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Machining of free-form surfaces. Part II: Calibration and forces
International Journal of Machine Tools & Manufacture 46 (2006) 736–746 www.elsevier.com/locate/ijmactool
Machining of free-form surfaces. Part II: Calibration and forces B. Ozturk, I. Lazoglu*, H. Erdim Manufacturing Automation and Research Center, Department of Mechanical Engineering, Koc University, Rumeli Feneri Yolu, Sariyer, Istanbul 34450, Turkey Received 16 February 2005; accepted 19 July 2005 Available online 12 September 2005
Abstract In the machining simulations of 3D free-form surfaces by ball-end milling, calibration coefficients play very critical role in force predictions. In other words, obtaining the calibration coefficients from calibration tests is a very influential process in the prediction of cutting forces. Accurately obtained calibration coefficients lead to better force predictions. In the literature, the calibration coefficients are assumed to be independent of start and exit angles of the engagement region, thus they are assumed to be identical for any engagement region. Calibration coefficients are obtained from the horizontal slot cutting tests and these tests are repeated for different depth of cuts. In this paper, in order to achieve more accurate force predictions in free-form machining simulations, a new modification algorithm for calibration coefficients is presented. In this research, with theoretical analysis and experimental force signals, it is shown that the inclination angle has a great importance in calibration and in force simulation of 3D free-form machining. q 2005 Elsevier Ltd. All rights reserved. Keywords: Calibration coefficients; Engagement region; Inclination angle; Sculptured surfaces
1. Introduction With the enlargement of the free-form surface machining fields in industry, fast and accurate process simulations of free-form surfaces are highly demanded. In the analysis of mechanics and dynamics of 3D free-form machining processes, it is the fact that the most critical parameter is usually the cutting force. Therefore, in the literature up to now, there are various simulation algorithms developed for ball-end milling process; Yang and Park [1] represented a ball-end mill with infinitesimal disc elements that is an approach from end mill force analysis. Lee and Altintas [2] predicted the cutting forces from orthogonal cutting data and Kim et al. [3] presented a force model with Kr/Kt cutting coefficients and include the effect of inclination angle to chip thickness by rotating the coordinate frame and evaluated the cutting coefficients at every rotation angle. In other words, they assume a polynomial like behavior of the calibration coefficients over the engagement region * Corresponding author. Tel.: C90 212 338 1587; fax: C90 212 338 1548. E-mail address: [email protected] (I. Lazoglu).
0890-6955/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2005.07.037
and find the coefficients of the polynomials empirically from horizontal slot cutting tests. Lazoglu [4] employed a Z-mapping and Boolean approach, and formed a model that determined surface topography in addition to predicting cutting forces. Lamikiz et al. [5] performed a similar approach to Lazoglu [4] and he assumed that the calibration coefficients are polynomial of the z-coordinate of the disc elements. They state that assuming friction coefficients as a constant is a valid assumption and assumed edge coefficients constant throughout the machining process. They state that obtained results were only valid for a specific testing case under a given set of conditions and only for the case of horizontal slot cutting, they do not include any explanation of the effect of inclination angle on the calibration coefficients. Moreover, they performed various tests however they do not include a free-form surface. In calibration process, Azeem et al. [6] proposed a simplified calibration technique for the mechanistic approach. Most of the force models in the literature are for 2 12 D machining but not for 3D free-form surfaces. In the analysis of literature, it is seen that those models can handle the 3D free-form surface machining simulations, which have deficiency of excluding inclination angle dependency of the calibration force coefficients by imposing the method stated in this paper.
B. Ozturk et al. / International Journal of Machine Tools & Manufacture 46 (2006) 736–746 200
Cutting Forces for 48 mm/min Fx[N] Fy[N] Fz[N]
100
Force [N]
0
-100
0.5 mm 1.0 mm 1.5 mm 2.0 mm
-200
3.0 mm 4.0 mm
-300 6.0 mm -400
0
100
200
300
400
500
600
700
800
737
Center (VMC). The cutter was carbide ball-end mill cutter from CoroMill Plura series of Sandvik with 12 mm diameter, 37 mm projection length and 308 helix angle. The workpiece materials were aluminum blocks (Al7039) of size 250!170!38 mm3. Kistler 3-component dynamometer (Model 9257B) was used for force data acquisition. Fig. 1 is the combination of forces at the selected depth of cuts which were later to be used for obtaining the calibration coefficients for a feedrate of 48 mm/min. The calibration algorithm of Guzel, Lazoglu [7] is used after this point for seven depths of cut: 0–0.5, 0.5–1, 1–1.5, 1.5–2, 2–3, 3–4, 4–6 mm for feedrates of 48, 96, 144 and 192 at a spindle speed of 600 rpm and force calibration coefficients are presented in Table 1.
Angle [deg] Fig. 1. Combination of forces at selected depth of cuts for a feedrate of 48 mm/m.
