- Email: [email protected]
Research on generation of three-dimensional surface with micro-electrolyte jet machining
CIRP Journal of Manufacturing Science and Technology 1 (2008) 27–34
Contents lists available at ScienceDirect
CIRP Journal of Manufacturing Science and Technology journal homepage: www.elsevier.com/locate/cirpj
Research on generation of three-dimensional surface with micro-electrolyte jet machining W. Natsu *, S. Ooshiro, M. Kunieda Department of Mechanical Systems Engineering, Tokyo University of Agriculture and Technology, 2-24-16 Nakacho, Koganei, Tokyo 184-8588, Japan
A R T I C L E I N F O
A B S T R A C T
Article history:
This paper proposes a method to generate three-dimensional surfaces with micro-electrolyte jet machining and discusses how to determine the nozzle path and scanning speed. In order to verify the effectiveness of the proposed method, calculated and machined results were compared. It was found that three-dimensional surfaces can be generated by superimposing elemental curved grooves, and good agreement between the cross-sectional shapes of the superimposed and machined surfaces with those of the target surface proves the effectiveness of the proposed method. In addition, layer-by-layer machining with optimized interval of the nozzle path is an effective way to reduce machining errors. ß 2008 CIRP.
Available online 11 July 2008 Keywords: ECM Electrolyte jet machining Micro-machining Micro-nozzle Surface generation Optimization
1. Introduction Electrolyte jet machining [1–5] is a method to remove or add material on the workpiece surface to obtain required shapes. The movement of a simple nozzle enables selective machining without the use of special masks, since the electrolytic current concentrates on the jet area [6]. Up to now, the following applications of electrolyte jet machining have been reported; removing process [1–5] by anodic ion dissolution from the anode, adding process [7] by metallic ion deposition on the cathode, and coloring process [8] by anodic oxidation film formation. Since electrolyte jet machining is an electrolytic process, metallic materials can easily be processed, regardless of their hardness. In addition, there are no burrs, cracks, and heat affected zones on the machined surface. Hence, applications to micro-precision machining of molds for micro-lenses and micro-grooves on bearing surface can be looked forward to. In order to accelerate practical application, laser assisted jet electrochemical machining was attempted to improve machining precision and efficiency [2,9,10]. In addition, high machining efficiency by using multiple nozzles [11] and micromachining by adopting filter and vacuuming devices [12] were successfully achieved. Meanwhile, by scanning the simple nozzle, complicated shapes could also be fabricated. To date, the authors have developed a fabricating method to obtain two-dimensional complicated surfaces by superimposing elemental linear grooves [13]. However, due to the strong demands for the machining of
* Corresponding author. Tel.: +81 42 388 7776; fax: +81 42 388 7776. E-mail address: [email protected] (W. Natsu). 1755-5817/$ – see front matter ß 2008 CIRP. doi:10.1016/j.cirpj.2008.06.006
complicated three-dimensional surface in practical use, in this study, the authors therefore attempted machining of the required micro three-dimensional surface by superimposing elemental curved grooves. A method to determine the scanning speed of the jet was developed when a tool path was given to obtain an arbitrary shape, and the influence of the tool path on the machining accuracy was discussed. 2. Principle and experimental setup for electrolyte jet machining Electrolyte jet machining is carried out by jetting electrolytic aqueous solution from a nozzle toward the workpiece while applying voltage between the nozzle and workpiece as shown in Fig. 1. When the electrolyte is supplied at a sufficient flow rate, a sudden change in thickness of the radial electrolyte flow occurs in areas far away from the nozzle due to the hydraulic jump phenomenon [14]. In this case, only the material directly under the jet is machined by electrolysis because the current is distributed in a narrow area [6]. Fig. 2 shows the experimental setup. The workpiece was installed on the machining table which can be positioned horizontally by an X–Y stage. The nozzle was installed on the slide table which can be moved in the Z direction to adjust the gap width between the workpiece and nozzle. The electrolyte was pressurized by an air compressor in the pressure tank and jetted from the nozzle toward the workpiece. A high speed bipolar power supply was used to supply the processing current. Current control was carried out with a control signal from a personal computer which simultaneously controls the X–Y stage. The nozzle was set to
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W. Natsu et al. / CIRP Journal of Manufacturing Science and Technology 1 (2008) 27–34
Fig. 3. Nozzle path for machining curved groove. Fig. 1. Schematic of electrolyte jet machining.
the cathode and the workpiece was set to the anode in the removing process. The removal rate of the workpiece material is generally determined by the current density in electrolyte jet machining. However, the generation of the oxide film becomes dominant in the area where the current density is lower than the threshold, when a passivation type electrolyte is used. Since the generated passive film prevents machining of the material, material removal is limited to a narrower area, thus improving machining precision. For this reason, a sodium nitrate aqueous solution, a passivation type electrolyte, was used in this study to realize precision machining. 3. Calculating shape of elemental curved groove 3.1. Difference between linear groove and curved groove The removal amount per unit area of the workpiece is proportional to the current density and the time period that the jet stays at that area. In the case that the nozzle is moved in a straight line, the current density and staying period of the jet is the same at the right and left sides of the nozzle center trajectory, as long as the distance from the nozzle center is equal. Hence, the machining amount on both sides is equal and the cross-sectional shape shows a line symmetry [13]. However, when the nozzle is moved circularly as shown in Fig. 3, the shape of the arched path along which a certain point on the workpiece follows in the jet differs between the right and left sides of the nozzle center trajectory, even if the distance from the nozzle center is equal. Hence, the current density is not equivalent. In addition, the central angle of the arc differs between the right and left sides of the nozzle center. Thus, the staying time of the jet differs according to the right or left side of the trajectory, which means that the
Fig. 2. Experimental equipment for electrolyte jet machining.
