an analysis and optimization of the geometrical inaccuracy in wedm rough corner cutting

Int J Adv Manuf TechnolDOI 10.1007/s00170-014-6002-5

ORIGINAL ARTICLE

An analysis and optimization of the geometrical inaccuracyin WEDM rough corner cutting

Zhi Chen · Yu Huang · Zhen Zhang · He Li ·Wu yi Ming · Guojun Zhang

Received: 17 December 2013 / Accepted: 22 May 2014© Springer-Verlag London 2014

Abstract Wire electrical discharge machining (WEDM)has occupied an important position in some high-precisionand high-performance manufacturing industries due to itscapability of accurate and efficient machining parts withvarying hardness or complex shapes. However, the high-machining precision and efficiency, especially at rough cor-ner cutting, cannot be satisfied simultaneously because ofsome phenomena such as wire rupture, deflection, vibration,etc. This paper aims to analyze and reduce the geometricalinaccuracy of rough corner cutting; first of all, the majorcauses of corner inaccuracy (45◦, 90◦, and 135◦ angle) areanalyzed in detail. Secondly, an elliptic fitting method isproposed to describe the trajectory of wire electrode center,and the feasibility of model is confirmed by measuring thecorner edge of workpiece. Moreover, three sets of Taguchiexperiments (L2737) are designed to investigate the maineffect and influence trends of control factors on cornererror. Eventually, some optimized control factor combina-tions are sought by generalized non-linear regression model.

Z. Chen · Y. Huang · Z. Zhang · H. Li · W. MingSchool of Mechanical Science and Engineering, HuazhongUniversity of Science & Technology, Wuhan, 430074, China

Z. Chene-mail: [email protected]

G. Zhang (�)State Key Lab of Digital Manufacturing Equipment &Technology, School of Mechanical Science and Engineering,Huazhong University of Science & Technology, Wuhan 430074,Chinae-mail: [email protected]

G. ZhangGuangdong Province Key Lab of Digital ManufacturingEquipment, Dongguan 523000, China

As a result of confirmatory experiments, more than 50 %decrease of corner error has been achieved at 5 mm/mincutting feedrate (a high-cutting feedrate of the machinetool used in this study) by the optimized control factorscombination in rough corner cutting.

Keywords Wire electrical discharge machining · Roughcorner cutting · Inaccuracy · Optimization · Control factor

1 Introduction

Wire electrical discharge machining (WEDM) is widelyused in high-precision molds, micro-mechanical compo-nents, electronics, and aerospace industries because of itscapability of machining parts with intricate shapes andhard materials [1, 2]. In modern manufacturing industry, itrequires some performance simultaneously, such as preci-sion, surface finish, and cutting time [3], while in actualrough corner-cutting process, it has to sacrifice machin-ing efficiency to satisfy high precision, so it is necessaryto understand the behavior of wire electrode and the fun-damental mechanism of geometrical inaccuracy in cornercutting.

A large number of previous researches about the processparameters to simultaneously obtain high-surface qualityand cutting speed which have been carried out [4–8]. Thereare also many researches on machining precision in thepast. Puri and Bhattacharyya, Sarkar et al., Mingqi et al.,and Zhang et al. [9–12] focused on improving accuracydue to wire lag in straight path cutting; therefore, somefitting methods are used to describe the wire center trajec-tory, such as quadratic regression fitting, arc fitting, and aparabolic fitting. Mohri et al., Puri and Bhattacharyya, andOkada et al. [13–15] investigated the characteristic effects

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Int J Adv Manuf Technol

of wire vibration and presented some methods of decreasingvibration amplitude; some analysis approaches for the solu-tion of the wire electrode vibration equation are presentedto investigate the characteristic effects of wire vibration.Yan and Huang [16] designed a closed loop wire tensioncontrol system to improve machining accuracy; accordingto the experiment result, approximately 50 % corner-errorhas been reduced and the vertical straightness significantlyimproved. Hsue et al. [17] proposed fundamental geome-try analysis and a model to calculate the metal removal ratein corner cutting, and an exponential function is used tofit the wire center trajectory. Han et al. [18] implementeda simulation method of corner error in WEDM rough cut-ting; it can provide some references for the study of acontrol method in improving the corner machining accu-racy. Lin et al. [19] developed a control strategy basedon fuzzy logic to increase precision in corner-cutting pro-cess, and the result shows that machining accuracy improvesmore than 50 % of those in normal machining, while themachining process time increases not more than 10 % of thenormal value. Sanchez et al. [20] studied the influence ofcutting speed limitation on the accuracy of WEDM cornercutting.

Some approaches are proposed to improve the machin-ing accuracy in the above researches, and the improvingaccuracy result is also significant. While the mechanism ofcorner-error is unknown, and the discharge concentrationphenomenon in acute rough corner cutting is rarely takeninto account; besides, optimizing process control factorsmay be another method to decreasing corner-error.

