optimization of process parameters of mechanical type advanced machining processes using genetic...

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doi:10.1016/j.ijm

Abbreviations

machining; AM

water jet machi

machining; ECM

machining; IBM

MAF, magneti

rate in mm3/s; P

WJM, water jet�Correspond

91 512 2597 408

E-mail addr

(N.K. Jain), vk

International Journal of Machine Tools & Manufacture 47 (2007) 900–919

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Optimization of process parameters of mechanical type advancedmachining processes using genetic algorithms

Neelesh K. Jaina, V.K. Jainb,�, Kalyanmoy Debb

aMechanical & Industrial Engineering Department, Indian Institute of Technology, Roorkee 247 667, IndiabMechanical Engineering Department, Indian Institute of Technology, Kanpur 208 016, India

Received 31 December 2005; received in revised form 9 June 2006; accepted 3 August 2006

Available online 26 September 2006

Abstract

Generally, unconventional or advanced machining processes (AMPs) are used only when no other traditional machining process can

meet the necessary requirements efficiently and economically because use of most of AMPs incurs relatively higher initial investment,

maintenance, operating, and tooling costs. Therefore, optimum choice of the process parameters is essential for the economic, efficient,

and effective utilization of these processes. Process parameters of AMPs are generally selected either based on the experience, and

expertise of the operator or from the propriety machining handbooks. In most of the cases, selected parameters are conservative and far

from the optimum. This hinders optimum utilization of the process capabilities. Selecting optimum values of process parameters without

optimization requires elaborate experimentation which is costly, time consuming, and tedious. Process parameters optimization of AMPs

is essential for exploiting their potentials and capabilities to the fullest extent economically. This paper describes optimization of process

parameters of four mechanical type AMPs namely ultrasonic machining (USM), abrasive jet machining (AJM), water jet machining

(WJM), and abrasive-water jet machining (AWJM) processes using genetic algorithms giving the details of formulation of optimization

models, solution methodology used, and optimization results.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: AJM; AWJM; AMPs; Genetic algorithms (GA); Optimization; USM; WJM

1. Introduction

Advanced engineering materials such as polymers,ceramics, composites, and superalloys play an everincreasing important role in modern manufacturing in-dustries, especially, in aircraft, automobile, cutting tools,die and mold making industries [1]. Higher costs associatedwith the machining of these materials, and the damage

e front matter r 2006 Elsevier Ltd. All rights reserved.

achtools.2006.08.001

: AFM, abrasive flow machining; AJM, abrasive jet

Ps, advanced machining processes; AWJM, abrasive-

ning; CHM, chemical machining; EBM, electron beam

, electro-chemical machining; EDM, electro-discharge

, ion beam machining; LBM, laser beam machining;

c abrasive finishing; MRR, volumetric material removal

AM, plasma arc machining; USM, ultrasonic machining;

machining.

ing author. Tel.: +91 512 2597 916; fax: +91 512 2590 007,

.

esses: [email protected], [email protected]

[email protected] (V.K. Jain).

caused during their machining are major impediments inthe processing and hence limited applications. Further,stringent design requirements also pose major challenges totheir manufacturing industries. These include precisemachining of complex and complicated shapes and/or sizes(i.e. an aerofoil section of a turbine blade, complex cavitiesin dies and molds, etc.), various hole-drilling requirements(i.e. non-circular, small or micro size holes, holes at shallowentry angles, very deep holes, and burr less curved holes),machining of low rigidity structures, machining at micro ornano levels with tight tolerances, machining of inaccessibleareas, machining of honeycomb structured materials,fabrication of micro-electro mechanical systems (MEMS),and nano-finish and surface integrity requirements. Un-conventional or advanced machining processes (AMPs)have been developed since the World War II largely inresponse to new, challenging, and unusual machining andor shaping requirements [2]. Alting [3] classified the AMPsinto four categories according to the type of energy used in

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Nomenclature

At cross-sectional area of cutting tool (mm2)Av amplitude of vibration (mm)C ratio indicating what portion of abrasive water

jet (AWJ) is involved in cutting wear mode andis given by C ¼ 1� ðat=aoÞ if atpao

CD coefficient of dischargeCav concentration of abrasive particles by volumeCfw drag or skin friction coefficient for the work-

piece materialdawn diameter of abrasive-water jet nozzle (mm)dm mean diameter of abrasive particles (mm)

ðdm ¼ 15:24M�1a Þ

dwn diameter of water jet nozzle (mm)EY Young’s modulus of elasticity (MPa or N/mm2)Fs static feed force (N)fn traverse rate of nozzle (mm/s)fr roundness factor of an abrasive particle, i.e.,

ratio between average diameter of particlecorners to diameter of maximum inscribed circle

fs sphericity factor of the abrasive particlesfv frequency of vibration (Hz or cycles/s)H Brinell hardness number (same as flow stress)

(MPa or N/mm2)Hd dynamic hardness, i.e., resistance with which

target material resists the indentation (MPa)h depth of indentation or penetration depth (mm)hc indentation depth of due to cutting wear (mm)hd indentation depth due to deformation wear

(mm)ht total depth of indentation in AWJM process

(mm) ðht ¼ hc þ hdÞ

ls string length (for real-coded genetic algorithms)mp mass of an abrasive particleMa abrasive mesh size_Ma mass flow rate of abrasive particles (kg/s)_Mw mass flow rate of water (kg/s) ½ _Mw ¼ CDðp=4Þ

d2wnvwrwater�

Ng number of generationsna number of abrasive particles striking the target

surface per unit time (per second)Pmax allowable power consumption value (kW)Ps population sizePw pressure of water jet at the nozzle exit (MPa or

N/mm2) ½Pw ¼ ð1=2000Þrwaterv2w�

pc crossover probabilitypm mutation probabilityRa surface roughness value (mm)Ro

a initial surface roughness value (mm)ðRaÞmax allowable surface roughness value (mm)Rl loading ratio, i.e., ratio of mass flow rate of

abrasive particles to that of water ½Rl ¼ _Ma= _Mw�

rm mean radius of the abrasive grains (mm), i.e.,ðrm ¼ 7:62M�1

a Þ

va velocity of the impacting abrasive particle(mm/s)

vac critical velocity of the abrasive particles (mm/s)vaw velocity of the abrasive water jet (mm/s) ½vaw ¼

ðxvwÞ=ð1þ RlÞ�

vw velocity of the water jet at the nozzle exit (mm/s) ðvw ¼

ffiffiffi2p

104:5ffiffiffiffiffiffiPw

pÞ

X stand-off-distance, i.e., distance between tip ofnozzle exit and surface being machined (mm)

Xi length of initial region of the water jet (mm)a angle of impingement or attack (degrees)ao angle of impingement at which maximum

erosion occurs (degrees)at angle of impingement at top of machined

surface (degrees)dc critical plastic strain or erosion ductilityZa proportion of abrasive particles effectively

participating in the erosion or machiningprocess

Zc simulated binary crossover (SBX) parameterZm parameter for polynomial mutationZw damping coefficient for the workpiece material

(kg/mm2 s)n Poisson’s ratiol indentation ratio, ½l ¼ ht=hw ¼ Hw=H t�

y half of mean angle of asperity of abrasivecutting edges (degrees)

r density (kg/mm3)rwater density of the water jet ( ¼ 10�6 kg/mm3)sc compressive yield strength (MPa or N/mm2)se elastic limit (MPa or N/mm2)sfw flow strength of the workpiece material (MPa

or N/mm2)sy tensile yield strength (MPa or N/mm2)x mixing efficiency between abrasive and waterz amount of indentation volume, which is plas-

tically deformed (dependent upon indentationgeometry, impact velocity, and the targetmaterial)

Subscripts

a abrasivec carrier mediumi inletmax maximum valuet toolw workpiece materialx at a distance from inlet or entrance

N.K. Jain et al. / International Journal of Machine Tools & Manufacture 47 (2007) 900–919 901

ARTICLE IN PRESSN.K. Jain et al. / International Journal of Machine Tools & Manufacture 47 (2007) 900–919902

material removal: chemical, electro-chemical, mechanicaland thermal.

