Dengyong WANG, Jinzheng LI, Bin HE, Di ZHU,*

a College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

b Jiangsu Key Laboratory of Precision and Micro-Manufacturing Technology, Nanjing 210016, China

KEYWORDS

Abstract The inter-electrode gap(IEG)is an essential parameter for the anode shaping process in electrochemical machining (ECM) and directly affects the machining accuracy. In this paper, the IEG during the leveling process of an oval anode workpiece in counter-rotating ECM (CRECM)is investigated.The variation of the minimum IEG is analyzed theoretically,and the results indicate that rather than reaching equilibrium, the minimum IEG in CRECM expands constantly when a constant feed speed is used for the cathode tool. This IEG expansion leads to a poor localization effect and has an adverse influence on the roundness of the machined workpiece. To maintain a small constant IEG in CRECM,a variable feed speed is used for the cathode based on a fitted equation.The theoretical results show that the minimum IEG can be controlled at a small value by using an accelerated feed speed. Experiments have been conducted using a specific experimental apparatus in which the cathode tool is designed as a combined structure of two sectors and a thin sheet.By detecting the machining currents flowing through the minimum IEG, how the latter varies is obtained indirectly.The results indicate that using an accelerated feed speed is effective for controlling the IEG, thereby improving the roundness of the machined workpiece.

1. Introduction

Electrochemical machining (ECM) is an anodic dissolution process that is efficient at removing materials regardless of their hardness.Compared with conventional mechanical methods, ECM involves no machining stress, tool wear, or recast layer and has therefore been used widely in the aerospace,automotive, defense, and medical industries.1-5

In ECM, a small inter-electrode gap (IEG) of around a fraction of a millimeter is generally used. The IEG is essential for ECM accuracy because the gap distribution determines directly the final geometric shape of the anode workpiece.Many studies have analyzed the IEG distribution rules.Rajurkar et al.6built a mathematical model of the IEG in pulsed ECM and developed an on-line monitoring system by detecting the current signal. Mount et al.7analyzed the IEG in a planar-workpiece-planar-tool configuration using finitedifference method. Rajurkar et al.8predicted the minimum machining allowance by calculating the IEG and found that using a pulsed current with a passivation electrolyte could mitigate the effect of memory errors and improved the machining accuracy.De Silva et al.9used a narrow IEG(＜50 μm)to analyze the copy accuracy in precision ECM. Riggs et al.10predicted the IEG and workpiece geometry in electrochemical cavity sinking by using a computer-implemented model.

Most previous studies of the IEG were concerned with the material dissolution process in sinking ECM, in which a preshaped cathode tool with constant feed speed is generally used.The IEG becomes steady after reaching an equilibrium state.11-13However, in some ECM methods, the IEG can hardly reach equilibrium but changes constantly.For example,(i) it varies periodically with the vibration of the cathode tool in vibrating ECM,14(ii) it expands with increasing machining time in through-mask ECM,15,16(iii) and it changes with the envelope movement of the universal tool electrode in electrochemical generating machining.17

The focus herein is on the IEG in counter-rotating ECM(CRECM), which is a new ECM method proposed by Zhu et al. to machine the complex outer surfaces of revolving casing parts.18,19As shown in Fig.1, a cylindrical cathode tool with hollow windows is typically used in CRECM. The anode workpiece and cathode tool rotate in opposite directions to each other at the same angular velocity w while simultaneously the cathode tool moves toward the anode workpiece at a constant feed speed f. As the materials on the anode surface are dissolved electrochemically,convex structures of various shape are fabricated gradually on the corresponding areas of the hollow windows. This method helps obtain a uniformly distributed wall thickness and a smooth surface for complex thin-walled revolving parts.20In CRECM, the position of a given point on the workpiece surface changes constantly as the electrodes rotate, resulting in a periodically variable IEG that differs distinctly from the steady IEG that arises after reaching equilibrium in sinking ECM.Therefore,the IEG that is studied herein is the minimum IEG Δ located on the center line of the electrodes (Fig.1).

Fig.1 Principle of counter-rotating electrochemical machining(CRECM).

Because of the machining deformation in the turning procedure,the initial blank of the thin-walled revolving part is more likely to be oval. Consequently, an elliptical anode workpiece is used herein. Because the material dissolution rate on the high point of the elliptical anode workpiece exceeds that on the low point, the anode workpiece is levelled gradually into a circular shape with the cathode feed.During the leveling process, the IEG is a critical factor for obtaining the desired roundness of the anode workpiece, with a small IEG helping to localize the anodic dissolution and improve the leveling ability.