In this paper, firstly, the major causes of corner inac-curacy are investigated to clearly understand formationmechanism of corner-error in different angles’ rough cornercutting and wire deflection; vibration and discharge concen-tration phenomenon is taken into consideration. Secondly,an elliptic model fitting method is proposed to describe thewire center trajectory, and it can be found that the wire cen-ter trajectory can be fitted to an elliptic theoretical modelpreferably in the right and obtuse angle corner cutting.Moreover, three sets of Taguchi experiments (L2737) aredesigned to analyze the main effect and influence trendsof control factors on corner-error in 45◦, 90◦,and 135◦angle rough corner cutting respectively. Finally, three gen-eralized non-linear regression models are built to seek theoptimal control factors’ combination in corner cutting (45◦,90◦, 135◦ angle) respectively. The result of confirmatoryexperiments demonstrates that more than 50 % decrease ofcorner error has been achieved at 5 mm/min cutting fee-drate (a high-cutting feedrate of the machine tool used inthis study) by the optimized control factors’ combinationin rough corner cutting; in addition, it can be also foundthat the generalized non-linear regression models have acapacity for predicting preferable corner-error.

2 Causes for corner inaccuracy

In order to better understand the forming mechanism of cor-ner error, three main causes, namely unbalance of externalload, wire deflection, and discharge concentration phe-nomenon, are presented and analyzed in this section.

2.1 The unbalance of external load on wire electrode

In the WEDM process, the wire electrode suffers wiretension, discharge spark force, hydraulic forces, electro-static force, electrodynamics force, etc. [13]; furthermore,all of them are of different natures and directions. Hydraulicforces push the wire electrode back of the CNC path dueto flushing, gas spreading, and bursting, while electrostaticforce attracts the wire electrode to discharge surface and off-sets each other on both sides of wire electrode. Besides, thedischarge spark force is a resultant force of material remov-ing force, discharge pressure, explosive force, etc., and itsvalue depends on the characteristic of material and the pro-cess parameters. In addition, the discharge spark force is themajor cause of wire electrode deflection and vibration. Asshown in Fig. 1, points A and B are the actual position andCNC path position of wire electrode, respectively, and thedistance between point A and point B is the wire lag (δ),Fr is the resultant force of the discharge spark forces andhydraulic forces, Ft is the lateral component force of wiretension, and Fr and Ft are a pair of balance forces in stablestraight path machining. However, as shown in Fig. 2, thedirections of Fr and Ft are no longer in the same line on cor-ner cutting, so an unbalance force (Fs) is produced becauseof the changing of discharge surface and discharge angle,then the machining inaccuracy increases sharply due to theunstable machining, wire deflection, and vibration varyingobviously.

Figure 3 is the top view of change of discharge angle incorner cutting, where O1 and O2 are the first and secondwire center points, and θ1 and θ2 are the first and seconddischarge angles, respectively, θ is corner-cutting angle, andd0 is the machining gap.

Fig. 1 Force analysis in stable straight path machining


Fig. 2 Force analysis in corner cutting

2.2 Wire deflection and vibration

Although the wire electrode is tiny and flexible, it is alsowith some rigidity under puny discharge spark force in fact.In the actual machining process, the wire center always ispulled back of the CNC path, the wire lag (δ) has non-significant impact on straight path machining precision,while deflection and vibration are the major causes of thecorner error [21]. As shown in Fig. 4, a collapsing phe-nomenon is formed in rough corner cutting due to the wirelag of the CNC path and actual path, and the wire lag canbe of the order of a hundred microns, which becomes unac-ceptable for most precise applications [3]. Meanwhile, thevibration amplitude may be up to 20 μm and plays a moreimportant role to decide the accuracy of thin and small parts[13, 14].

d0 = r + A (1)

In the above equation, d0 is the machining gap, and r andA are the discharge radius and wire vibration amplitude,respectively.

Fig. 3 A top view of change of discharge angle in corner cutting

Fig. 4 Schematic diagram of corner-error in right angle corner cutting

2.3 The discharge concentration phenomenon

The electric field intensity and electron emission can beenhanced obviously in an acute angle part machining dueto the accumulation of electric discharges and the changeof the discharge angle. Hence, the discharge concentrationphenomenon would have more outstanding performance,and the removed material cannot be washed away easily asthe line cutting. Besides, according to Eq. 1, the machin-ing gap (d0) increases visibly because of the sharp raisingof discharge radius (r). In the practical rough corner-cuttingprocess, the acute angle corner error is several times largerthan the obtuse and right angles, and the discharge concen-tration is a natural phenomenon which can just be decreasedby some approaches in acute angle corner cutting.

In addition, there are some non-significant influences ofcorner inaccuracy, such as material chip difficultly pushedand rise of cutting temperature.

According to the above analysis, a summary of majorcauses in different angle corner cutting can be obtained inTable 1.

3 Geometry model of wire electrode center trajectory

Although a large number of researchers have been carriedout to qualitatively analyze and quantitatively calculate thewire deflection and vibration, a very few researches aboutthe corner error have been performed due to numerous com-plexities, such as variation of wire lag, enhancement of wirevibration, complex discharge concentration mechanism, etc.