In this paper, a new algorithm is presented for the calibration coefficients which are evaluated from the mechanistic approach to simulate the force in free-form machining. Most of the researches about ball-end milling are verified for 2 12 D operations. Validations are performed for half immersion cutting; bisection cutting and slot cutting tests. Free-form surface cutting are not performed very commonly in the literature due to difficulties in contact region determination, the chip thickness determination and in calibration process. In part-I contact region determination is analyzed for monotonic free-form surfaces. In this part of the paper, a new modification algorithm is presented for accurate prediction of calibration coefficients in machining of free-form surfaces. Inclination dependency of calibration coefficients is stated mathematically, validation tests are performed for inclined surfaces and for 3D free-form surfaces.
2. Determination of calibration coefficients In order to determine the cutting force coefficients in this study, Guzel and Lazoglu’s mechanistic approach [7] is used. However, some modifications on cutting forces as detailed in the following sections were performed for better force predictions. The experiments for validation were performed on Mazak FJV-200 UHS Vertical Machining
3. Modification of calibration coefficients The calibration coefficients obtained from horizontal slot cutting tests give accurate results for 2 12 D cutting conditions, on the other hand in free-form surface machining, the measured and simulated results do not match very accurately if one tries to use the modified Martelotti’s [8] chip thickness given in Eq. (1) c Z h sinðqÞsinðjÞ
(1)
where h is feed per tooth per revolution, q is the rotation angle and j is the zenith angle (disc position angle) [7]. Martelotti’s approach is simply the kinematic analysis of a point on the cutting edge. It includes rotational and translational motion and is valid if rotation vector and translational vectors are perpendicular. On the other hand, in free-form surfaces these two vectors are not always perpendicular as shown in Fig. 2, therefore the calibration coefficients for different inclination angles are not valid. They need to be modified by including the inclination angles. In this paper, factors to modify the calibration coefficients obtained from horizontal slot cutting tests are evaluated from the end mill analytical equations. This is acceptable since the ball-end mill can be geometrically represented by end mills at different depths. In addition to that, if the terms which include zenith angle are evaluated at 908, the ball-end mill force equations [7] simplify to the end mill equations. Evaluation of zenith angle (j) at 908 is simply a geometrical conversion of a ball-end mill
Table 1 Calibration coefficients for selected depth of cuts Sections from tool tip coefficients
0–0.5 (mm)
0.5–1 (mm)
1–1.5 (mm)
1.5–2 (mm)
2–3 (mm)
3–4 (mm)
4–6 (mm)
Krc (N/mm2) Kre (N/mm) Kjc (N/mm2) Kje (N/mm) Ktc (N/mm2) Kte (N/mm)
2991.4 31.2 1113.4 11.7 5353.9 36.4
695.4 9.6 784.9 12.7 2549 17.3
22.1 9.0 752.7 3.6 1181.8 29.6
195.7 2.5 178.5 19.5 1429.1 3.1
105.1 7.2 101.5 13.5 971.4 16.3
483.3 14.8 30.8 8.7 847.6 16.9
353 11.9 5.6 0 932.1 10.7
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From Eq. (3), calibration coefficients can be written as: Ktc Z
4F yc 4F ye 4F ; Kte Z ; Krc ZK xc ; Na Na Na
4F pF zc 2F Kre ZK xc ; Kjc Z ; Kje Z ze Na Na Na
(4)
This slot cutting calibration coefficients can be used effectively in end milling processes. On the other hand, since calibration coefficients are dependent on the engagement angles, they need to be modified according to the inclination angle of the surface in ball-end milling. In order to analyze this situation, the calibration coefficients which can be derived from Eq. (2) need to be used instead of the calibration coefficients which are obtained from Eq. (4). However, the conflict to use Eq. (2) is that which start and exit angles are going to be taken since they are not constant throughout the engagement domains unlike horizontal slot cutting F xc C AA12 F yc F xc C AA21 F yc ; Ktcnew Z ; Krcnew Z A2 K AA12 A1 K AA21
Fig. 2. Feed direction in downward motion, rotational vector and inclination angle.
