished workpiece surface
5
(see an example to be presented in Section 3). It results in more air-cut, which could significantly reduce the cutting efficiency. Furthermore, when the tool’s feed direction is changed in the finishing path, it often leaves an abrupt cutter mark on the finished workpiece surface.
2.3 Proposed algorithm (a) Main algorithm for optimization of block removal sequence To address the issues discussed in Section 2.2, the algorithm is improved. The objective of the proposed algorithm is to find the sequence of removing the blocks B1 to BNb such that the workpiece displacement for all removal step, d(Bik, f, k), is always less than the given threshold value, dc, for any step k. Namely, the objective can be written as:
Find {𝑖𝑘 }𝑘=1,⋯,𝑁𝑏 such that 𝑑(B𝑖𝑘 , 𝑓, 𝑘 ) < 𝑑𝑐 for any 𝑘
(2)
The proposed algorithm is as follows: (a-1)
The volume to be removed is divided into blocks, Bi (i=1, …, Nb). The blocks are given in priori.
(a-2)
The algorithm starts from the finished workpiece geometry with all the blocks removed, i.e. k = {null} where k = 0.
(a-3)
Calculate the displacement, d ( Bi, f, k ), for all the “candidate” blocks, Bi.
(a-4)
Find all ik that meets d ( Bik , f, k ) < dc. If there exist multiple ik, then add them all. Their sequence can be arbitrarily chosen. Update k to k+1 by adding all the blocks Bik..
(a-5)
If there exists no ik, it means that there is no such block sequence that achieves the objective in Eq. (2). The calculation finishes without any solution. If it is acceptable to change the axial depth of cut to reduce the workpiece deformation, go to the algorithm (b).
(a-6)
kk+1 and repeat (a-3)-(a-5) until all the blocks are added.
(a-7)
After this algorithm is finished, the blocks can be modified to increase the radial depth of cut for
6
enhancing the machining efficiently. Go to the algorithm (c).
Unlike the past algorithm in Section 2.1, this improved algorithm guarantees that the workpiece displacement at each removal step, d ( Bi, f, k ), is always smaller than the threshold, dc. To obtain the minimum dc that can be achieved, the algorithm (a) should be iterated with reducing dc until there exists no solution. This iterative calculation gives a solution to the problem in Eq. (2). The proposed algorithm addresses the issue (a) in Section 2.2.
(b) Sub-division of blocks to reduce the axial depth of cut When there exists no block satisfying d ( Bik , f, k ) < dc in Step (a-5) in the algorithm (a), candidate blocks can be divided to reduce the axial depth cut: (b-1) All the candidate blocks, Bi, are divided into two sub-blocks in the axial direction. Denote them as B i1 (lower) and Bi2 (upper) (see Fig. 3). (b-2) Find all i1k from lower sub-blocks that meets d(Bi1k , f, k) < dc. If there exist multiple i1k, then add them all. Their sequence can be arbitrarily chosen. Update k to k+1 by adding all the blocks Bi1k.. (b-3) If lower sub-block B i1 is added at the k-th step, then upper sub-block B i2 will be in the candidate blocks in the (k+1)-th step. (b-4) Go back to the algorithm (a).
By sub-dividing each block, the axial depth of cut is reduced, which naturally reduces the workpiece deformation. This algorithm is added to address the issue (b) discussed in Section 2.2.
(c) Post-optimization modification to increase the radial depth of cut
7
In contrast to the case in (b), when the displacement at block Bik, d ( Bik , f, k ), is significantly smaller than the threshold dc, two blocks adjacent to each other in the direction normal to the feed direction can be combined, as illustrated in Fig. 4. It doubles the radial depth cut, and then enhances the machining efficiency. Here it is assumed that each block is removed by a single cut (see Fig. 2(a)). This can be done after the main algorithm (a) is finished as a post-optimization modification of the designed removal sequence. This modification is also to address the issue (b) discussed in Section 2.2.
(d) Design of blocks to be removed To apply the algorithm (a), the removed volume must be divided into the blocks in priori. It must be designed properly, taking the machining process into consideration. As discussed in Section 2.2 (c), when the tool’s feed direction is changed in the finishing path, it often leaves an abrupt cutter mark on the finished workpiece surface (see the case study to be presented in Section 3). When it is not acceptable, the blocks should not be divided in the feeding direction, as illustrated in Fig. 5. Then, the tool path will be parallel to the finished workpiece surface. The same algorithm (a) can be applied to determine the removal sequence. This addresses the issue (c) in Section 2.2.
