design and analysis of ecm toolings

Design and analysis of ECM toolings

V.K. Jain* and P.C. Pandey +

Existing analytical methods for tooling design in electrochemical machining (ecm) are inaccurate so tools have been designed by trial and error which is costly and time consuming. This paper presents a modified ecm theory which gives comparatively accurate results especially for simple shaped workpieces. For complex shaped workpieces and situations, a finite element formulation of the ecm problem has been developed. Results agree well with experimental values. A parametric study of the process which highlights certain guidelines for ecm tooling designers is also presented

Ecm tooling design is basically concerned with tool profile computation, which under given conditions, will produce the desired work-piece shape and size. Analytical solution of the problem is complicated by the number of inter- acting parameters. The tool shape required to obtain a work-piece of known profile can be computed by a rLumber of models 1-6. The existing models, however, over- simplify the situation and do not yield accurate results, especially when complex shaped workpieces are involved. Tipton 7-9 has used finite difference techniques to obtain the tool shape for a workpiece of specified geometry, but with little success. Problems related to ecm tooling design have also been investigated by K6nig et al. I°'11 and othersl2-1s.

Due to the poor accuracy of the analytical methods, the approximate shape ot an ecru tool is obtained in practice

*Research Scholar and +Professor, Department of Mechanical and Industrial Engineering, University of Roorkee, Roorkee-247 672, India

by a standard computatior~al technique to which certain corrections based on practical considerations and shape complexity are added. To date, a general method for analytical design of ecm toolings has not existed and the practice of designing such tools by trial and error is costly and time consuming.

This paper presents a comparative study of the following design analysis techniques for ecm toolings:

Conventional ecm theory (model-I) Modified conventional theory (model-2) Finite element technique (fet-1), and Cos '8' theory,

Based on a parametric study of the process, general guide lines for tooling design are given. For the sake of analysis, sinking of rectangular holes by ecm, for three different conditions of electrolyte flow, are considered (Figs 1-3). The electrolyte is assumed to behave like a pure ohmic resistance and the initial gap conditions are uniform throughout, ie YF=YI=YR=Y O. ecru theory ( m o d e l - I )

The classical method for obtaining the tool profile that would machine a work of given shape, is to compute the corresponding equilibrium gap. For the case of hole sinking by ecm, the equilibrium gap is given by:

Ye = E Ev K Fp m (1)

This has been found to give erroneous results because it is based on the following assumptions:

During machining, conductivity of the electrolyte remains constant The electrolyte within the frontal gap (between the

Nomenclatu re

A Ac, B,CI C E

Ev FF HL,HBL,HTL J K Kij,etc Km L n R R B,R L,etc RBL1,RTL1, RBL2and RTL2

RBL,RTL,RTR, RBR

Cross-sectional area of the electric resistance Constants Specific heat of the electrolyte Electrochemical equivalent of the work- material Applied voltage Front feed See Fig 1 Current density Electrolyte electrical conductivity Coefficients of stiffness matrix Stiffness matrix (oi" conductance matrix) Length of the resistance Exponent Electric resistance Resistances as shown in Fig 4

Radii of bottom left and top left corners on tool and work respectively (assumed constant)

Resistances contributed by tool-work- "corners having some assumed fixed radii

T t V x

Y YI F, YE F,YTF

ev Pe Pm

Subscripts

BL,BR,TL,TR

e i,j .... L,R,F,T,B

O t

Temperature, °C Time, s Electrolyte f low velocity Distance in 'x' direction (electrolyte f low direction) Interelectrode gap

Initial gap, equilibrium gap and interelectrode gap at any time t respectively Temperature coefficient of electrolyte conductivity Void fraction Electrolyte density Work density

Bottom left, bottom right, top left and top right respectively (Fig 1 ) Equilibrium condition Node numbers Left, right, front, top and bottom respectively Condition at the start of machining Condition at time t

PR EC ISION E N G I N E E R ING 0141-6359/79/040199-08 $02.00 © 1979 IPC Business Press 199