to an end mill. Nac Fx Z ðK cosð2fÞKKrc ð2fKsinð2fÞÞÞ 8p tc fex Na ðKKte sinðfÞ C Kre cosðfÞÞ C ; 2p fst Nac ðK ð2fKsinð2fÞÞ C Krc cosð2fÞÞ F y Z 8p tc fex Na K ðKte cosðfÞ C Kre sinðfÞÞ ; 2p fst fex Na ðKKac c cosðfÞ C Kae fÞ Fz Z 2p fst
F xe C AA34 F ye F new ; ZK zc ; Krenew Z Kjc A4 A C A3 4
Ktenew
(5)
A4
F xe K AA43 F ye F new ; Kje Z Z ze ; A5 A3 C AA43
where fex fex Na Na cosð2fÞ A1 Z ; A2 Z K ð2fKsinð2fÞÞ ; 8p 8p fst fst (2)
Eq. (2) is the analytical equation of average end mill forces stated by Altintas [9]. The calibration coefficients can be obtained by considering horizontal slot cutting tests which means start angles are 08 and exit angles are 1808. 2 3 Na Na F x Z 4K ðKrc cÞK ðKre Þ5 4 p 2 3 Na Na ðK Þ5 gF q Z F qc c C F qe F y Z 4 ðKtc cÞ C (3) 4 p te 2 3 Na Na ðKae Þ5 F z Z 4 ðKac cÞ C p 2
fex fex Na Na ðcosðfÞÞ A3 Z K sinðfÞ ; A4 Z ; 2p 2p fst fst
Na ðfÞ A5 Z 2p
fex fst
Second disadvantage is that the calibration process must be repeated for different inclination angles, thus the time and work spent for calibration increase. For this reason, a modification of calibration coefficient factor is introduced by considering Eqs. (2) and (3). This factor is going to be used to modify calibration coefficients, which are obtained by slot cutting calibration tests. The equations for new calibration coefficients are as given in Eq. (5). In order to handle the problem about the values of start and exit angles that are going to be put into Eq. (2) or Eq. (5), one can take the average of these angles, in whole spectrum with respect to all disc elements which are in contact at that depth of cut. More conveniently, average of start and exit angles can be taken in the intervals where
B. Ozturk et al. / International Journal of Machine Tools & Manufacture 46 (2006) 736–746 F P
f st Z iZ1
tanK1 F
y x
F P
; f ex Z iZ1
tanK1 Kyx F
739
;
qffiffiffiffiffiffiffiffiffiffiffiffiffi ri2 KR22 ; y Z ri Kx2 ; R 2 2 1K R qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ri Z R2 KðRKzi Þ2 ; zi Z i; R2 Z R sinðlÞ; F xZ
(6)
i Z 1; .; Nd
Fig. 3. Inclined slot cutting test.
calibration coefficients are valid (inclined slot cutting is assumed). In this paper, since calibration coefficients are taken for seven depths of cut, the average of start and exit angles are taken at these seven intervals. In part-I, the start and exit points are presented, in addition to that for an inclined slot cutting test shown in Fig. 3, start and exit angles are as follows
where l is the inclination angle, x, y are the intersection points of the boundaries with each disc element which are going to be used to define f st (average start angle) and f ex (average exit angle) and Nd is the total disc number in contact with that depth of cut. If calibration tests are proceeded with inclined slot cutting at different inclination angles, then fexZpKfst and thus makes some simplification in the terms given in Eq. (5). A1Z0 and A3Z0 at these test conditions, which means they are not inclination dependent terms for inclined slot cutting tests. From Eq. (5) evaluating the terms at the given condition leads to:
Fig. 4. Forces in 58 upward slot cutting tests. (a) Experimental and simulated Fx forces. (b) Detail view of Fx. (c) Experimental and simulated Fy forces. (d) Detail view of Fy.
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Na A2 ZK ððpK2fst Þ C sinð2fst ÞÞ; 4p Na Na ðpK2fst ÞÞ A4 ZK cosðfst Þ; A5 Z p 2p
Ktec Z (7)
new Kjc Ktcnew Krcnew c c ; K Z ; K Z ; rc jc old Ktcold Krcold Kjc
where Ktcold Z Krcold
F new 4F old yc yc / Ktcnew ZK ; Na A2
F new 4F old ZK xc / Krcnew Z xc ; Na A2
old Kjc Z
F new pF old zc new / Kjc ZK zc ; Na A4
where Kteold Z
If one defines the modification factor as a multiplier of the slot cutting calibration coefficients to be equal to the calibration coefficients of the machining inclination angles then below equations will state the modification of calibration coefficient factors. Ktcc Z
new Kje Ktenew Krenew c c ; K Z ; K Z ; re je old Kteold Kreold Kje
(8)
F new 4F old ye ye / Ktenew ZK ; Na A4
4F old F new Kreold ZK xc / Krenew Z xe ; Na A4 old Kje Z
2F old F new ze new / Kje Z ze Na A5
Eq. (7) simplifies to Eq. (8) and one can show that: Ktcc Z
F new p yc ; ððpK2fst Þ C sinð2fst ÞÞ F old yc
Krcc Z
p ððpK2fst Þ C sinð2fst ÞÞ
c Kjc
1 Z cosðfst Þ
Krec Z
F new xc ; F old xc
F new p zc ; Ktec Z old 4 cosðf F zc st Þ
p 4 cosðfst Þ
F new ye ; F old ye
F new p xe c ; Kje Z old ðpK2f F xe st Þ
(9)
F new ze ; F old ze
Fig. 5. Forces in 108 upward slot cutting tests. (a) Experimental and simulated Fx forces. (b) Detail view of Fx. (c) Experimental and simulated Fy forces. (d) Detail view of Fy.
B. Ozturk et al. / International Journal of Machine Tools & Manufacture 46 (2006) 736–746
4. Validations with inclined slot cutting tests In the validation of modified calibration coefficients method and analytical engagement region, slot cutting experiments are performed at different inclination angles which are K20, K10, K5, 5, 10, 208. Figs. 4–9 include measured force signals from slot cutting tests and modification algorithm force predictions. All slot validation tests and free-form tests are performed with the same workpiece and same cutter stated in Section 2. The engagement domains for all cutting tests are taken from ACRsim program [9] and simulations are performed with these domains. ACRsim gives an output in the form of number of CL-points, start angle, exit angle and inclination angle. Therefore, inclination angle information comes from the domain and force simulation code uses this information to select correct coefficient modification factors, it modifies calibration matrix accordingly. Figs. 4–9 are the slot cutting results for positively and negatively inclined surfaces for a feedrate of 48 mm/min at a spindle speed of 600 rpm. With the increase of the inclination angle simulated results with two methods are getting away from each other and from experimental data, however by using the modification of calibration method the experimental data is predicted with an error below 10%.
