a novel method of determination of wire lag for enhanced profile accuracy in wedm

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Precision Engineering 35 (2011) 339–347

Contents lists available at ScienceDirect

Precision Engineering

journa l homepage: www.e lsev ier .com/ locate /prec is ion

novel method of determination of wire lag for enhanced profile accuracy inEDM

. Sarkara,∗, M. Sekhb, S. Mitraa, B. Bhattacharyyaa

Production Engineering Department, Jadavpur University, Kolkata 700032, IndiaProduction Engineering Department, H.I.T., Midnapore(E) 721657, India

r t i c l e i n f o

rticle history:eceived 4 April 2010eceived in revised form 20 October 2010ccepted 6 January 2011

a b s t r a c t

Wire bending due to gap force is a major cause of imprecision in WEDM applications. To achieve higherprecision and accuracy the knowledge of gap force and wire lag is extremely essential. In the presentresearch, an in depth study on wire lag phenomenon has been carried out. A novel method to measuregap force intensity and wire lag under any given machining condition has been proposed by developing

vailable online 13 January 2011

eywords:EDMire offset

rofile accuracy

an analytical model. Experiments were carried out to verify the proposed model. Beside this, the impactof wire deflection on profile accuracy during cutting cylindrical job has been investigated. Based upon thedeveloped analytical model an effective method has been proposed to eliminate this inaccuracy usingwire lag compensation technique. The research finding will lead to better understanding of the gap forcephenomena and will promote significant development in the domain of high precision WEDM.

ire lag compensationap force

. Introduction

The cutting speed, surface finish and taper angle are improvingarkedly day by day since the beginning of wire electrical dis-

harge machining (WEDM). But, wire bending, which is a majorause of cutting imprecision, is still hampering the part accuracyor various applications. When cutting out a sharp corner or curvedrofile, the bending of the wire creates a geometrical error onhe workpiece. This error can be of the order of a few hundred

icrons; which for some applications becomes unacceptable [1].everal research efforts have been given for improving the accu-acy in WEDM. Dekeyser and Snoeys [2] proposed off-line pathodification strategy to enhance the profile accuracy. Later Dauw

nd Beltrami [1] developed on line wire path control system usingn on line optical wire position sensor. Hsue et al. [3] addressedhe unsteady phenomena of the machining parameters at the cor-er using the concept of discharge angle. To reduce the machiningrrors of corner parts, Lin et al. [4] developed a fuzzy control strat-gy to improve corner accuracy through reduction of wire lag nearorner by increasing the pulse off time. Using Taguchi method-
logy Puri and Bhattaharyya [5] minimized corner inaccuracy inrim cutting operation through optimization of process parame-ers. Sanchez et al. [6] proposed a technological data base systemn which experimental knowledge and numerical simulation of the
∗ Corresponding author.E-mail address: [email protected] (S. Sarkar).

141-6359/$ – see front matter © 2011 Elsevier Inc. All rights reserved.oi:10.1016/j.precisioneng.2011.01.001

© 2011 Elsevier Inc. All rights reserved.

process allow the user to select the optimum cutting strategy, eitherby wire path modification or by cutting regime modification. Yanet al. [7] claimed improvement in corner accuracy using on lineclosed loop wire tension control system. A corner error simulationmethod was proposed by Fuzhu et al. [8] to predict the actual cornerprofile under different cutting condition. Sanchez et al. [9] studiedhow the number of finishing cut and cutting speed limitation influ-ences the corner accuracy on different thickness workpiece. Linand Liao [10] proposed an effective-wire-compensation scheme tospecify the wire location matrices required to improve the precisionof the machining results while manufacturing components withtapered features or complex profile. Dodun et al. [11] investigatedthe error in machining of small corner angle in thin parts.

From the review of the past research, it is found that the pro-cedure to improve accuracy can broadly be classified in to twocategories. The first procedure is to modify the cutting parame-ters (pulse on time, pulse off time peak current etc.) in order toreduce the wire deflection [2,4–6,9]. This methods result in low-ering of cutting speed at the corner. The second procedure is tomodify the wire path to correct the geometrical inaccuracy in anon-line [1,6,11] manner. In the second procedure, although the cut-ting speed is not reduced but accuracy level is comparatively poorcompared to first procedure for smaller angle [6]. To implement
the either of the above procedure the knowledge of wire deflec-tion for a given machining parameter setting is extremely essential.Some researchers attempted to measure the wire displacementusing optical sensor [1] or electrical contact between wire andworkpiece [12]. But it is not an easy task to measure the wire posi-
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340 S. Sarkar et al. / Precision Engine