Generally AMPs are characterized by low value ofmaterial removal rate (MRR) and high specific energyconsumption. AMPs are used only when no othertraditional machining process can meet the necessaryrequirements efficiently and economically because mostof the AMPs are associated with relatively higher initialinvestment cost, power consumption and operating cost,tooling and fixture cost, and maintenance cost. Thereforeeffective, efficient, and economic utilization of capabilitiesof AMPs necessitates selection of optimum processparameters. Generally, values of process parameters ofAMPs are selected either based on the experience,expertise, and knowledge of the operator or from thepropriety machining handbooks. Selection of processparameters based on the operator experience does notcompletely satisfy the requirements of high efficiency andgood quality. While machining tables can be a better choicein a factory environment for one or two processes butcannot be used for a wide range of machining processesand their operating conditions. In most of the cases,selected parameters are conservative and far from theoptimum. This hinders optimum utilization of the processcapabilities. Selecting optimum values of process para-meters without optimization requires elaborate experimen-tation which is costly, time consuming, and tedious.Therefore, to exploit potentials and capabilities of AMPsto the fullest extent economically, their process parametersoptimization is essential.

In mechanical type AMPs, material is removed bymechanical means like abrasion, erosion, or shear depend-ing on the nature of workpiece material, and machiningconditions. Of the six main mechanical type AMPs,abrasive jet machining (AJM) and ultrasonic machining(USM) can be considered as material removal processesparticularly suitable for hard and/or brittle materials.Water jet machining (WJM) and abrasive-water jetmachining (AWJM) are generally used for cutting andcleaning purposes. While, abrasive flow machining (AFM)and magnetic abrasive finishing (MAF) are fine finishingprocesses. This paper describes details of process para-meters optimization of four mechanical type AMPsnamely, USM, AJM, WJM, and AWJM processes usinggenetic algorithms (GA).

Different researchers have carried out process para-meters optimization of different types of AMPs from timeto time using different optimization models and solutiontechniques. Table 1 presents the summary of such paststudies highlighting the decision variables, objective func-tions, constraints, variable bounds, remarks, and theirlimitations.

Chakravarthy and Babu [4] used combination of simplegenetic algorithms (SGA) and fuzzy logic for optimalselection of three AWJM parameters namely water jetpressure, jet traverse rate, and abrasive flow rate. SGAwas used to generate a set of strings of input parameters.

A fuzzy rule base was used to predict depth of cut usingthese parameters as input. Those parametric combinations,for which predicted depth of cut was equal to the desireddepth of cut within a specified error amount, wereidentified as feasible combinations. The feasible parametriccombinations were used for optimization to minimizetotal cost of production. Kovacevic and Fang [5] haveapplied fuzzy set theory for selecting (though not theoptimum values) four AWJM process parameters namelywater jet pressure, jet traverse rate, abrasive flow rate,and inside diameter of AWJ nozzle to achieve the desireddepth of cut. Universes of discourse for AWJM processparameters were discretized into 17 levels with 5 linguisticterms and triangular membership function was used foreach parameter. Five fuzzy rules were employed foreach of the four AWJM process variables. De [6] alsoused SGAs to optimize five process parameters of USMprocess with the objective of maximizing MRR subjectedto the surface roughness constraint as mentioned in theTable 1.

2. Formulation of optimization models

Possibility of determining a global optimum solution andits accuracy depends on the type of optimization modelingtechnique used to express the objective functions andconstraints in terms of the decision variables. Accurate andreliable models of the process can compensate for inabilityto completely understand and adequately describe theprocess mechanisms [7]. Hence, formulation of optimiza-tion model is the most important task in optimization. Itinvolves expressing optimization problem as a mathema-tical model in a standard format, which can be directlysolved by an optimization algorithm [8].For process parameters optimization of AMPs, type of

objective functions and constraints, number of objectives,and extent of the importance or priority to be given to eachobjective depend on: (i) type of the application (i.e. roughor finish machining), (ii) volume of production (i.e. mass,batch, job-shop), (iii) nature of the work material (i.e.metallic or non-metallic, brittle or ductile, electrically/thermally conductive or non-conductive, etc.), and (iv)shape to be produced. Main objective for the bulk material

removal processes is to maximize MRR subjected toconstraints on surface roughness produced, power con-sumption, and tool (or nozzle) wear. Following procedurewas used to formulate the optimization models in thepresent work:

�
Identification of important decision variables. � Formulation of objective functions. Following strategy
was adopted:J A state-of-the-art-survey [9] of modeling of measures

of mechanical AMPs (MAMPs) performance (i.e.MRR) was carried out in which each model wascritically examined on the basis of its assumptions,limitations, and applicability.

ARTICLE IN PRESSTable

1

Summary

ofpast

studiesofprocess

parametersoptimizationofmechanicaltypeadvancedmachiningprocesses

Researcher

(Year)

Decisionvariables

Objectivefunction(s)

Constraints

andvariable

bounds

Rem

arksandlimitations

Ab

rasi

ve-w

ate

rje

tm

ach

inin

g(A

WJ

M)

Chakravarthyand

RameshBabu(1998)[4]

�T

hre

edecision

variables,each

withfive

levelsofvariations

�M

ax

imiz

epro

duct

ion

rate

�Noconstraints

and

variable

boundsused

�Essentiallysingle

objectiveoptimization

�W

ate

rje

tp

ress

ure

(6

0,

13

0,

20

0,

27

0,

35

0M

Pa

)

�M

inim

ize

abra

sive

consu

mpti

on

�Fuzzyrule

base

contained

125rules,which

weredeveloped

basedon125experim

ents

perform

edonParadisoGranite

�Jettraverse

rate

(30,

70,150,230,325mm/m

in.)

Ass

ign

ing

suit

ab

lew

eig

hta

ges

toea

cho

fth

ese

obje

ctiv

esth

eyw

ere

com

bin

edin

asi

ngle

obje

ctiv

e

as

tota

lco

sto

fm

ach

inin

g,

wh

ich

isto

be

min

imiz

ed

�GA

parametersused:No.ofvariables¼

3;

Stringlength¼

26;Populationsize¼

50;

Crossover

probability¼

0.5;Mutation

Probability¼

0.2.

�A

bra

sive

flo

wra

te(

30

,

50

,9

0,

13

0,

17

0g

/min

)

Totalcostofmachining¼½W

prð

C1þ

C2þ

C3þ

C4þ

C5þ

C6Þþ

WacC

1�

�length

ofcut

Jettraverse

rate

�� ,

Wprand

Wacare

weightages

forproductionrate

andabrasiveconsumption,respectively,and

C1¼

abrasiveconsumptioncost

($/h);

C2¼

machinehourlycost

($/h);

C3¼

laborcost

per

hour($/h);

C4¼

cost

ofprimary

nozzle

($/

h);

C5¼

cost

ofsecondary

nozzle

($/h);

C6¼

cost

ofpower

consumption($/h)

Ult

raso

nic

ma

chin

ing(U

SM)

De(1997)[6]

Fiv

edecisionvariables

Ma

xim

ize

MR

R,

Su

rfa

cero

ug

hn

ess

constrainti.e.

Rap(R

a)m

ax,where;

�Single

objectiveoptimization.

�Amplitudeofvibration

‘Av’(m

m);

FollowingexpressionforMRR

given

byM.C.

Shaw

wasused:

Ra¼

8F

sAvdm

6A

tCavH

wð1þlÞ

�� 1=2

;�

GA

parametersused:No.ofvariables¼

5;

Populationsize¼

30;Number

of

generations¼

100;Crossover

probability¼

0.9;

MutationProbability¼

0.01;Convergence

epsilon¼

0.001.

�Frequency

ofvibration

‘fv’(kHz)

¼1:04

K0:75

3A

0:25

t

8F

sAv

Hwð1þlÞ

�� 3=4

C1=4av

dm

fv,

0.005p

Avp0.1(m

m);

20p

f vp30(kHz);

5p

Cavp30(%

);

1p

Fsp

10(N

)

�N

om

enti

on

about

met

hod

of

calc

ula

tion

of

con

sta

nts

of

pro

po

rtio

nali

tya

nd

stri

ng

len

gth

use

d

inG

A

�Abrasive

concentrationbyvolume

‘Cav’(%

)

where

K3isaconstantofproportionality

relatingmeandiameter

ofabrasivegrains,and

diameter

ofprojectionsontheabrasivegrains

(¼

K3dm2)

�Abrasivesize

‘dm’

(mm)

�Staticfeed

force‘F

s’

(N)



J The most realistic, reliable, and easier to solve (fromoptimization point of view) model was selected andwas made dimensionally consistent.