Etectro[yte

I I=----J,~l\ ' , . : -_----- I I Fro?t

"---:o:---'T A ~ t - B -~--

To

HL

i Left

ELectroLyte

I ° t

Left -~ HTL ~- Top right

- - ~ --- Right

Bottom Left _~ HB L Bottom right

Tool

Elect r°lyte I

| I:: ~-_- - - ;---7-~----±Front

Work '

I

Fig 1 ecm sinking of rectangular holes by outward electrolyte flow, Fig 2 inward and Fig 3 side f low

RTL

;

; /

¢

¢ / / / / / / / / /

NBL

Work R RTR / / / / / / / / / / / / / / / . ~ / / / , ' / / / / / / / /

/ / /

~ , IVVV~.__ / / / / / R.

/

/ /

I /

I / / / / / / / / / / / / / / / / / / ) / / / / / / / / R e RSI~

Fig 4 Equivalent resistance of the electrolyte within the working gap for a hole sinking operation using rectangular tools

tool and work) contributes to the electrical resistance of the tool-electrolyte-work system, whereas the resistance of the electrolyte within the side gaps is assumed to be zero Gases evolved during the operation have no effect on electrolyte parameters During machining, the temperature of the electrolyte remains constant

In order to improve upon the accuracy of the results obtained from Equation (1), the following procedure was adopted.

M o d i f i e d t h e o r y ( m o d e l - 2 )

For a hole sinking operation using rectangular tools the equivalent resistance of the electrolyte within the working

gap (for three different conditions of flow) is shown in Fig 4. For the idealised model the following relationships between the resistances are valid:

1

Rtot Ri where,

1 1 - + Pi RB

and

• 1 .1 ÷ (2) Rj

1 1 1 1 + _ _ + _ _ + _ _ RL RT RR RF

1 1 1 1 1 - + + + - - Ri AB, AB.

The individual resistance in Equation (2) is given by the general relationship:

L R - (3)

KA

If the interelectrode gap is curved, the gap-resistance can be calculated by numerical integration. The temperature gradient in the electrolyte during ecm is given by:

d T j2 dx - KV Pe C - AcK = C1 (4)

and solution of this equation yields:

A T = (T -To)= l [exp [(AcKoex)(1--~v) n] -1] (5)

where K is given by Equation (9) and

j2 K F A c = ~ and J - EV YIF (6)

"For the condition when feed rate is zero, YF is given by:

YF = (2CF t + Y~F) ½ (7)

where

C F = (E E v K)/F Pm

and for finite feed, YTF at the instant t can be computed from:

1 t = ~-F [YIF -- YTF + YEF In .(YTFYIF--YEF_ YEF ) ] (8)

200 PRECISION ENGINEERING

Substituting for temperature rise, AT, from Equation (5), the electrolyte conductivity at t can be obtained from Equation (9):

K = K0 (1 + .e AT)(1 - aV )n (9)

Equations(6)-(9) were used for the computation of linear as well as volumetric metal removal (or penetration) rate (mrr) (Figs 9, 15 and 16). The accuracy of the computed results (Fig 5) in this case depends on the time interval (At) ~elected for the computation of temperature rise AT and the related parameters. In this case a time interval of 1 s can be seen to yield reasonably accurate results.

F i n i t e e l e m e n t t e c h n i q u e

For a unidimensional case (Fig 6), electrolyte temperature at any point is given by Equation (4) and as an approxi- mation it can be assumed that temperature conditions in the side and the front gap (ie YL, YTL and YBL) are identical. For the case under consideration, we can write Equation (4) as follows 19p2-23.

~i Xj-dT (dT)~._ (1o1

On simplification this reduces to:

DISx HS e = K m • T e (11 )

where Km is the conductance matrix given by:

Km = xj BTBdx (12a) xi

= for an element - 1

DIS and HS e in Equation (11 ) are evaluated as follows (Fig 1):

1 DIS =

Electrolyte mass flow rate x specific'heat

= [(YL(HL + YBL + YTL) + HTL X YTL +

HBLX YBL) V Pe C]-1 (12b) E~/ K xii

HS e = (12c) Rtot

where total resistance Rto t for an element is given by:

conslx cons2 Rt°t - cons1 +cons2 (12d)

and, YTL X YBL X YL

cons 1 = (YL(YTL XYBL + YBL X HTL)+YTL X YBL XHL)