Nomenclature

ri input wire offset setting (including wire lag com-pensation component) (mm)

rw radius of the wire (Fig. 2) (mm)� radial spark gap (Fig. 2) (mm)εwl required wire lag compensation while cutting cylin-

drical shape job Fig. 2 (mm)rp programmed radius of the centre of the wire

[Figs. 2 and 4(a)] (mm)ra actual radius of the profile traced by the wire centre

at the top and bottom surface [Figs. 2 and 4(a)] (mm)r′a distance between centre of the cylindrical job and

wire centre at the point where the job radius is min-imum [Fig. 4(a)] (mm)

raj maximum radius of the job (mm)r′aj

minimum radius of the job (mm)R Radius pf curvature(mm) [Fig. 4(b)]�1 angle ∠AOB of Fig. 4(a) (radian)�2 angle ∠BOC of Fig. 4(a) (radian)� angle ∠BOE of Fig. 4(a) (radian)˛ projected value of inclination of the wire above the

job surface on XY plane with respect to XZ plane, i.e.angle ∠DCG of Fig. 4(a) (radian)

ˇ′0 true inclination of the straight portion of the wire

above the job surface with respect to YZ plane (Fig. 5)ˇ0 projected value of ˇ′

0 on XZ plane shown[Fig. 4(b) and 5] (radian)

ˇ true inclination of radius of curvature of thewire at the point E with respect to XY plane[Fig. 4(b)](radian)

T tension of the wire above the job surface (kg)t job thickness (mm)b gap between wire guide support and job surface

(mm)q gap force intensity, i.e. gap force per unit length of

the wire (kg/mm)E′F′ infinitesimally small element on the wire which

subtends an angle dˇ at the centre of radius of cur-vature [Fig. 4(b)]

K radial wire lag compensation constant (mm2)ıi wire deflection between top and bottom surface in

the direction opposite to the applied tension (Fig. 6)(mm)

ı0 wire deflection at the top and bottom surfaces of thejob (Fig. 6) (mm)

ıt total deflection of the wire at the middle point(Fig. 6) (mm)

ı lateral deflection at the middle point of the wire

tmiTnNtniwAi

in rough and finish cut operation in order to enhance the accu-

l(Fig. 6) (mm)

ion accurately by the sensor. To maintain adequate flushing in theachining zone the sensor must be very small and at the same time

t must be strong enough to withstand very high dielectric pressure.he presence of debris in the dielectric medium and electromag-etic disturbances by EDM process creates further difficulties [1].evertheless, this method of wire deflection measurement is bound

o be very costly and for these reasons the above methods areot at all successful in view of industrial application. Thus there

s no straight forward method for measurement of exact value ofire deflection or wire lag under different machining conditions.t present this is a major difficulty towards accuracy improvement

n WEDM.

ering 35 (2011) 339–347

Apart from corner accuracy, the knowledge of wire lag valueis extremely essential to achieve profile accuracy in case of (con-tinuous) curved profile cutting. In general, to achieve dimensionalaccuracy the most accepted solution is based on cutting square orrectangular test parts and then measuring the deviation betweenthe programmed path and actual profile the wire offset value iscalculated and then this wire offset value is used as the inputparameter [13,14]. This method only compensate for wire radiusand spark gap, but this method will not take care of the inaccu-racies arises out of wire deflection and hence there will be someinaccuracy during curved profile cutting in spite of adopting wireoffset value. For example, it will be seen in subsequent section thatit is not even possible to achieve an accurate dimension of cylin-drical job using the conventional wire-offset value determined bythe trial-cut. It must be noted that the method of cutting parame-ter modification may minimize this error due to wire lag but it isnot practicable in case of curved profile cutting because it results inhuge increase in cutting time due to reduced cutting speed throughout the curved profile (it may be noted that incase of corner cutting,reduction in cutting speed will take place at the corner only) never-theless it cannot completely eliminate the inaccuracy due to wirelag effect. Thus to improve precision there is a need for implementa-tion of additional wire path compensation strategy in the CNC partprogram to compensate the effect of wire lag during curved pro-file cutting. Due to the lack of knowledge of exact value of wire lagthe WEDM manufacturers proposes time consuming experimen-tal trial-and-error methodologies for the correction of the errors.To reduce the experimental load and to contribute a more generalapproach to the problem, it is very essential to determine the exactwire deflection value under any given machining condition. In thisconnection it must be observed that although lot of research workshas been carried out to improve the corner (sharp or round) accu-racy but till date no research work has been reported on impact ofwire lag phenomenon on continuous curved profile accuracy.

In view of the above fact it is obvious that remarkable improve-ment in respect of accuracy can only be achieved if the exact valueof wire deflection under different cutting condition is known. In thepresent research paper an analytical model has been developed todetermine the gap force and wire deflection. Using the proposedmodel it will become very easy to determine the wire lag under dif-ferent machining conditions. Besides, this paper also demonstrateshow the proposed model can be utilized to achieve enhanced pre-cision of the cylindrical job using wire lag compensation strategy.