J Constants of proportionality were minimized byevaluation or substitution.

J Finally models were simplified, and/or rearranged toexpress the objective functions and constraints asfunctions of decision variables in a standard formatthat can be solved by a standard optimizationsolution technique.

�
Formulation of constraints: For the processes such asUSM and AJM, a limiting value of surface roughnessproduced should be used as a constraint. No explicitanalysis to predict surface roughness produced by theseprocesses is available till date, making it difficult toinclude surface roughness either as objective function orconstraint in the parametric optimization. Therefore, forUSM and AJM processes, one-fourth of indentationdepth (derived from the definition of Ra value) has beenused as a measure of surface roughness. WJM andAWJM processes involve high power consumptiontherefore a constraint on power consumption wasformulated for these processes. � Setting up variable bounds: Variable bounds were
selected based upon the survey of practical range ofvalues of decision variables of these processes asmentioned in Table 2. But, the variable bounds forAMPs are dependent on the nature of application andcapacities of resources such as machine tool, cuttingtool, medium, etc. Therefore, the selected variablebounds can be changed accordingly.

3. Optimization models

Following subsections briefly describe process introduc-tion and literature survey of material removal modeling,and details of the formulated optimization models forUSM, AJM, WJM, and AWJM processes.

3.1. Ultrasonic machining (USM)

In USM process, high frequency electrical energy isconverted into mechanical vibrations through a transducer,which are transmitted to the abrasive particles in the slurryvia an energy-focusing device or horn/tool assembly. USMis characterized by low MRR and almost no surfacedamage to the work material. It can be used for machiningboth electrically conductive and non-conductive materialspreferably with low ductility and high hardness (above 40on Rockwell C-scale). Shaw [18] proposed a staticand analytical model giving relationship between MRRand vibration amplitude, frequency, abrasive grit size andconcentration, and feed force, which can be used for alltype of materials. Miller [19] developed MRR model basedon plastic deformation restricting its application to ductilematerials only. Rosenberg et al. [20] included the statistical

distribution of abrasive particle size in their computation-ally intensive model. Cook [21] proposed the simplestmodel to predict linear machining rate. Kainth et al. [22]proposed a model using abrasive particle size distributionas given in [20]. Nair and Ghosh [23] also proposed acomputation-intensive model simulating the principles ofelastic wave propagation. Wang and Rajurkar [24]suggested a more realistic model taking into account thestochastic and dynamic nature of the process. But, it isapplicable to perfectly brittle materials only. Lee and Chan[25] developed an analytical model to predict effects ofvibration amplitude, grit size, and feed force on MRR andsurface roughness for ceramic composites.

3.1.1. Optimization model

Material removal model proposed by Shaw [18] has beenused in the present work because it is a simple and easy-to-optimize model, and it can be used for all types ofmaterials.

�
Decision variables: Five, i.e., amplitude of vibration ‘Av’(mm); frequency of vibration ‘fv’ (Hz or cycles/s); meandiameter of abrasive grains ‘dm’ (mm); volumetricconcentration of abrasive particles in slurry ‘Cav’; andstatic feed force ‘Fs’ (N). � Objective functions: Mazimize MRR:
Max4:963A0:25

t K0:75usm

½sfwð1þ lÞ�0:75F 0:75

s A0:75v C0:25

av dm f vðmm3=sÞ, (1)

where Kusm is a constant of proportionality (mm�1)relating mean diameter of abrasive grains, and diameterof projections on an abrasive grain ( ¼ Kusm dm

2 ).
� Surface roughness constraint:
1:0�1154:7

½Atsfwð1þ lÞ�0:5ðRaÞmax

F s Av dm

Cav

� �0:5X0:0: (2)

�
Variable bounds: Following variable bounds were for-mulated based on the survey of range of values ofdecision variables presented in the Table 2:
0:005pAvp0:1 ðmmÞ; 10; 000pf vp40; 000 ðHzÞ.

0:007pdmp0:15 ðmmÞ; 0:05pCavp0:5;

4:5pF sp45:0 ðNÞ.

3.2. Abrasive jet machining (AJM)

In the AJM process, a high velocity jet of abrasive particlesand carrier gas coming out from a nozzle impinges on thetarget surface and erodes it. This process is characterized bya relatively low power consumption and small capital cost.It is suitable for hard and/or brittle metals/alloys, semi-conductors, and ceramics (particularly glass). Importantprocess parameters of the AJM are abrasive type, sizeand concentration, type of carrier gas, nozzle shape, size,and wear characteristics, jet velocity, nozzle pressure, and

ARTICLE IN PRESS

Table

2

Literature

survey

andselectionofthevariable

boundsforUSM,AJM

,WJM

,andAWJM

processes

AMP

Decisionvariables(units)

References

Selectedvariable

bound

(Unit)

[11]

[12]

[13]

[14]

[15]

[16]

[6]

[17]

USM

Amplitudeofvibrations‘A

v’

(mm)

0.02

0.01–0.1

—0.01

0.005–0.075

(Halfofthe

grain

size)

—0.005–0.1

0.013–0.065

(ffimeandia

of

abrasivegrain)

0.005–0.1

(mm)

Frequency

ofvibration‘fv’

(Hz)

16,000–30,000

15,000–30,000

—20,000–40,000

15,000–30,000

20,000

20,000–30,000

10,000–40,000

10,000–40,000(H

z)

Sizeormeandia.ofabrasive

grains‘d

m’(m

m)

200–2000Mesh

size

100–800Mesh

size

240–800Mesh

size

100–800Mesh

size

120–1200Mesh

size

0.01–0.15mm

240–800Mesh

size

—100–1200Mesh

size

0.006–0.15mm

100–2000Meshsize

0.007–0.15(m

m)

Volumetricconcentrationof

abrasivegrainsin

slurry‘C

av’

Upto

40%

Upto

30%

30-60%

(by

weight)

Upto

40%

20–60%

30–60%

(by

weight)

5–30%

20–60%

5–50%

Staticfeed

force‘F

s’(N

)Upto

30

——

—Upto

45

—1–10

4.5–45

4.5–45

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

AJM

Mass

flow

rate

ofabrasives

‘_ Ma’

2–20g/m

inNotmentioned

Notmentioned

14g/m

in—

1–20g/m

in—

Notmentioned

1–30g/m

in

0.0000167–0.0005(kg/s)

Meandiameter

ofabrasive

grains‘d

m’(m

m)

0.01–0.05

0.01–0.05

0.010.05

0.01–0.05

—0.01–0.15

0.01–0.05

0.01–0.15

0.01–0.15(m

m)

Velocity

ofabrasiveparticles

‘va’(m

m/s)

200,000–400,000

150,000

150,000–300,000

Upto

300,000

—15,000–335,000

200,000–400,000

Notmentioned

1,50,000–4,00,000(m

m/s)

[10,12,14]

[11]

[13]

[15]

[16]

[17]

WJM

Waterjetpressure

atthe

nozzle

exit‘P

w’(M

Pa)

Notmentioned

—Upto

400

100-1000

40times

ofoil

pressure

69–415(for

HDM)0.2–6.9

(forWJM

)

1–400(M

Pa)

Diameter

ofwaterjetnozzle

‘dwn’(m

m)

—0.05–0.35

0.07–0.5

0.075–0.4

0.07–0.5

0.075–0.38

0.05–0.5

(mm)

Traverse

orfeed

rate

ofthe

nozzle

‘fn’(m

m/s)

——

—1–5

2.5–3,000

2.5–30,500

1–300(m

m/s)

Stand-off-distance

‘X’(m

m)

——

3–25

2.5–50

3–25

2.5–50

2.5–50.0

(mm)

[10–12,14,15]

[13]

[16]

[88]

[89]

[90]

[91]

[4]

AWJM

Waterjetpressure

atthe

nozzle

exit‘P

w’(M

Pa)