741
In Figs. 4–9, there are dashed lines which indicate the approach, steady-state and retract regions. Part-I is the analytical solution of full contact cases (steady-state). Thus, the error percentage values are evaluated for the values between these dashed lines. In addition to that, if the factors of modification of calibration coefficient method are independent of the workpiece material, one experiment will be sufficient to predict the change in calibration coefficients for different types of materials. In addition to that, as shown in Fig. 10 there is an identical linear trend observed for the modified calibration coefficient matrices for all calibration feedrates. In this figure, U is the calibration coefficient matrix, where Ulambda is the coefficient matrix modified according to the inclination angle of l and Uslot is the calibration coefficient matrix obtained from horizontal slot cutting tests. Thus, there is no need for a discretization of inclination angles, a linear behavior can be assumed between the angles K20, K10, K5, 5, 10, 208. In the validation of the stated method for free-form surfaces, two tests are performed for a feedrate of 48 mm/min at a spindle speed of 600 rpm. The material used in these tests is the same material with the calibration tests and tests are performed with the same cutter of the same projection length and helix angle. The predictions in these
Fig. 6. Forces in 208 upward slot cutting tests. (a) Experimental and simulated Fx forces. (b) Detail view of Fx. (c) Experimental and simulated Fy forces. (d) Detail view of Fy.
742
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Fig. 7. Forces in 58 downward slot cutting tests. (a) Experimental and simulated Fx forces. (b) Detail view of Fx. (c) Experimental and simulated Fy forces. (d) Detail view of Fy.
Fig. 8. Forces in 108 downward slot cutting tests. (a) Experimental and simulated Fx forces. (b) Detail view of Fx. (c) Experimental and simulated Fy forces. (d) Detail view of Fy.
B. Ozturk et al. / International Journal of Machine Tools & Manufacture 46 (2006) 736–746
743
Fig. 9. Forces in 208 downward slot cutting tests. (a) Experimental and simulated Fx forces. (b) Detail view of Fx. (c) Experimental and simulated Fy forces. (d) Detail view of Fy.
figures are again valid for the portion between two dashed lines. In free-form tests, errors are below 10% at maximum. Calculation time of engagement domains for airfoil and for sinusoidal surface were 9.1 and 26.9 s, respectively, with
MATLAB (CPU time) on a Pentium M 1.50 GHz–512 MB RAM. Therefore, the algorithm developed here in this paper is quite fast, and computation time is very acceptable for industrial implementations (Figs. 11–13).
Fig. 10. Linear trend in the modified calibration coefficient matrices.
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B. Ozturk et al. / International Journal of Machine Tools & Manufacture 46 (2006) 736–746
Fig. 11. Airfoil after machining simulation.
5. Conclusions In this paper, in order to achieve more accurate force predictions in 3D free-form machining processes, a new modification algorithm for calibration coefficients were
stated. It was observed that calibration coefficients obtained from 2 12 D cutting tests were not sufficient in prediction of the forces for 3D free-form surface machining. It was shown that in 3D free-form machining, the instantaneous inclination angle has a quite significant role in cutting
Fig. 12. Simulated and experimental forces of the airfoil. (a) Experimental and simulated Fx forces. (b) Detail view of Fx. (c) Experimental and simulated Fy forces. (d) Detail view of Fy.
B. Ozturk et al. / International Journal of Machine Tools & Manufacture 46 (2006) 736–746
745
Fig. 13. Simulated and measured forces in a sinus wave form. (a) Experimental and simulated Fx forces. (b) Detail view of Fx. (c) Predicted surface topography. (d) Experimental and simulated Fy forces. (e) Detail view of Fy.
forces. In this paper, with the new calibration algorithm considering the instantaneous inclination angle at every cutter location point along the tool path, the cutting force predictions and 3D free-form process simulations were enhanced.
Acknowledgements The authors acknowledge Scientific and Technical Research Council of Turkey (TUBITAK) for its support with Young Scientists Career Development Program.