2. Wire lag and its impact on corner and curved profileaccuracy

Although wire tension is applied on the wire, since the wireis thin and flexible it deforms due to gap force and as such wiredeflects opposite to the cutting direction and the actual wire posi-tion is always behind the wire guide. It is believed that this gap forcedevelops mainly due to reaction of explosive force from gas bubbles.Hydraulic force, electromagnetic force, electrostatic force etc. areother contributing factors. Primarily due to unstable nature of theplasma channel and stochastic nature of sparking, the force exertedon the wire is not constant; instead it fluctuates with respect totime which in turn promotes continuous vibration in the wire. Thisvibration is one major source of inaccuracy in WEDM as it has stronginfluences on overcut and tolerance in thickness direction. Severalresearchers have investigated this dynamic behavior of the wire

racy in WEDM [12,15–18]. Although instantaneous position of thewire is function of time but the mean displacement is found to beconstant for a given machining condition and is termed as wiredeflection or wire lag [16]. Thus, in contrast to dynamic behavior

S. Sarkar et al. / Precision Engineering 35 (2011) 339–347 341

δr

ξ

desired profile of the workpiece

actual profile of the workpiece

wire

F

otahdctfitctditF(dtccaε

Fl

actual wire geometry

assumed wire geometry

wire guide

workpiece

programmed path of the wire centreactual path of the wire centre

ig. 1. Influence of wire lag on geometrical inaccuracy during sharp corner cutting.

f the wire, the wire lag phenomenon is basically static behavior ofhe wire. Wire lag or wire deflection is mainly determined by aver-ge gap force and other parameters like wire tension, workpieceeight and gap between the workpiece surface and wire guide. Thiseflection causes various types of geometrical inaccuracies. Whileutting straight profile the length of the profile is always less thenhe programmed path of the wire guide. For this reason it is very dif-cult to cut accurate length slit in WEDM process particularly whenhe wire deflection is large. During sharp corner or curved profileutting, this mismatch between the wire position and guide posi-ion leads to other types of geometrical inaccuracies. Figs. 1 and 2epicts how the wire lag influences the geometrical inaccuracy dur-

ng sharp corner and profile cutting. From Fig. 1 it is observed thathe wire deflection has a strong influence on the corner inaccuracy.rom Fig. 2 it is observed actual profile traced by the wire centrera) does not matches with the programmed profile of the wire (rp)ue to wire deflection (ı0) at the surface of the job. In Fig. 2, εwl ishe perpendicular distance between programmed path of the wire
entre and actual path of the wire centre during cutting a cylindri-al shape job and is termed as wire lag compensation value. Thus tochieve desired profile the required wire lag compensation valuewl is equal to (rp − ra). Wire lag compensation component must
rwwl

programmed path of the wire centre.actual path of the wire centre.

raj

ra

rp

guide position

δ 0

ξ

ig. 2. Influence of wire lag on geometrical inaccuracy during curved profile (circu-ar) cutting.

Fig. 3. . Actual and assumed wire geometry (top view) during cutting curved profile.

be added (εwl) with wire radius (rw) and spark gap (�) to get thetotal wire compensation value, required in the CNC part program.Thus in order to achieve full precision during circular job cutting theinput wire offset setting ri = rw + � + εwl. In the subsequent sectionthe systematic procedure to determine this wire lag compensationvalue εwl will be discussed.

3. Mathematical model for gap force intensity and wire lag

Although expression for wire lag value as a function of gap forceis available for straight profile cutting [1] but no such expressionis available for curve profile cutting. In the present research studyattempt has been made to calculate the wire lag value while cuttingcircular profile. Calculation of wire lag value during circular pro-file cutting is much more complicated compared to straight pathcutting because during any curved profile cutting the direction ofgap force is not unidirectional rather the direction of the gap forceon the wire varies with respect to job height. Due to this fact thewire will not have only deflection opposite to the applied tensionbut there will be some small amount of lateral deflection as well.However, by following reasonable assumptions, analytical modelfor wire lag will be developed.

(i) Gap between job surface and wire guide (b) are same; both atthe top and bottom side.

(ii) It is assumed that no time dependent phenomena are influ-encing the wire behavior and gap force has been assumed tobe constant for a given parameter setting and is uniformly dis-tributed over the length of the wire which is between top andbottom surface of the job, i.e. gap force intensity q (gap forceper unit length) is constant.

(iii) As already mentioned that there will be very small amount oflateral bending due to lateral gap force; but to simplify theanalysis this bending has been neglected. Fig. 3 shows theactual and assumed top view of the wire during curved profilecutting. Thus wire is assumed to be rigid against lateral gapforce.

(iv) Usually the deflection of the wire is in the order of few hundredmicrons [1] and this value is usually much less compared to
the radius of the job and as such �1 and �2 can reasonably beassumed to be very small [Fig. 4(a)]. As �1 and �2 are small,the amount of lateral gap force is small and hence ˛ is usuallyvery small.

342 S. Sarkar et al. / Precision Engineering 35 (2011) 339–347

A B C

D

O

E

θ1 θ2

θr'a

ra

F(qRdβ)

β0β

dβ

B'

E'F'

C'

D'

ß0

R

aX

Y

Z

G

a

b

α

β0

Rt/2

b

θ

(qRdßcosθ )

X

Fu

(

iaooocar

r

wto

wj

ß0

ß'0

C

D'

d

d'

b

c α

G'D''C''F''E''B''A'

ig. 4. (a) Top view of the wire during cutting circular profile. (b) Front view of thepper half of the wire during cutting circular profile.

