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The mathematical modelling of electrochemical machiningwith flat ended universal electrodesAdam Ruszaj*, Maria Zybura-SkrabalakThe Institute of Metal Cutting, Cracow, PolandAbstractFormer investigations have proved that it is possible to reach significantly higher accuracy in comparison to classical electrochemicalsinking when universal electrodes are applied. When the ball ended universal electrodes are applied the majority sculptured surfaces can bemachined using 3D electrode displacement control system. When flat ended universal electrodes are applied for sculptured surfacesmachining usually the 5D electrode displacement control system must be applied. However, the last case gives the possibility to achieve thehigher metal removal rate. In this paper the primary investigations of machining with flat rectangular universal electrode are presented.# 2001 Elsevier Science B.V. All rights reserved.Keywords: Mathematical modelling; Universal electrodes; Machining1. Problem formulationInvestigations in the field of electrochemical machiningwith ball ended electrode proved that this way of machiningis very useful, especially in sculptured surfaces finishing.The main disadvantage of machining with ball ended elec-trode is small metal removal rate 13. In order to increasethe metal removal rate the investigations with flat endedelectrode have been undertaken. The scheme of sculpturedsurface machining with ball ended and flat rectangularelectrodes are presented in Fig. 1. The condition whichshould be fulfilled for flat ended electrode is: electrode axisof symmetry should be perpendicular to machined sculp-tured surface.In order to fulfil this condition the electrode displacementshould be controlled at least in 45 axes, while in the case ofmachining with ball ended electrode in three axis.In order to prove that it is right to build a test standequipped with 5 axes control unit the primary investigationsinthecaseofmachiningflatsurfacehavebeenundertaken.Atfirst the mathematical model has been built and then experi-ments have been carried out for the case presented in Fig. 2.2. The mathematical modelThe scheme of machining process, which is being ana-lysed is presented in Figs. 2 and 3. The rectangular universalelectrode is displaced over the machined surface. Theelectrochemical machining action takes place only in thearea below the electrode.Electrolyte is supplied into the machining area by aspecialnozzle insidewhichtheelectrodeismounted.Duringoneelectrodepassthematerialexcessaisremoved(Eq.(1):ai? s ? s0(1)Time of machining of an optional point on machined surfaceduring one electrode pass t can be calculated from Eq. (2):t ?bvp(2)Accordingly 1,4, the interelectrode gap thickness is givenby Eq. (3):s ?Bt ? s20q?(3)From Eqs. (2) and (3) result that thickness of interelectrodegap after oneelectrode pass decreases together with increaseof velocity of the electrode displacement and the same withdecrease of machining time.Taking into account the relationships (1)(3), Eq. (4) canbe obtainedai?Bt ? s20q? s0?(4)where B ? 2Zkvk?U ? E? is the constant of the machiningprocess, Z the current efficiency of electrochemical dissolu-Journal of Materials Processing Technology 109 (2001) 333338*Corresponding author.0924-0136/01/$ see front matter # 2001 Elsevier Science B.V. All rights reserved.PII: S0924-0136(00)00816-5tion process, kvthe electrochemical equivalent of machinedmaterial, k the electrolyte electrical conductivity, U themean interelectrode voltage, E the mean drops of potentialin the layers adjacent to the electrode and workpiece, aithethickness of material excess removed during one electrodepass, b the electrode length, s0the distance between elec-trode face and machined material initial interelectrodegap thickness in successive electrode pass, s the interelec-trodegapthicknessaftereachelectrodepassandtthetimeofmachining during successive electrode pass.From Fig. 2 it results that in the casewhen c b the samearea of machined surface can be machined during a fewelectrode passes. In this case the total material excessremoved can be calculated from the relationship:at?Xi?ni?1ai(5)where atis the total thickness of the material excessremoved, aithe thickness of material excess removed duringith electrode pass calculated from relationship (4), n thenumber of electrode passes over taken into account area.Material removal rate:Vw?Flt? at?vp;s0;U;c?cvp(6)where Vwis the metal removal rate, F the surface of materialexcess removed cross-section in a direction perpendicular toelectrode displacement.From the above presented relationships it results that:? together with velocity of electrode displacement increasethickness of material excess removed decreases becausetime of machining during one electrode pass alsodecreases;? metal removal rate increases with velocity of electrodedisplacement, however, at the same time, thickness ofmaterial excess removed decreases what is the reason ofmetalremovalratedecrease; inotherwordswhenvelocityof electrode displacement is higher than optimal value,Fig. 1. Scheme of sculptured surface machining with ball ended and flatrectangular electrode.