Remark 1: application to workpieces of free-form surface Throughout this paper, as illustrated in Fig. 1(a), the finished workpiece is a thin vertical wall and the removed stock is divided into rectangular blocks. This simplification is to illustrate the algorithm, and this paper’s basic concept can be applied to workpieces of arbitrary free-form surface. Figure 6 illustrates such an example. The geometry of each block is designed by shifting the finished workpiece’s surface. Each block can be removed by contour-parallel or direction parallel tool paths.
Remark 2: calculation of cutting force and workpiece deformation
8
In end milling processes, the instantaneous cutting force varies with tool rotation. Furthermore, within each block, since the workpiece’s stiffness varies as the tool is fed, the cutting force may vary. In the algorithm (a), the cutting force when removing the block Bi is represented by a single value, f. This is to simplify the calculation of d ( Bi, f, k ), and this value must be chosen carefully. Figure 7 (a) illustrates the side milling with a helix mill. Figure 7 (b) shows an example of the measured cutting force profile in the direction normal to the feed direction. As has been discussed in well-established cutting mechanism models, e.g. [15], a cutter tooth immerses into workpiece at position P1, which corresponds to the force value F1, and it exits at position P2 which corresponds to the force F2 . To predict the deformation at the point shown in Fig. 1 (b), f = F2 is used in the calculation of d (Bi, f, k ). In the following case study, the value of f was determined by a cutting test. Within each block, the workpiece’s deformation can vary due to the change in the remained stock’s stiffness. This can be particularly significant when blocks are not divided in the feed direction as shown in Fig. 5. In such a case, the suggested procedure to calculate the deformation, d(Bi, f, k ), is as follows: sub-divide the block Bi in the feed direction as shown in Fig. 8 (denote sub-blocks by Bij). Then, calculate the deformation d ( Bij, f, k ) at each sub-block. Take the maximum deformation as the representative deformation at the block Bi, i.e. 𝑑 ( B𝑖 , 𝑓, 𝑘 ) = max(𝑑( B𝑖𝑗 , 𝑓, 𝑘 )). 𝑗
For the simplification, this paper assumes that the workpiece deformation during the cutting, d(Bi, f,
k ), is copied as the thickness error of the finished workpiece, as shown in Fig. 9. In other words, when the finished workpiece’s thickness at the top edge of the block Bi is denoted by t(Bi), it can be represented by
𝑡(Bi ) = 𝑡 ∗ (Bi ) + 𝑑( B𝑖 , 𝑓, 𝑘 )
(3)
where t*(Bi) represents the nominal thickness. For a vertical thin wall, it is a constant value for any blocks.
9
3
Case study
3.1 Test objective and common conditions The objective of the cutting experiments presented in this section is to study the effectiveness of the proposed tool path design schemes in reducing the geometric error of the machined thin wall workpiece. The geometry of the initial workpiece (part blank) and the finished thin-wall workpiece are shown in Fig. 10. A square end mill is used. The tool's axial direction is in the Z direction. The cutting conditions are shown in Table 1. The radial depth of cut, Rdn, and the axial depth of cut, Adn, shown in Table 1 are “nominal” values adopted in Strategies 1 and 2. They may be modified in Strategy 3 (further details will be given in Section 3.2). The cutting force was measured in advance by using a dynamometer in a cutting test under the cutting conditions shown in Table 1. The workpiece was sufficiently thick such that the influence of its deformation was negligible. The cutting force corresponding to the tool engagement’s top edge point (see Fig. 7) was measured as 𝐹 = (𝐹𝑥 , 𝐹𝑦 , 𝐹𝑧 ) = (−9, 20, 3)N. This value is used throughout the tool path design to be presented in Section 3.2. As the machining process proceeds, the remained workpiece becomes gradually thinner, which likely changes the cutting force. This change is, however, considered minor and ignored in the tool path design. The authors’ previous work [16] presented the modelling of the influence of the workpiece’s thickness on the cutting force.