(v) From Fig. 4(a) and (b) it is seen that gap force acting oninfinitesimally small element EF is qRdˇ. Component of thegap force qRdˇ sin � produces lateral load on the wire and com-ponent qRdˇ cos �, acting in the plane of wire profile producesbending in the wire. Due to variation of � with respect to thick-ness (t) of the job this force will also vary and as such therewill be variation in radius of curvature (R). But as the value of� is very small this variation in R has been neglected and it hasbeen assumed to be constant. Beside this the wire is assumedto be perfectly flexible.

(vi) Usually the wire deflection between the top and bottom sur-faces of job (ıi) is small and as such the radius of curvature ofthe wire is usually very large compared to the thickness of thejob. Hence it is reasonably to assume ˇ0 is small [Fig. 4(b)].

vii) Wear of the wire due to sparking has not been considered inthis analysis.

The X, Y and Z axes are shown in Fig. 4(a) and (b) and it is selectedn such an way that origin coincides with the centre of the wire, i.e.t point B in Fig. 4(a) and B′′ in Fig. 4(b) and Z axis is parallel to axisf the cylindrical job. In Fig. 4(a), ra is the distance between centref the cylindrical job and wire centre at the top and bottom surfacef the job and r′

a is the minimum distance between centre of theylindrical job and wire centre. The radius of the wire is rw and themount of radial spark gap is �. Now from Fig. 2 the expression fora may be written as follows

a = raj + rw + � (1)

here, raj maximum radius of the at the top and bottom surface of
he job, rw is the radius of the wire and � is radial spark gap or radialvercut.
Fig. 5 shows the position of the wire Cd′. (which is above theorkpiece surface) in three dimensional space. In Fig. 5, ˛ is pro-

ected value of inclination of the wire above the job surface on

D

Fig. 5. Position of the straight portion of wire above the job surface in three dimen-sional space.

XY plane with respect to XZ plane, i.e. angle ∠DCG [also shown inFig. 4(a)] and ˇ′

0 is true inclination of the straight portion of thewire above the job surface with respect to YZ plane (Fig. 5) and thisvalue is equal to angle between wire and wire guide axis above andbelow the job surface. ˇ0 is projected value of ˇ′

0 on XZ plane shown[Figs. 4(b) and 5]. It shall be noted that for small value of ˛, C′D′ istangential to the curve A′C′ and thus true inclination of radius ofcurvature at the point C is also ˇ0 [Fig. 4(b)].

CD is the projected length of the straight portion of the wirein XY plane and CD′ is the projected length of the straight portionof the wire in XZ plane above the job surface. From Fig. 5 it is seenthat, tan ˇ′

0 cos ˛ = tan ˇ0; and since ˛ is very small; cos ˛ ≈ 1. Thus,ˇ′

0 = ˇ0.In Fig. 5, Dd′ = b is the gap between wire guide support and job

surface and the wire deflection at the top and bottom surface ofthe job ı0 = CD = b tan ˇ0/cos ˛ and since ˇ0 and ˛ are small theexpression for ı0 becomes

ı0 = bˇ0 (2)

From Fig. 4(b) it is seen that sin ˇ0 = t/2R. Where, t is the thick-ness of the job and since ˇ0 is small sin ˇ0 ≈ ˇ0. Thus the expressionfor radius of curvature R = t/2ˇ0.

From Fig. 4(a) and (b) it is seen that

r′a tan �1 + r′

a tan� = R − R cos � (3)

For small values of �1, � and ˇ; tan �1 ≈ �1, tan � ≈ �,cos ˇ ≈ 1 − ˇ2/2 and substituting R = t/2ˇ0, the above expressionmay be rewritten as follows

� = ˇ2t

4r′aˇ0

− �1 (4)

Again from Fig. 4(a) and (b) it is seen thata = FC = F′′C′′ = R(cos ˇ − cos ˇ0) and since ˇ and ˇ0 are small, amay be expressed as follows

a = R

2(ˇ2

0 − ˇ2) (5)

Now the upper half of the wire which is between centre and topsurface of the job [i.e. AC in Fig. 4(a) and A′C′ in Fig. 4(b)] has beenconsidered as a free body. This free body is in equilibrium under theaction of two forces: tensile force T and the distributed gap force. InFig. 4(b) E′F′ is an infinitesimally small arc element and it subtendsan angle dˇ and its radius of curvature R makes an angle ˇ withnegative X axis as shown in Fig. 4(b). The gap force acting on thiselement is qRdˇ. Considering equilibrium of the upper half of the
wire (which is between centre and top surface of the job) alongX-axis the following equation is obtained.∫ ˇ0
0

(qRdˇ) cos � = T sin ˇ′0 cos ˛ (6)

Engineering 35 (2011) 339–347 343

cr

R

e

ˇ

(o∫

aaEl

˛

p[∫

�l

�

u

˛

F(

�

e

�

˛i

r

a

r

⇒

α

θ 1θ 2

BA C

D

δδi

δl

ra'

