The mathematical modelling of electrochemical machining with flat ended universal electrodes

Journal of Materials Processing Technology 109 (2001) 333±338

The mathematical modelling of electrochemical machining with ¯at ended universal electrodes Adam Ruszaj*, Maria Zybura-Skrabalak The Institute of Metal Cutting, Cracow, Poland

Abstract Former investigations have proved that it is possible to reach signi®cantly higher accuracy in comparison to classical electrochemical sinking when universal electrodes are applied. When the ball ended universal electrodes are applied the majority sculptured surfaces can be machined using 3D electrode displacement control system. When ¯at ended universal electrodes are applied for sculptured surfaces machining usually the 5D electrode displacement control system must be applied. However, the last case gives the possibility to achieve the higher metal removal rate. In this paper the primary investigations of machining with ¯at rectangular universal electrode are presented. # 2001 Elsevier Science B.V. All rights reserved. Keywords: Mathematical modelling; Universal electrodes; Machining

1. Problem formulation Investigations in the ®eld of electrochemical machining with ball ended electrode proved that this way of machining is very useful, especially in sculptured surfaces ®nishing. The main disadvantage of machining with ball ended electrode is small metal removal rate [1±3]. In order to increase the metal removal rate the investigations with ¯at ended electrode have been undertaken. The scheme of sculptured surface machining with ball ended and ¯at rectangular electrodes are presented in Fig. 1. The condition which should be ful®lled for ¯at ended electrode is: electrode axis of symmetry should be perpendicular to machined sculptured surface. In order to ful®l this condition the electrode displacement should be controlled at least in 4±5 axes, while in the case of machining with ball ended electrode in three axis. In order to prove that it is right to build a test stand equipped with 5 axes control unit the primary investigations in the case of machining ¯at surface have been undertaken. At ®rst the mathematical model has been built and then experiments have been carried out for the case presented in Fig. 2. 2. The mathematical model The scheme of machining process, which is being analysed is presented in Figs. 2 and 3. The rectangular universal *

Corresponding author.

electrode is displaced over the machined surface. The electrochemical machining action takes place only in the area below the electrode. Electrolyte is supplied into the machining area by a special nozzle inside which the electrode is mounted. During one electrode pass the material excess a is removed (Eq. (1)): ai ˆ s ÿ s0

(1)

Time of machining of an optional point on machined surface during one electrode pass t can be calculated from Eq. (2): tˆ

b vp

(2)

Accordingly [1,4], the interelectrode gap thickness is given by Eq. (3): q (3) sˆ Bt ‡ s20 From Eqs. (2) and (3) result that thickness of interelectrode gap after one electrode pass decreases together with increase of velocity of the electrode displacement and the same with decrease of machining time. Taking into account the relationships (1)±(3), Eq. (4) can be obtained q  Bt ‡ s20 ÿ s0 (4) ai ˆ where B ˆ 2Zkv k…U ÿ E† is the constant of the machining process, Z the current ef®ciency of electrochemical dissolu-

0924-0136/01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 0 ) 0 0 8 1 6 - 5

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Fig. 1. Scheme of sculptured surface machining with ball ended and ¯at rectangular electrode.

tion process, kv the electrochemical equivalent of machined material, k the electrolyte electrical conductivity, U the mean interelectrode voltage, E the mean drops of potential in the layers adjacent to the electrode and workpiece, ai the thickness of material excess removed during one electrode pass, b the electrode length, s0 the distance between electrode face and machined material Ð initial interelectrode gap thickness in successive electrode pass, s the interelectrode gap thickness after each electrode pass and t the time of machining during successive electrode pass. From Fig. 2 it results that in the case when c < b the same area of machined surface can be machined during a few electrode passes. In this case the total material excess removed can be calculated from the relationship: at ˆ

iˆn X

ai

(5)

iˆ1

where at is the total thickness of the material excess removed, ai the thickness of material excess removed during