3.2 Tool path design strategies The following three tool path design strategies are compared: (1) Strategy 1: Conventional tool path: The conventional contour-parallel path shown in Fig.11 is used. The radial depth of cut is 0.2 mm (see Table 1), which means total 20 paths in the X-direction. The axial depth of cut is 15 mm (see Table 1),
10
which means total 3 layers in the Z-direction. The feed direction is parallel to the finished workpiece’s surface (from Y- to Y+). The entire stock is removed by total 60 paths.
(2) Strategy 2: Tool path designed based on the block removal sequence optimized by the algorithm (a): The block removal sequence is optimized by using the algorithm (a) in Section 2.3. The volume to be removed is divided into 5 blocks in the feed (Y) direction, 3 blocks in the axial (Z) direction, and 4 blocks in the direction normal to the feed direction (X) (refer to Fig. 13). In other words, the size of each block is 12015 mm (XYZ). The radial depth of cut is 0.2 mm as shown in Table 1. In other words, each block is removed by 5 repetitive contour-parallel paths (see Fig. 2(b)). The algorithm (a) was iteratively performed, reducing the threshold value, dc, from 0.1 mm to 0.06 mm. To calculate the workpiece deformation d ( Bi, f, k ) in step (a-3) in the algorithm (a), the FEM model was built in COMSOL and the calculation was run with MATLAB. Figure 12 shows the block removal sequence calculated by applying the algorithm (a) for the case of dc = 0.08 mm. The circled numbers represent each block, and the blocks are removed from 1 to 60. Figure 13 illustrates the block removal sequence. There was no solution for dc = 0.06 mm. Remark: To enhance the efficiency, when there are two adjacent blocks to be removed consequentially, they are removed altogether as shown in Fig. 14, instead of removing them one by one.
(3) Strategy 3: Tool path designed with no block division in the feed direction ((d) in Section 2.3) and with the sub-division of blocks (b) and the post-optimization modification (c) For the reason discussed in (d) in Section 2.3, the volume to be cut is not divided in the feed direction as shown in Fig. 15(a). The algorithm (a) is applied to 12 blocks. In the application of the algorithm (a), the maximum displacement within each block is calculated by sub-dividing it into five blocks and
11
calculating the deformation at each sub-block, as discussed in Remark 2 of Section 2.3(d). In the application of the algorithm (a), the target maximum deformation was set at dc = 0.04 mm. This is lower than in Strategy (2). To achieve it, the sub-division of blocks to reduce the axial depth of cut, presented in the algorithm (b) in Section 2.3, and the post-optimization modification to increase the radial depth of cut, presented in (c) in Section 2.3, were applied. Figure 15(b) shows the designed block removal sequence. The following combinations of cutting depths are included in Fig. 15(c): 1) Ad = 3.75mm, Rd = 0.2 mm, 2) Ad = 7.5 mm, Rd = 0.2 mm, and 3) Ad = 15 mm, Rd = 0.4 mm; 4) Ad = 15 mm, Rd = 0.2 mm.
3.3 Comparison in simulated workpiece displacement profiles Figure 16 shows the simulated displacement histories for Strategies 1 to 3. The horizontal axis for Strategy 1 represents the number of removed layers. The horizontal axis for Strategies 2 and 3 represent the index number of blocks shown in Figs. 12 and 15. The vertical axis shows the simulated workpiece deformation, d(Bik , f, k) (calculated in Step (a-3) in the algorithm (a)). Figure 17 shows the predicted thickness profiles of the finished workpiece, calculated by Eq. (3), at lines L 1 to L3 (see Fig. 19 for the location of L1 to L3). They are calculated by using the same FEM model as the one used in the algorithm (a). For Strategy 2 in Fig. 16, it can be observed that the simulated workpiece deformation, d(Bik , f, k), is smaller than the given threshold, dc = 0.08 mm for all the blocks. In Strategy 3, the simulated deformation is smaller than dc = 0.04 mm for all the blocks.