Oguide

H

ra rp

from Eq. (1), q may be estimated as follows

q =(

2T

tb

)√{rp

2 − (rja + rw + �)2}(1 + (t/3b))

(23)

0.005

0.01

0.015

0.02

0.025

Req

uir

ed w

ire

lag

co

mp

ensa

tio

n

ε wl (

in m

m)

Experimental data

Regression model

S. Sarkar et al. / Precision

Since R and q are assumed to be constant; as �, ˛, ˇ are smallos � ≈ 1, cos ˛ ≈ 1 and sin ˇ′

0 ≈ ˇ′0 ≈ ˇ0, the above equation

educes to

= T

q(7)

Using the above expression and putting R = t/2ˇ0, ˇ0 may bevaluated as follows

0 = tq

2T(8)

Now, considering equilibrium of the upper half of the wirewhich is between centre and top surface of the job) along Y-axisne can obtain the following equation:

ˇ0

0

(qRdˇ) sin � = T sin ˇ′0 sin ˛ (9)

As already stated, the radius of curvature (R) and gap force (q) aressumed to be constant. Since �, ˛, ˇ are small; sin � ≈ �, sin ˛ ≈ ˛,nd sin ˇ′

0 ≈ ˇ′0 ≈ ˇ0 and substituting the values of � and R from

q. (4) and Eq. (7) respectively and finally integrating between theimits, ˛ can be expressed as

= tˇ0

12r′a

− �1 (10)

Again, moment about point C of the upper half of the wire in XYlane must be zero. Thus considering the moment about point CFig. 4(a)] one obtain the following equation:

ˇ0

0

a(qRdˇ) sin � = 0 (11)

Now using sin � ≈ �, since qR2 /= 0 and substituting the value ofand a from Eq. (4) and Eq. (5), and finally integrating between the

imits the expression for �1 becomes

1 = tˇ0

20r′a

(12)

Using the above value of �1, the value of ˛ can now be estimatedsing Eq. (10).

= tˇ0

30r′a

(13)

It is seen that at point C in Fig. 4(a) � = �2 and at point C′ inig. 4(b) ˇ = ˇ0. Substituting these boundary values of � and ˇ in Eq.4) the following relationship is obtained.

2 + �1 = tˇ0

4r′a

(14)

Again substituting the value of �1 from Eq. (12) in the abovexpression, �2 may be estimated as follows

2 = tˇ0

5r′a

(15)

In Fig. 6 considering �OCD, and substituting the value of �2 andusing Eq. (15) and Eq. (13) respectively, and remembering (�2 − ˛)

s very small the following relation is obtained.

p2 − ra

2 = b2ˇ20 + bˇ0

2tra

3r′a

(16)

As the value of �2 is small and as such r′a ≈ ra and hence the

bove expression becomes( )
p
2 − ra2 = b2ˇ0

2 1 + t

3b(17)

ra = rp

{1 − ˇ0

2b2

rp2

(1 + t

3b

)}1/2

(18)

Fig. 6. Wire lag deflection during cutting circular profile (top view).

Since ˇ0 is small, ˇ04 and higher order terms have been ignored

and thus the above expression becomes

ra = rp

{1 − ˇ0

2b2

2rp2

(1 + t

3b

)}(19)

Substituting the value of ˇ0 using the Eq. (8) it is seen that

rp(rp − ra) = 18

(btq

T

)2 (1 + t

3b

)(20)

⇒ rpεwl = K (21)

where

K = 18

(btq

T

)2 (1 + t

3b

)and εwl = (rp − ra) (22)

For a given machining condition all the parameters like b, t, q andT remain unchanged and hence K is a constant. Thus required wirelag compensation (εwl) is inversely proportional to programmedradius (rp). The constant K is termed as radial wire lag compensa-tion constant. Eq. (21) is very much useful to determine the amountwire lag compensation (εwl) for different radius cylindrical job. Thisvalue will be used as input parameter in the part program to obtainthe accurate dimension of the job. From the above expression it isobserved that inaccuracy due to wire lag depends upon the geom-etry of the job and is more significant in smaller radius job wherethe accuracy requirement is generally higher.

Now combining Eq. (8) and Eq. (17) and putting the value of ra

0

876543210

Programmed radius of the wire centre r p (in mm)

Fig. 7. Experimental data and regression model of wire lag compensation for dif-ferent program radius (job height = 23.1 mm).


0

0.005

0.01

0.015

0.02

0.025

0.03

86420

dius

Req

uir

ed w

ire

lag

co

mp

ensa

tio

n

ε wl

(in

mm

)

Experimental data

Theoretical curve

ompe

afrgtf

tc

ı

dd

ı

tb

ı

ı

sc

ewa�

ı

or

Programmed ra

Fig. 8. Experimental data and regression model of wire lag c

By cutting a square or rectangular test parts � can be determinednd by cutting a circular (cylindrical) test part raj can be determinedor a given rp. Beside this, other machining parameters, i.e. wireadius (rw), wire tension (T), job thickness (t) and gap between wireuide and job surface (b) are all known parameters. Thus by usinghe above equation average gap force intensity can be estimatedor any given machining condition.