Notmentioned

50–400

Upto

400

137–241

69–350

150–350

150–280

50–400

50–400(M

Pa)

Diameter

ofabrasivewaterjet

nozzle

‘dawn’(m

m)

——

—4.3

—0.8–1.1

0.8–2.4

—0.5–5(m

m)

Nozzle

feed

rate

‘fn’(m

m/s)

—2.5

3-10

0.2

-25

1.67–13.33

3.33–25

0.33–6.67

0.2–25(m

m/s)

Mass

flow

rate

ofwater‘_ Mw’

(Kg/s)

—Upto

0.2kg/s

—Waterjetdia

0.127-0.635mm

Waterjetdia

0.3mm

Waterjetdia

0.3mm

—0.01–0.2

(kg/s)

Mass

flow

rate

ofabrasives

‘_ Ma’(K

g/s)

—0.0075

0.00167–0.0833

0.033

0.0005–0.025

0.0075–0.013

0.0033–0.0125

0.00033–0.0033

0.0003–0.08(kg/s)



stand-off-distance (SOD). Material in AJM is removed dueto erosive action caused by impingement of high velocityabrasive jet on the workpiece surface. In the case of ductile

materials, material is removed by plastic deformation andcutting wear. In the case of brittle materials, it may take placedue to indentation rupture, elastic-plastic deformation,critical plastic strain theory, radial cracking and propagationor surface energy criterion.

Finnie [26] was the first researcher to analytically modelthe erosive wear of ductile materials by the impact of solidabrasive particles. But, predictions of this model do notagree well with the experimental results at higher impactangles and cannot be used for near-orthogonal impacts.Also, it does not take into account the effects of abrasiveparticle size and shape on the erosion and gives a velocityindex of 2, while experimental results predict it to be in therange of 2–3 (as reported in [45]). Model proposed byBitter [27] accounts for deformation wear (based on theplastic deformation) and predicts erosive wear of bothbrittle and ductile materials more accurately. Sheldon andFinnie [28] developed an analytical model for erosivecutting of brittle materials by normal impact of abrasives.But, the constants involved in this model require complexcalculations. Neilson and Gilchrist [29] simplified theBitter’s model and detailed the procedure to correlate theerosion relationships with the experimental results. Shel-don and Kanhere [30] analyzed erosion process ofrelatively soft and ductile materials (i.e. aluminum)considering the impact by a relatively large (E2500 mm)single abrasive particle and developed a simple analyticalmaterial removal model, which is applicable at relativelylow impact velocities (i.e.o450m/s). This model predicts avelocity exponent of 3 rather than 2 as given by energyconsiderations therefore its predictions agree more closelywith the experimental results as compared to the previousmodels. Lawn [31] proposed a simple model for the wear ofbrittle materials by the abrasives bonded tool. Sarkar andPandey [32], and Neema and Pandey [33] also proposedrelatively simpler models for brittle materials. The formerconsidered the effect of nozzle pressure on MRR, the latterqualitatively explained the nature of profile of the erodedsurface with increasing SOD using compressible fluid flowconditions. Moore and King [34] proposed two models topredict wear volume of brittle materials by plasticdeformation and indentation fracture. But there is aconsiderable scatter between prediction from these modelsand experimental results. Neglecting the effects of strainhardening and using critical plastic strain criterion,Hutching [35] proposed an easy-to-use model for theerosion of ductile materials by spherical particles at normalimpacts. The only model incorporating the effects ofstatistical distribution of abrasive particle size and velocityon the erosion of brittle materials was proposed byMarshall et al. [36]. Sundararajan [37] proposed anempirical model to predict volume of a crater formedduring the oblique impacts on ductile materials. This modelaccurately predicts the crater volume over the wide range

of abrasive velocity (50–360m/s) and up to impact angle of601. Based on Hertz’s contact stress, Murthy et al. [38]proposed another simple model to predict MRR of brittle

materials at normal impacts. It gives velocity exponentequal to 1.2, while experimental observations reveal it to beapproximately 3. Jain et al. [39] developed a simple model,identical to that of Sarkar and Pandey [32], for estimatingMRR of brittle materials by normal impacts.In the AJM process abrasive particles generally impinge

the target surface at normal impact angle, thereforematerial removal models of Sarkar and Pandey [32] forbrittle material and Hutching [35] for ductile material havebeen selected to formulate the following two optimizationmodels for the AJM process.

3.2.1. Optimization model for brittle materials at normal

impingement of abrasive particles

�
Decision variables: Three, namely mass flow rate ofabrasive particles ‘ _Ma’ (kg/s); mean radius of abrasiveparticles ‘rm’ (mm); and velocity of abrasive particles ‘va’(mm/s). � Objective function: Maximize MRR:
Max 0:0035Za

s0:75fw r0:25a

_Mav1:5a ðmm3=sÞ. (3)

�
Surface roughness constraint:
1:0�18:26

ðRaÞmax

rasfw

� �0:5rmvaX0:0. (4)

�
Variable bounds: Following variable bounds wereselected on the basis of the survey of range of valuesof decision variables presented in the Table 2:
0:0000167p _Map0:0005 ðkg=sÞ;

0:005prmp0:075 ðmmÞ;

150000pvap400000 ðmm=sÞ.

3.2.2. Optimization model for ductile materials at normal

impingement of abrasive particles

�
Decision variables: Three, namely mass flow rate ofabrasive particles ‘ _Ma’ (kg/s); mean radius of abrasiveparticles ‘rm’ (mm); velocity of abrasive particles ‘va’(mm/s). � Objective function: Maximize MRR:
Max 1:0436� 10�6 zrw

d2cwH1:5dwr0:5a

_Ma v3aðmm3=sÞ. (5)

�
Surface roughness constraint:
1:0�25:82

ðRaÞmax

raHdw

� �0:5rm vaX0:0. (6)

ARTICLE IN PRESSN.K. Jain et al. / International Journal of Machine Tools & Manufacture 47 (2007) 900–919 907

Variable bounds: Following variable bounds were
� selected on the basis of the survey of range of valuesof decision variables presented in the Table 2:
0:0000167p _Map0:0005 ðkg=sÞ;

0:005prmp0:075 ðmmÞ;

150000pvap400000 ðmm=sÞ.

3.3. Water jet machining (WJM)

The kinetic energy of water jet can be used either fordestructive or precision cutting and cleaning applications.Destructive applications of water jet include hydraulicmining, tunnel boring, cutting anti-skid grooves in airways,making trenches and laying cables, etc. While, precision

applications include hole-making, cleaning, descaling,deburring, cutting of printed circuit boards, profile cuttingof fiber-reinforced plastic aircraft structures, etc. Water jetcutting (WJC) normally involves penetration of the solidby a continuous jet while water jet cleaning (WJCl) involveserosion by discrete droplets. WJM is better suited for thematerials such as corrugated board, leather, kevlar,asbestos, glass epoxy, paper products, graphite, boron,FRP, and some brittle materials. Important processparameters of WJM are SOD, water pressure, travel speed(or feed rate) of jet, and nozzle diameter.

Hashish and duPlessis [40] carried out theoreticalinvestigations for a wide range of materials and developeda simple non-dimensional equation for the penetrationdepth, volume removal per unit length, and specific energyin terms of the four important process parameters. Sameauthors [41] modified this equation by combining itwith an empirical equation to take into account the jetspreading and velocity decay of the jet in the air, andused it to study the effects of SOD and number of passeson the penetration depth, MRR, and specific energyconsumption. The analytical predictions of this modelfor penetration depth and specific energy are in goodagreement with the published experimental results. Leuet al. [42] did mathematical modeling and experimentalverification for WJCl considering water jet as stationary

and normal impinging. They developed a simple analyticalexpression for cleaning width as a function of SOD, waterjet pressure, and nozzle radius. They also derived relationsfor optimum and critical SOD and maximum cleaningwidth. Meng et al. [43] developed a computationallyintensive and semi-empirical model for WJCl processconsidering water jet as moving and striking at normalincidence angle.

3.3.1. Optimization model

Following multi-objective optimization model for theWJC process was formulated using the expression ofspecific energy and penetration depth developed byHashish and duPlessis [41], and assuming that width ofcut equal to diameter of water jet to calculate MRR. But, it

has been solved as a single-objective optimization problemconsidering maximization of MRR as the only objectivefunction.