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), E1and E2: position of electrode in the first pass (E1), second pass (E2), and so on 1,2.Fig. 3. Scheme of ECM machining with rectangular universal electrodedisplaced over machined surface. vp: velocity of electrode displacement; 1:workpiece; 2: electrode tool; 3: nozzle for electrolyte supplying intointerelectrode gap; s0: thickness of initial interelectrode gap; s: thickness ofinterelectrode gap after time t; b: electrode length.334A. Ruszaj, M. Zybura-Skrabalak/Journal of Materials Processing Technology 109 (2001) 333338metal removal rate decreases together with velocity ofelectrode displacement increase;? together with interelectrode voltage increase thickness ofmaterial excess removed and metal removal rate increasesbecausetheintensityofdissolutionprocessalsoincreases;? together with initial interelectrode thickness decrease thecurrent density and intensity of dissolution processincreases which is the reason of thickness of materialexcess removed and metal removal rate increase; how-ever, for small interelectrode thickness values the hydro-dynamic conditions become worse which can limit theintensity of dissolution process by increase of hydrogenconcentration and electrolyte temperature;? thickness of material excess removed and metal removalrate increase together with electrode dimensions increase;however, electrode dimensions are limited because ofworse and worse hydrodynamic conditions into themachining area;? together with electrode cross feed increase the time ofmachining decreases which is the reason of thickness ofmaterial excess removed decrease and metal removal rateincrease.From the above presented model it is difficult to deducesurface waviness (shape errors on the border line betweensuccessive electrode passes); taking into account formerinvestigations 2,3 with ball ended electrode it is possibleto state that waviness should increase together with elec-trode cross feed; electrode cross feed should be chosen so asthe total time of machining was constant for each area onmachined surface; waviness will be also dependent onelectrode edges reproduction in machined material; a moredetailed explanation will be possible after experimental testresults analysis.3. Experimental testsExperiments have been carried out for the case presentedin Figs. 2 and 3. In the result of analysis of phenomenaoccurring in interelectrode gap the following factors havebeen distinguished.Input factors:vpvelocity of electrode tool displacement,vp? 1?59mm=minUinterelectrode voltage, U ? 8?20Vs0initial interelectrode gap thickness, s0? 0:1?1:3mmccross feed per electrode pass, c ? 0?5mm=passOutput factors:attotal thickness of material excess removed duringmachiningDmachined surface waviness (shape error on theborder line between successive electrode passes)Vwmetal removal rateConstant factors:peinlet electrolyte pressure, pe? 1MPabdimensions of the electrode, b ? 5mm; electrodematerial, copper Cu; machined material, hardenedsteel NC6CeNaNO3water solution concentration, Ce? 15%For experimental test results presentation the neural netshave been applied. The neural nets give significantly lowererrors of approximation in comparison to equations ofregression. In the presented investigations the three-layerneural nets have been applied.1Using these nets it is veryeasy tofindoutquickly machiningprocessindicatorsforanycombinations of investigated parameter values. Main tech-nologicalindicatorsofthe process: a,Dand F(necessaryformetal removal rate calculations) have been taken fromprofilograms of machined surface cross-section in the direc-tion perpendicular to electrode displacement. Examples oftheseprofilogramsarepresented inFigs.4and5.Someotherresults of experiments obtained from neural nets are pre-sented in Figs. 68.From Figs. 4 and 5 result that primary explanation ofrelation between D and electrode cross feed c was right.In the case presented in Fig. 4, c was too high incomparison to electrode dimensions and because of thisfact in the machined surface there are areas with differenttotal times of machining. As a result waviness D is sig-nificant. In the case presented in Fig. 5 cross feed c wassmaller and total time of machining was for the wholemachined surface constant. Waviness D was in this casecreated mainly as a result of electrode edges reproductionin machined material and its value is significantly smallerthan in the case presented in Fig. 4. Process of electrodeedges reproduction depends significantly on interelectrodevoltage U and velocity of electrode displacement vp(seeFig. 6).From Fig. 6 results that electrode edge reproduction inmachined material depends on thickness of material excessremoved and waviness D increase with decrease of velocityof electrode displacement and increase of interelectrodevoltage U. However, there is an optimal value of U forwhich the waviness reaches minimum. Relationships pre-sented in Figs. 7 and 8 can be explained using a mathema-tical model (as it has been done above in analysis ofmathematical model).Below, the comparison between experimental test result-sand results of theoretical calculation will be presented(Figs. 912). The theoretical calculations have been carriedoutunderassumptionthatZkv? 