cross-sectional shape will not be symmetrical in this case, and the shape changes with the radius of center trajectory. 3.2. Calculation of cross-sectional shape of elemental curved groove This section describes the calculation method of the crosssectional shape of an elemental curved groove from a pit shape which is produced by a stationary jet. Since the material in the area with low current density is not removed as discussed in Section 2 when sodium nitrate electrolyte is used, the produced shape differs from the current density distribution which can be obtained from analysis. Hence, in this study, the shape of the curved groove was calculated based on the pit shape obtained experimentally using a stationary jet. When other kinds of electrolyte, such as sodium chloride, were used in experiments, the shape of curved groove was calculated based on the corresponding pit shape obtained experimentally. The cross-sectional shape of a pit obtained under the conditions shown in Table 1 was measured and shown in Fig. 4. The machining time was 0.5 s, and a nozzle of 50 mm inner diameter was used. The radius of the obtained pit R0 was 37.5 mm. From this experimental Table 1 Conditions for pit machining Nozzle inner diameter (mm) Electrolyte Electrolyte concentration (wt%) Tank pressure (MPa) Gap width (mm) Machining current (mA) Machining time (s) Workpiece
50 Sodium nitrate aqueous solution 20 0.5 0.3 3 0.5 Stainless steel
Fig. 4. Cross-sectional shape of machined pit.
W. Natsu et al. / CIRP Journal of Manufacturing Science and Technology 1 (2008) 27–34
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Fig. 6. Machined area. (a) r0 R0, (b) r0 > R0.
Fig. 5. Method to calculate depth of curved groove.
result, the depth of the standard pit machined by a stationary jet per unit time, 1 s, was expressed as a function of the distance from the pit center, R mm: DðRÞ ðmmÞ ¼ ð6:0 106 ÞR4 þ ð3:2 103 ÞR2 14:8
r v0 r0
(2)
The time dt needed for the jet to pass the infinitesimal length dl on the path can be expressed as dt ¼
r0 r0 r0 dl ¼ r du ¼ du rv0 rv0 v0
(3)
Here, since the depth of the standard pit, D(R), is expressed with Eq. (1), the calculated depth of the curved groove can be expressed by Eq. (4) by integrating the machining depth of the standard pit with the time dt. dðrÞ ¼
Z
t0
DðRÞdt ¼ t0
Z u0 u 0
DðRÞ
r0 du v0
Here, R is the function of u and can be expressed as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ¼ r 2 þ r 0 2 2rr 0 cos u
u0 ¼ p
ðr 0 R0 and r R0 r 0 Þ
(6)
u0 ¼ cos1
r 2 þ r02 R20 2rr 0
(6 )
(1)
Then the cross-sectional shape of the curved groove obtained by scanning the nozzle circularly is calculated as follows. Fig. 5 shows the relation between the generated groove and trajectory of the electrolyte jet. To calculate the groove depth at a location with a distance r from the curvature center, the arched path in the jet shown with the bold line in Fig. 5 is considered. Suppose that the nozzle scanning speed at the nozzle center is v0 and the distance between the curvature center and nozzle center is r0, then the passing speed v of the electrolyte jet at the location distanced r from the curvature center can be expressed as v¼
In the case that the radius of nozzle center trajectory r0 is equal or smaller than the pit radius R0, a circle area whose radius is r0 + R0 as shown in Fig. 6(a) will be produced. The depth at position r is obtained by integral from u0 to u0 as shown in Eq. (4), where u0 is expressed by Eq. (6).
(4)
(5)
ðr 0 R0 and r > R0 r 0 Þ
0
When r0 is larger than R0, a ring area between the circles where radii are r0 R0 and r0 + R0, respectively, will be produced as shown in Fig. 6(b). The depth of the groove at r position is obtained by integrating from u0 to u0, where u0 is expressed by,
u0 ¼ cos1
r 2 þ r02 R20 2rr 0
ðr 0 > R0 Þ
(7)
3.3. Confirmation of calculated result In order to compare calculated groove depths with experimental ones, three curved grooves whose radius of nozzle trajectory r0 was 30, 100 and 500 mm were machined. The nozzle scanning speed was set to 0.25 mm/s, and the machining conditions shown in Table 1 were used. Fig. 7 shows the machined shapes of the curved grooves and Fig. 8 shows their cross-sectional shapes. The calculated results are also shown in the same figure for comparison. Since all calculated results were similar to the machined shapes, the reliability of the calculation method was confirmed. In addition, from Fig. 8(b) and (c) it can be found that the cross-sectional shape is not line symmetry. The reason is that the staying time of the jet at a unit area on the left side of the nozzle center trajectory is longer than that on the right side. As a result, the deepest position of the groove shifted to the left side of the nozzle center. For a curved groove with a radius of curvature of 100
Fig. 7. Curved grooves machined using micro-nozzle grooves. (a) r0 = 30 mm, (b) r0 = 100 mm, (c) r0 = 500 mm.