Herein, a numerical model is established to analyze the minimum IEG Δ during the leveling process in CRECM.The simulation results indicate that far from reaching equilibrium,Δ expands constantly when a constant feed speed is used for the cathode tool. This IEG expansion leads to poor localization and adversely affects the roundness of the machined workpiece. Therefore, to maintain a small constant IEG in CRECM,a variable feed speed is used for the cathode according to a fitted equation.Experiments are conducted by using a specific experimental set-up. How Δ varies is obtained indirectly by measuring the machining current. It is shown that Δ can be controlled effectively at a small value by using an accelerated feed speed, thereby improving the roundness of the machined workpiece.

2. Numerical simulation of leveling process in CRECM

Fig.2 shows the principle of the leveling process in CRECM.An oval workpiece and a circular cathode tool rotate synchronously in different directions while simultaneously the cathode feeds toward the workpiece at a constant speed f.The initial roundness deviation of the oval workpiece is δ0.The radius of the cathode tool is Rc. With high-speed electrolytes flushing the IEG,the materials of the anode workpiece within the narrow machining area are dissolved electrochemically. Because the material dissolution rate on the high point Khof the elliptical anode workpiece exceeds that on the low point Kl,the anode workpiece is levelled gradually into a circular shape with the cathode feed,and the roundness deviation is reduced to be δ.

Fig.2 Leveling process in CRECM.

To study how the workpiece profile evolves, the cathode tool is assumed to rotate around the elliptical workpiece as shown in Fig.3. Considering the changeable IEG for a given point on the anode surface during rotation, the variation of the minimum IEG is used for study. As shown in Fig.2, the initial minimum IEG for the high point Khon the elliptical anode surface is defined to be Δh0.

where a is the radius of the major axis of the ellipse,mm,and b is that of the minor axis, mm.

During the CRECM process, the coordinates (xO1,yO1) of the center point O1of the cathode tool at time t can be calculated using

where Rcis the radius of the cathode, mm, θ is the rotation angle at time t which is equal to θ=wt,rad,f is the feed speed of the cathode, mm/min, and w is the angular velocity,rad/min.

The distribution of the electric potential φ in the electrolyte domain satisfies the Laplace equation, namely

Fig.3 Equivalent movements of CRECM.

with boundary conditions

where U is the potential difference between the cathode tool and anode workpiece, V.

The total amount Mtof material dissolution in the nth time period in the mth circle can be calculated approximately using a discrete summation, namely

where Miis the total amount of material dissolution at point K in the mth circle, mm, and T1is the machining time for each circle, min.

The minimum IEG ΔKfor point K in the mth circle can be calculated according to

where ΔK0is the initial minimum IEG for point K,mm,F is the total amount of cathode feed,mm,and Mmis the total amount of material dissolution during m circles, mm.

3. Numerical simulation results

3.1. Shape evolution of anode workpiece

The electric field intensities on the anode surface in each short time period were calculated by using a finite-element method,and the evolution of the shape of the anode workpiece was obtained by combining Eqs. (1)-(10). The typical Inconel 718 super-alloy was studied,and the actual volume electrochemical equivalent ηω was calculated according to a fitted equation based on the weight-loss measurements.21The simulation conditions are listed in Table 1,and the feed speed of the cathode was kept constant at 0.06 mm/min during the simulation process.

Fig.4 shows the simulated evolution of profile of the anode workpiece in polar coordinates. The profile changes gradually from an initial oval shape to a rounded shape with 6 mm of cathode feed. The radius distributions for different points on the anode surface at different amounts of cathode feed F are shown in Fig.5. Initially, the radius distribution is obviously wavy because of a large roundness error of 1 mm for the oval anode workpiece.When the amount of cathode feed is 0.6 mm,the amplitude of the wavy line is measured to be around 0.67 mm and decreases with the cathode feed. Fig.6 shows the variation of the roundness errors calculated according to the deviation values between the peaks and troughs. A lineardecreasing trend in the roundness error can be observed in the initial stage, with the value is decreasing to 0.053 mm for F=2.4 mm. Thereafter, the roundness error decreases far slower. When the amount of cathode feed exceeds 3 mm, the roundness error tends to be relatively steady,and only a slight downward trend can be found.

Table 1 Relevant simulation conditions.

Fig.4 Simulated profile evolution of anode workpiece with 6 mm of cathode feed.

Fig.5 Radius distribution on anode surface at different amounts of cathode feed.

Fig.6 Variation of roundness error with amount of cathode feed.