It may be noted that due to symmetry the amount of wire deflec-ion at the top and bottom surface of the job are same. Now byombining Eq. (8) in Eq. (2), ı0 may be expressed as follows:

0 = btq

2T(24)

In Fig. 6 wire deflection between top and bottom surface in theirection of ı0, is HC(=ıi). Since �1, �2 and ˛ are small the wireeflection between the job surfaces ıi may be expressed as follows:

i = (r′a tan �1 + r′

a tan �2) cos ˛ = r′a(�1 + �2) (25)

Now putting the value of �1 + �2 = tˇ0/4r′a from Eq. (14) and

hen substituting the value of ˇ0 from Eq. (8) the expression for ıiecomes

i = t2q

8T(26)

So the expression for total deflection (ıt) becomes

t = ıi + ı0 = t2q

8T+ btq

2T(27)

It may now be noted that this expression of wire deflection isame as obtained by the previous researcher during straight pathutting [1].

As already mentioned, that there will be certain amount of lat-ral deflection (ıl) in the wire. The maximum lateral wire deflectionill occur at the centre of the wire. From Fig. 6 it is seen that the

mount of lateral deflection at the centre of the wire is AH and since1, �2 and ˛ are small ıl may expressed as follows:

i = (r′a tan �1 + r′

a tan �2) sin ˛ = r′a(�1 + �2)˛ (28)

Now putting the value of �1 + �2 = tˇ0/4r′a from Eq. (14), value

f ˛ from Eq. (13), then substituting the value of ˇ0 from Eq. (8) andemembering r′

a ≈ ra the expression for ıl becomes

ıl = q2t4

480raT2= q2t4

480(raj + rw + �)T2(29)

of the wire centre rp (in mm)

nsation for different program radius (job height = 37.5 mm).

From the above expression it is clear that lateral deflection isstrongly influenced by job thickness and wire tension. It is alsoobserved that amount of lateral deflection reduces with increase injob radius (raj). It may also be noted that during straight path cut-ting raj is infinite and hence ıl = 0. Thus as expected, during straightpath cutting amount of lateral deflection is zero.

4. Results and discussion

To verify the proposed model Eq. (21) has been used. Three dif-ferent jobs having three different heights, e.g. 23.1 mm, 37.5 mmand 62.6 mm were machined. For each height four cylindrical jobshaving different values of programmed radius (with zero offsetvalue) as shown in Tables 1–3 were cut. A typical wire-cut EDMmachine, supercut-734, was used for conducting the experimentswith a typical die steel (M2-hardened and annealed: 0.85% C, 4%Cr, 6.25% W, 5% Mo, 2% V) plate as the workpiece (anode). A brasswire of 125 �m radius (rw) was used as the tool electrode (cath-ode). Besides, another rectangular job were cut out from all thethree die steel plates in the same parameter setting in order todetermine the radial overcut or spark gap (�). The value of � wasfound to be 146 �m, 151 �m and 153 �m for 23.1 mm, 37.5 mmand 62.6 mm thickness workpiece respectively. The job dimen-sions were measured by a digital micrometer having a least countof 0.001 mm. In all cases the gap between wire guide and work-piece surface (b) was 15.37 mm and wire tension (T) was kept at0.9 kg. Other machining parameters which have been kept con-stant are as follows: pulse on time (1.2 �s), pulse off time (20 �s),peak current (180 Amp), open-circuit voltage (100 V), water pres-sure (15 kg/cm2), servo voltage (3 V), servo feed gain (0.150). Thewire is fed with a servo feed control mode. Under this cuttingcondition, cutting speed were 1.39 mm/min, 0.92 mm/min and0.6 mm/min for 23.1 mm, 37.5 mm and 62.6 mm thick job respec-tively.

Actual measured radius of the job at the top surface (raj) hasbeen shown in Tables 1–3. In this connection it may be noted outthat ideally top and bottom radius of the job should be same dueto symmetry but due to wear of the wire the bottom radius is usu-ally a little bit higher. However, to avoid the effect of the wear of
the wire, only the radius at the top surface of the job has beenconsidered. It is observed in Tables 1–3 that the value of radialwire lag compensation constant K (e.g. product of wire lag com-pensation and programmed radius of the wire centre) is invariantwith radius change and remain almost constant for a particular job

S. Sarkar et al. / Precision Engineering 35 (2011) 339–347 345

Table 1Summary of the experimental result for 23.1 mm thick workpiece.