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Fig. 8. Cross-sectional shape of elemental curved grooves. (a) r0 = 30 mm, (b) r0 = 100 mm, (c) r0 = 500 mm.
and 500 mm, the deepest position shifted to the left by 12.7 and 2.7% of the groove width, respectively. 4. Surface generation by superimposing elemental curved grooves
Since a curved groove k times deeper than that of the elemental groove can be obtained by reducing the scanning speed to 1/k of v0 which was used to obtain the elemental curved groove, the scanning speed to obtain the depth aiyi(r) is expressed by, vi ¼
Two-dimensional complicated surface can be generated by superimposing linear grooves [13]. However, to obtain a threedimensional surface such as spherical surface, superimposition of curved grooves is necessary. In this section, the method to generate a three-dimensional surface is explained. 4.1. Theory for 3D surface generation by superimposing elemental curved grooves There are different ways to generate a 3D surface shape by superimposing curved grooves. In this study, it is carried out by controlling the path and scanning speed of the nozzle. The crosssectional shape of the ith elemental curved groove with the nozzle scanning speed of v0 is obtained by the method explained in Section 3 and expressed as yi(r). Then, the cross-sectional shape of the ith curved groove with the nozzle scanning speed of v0 =ai and the same trajectory as the ith elemental curved groove is aiyi(r). Suppose the target shape YT is approximated by superimposition of n elemental curved grooves (see Fig. 9), the superimposed groove shape Ys can then be expressed by summation of n curved grooves as, Y s ðrÞ ¼
n X ai yi ðrÞ
(8)
1
To evaluate the difference between the target and superimposed shapes, an evaluation function f is adopted as follows:
f¼
n X ½Y T ðr j Þ Y s ðr j Þ2
v0 ai
4.2. Example of groove generation In order to confirm the validity of the above method, a target curved groove whose cross-sectional shape is 375 mm wide and 20 mm deep parabolic curve, expressed by the following equation, was generated. Y T ðrÞ ðmmÞ ¼ ð5:68 104 Þðr 350Þ2 20
(9)
Here, rj represents the r coordinate value of the evaluated point, where the difference between the target and superimposed shape is evaluated. Since the difference was evaluated on the trajectory of the nozzle center, the total number of evaluated points was n. It is clear that other variables except ai in the evaluation function are known. Therefore, in order to obtain the superimposed shape which best fits the target one, there is only a need to find out a set of superimposed coefficients ai (i = 1 to n) which minimizes the evaluating function. This means that the optimized superimposed coefficients can be obtained by solving the n simultaneous equations derived by equating the derivative of the evaluation function with respect to ai to 0 as follows: ði ¼ 1; 2; . . . ; nÞ
(12)
Twenty-five elemental curved grooves whose radius of curvature ranges from 200 to 500 mm were superimposed at a radial interval of 12.5 mm, which is one-quarter of the nozzle inner diameter. Fig. 10(a) shows the cross-sectional shape of the superimposed groove obtained by the calculation and the target shape, while Fig. 10(b) shows the difference between the results of the superimposed and target shapes. In this study, the difference between the superimposed and target shapes is called superimposing error, while the difference between the machined and target shapes is called machining error. Fig. 10 shows that the shape of the superimposed groove is nearly the same as that of the target curved groove, although there is some error on the edge area of the groove. The superimposed curved groove was machined using the nozzle scanning speeds obtained from the above method. Other machining conditions were the same as those in Table 1.
1
@f ¼0 @ai
(11)
(10) Fig. 9. Shape generation by superimposing elemental curved grooves.
W. Natsu et al. / CIRP Journal of Manufacturing Science and Technology 1 (2008) 27–34
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Fig. 10. Comparison of superimposed and target shapes. (a) Cross-sectional shape, (b) superimposing error.
Fig. 11. Groove shape machined by superimposing elemental curved grooves.
Fig. 11 shows the three-dimensional shape measured using a laser confocal microscope. Furthermore, Fig. 12(a) shows the target, superimposed, and machined cross-sectional shape, while Fig. 12(b) shows the superimposing and machining errors. It can be seen from Fig. 12 that the machined shape more or less corresponds with the target one except at the edge area of the groove. This fact proves that complicated shapes can be generated by superimposing the elemental curved grooves. 5. Generation of spherical surface In the above section, a parabolic shaped groove was generated by superimposing the elemental curved grooves whose curvature radii were larger than the pit radius. However, in order to obtain an arbitrary surface, superimposing the elemental curved grooves
whose radii of curvature of the nozzle center trajectory are smaller than the pit radius is necessary. In this section, a spherical surface which needs superimposition of these two kinds of elemental grooves is generated and the results are discussed. Also described are superimposing errors which occur in three kinds of superimposing methods; namely one-step superimposing, layer-bylayer superimposing, and layer-by-layer superimposing with interval optimization. Here, one-step superimposing means to reach the target depth by superimposing each elemental curved groove only once. In the case of layer-by-layer superimposing, the target depth is obtained by multiple steps of superimposition in order to reduce the accumulated error and increase superimposing precision. The target surface is a portion of spherical surface shown in Fig. 13. The radius of the spherical surface is 1100 mm and the depth is 120.2 mm. Such a target surface was chosen because the surface could be mathematically expressed easily. For other shapes, if the surfaces can be expressed mathematically or numerically, the scanning speed to generate the surface will be obtained with Eqs. (8)–(11) in a similar way. However, for the complicated shapes with markedly uneven surface, machining error will increase. 5.1. One-step superimposing In the case of one-step superimposing, one pit and 37 elemental curved grooves were used to generate the spherical surface. The pit was produced at the center first. Then, 37 concentric curved grooves were superimposed with a uniform radial interval of 12.5 mm. Under this condition, the superimposed coefficients for the pit and each elemental groove were obtained with Eq. (10), and the superimposed result was calculated with Eq. (8). The target and calculated cross-sectional shapes are shown in Fig. 14. The results showed that there exists considerable superimposing error around
Fig. 12. Comparison of target, superimposed and machined shapes. (a) Cross-sectional shape, (b) superimposing and machining error.