Even though a system based on the on-line monitoringof the wire position has been designed by method of anoptical sensor [3], the real-time control of wire position can-not be easily carried out, especially in rough corner cutting.It is necessary to propose a geometry model of wire cen-ter trajectory to quantitatively study and predict the cornererror.


Table 1 Major causes indifferent angle corner cutting Angle Angle≥ 135◦ 45◦ ≤ Angle≤ 135◦ Angle≤ 45◦

Major causes Lag and vibration Lag, vibration and discharge concentration Discharge concentration

3.1 Geometrical modeling of wire lag

The wire electrode suffers bending deformation in naturebecause of the wire tension, explosive force from gas bub-bles, plasma of the erosion mechanism, hydraulic forces dueto flushing, electrostatic force, electrodynamic force, etc. Inaddition, the actual mechanical behavior of the wire centerelectrode is very complicated in cutting process due to thevariation of forces, mechanical capacity of materials, highlyrandomness of discharge sparks, etc. wherein this stochasticnature is closely related to a lot of factors, such as fluctu-ation of pulse voltage and current, random ionic migration,randomness of discharge times and point, different purityand electric conductivity of materials, and so on [9]. In thissubsection, a quantitative model of wire lag (δ) is proposedby the geometric analysis, for simplification, the followingbasic assumptions are required in the straight line cutting:

a. Workpiece is symmetrically set between two guidewheels.

b. The wire axial tensile force is constant, and the wire isstatic (not moving) [9, 10, 18].

c. The wire electrode is thin and flexible, and wire mass isuniformly distributed along its length [9, 10, 18].

d. The wire vibration can be ignored because the dampingcoefficient is appreciable in a thick workpiece machin-ing process; a 25-mm-thick block of SKD-11 steel isused in this study [22].

e. The discharge resultant force, which is acting on thewire perpendicular to the axial force, can be regarded asdistributed spark force acting per unit length of the wire,and the direction of this force is opposite to the cuttingdirection, as shown in Fig. 4 [3, 9, 10, 23].

As shown in Fig. 5, the wire trajectory in the dischargearea and non-discharge area can be simply divided into threelines; the former simplification is derived from the fact thatwire lag (δ) is 10 ∼ 30 times bigger than the deflection inthe discharge area [11], and the latter one is due to the factthat no extra load acts on the wire in the non-discharge area.The wire mechanical analysis can be proposed as Eq. 2,

T∂2y

∂z2− EI

∂4y

∂z4= ρ

∂2y

∂zt+ β

∂y

∂t+ q(z,t) (2)

where L is the span of two guide wheels (m), h is the work-piece thickness (m), T is the wire tension (N), y is the wirelag function(m), t is the time (s), ρ is the wire mass (kg/m3),β is the damping coefficient (Ns/m2), E is the Young’smodulus (N/m2), I is the area moment of inertia (m4), d

the wire diameter (m), and q is the distributed spark force(N/m2).

Through the above simplification, the wire mass, inertiaforce, and vibration are ignored because of their non-significant influences on wire lag function, so Eq. 2 can alsobe simplified as Eq. 3.

q(z,t)h

2≈ 2T δ

L − h(3)

Then, the wire lag δ can be calculated by Eq. 4.

δ = qh(L − h)

4T(4)

Equation 4 describes a simplified formula of wire lag (δ)in statical state, and the wire lag has been found to have anincreasing trend with the increase of distribute spark forces(q), workpiece thickness (h), and the span of two guide

Fig. 5 Schematic diagram of static deflection of wire tool in wireEDM process


wheels (L); on the contrary, there is an inverse relationshipbetween the wire lag (δ) and the wire tension (T ).

3.2 Geometrical modeling of corner error

In this subsection, three elliptic models are proposed to fitthe wire center trajectory in the right, acute, and obtuseangle corner cutting, respectively. As shown in Fig. 6, theactual wire center position asymmetrically deviates from theCNC path. The error of cut-in part increases quickly andmay be decided by the wire lag; however, the error of cut-out part decreases slowly and may be decided by machiningconditions.

Moreover, each mass point of wire is also in a plane underthe deformation phenomenon, and mechanical analysis canbe simplified as Eq. 5.

T

(∂2x

∂z2+ ∂2y

∂z2

)− EI

(∂4x

∂z4+ ∂4y

∂z4

)

= ρ

(∂2x

∂zt+ ∂2y

∂zt

)+ β

(∂x

∂t+ ∂y

∂t

)+ q(z,t) (5)

In Fig. 7, the cutting feedrate (v) is set as a constant in thestable cutting, and the CNC path is A1 − A2 − A3 − A4,while the actual wire trajectory is B1 − B2 − B3 − B4 dueto the wire lag (δ). The wire trajectory plane turns 90◦ inright angle corner cutting, and the wire lag (δ) is changed alittle because machining parameters remain unchanged, sothe actual wire center trajectory in the corner cutting can befitted to an elliptic model according to the composition ofdimensional motions and the simulation result in ADMASsimulation software.