("1 RBL2 RTL2 ",

cons2= 2 , / I n ( ~ ) ×/n(R--T'-L-T)-I I RBL2 - ~ I (12f) LIn(R--~-~-) + In(R---T-ZT) J

'X' = (DIS HS e) in Equation (11 ) is thus calculated for each node. Similarly 'X' can be calculated for the case of model-1 :

X = DI H e (12g)

where,

E ~ / K xji 1 H e = and DI -

YL Pe YL V C

TE EQI34 1 3

c] QI33 o

~ Ess3 YE ¢ QI32 ,~ O O t .

~ ~6.7 -~

al31 4 ss2 -'= ~,

0.130 - 6.6 551

Zero feed rate -- Results by fet-I

Machining.time 20s o At 2s /

&t 20s / . . . . Temperature / j - - - - Current density / /

-- - - - - - Interelectrode gap / / Conductivity / / ix-" ~

/ / / / . f_.oC;--'G

- / / . ~ Z ~ - - . ~ o > L f - .~..-

.,('/_~c.~. f . . . ~ . . . f o.,.~,"

,,~,.o" --;.

2 4 6 8 I0 12 14

Distance, m = 10 -2

Fig 5 Effect of time interval, At, on ecm parameters

~---=-_ ~ - - ~ Tool

i j k work

Fig 6 Unidimensional case

For this purpose the heat flowing towards a node (Fig 6) is treated as positive while that flowing out of the node is treated as negative. Assembly 22'23 of the equations of the elements, say two, will result in a symmetric, sparse and banded matrix of the type:

Ix l Ii 1 !1 IT I X2 = - 2 - T2 (13)

X3 - 1 T3

Temperature change of the electrolyte at entry can be taken as zero. Therefore,

T1 = 30°C (ie, room temperature) (14)

Equations (13) and (14) would enable us to obtain the unknown nodal temperatures T 2 and ?'3. Results obtained by finite.element analysis using the classical theory of ecm are described as model fe t -1 . Theoretical results obtained by fet-1 (Fig 7) have been compared with experimental results 14 for the machining conditions given in Table 1 and found to agree well.

(12e) Cos '~' t h e o r y

Results obtained from fet-1 have been compared (Fig 8) with those given by the cosine theory 20'21 which is based on the following assumptions:

For a tool having its face normal to the feed direction, the equilibrium gap is given by Equation(1 ). If the feed direction is inclined at an angle ~ to the normal to the tool face, or vice versa, the equilibrium igap would be equal to Ye/cos~.

Cosine theory has limited applications and the following drawbacks:

It does not yield valid results for regions with sharp corners. It is only applicable for 'e' < 45 °,

PRECISION E N G I N E E R I N G 201

It does not account for the effects of electrolyte f low, void fraction, change in electrolyte temperature, etc.

From Fig 8, clearly, discontinuit ies in the tool shape increase local metal removal (or penetration) and act as a barrier in obtaining the mirror image of the tool on the workpiece.

Parametric study of ecru

Resul ts a n d d iscuss ion

A parametric study of ecm could help in better understand- ing of the interrelationship between tool-feed rate, electrolyte f low velocity, interelectr0de gapl etc. Such infor- mation is useful for better design of ecm toolings. The analytical results presented in this section have been evaluated for the condit ions given in Table 2 and have been discussed for zero feed and f ini te feed condit ions.

Generation of side surfaces in ecm can be considered to be equivalent to machining under zero feed condit ions

and hence a discussion of the process parameters governing the metal removal under zero feed condit ions is included here.

Zero feed

Ef fec t s o f e l e c t r o l y t e s u p p l y c o n d i t i o n s

Increase in electrolyte f low velocity lowers the temperature rise wi th in the interelectrode gap, consequently the electrolyte conduct iv i ty and the current density decrease which f inal ly results in low metal removal (or penetration) and small overcut (Fig 9).