W. Natsu et al. / CIRP Journal of Manufacturing Science and Technology 1 (2008) 27–34
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each layer was 40, 40 and 40.2 mm, respectively. While in the case of six layers, the target depth for the first five layer was 20 mm and for the last layer 20.2 mm. The superimposing method for each layer was the same as that used in one-step superimposing. That is, a pit was produced at the center first, and then 37 concentric curved grooves were superimposed with an uniform interval of 12.5 mm. The superimposed coefficients for the pit and each elemental groove were obtained from Eq. (10). The target and calculated cross-sectional shapes in the case of 6-layer superimposing are shown in Fig. 15. The peak-to-valley value of the error for one-step superimposing was 57 mm (see Fig. 14(b)), while that for the 3-layer superimposing was 27 mm and that for the 6-layer superimposing was about 14 mm. From these results, it can be concluded that the layer-by-layer superimposing is an effective way to reduce the superimposing error. Although further increase in the number of layers may decrease superimposing error, excessive number will result in an increased machining time because the scanning speed of the nozzle has a limit. 5.3. Layer-by-layer superimposing with interval optimization Fig. 13. Target shape.
the center of the spherical cave, although the error is insignificant at other locations. 5.2. Layer-by-layer superimposing If the target spherical surface is generated by several steps, the superimposing error caused by the former step may be compensated by the following step to some degree. Therefore, the layerby-layer superimposing was attempted. The spherical cave 120.2 mm in depth was generated with three layers and six layers. In the case of three layers, the target depth for
Although it was found that layer-by-layer superimposing could reduce the superimposing error greatly, there still remains some superimposing error around the center. Hence, the influence of radial interval on superimposing error was investigated. A 6-layer superimposing was carried out. Since the depth of the superimposed surface was deeper than the target shape at the center, the pit at the center was not produced as shown in Fig. 15. Thus, the spherical surface was machined by superimposing 38 concentric elemental curved grooves. Besides, since the superimposing error near the center is remarkable, the interval of the path was optimized at the center area only. As shown in Fig. 16, the radial interval for the first four curved grooves were changed from 1 to 35 mm, while the total of them were fixed as expressed by the
Fig. 14. Cross-sectional shape and superimposing error in the case of one-step superimposing. (a) Cross-sectional shape, (b) superimposing error.
Fig. 15. Cross-sectional shape and superimposing error in the case of layer-by-layer superimposing. (a) Cross-sectional shape, (b) superimposing error.
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Fig. 18. Machined spherical surface. Fig. 16. Optimization of radial interval of elemental curved grooves.
Fig. 17. Cross-sectional shape and superimposed error in the case of layer-by-layer superimposing with interval optimization. (a) Cross-sectional shape, (b) superimpose error.
following equation.
Dr1 þ Dr2 þ Dr3 þ Dr4 ¼ 37:5 ðmmÞ
(13)
For the other 34 grooves, the interval was fixed at 12.5 mm. The optimal superimposed coefficients for the 38 circular grooves were obtained from Eq. (10), under the different combinations for Dr1 to Dr4. The superimposed cross-sections for different combinations were then compared, and the cross-sectional shape with the smallest superimposing error is shown in Fig. 17. Comparing Fig. 15 with Fig. 17, it can be seen that optimizing the radial interval reduces the superimposing error not only at the center area but also in the peripheral area. 5.4. Comparison of machined and calculated results Using the optimized interval and scanning speed obtained in Section 5.3, the layer-by-layer machining was carried out. The machining conditions were the same as those in Table 1. The measured three-dimensional shape and the cross-sectional shape are shown in Figs. 18 and 19, respectively. From Fig. 19, it can be seen that by superimposing elemental circular grooves, a spherical shape could be generated. However, the machining error of 30 mm in maximum was generated. The reason for this machining error is considered to be due to the change in the gap width during machining. In order to investigate the influence of the gap width, two grooves were machined at the gap width 0.3 and 0.42 mm, which were the same as the gap width before and after the spherical surface generation. The scanning speed for the two grooves was 0.25 mm/s. The cross-sectional shapes of the obtained grooves are shown in Fig. 20. It was found from the results that the machined
Fig. 19. Cross-sectional shape of machined spherical surface.
Fig. 20. Influence of change in gap width.
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W. Natsu et al. / CIRP Journal of Manufacturing Science and Technology 1 (2008) 27–34
area broadened with the increase of the gap width. Hence, for precision machining, the change in the gap width should be taken into consideration. 6. Conclusions In this study, a method to generate a three-dimensional surface with micro-electrolyte jet machining was proposed and verified theoretically and experimentally. From the calculations and experiments, the following conclusions were drawn. 1. Three-dimensional surface could be generated by superimposing elemental curved grooves. 2. Good agreement of the cross-sectional shapes of the calculated and superimposed surface with the target surface proves the effectiveness of the proposed method. 3. Layer-by-layer superimposing with interval optimization is an effective way to reduce superimposing error. 4. For precise surface generation, change in gap width during machining should be taken into consideration.
Acknowledgement The research presented in this paper is partially funded by Japan Society for the Promotion of Science (Grants-in-Aid for Scientific Research (C), Project No. 18560098).