As shown Fig. 8, a fitting elliptic model can be derivedin right angle corner cutting as Eq. 6,

(x − a)2

a2+ (y − b)2

b2= 1 (6)

where b (minor axis of the ellipse) may be equal to wirelag, a (major axis of the ellipse) is a parameter relating to

Fig. 6 Wire center trajectory in right angle rough corner cutting

Fig. 7 The top view of wire deflection in right angle corner cutting

machining conditions and wire mass (m0), and it can beacquired by Eq. 7. The independent variable (x) representsthe displacement of guides relative to the apex of the corneron the workpiece. Obviously, there is another mathematicalexpression of Eq. 6 which can be worked out as Eq. 8.

a = v

√m0(L − h)

T(7)

y = b − b

√1 − (x − a)2

a2(8)

In a similar way, as shown in Figs. 9 and 10, two fittingelliptic models may be derived as Eqs. 9 and 10 in acute andobtuse angle corner cutting.

(x − a − x0)2

a2+ (y − b)2

b2= 1 (9)

y = b − b

√1 − (x − a − x0)2

a2(10)

The constraint conditions may be obtained as the CNCpath that is tangent to the fitting elliptic model at thepoint (δ cos θ , δ sin θ ), then the boundary conditions can beproposed as Eqs. 11, 12, and 13.

(δcosθ − a − x0)2

a2+ (δsinθ − b)2

b2= 1 (11)

Fig. 8 Fitting elliptic model in right angle corner cutting


Fig. 9 Fitting elliptic model in acute angle corner cutting

There is just one intersection point between Eqs. 12 and 13,

y = xtanθ (12)

(x − a − x0)2

a2+ (y − b)2

b2= 1 (13)

where b may be a decision for wire lag (δ), a and x0 areparameters relating to machining conditions, and θ is thedischarge angle and ranges from acute angle to obtuse angle.

3.3 Confirmatory experiment of the geometry model

Through the above modeling process, there are two orthree parameters related to machining conditions due to theimmeasurability of the CNC traction and the distributedspark force, so the feasibility of fitting elliptic model shouldbe confirmed by measuring the geometrical edge of theworkpiece.

The confirmatory experiment is carried out on a W-A530 WEDM machine with iso-energy-type plus generator,which is made by Dongguan Hustinova Precision Machin-ery Co., Ltd., and it is equipped with deionized wateras dielectric fluid (resistance value 80 �) to obtain highefficiency and accuracy [24]. A brass wire of 0.25-mmdiameter is used as tool electrode in all experiments, andthe workpiece is a 25-mm-thick block of SKD-11 steel(C, 1.50 %; Cr, 12.0 %; Mo, 0.80 %; V, 0.7 %; Mn,

Fig. 10 Fitting elliptic model in obtuse angle corner cutting

0.45 %; and Si, 0.25 %), which is a high-carbon and high-chromium alloy tool steel used in the production of dies,plastic injection molding dies, precision gauges, spindle,jigs, fixtures, etc. Furthermore, the geometrical edge is mea-sured by KEYENCE VH-Z500R (a high-accuracy digitalmicroscope) at ×500 magnification.

The machining parameter condition is set as follows:water pressure (5 Kg/cm2), pulse on time (12 μs), pulse offtime (12 μs), pulse peak voltage (40 V), cutting feedrate(4 mm/min), wire tension (10 Kgf), wire speed (10 m/min),pulse average current (10 A), and the span of two guidewheels is 45 mm.

Firstly, the point coordinates of workpiece edge are mea-sured by digital microscope. Secondly, a fitting ellipticequation is deduced by the least square method. Then, theparameters (a, b, x0), which are related to machining con-ditions in the elliptic theoretical model, can be calculated.Eventually, the wire CNC path, actual trajectory, and modeltrajectory are drawn in the same figure.

Three confirmatory experiments are carried out in thissubsection, and the cutting angles are 45◦, 90◦, 135◦ respec-tively. In addition, the parameters (a, b, x0) of three elliptictheoretical models are listed in Table 2, and the wire CNCpath, actual trajectory, and model trajectory are drawn inFigs. 11, 12, and 13, respectively.

In conclusion, the wire actual trajectory can be fittedto an elliptic theoretical model preferably in the right andobtuse angle corner cutting. While there is deviation of 0 ∼8 μm between elliptic model and actual trajectory of wireelectrode, this phenomenon maybe a result in the rise of dis-charge radius due to discharge concentration phenomenonin acute angle corner cutting.