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References [1] M. Yang, H. Park, The prediction of cutting force in ball-end milling, International Journal of Machine Tools and Manufacture 31 (1991) 45–54. [2] P. Lee, Y. Altintas, Prediction of ball-end milling forces from orthogonal cutting data, International Journal of Machine Tools and Manufacture 36 (1996) 1059–1072. [3] G.M. Kim, P.J. Cho, C.N. Chu, Cutting force prediction of sculptured surface ball-end milling using Z-map, International Journal of Machine Tools and Manufacture 40 (2000) 277–291. [4] I. Lazoglu, Sculpture surface machining, a generalized model of ballend milling force system, International Journal of Machine Tools and Manufacture 43 (2003) 453–462. [5] A. Lamikiz, L.N. Lo’pez, J.A. de Lacalle, M.A. Salgado, Cutting force estimation in sculptured surface milling, International
[6]
[7]
[8] [9]
[10]
Journal of Machine Tools and Manufacture 44 (2004) 1511– 1526. A. Azeem, H.Y. Feng, L. Wang, Simplified and efficient calibration of a mechanistic cutting force model for ball-end milling, International Journal of Machine Tools and Manufacture 44 (2004) 291–298. B.U. Guzel, I. Lazoglu, Increasing productivity in sculpture surface machining via off-line piecewise variable feedrate scheduling based on the force system model, International Journal of Machine Tools and Manufacture 44 (2004) 21–28. M.E. Martelotti, An analysis of the milling process, Transactions of the ASME 63 (1941) 677–700. B. Ozturk, I. Lazoglu, Machining of free-form surfaces. Part I: analytical chip load, International Journal of Machine Tools and Manufacture. doi:10.1016/j.Machtools.2005.07.038. Y. Altintas, Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design, Cambridge University Press, 2000.
Machining of free-form surfaces. Part II: Calibration and forces B. Ozturk, I. Lazoglu*, H. Erdim Manufacturing Automation and Research Center, Department of Mechanical Engineering, Koc University, Rumeli Feneri Yolu, Sariyer, Istanbul 34450, Turkey Received 16 February 2005; accepted 19 July 2005 Available online 12 September 2005
Abstract In the machining simulations of 3D free-form surfaces by ball-end milling, calibration coefficients play very critical role in force predictions. In other words, obtaining the calibration coefficients from calibration tests is a very influential process in the prediction of cutting forces. Accurately obtained calibration coefficients lead to better force predictions. In the literature, the calibration coefficients are assumed to be independent of start and exit angles of the engagement region, thus they are assumed to be identical for any engagement region. Calibration coefficients are obtained from the horizontal slot cutting tests and these tests are repeated for different depth of cuts. In this paper, in order to achieve more accurate force predictions in free-form machining simulations, a new modification algorithm for calibration coefficients is presented. In this research, with theoretical analysis and experimental force signals, it is shown that the inclination angle has a great importance in calibration and in force simulation of 3D free-form machining. q 2005 Elsevier Ltd. All rights reserved. Keywords: Calibration coefficients; Engagement region; Inclination angle; Sculptured surfaces
1. Introduction With the enlargement of the free-form surface machining fields in industry, fast and accurate process simulations of free-form surfaces are highly demanded. In the analysis of mechanics and dynamics of 3D free-form machining processes, it is the fact that the most critical parameter is usually the cutting force. Therefore, in the literature up to now, there are various simulation algorithms developed for ball-end milling process; Yang and Park [1] represented a ball-end mill with infinitesimal disc elements that is an approach from end mill force analysis. Lee and Altintas [2] predicted the cutting forces from orthogonal cutting data and Kim et al. [3] presented a force model with Kr/Kt cutting coefficients and include the effect of inclination angle to chip thickness by rotating the coordinate frame and evaluated the cutting coefficients at every rotation angle. In other words, they assume a polynomial like behavior of the calibration coefficients over the engagement region * Corresponding author. Tel.: C90 212 338 1587; fax: C90 212 338 1548. E-mail address: [email protected] (I. Lazoglu).
0890-6955/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2005.07.037
and find the coefficients of the polynomials empirically from horizontal slot cutting tests. Lazoglu [4] employed a Z-mapping and Boolean approach, and formed a model that determined surface topography in addition to predicting cutting forces. Lamikiz et al. [5] performed a similar approach to Lazoglu [4] and he assumed that the calibration coefficients are polynomial of the z-coordinate of the disc elements. They state that assuming friction coefficients as a constant is a valid assumption and assumed edge coefficients constant throughout the machining process. They state that obtained results were only valid for a specific testing case under a given set of conditions and only for the case of horizontal slot cutting, they do not include any explanation of the effect of inclination angle on the calibration coefficients. Moreover, they performed various tests however they do not include a free-form surface. In calibration process, Azeem et al. [6] proposed a simplified calibration technique for the mechanistic approach. Most of the force models in the literature are for 2 12 D machining but not for 3D free-form surfaces. In the analysis of literature, it is seen that those models can handle the 3D free-form surface machining simulations, which have deficiency of excluding inclination angle dependency of the calibration force coefficients by imposing the method stated in this paper.