References [1] Chase, B.D., Holt, D.B., Unvala, B.A., 1972, Jet Polishing of Semiconductors, Journal of Electrochemical Society, 119/3: 310–313. [2] Datta, M., Romankiw, L.T., Vigliotti, D.R., von Gutfeld, R.J., 1989, Jet and LaserJet Electrochemical Micromachining of Nickel and Steel, Journal of Electrochemical Society, 136/8: 2251–2256. [3] Kozak, J., 1989, Some Aspects of Electro Jet Drilling, Developments in Production Engineering Design & Control, 363–369. [4] Kunieda, M., Yoshida, M., Yoshida, H., Akamatsu, Y., 1993, Influence of Micro Indents Formed by Electro-Chemical Jet Machining on Rolling Bearing Fatigue Life, ASME PED, 64:693–699. [5] Kozak, J., Rajurkar, K.P., Balkrishna, R., 1996, Study of Electrochemical Jet Machining, Process Transactions of the ASME-Journal of Manufacturing Sciences and Engineering, 118/4: 490–498. [6] Yoneda, K., Kunieda, M., 1996, Numerical Analysis of Cross Section Shape of Micro-Indents Formed by the Electrochemical Jet Machining, Journal of JSEME, 29/63: 1–8. (in Japanese). [7] Kunieda, M., Katoh, R., Mori, Y., 1998, Rapid Prototyping by Selective Electro Deposition using Electrolyte Jet, Annals of the CIRP, 47/1: 161–164. [8] Mori, Y., Kunieda, M., 1997, Maskless Coloring of Titanium Alloy using Electrolyte Jet, Proceedings of JSEME Annual Meeting, 13–16. (in Japanese). [9] Rajurkar, K.P., Kozak, J., 2001, Laser Assisted Electrochemical Machining, Transactions of NAMRI, 29:421–427. [10] Pajak, P.T., Desilva, A.K.M., Harrison, D.K., Mcgeough, J.A., 2006, Precision and Efficiency of Laser Assisted Jet Electrochemical Machining, Precision Engineering, 30:288–298. [11] Ikeda, T., Natsu, W., Kunieda, M., 2006, Electrolyte Jet Machining using Multiple Nozzles, International Journal of Electrical Machining, 11:25–31. [12] Ooshiro, S., Kunieda, M., Natsu, W., 2007, Study on Micro Electrolyte Jet Machining, in: Proceedings of the 15th ISEM, pp.387–392. [13] Natsu, W., Ikeda, T., Kunieda, M., 2007, Generating Complicated Surface with Electrolyte Jet Machining, Precision Engineering, 31/1: 33–39. [14] Bryson, A.E., 1972, Waves in Fluids, Illustrated Experiments in Fluid Mechanics, 105–112.
Contents lists available at ScienceDirect
CIRP Journal of Manufacturing Science and Technology journal homepage: www.elsevier.com/locate/cirpj
Research on generation of three-dimensional surface with micro-electrolyte jet machining W. Natsu *, S. Ooshiro, M. Kunieda Department of Mechanical Systems Engineering, Tokyo University of Agriculture and Technology, 2-24-16 Nakacho, Koganei, Tokyo 184-8588, Japan
A R T I C L E I N F O
A B S T R A C T
Article history:
This paper proposes a method to generate three-dimensional surfaces with micro-electrolyte jet machining and discusses how to determine the nozzle path and scanning speed. In order to verify the effectiveness of the proposed method, calculated and machined results were compared. It was found that three-dimensional surfaces can be generated by superimposing elemental curved grooves, and good agreement between the cross-sectional shapes of the superimposed and machined surfaces with those of the target surface proves the effectiveness of the proposed method. In addition, layer-by-layer machining with optimized interval of the nozzle path is an effective way to reduce machining errors. ß 2008 CIRP.
Available online 11 July 2008 Keywords: ECM Electrolyte jet machining Micro-machining Micro-nozzle Surface generation Optimization
1. Introduction Electrolyte jet machining [1–5] is a method to remove or add material on the workpiece surface to obtain required shapes. The movement of a simple nozzle enables selective machining without the use of special masks, since the electrolytic current concentrates on the jet area [6]. Up to now, the following applications of electrolyte jet machining have been reported; removing process [1–5] by anodic ion dissolution from the anode, adding process [7] by metallic ion deposition on the cathode, and coloring process [8] by anodic oxidation film formation. Since electrolyte jet machining is an electrolytic process, metallic materials can easily be processed, regardless of their hardness. In addition, there are no burrs, cracks, and heat affected zones on the machined surface. Hence, applications to micro-precision machining of molds for micro-lenses and micro-grooves on bearing surface can be looked forward to. In order to accelerate practical application, laser assisted jet electrochemical machining was attempted to improve machining precision and efficiency [2,9,10]. In addition, high machining efficiency by using multiple nozzles [11] and micromachining by adopting filter and vacuuming devices [12] were successfully achieved. Meanwhile, by scanning the simple nozzle, complicated shapes could also be fabricated. To date, the authors have developed a fabricating method to obtain two-dimensional complicated surfaces by superimposing elemental linear grooves [13]. However, due to the strong demands for the machining of
* Corresponding author. Tel.: +81 42 388 7776; fax: +81 42 388 7776. E-mail address: [email protected] (W. Natsu). 1755-5817/$ – see front matter ß 2008 CIRP. doi:10.1016/j.cirpj.2008.06.006
complicated three-dimensional surface in practical use, in this study, the authors therefore attempted machining of the required micro three-dimensional surface by superimposing elemental curved grooves. A method to determine the scanning speed of the jet was developed when a tool path was given to obtain an arbitrary shape, and the influence of the tool path on the machining accuracy was discussed. 2. Principle and experimental setup for electrolyte jet machining Electrolyte jet machining is carried out by jetting electrolytic aqueous solution from a nozzle toward the workpiece while applying voltage between the nozzle and workpiece as shown in Fig. 1. When the electrolyte is supplied at a sufficient flow rate, a sudden change in thickness of the radial electrolyte flow occurs in areas far away from the nozzle due to the hydraulic jump phenomenon [14]. In this case, only the material directly under the jet is machined by electrolysis because the current is distributed in a narrow area [6]. Fig. 2 shows the experimental setup. The workpiece was installed on the machining table which can be positioned horizontally by an X–Y stage. The nozzle was installed on the slide table which can be moved in the Z direction to adjust the gap width between the workpiece and nozzle. The electrolyte was pressurized by an air compressor in the pressure tank and jetted from the nozzle toward the workpiece. A high speed bipolar power supply was used to supply the processing current. Current control was carried out with a control signal from a personal computer which simultaneously controls the X–Y stage. The nozzle was set to
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W. Natsu et al. / CIRP Journal of Manufacturing Science and Technology 1 (2008) 27–34
Fig. 3. Nozzle path for machining curved groove. Fig. 1. Schematic of electrolyte jet machining.