4 Taguchi experiment and optimization of processparameters

In consideration of so many disadvantages of the corner-error, there are some traditional approaches to solve thisproblem, such as stoping machining at corner point, reduc-ing the cutting feedrate, adapting the programmed path [19,25], etc. The former two methods sacrifice the machiningefficiency to meet high accuracy, so the advantage of the

Table 2 Parameters (a, b, x0) of three elliptic theoretical models

Parameters 45◦ 90◦ 135◦

Major axis a (μm) 39.5 76.5 86.1

Minor axis b (μm) 15.8 30.6 34.4

x0 (μm) 18.8 0 −27.8

Wire lag δ (μm) 30.4 30.7 30.9

Error (μm) 26.5 18.4 8.6


Fig. 11 Wire center trajectory in acute angle (45◦) corner cutting

high precision and efficient machining technique cannot beperfectly presented, while the last one is just applied insome manufacturing area, and it increases the cut-time too.Meanwhile, on-line modifying the wire path is proposed toimprove accuracy at some degree in the advanced machinetool [3], whereas this method cannot improve machiningaccuracy under large deflection and big discharge radiusvery well.

According to the elliptic theoretical model, the corner-cutting error has a direct relationship to wire lag, dischargeradius, cutting feedrate, and so on, and since the wire lagmay be decided by distributed spark force, the span of twoguide wheels, wire tension, and workpiece thickness. More-over, the discharge radius may depend on the dischargevoltage, cutting angle, workpiece material, and dischargeconcentration phenomenon. In a word, the accurate calcu-lation model of corner-cutting error is too complex to bederived. However, some decreasing corner-error approachescan be proposed by using experiment method althoughaccurate calculation model is unknown.

Taguchi experiment has been widely applied in DOE toobtain the characteristic data by using orthogonal arrays, toanalyze the main effects of objective variables, and to seek

Fig. 12 Wire center trajectory in right angle (90◦) corner-cutting

Fig. 13 Wire center trajectory in obtuse angle (135◦) corner cutting

the optimal process parameters in an extremely complicatedprocess [9, 26–28]. In this section, three sets of experi-ments (L2737) are designed based on the Taguchi method ofexperimental design to evaluate the main effect factors andinfluence trends on corner error in 45◦, 90◦, and 135◦ anglerough corner cutting respectively.

4.1 Experimental design and results

The flowchart of optimizing parameters is drawn as inFig. 14, and seven process parameters (water pressure (A),pulse on time (B), pulse off time (C), pulse peak voltage(D), cutting feedrate (E), wire tension (F ), and wire speed(G)) are selected as the control factors wherein each factoris designed to have three levels as given in Table 3. Besides,the pulse average current maintains as a constant value(10 A) because it cannot be set in rough cutting, and theother machining conditions are the same with Section 3.3.

Taguchi experiments (L2737) are designed by MINITAB16 (a statistical analysis software), and the measured data ofcorner-error are listed in Table 4.

4.2 Main effects of analysis and discussion of corner error

Analysis of variance (ANOVA) is a method used for signifi-cance analysis of two or more factors [29, 30], and variance

Table 3 Factors and their levels on rough corner cutting

Factors Levels Unit

Water pressure (A) 3 5 7 Kg/cm2

Pulse on time (B) 8 12 16 μs

Pulse off time (C) 8 12 16 μs

Pulse peak voltage (D) 30 40 50 V

Cutting feedrate (E) 3 4 5 mm/min

Wire tension (F ) 5 10 15 Kgf

Wire speed (G) 5 10 15 m/min


Fig. 14 The flowchart of optimizing parameters

ratios (F values) of factors are calculated by Eq. 14,

SSA = m

n∑i=1

xi − x2 (14)

where SSA is the effect quadratic sum of factor A, n is thelevel of factor A, m is the level of other parameter, xi is theaverage of factor A, and x is the average of total factor.

In this subsection, analysis of variance is applied to ana-lyze the main effects of corner-error, and F values are a keyindicator which presents the magnitude of each control fac-tor, then a unitary processing for each F value is made toevaluate synthetic effect of each factor. Figure 15 shows thevariance ratios (F values) for water pressure (A), pulse ontime (B), pulse off time (C), pulse peak voltage (D), cuttingfeed rate (E), wire tension (F ), and wire speed (G).

In three sets of experiments (45◦, 90◦, and 135◦), com-pared with the variances and degrees of contribution foreach control factor from Fig. 14, it is found that the influ-ence trends of each process factor to corner-error (45◦,

Table 4 Experimental design and data

No. A (Kg/cm2) B (μs) C (μs) D (V ) E (mm/min) F (Kgf) G (m/min) e − 45◦ (μm) e − 90◦ (μm) e − 135◦ (μm)