The ecm tool designer aims to achieve maximum mrr wi th best possible dimensional control. Min imum deviations, however, are possible at high electrolyte f low velocity, whereas the maximum mrr occurs at low f low velocities. In view of the conf l ict ing requirements the fo l lowing guidelines for ecm tool designers are suggesIed.

Length of the path travelled by the electrolyte between inlet and exi t should be minimised 27, in

'~D [ Experiment o._.o ~x [ 0 10 t x = 84 mm Experiment ------ 10[ fat-1 t=20s • 10 fat-1 t=140s • ~ ~ - Current 14

k t=32Os • ~E E < t 260s • L.

" f I ~'ci" -''°-- "e-''~" ~ 5

b b J ~ , , , , , ~ , , , , , i , Q ) t t ~ i t t

0 50 100 150 0 5 10 15 0 100 ' ' '2C)0' ' ' '3C)0 Distance x, mm Distance x,mm Machining time, s

x-2Smm ]O 1 ' - - 1% , I I ~ -E × ,_~

o < x Q - ° ° o ~ x

× X ¢" ~ ° °°° ° ° °° o o o o ~ 5 _ X x x 0

x X ~:

202 A •

. . . . 160 . . . . 2 6 d ' ' '3(~::) ' u ' ' I . . . . . . . . . . . . . . . I ~ 20.0 /

o 0 0 100 200 300 ~- O Machining time, s Machining time, s

Zero feed rote, Experiment A, x, o, m, fat-1 e---.4, x = Distance

Machining time =32Os

A & & & & & A & & & &

'sb . . . . 6 6 " 5'o Distance x,mm

Fig 7 Comparison o f theoretical, f e t - I, and experimental results

0

4

E 5 E d 6 ~ 0 '

1

2

3 4

5

%

Feed

x Too / Etectroty_~te ~ ~"-' J ~ . _ ~

Work )

: \ / l i i i I

5 ' 4 6 8 10

IX , L i i J i i i i i i i i LK~ '

P : x , , \ t=24o/

' 1"2"0 ' 2 ' 4 ' L ' 13 ' lb ' 1~2 DtstGnce, mm

i i

' 2 ' 4 ' 6 ' 8 1'0 Distance, mm

Fig 8 Progressive surface generation in ecru with mi ld steel, A = 0.015 totals, Ko = 0.02 ~-1 rnm-1

202 P R E C I S I O N E N G I N E E R I N G

Table 1 Experimental conditions 14

Initial interelectrode gap Width of work Length of work Electrolyte f low velocity Initial temperature of electrode Electrolyte conductivity Applied voltage Wor[<piece thickness Machining time

0.5 mm 40.0 mm

140.0 mm 10.0 m/s

20 -+ 1°C 0.013 ~,-i mm-I 2.8 volts 20.9-+ 1 x 10 -2 mm 300 s

Table 2 Conditions used for the analytical rest,Its

Feed rate, A 0.0, 0.02, 0.03, 0.04, 0.05, 0.06, 0.08, 0.10 m m / s

0.02 ~-z mm-1 Electrolyte conductivity

Temperature coefficient of specific conductivity,

Specific heat, C

Electrolyte density, DE

Work material density, DW

Electrochemical equivalent of work material, E

Effective applied voltage, E V

Faraday's constant, F

Time of each computation cycle

Initial temperature of electrolyte

Elec¼rolyte f low velocity, V

Initial electrode gap, YI

0.02/°C

4.18 W/g ° C

1.0X 103 kg/m 3

8.93 x 103 kg/m 3

63.57 x 10 -3 kg

10.0 V

96500.0 C

1.0s

25.0°C

10.0, 9.0, 8.0, 7.0, 6.0, 5.0, 4.0 m/s

0.2, 0.3, 0.9, 0.5, 0.6, 0.7, 0.8, 0.9, 1 •0 mm.

order to limit the variations in electrolyte conductivity and related parameters. Electrolyte with low temperature coefficient, ~, should be used. This can be achieved by the use of certain additives 24 . The electrolyte should have high electrical conductivity. This can be acl~ieved by using electrolytes at elevated temperature or by the use of certain additives 24 .