�
Decision variables: Water jet pressure at the nozzle exit‘Pw’ (MPa or N/mm2); diameter of water jet nozzle ‘dwn’(mm); traverse rate of the nozzle ‘fn’ (mm/s); SOD ‘X’(mm). � Objective functions:
(1) Maximize MRR:

Max0:297

Cfwd1:5wnf nX 0:5c

23 1�

syw2Pwf

� �

� 1� e�2256:76CfwZw

Pwff n

h iðmm3=sÞ: ð7Þ

(2) Minimize specific energy:

Min3:74� 101:5 Cfw

c2=3 1� ðsyw=2PwfÞ� �

1� e�2256:76Cfw=Zw ðPwf=f nÞ

h i

�d0:5wnP1:5

w

f nX 0:5ðJ=mm3Þ. ð8Þ

Here

f ¼2

K1½0:5� 0:57cþ 0:2c2

�;

in which c ¼ 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

Pw

scwK1

2

s !;

K1 ¼X

X i

; and CD � 0:7.

�
Power consumption constraint:
1:0�1:11� 10�1:5CD d2

wn P1:5w

PmaxX0:0 (9)

�
Variable bounds: Following variable bounds wereselected on the basis of the survey of range of valuesof decision variables presented in the Table 2: �
1:0pPwp400:0 ðMPaÞ; 0:05pdwnp0:5 ðmmÞ;

1:0pf np300:0 ðmm=sÞ; 2:5pXp50:0 ðmmÞ.

3.4. Abrasive-water jet machining (AWJM)

The best of AJM and WJM processes have beencombined to create a process known as AWJM. Thisprocess relies on erosive action of abrasive laden water jetfor applications of cutting, drilling, cleaning, and descalingof thick sections of very soft to very hard materials athigher rates. A stream of small abrasive particles isintroduced and entrained in the water jet in such a mannerthat water jet’s momentum is partly transferred to theabrasive particles. Role of carrier fluid (water) is primarily


to accelerate large quantities of abrasive particles to a highvelocity and to produce a highly coherent jet [45].Important process parameters of AWJM can be categor-ized as hydraulic parameters: water pressure, and waterflow rate (or water jet nozzle diameter); abrasive para-meters: type, size, shape, and flow rate of abrasive particles;cutting parameters: traverse rate, SOD, number of passes,angle of attack, and target material; and mixing para-meters: mixing method (forced or suction), abrasivecondition (dry or slurry) and mixing chamber dimensions.Variety of materials that can be machined by AWJMinclude copper and its alloys, aluminum, lead, steel,tungsten carbide, titanium, ceramics, composites, acrylic,concrete, rocks, graphite, silica glass, etc. Most promisingapplication includes machining of sandwiched honeycombstructural materials frequently used in aerospace industries.Visual examination of the cutting process in AWJMsuggests two dominant modes of material removal. Firstis erosion by cutting wear due to particle impact at shallowangles on the top surface of the kerf. Second is deformation

wear due to excessive plastic deformation caused byparticle impact at large angles, deeper into the kerf [44,45].

Hashish [44] used erosion model of Finnie [26] todevelop a model to predict combined depth of cut due tocutting and deformation wear for ductile materials only.Hashish [45] developed an improved erosion model byexpanding Finnie’s model [26] to include the effects ofabrasive particle size and shape (expressed by sphericityand roundness numbers), which was weakness of theearlier model [44]. Hashish [45] used this model to predictdepth of cut due to cutting wear, while the predictionof depth of cut due to deformation wear was based onBitter’s model [27]. But, this model neglected the variationin kerf width along the depth of cut. Using the sameimproved erosion model, Paul et al. [46] developedanalytical model of generalized kerf shape for ductile

materials considering variation in kerf width along itsdepth. Same authors [47] also developed a complexmathematical model for total depth of cut for polycrystal-

line brittle materials accounting for the effects of abrasiveparticle size and shape, but neglecting variation of kerfwidth along the depth of cut. Choi and Choi [48] developedan analytical model for AWJM of brittle materials.Expression developed by them to predict volume of workmaterial removed by a single abrasive particle is not interms of process parameters, and moreover involvesconstant of proportionality.

3.4.1. Optimization model for ductile materials

Following optimization model was developed usinganalysis of Hashish [45] for predicting depth of cut dueto cutting and deformation wear and assuming width of cutequal to diameter of the abrasive water jet. In this model,variation in the velocity of abrasive water jet (which is truefor shallow depth of cut), and its effects on the kerf walldrag have been neglected for both cutting wear anddeformation wear zones. Also, threshold velocity concept

has not been considered for the depth of cut due to cuttingwear.

�
Decision variables: Five, namely water jet pressure at thenozzle exit ‘Pw’ (MPa); diameter of abrasive-water jetnozzle ‘dawn’ (mm); traverse or feed rate of the nozzle ‘fn’(mm/s); mass flow rate of water ‘ _Mw’ (kg/s) and massflow rate of abrasives ‘ _Ma’ (kg/s). � Objective functions: Maximize MRR:
Max dawnf nðhc þ hdÞ ðmm3=sÞ, (10)

where indentation depth due to cutting wear ‘hc’ andindentation depth due to deformation wear ‘hd’ arecalculated using the expressions mentioned in theAppendix A.
� Power consumption constraint:
1:0�Pw

_Mw

PmaxX0:0. (11)

�
Variable bounds: Based upon the survey of range ofvalues of decision variables presented in the Table 2,following variable bounds were formulated:
50:0pPwp400:0 ðMPaÞ; 0:5pdawnp5:0mmÞ;

0:2pf np25:0 ðmm=sÞ; 0:02p _Mwp0:2 ðkg=sÞ;

and 0:0003p _Map0:08ðkg=sÞ.

4. Solution methodology

Optimization models formulated in the present work arenon-linearly constrained single and multi-objective optimi-zation problems. Some of these models (those of WJM,AWJM) involve implicit functions of the decision vari-ables. Such optimization models are too difficult to solveusing the traditional optimization techniques. Evolutionary

algorithms (EA) in general and genetic algorithms (GA) inparticular have proven to be powerful tools to solvecomplex optimization problems, and generally do notdepend on the initial solution. Hence, GA were identifiedas the most suitable solution technique to solve theformulated optimization models.GA are computerized search and optimization algo-

rithms belonging to the class of EA and work with a set orpopulation of solutions as opposed to traditional optimiza-tion technique and evolve the set of optimum solutionsclustered around the global optimum solution using theprinciples of natural genetics and natural selection [49].Operation of GA begins with generation of a set of randomsolutions (known as population). Each solution is evaluatedto find its fitness value. Higher fitness value indicatesgoodness of the solution. The generated population is thenoperated by the reproduction, crossover, and mutationoperators to create the new population which is evaluatedand tested for the termination criterion. One cycle of three

ARTICLE IN PRESS

1As a representative sample, such summary of optimization results has

been presented for the USM process only keeping in view the limitation on

the paper length.2Normalized constraint ¼ 1:0� obtained value of the constraint

allowable value of the constraint


GA operations and the subsequent evaluation constitute ageneration in the GA terminology. Reproduction orselection operator selects good solutions from the currentgeneration in proportionate to their fitness value to form amating pool. Crossover operator creates new and hopefullybetter solutions by crossing over the solutions selectedfrom the mating pool according to the crossover prob-ability. Mutation operator alters a good solution locally tohopefully create a better solution and helps in maintainingthe diversity in the population. This procedure of GAoperations is continued until the termination criterion ismet or for a specified number of generations.

GA are naturally suitable for solving maximizationproblems. But, minimization problems can be easilytransformed into maximization problems using somesuitable transformation, therefore GA can also solveminimization problems in an equally effective manner.GA can be either binary or real-coded. Binary-coded GAdiscretize the search space and accuracy of the optimumsolution depends on the string length of the decisionvariables, therefore real-coded GA with following para-meters were used: Reproduction operator: Tournament

selection, crossover operator: simulated binary crossover

(SBX), mutation operator: polynomial mutation, popula-tion size ‘Ps’ ¼ 15, 20, and 25 times number of decision

variables, number of generations ‘Ng’ ¼ 100, crossoverprobability ‘pc’ ¼ 0.9, SBX parameter ‘Zc’ ¼ 2 and 10,parameter for polynomial mutation ‘Zm’ ¼ 10 and 50,mutation probability ‘pm’ ¼ ð1:0=number of decisionvariablesÞ.