3.4 Experimental setup The part blank shown in Fig. 10(a) was machined with Strategies 1 to 3. The experimental setup is shown in Fig. 18. A five axis machining center, NMV3000 DCG by DMG MORI Co., Ltd., and a straight end mill, CA-MFE14 by OSG Corp., were used. The specifications of the tool and cutting conditions are
12
shown in Table 1. Measurement of the workpiece displacement during the cutting operation: During the cutting process, the workpiece displacement was continuously measured by a triangulation-based laser displacement sensor (LK-G10 by Keyence Corp.) at three positions P1, P2, and P3 as shown in Fig. 19. Measurement of the finished workpiece’s thickness profile: The thickness profiles of the finished workpieces were also measured on-the-machine. An interferometry-based laser displacement sensor (SI-F10 by Keyence Corp.) was used. The sensor mounted on the spindle scans the surface of a gauge block and the finished workpiece surface from both sides, as shown in Fig. 20. The thickness of the block gauge was pre-calibrated. By the measured profiles on the finished workpiece, D1 and D2 in Fig. 20, and those on the gauge block, d1 and d2, the thickness of the finished workpiece, T, can be measured. X1, X2, x1, x2 are spindle positions in X in the machine tool coordinates. Three lines on the finished workpiece, L1 to L3 in Fig. 19, were scanned by feeding the sensor to the Y-direction.
3.5 Measured and thickness profiles of the finished workpieces Figure 21 shows the thickness profiles of the finished workpieces measured as shown in Section 3.4. In Fig. 21(a), three profiles measured at the lines L1 to L3 are plotted in the 3D view, and they are compared between Strategies 1 to 3. In Fig. 21(b-1) to (b-3), they are presented in the XY plane for clearer comparison. In Fig. 21(b-1) to (b-3), the workpiece’s displacements (the values are added with nominal thickness 1 mm) when the tool pass the measured points P1 to P3 (in the opposite side) in the finishing cut are also shown by the marks ●▲■. The displacement in Strategy 2 is missing in Fig. 21(b-3) because severe chatters happened there. An example displacement profile measured at P 1 in the finishing process in Strategy 1 is shown in Fig. 22. The tool passes the same Y position as the sensor at 14.4 sec after the finishing process starts. The peak displacement within 0.01 sec around this time is taken and shown in the
13
mark '●' in Fig. 21(b) (Strategy 1).
3.6 Discussion on experimental results (1) On the comparison in thickness profiles in Strategies 1 to 3: Figure 21 clearly shows that Strategy 3 minimized the error in the finished workpiece’s thickness profile. From the nominal thickness 1 mm, Strategy 1 resulted in the geometric error at maximum 0.2 mm (see Fig. 21 (b-1)). In Strategy 3, the maximum error was reduced to 0.036 mm. Strategy 2 also reduced the workpiece’s average geometric error significantly compared to Strategy 1. However, the entry and the exit of the tool to the finishing path caused significant “bump” in the surface profile, as can be observed e.g. at Y=20 and 80 mm in Fig. 21(b-1). This is predicted in the discussion in Section 2.3 (d). Furthermore, the workpiece finished by Strategy 2 was partially subject to the chattering (see also (5) in this subsection). In Strategy 3, the thickness profiles are partially below the nominal thickness 1 mm, which means the workpiece is over-cut. This is potentially caused by the workpiece’s vibration.
(2) On the difference between the measured surface profile and the workpiece’s displacement during the cutting: Figure 21 (b-1) to (b-3) also show the workpiece’s displacement measured during the cutting. It can be observed that Strategies 2 and 3 reduced this displacement compared to Strategy 1. However, they are significantly different from the surface profiles of the finished workpieces. This difference can be potentially caused by 1) the tool deflection caused by the cutting force, 2) the measurement error of the laser displacement sensor, including the error caused by the limited measurement bandwidth, and 3) the contribution of the material removal process, including the “spring-back” of the material which typically occurs when the radial depth of cut is smaller.
14
(3) On the difference between measured and simulated surface profiles of the finished workpieces: The FEM simulation shown in Fig. 17 predicts that Strategies 2 and 3 can reduce the maximum deformation by about 20 % and 63 %, respectively, compared to Strategy 1. This is similar in actual surface profiles shown in Fig. 21(b-1), where 21% and 82% reduction can be observed. However, the difference between simulated and measured surface profiles are quantitatively large. This difference can be potentially caused by 1) the difference in the cutting force in simulation and actual cutting. In simulation the force was taken as a constant value (see Section 3.1), while in the actual cutting the force varies as was discussed in Remark 2 of Section 2.3 (see also [16]), 2) The contribution of the material removal process, including the “spring-back” of the material, or the tool’s deflection due to the cutting force. In other words, Eq. (3), assuming that the workpiece instantaneous deformation is copied as the thickness error of the finished workpiece, may have significant error. The causes for the estimation error of the finished workpiece’s thickness (the difference between Figs. 17 and 21) are not fully investigated yet. Further investigation on such issues will be planned to build a more accurate simulation model for predicting the workpiece profile.