1 3 8 8 30 3 5 5 22.6 14.5 8.2

2 3 8 8 30 4 10 10 34.9 20.6 11.7

3 3 8 8 30 5 15 15 50.8 28.3 18.7

4 3 12 12 40 3 5 5 40.1 28.1 13.3

5 3 12 12 40 4 10 10 58.9 38 22.8

6 3 12 12 40 5 15 15 79.9 48.9 36.1

7 3 16 16 50 3 5 5 77.6 53.9 37.2

8 3 16 16 50 4 10 10 86.9 59.7 30.7

9 3 16 16 50 5 15 15 114.7 74.5 53.6

10 5 8 12 50 3 10 15 35.1 19.4 8.7

11 5 8 12 50 4 15 5 50.7 28.3 18.9

12 5 8 12 50 5 5 10 76.6 46.6 32.4

13 5 12 16 30 3 10 15 23.3 13.4 8.7

14 5 12 16 30 4 15 5 44.5 28 15.4

15 5 12 16 30 5 5 10 62.9 43 26

16 5 16 8 40 3 10 15 56.3 41.3 24.3

17 5 16 8 40 4 15 5 71.5 51 31

18 5 16 8 40 5 5 10 114.1 78.6 62.6

19 7 8 16 40 3 15 10 18.6 10.1 10.7

20 7 8 16 40 4 5 15 39.7 22.5 9.1

21 7 8 16 40 5 10 5 55.9 32.8 14.2

22 7 12 8 50 3 15 10 43.3 28.8 12.7

23 7 12 8 50 4 5 15 78.5 49.7 37.8

24 7 12 8 50 5 10 5 88.9 57.7 39.4

25 7 16 12 30 3 15 10 49.6 35.5 24.5

26 7 16 12 30 4 5 15 72.2 50.7 35.7

27 7 16 12 30 5 10 5 92.2 63.8 42.7

e − 45◦ error in 45◦ angle corner cutting, e − 90◦ error in 90◦ angle corner cutting, e − 135◦ error in 135◦ angle corner cutting


Fig. 15 Analysis of variance of control factors in 45◦, 90◦, and 135◦corner-error

90◦, and 135◦) is roughly the same, more specifically, thecorner-errors are found to have an increasing trend withthe increase of pulse on time (B), pulse peak voltage (D),and cutting feedrate (E). This establishes the fact that thedistributed spark force is positively associated with the dis-charge power, and the discharge power can be calculated byEq. 15. On the contrary, the corner-errors have a decreas-ing trend with the increasing of pulse off time (C) andwire tension (F ). Influence trend of the former factor canbe explained by Eq. 15, while the influence trend of thelatter one is attributed to the reason that the wire lag isin inverse proportion to the wire tension (F ). In addition,water pressure (A) and wire speed (G) have a little influenceon corner-error. The little influence of water pressure (A)indicates that hydraulic force is also small compared withdistributed spark force, and the little influence of wire speed(G) shows that the assumption (wire is static) is feasible incalculating corner-error,

E(t) =∫ tp

0U(t)I(t)dt (15)

where E(t) is the discharge power, U(t) is the pulse peakvoltage, I(t) is the pulse peak voltage current, tON is thepulse on time, tOFF is the pulse off time, and tp is the pulseperiod.

Although the influence trends of each process factor forcorner-errors (45◦, 90◦, and 135◦) are similar, each factorhas totally different impact on corner-errors as shown inTable 5.

a. In acute angle (45◦) corner cutting, cutting feedrate (E)is the most important factor for corner-error, and thisis because of the positive correlation between cuttingfeedrate (E) and major axis (a) of the ellipse fittingequation according to Eq. 7. Moreover, the dischargecontrol factors (pulse on time (B), pulse off time (C),and pulse peak voltage (D)) are also important controlfactors due to the discharge concentration phenomenonalways existing significantly. However, the wire tension(F ) is relatively insignificant for corner-error.

b. In contrast, wire tension (F ) is the main factor forcorner-error, and the discharge control factors (pulseon time (B), pulse off time (C), and pulse peak volt-age (D)) become less important in obtuse angle (135◦)corner cutting. This is in support of the fact that the dis-charge concentration phenomenon almost disappears,so the wire deflection rises to the main effect on corner-error, and there is a close correlation between wiredeformation phenomenon and wire tension.

c. On the other hand, it is found that there are no verysignificant differences of the importance of dischargecontrol factors (pulse on time (B), pulse off time (C),and pulse peak voltage (D)), cutting feedrate (E), andwire tension (F ) in the right angle corner cutting. Thiscan be explained that corner-error is affected by a com-bination of wire deflection, cutting feedrate (E), and thedischarge concentration phenomenon.

4.3 Generalized non-linear regression model

Generalized non-linear regression model is a statistical anal-ysis technique which combines the mechanism of varianceanalysis with the mechanism of regression analysis, and itis widely used in analyzing the relationship between severaldependent variables and independent multi-variable. More-over, it has enough capacity to solve complex non-linearproblems such as modeling, prediction, optimization, andadaptive control [31–33]. In this subsection, three gener-alized non-linear regression models are built to obtain therelationship between corner-error (45◦, 90◦, and 135◦) andcontrol factors (water pressure (A), pulse on time (B), pulse

Table 5 The order of importance of each factor

Angles Orders

A B C D (V) E F G

(Kg/cm2) (μs) (μs) (V) (mm/min) (Kgf) (m/min)

45◦ 6 2 3 4 1 5 7

90◦ 6 1 3 5 2 4 7

135◦ 7 3 2 5 4 1 6


Fig. 16 Normal probability diagram in 45◦ angle corner-error

off time (C), pulse peak voltage (D), cutting feed rate (E),wire tension (F ), and wire speed (G)) respectively.