B. Ozturk et al. / International Journal of Machine Tools & Manufacture 46 (2006) 736–746 200
Cutting Forces for 48 mm/min Fx[N] Fy[N] Fz[N]
100
Force [N]
0
-100
0.5 mm 1.0 mm 1.5 mm 2.0 mm
-200
3.0 mm 4.0 mm
-300 6.0 mm -400
0
100
200
300
400
500
600
700
800
737
Center (VMC). The cutter was carbide ball-end mill cutter from CoroMill Plura series of Sandvik with 12 mm diameter, 37 mm projection length and 308 helix angle. The workpiece materials were aluminum blocks (Al7039) of size 250!170!38 mm3. Kistler 3-component dynamometer (Model 9257B) was used for force data acquisition. Fig. 1 is the combination of forces at the selected depth of cuts which were later to be used for obtaining the calibration coefficients for a feedrate of 48 mm/min. The calibration algorithm of Guzel, Lazoglu [7] is used after this point for seven depths of cut: 0–0.5, 0.5–1, 1–1.5, 1.5–2, 2–3, 3–4, 4–6 mm for feedrates of 48, 96, 144 and 192 at a spindle speed of 600 rpm and force calibration coefficients are presented in Table 1.
Angle [deg] Fig. 1. Combination of forces at selected depth of cuts for a feedrate of 48 mm/m.
In this paper, a new algorithm is presented for the calibration coefficients which are evaluated from the mechanistic approach to simulate the force in free-form machining. Most of the researches about ball-end milling are verified for 2 12 D operations. Validations are performed for half immersion cutting; bisection cutting and slot cutting tests. Free-form surface cutting are not performed very commonly in the literature due to difficulties in contact region determination, the chip thickness determination and in calibration process. In part-I contact region determination is analyzed for monotonic free-form surfaces. In this part of the paper, a new modification algorithm is presented for accurate prediction of calibration coefficients in machining of free-form surfaces. Inclination dependency of calibration coefficients is stated mathematically, validation tests are performed for inclined surfaces and for 3D free-form surfaces.
2. Determination of calibration coefficients In order to determine the cutting force coefficients in this study, Guzel and Lazoglu’s mechanistic approach [7] is used. However, some modifications on cutting forces as detailed in the following sections were performed for better force predictions. The experiments for validation were performed on Mazak FJV-200 UHS Vertical Machining
3. Modification of calibration coefficients The calibration coefficients obtained from horizontal slot cutting tests give accurate results for 2 12 D cutting conditions, on the other hand in free-form surface machining, the measured and simulated results do not match very accurately if one tries to use the modified Martelotti’s [8] chip thickness given in Eq. (1) c Z h sinðqÞsinðjÞ
(1)
where h is feed per tooth per revolution, q is the rotation angle and j is the zenith angle (disc position angle) [7]. Martelotti’s approach is simply the kinematic analysis of a point on the cutting edge. It includes rotational and translational motion and is valid if rotation vector and translational vectors are perpendicular. On the other hand, in free-form surfaces these two vectors are not always perpendicular as shown in Fig. 2, therefore the calibration coefficients for different inclination angles are not valid. They need to be modified by including the inclination angles. In this paper, factors to modify the calibration coefficients obtained from horizontal slot cutting tests are evaluated from the end mill analytical equations. This is acceptable since the ball-end mill can be geometrically represented by end mills at different depths. In addition to that, if the terms which include zenith angle are evaluated at 908, the ball-end mill force equations [7] simplify to the end mill equations. Evaluation of zenith angle (j) at 908 is simply a geometrical conversion of a ball-end mill
Table 1 Calibration coefficients for selected depth of cuts Sections from tool tip coefficients
0–0.5 (mm)
0.5–1 (mm)
1–1.5 (mm)
1.5–2 (mm)
2–3 (mm)
3–4 (mm)
4–6 (mm)
Krc (N/mm2) Kre (N/mm) Kjc (N/mm2) Kje (N/mm) Ktc (N/mm2) Kte (N/mm)
2991.4 31.2 1113.4 11.7 5353.9 36.4
695.4 9.6 784.9 12.7 2549 17.3
22.1 9.0 752.7 3.6 1181.8 29.6
195.7 2.5 178.5 19.5 1429.1 3.1
105.1 7.2 101.5 13.5 971.4 16.3
483.3 14.8 30.8 8.7 847.6 16.9
353 11.9 5.6 0 932.1 10.7
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From Eq. (3), calibration coefficients can be written as: Ktc Z
4F yc 4F ye 4F ; Kte Z ; Krc ZK xc ; Na Na Na
4F pF zc 2F Kre ZK xc ; Kjc Z ; Kje Z ze Na Na Na
(4)
This slot cutting calibration coefficients can be used effectively in end milling processes. On the other hand, since calibration coefficients are dependent on the engagement angles, they need to be modified according to the inclination angle of the surface in ball-end milling. In order to analyze this situation, the calibration coefficients which can be derived from Eq. (2) need to be used instead of the calibration coefficients which are obtained from Eq. (4). However, the conflict to use Eq. (2) is that which start and exit angles are going to be taken since they are not constant throughout the engagement domains unlike horizontal slot cutting F xc C AA12 F yc F xc C AA21 F yc ; Ktcnew Z ; Krcnew Z A2 K AA12 A1 K AA21
Fig. 2. Feed direction in downward motion, rotational vector and inclination angle.
