c Allerton Press, Inc., 2010. ISSN 1066-369X, Russian Mathematics (Iz. VUZ), 2010, Vol. 54, No. 10, pp. 67–70. c V.P. Zhitnikov, E.M. Oshmarina, and G.I. Fyodorova, 2010, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2010, No. 10, Original Russian Text pp. 77–81.
The Use of Discontinuous Functions for Modeling the Dissolution Process of Steady-State Electrochemical Shaping V. P. Zhitnikov* , E. M. Oshmarina**, and G. I. Fyodorova*** (Submitted by D. V. Maklakov) Ufa State Aviation Technical University, ul. K. Marksa 12, Ufa, 450000 Russia Received March 19, 2010
Abstract—We consider a modified statement for the electrochemical shaping problem. In modeling the process of the anodic dissolution we use a jump-like current efficiency function that defines the rate of the movement of the anode border. The anode surface is divided into three parts, namely, that of the active dissolution, that of no dissolution (with a small current density), and a transient part, where the current density equals the critical value. DOI: 10.3103/S1066369X10100099 Key words and phrases: electrochemical shaping, jump-like current efficiency function, hodograph method.
The study of electrochemical shaping is of great practical interest, because the technologies of electrochemical machining (ECM) are widely used in various branches of industry. Works [1–4] are devoted to the mathematical modeling of ideal ECM processes by means of the theory of functions of complex variables. The dissolution rate Vecm is defined by Faraday’s law Vecm = kηj,
k = ε/ρ,
where ε and ρ are the electrochemical equivalent and the density of the dissolved material; j is the current density on the anodic boundary; η = η(j) is the current efficiency (a portion of the current that takes part in the reaction of the metal dissolution). Earlier in solving the ECM problems the dependence η (j) of the current efficiency on the current density either was assumed to be constant [1, 4] or was approximated by hyperbolic [2, 3] or homographic functions [4]. As follows from the experiment [4], in the case of intensive ECM by the pulse current the current efficiency drops abruptly in passive electrolytes as the current density j decreases and j is large enough. This leads to the idea of approximating the dependence η (j) by functions that have a discontinuity of the first kind. In this paper we approximate η (j) by the step function η0 , j > j1 ; (1) η (j) = 0, j < j1 . We consider the problem of machining by the flat semi-infinite electrode tool (ET) that moves vertically downward with a constant speed Vet (Fig. 1, a). This problem has been solved in [2] under the assumption that the current efficiency is a square-fractional function. With the step function η(j) three zones with three types of boundary conditions are formed on the workpiece surface AC. The first zone AE, where the intensity of the electric field exceeds E1 = j1 /κ (κ is the electrical conductivity of the electrolyte), is characterized by the steady-state condition [4] |E| = −E0 sin θ, where θ is the angle of the inclination of the intensity vector to the X-axis and E0 is *
E-mail:
[email protected]. E-mail:
[email protected]. *** E-mail:
[email protected]. **
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Fig. 1. Shapes of the inter-electrode gap: the physical plane (a), the plane of the complex potential (b), AE is the active dissolution zone, ED is the zone of the constant (critical) intensity of the electric field, DC is the zone of the undissolved (rectilinear) boundary.