the cathode and the workpiece was set to the anode in the removing process. The removal rate of the workpiece material is generally determined by the current density in electrolyte jet machining. However, the generation of the oxide film becomes dominant in the area where the current density is lower than the threshold, when a passivation type electrolyte is used. Since the generated passive film prevents machining of the material, material removal is limited to a narrower area, thus improving machining precision. For this reason, a sodium nitrate aqueous solution, a passivation type electrolyte, was used in this study to realize precision machining. 3. Calculating shape of elemental curved groove 3.1. Difference between linear groove and curved groove The removal amount per unit area of the workpiece is proportional to the current density and the time period that the jet stays at that area. In the case that the nozzle is moved in a straight line, the current density and staying period of the jet is the same at the right and left sides of the nozzle center trajectory, as long as the distance from the nozzle center is equal. Hence, the machining amount on both sides is equal and the cross-sectional shape shows a line symmetry [13]. However, when the nozzle is moved circularly as shown in Fig. 3, the shape of the arched path along which a certain point on the workpiece follows in the jet differs between the right and left sides of the nozzle center trajectory, even if the distance from the nozzle center is equal. Hence, the current density is not equivalent. In addition, the central angle of the arc differs between the right and left sides of the nozzle center. Thus, the staying time of the jet differs according to the right or left side of the trajectory, which means that the
Fig. 2. Experimental equipment for electrolyte jet machining.
cross-sectional shape will not be symmetrical in this case, and the shape changes with the radius of center trajectory. 3.2. Calculation of cross-sectional shape of elemental curved groove This section describes the calculation method of the crosssectional shape of an elemental curved groove from a pit shape which is produced by a stationary jet. Since the material in the area with low current density is not removed as discussed in Section 2 when sodium nitrate electrolyte is used, the produced shape differs from the current density distribution which can be obtained from analysis. Hence, in this study, the shape of the curved groove was calculated based on the pit shape obtained experimentally using a stationary jet. When other kinds of electrolyte, such as sodium chloride, were used in experiments, the shape of curved groove was calculated based on the corresponding pit shape obtained experimentally. The cross-sectional shape of a pit obtained under the conditions shown in Table 1 was measured and shown in Fig. 4. The machining time was 0.5 s, and a nozzle of 50 mm inner diameter was used. The radius of the obtained pit R0 was 37.5 mm. From this experimental Table 1 Conditions for pit machining Nozzle inner diameter (mm) Electrolyte Electrolyte concentration (wt%) Tank pressure (MPa) Gap width (mm) Machining current (mA) Machining time (s) Workpiece
50 Sodium nitrate aqueous solution 20 0.5 0.3 3 0.5 Stainless steel
Fig. 4. Cross-sectional shape of machined pit.
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Fig. 6. Machined area. (a) r0 R0, (b) r0 > R0.
Fig. 5. Method to calculate depth of curved groove.
result, the depth of the standard pit machined by a stationary jet per unit time, 1 s, was expressed as a function of the distance from the pit center, R mm: DðRÞ ðmmÞ ¼ ð6:0 106 ÞR4 þ ð3:2 103 ÞR2 14:8
r v0 r0
(2)
The time dt needed for the jet to pass the infinitesimal length dl on the path can be expressed as dt ¼
r0 r0 r0 dl ¼ r du ¼ du rv0 rv0 v0
(3)
Here, since the depth of the standard pit, D(R), is expressed with Eq. (1), the calculated depth of the curved groove can be expressed by Eq. (4) by integrating the machining depth of the standard pit with the time dt. dðrÞ ¼
Z
t0
DðRÞdt ¼ t0
Z u0 u 0
DðRÞ
r0 du v0
Here, R is the function of u and can be expressed as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ¼ r 2 þ r 0 2 2rr 0 cos u
u0 ¼ p
ðr 0 R0 and r R0 r 0 Þ
(6)
u0 ¼ cos1
r 2 þ r02 R20 2rr 0
(6 )
(1)
Then the cross-sectional shape of the curved groove obtained by scanning the nozzle circularly is calculated as follows. Fig. 5 shows the relation between the generated groove and trajectory of the electrolyte jet. To calculate the groove depth at a location with a distance r from the curvature center, the arched path in the jet shown with the bold line in Fig. 5 is considered. Suppose that the nozzle scanning speed at the nozzle center is v0 and the distance between the curvature center and nozzle center is r0, then the passing speed v of the electrolyte jet at the location distanced r from the curvature center can be expressed as v¼
In the case that the radius of nozzle center trajectory r0 is equal or smaller than the pit radius R0, a circle area whose radius is r0 + R0 as shown in Fig. 6(a) will be produced. The depth at position r is obtained by integral from u0 to u0 as shown in Eq. (4), where u0 is expressed by Eq. (6).
(4)
(5)
ðr 0 R0 and r > R0 r 0 Þ
0
When r0 is larger than R0, a ring area between the circles where radii are r0 R0 and r0 + R0, respectively, will be produced as shown in Fig. 6(b). The depth of the groove at r position is obtained by integrating from u0 to u0, where u0 is expressed by,
u0 ¼ cos1
r 2 þ r02 R20 2rr 0
ðr 0 > R0 Þ
(7)
3.3. Confirmation of calculated result In order to compare calculated groove depths with experimental ones, three curved grooves whose radius of nozzle trajectory r0 was 30, 100 and 500 mm were machined. The nozzle scanning speed was set to 0.25 mm/s, and the machining conditions shown in Table 1 were used. Fig. 7 shows the machined shapes of the curved grooves and Fig. 8 shows their cross-sectional shapes. The calculated results are also shown in the same figure for comparison. Since all calculated results were similar to the machined shapes, the reliability of the calculation method was confirmed. In addition, from Fig. 8(b) and (c) it can be found that the cross-sectional shape is not line symmetry. The reason is that the staying time of the jet at a unit area on the left side of the nozzle center trajectory is longer than that on the right side. As a result, the deepest position of the groove shifted to the left side of the nozzle center. For a curved groove with a radius of curvature of 100
Fig. 7. Curved grooves machined using micro-nozzle grooves. (a) r0 = 30 mm, (b) r0 = 100 mm, (c) r0 = 500 mm.