Selection of optimum GA parameters is necessary toensure rapid on-line and off-line convergence of GA.Objective of optimization (also known as calibration orparameterization) of GA parameters is to identify theirvalues for the problems involving fixed string length suchthat the value of objective function close to the true optimacan be reached by GA using minimum number of functionevaluations. Presently no general methodology is availablefor optimum selection of GA parameters. Bagchi [50] hasmentioned a parameterization methodology based on theprinciples of statistical design of experiments (DOE), whichis effective and efficient in evaluating the effects of multiplefactors on a process. DOE techniques such as full factorialdesign, orthogonal arrays, latin squares, etc., can be usedfor this purpose. Of the different parameters mentionedabove for real-coded GA, population size ‘Ps’, SBXparameter ‘Zc’, and polynomial mutation parameter ‘Zm’are considered to be the most influential parameters. Tooptimize the values of these three parameters total 12combinations obtained by three values of population size(15, 20, and 25 times of number of decision variables but aneven number), two values of SBX parameter ‘Zc’ (2 and 10),and two values of polynomial mutation parameter ‘Zm’ (10and 50) were used. For each combination of theseparameters 10 runs of GA were made, i.e., optimizationproblem was solved 10 different times, each time startingfrom the different random solution. And of these 120

solutions, the best solution in terms of numerical values ofthe objective function and constraint was considered asthe optimum solution. The optimum solution thusobtained is specific to the values of the constants used inan optimization model.Obtained optimization results were analyzed by means

of the graphs showing dependence of the objective functionand constraints on the decision variables. These graphswere plotted by varying single decision variable at a timeand keeping the values of other decision variables constantthat were selected on the basis of one of the optimumsolutions.

5. Results and discussion

5.1. Ultrasonic machining (USM)

Table 3 presents the values of the constants and decisionvariables used during optimization. Summary of theoptimization results for different combinations of popula-tion size, SBX parameter, and polynomial mutationparameter is presented in Table 4.1 Based on these results,following overall optimum solution (the exact values of theoptimization solution have been rounded-off only, but inpractice the nearest standard values should be used) wasobtained in the 7th run for the population size ‘Ps’ ¼ 126;SBX parameter ‘Zc’ ¼ 2; and polynomial mutation para-meter ‘Zm’ ¼ 10.

Amplitude of ultrasonicvibrations (mm)

¼ 0.0263

Frequency of ultrasonicvibrations (Hz)

¼ 39333.9

Mean diameter of abrasiveparticles (mm)

¼ 0.1336

Volumetric concentrationof abrasive particles

¼ 0.479 (or 47.9%)

Static feed force (N)
¼ 10.8 Optimum value of MRR(mm3/s)
¼ 3.553

Value of the normalizedconstraint2

¼ 0.0214

Average surface roughnessachieved

¼ 0.783 mm (Allowable

0.8 mm)

Optimality of this solution can be confirmed with thehelp of the graphs, depicted in Figs. 1(a)–(e), showingdependence of the objective function (MRR, Eq. (1)) andthe constraint (surface roughness produced, Eq. (2)) on thedecision variables as discussed below.As depicted in the Figs. 1(a)–(e) and evident from the

Eqs. (1) and (2), MRR in USM process increases with an

ARTICLE IN PRESS

Table 3

Values of the constants and decision variables used in the process parameters optimization of USM

Constants

Notation Details Units Values

At Cross-sectional area of cutting tool mm2 20

sft Flow stress of abrasive particles MPa 28,000 (SiC)

sfw Flow stress of the work material MPa 6900 (WC)

Kusm A constant of proportionality relating mean

diameter of abrasive grains, and diameter of

projections on an abrasive grain ( ¼ Kusm dm2 )

mm�1 0.1

(Ra)max Allowable surface roughness value mm 0.8

Decision variables

Notation Details Units Range (When used as a

variable)

Value (When used

as a constant to plot

the graphs)

Av Amplitude of vibration (mm) 0.005–0.1 0.02

fv Frequency of vibration (Hz) 10,000–40,000 36,000

dm Mean diameter of abrasive grains (mm) 0.007–0.15 0.1

Cav Volumetric concentration of abrasive grains

in slurry

0.05–0.5 0.48

Fs Static feed force (N) 4.5–45.0 20

Table 4

Summary of the optimization results for different combinations of real-coded GA parameters for the USM

Population size, Ps SBX parameter, Zc Polynomial

mutation

parameter, Zm

Optimum solution

Run no. Objective function

MRR (mm3/s)

Normalized

constraint

Generation

no.

76 2 10 10 3.036 0.0047 2

76 2 50 2 2.966 0.0156 11

76 10 10 1 3.264 0.0183 4

76 10 50 9 3.146 0.0567 11

100 2 10 3 3.324 0.0537 41

100 2 50 5 3.177 0.0067 85

100 10 10 9 3.296 0.0218 14

100 10 50 6 3.362 0.0202 57

126 2 10 7 3.553 0.0214 4

126 2 50 5 3.276 0.0240 7

126 10 10 10 3.510 0.0131 4

126 10 50 10 3.116 0.0166 9

N.K. Jain et al. / International Journal of Machine Tools & Manufacture 47 (2007) 900–919910

increase in amplitude of vibrations (Fig. 1(a)), frequency ofvibrations (Fig. 1(b)), mean diameter of abrasive particles(Fig. 1(c)), volumetric concentration of abrasive particles(Fig. 1(d)), and static feed force (Fig. 1(e)). While surfaceroughness increases with increase in amplitude of vibra-tions (Fig. 1(a)), but it does not depend on frequency ofvibrations (Fig. 1(b)). Ra value increases with meandiameter of abrasive particles (Fig. 1(c)), but decreaseswith volumetric concentration of abrasive particles(Fig. 1(d)). However, it increases with static feed force(Fig. 1(e)). It establishes the validity of the model.

Therefore, values of amplitude of ultrasonic vibrations,mean diameter of abrasive particles, and static feed forcecorresponding to the allowable surface roughness value willbe their optimum values as they will give the maximumvalue of MRR that can be achieved while satisfying thesurface roughness constraint. While maximum possiblevalues of frequency of ultrasonic vibration and volumetricconcentration of abrasive particles will be the optimumvalues. Obtained optimum values of the decision variablesare in agreement with these analytical and graphicalobservations. These agreements and value of the surface

ARTICLE IN PRESS

Fig. 1. Variation of objective function (MRR) and surface roughness constraint with (a) amplitude of vibration; (b) frequency of vibration; (c) mean

diameter of abrasive particles; (d) volumetric concentration of abrasive particles; (e) static feed force.


roughness achieved (Eequal to the constraint limit)confirm the optimality of the solution.

5.2. Abrasive jet machining (AJM)

5.2.1. Brittle materials

Table 5 presents the values of the constants and decisionvariables used in the process parameters optimization ofAJM of brittle materials at normal impact of abrasive

particles using the formulated optimization model. Follow-ing overall optimum solution was obtained in the 1st runfor the population size ‘Ps’ ¼ 76; SBX parameter ‘Zc’ ¼ 10;and polynomial mutation parameter ‘Zm’ ¼ 10.