/nterelectrode gap

With a decrease in interelectrode gap, temperature rise and electrolyte conductivity have both been found to increase (Fig 10)• This is because of increased current f low and the heat generation. Linear and volumetric metal removal also increase with a decrease in interelectrode gap.

Side surface generation

In a hole sinking operation, as in Figs 11-13, side surfaces are theoretically generated under conditions of zero feed (tool side surfaces being parallel to the feed direction). Accurat6 prediction of the geometry of a generated work- surface, when complex shapes are involved, is possible only when an accurate estimate of the current densiW at different points is available.

Considerable over-cut on the holes produced during sinking has been observed in the case of bare as well as coated tools. Generation of surfaces that are straight, flat and parallel to the feed direction is governed by Equation (7). This eauation would yield accurate results onlv when simultaneous effects of different parameters are accounted for. This problem of side surface generation has been studied by Collett et al s and Hewson Browne 6 by the complex variable technique. Tsuei I and Nilson have analysed this problem but report considerable deviation between analytical and experimental results. Kawafune 14 has used numerical analysis for analytical determination of the hole shape produced by an ecru tool, whereas, KSnig 1°'11 has developed extensive empirical equations for this purpose. The existing methods for the computation of side surface profile in ecru do not yield satisfactory results for workpieces with complicated profiles.

One of the practical methods suggested for mini- raising the overcut in ecru is to insulate the sides of the tool.

As an alternative to coating, the authors recommend the use of tool bits properly mounted in tool holders made of plastics or other similar insulating material. To make them more effective, sides of the tool bits could be insulated which leads to practically zero overcut.

Figs 11-13 show the side surface generated by different types of tools and electrolyte f low for the condition YTL=YBL=COnSt. It is evident that coated tool bits result in no overcut whereas bare tool bits would lead to small overcut compared to that with bare tools.

12 - Zero feed rate Results by let -I / Machining time = 4s /

iO - ~ Current density 4 / ' . . . . Penetration / "

0.33 - Temperature r ise/" / /

. , . , 8 -

= o . ~ . - / ,, 6 , 4 ~, / . / / /

• / / " /-~.I~o. / / , ~ , . _ , ~ ' z , /

/ . . . ///" ~" / . - / / / . : . / / ' / . / . /, ,///~ . ll~, ~" "g / / /

. . . ~ : Y / i i / / /

T o 0.31 - R E 2 4 -~

~Q30 0.36

2 - i 0.29 0.35 V /

0 ' r / / / S /

o.,, I- I x,, / / > ~ / /

0.26 Q32

Q31

I 1 0.30 [ I 0 I0 20 30

Distance, mm

Fig 9 Effect o f electrolyte flow velocity on current density, metal removal and temperature rise

P R E C I S I O N E N G I N E E R I N G 203

0.27

0.26

'o 0.25 =¢ T

E E

~0.24

~n 0.23

022

Q21

4 E E

~ 3 -g t- t- u O

2

O 2'366

~-3 / ~ Zero feed rote ~-2 ~-- \ Results by fet-I

L k Distance = 50mm 2' 1 \ V= I0 m/s -~0 ~-- k Temperature --

/ \Conductivity -- --.

\ ,~ ~- ' , ~

',, / \ \ \ ) _

,5 ~ \ \ \,." ,\ ", \ \:~ ,, k "\ \ I \ ", \ ,"

"1- \ "-, \ ,21- \ ",, \ "L \ "-..

i

5 F -

4 } -

3 ~ - . . . . . . . . . . . . . . . °-Z k 0"!

O£ I0

I I d I 2 3 4

Machining time, s

+-, Drifting operation " ~ Machining time =180s

+ ~ Feed = O 03 m m/s " ~ + . V=80 m/s

~+ YF=YL=YR'OSmm YL=YR - const

+~*~*~Ba re too[ \

Fe~d TooL X

I I I I I I / / 2365 2364 2363 2362 2361 2.360

Over cut on left side, mm

-5 o

E~ O u / I (

Machining with feeding electrode

Effect of feed With increasing feed velocity the electrolyte temperature increases. It was observed that the rate of temperature rise beyond a feed rate of 0.1 mm/s was high because of decrease in interelectrode gap at the faster rate. A similar effect has also been observed on the electrolyte conduc- t iv i ty and current density (Fig 14). High current density and higher feed rate result in increased mrr but lead to non-uniform metal removal. For higher machining accuracy, therefore, the feed rate should be such that the equilibrium conditions are achieved in minimum possible time. Further, with high feed rate, sparking may damage the workpiece and tool 2s-26 .