1:7mm3=Amin,k ? 0:13O?1cm?1and E ? 0. In reality, above specifiedcoefficients, are not constant and change together withprocess parameters (especially current density: j ? f?U;vp?.1The neural nets have been built and taught by Dr. Inz . KrzysztofKarbowski from Cracow University of Technology.A. Ruszaj, M. Zybura-Skrabalak/Journal of Materials Processing Technology 109 (2001) 333338335From Figs. 911 result that generally the differencesbetween results of experiments and theoretical calculationare not significant, but there are some exceptions. Forinstance, for small values of interelectrode voltage (Fig. 9)and velocity of electrode displacement (Fig. 12). In this casebecause of high electrodes polarisation and passivationphenomena the real process is stopped for same values ofinterelectrode gap thickness while the theoretical processcarried out according to the above presented mathematicalmodel does not taken into account this fact.This is the reason for significant differences betweenexperimental tests and theoretical calculation results forsmall values of U and vp.Mathematical model can also be used for waviness cal-culation. But only waviness resulted from differences inmachining time for different areas of machined surface canFig. 4. Machined surface cross-section in the direction perpendicular to electrode displacement for process parameters: U ? 17V, vp? 15:5mm=min,s0? 0:4mm, c ? 3:75mm=pass, at? 0:371mm, D ? 0:109mm (from experimental tests); at? 0:403mm, D ? 0:169mm (from theoretical calculations).Fig. 5. Machined surface cross-section in the direction perpendicular to electrode displacement for process parameters: U ? 17V, vp? 44:5mm=min,s0? 0:4mm, c ? 1:25mm=pass, at? 0:299mm, D ? 0:008mm (from experimental tests); at? 0:304mm, D ? 0mm (from theoretical calculations).Fig. 6. Relationship D ? f?U;vp? for s0? 0:1mm and c ? 1:25mm=pass (according to the mathematical model, D ? 0).336A. Ruszaj, M. Zybura-Skrabalak/Journal of Materials Processing Technology 109 (2001) 333338be calculated (see Fig. 4). Using this model, it is impossibleto calculate the waviness resulting from electrode edgesreproduction in machined area. But this component ofwaviness is not significant in the analysed case (Figs. 5and 6) on condition that time of machining is constant foreach point of machined surface.4. RecapitulationTaking into account results of former investigations 13and above presented considerations it is right to state thatwhen machining with flat electrode it is possible to reachhigher metal removal rate and smaller machined surfacewaviness than in the case of machining with ball endedelectrode. This statement is true for the electrodes with theFig. 7. Relationship at? f?U;vp? for s0? 0:1mm and c ? 1:25mm=pass.Fig.8. RelationshipVw? f?U;vp?fors0? 0:1mmandc ? 1:25mm=pass.Fig. 9. Relationship at? f?U?. 1: experimental tests results, 2: results ofcalculations when using the above presented mathematical model, otherparameters: c ? 2:5mm=pass, U ? 14V, s0? 0:7mm, vp? 30mm=min.Fig. 10. Relationship at? f?s0?. 1: experimental tests results, 2: results ofcalculation when using the above presented mathematical model, otherparameters: c ? 2:5mm=pass, U ? 14V, vp? 30mm=min.Fig. 11. Relationship at? f?c?. 1: experimental tests results, 2: results ofcalculations when using the above presented mathematical model, otherparameters: U ? 14V, s0? 0:7mm, vp? 30mm=min.Fig. 12. Relationship at? f?vp?. 1: experimental tests results, 2: results ofcalculations when using the above presented mathematical model, otherparameters: c ? 2:5mm=pass, U ? 14V, s0? 0:7mm.A. Ruszaj, M. Zybura-Skrabalak/Journal of Materials Processing Technology 109 (2001) 333338337samemachiningsurface.Itmeansthattheconditionbelowisfulfilled:F1? ab ? F2? pR2(7)where F1is the surface of flat electrode, F2the surface of theball ended electrode main cross-section.The increase of metal removal rate in the case of machin-ing with flat electrode results from the fact that the meaninterelectrodegapthicknessishigherthanincaseofmachin-ing with ball ended one. But increase of the flat endedelectrode surface is limited by hydrodynamic conditionsinto interelectrode area and radius of machined surfacecurvature. For electrode surface higher than in the caseof presented experiments the electrolyte should be putinto interelectrode area through the hole made in theelectrode.The decrease of machined surface waviness takes placewhen time of machining for each point on machined surfaceis constant. This condition is fulfilled when b/c is an integernumber and machined surface waviness is created only as aresult of electrode edges reproduction in machined materialwhile in case of machining with ball ended electrode wavi-ness is created as a reproduction of electrode shape.Abovepresentedconclusionsare onlythepremisesforthestatement that in the case of sculptured surfaces machiningwith flat ended e
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