Mass flow rate of abrasive particles(kg/s)

¼ 0.0005

Mean radius of abrasive particles(mm)

¼ 0.005

ARTICLE IN PRESS

Table 5

Values of the constants and decision variables used in the process parameters optimization of AJM using the formulated optimization model based on the

analysis of Sarkar and Pandey [32]

Constants

Notation Details Units Value

ra Density of abrasive particles kg/mm3 3.85� 10�6 (Al2O3)

Za Proportion of abrasive particles effectively

participating in the machining process

0.7

sfw Flow stress of work material MPa 5,000 (Glass)


Decision Variables


variable)

Value (When used as a

constant to plot graphs)

_Ma Mass flow rate of abrasive particles (kg/s) 0.0,000,167–0.0,005 0.0005

rm Mean radius of abrasive particles (mm) 0.005–0.075 0.005

va Velocity of abrasive particles (mm/s) 1,50,000–4,00,000 3,00,000


Velocity of abrasive particles (mm/s)
¼ 315504.3 Optimum value of MRR (mm3/s) ¼ 8.236 Value of the normalized constraint ¼ 0.0003 Average surface roughness achieved ¼ 0.7998 mm
(Allowable 0.8 mm)

Figs. 2(a)–(c) show the variation of the objectivefunction (Eq. (3)) and the constraint (Eq. (4)) with thedecision variables (i.e. mass flow rate, mean radius, andvelocity of the abrasive particles). According to theformulated optimization model, MRR increases linearlywith mass flow rate of abrasive particles while surfaceroughness produced does not depend on it (Fig. 2(a)).Therefore, maximum possible value of mass flow rate ofabrasive particles will be the optimum solution. Theobtained value (0.0005 kg/s, same as its upper bound) isin agreement with this fact. MRR does not depend on themean radius of the abrasive particles while surface rough-ness produced increases linearly with it (Fig. 2(b)). There-fore, minimum value of mean radius of abrasive particlesshould be used to get optimum results (better surfacefinish). Thus its value obtained (0.005mm, same as itslower bound) is the optimum. MRR increases with velocityof abrasive particles with an exponent of 1.5, while surfaceroughness produced increases linearly with it (Fig. 2(c)).Therefore, the value of velocity of abrasive particlescorresponding to the acceptable surface roughness valuewill be the optimum. Its obtained value (315504.3mm/s) isin agreement with this. These agreements and the fact thatthe value of the surface roughness achieved is approxi-mately equal to the constraint limit confirm the optimalityof the above-mentioned optimum solution.

5.2.2. Ductile materials

Table 6 presents the values of the constants and decisionvariables used in the process parameters optimization of

AJM of ductile materials at normal impact or impingement

of abrasive particles using the formulated optimizationmodel based on analysis of Hutching [35]. Followingoptimum solution is obtained in the 1st run for thepopulation size ‘Ps’ ¼ 46; SBX parameter ‘Zc’ ¼ 10; andpolynomial mutation parameter ‘Zm; ¼ 10.


¼ 0.0005

Mean radius of abrasive particles(mm)

¼ 0.005

Velocity of abrasive Particles (mm/s)

¼ 333214.7

Optimum value of MRR (mm3/s)
¼ 0.6025 Value of the normalized constraint ¼ 0.0011 Average surface roughnessachieved
¼ 1.9978 mm(Allowable 2.0 mm)

Figs. 3(a)–(c) show the variation of the objectivefunction (Eq. (5)) and the constraint (Eq. (6)) with thedecision variables. MRR increases linearly with mass flowrate of abrasive particles while surface roughness produceddoes not depend on it (Fig. 3(a)). Therefore, maximumpossible value of mass flow rate of abrasive particles will bethe optimum solution. The obtained value (0.0005 kg/s,same as its upper bound) is in agreement with this fact.MRR does not depend on the mean radius of the abrasiveparticles while surface roughness produced increaseslinearly with it (Fig. 3(b)). Therefore, minimum value ofmean radius of abrasive particles should be used to get theoptimum results, i.e., better surface finish. Thus its valueobtained (0.005mm, same as its lower bound) is theoptimum. MRR increases non-linearly with velocity ofabrasive particles while surface roughness producedincreases linearly with it (Fig. 3(c)). Therefore, value ofvelocity of abrasive particles corresponding to the accep-table surface roughness value will be the optimum. Its

ARTICLE IN PRESS

Fig. 2. Variation of MRR and surface roughness with (a) mass flow rate of abrasive particles; (b) mean radius of abrasive particles; (c) velocity of abrasive

particles.

Table 6

Values of the constants and decision variables used in the process parameters optimization of AJM using the formulated optimization model based on the

analysis of Hutching [35]

Constants

Notation Details Units Value [35]

ra Density of abrasive particles kg/mm3 2.48� 10�6 (Glass

bead)

rw Density of work material kg/mm3 2.7� 10�6 (Al-

6061–T6)

dcw Critical plastic strain or erosion ductility

of work material

1.5

Hdw Dynamic hardness of work material MPa 1150 (Al-6061-T6)

z Amount of indentation volume

plastically-deformed

1.6


Decision Variables


variable)

Value (when used as a

constant to plot

graphs)

_Ma Mass flow rate of abrasive particles (kg/s) 0.0,000,167–0.0,005 0.0005

rm Mean radius of abrasive particles (mm) 0.005–0.075 0.005

va Velocity of abrasive particles (mm/s) 1,50,000–4,00,000 3,25,000


ARTICLE IN PRESS

Fig. 3. Variation of MRR and surface roughness with (a) mass flow rate of abrasive particles; (b) mean radius of abrasive particles; (c) velocity of abrasive

particles.


obtained value (333214.7mm/s) is in agreement with this.These agreements and the fact that the value of the surfaceroughness achieved is approximately equal to the con-straint limit confirm the optimality of the above-mentionedsolution for the AJM of ductile materials at normal impactof abrasive particles.

5.3. Water jet machining (WJM)

Table 7 presents the values of the constants and decisionvariables used in the parametric optimization of WJM ofMaple (a type of wood) using the formulated optimizationmodel based on the analysis of Hashish and duPlessis [41].Following optimum solution was obtained in the 10th runfor the population size ‘Ps’ ¼ 80; SBX parameter ‘Zc’ ¼ 10;and polynomial mutation parameter ‘Zm’ ¼ 10.

Water jet pressure at nozzle exit(MPa)

¼ 397.00

Diameter of water jet nozzle (mm)
¼ 0.5 Traverse or feed rate of the nozzle(mm/s)
¼ 214.41

Stand-off distance (mm)
¼ 2.54 Optimum value of MRR (mm3/s) ¼ 139.79
Value of the normalized constraint
¼ 0.0331 Power consumption achieved ¼ 48.345 kW
(Allowable 50 kW)

Figs. 4(a)–(d) show the variation of the objectivefunction (Eq. (7)) and the constraint (Eq. (9)) with thedecision variables. MRR in WJM increases with water jetpressure at nozzle exit but power consumption alsoincreases (Fig. 4(a)). It is also found that MRR increaseswith the diameter of water jet nozzle but power consump-tion increases at relatively faster rate (Fig. 4(b)) as it varieswith the square of diameter of water jet nozzle (Eq. (9)).Therefore, values of the water jet pressure and diameter ofwater jet nozzle corresponding to the limiting powerconsumption value will be the optimum. In the presentcase their optimum values have been obtained as 397MPaand 0.5mm, respectively, for an allowable power con-sumption of 50 kW. Initially, MRR increases very rapidlywith feed or traverse rate of the nozzle but after a certainvalue it becomes more or less steady while powerconsumption is independent of it (Fig. 4(c)). Therefore,the value of the nozzle feed rate beyond which there isno significant increase in MRR will be the optimum value.

ARTICLE IN PRESS

Table 7

Values of the constants and decision variables used in the process parameters optimization of WJM

Constants

Notation Details Units Value for Maple [41]

Cfw Drag or skin friction coefficient for work

material

0.005

Zw Damping coefficient of the work material Kgmm�2 s�1 2357.3

scw Compressive yield strength of work

material

MPa 26.2

syw Tensile yield strength of the work material MPa 3.9

Xi Length of initial region of the water jet mm 20

Pmax Allowable power consumption value kW 50

Decision Variables

Notation Details Units Range (when used as a

variable)



Pw Water jet pressure at the nozzle exit (MPa or N/mm2) 1.0–400.0 390.0

dwn Diameter of water jet nozzle (mm) 0.05–0.5 0.5

fn Traverse or feed rate of the nozzle (mm/s) 1.0–300.0 75.0

X Stand-off-distance (mm) 2.5–50.0 5.0

Fig. 4. Variation of MRR and power consumption with (a) water jet pressure at nozzle exit; (b) diameter of water jet nozzle; (c) traverse or feed rate of the

nozzle; (d) the stand-off-distance.