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Fig. 8. Cross-sectional shape of elemental curved grooves. (a) r0 = 30 mm, (b) r0 = 100 mm, (c) r0 = 500 mm.
and 500 mm, the deepest position shifted to the left by 12.7 and 2.7% of the groove width, respectively. 4. Surface generation by superimposing elemental curved grooves
Since a curved groove k times deeper than that of the elemental groove can be obtained by reducing the scanning speed to 1/k of v0 which was used to obtain the elemental curved groove, the scanning speed to obtain the depth aiyi(r) is expressed by, vi ¼
Two-dimensional complicated surface can be generated by superimposing linear grooves [13]. However, to obtain a threedimensional surface such as spherical surface, superimposition of curved grooves is necessary. In this section, the method to generate a three-dimensional surface is explained. 4.1. Theory for 3D surface generation by superimposing elemental curved grooves There are different ways to generate a 3D surface shape by superimposing curved grooves. In this study, it is carried out by controlling the path and scanning speed of the nozzle. The crosssectional shape of the ith elemental curved groove with the nozzle scanning speed of v0 is obtained by the method explained in Section 3 and expressed as yi(r). Then, the cross-sectional shape of the ith curved groove with the nozzle scanning speed of v0 =ai and the same trajectory as the ith elemental curved groove is aiyi(r). Suppose the target shape YT is approximated by superimposition of n elemental curved grooves (see Fig. 9), the superimposed groove shape Ys can then be expressed by summation of n curved grooves as, Y s ðrÞ ¼
n X ai yi ðrÞ
(8)
1
To evaluate the difference between the target and superimposed shapes, an evaluation function f is adopted as follows:
f¼
n X ½Y T ðr j Þ Y s ðr j Þ2
v0 ai
4.2. Example of groove generation In order to confirm the validity of the above method, a target curved groove whose cross-sectional shape is 375 mm wide and 20 mm deep parabolic curve, expressed by the following equation, was generated. Y T ðrÞ ðmmÞ ¼ ð5:68 104 Þðr 350Þ2 20
(9)
Here, rj represents the r coordinate value of the evaluated point, where the difference between the target and superimposed shape is evaluated. Since the difference was evaluated on the trajectory of the nozzle center, the total number of evaluated points was n. It is clear that other variables except ai in the evaluation function are known. Therefore, in order to obtain the superimposed shape which best fits the target one, there is only a need to find out a set of superimposed coefficients ai (i = 1 to n) which minimizes the evaluating function. This means that the optimized superimposed coefficients can be obtained by solving the n simultaneous equations derived by equating the derivative of the evaluation function with respect to ai to 0 as follows: ði ¼ 1; 2; . . . ; nÞ
(12)
Twenty-five elemental curved grooves whose radius of curvature ranges from 200 to 500 mm were superimposed at a radial interval of 12.5 mm, which is one-quarter of the nozzle inner diameter. Fig. 10(a) shows the cross-sectional shape of the superimposed groove obtained by the calculation and the target shape, while Fig. 10(b) shows the difference between the results of the superimposed and target shapes. In this study, the difference between the superimposed and target shapes is called superimposing error, while the difference between the machined and target shapes is called machining error. Fig. 10 shows that the shape of the superimposed groove is nearly the same as that of the target curved groove, although there is some error on the edge area of the groove. The superimposed curved groove was machined using the nozzle scanning speeds obtained from the above method. Other machining conditions were the same as those in Table 1.
1
@f ¼0 @ai
(11)
(10) Fig. 9. Shape generation by superimposing elemental curved grooves.
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Fig. 10. Comparison of superimposed and target shapes. (a) Cross-sectional shape, (b) superimposing error.
Fig. 11. Groove shape machined by superimposing elemental curved grooves.
Fig. 11 shows the three-dimensional shape measured using a laser confocal microscope. Furthermore, Fig. 12(a) shows the target, superimposed, and machined cross-sectional shape, while Fig. 12(b) shows the superimposing and machining errors. It can be seen from Fig. 12 that the machined shape more or less corresponds with the target one except at the edge area of the groove. This fact proves that complicated shapes can be generated by superimposing the elemental curved grooves. 5. Generation of spherical surface In the above section, a parabolic shaped groove was generated by superimposing the elemental curved grooves whose curvature radii were larger than the pit radius. However, in order to obtain an arbitrary surface, superimposing the elemental curved grooves
whose radii of curvature of the nozzle center trajectory are smaller than the pit radius is necessary. In this section, a spherical surface which needs superimposition of these two kinds of elemental grooves is generated and the results are discussed. Also described are superimposing errors which occur in three kinds of superimposing methods; namely one-step superimposing, layer-bylayer superimposing, and layer-by-layer superimposing with interval optimization. Here, one-step superimposing means to reach the target depth by superimposing each elemental curved groove only once. In the case of layer-by-layer superimposing, the target depth is obtained by multiple steps of superimposition in order to reduce the accumulated error and increase superimposing precision. The target surface is a portion of spherical surface shown in Fig. 13. The radius of the spherical surface is 1100 mm and the depth is 120.2 mm. Such a target surface was chosen because the surface could be mathematically expressed easily. For other shapes, if the surfaces can be expressed mathematically or numerically, the scanning speed to generate the surface will be obtained with Eqs. (8)–(11) in a similar way. However, for the complicated shapes with markedly uneven surface, machining error will increase. 5.1. One-step superimposing In the case of one-step superimposing, one pit and 37 elemental curved grooves were used to generate the spherical surface. The pit was produced at the center first. Then, 37 concentric curved grooves were superimposed with a uniform radial interval of 12.5 mm. Under this condition, the superimposed coefficients for the pit and each elemental groove were obtained with Eq. (10), and the superimposed result was calculated with Eq. (8). The target and calculated cross-sectional shapes are shown in Fig. 14. The results showed that there exists considerable superimposing error around
Fig. 12. Comparison of target, superimposed and machined shapes. (a) Cross-sectional shape, (b) superimposing and machining error.