to an end mill. Nac Fx Z ðK cosð2fÞKKrc ð2fKsinð2fÞÞÞ 8p tc fex Na ðKKte sinðfÞ C Kre cosðfÞÞ C ; 2p fst Nac ðK ð2fKsinð2fÞÞ C Krc cosð2fÞÞ F y Z 8p tc fex Na K ðKte cosðfÞ C Kre sinðfÞÞ ; 2p fst fex Na ðKKac c cosðfÞ C Kae fÞ Fz Z 2p fst
F xe C AA34 F ye F new ; ZK zc ; Krenew Z Kjc A4 A C A3 4
Ktenew
(5)
A4
F xe K AA43 F ye F new ; Kje Z Z ze ; A5 A3 C AA43
where fex fex Na Na cosð2fÞ A1 Z ; A2 Z K ð2fKsinð2fÞÞ ; 8p 8p fst fst (2)
Eq. (2) is the analytical equation of average end mill forces stated by Altintas [9]. The calibration coefficients can be obtained by considering horizontal slot cutting tests which means start angles are 08 and exit angles are 1808. 2 3 Na Na F x Z 4K ðKrc cÞK ðKre Þ5 4 p 2 3 Na Na ðK Þ5 gF q Z F qc c C F qe F y Z 4 ðKtc cÞ C (3) 4 p te 2 3 Na Na ðKae Þ5 F z Z 4 ðKac cÞ C p 2
fex fex Na Na ðcosðfÞÞ A3 Z K sinðfÞ ; A4 Z ; 2p 2p fst fst
Na ðfÞ A5 Z 2p
fex fst
Second disadvantage is that the calibration process must be repeated for different inclination angles, thus the time and work spent for calibration increase. For this reason, a modification of calibration coefficient factor is introduced by considering Eqs. (2) and (3). This factor is going to be used to modify calibration coefficients, which are obtained by slot cutting calibration tests. The equations for new calibration coefficients are as given in Eq. (5). In order to handle the problem about the values of start and exit angles that are going to be put into Eq. (2) or Eq. (5), one can take the average of these angles, in whole spectrum with respect to all disc elements which are in contact at that depth of cut. More conveniently, average of start and exit angles can be taken in the intervals where
B. Ozturk et al. / International Journal of Machine Tools & Manufacture 46 (2006) 736–746 F P
f st Z iZ1
tanK1 F
y x
F P
; f ex Z iZ1
tanK1 Kyx F
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;
qffiffiffiffiffiffiffiffiffiffiffiffiffi ri2 KR22 ; y Z ri Kx2 ; R 2 2 1K R qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ri Z R2 KðRKzi Þ2 ; zi Z i; R2 Z R sinðlÞ; F xZ
(6)
i Z 1; .; Nd
Fig. 3. Inclined slot cutting test.
calibration coefficients are valid (inclined slot cutting is assumed). In this paper, since calibration coefficients are taken for seven depths of cut, the average of start and exit angles are taken at these seven intervals. In part-I, the start and exit points are presented, in addition to that for an inclined slot cutting test shown in Fig. 3, start and exit angles are as follows
where l is the inclination angle, x, y are the intersection points of the boundaries with each disc element which are going to be used to define f st (average start angle) and f ex (average exit angle) and Nd is the total disc number in contact with that depth of cut. If calibration tests are proceeded with inclined slot cutting at different inclination angles, then fexZpKfst and thus makes some simplification in the terms given in Eq. (5). A1Z0 and A3Z0 at these test conditions, which means they are not inclination dependent terms for inclined slot cutting tests. From Eq. (5) evaluating the terms at the given condition leads to:
Fig. 4. Forces in 58 upward slot cutting tests. (a) Experimental and simulated Fx forces. (b) Detail view of Fx. (c) Experimental and simulated Fy forces. (d) Detail view of Fy.
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Na A2 ZK ððpK2fst Þ C sinð2fst ÞÞ; 4p Na Na ðpK2fst ÞÞ A4 ZK cosðfst Þ; A5 Z p 2p
Ktec Z (7)
new Kjc Ktcnew Krcnew c c ; K Z ; K Z ; rc jc old Ktcold Krcold Kjc
where Ktcold Z Krcold
F new 4F old yc yc / Ktcnew ZK ; Na A2
F new 4F old ZK xc / Krcnew Z xc ; Na A2
old Kjc Z
F new pF old zc new / Kjc ZK zc ; Na A4
where Kteold Z
If one defines the modification factor as a multiplier of the slot cutting calibration coefficients to be equal to the calibration coefficients of the machining inclination angles then below equations will state the modification of calibration coefficient factors. Ktcc Z
new Kje Ktenew Krenew c c ; K Z ; K Z ; re je old Kteold Kreold Kje
(8)
F new 4F old ye ye / Ktenew ZK ; Na A4
4F old F new Kreold ZK xc / Krenew Z xe ; Na A4 old Kje Z
2F old F new ze new / Kje Z ze Na A5
Eq. (7) simplifies to Eq. (8) and one can show that: Ktcc Z
F new p yc ; ððpK2fst Þ C sinð2fst ÞÞ F old yc
Krcc Z
p ððpK2fst Þ C sinð2fst ÞÞ
c Kjc
1 Z cosðfst Þ
Krec Z
F new xc ; F old xc
F new p zc ; Ktec Z old 4 cosðf F zc st Þ
p 4 cosðfst Þ
F new ye ; F old ye
F new p xe c ; Kje Z old ðpK2f F xe st Þ
(9)
F new ze ; F old ze
Fig. 5. Forces in 108 upward slot cutting tests. (a) Experimental and simulated Fx forces. (b) Detail view of Fx. (c) Experimental and simulated Fy forces. (d) Detail view of Fy.