Variable feed rates are common during machining of curved surfaces. Higher feed rates with the same machining accuracy can be used by either applying higher voltage or using an electrolyte with higher conductivity.

Interelectrode gap

Fig 15 shows the variation in electrolyte temperature, penetration and volumetric metal removal with time. Variations in electrolyte conductivity and current density were als O computed. If the initial interelectrode gap (0.7 mm) is larger than the corresponding equilibrium value, the gap tends to decrease and, therefore, temperature, electrolyte conductivity and current density increase. For the case when the initial gap is less than the equilibrium value the gap would tend to increase with time, so the temperature, conductivity~ and current density would decrease.

Electrolyte flow velocity At high electrolyte f low velocities, the temperature and interelectrode gap both decrease. A similar effect has also been observed with current density and electrolyte con-

E E

I ;

"(3

t - t - u 0

0 Drilling operation / + Machining time =90s / + " Feed = O 003 m m/s . / B o re toot V = 8 0 m/s / YF.YL. YR=O,Smm / * YL =YR = const. /

/ + F / eed

+,/ ~ \ Toot -/

1~ ~ Machined t ~ ~/~----~-'~ profile

~5

P

\ 4 i i i

1"700 1"7'08 17C25 # 0 I" 5 6 1707 1706 Over cut on left side, mm

Fig 10 Effect o f inter-electrode gap on electrulyte temperature rise and conductivity, above left

Fig 11 Side surface generation during ec dri l l ing with outward f low of electrolyte, left

FLq 12 Side surface generation during ec drilling, with inward electrolyte flow, above


/ / !

Machined p r o f i t e

" / ,o

_ _ N I_ , ~,-n .

q '=

l i

t ~

-5 £

o / \

' ' ' b ~ o ' ' ~ o ' s ' 1.709 1 7 0 8 1'707 17 6 1 5 1 7 0 4 0 Over cut [eft side, mm

1 3

i

EE E/ I E/ I

°

8 ~3 7 ~I I u \

/_,.,/

/ 4 6 & ~ . 5 / ,

Drifting operot ion Machining time - 180s Feed - 0 0 3 mm/s V = 8 O m / s YF =YL =YR = C' 5 mm YL" Yp= const.

B o r e t o o t / / d /

/ /

/

/ /

/ /

/ /

/ /

/ /

/ /

I I I I I I I

1712 1713 1714 1715 1'716 1717 1'718

Over c u t right side, mm

025 -

0.84

0.80

0.24 - -

o.m i TC~ 0.72

0.2:5

0.68

~ 0 , 6 4

0.22 "~

0.60

0.56

0.21 --

0.52

0.48

0 .44

0.40

Results by fet - I Distance = 30ram

I I 0.O. 6

l ' :21 /

/ / ! /

@! ," /

/ / /

/ / ~e~/

/ / /

/ / / ogg / /

/ / i I / / . i

/ I / / i 1 1 0.0~

1 / .~ ~ ...- ~,=O.O4 m...m~

I I I 2 3 4

Machining t ime, s

Fig 13 Side surface generation during ec drilling with side electrolyte flow, above

Fig 14 Effect of feed rate on current density and electrolyte conductivity,/eft

Fig 15 Effect of in.terelectrode gap on metal removal and temperature rise, b e l o w

==

~ 3

E

1.05

0.95

o - r -

0.85

0.75

0.65

0.28

)26

' 0 ).24

i g ).22

,~' ) .2¢

0.18

O.IE

0.14

Results by fet - I Distance = 30 .Omm

- A = O . O 3 m m / s , V= lO.Om/s Penetration

. . . . Volumetric mrr _ - - . . . . Temperature rise

YI = O.Tmm

~ .N.- . . . .