Fig. 3. Scheme of ECM machining with rectangular universal electrode displaced over machined surface. vp : velocity of electrode displacement; 1: workpiece; 2: electrode tool; 3: nozzle for electrolyte supplying into interelectrode gap; s0: thickness of initial interelectrode gap; s: thickness of interelectrode gap after time t; b: electrode length.

ith electrode pass calculated from relationship (4), n the number of electrode passes over taken into account area. Material removal rate: Vw ˆ

Fl  at …vp ; s0 ; U; c†cvp t

(6)

where Vw is the metal removal rate, F the surface of material excess removed cross-section in a direction perpendicular to electrode displacement. From the above presented relationships it results that:  together with velocity of electrode displacement increase thickness of material excess removed decreases because time of machining during one electrode pass also decreases;  metal removal rate increases with velocity of electrode displacement, however, at the same time, thickness of material excess removed decreases what is the reason of metal removal rate decrease; in other words when velocity of electrode displacement is higher than optimal value,

Fig. 2. Scheme of electrochemical machining with universal rectangular electrode moving above the machined surface. E: electrode made of M1 copper, P: workpiece made of NC6 steel (hardness 64 HRC), E1 and E2: position of electrode in the ®rst pass (E1), second pass (E2), and so on [1,2].

A. Ruszaj, M. Zybura-Skrabalak / Journal of Materials Processing Technology 109 (2001) 333±338









metal removal rate decreases together with velocity of electrode displacement increase; together with interelectrode voltage increase thickness of material excess removed and metal removal rate increases because the intensity of dissolution process also increases; together with initial interelectrode thickness decrease the current density and intensity of dissolution process increases which is the reason of thickness of material excess removed and metal removal rate increase; however, for small interelectrode thickness values the hydrodynamic conditions become worse which can limit the intensity of dissolution process by increase of hydrogen concentration and electrolyte temperature; thickness of material excess removed and metal removal rate increase together with electrode dimensions increase; however, electrode dimensions are limited because of worse and worse hydrodynamic conditions into the machining area; together with electrode cross feed increase the time of machining decreases which is the reason of thickness of material excess removed decrease and metal removal rate increase.

From the above presented model it is dif®cult to deduce surface waviness (shape errors on the border line between successive electrode passes); taking into account former investigations [2,3] with ball ended electrode it is possible to state that waviness should increase together with electrode cross feed; electrode cross feed should be chosen so as the total time of machining was constant for each area on machined surface; waviness will be also dependent on electrode edges reproduction in machined material; a more detailed explanation will be possible after experimental test results analysis. 3. Experimental tests Experiments have been carried out for the case presented in Figs. 2 and 3. In the result of analysis of phenomena occurring in interelectrode gap the following factors have been distinguished. Input factors: velocity of electrode tool displacement, vp vp ˆ 1ÿ59 mm=min U interelectrode voltage, U ˆ 8ÿ20 V initial interelectrode gap thickness, s0 ˆ 0:1ÿ1:3 mm s0 c cross feed per electrode pass, c ˆ 0ÿ5 mm=pass Output factors: total thickness of material excess removed during machining D machined surface waviness (shape error on the border line between successive electrode passes) Vw metal removal rate

at

335

Constant factors: inlet electrolyte pressure, pe ˆ 1 MPa dimensions of the electrode, b ˆ 5 mm; electrode material, copper Cu; machined material, hardened steel NC6 NaNO3 water solution concentration, Ce ˆ 15% Ce pe b