Exp. no. Program-medradius of thewire centre rp

(mm)

Actual radius ofthe job at thetop surface raj

(in mm)

Actual radius ofthe profiletraced by thewire centre atthe top surfacera

(ra = rja + rw + �)

Required wirelagcompensationεwl

(εwl = rp − ra)(in mm)

Radial wire lagcompensationconstant(K = rpεwl)(mm2)

Gap forceintensity q(kg/mm)

1 1.5 1.337 1.483 0.018 0.0263 10.71 × 10−4

2 3 2.846 2.992 0.009 0.0255 10.58 × 10−4

3 5 4.849 4.995 0.005 0.0250 10.48 × 10−4

4 7 6.851 6.997 0.004 0.0245 10.38 × 10−4



(mm)


(in mm)


(ra = rja + rw + �)





1 1.5 1.330 1.480 0.020 0.0300 6.42 × 10−4

2 3 2.840 2.990 0.011 0.0315 6.59 × 10−4

3 5 4.844 4.994 0.006 0.0300 6.44 × 10−4

4 7 6.846 6.996 0.005 0.0315 6.60 × 10−4



(mm)


(in mm)


(ra = rja + rw + �)





74 −4

879294

ht

pfa6

r

f

r

a

r

etoma(ot

wlimprecision can be avoided very easily. For example if the existingstandard method is used in case of 62.6 mm thick and 3 mm diam-eter job it results in undersize job and its diameter will be (2εwl=)54 �m less than the desired value. This type of error can easily be

0.005

0.01

0.015

0.02

0.025

0.03

0.035

equ

ired

wir

e la

g c

om

pen

sati

on

ε w

l (

in m

m)

Experimental data

Theoretical curve

1 1.5 1.321 1.42 3 2.835 2.93 5 4.840 4.94 7 6.842 6.9

eight as predicted by the analytical model. Thus it is obvious thathe proposed analytical model is quite valid in practical machining.

Using the experimental data shown in Tables 1–3, wire lag com-ensation constant K has been calculated using regression analysisor each job height and the R2 values are very close to 1 and theyre 0.987, 0.978 and 0.997 for job height 23.1 mm, 37.5 mm and2.6 mm respectively. The regression equations are as follows:

For 23.1 mm job

pε = 0.027 (30a)

or 37.5 mm job

pε = 0.033 (30b)

nd for 62.6 mm job

pε = 0.027 (30c)

To further verify the proposed regression model another set ofxperiment has been carried out for each job thickness. Results ofhese verification experiments are shown in Tables 4–6. The resultsf verification experiment and curves of corresponding regression
odel based upon Eq. (21) are shown in Figs. 7–9. As expected from
nalytical model it is observed that required wire lag compensationεwl) is inversely proportional to programmed radius (rp) and it isbserved that the developed model matches extremely well withhe experimental results for each job thickness.

0.027 0.0398 3.88 × 100.013 0.0390 3.85 × 10−4

0.008 0.0400 3.90 × 10−4

0.006 0.0420 4.00 × 10−4

It is further observed from Tables 1–3; that the level of inaccu-racy due to wire lag was in the order of 18 �m to 27 �m particularlyfor smaller radius job where the accuracy requirement is generallyhigher. However with the use of suitable wire lag compensationvalue (ε ) in the wire offset setting in CNC part program this

0

86420

Programmed radius of the wire centre r (in mm)

R

Fig. 9. Experimental data and regression model of wire lag compensation for dif-ferent program radius (job height = 62.6 mm).


Table 4Summary of the result of verification experiment for 23.1 mm thick workpiece.


(mm)


(in mm)


(ra = rja + rw + �)





1 2 1.841 1.987 0.013 0.0260 10.68 × 10−4

2 4 3.848 3.994 0.006 0.0240 10.27 × 10−4

3 6 5.850 5.996 0.004 0.0240 10.27 × 10−4



(mm)


(in mm)


(ra = rja + rw + �)





1 2 1.835 1.985 0.015 0.0300 6.43 × 10−4

2 4 3.843 3.993 0.008 0.0300 6.44 × 10−4

3 6 5.845 5.995 0.005 0.0300 6.44 × 10−4



(mm)


(in mm)


(ra = rja + rw + �)





819094

e1bdbajjgbbf

(a

TG

1 2 1.828 1.92 4 3.838 3.93 6 5.841 5.9

liminated by adopting suitable wire offset setting (i.e. instead of53 �m it would be 207 �m) in CNC part program. It may furthere observed from Tables 1–3 that with increase in radius, the errorue to wire lag decreases. Using Eq. (23) gap force intensity haseen calculated for each height and also shown in Tables 1–3 ands expected they are very close and almost constant for a particularob height. The average value of gap force intensity for a particularob height has been calculated and shown in Table 7. Apart fromap force intensity (q) Table 7 shows total gap force (tq), deflectionetween top and bottom surface (ıi), wire deflection at the top and
ottom surfaces of the job (ı0) and total deflection of the wire (ıt)or various job heights.
From Table 7 it is most interesting to note that the total gap forceqt), i.e. product of job thickness and gap force intensity remainlmost constant (i.e. independent of job height) for a given machin-

able 7ap force and deflection for various job heights.