(4) On the efficiency: The efficiency of three cutting strategies is also compared as shown in Fig. 23. The cutting time is simulated by using a machining simulator with the NC programs used in the experiments. The cutting time for Strategy 2 is longer than Strategy 1 by about 11 min. This is predicted in the discussion in Section 2.3 (d). On the other hand, the cutting time for Strategy 3 is longer only by 2 min.
(5) On the occurrence of the chatter:
15
Figure 24 shows the appearance of the finished surfaces. The workpiece by Strategy 2 shows clear chatter marks. It also shows abrupt cutter marks (“bump”) caused by the entry and the exit of the tool to each block. Interestingly, the chatter mark on the Strategy 2 workpiece can be observed on the third (lowermost) layer, where the workpiece stiffness is predicted higher than the first layer. At the 3 rd layer, the chatter seemed to start when the cutter passes this “bump” in higher (2nd or 1st) layers. This can be a critical issue for Strategy 2, as was discussed in Section 2.3 (d).
(6) On the measurement of workpiece displacement during the cutting process In Fig. 22, the workpiece’s displacement profile measured by the triangulation-based laser displacement sensor shows low-frequency periodical variation (its period is about 1.2 sec). This may be caused by the beat of the vibrations of slightly different frequencies, or the aliasing. The workpiece’s displacements shown in Fig. 21 (b-1) to (b-3) may contain the measurement error caused by this (up to 15%).
(7) On the calculation time of block removal sequence The calculation time to obtain the optimal block removal sequence for Strategy 2 was 20 minutes (1229 seconds) while for Strategy 3 it was 97 minutes (5860 seconds). Most of the calculation time was consumed for the calculation of the workpiece displacement, d ( Bi, f, k ) in Step (a-3) in Section 2.3 (a), by the FEM calculation. If this calculation can be simplified by e.g. using an analytical model, then the entire calculation time would be significantly reduced.
4
Conclusion The proposed algorithm optimizes the sequence to remove blocks in cutting operations, such that the
maximum deformation of the workpiece in the finishing process is minimized (Strategy 2 in Section 3). If
16
the blocks are divided in the feed direction, the tool must change its direction on the finishing path to remove each block. This may cause abrupt cutter marks or severe vibration. To address this issue, Strategy 3 does not divide the blocks in the feed direction, which gives the tool path parallel to the finished workpiece surface. To reduce the workpiece deformation, sub-division of the blocks in the tool’s axial direction can be done to reduce the axial depth of cut. The post-optimization modification can increase the radial cutting depth to improve the cutting efficiency. The effectiveness of the proposed tool path design strategies in reducing the geometric error of the finished thin-wall workpiece was studied by cutting experiments.
17
References [1]
Ratchev S, Liu S, Huang W, Becker AA. An advanced FEA based force induced error compensation strategy in milling. Int J Mach Tools Manuf 2006;46:542–51.
[2]
Toride A, Yamada T, Araki S. Study on the generation of micro shafts by turning operation (2nd report) Discussion on the geometrical shapes of cutting edges. J Japan Soc Abras Technol 2008;52:466–71.
[3]
Smith S, Dvorak D. Tool path strategies for high speed milling aluminum workpieces with thin webs. Mechatronics 1998;8:291–300.
[4]
Koike Y, Matsubara A, Nishiwaki S, Izui K, Yamaji I. Cutting Path Design to Minimize Workpiece Displacement at Cutting Point: Milling of Thin-Walled Parts. Int J Autom Technol 2012;6:638–47.
[5]
Koike Y, Matsubara A, Yamaji I. Design method of material removal process for minimizing workpiece displacement at cutting point. CIRP Ann - Manuf Technol 2013;62:419–22.
[6]
Rai JK, Xirouchakis P. Finite element method based machining simulation environment for analyzing part errors induced during milling of thin-walled components. Int J Mach Tools Manuf 2008;48:629–43.
[7]
Kolluru K, Axinte D, Becker A. A solution for minimising vibrations in milling of thin walled casings by applying dampers to workpiece surface. CIRP Ann - Manuf Technol 2013;62:415–8.