B. Ozturk et al. / International Journal of Machine Tools & Manufacture 46 (2006) 736–746
4. Validations with inclined slot cutting tests In the validation of modified calibration coefficients method and analytical engagement region, slot cutting experiments are performed at different inclination angles which are K20, K10, K5, 5, 10, 208. Figs. 4–9 include measured force signals from slot cutting tests and modification algorithm force predictions. All slot validation tests and free-form tests are performed with the same workpiece and same cutter stated in Section 2. The engagement domains for all cutting tests are taken from ACRsim program [9] and simulations are performed with these domains. ACRsim gives an output in the form of number of CL-points, start angle, exit angle and inclination angle. Therefore, inclination angle information comes from the domain and force simulation code uses this information to select correct coefficient modification factors, it modifies calibration matrix accordingly. Figs. 4–9 are the slot cutting results for positively and negatively inclined surfaces for a feedrate of 48 mm/min at a spindle speed of 600 rpm. With the increase of the inclination angle simulated results with two methods are getting away from each other and from experimental data, however by using the modification of calibration method the experimental data is predicted with an error below 10%.
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In Figs. 4–9, there are dashed lines which indicate the approach, steady-state and retract regions. Part-I is the analytical solution of full contact cases (steady-state). Thus, the error percentage values are evaluated for the values between these dashed lines. In addition to that, if the factors of modification of calibration coefficient method are independent of the workpiece material, one experiment will be sufficient to predict the change in calibration coefficients for different types of materials. In addition to that, as shown in Fig. 10 there is an identical linear trend observed for the modified calibration coefficient matrices for all calibration feedrates. In this figure, U is the calibration coefficient matrix, where Ulambda is the coefficient matrix modified according to the inclination angle of l and Uslot is the calibration coefficient matrix obtained from horizontal slot cutting tests. Thus, there is no need for a discretization of inclination angles, a linear behavior can be assumed between the angles K20, K10, K5, 5, 10, 208. In the validation of the stated method for free-form surfaces, two tests are performed for a feedrate of 48 mm/min at a spindle speed of 600 rpm. The material used in these tests is the same material with the calibration tests and tests are performed with the same cutter of the same projection length and helix angle. The predictions in these
Fig. 6. Forces in 208 upward slot cutting tests. (a) Experimental and simulated Fx forces. (b) Detail view of Fx. (c) Experimental and simulated Fy forces. (d) Detail view of Fy.
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Fig. 7. Forces in 58 downward slot cutting tests. (a) Experimental and simulated Fx forces. (b) Detail view of Fx. (c) Experimental and simulated Fy forces. (d) Detail view of Fy.
Fig. 8. Forces in 108 downward slot cutting tests. (a) Experimental and simulated Fx forces. (b) Detail view of Fx. (c) Experimental and simulated Fy forces. (d) Detail view of Fy.
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Fig. 9. Forces in 208 downward slot cutting tests. (a) Experimental and simulated Fx forces. (b) Detail view of Fx. (c) Experimental and simulated Fy forces. (d) Detail view of Fy.
figures are again valid for the portion between two dashed lines. In free-form tests, errors are below 10% at maximum. Calculation time of engagement domains for airfoil and for sinusoidal surface were 9.1 and 26.9 s, respectively, with
MATLAB (CPU time) on a Pentium M 1.50 GHz–512 MB RAM. Therefore, the algorithm developed here in this paper is quite fast, and computation time is very acceptable for industrial implementations (Figs. 11–13).
Fig. 10. Linear trend in the modified calibration coefficient matrices.
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Fig. 11. Airfoil after machining simulation.
5. Conclusions In this paper, in order to achieve more accurate force predictions in 3D free-form machining processes, a new modification algorithm for calibration coefficients were
stated. It was observed that calibration coefficients obtained from 2 12 D cutting tests were not sufficient in prediction of the forces for 3D free-form surface machining. It was shown that in 3D free-form machining, the instantaneous inclination angle has a quite significant role in cutting
Fig. 12. Simulated and experimental forces of the airfoil. (a) Experimental and simulated Fx forces. (b) Detail view of Fx. (c) Experimental and simulated Fy forces. (d) Detail view of Fy.
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Fig. 13. Simulated and measured forces in a sinus wave form. (a) Experimental and simulated Fx forces. (b) Detail view of Fx. (c) Predicted surface topography. (d) Experimental and simulated Fy forces. (e) Detail view of Fy.
forces. In this paper, with the new calibration algorithm considering the instantaneous inclination angle at every cutter location point along the tool path, the cutting force predictions and 3D free-form process simulations were enhanced.
Acknowledgements The authors acknowledge Scientific and Technical Research Council of Turkey (TUBITAK) for its support with Young Scientists Career Development Program.
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