For experimental test results presentation the neural nets have been applied. The neural nets give signi®cantly lower errors of approximation in comparison to equations of regression. In the presented investigations the three-layer neural nets have been applied.1 Using these nets it is very easy to ®nd out quickly machining process indicators for any combinations of investigated parameter values. Main technological indicators of the process: a, D and F (necessary for metal removal rate calculations) have been taken from pro®lograms of machined surface cross-section in the direction perpendicular to electrode displacement. Examples of these pro®lograms are presented in Figs. 4 and 5. Some other results of experiments obtained from neural nets are presented in Figs. 6±8. From Figs. 4 and 5 result that primary explanation of relation between D and electrode cross feed c was right. In the case presented in Fig. 4, c was too high in comparison to electrode dimensions and because of this fact in the machined surface there are areas with different total times of machining. As a result waviness D is signi®cant. In the case presented in Fig. 5 cross feed c was smaller and total time of machining was for the whole machined surface constant. Waviness D was in this case created mainly as a result of electrode edges reproduction in machined material and its value is signi®cantly smaller than in the case presented in Fig. 4. Process of electrode edges reproduction depends signi®cantly on interelectrode voltage U and velocity of electrode displacement vp (see Fig. 6). From Fig. 6 results that electrode edge reproduction in machined material depends on thickness of material excess removed and waviness D increase with decrease of velocity of electrode displacement and increase of interelectrode voltage U. However, there is an optimal value of U for which the waviness reaches minimum. Relationships presented in Figs. 7 and 8 can be explained using a mathematical model (as it has been done above in analysis of mathematical model). Below, the comparison between experimental test resultsand results of theoretical calculation will be presented (Figs. 9±12). The theoretical calculations have been carried out under assumption that Zkv ˆ 1:7 mm3 =A min, k ˆ 0:13 Oÿ1 cmÿ1 and E ˆ 0. In reality, above speci®ed coef®cients, are not constant and change together with process parameters (especially current density: j ˆ f …U; vp †.

1 The neural nets have been built and taught by Dr. InzÇ. Krzysztof Karbowski from Cracow University of Technology.

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Fig. 4. Machined surface cross-section in the direction perpendicular to electrode displacement for process parameters: U ˆ 17 V, vp ˆ 15:5 mm=min, s0 ˆ 0:4 mm, c ˆ 3:75 mm=pass, at ˆ 0:371 mm, D ˆ 0:109 mm (from experimental tests); at ˆ 0:403 mm, D ˆ 0:169 mm (from theoretical calculations).

Fig. 5. Machined surface cross-section in the direction perpendicular to electrode displacement for process parameters: U ˆ 17 V, vp ˆ 44:5 mm=min, s0 ˆ 0:4 mm, c ˆ 1:25 mm=pass, at ˆ 0:299 mm, D ˆ 0:008 mm (from experimental tests); at ˆ 0:304 mm, D ˆ 0 mm (from theoretical calculations).

Fig. 6. Relationship D ˆ f …U; vp † for s0 ˆ 0:1 mm and c ˆ 1:25 mm=pass (according to the mathematical model, D ˆ 0).

From Figs. 9±11 result that generally the differences between results of experiments and theoretical calculation are not signi®cant, but there are some exceptions. For instance, for small values of interelectrode voltage (Fig. 9) and velocity of electrode displacement (Fig. 12). In this case because of high electrodes polarisation and passivation phenomena the real process is stopped for same values of interelectrode gap thickness while the theoretical process

carried out according to the above presented mathematical model does not taken into account this fact. This is the reason for signi®cant differences between experimental tests and theoretical calculation results for small values of U and vp . Mathematical model can also be used for waviness calculation. But only waviness resulted from differences in machining time for different areas of machined surface can

A. Ruszaj, M. Zybura-Skrabalak / Journal of Materials Processing Technology 109 (2001) 333±338

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Fig. 10. Relationship at ˆ f …s0 †. 1: experimental tests results, 2: results of calculation when using the above presented mathematical model, other parameters: c ˆ 2:5 mm=pass, U ˆ 14 V, vp ˆ 30 mm=min.

Fig. 7. Relationship at ˆ f …U; vp † for s0 ˆ 0:1 mm and c ˆ 1:25 mm=pass.

Fig. 8. Relationship Vw ˆ f …U; vp † for s0 ˆ 0:1 mm and c ˆ 1:25 mm=pass.