[8]
Liu S, Zheng L, Zhang ZH, Wen DH. Optimal fixture design in peripheral milling of thin-walled workpiece. Int J Adv Manuf Technol 2006;28:653–8.
[9]
Budak E, Comak A, Ozturk E. Stability and high performance machining conditions in simultaneous milling. CIRP Ann - Manuf Technol 2013;62:403–6.
[10]
Aijun T, Zhanqiang L. Deformations of thin-walled plate due to static end milling force. J Mater
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Process Technol 2008;206:345–51. [11]
Tsai JS, Liao CL. Finite-element modeling of static surface errors in the peripheral milling of thin-walled workpieces. J Mater Process Technol 1999;94:235–46.
[12]
Sagherian R, Elbestawi MA. A simulation system for improving machining accuracy in milling. Comput Ind 1990;14:293–305.
[13]
http://www.malinc.com/wp-content/uploads/2014/05/CutPro_Guide.pdf
[14]
Smith S, Wilhelm R, Dutterer B, Cherukuri H, Goel G. Sacrificial structure preforms for thin part machining. CIRP Ann - Manuf Technol 2012;61:379–82.
[15]
Budak E, Altintas Y. Modeling and avoidance of static form errors in peripheral milling of plates. Int J Mach Tools Manuf 1995;35:459–76.
[16]
Wang J, Ibaraki S, Matsubara A, Shida K, Yamada T. FEM-Based Simulation for Workpiece Deformation in Thin-Wall Milling. Int J Autom Technol 2015;9:122–8.
19
Displacement d (B5 , f, Γ )
B4
Force f
B5
B1
B6 B2 B3
B10
B7
B11
B8
B12
B9 (a)
(b)
Γi = { B10 , B11 , B12 }
Fig. 1. An example problem setup: (a) the volume to be removed is divided into a set of blocks, B1 to B12, (b) the definition of the simulated workpiece displacement, d(Bi, f, k).
20
Tool
(a)
(b)
Fig. 2. Possible removal of each block. The tool path to remove each block can be designed arbitrarily: (a) a block removed by a single cut; (b) a block removed by a repetitive contour-parallel path.
21
B42 B62
B41
B61
Fig. 3. Sub-division of blocks when there exists no candidate block that can be added with d ( Bik , f, k)
< dc
22
Fig. 4. Post-optimization modification to increase the radial depth of cut
23
Feeding
Fig. 5. Blocks are not divided in the feeding direction
24
Thin wall
Fig. 6. Example block division for a free-form surface workpiece
25
Tool P2 P1
Last cut-layer of block
F1
F2
60
Simulation Experimental
50 40
Cutting force (N)
Block
30 20 10 0 -10 -20
(a)
Rotation angle (104 deg)
(b)
Fig. 7. Cutting force value used in simulation: (a) geometry of milling; (b) force pattern
26
d( Bi5, f, ) Bi5
d( Bi1, f, )
Bi1
Bi
Fig. 8. Sub-division of a block to calculate the maximum deformation in the block Bi
27
End mill
Deformation d( Bi, f, Γk )
Nominal thickness t*(Bi)
Predicted Thickness t(Bi)
Workpiece
(a)
(b)
Fig.9. The relationship of the deformation in finishing cut and the thickness of the finished workpiece: (a) the deformation in the cutting; (b) the finished workpiece
28
5
1
unit : mm
100
45
65
z (a)
y x (b)
Fig. 10. Geometry of workpiece: (a) initial shape; (b) final shape
29
z
y x
(a)
(b)
(c)
(d)
Fig.11 Strategy 1: conventional contour-parallel path: (a) 2 layers are removed; (b) 20 layers are removed; (c) 22 layers are removed; (d) 60 layers are removed
30
Block layer in Z
Sequence when e080 22 17 21 15 1210 7 20 14 16 5 2 11 13 6 8 9 4 1 3 50 4240 32 49 30 2725 19 41 4831 38 39 29 26 2818 23 24 59 6054 55 58 46 4736 37 56 5751 52 53 45 43 4433 34 35 z y x
Block layer Y mm; Fig. 12. The calculated block removal sequence of Strategy 2, dc =in 0.08 Block layer in X
31
z
y x
(a)
(b)
(c)
(d)
Fig. 13. Strategy 2: the block removal sequence designed by the proposed algorithm. The sequence corresponds to Fig. 12: (a) 1st to 2nd blocks are removed; (b) 1st to 19th blocks are removed; (c) 1st to 42nd blocks are removed; (d) 1st to 60th blocks are removed.