the intensity at the point A. The value E0 is determined from the condition that the dissolution rate at the point A equals the ET feed rate: E0 = U/s = Vet /kη0 κ. With |E| < E1 we have no dissolution, hence, in this zone the anodic boundary is the vertical segment DC. From the condition of smoothness of the steady-state boundary we conclude that the only value |E| = E1 corresponds to the boundary region ED and in this region the conduction efficiency changes continuously and monotonically from η0 to 0. Represent dependence (1) in the parametric form η = η0 f1 (σ), j = κ |E| = κE1 f2 (σ), so that the equality |E| = E1 is fulfilled on a finite segment of the variation range of the parameter σ and both functions f1 (σ) and f2 (σ) are continuous. Then on all segments the steady-state condition takes the form E0 sin θ (σ) . (2) f1 (σ) f2 (σ) = − E1 To solve the problem means to find either the analytic function (the complex potential) W (Z) = ϕ(X, Y ) + iψ(X, Y ) or two functions Z(ζ) = X + iY and W (ζ) of the parametric complex variable ζ. On the boundaries that correspond to the ET and the workpiece surface the potential ϕ is assumed to be constant and equal to −U and zero, respectively. Because of this in the plane of the complex potential the domain that corresponds to the interelectrode gap (IEG) is a strip (Fig. 1, b). In the plane of the intensity hodograph E = dW /dZ = |E| e−iθ the arc of the circumference of radius E0 /2 centered at the point iE0 /2 corresponds to the part of the steady-state surface with |E| > E1 (Fig. 2 a). The segment DC of the real axis corresponds to the region without dissolution. The arc of the circumference of radius E1 ≤ E0 centered at the origin of coordinates corresponds to the transient region. On the boundary regions A F and C F that correspond to the upper and lower surfaces of the ET, the intensity vector is inclined to the X-axis at angles θ = −π/2 and θ = π/2, respectively. Because of this in the hodograph plane vertical half-lines correspond to these regions. The existence of a split that starts from the point D and continues the boundary regions CD or ED is in contradiction with (2). In [5] one shows that the absence of the splits is equivalent in a sense to the Brillouin–Villat condition. Thus, the shape of the domain in the hodograph plane is known and uniquely determined. We introduce the parametric planes t1 = E0 /E and ζ (Fig. 2, b, c) and represent the function t1 (ζ) as the sum ∞ 1 ζ +1 + cm ζ m − ζ −m . (3) t1 = − ln ζ + D1 π ζ −1 m=1
2
d t1 1 Since t1 − α ∼ (ζ − p)3 , α = E0 /E1 , we have dt dζ (p) = 0, dζ 2 (p) = 0. This allows us to express c1 and D1 . The coefficients cm , m > 1, should be chosen so as to fulfill the equality iσ t1 pe = 1/α, 0 ≤ σ ≤ π.
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Fig. 2. Shapes of domains in the plane of the intensity hodograph (a) and in parametric planes (b and c).
Fig. 3. Results of the solution: a indicates the dependence of the lateral gap on the ratio 1/α = E1 /E0 ; b shows shapes of workpiece surfaces (curves 1–6 correspond to α = 1.0; 1.5; 2; 3; 5; ∞).
We represent the conformal mapping W (ζ) by the function ∞ 2m−1 ζ+1 4U p2m−1 1 2U −2m+1 ln + iπ − i + ζ ζ . W (ζ) = i π ζ−1 π m=1 2m − 1 p2m−1 − p−2m+1 The function Z(ζ) can be found by numerical integration ζ dW 1 dζ, t1 (ζ) Z (ζ) = E0 eiσ0 dζ where ζ = eiσ0 is the image of the point F . The value σ0 is determined from the equation obtained by equating expression (3) to zero. In Fig. 3, a we show the dependence of the ratio of the lateral gap SL = XF to the end-face gap S on 1/α. In Fig. 3, b one can see that as α → ∞ the zone of increases and the solution Yactive dissolution S π tends to the known steady-state solution X = π 2 ln S + 1 + 2 ln 2 + 2 [2] that has no vertical asymptote. As α →1 the zone of active dissolution disappears, the constant intensity zone increases and at the limit we have a regime more intrinsic for electroerosive machining. RUSSIAN MATHEMATICS (IZ. VUZ) Vol. 54 No. 10 2010
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Thus, the proposed model enables us to formulate and solve new problems of electrochemical machining, to investigate the dependence of the workpiece shape and its parameters on the maximumto-critical ratio of the intensity for all possible values of this ratio. The application of this model allows us to analyze the main characteristic zones of the workpiece surface. REFERENCES 1. V. V. Klokov, Electrochemical Shaping (Kazansk. Gos. Univ., Kazan, 1984) [in Russian]. 2. A. Kh. Karimov, V. V. Klokov, and E. I. Filatov, Methods of Calculating Electrochemical Shaping (Kazansk. Gos. Univ., Kazan, 1990) [in Russian]. 3. L. M. Kotlyar and N. M. Minazetdinov, Modeling of the Process of Electrochemical Machining of Metals for the Technological Preparation to Manufacturing on CNC Machines (Academia, Moscow, 2005) [in Russian]. 4. V. P. Zhitnikov and A. N. Zaitsev, Pulse Electrochemical Machining (Mashinostroenie, Moscow, 2008) [in Russian]. 5. N. M. Minazetdinov, “One Problem of Electrochemical Machining,” Zhurn. Prikl. Mekh. i Tekhn. Fiz. 50 (3), 214-220 (2009).
Translated by D. V. Maklakov
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