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each layer was 40, 40 and 40.2 mm, respectively. While in the case of six layers, the target depth for the first five layer was 20 mm and for the last layer 20.2 mm. The superimposing method for each layer was the same as that used in one-step superimposing. That is, a pit was produced at the center first, and then 37 concentric curved grooves were superimposed with an uniform interval of 12.5 mm. The superimposed coefficients for the pit and each elemental groove were obtained from Eq. (10). The target and calculated cross-sectional shapes in the case of 6-layer superimposing are shown in Fig. 15. The peak-to-valley value of the error for one-step superimposing was 57 mm (see Fig. 14(b)), while that for the 3-layer superimposing was 27 mm and that for the 6-layer superimposing was about 14 mm. From these results, it can be concluded that the layer-by-layer superimposing is an effective way to reduce the superimposing error. Although further increase in the number of layers may decrease superimposing error, excessive number will result in an increased machining time because the scanning speed of the nozzle has a limit. 5.3. Layer-by-layer superimposing with interval optimization Fig. 13. Target shape.
the center of the spherical cave, although the error is insignificant at other locations. 5.2. Layer-by-layer superimposing If the target spherical surface is generated by several steps, the superimposing error caused by the former step may be compensated by the following step to some degree. Therefore, the layerby-layer superimposing was attempted. The spherical cave 120.2 mm in depth was generated with three layers and six layers. In the case of three layers, the target depth for
Although it was found that layer-by-layer superimposing could reduce the superimposing error greatly, there still remains some superimposing error around the center. Hence, the influence of radial interval on superimposing error was investigated. A 6-layer superimposing was carried out. Since the depth of the superimposed surface was deeper than the target shape at the center, the pit at the center was not produced as shown in Fig. 15. Thus, the spherical surface was machined by superimposing 38 concentric elemental curved grooves. Besides, since the superimposing error near the center is remarkable, the interval of the path was optimized at the center area only. As shown in Fig. 16, the radial interval for the first four curved grooves were changed from 1 to 35 mm, while the total of them were fixed as expressed by the
Fig. 14. Cross-sectional shape and superimposing error in the case of one-step superimposing. (a) Cross-sectional shape, (b) superimposing error.
Fig. 15. Cross-sectional shape and superimposing error in the case of layer-by-layer superimposing. (a) Cross-sectional shape, (b) superimposing error.
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Fig. 18. Machined spherical surface. Fig. 16. Optimization of radial interval of elemental curved grooves.
Fig. 17. Cross-sectional shape and superimposed error in the case of layer-by-layer superimposing with interval optimization. (a) Cross-sectional shape, (b) superimpose error.
following equation.
Dr1 þ Dr2 þ Dr3 þ Dr4 ¼ 37:5 ðmmÞ
(13)
For the other 34 grooves, the interval was fixed at 12.5 mm. The optimal superimposed coefficients for the 38 circular grooves were obtained from Eq. (10), under the different combinations for Dr1 to Dr4. The superimposed cross-sections for different combinations were then compared, and the cross-sectional shape with the smallest superimposing error is shown in Fig. 17. Comparing Fig. 15 with Fig. 17, it can be seen that optimizing the radial interval reduces the superimposing error not only at the center area but also in the peripheral area. 5.4. Comparison of machined and calculated results Using the optimized interval and scanning speed obtained in Section 5.3, the layer-by-layer machining was carried out. The machining conditions were the same as those in Table 1. The measured three-dimensional shape and the cross-sectional shape are shown in Figs. 18 and 19, respectively. From Fig. 19, it can be seen that by superimposing elemental circular grooves, a spherical shape could be generated. However, the machining error of 30 mm in maximum was generated. The reason for this machining error is considered to be due to the change in the gap width during machining. In order to investigate the influence of the gap width, two grooves were machined at the gap width 0.3 and 0.42 mm, which were the same as the gap width before and after the spherical surface generation. The scanning speed for the two grooves was 0.25 mm/s. The cross-sectional shapes of the obtained grooves are shown in Fig. 20. It was found from the results that the machined
Fig. 19. Cross-sectional shape of machined spherical surface.
Fig. 20. Influence of change in gap width.
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area broadened with the increase of the gap width. Hence, for precision machining, the change in the gap width should be taken into consideration. 6. Conclusions In this study, a method to generate a three-dimensional surface with micro-electrolyte jet machining was proposed and verified theoretically and experimentally. From the calculations and experiments, the following conclusions were drawn. 1. Three-dimensional surface could be generated by superimposing elemental curved grooves. 2. Good agreement of the cross-sectional shapes of the calculated and superimposed surface with the target surface proves the effectiveness of the proposed method. 3. Layer-by-layer superimposing with interval optimization is an effective way to reduce superimposing error. 4. For precise surface generation, change in gap width during machining should be taken into consideration.
Acknowledgement The research presented in this paper is partially funded by Japan Society for the Promotion of Science (Grants-in-Aid for Scientific Research (C), Project No. 18560098).
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