Job height(mm)

Parameters

Average gapforce intensity(q) (in kg/mm)

Total gap force(tq) (in kg)

23.1 10.54 × 10−4 0.02434237.5 06.51 × 10−4 0.02442262.6 03.91 × 10−4 0.024461

0.020 0.0390 3.85 × 10−4

0.010 0.0400 3.90 × 10−4

0.007 0.0390 3.86 × 10−4

ing parameter setting and it is independent of geometry of the job,i.e. radius or height of the job. Using experimental data and regres-sion analysis the relationship between job thickness and sparkingforce intensity may be expressed as follows:

qt = 24.372 × 10−3 (31)

Thus it can be concluded that gap force intensity is inversely pro-portional to job height under servo feed control mode. This findingis of great importance in view of the fact that the result obtainedfor a particular job height is also useful for other job heights and
this in turn can reduces huge amount of experimental load.
Fig. 10 shows variation of gap force intensity with respect to jobheight using regression Eq. (31). Apart from this average gap forceintensity has been calculated for each job height from the result ofverification experiment (Tables 4–6) and also shown in Fig. 10. It is

Deflectionbetween topand bottomsurface (ıi) (inmm)

Wire deflectionat the top andbottomsurfaces of thejob (ı0) (inmm)

Total deflectionof the wire (ıt)(in mm)

0.078096 0.20785 0.2859460.127197 0.208536 0.3357330.212674 0.208869 0.421544

S. Sarkar et al. / Precision Engine

0.0003

0.00045

0.0006

0.00075

0.0009

0.00105

Gap

fo

rce

inte

nsi

ty q

(in

Kg

/mm

)Experimental Data

Reression Model

owadaautttwi

5

aEvmdipojiphdrasBrjm

[

[

[

[

[

[

655545352515

Job height t (in mm)

Fig. 10. Variation of gap force intensity with job height.

bserved that the data of verification experiment matches very wellith the regression equation. It is also observed from Table 7 that

lthough total wire deflection (ıt) increases with job height the wireeflection at the surface of the job (ı0 = 183 �m) is almost constantnd is independent of the job height. The same phenomenon waslso noticed by previous researcher during online measurement,sing optical sensor [1]. This can be explained by the fact that ashe wire deflection at the surface of the job (ı0) is a function ofotal gap force (Eq. (25)) and again as already mentioned that theotal gap force is constant under servo feed control mode and henceire deflection at the surface of the job is almost constant and is

ndependent of the job height.

. Conclusions

An analytical model has been developed to understand andnalyze the wire lag phenomenon during cutting cylindrical job.xperimental result demonstrates that the proposed model is quitealid in practical machining situation. Based upon the analyticalodel, a novel method to measure gap force intensity and wire

eflection has been introduced. Once the wire lag value is knownt will be possible to modify the path to generate high precisionrofile. In the present research paper the impact of wire deflectionn profile inaccuracy has been investigated on a set of cylindricalob having different values of radius and height. To eliminate thismprecision an effective method to determine the wire lag com-ensation component for cylindrical job having any arbitrary radiusas been demonstrated. It is observed that the level of inaccuracyue to wire lag is higher for smaller radius job, where the accu-acy requirement is generally higher. It is analytically developednd also experimentally proved that the required wire lag compen-
ation (εwl) is inversely proportional to programmed radius (rp).eside this, another interesting outcome from the experimentalesult is that the gap force intensity is inversely proportional toob height for a given machining condition with servo feed control
ode. This result is of great significance in view of the fact that the

[

[

[

ering 35 (2011) 339–347 347

result obtained for a particular job height can also be used for otherjob heights and this in turn can reduces a lot of experimental load.Using the analytical model and by just two test cuts (one square orrectangular piece and another circular test piece) it is now easilypossible to measure the gap force intensity (and wire deflection)for any given machining condition. The present research findingswill open up a new avenue in the field of high precision WEDM inmodern manufacturing industry.

Acknowledgement

Authors acknowledge assistance provided by the CAS Ph-IIIProgramme of Production Engineering Department, Jadavpur Uni-versity under University Grants Commission, New Delhi.

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[3] Hsue WJ, Liao YS, Lu SS. Fundamental geometry analysis of wire electrical dis-charge machining in corner cutting. International Journal of Machine Tools &Manufacture 1999;39:651–67.

[4] Lin CT, Chung IF, Huang SY. Improvement of machining accuracy by fuzzy logicat corner parts for wire-EDM. Fuzzy Sets and Systems 2001;122:499–511.

[5] Puri AB, Bhattaharyya B. An analysis and optimisation of the geometrical inac-curacy due to wire lag phenomenon in WEDM. International Journal of MachineTools & Manufacture 2003;43:151–9.

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[9] Sanchez JA, Rodil JL, Herrero A, De Lacalle LNL, Lamikiz A. On the influence ofcutting speed limitation on the accuracy of wire-EDM corner-cutting. Journalof Material Processing Technology 2007;182(1–2):574–9.

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14] Sarkar S, Mitra S, Bhattacharyya B. Modeling and optimization of wire electricaldischarge machining of �-TiAl in trim cutting operation. Journal of MaterialProcessing Technology 2008;205:376–87.

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a novel method of determination of wire lag for enhanced profile accuracy in wedm

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