In the present case, an optimum value of 214.41mm/s hasbeen obtained. MRR decreases with increasing SODinitially at faster rate and then slowly, while powerconsumption is independent of SOD (Fig. 4(d)). Thereforea smaller value of SOD will be desirable. Thus the valueobtained ( ¼ 2.54mm while lower limit used was 2.5mm)of SOD is the optimum. These arguments and the value ofthe power consumption achieved being very near to that ofthe constraint limit, confirm the optimality of the above-mentioned solution for the WJM process.

5.4. Abrasive-water jet machining (AWJM)

Table 8 presents the values of the constants and decisionvariables used during optimization. Following optimum

solution was obtained in the 10th run for the populationsize ‘Ps’ ¼ 126; SBX parameter ‘Zc’ ¼ 2; and polynomialmutation parameter ‘Zm’ ¼ 50.

Table 8

Values of the constants and decision variables u

Constants

Notation Details

ra Density of abrasive part

va Poisson’s ratio of abrasi

EYa Young’s modulus of elas

particles

fr Roundness factor of the

fs Sphericity factor of the a

Za Proportion of abrasive g

participating in machinin

vw Poisson’s ratio of work m

EYw Young’s modulus of elas

material

sew Elastic limit of work ma

sfw Flow stress of the work

Cfw Drag or skin friction coe

material

x Mixing efficiency betwee

water

Pmax Allowable power consum

Decision Variables

Notation Details

Pw Water jet pressure at the

dawn Diameter of abrasive-wa

fn Traverse or feed rate of_Mw Mass flow rate of water

_Ma Mass flow rate of abrasi

Water jet pressure at nozzle exit(MPa)

¼ 398.3

Diameter of abrasive-water jetnozzle (mm)

¼ 3.726

Traverse or feed rate of the nozzle(mm/s)

¼ 23.17

Mass flow rate of water (kg/s)
¼ 0.141
sed in the parametric optimiz

Units

icles kg/mm

ve particles

ticity of abrasive MPa

abrasive particles

brasive particles

rains effectively

g

aterial

ticity of work MPa

terial MPa

material MPa

fficient for work

n abrasives and

ption value kW

Units

nozzle exit (MPa)

ter jet nozzle (mm)

the nozzle (mm/s)

(kg/s)

ves (kg/s)


ation of AWJM

Value [45]

3 3.95� 10�6 (For Al2O3)

0.25

3,50,000

0.35

0.78

0.7

0.20 (For Ti)

1,14,000

883

8142

0.002

0.8

56

Range (when used as a

variable)

50.0–400.0

0.5–5.0

0.2–25.0

0.01–0.2

0.0,003–0.08

¼ 0.079

Optimum value of MRR (mm3/s)
¼ 90.28 Value of the normalized constraint ¼ 0.0005 Power consumption achieved ¼ 55.972 kW
(Allowable 56 kW)

Figs. 5(a)–(e) show the variation of the objective function(i.e. MRR, Eq. (10)) and the constraint (Eq. (11)) with thedecision variables. The optimality of the solution mentionedabove can be confirmed with the help of these figures asdiscussed below. MRR in AWJM increases with water jetpressure at nozzle exit and so as the power consumption(Eq. (11) and Fig. 5(a)). Therefore, value of the water jetpressure corresponding to the limiting power consumptionvalue will be its optimum. In the present case, an optimumvalue of 398.3MPa (upper bound used 400MPa) for anallowable power consumption of 56kW has been obtained.MRR is found to increase with diameter of abrasive-water jetnozzle (Fig. 5(b)) and feed rate of nozzle (Fig. 5(c)) initiallyvery rapidly but after a certain value it becomes more or lesssteady, while power consumption is independent of theseparameters. Therefore, the values of diameter of abrasive-water jet nozzle and nozzle feed rate beyond which there isno significant increase in MRR will be their optimum values.



395.0

1.75

16.0

0.141

0.075

ARTICLE IN PRESS

Fig. 5. Variation of MRR and power consumption with (a) water jet pressure at nozzle exit; (b) diameter of abrasive-water jet nozzle; (c) traverse or feed

rate of the nozzle; (d) mass flow rate of water; (e) mass flow rate of abrasives.


In the present case, their optimum values of 3.726mm (upperbound used for abrasive-water jet nozzle diameter was5.0mm) and 23.17mm/s (upper bound used for nozzle feedrate being 25mm/s), respectively, have been obtained. Asdepicted in the Fig. 5(d), both MRR and power consumptionincrease with increasing mass flow rate of water. Therefore,value of the mass flow rate of water corresponding to thelimiting power consumption value will be its optimum. In the

present case, an optimum value of 0.141kg/s (upper boundused 0.2 kg/s) has been obtained. MRR also increases withmass flow rate of abrasive particles, while power consump-tion is independent of it (Fig. 5(e)). Therefore its maximumpossible value will be the optimum. Thus, the value obtainedin the present case ( ¼ 0.079kg/s while upper limit used was0.08kg/s) is the optimum. From this discussion, and the factthat the value of the power consumption obtained is


approximately equal to the constraint limit, it can beconcluded that the obtained solution for AWJM processparameters optimization is the optimum solution.

6. Conclusions

The formulated optimization models are multi-variablenon-linearly constrained single and multi-objective (forWJM only) optimization problems. For WJM and AWJMprocesses, the formulated objective functions and con-straints are very complicated and implicit functions of thedecision variables. Traditional optimization methods werefound unsuitable to solve such problems therefore real-coded version of the GA was used for solving theformulated optimization models. To ensure that theobtained optimum solution is global-optimum or nearglobal-optimum, concept of statistical DOEs was used tooptimize three most influential and important parametersof real-coded GA namely population size, SBX parameter,and polynomial mutation parameter. The present optimum

hd ¼Zadawn

_Ma½K1_MwP0:5

w � ð_Ma þ _MwÞvac�

2

ð1570:8sfwÞd2awnf nð

_Ma þ _MwÞ2þ ðK1Cfw ZaÞ½K1

_MwP0:5w � ð

_Ma þ _MwÞvac� _Ma_MwP0:5

w

(A.4)

solutions are specific to the values of the constants used inthe optimization models. The optimization results wereconfirmed graphically with the help of the graphs showingdependence of the objective function and constraint on thedecision variables. Only single objective optimization wasdone to check the suitability and validity of the materialremoval models, on the basis of which optimization modelswere formulated.

Acknowledgements

Authors acknowledge the financial support provided bythe Department of Science and Technology (DST), Govern-ment of India, New Delhi, under the grant no. III .5(199)/99-ET for the project entitled ‘‘Automated Process

Selection and Optimization of Advanced Machining Pro-

cesses’’.

Appendix A

Indentation depth of cutting wear ‘hc’ can be calculatedusing

hc ¼1:028� 104:5x

CKr0:4a

� �d0:2awn

_M0:4a

f 0:4n

!_MwP0:5

w

_Ma þ _Mw

� �

�18:48K2=3

a x1=3

C1=3K f 0:4

r

!_MwP0:5

w

_Ma þ _Mw

� �1=3

; if atpao.

hc ¼ 0; Otherwise. ðA:1Þ

Here ao is angle of impingement at which maximumerosion occurs and is given by

ao �0:02164 C

1=3K f 0:4

r

K2=3a x1=3

!_Ma þ _Mw

_MwP0:5w

!1=3

ðdegreesÞ. (A.2)

And, at is the angle of impingement at top of machinedsurface, which is approximately given by

at �0:389� 10�4:5r0:4a CK

x

� �d0:8awnf 0:4

n_Ma þ _Mw

_M0:4a

_MwP0:5w

!ðdegreesÞ.

(A.3)

where CK¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3000sfwf 0:6

r =raq

ðmm=sÞ and Ka¼1þmpr2m=Ipin which mp is mass of an abrasive particle and Ip ismoment of inertia of an abrasive particle about its center ofgravity. Ip ¼ 0:5 mpr2m for the particles with high sphericityvalue (40.7), and Ip ¼ 0:3 mpr2m for particles with lowersphericity value.Indentation depth due to deformation wear ‘hd’ can be

found using

where vac is critical velocity of the abrasive particles,

vac ¼ 5p2s2:5ew

r0:5a

1� n2aEYa

þ1� n2wEYw

� �2ðmm=sÞ; and K1 ¼

ffiffiffi2p� 104:5x.

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