_ ~ - - ~ # 0 . g m m

~ - ' ~ "

I I 1 2 3 4 Machining t ime, S

PRECISION ENGINEERING 205

0.16 [ ....

i i i

I Results by fet- I Feed rate, A = OO3mm/s

0.38 ~ 4 ~ = 05mm

0 3 7

0

x

E 015 -- 7 E 0

x

E

° o., t L5

0.13

0.36

0.35

0.34

033

032

0.31

0 .30

029 0 I0 20 30

Distance, mm I I I

2 3 Machining time, s

Fig 16 Ef fec t o f e lect ro ly te f l o w veloci ty on l inear and vo lumet r ic meta l removal

duct iv i ty . With higher f l ow velocit ies, however, the metal removal (Fig 16) decreases in a manner simi lar to temperature. A t lower f l ow ve loc i ty the machining accuracy has been found to be poor. An o p t i m u m value of f l ow ve loc i ty should therefore be selected so that good dimensional cont ro l as well as a high mrr can be achieved.

C o n c l u s i o n s

This paper has demonstrated the use of modi f ied ecm theory for the compu ta t i on of metal removal rates and the product ion of an equ i l ib r ium wo rk surface. For cor r~ lex shaped workpieces, however, i t is no t possible to compute the gap resistances accurately and hence the use of l e t has been recommended. S implex type elements for f in i te e lement analysis of the gap condi t ions in ecm y ie ld results that compare favourab ly w i th the exper imenta l data. Results f rom f in i te e lement analysis also give an insight of the nature of in teract ion between the process parameters in ecru and wou ld help in better understand- ing the problems associated w i th ecm too l ing design.

Work is cur ren t ly in progress on the appl icat ion of l e t to two and three dimensional machin ing problems by ecrn.

References 1. Tsuei Y.G. et al Theoretical and Experimental Study of Work-

piece Geometry in Electrochemical Machining. ASME Paper No. 76 WA/Prod 1--5

2. Nilson R.H. and Tsuei Y.G. Inverted Cauchy Problems for the Laplace Equation in Engineering Design. J. Engg. Math., 1974, 8 329-337

3. Nilson R.H. and Tsuei Y.G. Free Boundary Problem for the Laplace Equation with Application to ECM Tool Design. Trans ASME (App. Mech.) 1976, 98, 54-58

4. Nilson R.H. and Tsuei Y.G. Free Boundary Problem of ECM By Alternating Field Technique on Inverted Plane Computer Methods. App. Mech. and Engg., 1975, 6 265-282

5. Collett D.E., Hewson-Browne R.C. and Windle W. A Complex Variable Approach to Electrochemical Machining Problems. J. Engg, Math, 1970, 4 29 -37

6. Hewson-Browne R.C. Further Applications of Complex Variable Methods to Electrochemical Machining Problems J. Engg. Mat., 1971, 5 233--240

7. Tipton H. The Determination of the Shape of TooLs for Use in Electrochemical Machining, Research Report No. Forty, 1971, 9 -42 The Machine Tool Industry Research Association, UK

8. Tipton H. The Calculation of Tool Shapes for Electrochemical Machining. Fundamentals of Electrochemical Machining, Edited by C.L. Faust, 1971, 87--102

9. Tipton H. The Dynamics of Electrochemical Machining Proc. 5th Int. MTDR, 1964, 509-522

10. Ki~nig W. and Humb H.J. Mathematical Models for the Calcu- lation of the Contour of the Anode in ECM. Annals CIRP, 1977, 25 ( I ) 83

11. K6nig W. and Pahl D. Accuracy and Optimal Working Con- ditions in Electrochemical Machining Annals CIRP, 1970, 18 223-230

12. Loutrel S.P. and Cook N.H. A Theoretical Model for High Rate E!ectrochemical Machining Trans ASME Paper No. 13- Prod-2 (1973) 1 -6

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19. 3ain V.K. and Pandey P.C. Application of Finite Element Technique for the Analysis of Electrochemical Wire Cutting Process. To be published

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