32
block to be removed after NO. k block NO. k
Remove two blocks together
Fig. 14. Example of adjacent blocks that are removed together
33
z
Ad = 3.75 mm Rd = 0.2mm
y x 4
Ad = 15 mm Rd = 0.2mm
5
6
32
10
1
98
7
11
Ad = 7.5 mm Rd = 0.2mm
15
16 14
(a)
13
12
(b)
(c)
Ad = 15 mm Rd = 0.4mm
Fig. 15. Strategy 3: tool path designed with no block division in the feed direction: (a) initial block division; (b) finished block division with removal sequence index; (c) the axial depth of cut, Ad, and the radial depth of cut, Rd, for each block
34
Fig. 16. Simulated workpiece displacements at each block in Strategies 1 to 3
35
Strategy 1
Strategy 2
Strategy 3
1.1
20%
(a)
63%
1.05
Thickness mm
1 0
20
40
60
80
100
20
40
60
80
100
20
40
60
80
100
1.1 (b) 1.05 1 0 1.1 (c) 1.05 1 0
Y mm
Fig. 17. Simulated thickness profiles of the finished workpiece at lines L1 to L3: (a) at the line L1; (b) at the line L2; (c) at the line L3
36
Laser sensor LK-G10 Tool
Z Workpiece
Y X
Fig. 18. Experimental setup
37
10 P1 P2
1 L1
16 31
P3
L2 L3 z
unit: mm y
Fig. 19.Positions on the workpiece’s surface measured by the laser displacement sensor (P 1 to P3) and the scanned lines to measure the finished workpiece’s thickness profiles (L1 to L3)
38
X2
X1 T D2
lasor sensor SI-F10
D1
workpiece d2 d1
Z X
x2 gauge block
t
Spindle P
x1
Fig. 20. Setup for the thickness error measurement of finished workpieces
39
Strategy 1 Strategy 2 Strategy 3
1.2 0 1.15
L1 Thickness mm
Z mm
-10
-20
L2 -30
-40
1
1
0
10
Y mm
Strategy 1 Strategy 2 Strategy 3
30
40 50 60 Y coordinate mm
70
80
90
70
80
90
Strategy 1 Strategy 2 Strategy 3
1.15
1.15
1.1
Thickness mm
Thickness mm
20
(b-1)
Thickness mm
1.2
Displacement in cutting
100
50
1.1 1.2
(a)
82% 1.1
1.05
Strategy 1 Strategy 2 Strategy 3
L3
0.9
21%
1.1
Displacement in cutting
Displacement in cutting 1.05
1
1.05 0.95 1
0.9 10
(b-2)
20
30
40 50 60 Y coordinate mm
70
80
90
10
(b-3)
20
30
40 50 60 Y coordinate mm
Fig. 21. Measured thickness profiles of the finished workpieces. The marks, ●▲■, represents the workpiece displacement measured during the cutting in Strategies 1 to 3 (at the points P 1 to P3 in Fig. 19): (a) 3D view; (b-1) at the line L1; (b-2) at the line L2; (b-3) at the line L3
40
70 16 s
60 Peak value at 14.4s from cutting starts.
Displacement m
50 40
Cutting finishes
30
Cutting starts
20 10 0 -10
6
8
10
12
14 Time s
16
18
20
22
Fig. 22. The workpiece displacement measured at the point P 1 during the finishing process in Strategy 1 (an example).
41
34 min
25 min
23 min
Strategy 1
Strategy 2
Strategy 3
Fig. 23. The overall cutting time in three strategies
42
Boundaries in Y direction
Boundaries between layers in Z “bump” Z Y (a)
(b)
(c)
Fig. 24. Appearances of finished workpiece: (a) Strategy 1; (b) Strategy 2; (c) Strategy 3
43
Table 1
Cutting conditions
Radial depth of cut (nominal), Rdn
0.2 mm
Axial depth of cut (nominal), Adn
15 mm
Cutting direction
up-cut
Spindle speed
2275 min-1
Feed per tooth
0.055 mm
Material of workpiece
Aluminum alloy, JIS A5052
Cutting fluid
None
Tool diameter Φ
14 mm
Tool helix angle β
40°
Tooth number z
3
Tool flute length
21 mm
44