Volume 13, no. 1Pages 39 - 51
Modelling of the Axisymmetric Precision Electrochemical ShapingV.P. Zhitnikov, N.M. Sherykhalina, S.S. Porechny, A.A. Sokolova
The problem on modelling of a precision shaping and boundary conditions are formulated according to Faraday's law and with applying of stepwise dependence current efficiency on current density. The problem is reduced to the solution of a boundary problem for definition of two analytical functions of the complex variable. The first function is a conformal mapping of region of parametrical variable on the physical plane. In order to determine this function we use the Schwartz's integral and a spline interpolation. Unlike a plane problem for determination of potential and stream function of an axisymmetric field, the integration transformations of the second analytical function are used. The analytical function is defined in the form of a sum of two addends. The first addend takes into account the singularities of the function so that the second addend has no singularities. The second function is defined by the Schwartz's integral. Interpolation by spline functions is carried out, where the spline coefficients are derivatives of these functions by means of which the intensity vector components are calculated. We propose the method to solve the axisymmetric stationary problems, which differs from the known methods by the accuracy. By means of the method, we obtain the numerical results, describing the workpiece form. The error estimation of the obtained results is carried out. Also, we show qualitative coincidence with results of plane problem solution. Full text
- electrochemical shaping; stepwise function; precision model; error estimation.
- 1. Zhitnikov V.P., Sherykhalina N.M., Porechny S.S. Comparison of Quasi-Stationary and Non-Stationary Solutions of Electrochemical Machining Problems Applying to Precision Cutting with Plate Electrode-Tool. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2019, vol. 12, no. 1, pp. 5-19. DOI: 10.14529/mmp190101
2. Christiansen S., Rasmussen H. Numerical Solutions for Two-Dimensional Annular Electrochemical Machining Problems. Journal of the Institute of Mathematics and Its Applications, 1976, no. 18, pp. 295-307. DOI: 10.1093/imamat/18.3.295
3. Klokov V.V. Influence of Alternating Current Output on Stationary Anode Shaping. Workshop of the Seminar on Boundary Value Problems, 1979, vol. 16, pp. 94-102. (in Russian)
4. Gazizov E.R., Maklakov D.V. A Method for Calculating Anodic Shaping by a Dihedral Cathode for an Arbitrary Current-Output Dependence. Teorija i praktika elektrofizikokhimicheskikh metodov obrabotki detaley v aviastrojenii [Theory and Practice of Electrophysico-Chemical Methods of Processing Details in Aircraft Building], 1994, pp. 32-35. (in Russian)
5. Datta M., Landolt D. Fundamental Aspects and Applications of Electrochemical Microfabrication. Electrochimica Acta, 2000, vol. 45, pp. 2535-2558. DOI: 10.1016/S0013-4686(00)00350-9
6. Hong Shik Shin, Bo Hyun Kim, Chong Nam Chu. Analysis of the Side Gap Resulting from Micro Electrochemical Machining with a Tungsten Wire and Ultrashort Voltage Pulses. Journal of Micromechanics and Microengineering, 2008, vol. 18, pp. 1-6. DOI: 10.1088/0960-1317/18/7/075009.
7. Shaohua Wang, Di ZhuYongbin, ZengYong Liu. Micro Wire Electrode Electrochemical Cutting with Low Frequency and Small Amplitude Tool Vibration. International Journal of Advanced Manufacturing Technology, 2011, vol. 53, no. 5-8, pp. 535-544. DOI: 10.1007/s00170-010-2835-8
8. Ningsong Qu, Xiaolong Fang, Weidong Li, Yongbin Zeng, Di Zhu. Wire Electrochemical Machining with Axial Electrolyte Flushing for Titanium Alloy. Chinese Journal of Aeronautics, 2013, vol. 26, no. 1, pp. 224-229. DOI: 10.1016/j.cja.2012.12.026
9. Kotlyar L.M., Minazetdinov N.M. Modeling of Electrochemical Machining with the Use of a Curvilinear Electrode and a Stepwise Dependence of the Current Efficiency on the Current Density. Journal of Applied Mechanics and Technical Physics, 2016, vol. 57(
efeq1), pp. 127-135. DOI: 10.1134/S0021894416010144
10. Volgin V.M., Lyubimov V.V., Gnidina I.V., Davydov A.D., Kabanova T.B. Effect of Current Efficiency on Electrochemical Micromachining by Moving Electrode. Procedia CIRP, 2016, vol. 55, pp. 65-70. DOI: 10.1016/j.procir.2016.08.031
11. Yuanlong Chen, Ming Fang, Lijun Jiang. Multiphysics Simulation of the Material Removal Process in Pulse Electrochemical Machining. International Journal of Advanced Manufacturing Technology, 2017, vol. 91, pp. 2455-2465. DOI: 10.1007/s00170-016-9899-z
12. Purcar M., Bortels L., Van den Bossche B., Deconinck J. 3D Electrochemical Machining Computer Simulations. Journal of Materials Processing Technology, 2004, vol. 149, no. 1-3, pp. 472-478. DOI: 10.1016/j.jmatprotec.2003.10.050
13. Zhouzhi Gu, Weiguo Zhu, Xiaohu Zheng, Xiaomin Bai. Cathode Design Investigation Based on Iterative Correction of Predicted Profile Errors in Electrochemical Machining of Compressor Blades. Chinese Journal of Aeronautics, 2016, vol. 29, no. 4, pp. 1111-1118. DOI: 10.1016/j.cja.2016.01.003
14. Cheng Guo, Jun Qian, Dominiek Reynaerts. Electrochemical Machining with Scanning Micro Electrochemical Flow Cell. Journal of Materials Processing Technology, 2017, vol. 24, no. 7, pp. 171-183. DOI: 10.1016/j.jmatprotec.2017.04.017
15. Lavrentiev M.A., Shabat B.V. Metody teorii funktsiy kompleksnogo peremennogo [Methods of the Theory of Functions of a Complex Variable]. Moscow, Nauka, 1987. (in Russian)
16. Henrici P. Computational Complex Analysis. N.-Y., Wiley Classic Library, 1993.
17. Polozhiy G.N. Obobshchenie teorii analiticheskih funkcij kompleksnogo peremenogo [Generalization of Analytic Functions of Complex Variable Theory]. Kiev: Kiev University, 1965.
18. Zhitnikov V.P., Zinnatullina O.R., Porechny S.S., Sherykhalina N.M. Determining the Limiting Solutions of Nonstationary Axisymmetric Hele-Shaw Problems. Journal of Applied Mechanics and Technical Physics, 2009, vol. 50, no. 4, pp. 617-627. DOI: 10.1007/s10808-009-0083-1
19. Zhitnikov V.P., Sherykhalina N.M., Zaripov A.A. Modelling of Precision Steady-State and Non-Steady-State Electrochemical Machining by Wire Electrode-Tool. Journal of Materials Processing Technology, 2016, vol. 235, pp. 49-54. DOI: 10.1016/j.jmatprotec.2016.03
20. Zhitnikov V.P., Sherykhalina N.M., Sokolova A.A. Problem of Reliability Justification of Computation Error Estimates. Mediterranean Journal of Social Sciences, 2015, vol. 6, no. 2, pp. 65-78. DOI: 10.5901/mjss.2015.v6n2s4p65