3.2. Variation of minimum IEG

Fig.7 shows the minimum IEG for the high point Khand the low point Klon the oval anode surface at a constant feed speed.For Kh,the IEG decreases slightly from an initial value of 0.2 mm in the transition stage, after which there is a linear rising trend, with the IEG increasing to around 0.33 mm after 10 mm of cathode feed. For Kl, there is an obvious initial downward trend due to the large initial IEG of 1.2 mm.However, after reaching a certain amount of cathode feed, a linearly increasing trend can also be observed. The increasing lines for high point Khand low point Klnearly overlap when the amount of cathode feed exceeds 3.8 mm.

The increase of the minimum IEG can be attributed to the effect of the decreasing diameter of the anode workpiece.Unlike the unchanged profile in sinking ECM after reaching an equilibrium state, the diameter of the anode workpiece becomes ever smaller in CRECM,thereby resulting in a higher material dissolution rate in the radial direction. Fig.8 shows the material dissolution rate Viin the radial direction at the high point Kh. The material dissolution rate increases initially and reaches 0.06 mm/min after 1.2 mm of cathode feed, after which it grows approximately linearly.

4. Control of minimum IEG using variable feed speed

Fig.7 Minimum inter-electrode gap (IEG) for high and low points using a constant feed speed for the cathode.

Fig.8 Material dissolution rate at high point Kh using a constant feed speed for cathode.

It is well known that the feed speed of the cathode and the material dissolution rate eventually reach equilibrium in sinking ECM, whereupon the IEG becomes steady. A small balanceable IEG can help localize the anodic dissolution and improve the leveling ability.However,according to the results shown in Figs. 7 and 8, the IEG in CRECM expands constantly with the cathode feed, which is disadvantageous for obtaining the desired roundness.

To maintain a constant small IEG, a variable feed speed is adopted for the cathode. According to the variation of the material dissolution rate shown in Fig.8,a constant feed speed of 0.06 mm/min is used in the transition stage up to 1.2 mm.Thereafter,the feed speed of cathode is set to equal to the calculated dissolution rate in each circle so that the IEG can be controlled to be constant. According to Eq. (11), the material dissolution rate Vican be calculated at the high point Khin each circle for a variable feed speed, as shown in Fig.9.Fig.10 shows the corresponding cathode feed speed for different cathode feed amounts. The cathode feed speed increases from an initial value of 0.06 mm/min to 0.08 mm/min when the feed amount is 10 mm.

Fig.11 shows how the minimum IEG at high point Khand low point Klon the oval anode surface varies when using a variable cathode feed speed.Compared with the obvious rising tendencies with a constant feed speed, the minimum IEG for both the high and low point can be controlled effectively.When the cathode feed amount exceeds 1.2 mm,the minimum IEG for high point Khis maintained to be about 0.19 mm(Fig.11(a)). In addition, the use of a variable feed speed can speed up the decrease of the IEG for Klin the transition stage.As shown in Fig.11(b), the cathode feed amount at which an IEG of 0.22 mm is reached can be reduced from 3.45 mm to 2.65 mm by using a variable feed speed, and this helps to improve the leveling ability.

Fig.9 Material dissolution rate at high point Kh using variable feed speed.

Fig.10 Variation of feed speed for cathode.

Fig.11 Variations of minimum IEG for high and low points using variable cathode feed speed.

5. Experiments and discussions

5.1. Experimental procedure

As shown in Fig.12,a specific experimental apparatus is developed to investigate the leveling process. An oval Inconel 718 workpiece is prepared and rotates clockwise at a constant angular speed. Because the cathode tool is circular, it does not rotate but merely moves toward the workpiece. The cathode tool is designed as a combined structure of two sectors and a thin sheet, the latter being insulated from the former.According to the current lines distribution in the IEG,the current line in pink color in the minimum IEG travels from the anode workpiece to the sheet cathode. Based on Ohm’s law,the minimum IEG Δ is calculated approximately as

where S is the size of the machining area, mm2, and I is the machining current flowing through the sheet cathode, A.

Fig.12 Experimental apparatus for leveling process in CRECM.

According to Eq. (13), the minimum IEG is inversely proportional to the machining current. The current signal can therefore be used to reflect indirectly how the minimum IEG varies while the anode workpiece rotates. A Hall current sensor is used to detect the current flowing through the sheet cathode.