The steps of the modeling process are as follows:

1. Identify the main goal. The main goal is to obtain a min-imum corner-error (45◦, 90◦, and 135◦) at high-cuttingfeedrate (E).

2. Variance analysis. It can be found out that whether theexperimental data conform to normal distribution lawor not, and the main effect and influence trends of con-trol factors on corner-error (45◦, 90◦, and 135◦) can beworked out in this step.

3. Full-factor quadratic regression. All monomial,quadratic term, and cross term of each factor can beconsidered in this regression analysis.

4. Optimizing the regression analysis. Some items shouldbe removed separately to obtain an optimal general-ized non-linear regression model according to the fittingratio of the model and p value (it is suitable that the pvalue is near 0.05 and fitting ratio is more than 90 %).

On the basis of step 2, the results of variance analy-sis about corner-error (45◦, 90◦, and 135◦) are shown inFigs. 16, 17, 18, 19, 20, and 21 respectively, it can befound that the experimental data of corner-error (45◦, 90◦,and 135◦) conform to normal distribution law as well, in

Fig. 17 Normal P-P in 45◦ angle corner-error (X∼ N(0, 1.22))


Fig. 19 Normal P-P in 90◦ angle corner-error (X∼ N(0, 12))


Fig. 21 Normal P-P in 135◦ angle corner-error (X∼ N(0,0.82))


Table 6 The summary of three regression models

Index Corner-error model

45◦ 90◦ 135◦

S 4.45496 2.48689 3.41324

R − Sq(R2) 98.29 % 98.16 % 97.43 %

DOF 11 12 14

Seq SS 17162.5 8631.80 5309.04

Adj SS 17162.5 8631.80 5309.04

Adj MS 1560.23 719.317 379.217

F value 78.6144 116.307 32.5502

P value 0.000000 0.000000 0.000000

S total variance, R−Sq(R2) goodness-of-fit, DOF degrees of freedom,Seq SS sequential sum of squares, Adj SS adjusted sum of squares, AdjMS adjusted mean squares, F value and p value T checking index

other words, three generalized non-linear regression mod-els can be proposed to optimize the control factors. Threegeneralized non-linear regression models are derived outby the means of the third and fourth steps. The regres-sion models of corner-error (45◦, 90◦, and 135◦) are shownin Eqs. 16, 17, and 18, respectively. The high coincidencedegrees between experimental data and regression modelscan be confirmed by high-fitting ratio and p values (asshown in Table 5).

e45◦ = 7.69 − 0.76A − 4.29B − 0.51C − 0.92D + 8.30E

−0.67E + 0.07G + 0.28B2 + 0.64B × E + 0.02

D2 + 0.12D × E (16)

e90◦ = 4.6 − 0.42A − 2.95B − 0.45C + 0.51D − 4.43E

0.60F − 0.10G + 0.21B2 − 0.01B × D + 0.47B

×E + 0.04D × E + 1.23E2 (17)

e135◦ = 81.33 − 0.34A − 3.06B − 0.56B − 0.20D

−38.53E + 0.14G + 0.18B2 − 0.02B × D

+1.08B × E − 0.18B × F + 0.35B × E

−0.06D × F + 2.66E2 + 0.20F 2 (18)

As Table 6 illustrated, the goodness-of-fit of three gener-alized non-linear regression models are all more than 90 %,and the p values of three generalized non-linear regressionmodels are all less than 0.05, which means that the threeregression models are significant to the experimental data,and these three models can be used for predicting preferablecorner-error.

4.4 Optimization of control factors

In the actual manufacturing process, obtaining high preci-sion and efficiency simultaneously could be ideal, while thetask of determining the optimal setting for each control fac-tor is complicated when multiple characteristics are to beoptimized. According to Fig. 15 and main effect analysisin Section 4.2, the ideal control factors may be followingcombination of factors (A2, B1, C3, D1, E1, F3, G1). How-ever, the combination of the factors cannot be realized inactual machining process because the discharge control fac-tors (pulse on time (B), pulse off time (C), and pulse peakvoltage (D)) are so conservative that the discharge power istoo small; therefore, material removing cannot be realized.In addition, there is another complicated problem which isthe interaction effects of control factors in the optimizationprocess.