Fig. 9. Relationship at ˆ f …U†. 1: experimental tests results, 2: results of calculations when using the above presented mathematical model, other parameters: c ˆ 2:5 mm=pass, U ˆ 14 V, s0 ˆ 0:7 mm, vp ˆ 30 mm=min.

Fig. 11. Relationship at ˆ f …c†. 1: experimental tests results, 2: results of calculations when using the above presented mathematical model, other parameters: U ˆ 14 V, s0 ˆ 0:7 mm, vp ˆ 30 mm=min.

Fig. 12. Relationship at ˆ f …vp †. 1: experimental tests results, 2: results of calculations when using the above presented mathematical model, other parameters: c ˆ 2:5 mm=pass, U ˆ 14 V, s0 ˆ 0:7 mm.

4. Recapitulation be calculated (see Fig. 4). Using this model, it is impossible to calculate the waviness resulting from electrode edges reproduction in machined area. But this component of waviness is not signi®cant in the analysed case (Figs. 5 and 6) on condition that time of machining is constant for each point of machined surface.

Taking into account results of former investigations [1±3] and above presented considerations it is right to state that when machining with ¯at electrode it is possible to reach higher metal removal rate and smaller machined surface waviness than in the case of machining with ball ended electrode. This statement is true for the electrodes with the

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same machining surface. It means that the condition below is ful®lled: F1 ˆ ab ˆ F2 ˆ pR2

(7)

where F1 is the surface of ¯at electrode, F2 the surface of the ball ended electrode main cross-section. The increase of metal removal rate in the case of machining with ¯at electrode results from the fact that the mean interelectrode gap thickness is higher than in case of machining with ball ended one. But increase of the ¯at ended electrode surface is limited by hydrodynamic conditions into interelectrode area and radius of machined surface curvature. For electrode surface higher than in the case of presented experiments the electrolyte should be put into interelectrode area through the hole made in the electrode. The decrease of machined surface waviness takes place when time of machining for each point on machined surface is constant. This condition is ful®lled when b/c is an integer number and machined surface waviness is created only as a result of electrode edges reproduction in machined material while in case of machining with ball ended electrode waviness is created as a reproduction of electrode shape. Above presented conclusions are only the premises for the statement that in the case of sculptured surfaces machining with ¯at ended electrode will be possible to reach higher metal removal rate and smaller machined surface waviness than with ball ended one. In order to prove these promises, further investigations should be carried out.

Acknowledgements Authors wish to thank the Polish Science Research Committee for ®nancial support (Research Project No. 8 8534 91 02), Directors of the Institute of Metal Cutting for creating good conditions for research and Colleagues from The Electrochemical Machining Department for helping in carrying out experimental tests. References [1] J. Kozak, A. Ruszaj, R. SøawinÂski, L. DaÎbrowski, Computer simulation of 3D numerically controlled electrochemical machining (ECM-NC), in: Proceedings of the 11th International Conference on Computeraided Production Engineering, London, UK, September 20±21, 1995, pp. 205±210. [2] A. Ruszaj, Procesy obroÂbek elektrochemicznej i elektroerozyjnej w roÂzÇnych odmianach kinematycznych (The processes of electrochemical and electrodischarge machining in different kinematic varieties). Prace Instytutu ObroÂbki Skrawaniem, seria: Zeszyty Naukowe, Nr. 76, 1989, 157 s. [3] A. Ruszaj, i inni, Sprawozdanie z projektu badawczego ®nansowanego przez KBN pt. ``System komputerowego projektowania i realizacji procesu obroÂbki elektrochemicznej niepro®lowanaÎ elektrodaÎ'' (niepublikowane). (Report of the research project ®nanced by KBN titled ``System of computer designing and carrying out the electrochemical machining process with universal electrode'' (nonpublished)), 1994. [4] A. Davydov, J. Kozak, Vysokoskorostnoe Elektrochimiczeskoje Formoobrazovanie (High Speed Electrochemical Shaping), Izd. Nauka, Moskva, 1990.