The initial oval anode workpiece is fabricated using wirecutting electrical discharge machining, and the radii of the major axis (a) and minor axis (b) are 26 mm and 25 mm,respectively. The radius of the cathode tool is 25 mm, and the thin sheet is 1 mm thick.The initial minimum IEG for high point Khis 0.2 mm,and the cathode feed amount is controlled to be 6 mm. Both constant and variable cathode feed speeds are used. According to the theoretical results shown in Fig.10, the variable feed speed presents an approximately linear growth trend when the amount of cathode feed ranges from 1.2 to 6 mm. Consequently, the variable feed speed satisfies

5.2. Results and discussions

Fig.13 shows the measured current signal flowing through the minimum IEG. The machining current fluctuates because of the non-uniform minimum IEG while the oval anode workpiece rotates. The upper and lower boundaries represent the variation tendencies of the current signals for high and low points Khand Kl, respectively. For a constant cathode feed speed as shown in Fig.13(a), the machining currents increase in the initial transition stages.Thereafter,there are obvious linear downward trends for both upper and lower boundaries.The descending slopes of the machining currents tend to be equal in the final stage. Fig.13(b) shows the variation of the machining currents using a variable cathode feed speed,showing that the descending slopes of the machining currents are reduced considerably.

Fig.13 Measured current signals flowing through minimum IEG.

According to Eq. (13), the minimum IEG for different points on the anode workpiece can be calculated approximately from the measured current signals. Even though there will be errors in the values because of the existence of stray currents and measuring errors, how the IEG varies will be instructive nevertheless.Fig.14 shows the overall variations of the calculated minimum IEG using the constant and variable feed speeds. For a constant cathode feed speed as shown in Fig.14(a),the IEG varies in a manner that is similar to the theoretical results in Fig.7.The IEG for both high and low points decreases initially in the transition stage, after which there are obvious rising trends with the amount of cathode feed.Fig.14(b) shows the calculated minimum IEG using a variable cathode feed speed.The IEG tends to reach a relatively steady state with very slight upward trends.

Fig.14 Overall variations of calculated minimum IEG.

Fig.15 Comparison of minimum IEG.

Fig.15 compares the minimum IEG in the initial and final processing stages between constant and variable cathode feed speeds. In the initial stage as shown in Fig.15(a), there are two overlapping periodic waveforms, which corresponds to the large roundness error of the initial oval anode workpiece.In the final stage (Fig.15(b)), the variations of the minimum IEG become approximately two straight lines. The minimum IEG for high point Khis expanded from 0.2 mm to 0.32 mm at a constant feed speed. By contrast, when a variable feed speed is used, the finial minimum IEG can be maintained at a smaller value of 0.22 mm. In addition, the fluctuation range of the IEG can be reduced from 0.06 mm to 0.03 mm. This indicates that a better roundness of the anode workpiece can be achieved by using a variable cathode feed speed.

Fig.16 Photographs of Inconel 718 workpieces.

Fig.17 Radius distributions of different points on anode surface.

Fig.16 shows the photographs of anode workpieces machined using constant and variable cathode feed speeds with the same amount of cathode feed of 6 mm. Clearly, the anode workpieces can be machined successfully from an oval shape to a rounded circle. The profiles of cylindrical surface of the machined anode workpieces were measured by using a threecoordinate measuring machine.The radii for points P at different angles θ on the anode surface are shown in Fig.17.For the machined workpiece using a constant feed speed (Fig.16(b)),the final average radius is measured to be around 19.91 mm.According to the amount of cathode feed and the initial minimum IEG, the final minimum IEG can be calculated as 0.29 mm, which is obviously larger than the initial value of 0.2 mm. The vibration of the radius is detected to be as high as 0.06 mm.The average radius of the workpiece using a variable feed speed is around 20.02 mm, and the minimum IEG is calculated to be around 0.18 mm, which is approximately equal to the initial value.Compared with that using a constant cathode feed speed, the fluctuation of the radius is within a smaller value of 0.02 mm, which indicates a better roundness.This verifies that using a variable feed speed is effective for controlling the IEG, thereby improving the leveling ability.

6. Conclusions

In this paper, the leveling process of an oval anode workpiece in CRECM is simulated based on an established numerical analysis model.How the minimum IEG varies during the leveling process is analyzed both theoretically and experimentally.The conclusions are summarized as follows.

(1) Unlike in sinking ECM, the minimum IEG in CRECM does not reach equilibrium but rather expands constantly at a constant feed speed, which is disadvantageous for the leveling process.

(2) To control the IEG in CRECM,a variable cathode feed speed is used according to a fitted equation.The theoretical results show that the minimum IEG can be controlled at a small value by using an accelerated feed speed.

(3) By detecting the machining current flowing through the minimum IEG, how the latter varies is obtained indirectly. The experimental results indicate that the IEG can be controlled effectively by using an accelerated feed speed,thereby improving the roundness of the machined workpiece.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (51535006, 51805259), and Natural Science Foundation of Jiangsu Province of China(BK20180431),and Fundamental Research Funds for the Central Universities of China (3082018NP2018406), and Young Elite Scientists Sponsorship Program by CAST of China,and Jiangsu Key Laboratory of Precision and Micro-Manufacturing Technology of China.