In this study, to obtain high efficiency, the optimizationprocess of control factors is carried out at 5 mm/min cut-ting feedrate. Above all, it must be found that how muchdischarge power can maintain 5 mm/min cutting feedratecontinuously, and the discharge power can be calculated byEq. 15. A large number of tests have been conducted toget the relationship between the discharge power and cut-ting feedrate. Hence, it is found that the discharge power(more than 125 W) can keep up 5 mm/min cutting feedrate.Secondly, based on three generalized non-linear regression

Table 7 Optimum factors’ combination models at 5 mm/min cutting feedrate

Angle A (Kg/cm2) B (μs) C (μs) D (V) E (mm/min) F (Kgf) G (m/min) Error (μm)

Optimal factors 45◦ 7 8 11 30 5 15 10 44.8

90◦ 7 8 11 30 5 15 15 24.97

135◦ 7 8 10 31 5 15 10 18.04

Preferable factors 45◦ 7 8 10 ∼ 11 30 ∼ 32 5 15 10 ∼ 15 44.80 ∼ 45.50

90◦ 7 8 10 ∼ 11 30 ∼ 32 5 15 12 ∼ 15 24.97 ∼ 25.87

135◦ 5 ∼ 7 8 10 ∼ 11 30 ∼ 31 5 15 10 ∼ 15 18.05 ∼ 19.20


Table 8 Confirmatory experiment data

No. A (Kg/cm2) B (μs) C (μs) D (V) E (mm/min) F (Kgf) G (m/min) Predicted results Experimental results Average relative error (%)

45◦ 90◦ 135◦ 45◦ 90◦ 135◦

1 7 10 10 45 4 15 10 50 29.8 19.3 52.4 31.2 20.9 5.6

2 7 10 14 35 5 15 15 55.9 33.5 24.3 54.3 32.7 25.6 3.5

3 7 14 10 35 5 10 10 83 56.7 36.4 79.4 54.1 31.5 8.3

4 7 8 8 45 4 10 15 48.4 27.8 14.6 46.2 24.3 18.5 13.4

5 7 16 16 45 3 10 10 59.2 42.8 18.6 62.3 46.3 22.9 10.4

6 7 8 11 30 5 15 10 44.8 25.5 18.1 48.4 27.8 22.1 11.3

7 7 8 11 30 5 15 15 45.2 24.8 18.7 49.3 28.3 22.7 12.5

8 7 8 10 32 5 15 10 47.1 27.4 18 46.7 29.3 21.8 8.3

models, optimal factors and preferable combination of fac-tors can be proposed by MATLAB mathematical software,and they are shown in Table 7.

4.5 Confirmatory experiment of Taguchi method

Carrying out a confirmatory experiment is of great impor-tance to experiment design and analysis, and it is an indis-pensable part of Taguchi method. Its aim is to find outcloseness of estimation between actual corner-errors andpredicted results of three generalized non-linear regres-sion models and insuring optimization process is effectiveand significant. In this research, some confirmatory experi-ments have been conducted; furthermore, the confirmatoryexperiment data and the comparison of the predicted andexperimental results are displayed in Table 8 and Fig. 22,respectively.

According to Table 8 and Fig. 22, it can be found thataverage relative error between predicted and experimen-tal results is near 10 %, and relative error is of randomdistribution. Hence, it means that three generalized non-linear regression models are of high predicted accuracy, andit is also found that the almost 50 % decrease of corner

Fig. 22 Comparison of the predicted and experimental results

error has been achieved by the optimized control factors’combination.

5 Conclusions

Geometry model of wire electrode center trajectory has beenproposed to quantificational calculate corner-error (45◦,90◦, 135◦) respectively. In addition, three sets of exper-iments (L2737) are designed to evaluate the main effectand influence trends of control factors on corner-errorin 45◦, 90◦, and 135◦ angle rough corner-cutting respec-tively. The following conclusions are drawn from the aboveinvestigation:

1. The major causes of corner-error in different anglesrough corner cutting are presented and analyzed,respectively. The wire deflection and vibration are themain causes of right and obtuse angle corner cut-ting. On the other hand, the discharge concentrationphenomenon plays an important role in acute anglecorner-cutting as well.

2. According to comparative results of the elliptic modeland actual trajectory of wire electrode, the wire centertrajectory can be fitted to an elliptic theoretical modelpreferably in the right and obtuse angle corner cutting,while there is deviation of 0 ∼ 8 μm between ellipticmodel and actual trajectory of wire electrode due to theobvious rising of discharge radius in acute angle cornercutting.

3. Three sets of Taguchi experiments (L2737) are designedto evaluate the main effect and influence trends of con-trol factors on corner-error in 45◦, 90◦, and 135◦ anglerough corner-cutting respectively. By variance analy-sis, it is found that the influence trends of each controlfactors on corner-error (45◦, 90◦, and 135◦) is approxi-mately the same; however, the orders of importance ofeach factor are somewhat different in the three sets ofexperiments (45◦, 90◦, and 135◦).


4. Three generalized non-linear regression models are pro-posed to optimize the control factors’ combination incorner cutting (45◦, 90◦, and 135◦) respectively, andthe feasibilities of three regression models are provedby the high predicted accuracy. In addition, it is alsofound that the optimal control factors are significantand preferable according to tiny average relative errorbetween predicted results and confirmatory experimentdata, and experimental result shows that almost 50 %decrease of corner error has been achieved by theoptimized control factors’ combination.

Acknowledgments This study is mainly supported by the NationalNatural Science Foundation of China (NSFC) under Grant No.51175207. In addition, National Key Technology R & D ProgramNo.2012BAF13B07 and Science and Technology Planning Project ofGuangdong Province No.2012B011300015 both aid this research.

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